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mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman"; mso-ansi-language:#0400; mso-fareast-language:#0400; mso-bidi-language:#0400;} </style> <![endif]--> </head> <body lang=3DRU link=3D"#006600" vlink=3Dpurple style=3D'tab-interval:35.4p= t'> <div class=3DSection1> <p class=3Dcaption4>Ивановский государственный энергетический университет</= p> <p> </p> <p class=3Dcaption4>Кафедра Электропривод и автоматизация промышленных уста= новок</p> <p> </p> <p class=3Dcaption1>Электронный конспект лекций</p> <p> </p> <p class=3Dcaption1>ОСНОВНЫЕ РАЗДЕЛЫ СОВРЕМЕННОЙ ТЕОРИИ АВТОМАТИЧЕСКОГО УПРАВЛЕНИЯ</p> <p> </p> <p> </p> <p> </p> <p> </p> <p class=3Dmain>Автор: КОЛГАНОВ Алексей Руфимович, доктор технических наук, профессор кафедры «Электропривод и автоматизация промышленных установок»</p> <p class=3Dmain>Тел. (0932) 385795, (0932) 419063</p> <p class=3Dmain><a href=3D"mailto:gom@cterra.ru">klgn@drive.ispu.ru</a></p> <p class=3Dmain><a href=3D"mailto:gom@cterra.ru">klgn@indi.ru</a></p> <p> </p> <h3>ВВЕДЕНИЕ</h3> <p>До пятидесятых годов ХХ века классической <i>теорией автоматического регулирования</i> (ТАР) было принято называть базирующуюся на рассмотрении линейных дифференциальных уравнений теорию устойчивости и качества процессо= в в системе объект - регулятор. Основы этой теории были заложены И.А. Вышнеград= ским и Дж. Максвеллом. По существу она тесно соприкасалась с теорией устойчивост= и «в <span class=3DGramE>малом</span>» А.М. Ляпунова, но имела ярко выраженную инженерную направленность.</p> <p>Под влиянием потребностей автоматизации управления технологическими проц= ессами и движущимися объектами <span class=3DGramE>в сороковых</span> и пятидесятых годах прошлого века интенсивно развивалась <i>теория автоматического управл= ения</i> (ТАУ). Она впитала существующие в то время методы теории связи теории колеб= аний и создала собственные методы анализа и синтеза систем с обратной связью. Эта прикладная теория автоматического управления именовалась в то время <span class=3DGramE><i>современной</i></span>. Получив во многом завершенные форм= ы, она составила предмет учебных дисциплин технических вузов, многочисленных учебн= иков и учебных пособий. До настоящего времени она является основным инструментом предварительного анализа и синтеза локальных систем стабилизации на стадии = их проектирования.</p> <p>В рамках этой инженерной теории использовались методы, основанные на частотном анализе, алгебре передаточных функций, преобразовании Лапласа. За= дача управления технологическими процессами и движущимися объектами решалась в «малом».</p> <p>Таким образом, предметом этой теории для сложных объектов являлось решен= ия множества частных задач на каждом этапе или режиме технологического процесс= а и движущегося объекта. Увязка всех этих частных задач для достижения конечной цели управления производилась на стадии проектирования системы на основе априорной информации с помощью методов, внешних по отношению к данной теори= и.</p> <p>В конце пятидесятых - начале шестидесятых годов, когда Л.С. Понтрягиным = была создана математическая теория оптимальных процессов, Р. Беллман предложил м= етод динамического программирования, а Р. Калман разработал общую теорию фильтра= ции и управления, были заложены основы <i>современной теории автоматического управления. (СТАУ)</i></p> <p>Основным характерными признаками СТАУ является описание процессов в <i>п= ространстве состояний</i> и применения для решения задач анализа и синтеза систем <i>ме= тодов пространства состояний</i>.</p> <p>В курсе лекций, который перед Вами, мы представим:</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l6 level1 lfo1;tab-stops:list 36.0pt'>способы описания динамических объектов управления в пространстве состояний с помощью векторно-матричных моделей в непрерывном и дискретном времени; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l6 level1 lfo1;tab-stops:list 36.0pt'>методы и алгоритмы анал= иза основных свойств объектов и систем управления на основе их векторно-матричных моделей; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l6 level1 lfo1;tab-stops:list 36.0pt'>методы и алгоритмы синт= еза систем управления с регуляторами и наблюдателями состояния. </li> </ul> <h3>1. Цель преподавания дисциплины</h3> <p>Курс имеет целью изучение современных методов теории пространства состоя= ния для решения задач анализа и синтеза систем автоматического управления. </p> <p>Полученные в курсе сведения используются в дальнейшем при изучении практически всех профилирующих дисциплин специальности, выполнении курсовых= и дипломных проектов.</p> <h3>2. Задачи изучения курса</h3> <p>Основной задачами курса являются:</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l12 level1 lfo2;tab-stops:list 36.0pt'>формирование навыков построения векторно-матричных моделей объектов и систем управления в непрерывном и дискретном времени; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l12 level1 lfo2;tab-stops:list 36.0pt'>формирование устойчиво= го представления об использовании математического аппарата матричной алге= бры для анализа и синтеза систем управления; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l12 level1 lfo2;tab-stops:list 36.0pt'>формирования <span class=3DGramE>навыков решения практических задач построения современных систем управления</span> с наблюдателями состояния полного и пониженно= го порядка; </li> </ul> <h3>3. Перечень дисциплин, усвоение которых необходимо для изучения курса</= h3> <p>а) <b>высшая математика</b> - дифференциальное, интегральное и операцион= ное исчисление, матрицы, функции от матриц, матричная алгебра, теория дифференциальных уравнений.</p> <p>б) <b>теория автоматического управления (классическая)</b> - принципы построения и элементный состав систем автоматического управления, преобразование Лапласа, передаточные функции, графы и структурные схемы.</p> <p>в) <b>теоретические основы электротехники</b> - математическое описание электрических цепей, переходные процессы в линейных и нелинейных цепях.</p> <h3>4. Общая характеристика и рекомендации по изучению материала</h3> <p>Материал электронного курса «Основные разделы современной теории автоматического управления» посвящен теоретическим и практическим вопросам изучения и применения методов пространства состояния для конструирования и изучения систем автоматического управления динамическими объектами на уровн= е их векторно-матричных моделей в непрерывном и дискретном времени.</p> <p>Теоретическая часть курса дает представление о методах и алгоритмах анал= иза и синтеза систем управления с использованием современной теории пространства состояния.</p> <p>Изучение практических вопросов предлагаемого материала позволяет сформировать начальные навыки по построению и использованию векторно-матрич= ных моделей электромеханических систем в задачах автоматического управления. Эт= от материал в первую очередь ориентирован на использования компьютерного компл= екса функционального проектирования систем управления динамическими объектами FuncPro 1.0 [6]. Приведенные здесь схемы моделей созданы и апробированы в с= реде этого комплекса. </p> <p>Программное обеспечение компьютерного комплекса FuncPro 1.0, электронное руководство по его применению, практическое пособие [6] могут быть <span class=3DGramE>поставлены</span> Вам при дополнительном обращении.</p> <h3>Лекция № 1.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема<span class=3DGramE>:«</span>Векторно-матричные= модели систем управления в непрерывном времени»<o:p></o:p></span></b></p> <p><b>Понятие пространства состояний</b></p> <p>Современная теория автоматического управления оперирует с векторно-матричными моделями динамических систем. При этом рассматриваются в общем случае многомерные системы, т.е. системы произвольного порядка со мно= гими входами и многими выходами, в связи, с чем широко используются векторно-матричные уравнения и аппарат векторной алгебры. Для получения векторно-матричной модели (ВММ) исследуемая динамическая система представля= ется в виде “черного ящика” с некоторым числом входных и выходных каналов (рис. = 1.1, а).</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shapetyp= e id=3D"_x0000_t75" coordsize=3D"21600,21600" o:spt=3D"75" o:preferrelative=3D"t" path=3D"m@4@= 5l@4@11@9@11@9@5xe" filled=3D"f" stroked=3D"f"> <v:stroke joinstyle=3D"miter"/> <v:formulas> <v:f eqn=3D"if lineDrawn pixelLineWidth 0"/> <v:f eqn=3D"sum @0 1 0"/> <v:f eqn=3D"sum 0 0 @1"/> <v:f eqn=3D"prod @2 1 2"/> <v:f eqn=3D"prod @3 21600 pixelWidth"/> <v:f eqn=3D"prod @3 21600 pixelHeight"/> <v:f eqn=3D"sum @0 0 1"/> <v:f eqn=3D"prod @6 1 2"/> <v:f eqn=3D"prod @7 21600 pixelWidth"/> <v:f eqn=3D"sum @8 21600 0"/> <v:f eqn=3D"prod @7 21600 pixelHeight"/> <v:f eqn=3D"sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok=3D"f" gradientshapeok=3D"t" o:connecttype=3D"rect"/> <o:lock v:ext=3D"edit" aspectratio=3D"t"/> </v:shapetype><v:shape id=3D"_x0000_i1025" type=3D"#_x0000_t75" alt=3D"" st= yle=3D'width:409.5pt; height:142.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge001.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image001.gif"/> </v:shape></p> <p class=3Dimage>Рис.1.1. Скалярное (а) и векторное (б) представления динамической системы в виде "черного ящика"</p> <p>Все переменные, характеризующие систему, можно разделить на три группы.<= /p> <p>1. <i>Входные переменные</i> или входные воздействия, генерируемые системами, внешними по отношению к исследуемой системе. Они характеризуются вектором входа.</p> <p><v:shape id=3D"_x0000_i1026" type=3D"#_x0000_t75" alt=3D"" style=3D'widt= h:90pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge002.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image002.gif"/> </v:shape><i>r</i> - число входов</p> <p>2. <i>Выходные переменные</i>, характеризующие реакцию системы на указан= ные входные воздействия. Представляются вектором выхода </p> <p><v:shape id=3D"_x0000_i1027" type=3D"#_x0000_t75" alt=3D"" style=3D'widt= h:97.5pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge003.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image003.gif"/> </v:shape><i>m</i> - число выходов.</p> <p>3. Промежуточные переменные, характеризующие внутреннее состояние систем= ы, - <i>переменные состояния</i>, представляются вектором</p> <p><v:shape id=3D"_x0000_i1028" type=3D"#_x0000_t75" alt=3D"" style=3D'widt= h:91.5pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge004.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image004.gif"/> </v:shape><i>n</i> - число переменных состояния.</p> <p>Таким образом, совокупность входов можно рассматривать как один обобщенн= ый вход, на который воздействует вектор входа <b>u</b>, совокупность выходов к= ак вектор <b>y</b>, а совокупность промежуточных координат, характеризующих состояние системы, - в виде вектора состояния <b>x</b> (см. рис. 1.1, б).</= p> <p><i>Состояние системы</i> - это та минимальная информация о прошлом, кото= рая необходима для полного описания будущего поведения (т.е. выходов) системы, = если поведение ее входов известно.</p> <p>Собственно система, ее входы и выходы - это три взаимосвязанных объекта, которые в каждой конкретной ситуации определяются соответственно математиче= ской моделью системы, заданием множе<span class=3DGramE>ств вх</span>одных и вых= одных переменных.</p> <p>Решение задач анализа и синтеза связано с исследованием состояний систем= ы, множество которых образует <i>пространство состояний</i>,<v:shape id=3D"_x0= 000_i1029" type=3D"#_x0000_t75" alt=3D"" style=3D'width:34.5pt;height:15pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge005.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image005.gif"/> </v:shape>.</p> <p><b>Векторно-матричные модели в непрерывном времени</b></p> <p>В общем случае динамическая система <span class=3DGramE>в</span> непреры= вном может быть описана парой матричных уравнений:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1030" type=3D"#_x0000_t75" alt=3D"" style=3D'width:102pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage006.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image006.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(1.1)</p> <p align=3Dright style=3D'text-align:right'>(1.2)</p> </td> </tr> </table> <p>где <b>F</b> - <i>n</i>-мерная вектор-функция системы; <b>Q</b> - <i>m</= i>-мерная вектор-функция выхода.</p> <p>Матричное уравнение (1.1) называют <i>уравнением состояния</i> системы. = Его решение, удовлетворяющее начальному условию <v:shape id=3D"_x0000_i1031" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:51pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge007.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image007.gif"/> </v:shape>, дает вектор состояния системы</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1032" type=3D"#_x0000_t75" alt=3D"" style=3D'width:107.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage008.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image008.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(1.3)</p> </td> </tr> </table> <p>Матричное уравнение (1.2), определяющее выходные переменные в зависимост= и от x(t) и u(t), называют <i>уравнением выхода</i>.</p> <p>В частном случае зависимости <v:shape id=3D"_x0000_i1033" type=3D"#_x000= 0_t75" alt=3D"" style=3D'width:153.75pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge009.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image009.gif"/> </v:shape>могут быть линейными комбинациями переменных состояния <i>x<sub>i= </sub></i> и входных переменных <i>u<sub>q</sub></i>. При этом динамическая система описывается в векторно-матричной форме:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1034" type=3D"#_x0000_t75" alt=3D"" style=3D'width:96.75pt;height:39.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage010.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image010.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(1.4)</p> <p align=3Dright style=3D'text-align:right'>(1.5)</p> </td> </tr> </table> <p>Переход к стационарным моделям позволяет оперировать с коэффициентными матрицами, т.е. со стационарными уравнениями</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1035" type=3D"#_x0000_t75" alt=3D"" style=3D'width:71.25pt;height:39.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage011.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image011.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(1.6)</p> <p align=3Dright style=3D'text-align:right'>(1.7)</p> </td> </tr> </table> <p>где:</p> <p><b>А</b> - функциональная матрица размером <i>n x n</i>, называемая <i>м= атрицей состояния системы</i> (объекта);</p> <p><span class=3DGramE><b>В</b></span> - <span class=3DGramE>функциональная= </span> матрица размером <i>n x r</i>, называемая <i>матрицей управления</i> (входа= );</p> <p><span class=3DGramE><b>С</b></span> - <span class=3DGramE>функциональная= </span> матрица размером <i>m x n</i>, называемая <i>матрицей выхода по состоянию</= i>;</p> <p><b>D</b> - функциональная матрица размером <i>m x r</i>, называемая <i>м= атрицей выхода по управлению</i>.</p> <p>Очень часто <b>D</b>=3D0, т.е. выход непосредственно не зависит от входа= .</p> <p>В дальнейшем под векторно-матричной моделью объекта (системы) будем пони= мать описание ее динамического поведения в классе стационарных непрерывных линей= ных систем, представленное в виде уравнений (1.6), (1.7).</p> <p>Таким образом, ВММ имеет единую форму представления, что значительно облегчает алгоритмизацию и компьютерную реализацию проектных процедур и проектных операций структурно-параметрического синтеза и анализа систем управления. Однако с использованием ВММ может быть получено лишь приближенн= ое проектное решение, которое потребует дальнейшего уточнения, так как такие модели отображают динамическое поведение реального объекта лишь в классе стационарных линейных систем.</p> <p>Построение ВММ реального объекта сопряжено с проблемами линеаризации исходного математического описания и приведения его к структурированному ви= ду - форме Коши.</p> <p>Если мы знаем физическое описание системы и можем записать уравнения, описывающие поведен<span class=3DGramE>ия ее о</span>тдельных частей, то по= лучить уравнения состояния системы обычно сравнительно не трудно. Покажем эту процедуру на нескольких примерах.</p> <p><b><i>Пример 1.1.</i></b> Получим уравнения состояния для <span class=3D= GramE>простейшей</span> <i>RLC</i>-цепи, показанной на рис 1.2.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1036" type=3D"#_x0000_t75" alt=3D"" style=3D'width:162.75pt;height:112.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge012.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image012.gif"/> </v:shape></p> <p><span class=3DGramE>Динамическое поведение этой системы при <v:shape id= =3D"_x0000_i1037" type=3D"#_x0000_t75" alt=3D"" style=3D'width:26.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge013.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image013.gif"/> </v:shape>полностью определяется, если известны начальные значения <v:shape id=3D"_x0000_i1038" type=3D"#_x0000_t75" alt=3D"" style=3D'width:57.75pt;h= eight:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge014.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image014.gif"/> </v:shape>и входное напряжение <i>U(t)</i> при <v:shape id=3D"_x0000_i1039" type=3D"#_x0000_t75" alt=3D"" style=3D'width:26.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge013.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image013.gif"/> </v:shape>. Следовательно, <v:shape id=3D"_x0000_i1040" type=3D"#_x0000_t75= " alt=3D"" style=3D'width:45pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge015.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image015.gif"/> </v:shape>можно выбрать в качестве переменных состояния, то есть <v:shape i= d=3D"_x0000_i1041" type=3D"#_x0000_t75" alt=3D"" style=3D'width:150pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge016.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image016.gif"/> </v:shape></span></p> <p>Для указанных переменных состояния можно записать дифференциальные уравн= ения</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1042" type=3D"#_x0000_t75" alt=3D"" style=3D'width:117.75pt;height:59.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge017.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image017.gif"/> </v:shape></p> <p>или в векторно-матричной форме</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1043" type=3D"#_x0000_t75" alt=3D"" style=3D'width:208.5pt;height:45.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage018.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image018.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(1.8)</p> </td> </tr> </table> <p>Таким образом для рассматриваемой системы матрицы<span class=3DGramE> <b= >А</b></span>, <b>В</b>, <b>С</b> векторно-матричной модели будут иметь следующий вид:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1044" type=3D"#_x0000_t75" alt=3D"" style=3D'width:237pt;height:45.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage019.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image019.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'> </p> </td> </tr> </table> <p><b><i>Пример 1.2.</i></b> На рис. 1.3. показан электродвигатель постоянн= ого тока независимого возбуждения, работающий при постоянном магнитном потоке (= <span class=3DGramE>Ф</span>=3D<i>const</i>).</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1045" type=3D"#_x0000_t75" alt=3D"" style=3D'width:162pt;height:174pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge020.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image020.gif"/> </v:shape></p> <p>Дифференциальные уравнения для такого объекта могут быть записаны относительно следующих переменных состояния: <v:shape id=3D"_x0000_i1046" t= ype=3D"#_x0000_t75" alt=3D"" style=3D'width:24.75pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge021.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image021.gif"/> </v:shape>- скорости вращения ротора, тока якоря <i>i(t)</i>, углового перемещения ротора <v:shape id=3D"_x0000_i1047" type=3D"#_x0000_t75" alt=3D= "" style=3D'width:23.25pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge022.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image022.gif"/> </v:shape>. При использовании знакомых зависимостей для электродвижущей сил= ы <v:shape id=3D"_x0000_i1048" type=3D"#_x0000_t75" alt=3D"" style=3D'width:78.75pt;h= eight:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge023.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image023.gif"/> </v:shape>и вращающего момента двигателя <v:shape id=3D"_x0000_i1049" type= =3D"#_x0000_t75" alt=3D"" style=3D'width:75.75pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge024.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image024.gif"/> </v:shape>получим уравнение электрической цепи</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1050" type=3D"#_x0000_t75" alt=3D"" style=3D'width:140.25pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge025.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image025.gif"/> </v:shape></p> <p>и уравнения вращающейся части</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1051" type=3D"#_x0000_t75" alt=3D"" style=3D'width:144.75pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge026.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image026.gif"/> </v:shape></p> <p>где <i>J</i> – приведенный момент инерции электродвигателя.</p> <p>Представляя векторы состояния, входа и выхода как <v:shape id=3D"_x0000_= i1052" type=3D"#_x0000_t75" alt=3D"" style=3D'width:270pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge027.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image027.gif"/> </v:shape>получим следующую векторно-матричную модель электродвигателя постоянного тока</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1053" type=3D"#_x0000_t75" alt=3D"" style=3D'width:267pt;height:130.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage028.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image028.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>1.9</p> </td> </tr> </table> <p>То есть для рассматриваемой системы матрицы<span class=3DGramE> <b>А</b>= </span>, <b>В</b>, <b>С</b> векторно-матричной модели будут иметь следующий вид:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1054" type=3D"#_x0000_t75" alt=3D"" style=3D'width:303.75pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage029.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image029.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'> </p> </td> </tr> </table> <p><i>Пример</i> <b>1.3.</b> Построим векторно-матричную модель электромеханического объекта - электропривода постоянного тока, приводящего= в движение через механический редуктор тяжелую платформу. Функциональная схема такого объекта приведена на рис. 1.4. </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1055" type=3D"#_x0000_t75" alt=3D"" style=3D'width:404.25pt;height:318pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage030.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image030.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'> </p> </td> </tr> </table> <p>Здесь легко выделить три функциональных элемента, соответствующие трем в= идам преобразования энергии:</p> <p><i>преобразователь</i>, осуществляющий управляемое преобразование электрической энергии;</p> <p><i>двигатель</i>, выполняющий преобразование электрической энергии в <sp= an class=3DGramE>механическую</span>, - электромеханический преобразователь;</= p> <p><i>механизм</i>, осуществляющий передачу механической энергии от вала двигателя через редуктор к рабочему органу - платформе.</p> <p>При использовании общеизвестных допущений [5] и обозначений координат и параметров такого объекта его динамическое поведение <span class=3DGramE>пр= и</span> МС=3D0 описывается следующей системой линейных дифференциальных уравнений:<= /p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1056" type=3D"#_x0000_t75" alt=3D"" style=3D'width:169.5pt;height:173.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage031.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image031.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>1.10</p> </td> </tr> </table> <p class=3DMsoNormal>Если компонентами вектора состояния выбрать <v:shape i= d=3D"_x0000_i1057" type=3D"#_x0000_t75" alt=3D"" style=3D'width:138.75pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge032.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image032.gif"/> </v:shape>, где <span class=3DGramE><i>U</i></span><i><sub>п</sub></i> – напряжение преобразователя, <i>i<sub>я</sub></i> - ток электродвигателя, <v= :shape id=3D"_x0000_i1058" type=3D"#_x0000_t75" alt=3D"" style=3D'width:15.75pt;h= eight:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge033.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image033.gif"/> </v:shape>- скорость вращения электродвигателя, <i>М<sub>У</sub></i> - моме= нт упругости механизма, <v:shape id=3D"_x0000_i1059" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:17.25pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge034.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image034.gif"/> </v:shape>- скорость вращения механизма, то элементы векторно-матричной мод= ели </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1060" type=3D"#_x0000_t75" alt=3D"" style=3D'width:96.75pt;height:34.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage035.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image035.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>1.11</p> </td> </tr> </table> <p>принимают следующий вид:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1061" type=3D"#_x0000_t75" alt=3D"" style=3D'width:286.5pt;height:170.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage036.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image036.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>1.12</p> </td> </tr> </table> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1062" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:261pt;height:156pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge037.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image037.gif"/> </v:shape></p> <p>После подстановки реальных значений параметров объекта, которые приведен= ы в табл. 1.1, компоненты матриц состояния<span class=3DGramE> <b>А</b></span> и управления <b>В</b> принимают вид (1.13).</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1063" type=3D"#_x0000_t75" alt=3D"" style=3D'width:284.25pt;height:117pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage038.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image038.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>1.13</p> </td> </tr> </table> <p>На рис. 1.5. приведено окно редактирования векторно-матричной модели (1.= 13) в среде <i>Компьютерного комплекса функционального проектирования динамичес= ких систем.</i></p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1064" type=3D"#_x0000_t75" alt=3D"" style=3D'width:450.75pt;height:349.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge039.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image039.gif"/> </v:shape></p> <h3>Контрольные вопросы к лекции № 1.</h3> <p>1. Какие переменные при построении математического описания системы прин= ято называть</p> <p>a) входными переменными;</p> <p>b) выходными переменными;</p> <p>c) переменными состояния?</p> <p>2. Математическое описание объекта с одним входом и одним выходом представлено структурной схемой, содержащей q элементов, представленных передаточной функцией общего вида</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1065" type=3D"#_x0000_t75" alt=3D"" style=3D'width:140.25pt;height:35.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge040.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image040.gif"/> </v:shape></p> <p>Как в этом случае можно определить размерность пространства состояния <v= :shape id=3D"_x0000_i1066" type=3D"#_x0000_t75" alt=3D"" style=3D'width:17.25pt;h= eight:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge041.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image041.gif"/> </v:shape>для описания этого объекта?</p> <p>3. Математическое описание объекта с двумя входами <v:shape id=3D"_x0000= _i1067" type=3D"#_x0000_t75" alt=3D"" style=3D'width:57.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge042.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image042.gif"/> </v:shape>и одним выходом <i>y(t)</i> представлено следующим уравнением в операторной форме</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1068" type=3D"#_x0000_t75" alt=3D"" style=3D'width:252.75pt;height:35.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge043.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image044.gif"/> </v:shape></p> <p>Какова в этом случае будет размерность пространства состояния n для опис= ания этого объекта?</p> <p>4. Выберите из приведенных ниже записей возможные формы представления <i= >уравнения состояния</i> для непрерывных систем.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1069" type=3D"#_x0000_t75" alt=3D"" style=3D'width:344.25pt;height:51.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge044.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image045.gif"/> </v:shape></p> <p>5. Объект управления имеет <i>r</i> – входов, <i>m</i> - выходов, его математическое описание в непрерывном времени содержит n дифференциальных уравнений первого порядка. Какова в этом случае будет размерность <i>матрицы состояния?</i></p> <p>6. Сформируйте векторно-матричную модель фильтра, электрическая схема которого представлена на рис. 1.6.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1070" type=3D"#_x0000_t75" alt=3D"" style=3D'width:227.25pt;height:142.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge045.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image046.gif"/> </v:shape></p> <p>Здесь следует учесть, что</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l8 level1 lfo3;tab-stops:list 36.0pt'>объект имеет один вход = - <i>U<sub>1</sub></i> один выход - <i>i<sub>H<span class=3DGramE> <span style=3D'font-style:= normal; vertical-align:baseline'>;</span></span></sub></i></li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l8 level1 lfo3;tab-stops:list 36.0pt'>все параметры электриче= ской схемы <i>R<sub>1</sub>, R<sub>2</sub>, L, C<sub>1</sub>, C<sub>2</sub>= , R<sub>H</sub></i> известны и являются постоянными; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l8 level1 lfo3;tab-stops:list 36.0pt'>могут быть использованы следующие обозначения <v:shape id=3D"_x0000_i1071" type=3D"#_x0000_t75= " alt=3D"" style=3D'width:155.25pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image046.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image048.g= if"/> </v:shape></li> </ul> <p>7.При составлении математического описания динамических процессов в упру= гом электромеханическом объекте, влючающем в себя электродвигатель постоянного = тока независимого возбуждения (<span class=3DGramE>Ф</span>=3D<i>const</i>) и ме= ханизм, модель которого представляется двухмассовой системой (см. пример 1.3), могут быть использованы следующие переменные:</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l11 level1 lfo4;tab-stops:list 36.0pt'><span class=3DGramE><i= >i</i></span><i><sub>я</sub></i> - ток электродвигателя, </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l11 level1 lfo4;tab-stops:list 36.0pt'><v:shape id=3D"_x0000_= i1072" type=3D"#_x0000_t75" alt=3D"" style=3D'width:15.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image033.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image033.g= if"/> </v:shape>- скорость вращения электродвигателя, </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l11 level1 lfo4;tab-stops:list 36.0pt'><i>М<sub>у</sub></i> – упругий момент механизма, </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l11 level1 lfo4;tab-stops:list 36.0pt'><v:shape id=3D"_x0000_= i1073" type=3D"#_x0000_t75" alt=3D"" style=3D'width:17.25pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image034.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image034.g= if"/> </v:shape>- скорость вращения механизма, </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l11 level1 lfo4;tab-stops:list 36.0pt'><v:shape id=3D"_x0000_= i1074" type=3D"#_x0000_t75" alt=3D"" style=3D'width:11.25pt;height:12.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image047.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image049.g= if"/> </v:shape>- угол поворота ротора электродвигателя, </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l11 level1 lfo4;tab-stops:list 36.0pt'><i>l</i> – линейное перемещение механизма. </li> </ul> <p>Какие из этих переменных, и в какой последовательности включены в состав вектора состояния <v:shape id=3D"_x0000_i1075" type=3D"#_x0000_t75" alt=3D"= " style=3D'width:114pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge048.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image050.gif"/> </v:shape>приведенной ниже векторно-матричной модели?</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1076" type=3D"#_x0000_t75" alt=3D"" style=3D'width:266.25pt;height:149.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge049.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image051.gif"/> </v:shape></p> <h3>ОТВЕТЫ</h3> <div align=3Dcenter> <table class=3DMsoNormalTable border=3D1 cellspacing=3D0 cellpadding=3D0 style=3D'mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>1.</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>a) переменные, характеризующие реакцию системы на входные воздействия;= </p> <p>b) переменные, генерируемые системами, внешними по отношению к исследу= емой системе;</p> <p>c) промежуточные переменные, характеризующие внутреннее состояние сист= емы.</p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>2.</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal><v:shape id=3D"_x0000_i1077" type=3D"#_x0000_t75" al= t=3D"" style=3D'width:57.75pt;height:35.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage050.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image052.gif"= /> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:3'> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>3.</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>n=3D2</p> </td> </tr> <tr style=3D'mso-yfti-irow:4'> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>4.</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>a, e, f, g</p> </td> </tr> <tr style=3D'mso-yfti-irow:5'> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>5.</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>(n x n): n – строк и n – столбцов</p> </td> </tr> <tr style=3D'mso-yfti-irow:6'> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>6.</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal><v:shape id=3D"_x0000_i1078" type=3D"#_x0000_t75" al= t=3D"" style=3D'width:254.25pt;height:156pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage051.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image053.gif"= /> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:7;mso-yfti-lastrow:yes'> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>7.</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal><v:shape id=3D"_x0000_i1079" type=3D"#_x0000_t75" al= t=3D"" style=3D'width:132pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage052.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i1/image054.gif"= /> </v:shape></p> </td> </tr> </table> </div> <p class=3DMsoNormal><o:p> </o:p></p> <h3>Лекция № 2.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Векторно-матричные модели систем управления в дискретном времени»<o:p></o:p></span></b></p> <p>Широкое применение в теории и практике автоматического управления цифров= ых управляющих устройств и систем управления с ЭВМ обусловливает необходимость рассмотрения вопросов построения ВММ непрерывных объектов в дискретном врем= ени.</p> <p>Цифровые системы управления содержат как непрерывные, так и квантованные= или дискретные сигналы. Наличие сигналов различного типа затрудняет описание динамического поведения системы. Однако часто можно ограничиться описанием поведения системы в моменты квантования. В этом случае сигналы выделяются только в дискретные моменты времени. Такие системы называют системами дискретного времени, они оперируют с последовательностями чисел, и, следовательно, для их описания естественно использовать разностные уравнени= я.</p> <p>Проблема заключается в нахождении способа описания непрерывной динамичес= кой системы, связанной с ЭВМ посредством аналого-цифрового (АЦП) и цифроаналого= вого (ЦАП) преобразователей. Рассмотрим схему, показанную на рис. 2.1. Сигналы в= ЭВМ представляют собой последовательности <v:shape id=3D"_x0000_i1080" type=3D"= #_x0000_t75" alt=3D"" style=3D'width:88.5pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge053.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i1.gif"/> </v:shape>, необходимо определить зависимости между ними.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1081" type=3D"#_x0000_t75" alt=3D"" style=3D'width:462pt;height:180pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge054.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i2.gif"/> </v:shape></p> <p>Построение дискретного эквивалента непрерывной системы называется <i>ква= нтованием непрерывной системы.</i></p> <p>В дальнейшем будем считать, что непрерывная система описывается уравнени= ями</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1082" type=3D"#_x0000_t75" alt=3D"" style=3D'width:74.25pt;height:39.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge055.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i3.gif"/> </v:shape></p> <p>Наиболее распространенная ситуация в цифровом управлении состоит в том, = что АЦП сохраняет уровень аналогового сигнала постоянным до тех пор, пока не потребуется новое преобразование</p> <p>Так как управляющий сигнал прерывист, необходимо установить его поведени= е в точках разрыва. Допустим, что сигнал непрерывен справа и представляется дискретным процессом</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1083" type=3D"#_x0000_t75" alt=3D"" style=3D'width:114pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage056.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i4.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.1)</p> </td> </tr> </table> <p>Определим связь между переменными системы в моменты квантования. При заданном состоянии в момент квантования <i>t<sub>k</sub></i> состояние в некоторый момент <i>t</i> можно получить, решив систему (2.1).</p> <p>Первому уравнению системы (2.1) соответствует однородное уравнение</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1084" type=3D"#_x0000_t75" alt=3D"" style=3D'width:63.75pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage057.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i5.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.2)</p> </td> </tr> </table> <p><span class=3DGramE>решение</span> которого равно</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1085" type=3D"#_x0000_t75" alt=3D"" style=3D'width:89.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage058.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i6.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.3)</p> </td> </tr> </table> <p>Обозначим <v:shape id=3D"_x0000_i1086" type=3D"#_x0000_t75" alt=3D"" sty= le=3D'width:91.5pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge059.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i7.gif"/> </v:shape>тогда решение уравнения можно представить как</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1087" type=3D"#_x0000_t75" alt=3D"" style=3D'width:101.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage060.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i8.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.3)</p> </td> </tr> </table> <p>Допустим, что решение неоднородного уравнения имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1088" type=3D"#_x0000_t75" alt=3D"" style=3D'width:170.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage061.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i9.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.4)</p> </td> </tr> </table> <p>Дифференцируя (2.4) по <i>t</i>, получим </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1089" type=3D"#_x0000_t75" alt=3D"" style=3D'width:140.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage062.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i10.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.5)</p> </td> </tr> </table> <p>Сравнение первого уравнения системы (2.1) и уравнения (2.5) позволяет получить следующее соотношение:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1090" type=3D"#_x0000_t75" alt=3D"" style=3D'width:108.75pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage063.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i11.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.6)</p> </td> </tr> </table> <p>откуда</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1091" type=3D"#_x0000_t75" alt=3D"" style=3D'width:162.75pt;height:39pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage064.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i12.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.7)</p> </td> </tr> </table> <p>Подставим (2.7) в (2.4), тогда</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1092" type=3D"#_x0000_t75" alt=3D"" style=3D'width:184.5pt;height:39pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage065.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i13.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.8)</p> </td> </tr> </table> <p>Здесь учтено, что</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1093" type=3D"#_x0000_t75" alt=3D"" style=3D'width:268.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge066.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i14.gif"/> </v:shape></p> <p>Вектор <b>С<span class=3DGramE><sub>2</sub></span></b> определяется по начальным условиям. При <i>t=3Dt<sub>k</sub></i></p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1094" type=3D"#_x0000_t75" alt=3D"" style=3D'width:103.5pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge067.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i15.gif"/> </v:shape></p> <p>Таким образом,</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1095" type=3D"#_x0000_t75" alt=3D"" style=3D'width:195.75pt;height:39pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage068.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i16.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.9)</p> </td> </tr> </table> <p>На основании решения (2.22) определим состояние и выход системы в следую= щий момент квантования<i> t<sub>k+1</sub><span class=3DGramE> <span style=3D'fo= nt-style: normal'>:</span></span></i></p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1096" type=3D"#_x0000_t75" alt=3D"" style=3D'width:198pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage069.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i17.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.10)</p> </td> </tr> </table> <p>где</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1097" type=3D"#_x0000_t75" alt=3D"" style=3D'width:104.25pt;height:57.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage070.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i18.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.11)</p> </td> </tr> </table> <p>Заметим, что характеристики системы между моментами ее квантования дают представление о реакции системы на ступенчатые воздействия с начальными условиями. Это означает, что между моментами квантования система функционир= ует как разомкнутая.</p> <p>Для периодического квантования с периодом <i>Т, t<sub>k</sub>=3DkT</i>, = модель (2.10) сводится к стационарной системе, которую будем рассматривать в дальнейшем. Для простоты записи примем <i>Т</i>=3D1, тогда</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1098" type=3D"#_x0000_t75" alt=3D"" style=3D'width:124.5pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage071.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i19.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.12)</p> </td> </tr> </table> <p>где</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1099" type=3D"#_x0000_t75" alt=3D"" style=3D'width:120pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage072.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i20.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.13)</p> </td> </tr> </table> <p>Если матрица<span class=3DGramE> <b>А</b></span> не особая, то возможна = запись</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1100" type=3D"#_x0000_t75" alt=3D"" style=3D'width:90.75pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage073.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i21.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.14)</p> </td> </tr> </table> <p>где <b>I</b>- единичная матрица.</p> <p>Таким образом, для получения ВММ непрерывной системы в дискретном времени требуется вычислить матричную экспоненту и проинтегрировать ее.</p> <p>Вначале рассмотрим простейший случай. Используя разложение матричной экспоненты в ряд, получим следующие выражения:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1101" type=3D"#_x0000_t75" alt=3D"" style=3D'width:120pt;height:69.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage074.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i22.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.15)</p> </td> </tr> </table> <p>При <i>i</i>=3D1 выражения (2.15) принимают вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1102" type=3D"#_x0000_t75" alt=3D"" style=3D'width:110.25pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage075.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i23.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.16)</p> </td> </tr> </table> <p>По существу такая аппроксимация совпадает с простейшим методом решения д= ифференциальных уравнений по Эйлеру.</p> <p>Модель (2.16) будем называть <i>дискретной моделью по Эйлеру.</i></p> <p>Более точное нахождение матриц <span class=3DGramE><b>Ф</b></span> и <b>= Г</b> может быть осуществлено различными способами, в том числе такими, как:</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l9 level1 lfo5;tab-stops:list 36.0pt'>разложение матричной экспоненты в ряд; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l9 level1 lfo5;tab-stops:list 36.0pt'>применение преобразован= ия Лапласа </li> </ul> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1103" type=3D"#_x0000_t75" alt=3D"" style=3D'width:91.5pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage076.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i24.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.17)</p> </td> </tr> </table> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l13 level1 lfo6;tab-stops:list 36.0pt'>использование теоремы Гамильтона - Кэли. </li> </ul> <p>Применение указанных способов проиллюстрируем примерами.</p> <p><b>Разложение матричной экспоненты в ряд</b></p> <p><b><i>Пример 2.1.</i></b> Получим ВММ в дискретном времени для интеграто= ра второго порядка, структурная схема которого приведена на рис. 2.2. Динамиче= ский процесс для такого объекта <span class=3DGramE>при</span> <v:shape id=3D"_x= 0000_i1104" type=3D"#_x0000_t75" alt=3D"" style=3D'width:51pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge077.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/image026.gif"/> </v:shape>описывается дифференциальным уравнением </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1105" type=3D"#_x0000_t75" alt=3D"" style=3D'width:42.75pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage078.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i26.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.18)</p> </td> </tr> </table> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1106" type=3D"#_x0000_t75" alt=3D"" style=3D'width:153pt;height:98.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge079.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i27.gif"/> </v:shape></p> <p>Вводя <v:shape id=3D"_x0000_i1107" type=3D"#_x0000_t75" alt=3D"" style= =3D'width:32.25pt; height:12pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge080.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i28.gif"/> </v:shape>как состояния системы, получим векторно-матричную запись уравнения (2.18)</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1108" type=3D"#_x0000_t75" alt=3D"" style=3D'width:132.75pt;height:75.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage081.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i29.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.19)</p> </td> </tr> </table> <p>Определим матрицы <span class=3DGramE><b>Ф</b></span> и <b>Г</b> разложе= нием матричной экспоненты в ряд.</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1109" type=3D"#_x0000_t75" alt=3D"" style=3D'width:294pt;height:86.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage082.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i30.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.20)</p> </td> </tr> </table> <p>Так как <b>А</b><span class=3DGramE><sup>2</sup></span>=3D0, приведенный= ряд сходится точно. </p> <p>Таким образом, векторно-матричная модель двойного интегратора в дискретн= ом времени имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1110" type=3D"#_x0000_t75" alt=3D"" style=3D'width:198.75pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage083.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i31.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.21)</p> </td> </tr> </table> <p><b>Использование преобразования Лапласа.</b></p> <p><b><i>Пример 2.2.</i></b> Для непрерывной модели электродвигателя постоянного тока будем считать, что <i>L<sub>я</sub></i>=3D0, <span class= =3DGramE><i>M</i></span><i><sub>с</sub></i>=3D0. Тогда уравнения электродвигателя будут иметь вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"45%" style=3D'width:45.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1111" type=3D"#_x0000_t75" alt=3D"" style=3D'width:115.5pt;height:80.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage084.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i32.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>или</p> </td> <td width=3D"45%" style=3D'width:45.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1112" type=3D"#_x0000_t75" alt=3D"" style=3D'width:150pt;height:65.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage085.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i33.gif"/> </v:shape></p> </td> </tr> </table> <p>В нормализованной форме записи значений параметров электродвигателя, то = есть <span class=3DGramE>при</span> <v:shape id=3D"_x0000_i1113" type=3D"#_x0000= _t75" alt=3D"" style=3D'width:103.5pt;height:16.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge086.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i34.gif"/> </v:shape>в пространстве состояния <v:shape id=3D"_x0000_i1114" type=3D"#_x= 0000_t75" alt=3D"" style=3D'width:210.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge087.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i35.gif"/> </v:shape>получим следующую запись ВММ в непрерывном времени.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1115" type=3D"#_x0000_t75" alt=3D"" style=3D'width:102pt;height:54pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge088.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i36.gif"/> </v:shape></p> <p>Найдем изображение матричной экспоненты по Лапласу:</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1116" type=3D"#_x0000_t75" alt=3D"" style=3D'width:332.25pt;height:63.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge089.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i37.gif"/> </v:shape></p> <p>Вычисляя обратное преобразование элементов полученной матрицы, получим матрицу</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1117" type=3D"#_x0000_t75" alt=3D"" style=3D'width:114pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge090.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i38.gif"/> </v:shape></p> <p>Для нахождения матрицы <b>Г</b> используем выражение (2.32):</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1118" type=3D"#_x0000_t75" alt=3D"" style=3D'width:239.25pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge091.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i39.gif"/> </v:shape></p> <p>Таким образом, векторно-матричная модель рассматриваемой системы в дискретном времени имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1119" type=3D"#_x0000_t75" alt=3D"" style=3D'width:255.75pt;height:63pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage092.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i40.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.22)</p> </td> </tr> </table> <p><b>Использование теоремы Гамильтона - Кэли.</b></p> <p>Для систем с реальными параметрами чаще всего применяют <i>теорему Гамильтона - Кэли</i>, согласно которой любую функцию квадратной матрицы </= p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1120" type=3D"#_x0000_t75" alt=3D"" style=3D'width:153.75pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge093.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i41.gif"/> </v:shape></p> <p>можно вычислить через характеристический полином </p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1121" type=3D"#_x0000_t75" alt=3D"" style=3D'width:154.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge094.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i42.gif"/> </v:shape></p> <p>где <v:shape id=3D"_x0000_i1122" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:74.25pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge095.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i43.gif"/> </v:shape>- корни характеристического уравнения матрицы<span class=3DGramE>= <b>А</b></span> <v:shape id=3D"_x0000_i1123" type=3D"#_x0000_t75" alt=3D"" style=3D'width:8= 1.75pt; height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge096.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i44.gif"/> </v:shape></p> <p><b><i>Пример 2.3.</i></b> Применяя указанную теорему, построим ВММ в дискретном времени для непрерывной системы</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1124" type=3D"#_x0000_t75" alt=3D"" style=3D'width:111.75pt;height:54pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge097.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i45.gif"/> </v:shape></p> <p>Первоначально определим корни характеристического уравнения</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1125" type=3D"#_x0000_t75" alt=3D"" style=3D'width:222pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge098.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i46.gif"/> </v:shape></p> <p><span class=3DGramE>решение</span> которого дает следующие значения:<v:s= hape id=3D"_x0000_i1126" type=3D"#_x0000_t75" alt=3D"" style=3D'width:97.5pt;he= ight:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge099.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i47.gif"/> </v:shape>.</p> <p>Так как порядок матрицы равен 2, имеем<v:shape id=3D"_x0000_i1127" type= =3D"#_x0000_t75" alt=3D"" style=3D'width:79.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge100.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i48.gif"/> </v:shape></p> <p>Коэффициенты <v:shape id=3D"_x0000_i1128" type=3D"#_x0000_t75" alt=3D"" = style=3D'width:29.25pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge101.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i49.gif"/> </v:shape>определяем из системы уравнений</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1129" type=3D"#_x0000_t75" alt=3D"" style=3D'width:106.5pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge102.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i50.gif"/> </v:shape></p> <p><span class=3DGramE>решение</span> которой имеет вид</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1130" type=3D"#_x0000_t75" alt=3D"" style=3D'width:141pt;height:55.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge103.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i51.gif"/> </v:shape></p> <p>Следовательно,</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1131" type=3D"#_x0000_t75" alt=3D"" style=3D'width:442.5pt;height:122.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage104.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i52.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(2.23)</p> </td> </tr> </table> <h3>Контрольные вопросы к лекции № 2.</h3> <p>1. Запишите <i>дискретную модель по Эйлеру</i> для непрерывного объекта представленного следующей векторно-матричной моделью в непрерывном времени<= /p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1132" type=3D"#_x0000_t75" alt=3D"" style=3D'width:168pt;height:36.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge105.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i53.gif"/> </v:shape></p> <p>2. Для непрерывного объекта, представленного на рис. 2.3 в виде структур= ной схемы, определите внутреннее содержание матриц <span class=3DGramE><b>Ф</b>= </span>, <b>Г</b>, <b>С</b> векторно-матричной модели в дискретном времени с периодом квантования <i>Т</i>.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1133" type=3D"#_x0000_t75" alt=3D"" style=3D'width:153pt;height:99pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge106.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i54.gif"/> </v:shape></p> <p>3. Непрерывный объект в дискретном времени с периодом квантования <i>Т</= i> представлен векторно-матричной моделью следующего вида</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1134" type=3D"#_x0000_t75" alt=3D"" style=3D'width:258.75pt;height:36.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge107.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i55.gif"/> </v:shape></p> <p>Запишите уравнение состояния этого объекта в <span class=3DGramE>непреры= вном</span> времении.</p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"15%" style=3D'width:15.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"15%" style=3D'width:15.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1135" type=3D"#_x0000_t75" alt=3D"" style=3D'width:252.75pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage108.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i56.gif"/> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"15%" style=3D'width:15.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1136" type=3D"#_x0000_t75" alt=3D"" style=3D'width:254.25pt;height:68.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage109.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i57.gif"/> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:3;mso-yfti-lastrow:yes'> <td width=3D"15%" style=3D'width:15.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1137" type=3D"#_x0000_t75" alt=3D"" style=3D'width:141.75pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage110.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i2/i58.gif"/> </v:shape></p> </td> </tr> </table> <p class=3DMsoNormal><o:p> </o:p></p> <h3>Лекция № 3.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Вычислительные алгоритмы формирования векторно-матричных моделей в дискретном времени»<o:p></o:p></span></b></p> <p>Рассмотренные примеры формирования векторно-матричных моделей показывают, что указанная процедура является трудоемкой и практически не может быть выполнена для реальных объектов. Поэтому представляют определенный интерес вычислительные алгоритмы построения дискретных моделей. Рассмотрим некоторы= е из них.</p> <p><b><i>Алгоритм 1.</i></b><i> Вычисление матричной экспоненты с помощью степенного ряда при заданном значении периода квантования Т.</i></p> <p>Получение относительно точного решения с помощью приведенного в предыдущ= ей лекции степенного ряда</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1138" type=3D"#_x0000_t75" alt=3D"" style=3D'width:120pt;height:69.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage074.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i1.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.1)</p> </td> </tr> </table> <p>сопряжено с необходимостью вычисления высоких степеней матрицы <span class=3DGramE><b>А</b>.</span> Однако с помощью алгоритма, построенного на основании теоремы Кели-Гамильтона матричная экспонента может быть вычислена= с помощью (<i>n</i>-1) степеней матрицы <b>А</b>.</p> <p>Согласно этому алгоритму вычисление матриц <span class=3DGramE><b>Ф</b><= /span> и <b>Г</b> производится в несколько этапов (шагов).</p> <p><i>Шаг 1</i>: вычисляют первые (<i>n</i> - 1) степеней матрицы <b>A</b>.= </p> <p><i>Шаг 2</i>: вычисляют коэффициенты характеристического уравнения </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1139" type=3D"#_x0000_t75" alt=3D"" style=3D'width:177pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage111.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i2.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.2)</p> </td> </tr> </table> <p>по следующим формулам:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1140" type=3D"#_x0000_t75" alt=3D"" style=3D'width:202.5pt;height:136.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage112.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i3.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.3)</p> </td> </tr> </table> <p>где <b>T<sub>k</sub></b>=3D<i>t<sub>r</sub>(A<sup>k</sup>)</i> - след ма= трицы <b><i>A<sup>k</sup></i></b>. </p> <p><i>Шаг 3</i>: согласно теореме Кели-Гамильтона <i>n</i> - ая степень мат= рицы вычисляется через коэффициенты характеристического уравнения как</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1141" type=3D"#_x0000_t75" alt=3D"" style=3D'width:171pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage113.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i4.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.4)</p> </td> </tr> </table> <p><i>a (n + m)</i>-я степень матрицы <b>A</b> находится с помощью последов= ательного умножения этого соотношения на матрицу <b>A</b>.</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1142" type=3D"#_x0000_t75" alt=3D"" style=3D'width:205.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage114.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i5.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.5)</p> </td> </tr> </table> <p>для <i>m</i>=3D0,1,2...; где</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1143" type=3D"#_x0000_t75" alt=3D"" style=3D'width:75.75pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage115.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i6.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.6)</p> </td> </tr> </table> <p>а остальные коэффициенты определяются из рекуррентных соотношений:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1144" type=3D"#_x0000_t75" alt=3D"" style=3D'width:143.25pt;height:95.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage116.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i7.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.7)</p> </td> </tr> </table> <p><i>Шаг 4</i>: для любого заданного <i>T</i> функцию <i>e</i><b><sup>A</s= up></b><i><sup>T</sup></i> можно записать как</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1145" type=3D"#_x0000_t75" alt=3D"" style=3D'width:397.5pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage117.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i8.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.8)</p> </td> </tr> </table> <p>Таким образом, функцию <i>e</i><b><sup>A</sup></b><i><sup>T</sup></i> мо= жно определить с любой заданной точностью с помощью уже вычисленных матриц <b>A= </b><sup>2</sup>, <b>A</b><sup>3</sup>,..., <b>A</b> <sup>n-1</sup> и коэффициентов q<sub>ij<= /sub> без вычисления и суммирования степеней матрицы <b>A</b> выше <i>(n-1)</i>.<= /p> <p>Интегрирование матричной экспоненты при разложении в ряд заменяется взвешенной суммой</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1146" type=3D"#_x0000_t75" alt=3D"" style=3D'width:275.25pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage118.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i9.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.9)</p> </td> </tr> </table> <p><span class=3DGramE>которая</span> вычисляется аналогично матричной эксп= оненте</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1147" type=3D"#_x0000_t75" alt=3D"" style=3D'width:285.75pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage119.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i10.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.10)</p> </td> </tr> </table> <p>Применение соотношений (3.8-3.10) возможно только в том случае, если зар= анее определена величина периода квантования.</p> <p>К преимуществам данного метода можно отнести простоту алгоритма построен= ия, быструю сходимость и малое время расчета при малых значениях <i>Т</i>. А то обстоятельство, что алгоритм не требует вычисления высоких степеней матрицы, позволяет избежать зацикливания исполняющей программы, если заданная <span class=3DGramE>точность не</span> достигается в результате плохой сходимости степенного ряда. Однако алгоритм усложняется этапами вычисления коэффициент= ов характеристического уравнения и рекуррентных соотношений, требующих введени= я в вычислительный процесс дополнительных переменных.</p> <p>Методы вычисления матричной экспоненты разложением в степенной ряд имеют существенный недостаток - изменение периода квантования требует повторения = всех этапов алгоритма.</p> <p>Кроме того, сходимость алгоритма и точность решения для конкретной модели определятся значением периода квантования <i>Т</i>. Определение критического значения <span class=3DGramE><i>Т</i></span><i><sub>max</sub></i> является проблемой, аналогичной проблеме выбора шага численного решения дифференциал= ьных уравнений с использованием одношаговых методов. Это обстоятельство существе= нно ограничивает использование данного алгоритма конструирования ВММ в дискретн= ом времени.</p> <p><b><i>Алгоритм</i> 2.</b><i> Вычисление матричной экспоненты при неизвес= тном значении периода квантования Т.</i></p> <p>Особую актуальность представляют <i>символьно-численные алгоритмы</i> формирования дискретной ВММ, когда величина периода квантования заранее неизвестна.</p> <p>Для этих целей могут быть использованы представление <i>e</i><b><sup>A</= sup></b><i><sup>T</sup></i> в виде функции от матрицы. Анализ известного математического аппарата вычисления функций от матриц показывает, что наиболее приемлемым методом вычисления матричной экспоненты является формула Сильвестра [12], согласно которой </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1148" type=3D"#_x0000_t75" alt=3D"" style=3D'width:109.5pt;height:35.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage120.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i11.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.11)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1149" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:51.75pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge121.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i22.gif"/> </v:shape>- различные собственные значения матрицы <b>A</b> (то есть <v:sha= pe id=3D"_x0000_i1150" type=3D"#_x0000_t75" alt=3D"" style=3D'width:101.25pt;= height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge122.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i23.gif"/> </v:shape>), <i>m<sub>k</sub></i> - кратность собственного значения <v:shape id=3D"_x0000_i1151" type=3D"#_x0000_t75" alt=3D"" style=3D'width:14.25pt;h= eight:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge123.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i24.gif"/> </v:shape>как нуля минимального многочлена матрицы <b>A</b>, <b>Z</b><i><su= b>kl</sub></i> - компоненты матрицы <b>A</b>, определяемые через значения приведенной присоединенной матрицы.</p> <p>Если характеристический многочлен матрицы<span class=3DGramE> <b>А</b></= span> <v:shape id=3D"_x0000_i1152" type=3D"#_x0000_t75" alt=3D"" style=3D'width:93.75pt;h= eight:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge124.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i13.gif"/> </v:shape>имеет все различные нули, то <i>m<sub>k</sub></i>=3D1 для <i>k</i= >=3D1,2, … ,n и минимальный многочлен совпадает с характеристическим. При этом основ= ная формула теоремы Сильвестра приводится к следующему виду:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1153" type=3D"#_x0000_t75" alt=3D"" style=3D'width:197.25pt;height:52.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage125.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i14.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.12)</p> </td> </tr> </table> <p>Если первоначально считать период квантования неизвестным, то будет целесообразным представить матрицы <span class=3DGramE><b>Ф</b></span> и <b= >Г</b> формируемой ВММ трехмерными <b>Ф</b>(<i>n</i> x <i>n</i> x <i>n</i>), <b>Г<= /b>(<i>n</i> x <i>m</i> x <i>n</i>).</p> <p>В этом случае матрицы дискретной модели записывается как произведение некоторых матричных коэффициентов, умноженных на собственные моды <v:shape id=3D"_x0000_i1154" type=3D"#_x0000_t75" alt=3D"" style=3D'width:101.25pt;= height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge122.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i23.gif"/> </v:shape>, то есть</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1155" type=3D"#_x0000_t75" alt=3D"" style=3D'width:177pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage126.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i15.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.13)</p> </td> </tr> </table> <p>Таким образом, трехмерные матрицы дискретных моделей содержат <i>n</i> квадратных коэффициентных матриц <b>F<i><sub>i</sub></i></b> или <b>G<i><su= b>i</sub></i></b> и при выбранном значении периода квантования <i>Т</i> численное значение каждого элемента матриц вычисляется по формулам</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1156" type=3D"#_x0000_t75" alt=3D"" style=3D'width:254.25pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage127.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i16.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.13)</p> </td> </tr> </table> <p>Причем моды комплексных собственных значений записываются через синус и косинус, например, для <v:shape id=3D"_x0000_i1157" type=3D"#_x0000_t75" al= t=3D"" style=3D'width:318.75pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge128.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i17.gif"/> </v:shape>значения корней характеристического полинома определяются численн= ым методом по алгоритму, в основу которого положен метод нахождения комплексных корней многочлена Берстоу [12].</p> <p>Рациональность такого способа формирования дискретных ВММ объясняется, прежде всего, тем, что наиболее трудоемкие вычислительные операции по определению постоянных матричных коэффициентов выполняются один раз для люб= ого количества исследуемых значений периода квантования. Такого никак нельзя добиться при использовании степенного ряда.</p> <p>Для подтверждения преимуществ символьно-численного алгоритма рассмотрим варианты формирования дискретной модели непрерывного объекта 3-го порядка, представляющего собой линеаризованную модель сериесного электродвигателя.</= p> <p>В непрерывном времени объект описывается векторно-матричной моделью вида= :</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1158" type=3D"#_x0000_t75" alt=3D"" style=3D'width:240pt;height:56.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage129.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i19.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(3.14)</p> </td> </tr> </table> <p>Компонентами вектора состояния здесь являются: магнитный поток - <span class=3DGramE><i>Ф</i></span>, ток - <i>i</i> , скорость вращения ротора -<= v:shape id=3D"_x0000_i1159" type=3D"#_x0000_t75" alt=3D"" style=3D'width:12.75pt;h= eight:12.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge130.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i26.gif"/> </v:shape> .</p> <p>В результате вычислительных экспериментов сформировано выражение для вычисления матричной экспоненты</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1160" type=3D"#_x0000_t75" alt=3D"" style=3D'width:448.5pt;height:119.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge131.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i20.gif"/> </v:shape></p> <p>Многочисленные эксперименты по оценки достоверности результатов конструирования ВММ в дискретном времени показывают, что значения переменных состояния, вычисленные с использование построенной дискретной модели, практически не отличаются от значений координат непрерывной модели. Результ= аты единичного эксперимента приведены на рис. 3.1.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1161" type=3D"#_x0000_t75" alt=3D"" style=3D'width:455.25pt;height:378pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge132.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i18.gif"/> </v:shape></p> <p>На основе вышесказанного можно сделать вывод о предпочтительном использовании символьно-численного алгоритма формирования ВММ в дискретном времени, основанного на теореме Сильвестра, что дает возможность анализиров= ать зависимость результатов вычисления от периода квантования, а вычисленные собственные значения и коэффициенты характеристического уравнения могут быть использованы в дальнейших расчетах.</p> <p>Заканчивая рассмотрение векторно-матричных моделей, целесообразно сформулировать следующие выводы.</p> <p>1. Векторно-матричные модели являются высокоформализованным средством математического описания систем управления, общая структура которого не зав= исит от сложности объекта.</p> <p>2. Проблемы автоматизации проектных процедур анализа и синтеза систем управления на основе их векторно-матричных моделей могут быть решены с помо= щью единого математического аппарата матричной алгебры.</p> <p>3. Проектные операции построения ВММ в непрерывном времени для объектов высокого порядка, сопряженные со значительными временными затратами и определенными математическими затруднениями, должны быть автоматизированы.<= /p> <p>4. Проектные операции построения ВММ в дискретном времени практически не осуществимы без применения ЭВМ.</p> <h3>Контрольные вопросы к лекции № 3.</h3> <p>1. Для построения дискретной модели непрерывного объекта 5-го порядка используется алгоритм вычисления матричной экспоненты с помощью степенного = ряда 25 степени (<i>i</i>=3D0, 1, …, 25). Какую максимальную степень (<i>k</i>)м= атрицы<span class=3DGramE> <b>А</b></span> (<b>A<i><sup>k</sup></i></b>) при этом необх= одимо вычислить?</p> <p>2. Укажите основной недостаток алгоритма вычисления матричной экспоненты= с помощью степенного ряда.</p> <p>3. Назовите главное достоинство символьно-численного алгоритма формирова= ния дискретной векторно-матричной модели.</p> <p>4. При каких условиях для вычисления матрицы состояния <span class=3DGra= mE><b>Ф</b></span> дискретной ВММ с помощью символьно-численного алгоритма дважды используются одинаковые матричные коэффициенты, т.е. <b>F</b><i><sub>k</sub></i>=3D <b>F= <i><sub>k+1</sub></i>.<o:p></o:p></b></p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p><i>k</i>=3D4</p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Изменение периода <i>T</i> квантования требует повторения всех этапов алгоритма</p> </td> </tr> <tr style=3D'mso-yfti-irow:3'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Наиболее трудоемкие вычислительные операции по определению постоянных матричных коэффициентов выполняются один раз для любого количества исследуемых значений периода квантования</p> </td> </tr> <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>4</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>В том случае, если имеются комплексные собственные значения матрицы<sp= an class=3DGramE> А</span> комплексно, т.е.<v:shape id=3D"_x0000_i1162" type= =3D"#_x0000_t75" alt=3D"" style=3D'width:150pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage133.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i3/i21.gif"/> </v:shape> </p> </td> </tr> </table> <h3>Лекция № 4.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Анализ свойств объектов и систем управления»= <o:p></o:p></span></b></p> <p>Здесь мы рассмотрим вопросы анализа специфических свойств объектов и сис= тем управления, представленных векторно-матричными моделями в непрерывном време= ни</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1163" type=3D"#_x0000_t75" alt=3D"" style=3D'width:71.25pt;height:42.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage134.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i1.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.1)</p> </td> </tr> </table> <p>моделями типа "вход - выход"</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1164" type=3D"#_x0000_t75" alt=3D"" style=3D'width:114pt;height:23.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage135.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i2.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.2)</p> </td> </tr> </table> <p>где</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1165" type=3D"#_x0000_t75" alt=3D"" style=3D'width:147.75pt;height:45.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage136.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i3.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>     = ;</p> </td> </tr> </table> <p>а также векторно-матричными моделями в дискретном времени</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1166" type=3D"#_x0000_t75" alt=3D"" style=3D'width:130.5pt;height:39.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage137.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i4.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.3)</p> </td> </tr> </table> <p>Полную картину динамического поведения объектов и систем управления, как= в непрерывном, так и в дискретном времени можно оценить по результатам решения уравнений состояния (4.1) или (4.3).</p> <p>Общее решение уравнений состояния в непрерывном времени было выполнено п= ри рассмотрении алгоритмов вычисления матриц состояния (<b>Ф</b>) и управления= (<b>Г</b>) векторно-матричной модели непрерывного объекта в дискретном времени.</p> <h4>Решение уравнения состояния в дискретном времени.</h4> <p>Дискретная стационарная система может быть описана разностными уравнения= ми (4.3), если значение периода квантования для простоты записи предварительно принято равным <i>Т</i>=3D1.</p> <p>Предположим, что известны начальный вектор <b>x</b>(<i>k<sub>0</sub></i>= ) и входные сигналы: <i>u(k0), u(k0+1), u(k0+2),... .</i></p> <p>Систему уравнений (4.3) можно решить просто, выполнив следующие итерации= :</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1167" type=3D"#_x0000_t75" alt=3D"" style=3D'width:411.75pt;height:1in'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge138.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i5.gif"/> </v:shape></p> <p>или для любого значения <i>k</i> имеем</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1168" type=3D"#_x0000_t75" alt=3D"" style=3D'width:397.5pt;height:79.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage139.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i6.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.4)</p> </td> </tr> </table> <p>Полученное решение (4.4) состоит из двух частей: одна зависит от начальн= ых условий, другая является взвешенной суммой входных сигналов.</p> <h4>Достижимость и управляемость</h4> <p>При решении задач управления методами теории пространства состояний предварительно рассматриваются некоторые фундаментальные свойства динамичес= ких систем, которые не встречаются в классической теории управления, оперирующей только входными и выходными сигналами элементов рассматриваемой системы. Та= кими свойствами являются <i>достижимость, управляемость и наблюдаемость.</i> Нал= ичие этих свойств у объектов управления позволяет рассчитывать оптимальное управление с помощью простых математических операций.</p> <p>Сформулируем определения понятий достижимости и управляемости.</p> <p><b>Определение 4.1.</b> Состояние x(<i>t<sub>1</sub></i>) линейной систе= мы достижимо, если существует момент времени <i>t<sub>0</sub> < t<sub>1</su= b></i> и такой вход, который переводит начальное состояние системы x(<i>t<sub>0</s= ub></i>)=3D0 в желаемое состояние x(<i>t<sub>1</sub></i>), при условии, что интервал <sp= an class=3DGramE>(<i> </i></span><i>t<sub>0</sub> - t<sub>0</sub></i>) конечен= . </p> <p><b>Определение 4.2.</b> Состояние x(<i>t<sub>1</sub></i>) линейной систе= мы управляемо, если существует момент времени <i>t<sub>2</sub> > t<sub>1</s= ub></i> и такой вход, который переводит состояние системы x(<i>t<sub>1</sub></i>) в состояние x(<i>t<sub>2</sub></i>)=3D0 (начало координат), при условии, что интервал (t<sub>2</sub> - t<sub>1</sub>) конечен.</p> <p>Для непрерывных систем вида (4.1) каждое <i>достижимое</i> состояние <i>= управляемо</i>. Поэтому при анализе непрерывных систем говорят только об управляемости.</p> <p>Для исследования достижимости используем векторно-матричную модель объек= та управления (ОУ) в дискретном времени при <i>Т</i>=3D1</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1169" type=3D"#_x0000_t75" alt=3D"" style=3D'width:126pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage140.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i7.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.5)</p> </td> </tr> </table> <p><b>Теорема 4.1.</b> Состояние системы x(<i>n</i>) <i>достижимо</i>, если= и только если ранг матрицы достижимости <v:shape id=3D"_x0000_i1170" type=3D"= #_x0000_t75" alt=3D"" style=3D'width:122.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge141.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i8.gif"/> </v:shape>равен размерности пространства состояний n.</p> <p>Предположим, что задано начальное состояние x(0). Тогда состояние в моме= нт времени <i>n</i> <span class=3DGramE>( </span><i>n</i> - порядок системы) определяе= тся соотношением</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1171" type=3D"#_x0000_t75" alt=3D"" style=3D'width:291.75pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage142.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i9.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.6)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1172" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:114pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge143.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i10.gif"/> </v:shape></p> <p>Если матрица <b>W</b><sub>D</sub> имеет ранг <i>n</i> , то можно найти <= i>n</i> уравнений, решением которых является такой управляющий сигнал, что из начального состояния x(0) система перейдет в желаемое конечное состояние x(= <i>n</i>).</p> <p>Условия управляемости могут быть получены также из выражения (4.6). Из определения 4.2 следует, что <v:shape id=3D"_x0000_i1173" type=3D"#_x0000_t= 75" alt=3D"" style=3D'width:93pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge144.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i11.gif"/> </v:shape>. Тогда</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1174" type=3D"#_x0000_t75" alt=3D"" style=3D'width:216.75pt;height:20.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage145.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i12.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.7)</p> </td> </tr> </table> <p><span class=3DGramE>Для того чтобы система была управляемой, т. е. могла= быть переведена из состояния x(0)<v:shape id=3D"_x0000_i1175" type=3D"#_x0000_t7= 5" alt=3D"" style=3D'width:11.25pt;height:11.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge146.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i39.gif"/> </v:shape> 0 с помощью входной последовательности u(0), ... , u(n-1) в сост= ояние x(n)=3D 0, необходимо, чтобы состояние x(0) принадлежало пространству, натя= нутому на векторы <v:shape id=3D"_x0000_i1176" type=3D"#_x0000_t75" alt=3D"" style= =3D'width:123.75pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge147.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i13.gif"/> </v:shape>Векторы должны быть линейно независимы, так как в противном случае состояние x(n)=3D 0 не может быть достигнуто.</span> Из этого следует:</p> <p>Теорема 4.2. Состояние x(0) <v:shape id=3D"_x0000_i1177" type=3D"#_x0000= _t75" alt=3D"" style=3D'width:11.25pt;height:11.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge146.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i39.gif"/> </v:shape>0 системы (4.5) управляемо, если и только если ранг матрицы <v:sh= ape id=3D"_x0000_i1178" type=3D"#_x0000_t75" alt=3D"" style=3D'width:117.75pt;= height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge148.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i14.gif"/> </v:shape>равен размерности пространства состояний <i>n</i>.</p> <p>Очевидно, что теорема справедлива только при невырожденной матрице Ф. Известно, что ранг матрицы останется неизменным, если ее умножить на невырожденную матрицу. Поэтому, если матрицу управляемости умножить слева н= а Фn , получим матрицу достижимости, то есть при <i>det</i> (<span class=3DGramE= >Ф</span> )<v:shape id=3D"_x0000_i1179" type=3D"#_x0000_t75" alt=3D"" style=3D'width:= 11.25pt; height:11.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge146.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i39.gif"/> </v:shape> 0</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1180" type=3D"#_x0000_t75" alt=3D"" style=3D'width:63.75pt;height:20.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage149.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i15.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.8)</p> </td> </tr> </table> <p>Если матрица <span class=3DGramE>Ф</span> не вырождена, условия достижим= ости и управляемости эквивалентны.</p> <p>В непрерывных системах требование достижимости совпадает с требованием управляемости. Поэтому здесь используют только понятие управляемости, замен= яя его в большинстве случаев понятием достижимости.</p> <p>Теорема 4.3. Состояние непрерывной системы управляемо, если и только если ранг матрицы <v:shape id=3D"_x0000_i1181" type=3D"#_x0000_t75" alt=3D"" sty= le=3D'width:2in; height:20.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge150.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i16.gif"/> </v:shape>равен размерности пространства состояний.</p> <p>Дополнительно можно ввести понятие индекса управляемости системы (4.1). Индексом управляемости системы называется такое минимальное целое число y, = при котором матрица Q<sub>y</sub> ,определяемая выражением <v:shape id=3D"_x000= 0_i1182" type=3D"#_x0000_t75" alt=3D"" style=3D'width:11.25pt;height:11.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge146.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i39.gif"/> </v:shape>, имеет ранг, равный размерности пространства состояний n. В общем случае <v:shape id=3D"_x0000_i1183" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:100.5pt; height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge151.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i18.gif"/> </v:shape>.</p> <p>Если индекс управляемости системы равен рангу матрицы Q<sub>y</sub> (y= =3Dn), в этом случае речь может идти о полной управляемости динамической системы. В противном случае (y < n) система характеризуется неполной управляемостью= .</p> <p>Рассмотрим два характерных случая:</p> <p>а) <span class=3DGramE><b>В</b></span> - матрица-столбец размером n x 1. Матрица управляемости Q<sub>y</sub> в этом случае - квадратная. Вычисляется определитель. Если определитель det(Qy)<v:shape id=3D"_x0000_i1184" type=3D= "#_x0000_t75" alt=3D"" style=3D'width:11.25pt;height:11.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge146.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i39.gif"/> </v:shape> 0, матрица имеет ранг, равный порядку системы. В этом случае сис= тема полностью управляема;</p> <p>б) <span class=3DGramE><b>В</b></span> - <span class=3DGramE>матрица</sp= an> размером n x m , то есть система имеет m каналов управления. Матрица управляемости Q<sub>y</sub> имеет размер n x(n x m): n- строк и n x m столб= цов. В общем случае вычисляются n (m-1)+1 определителей матрицы управляемости порядка n каждый. Если хотя бы один из определителей отличен от нуля, систе= ма будет полностью управляемой и ранг матрицы управляемости равен порядку сист= емы n. </p> <p>Пример 4.1. Определим управляемость системы</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1185" type=3D"#_x0000_t75" alt=3D"" style=3D'width:114.75pt;height:1in'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage152.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i19.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(4.9)</p> </td> </tr> </table> <p>a)</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1186" type=3D"#_x0000_t75" alt=3D"" style=3D'width:339pt;height:1in'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage153.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i20.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'> </p> </td> </tr> </table> <p>матрица управляемости</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1187" type=3D"#_x0000_t75" alt=3D"" style=3D'width:348.75pt;height:56.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge154.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i21.gif"/> </v:shape></p> <p>система управляема.</p> <p>б)</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1188" type=3D"#_x0000_t75" alt=3D"" style=3D'width:472.5pt;height:129pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge155.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i22.gif"/> </v:shape></p> <p>Вычисляются n (m-1)+1=3D4 определителя для матриц Q1, ..., Q4 порядка n= =3D3. Матрица для расчета каждого последующего определителя формируется путем отб= роса первого (в матрицах Q1, ..., Q3) и захвата следующего столбца матрицы управляемости <span class=3DGramE>Q</span>у. Если хотя бы один из определит= елей отличен от нуля, то система управляема. Det(Q1)=3D -16, значит, система, рассматриваемая в примере, управляема, и другие определители можно не вычислять.</p> <p>В случае представления объекта управления моделью типа “вход - выход” (4= .2) условием его управляемости является отсутствие общих корней полиномов<span class=3DGramE> А</span>(s) и B(s), то есть система (4.2) управляема, если и только если алгебраические уравнения</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1189" type=3D"#_x0000_t75" alt=3D"" style=3D'width:161.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage156.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i23.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>4.10</p> </td> </tr> </table> <p class=3DMsoNormal><span style=3D'display:none;mso-hide:all'><o:p> <= /o:p></span></p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1190" type=3D"#_x0000_t75" alt=3D"" style=3D'width:163.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage157.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i24.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>4.11</p> </td> </tr> </table> <p>не имеют общих корней.</p> <p>Данное условие может быть проверено как непосредственным вычислением кор= ней полиномов, так и косвенным путем.</p> <p>Многочлены A(s) и B(s) передаточной функции H(s)=3DB(s)/A(s) имеют, по к= райней мере, один общий корень, если их результант, то есть определитель порядка (m+n), det(R)=3D0, где матрица</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1191" type=3D"#_x0000_t75" alt=3D"" style=3D'width:302.25pt;height:128.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage158.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i25.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>4.12</p> </td> </tr> </table> <p>Таким образом, система, описываемая передаточной функцией H(s) считается управляемой, если ее результант отличен от нуля. Результант имеет порядок m= +n, где m- порядок числителя, n- порядок знаменателя.</p> <p>Результант формируется следующим образом:</p> <p>а) первые m строк результанта - коэффициенты полинома знаменателя ak (k=3D0,1,...,n). При этом коэффициенты rii=3Da0 (i=3D1,2...,m); вправо от а= <span class=3DGramE>0</span> по строке записываются коэффициенты в строке - нулев= ые. Общее число коэффициентов в строке - (n+m).</p> <p>б) следующие n строк результанта формируются аналогично с использованием коэффициентов полинома числителя bk (k=3D0,1,...,m). </p> <p>Пример. 4.2. Определим управляемость системы, представленной передаточной функцией</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1192" type=3D"#_x0000_t75" alt=3D"" style=3D'width:273.75pt;height:1in'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage159.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i26.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>4.13</p> </td> </tr> </table> <p class=3DMsoNormal><span style=3D'display:none;mso-hide:all'><o:p> <= /o:p></span></p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"50%" style=3D'width:50.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1193" type=3D"#_x0000_t75" alt=3D"" style=3D'width:184.5pt;height:144.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage160.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i27.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>, det(R)=3D0, система не управляема!</p> </td> </tr> </table> <p>Прямой расчет корней числителя и знаменателя дает аналогичные результаты, приведенные в табл. 4.1</p> <p align=3Dright style=3D'text-align:right'>Таблица 4.1. Результаты расчета= корней полиномов числителя и знаменателя передаточной функции</p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1194" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:444.75pt;height:102.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge161.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/image028.gif"/> </v:shape></p> <p>Таким образом, числитель и знаменатель передаточной функции H(s) имеют д= ва общих корня (-1.000, -1.414 и -1.000, 1.414). Значит, система не управляема. Изменение значений корней для этих пар в числителе или знаменателе переведет систему в ранг управляемых, а взаимное расположение корней на комплексной плоскости позволит судить о степени управляемости. Естественно, что изменен= ие корней приведет к некоторому изменению самой передаточной функции H(s). Так, для корней числителя, приведенных ниже в таблице, передаточная функция запишется в виде (4.14)</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1195" type=3D"#_x0000_t75" alt=3D"" style=3D'width:481.5pt;height:64.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge162.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/image029.gif"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Наблюдаемость<o:p></o:p></span></b></p> <p>Для осуществления управления необходимо иметь информацию о текущем состо= янии системы, то есть о значениях вектора состояния x(t) в каждый момент времени. Однако некоторые из переменных состояния являются абстрактными, не имеют физических аналогов в реальной системе или же не могут быть измерены. Измеряемыми и наблюдаемыми являются физические выходные переменные y(t).</p> <p>Таким образом, возникает вопрос: можно ли определить вектор состояния по измеряемому вектору выхода и вектору входа?</p> <p>Определение 4.3. Состояние x(t) называется наблюдаемым, если в момент времени наблюдения t=3Dt0 можно однозначно определить x(t0) по данным измер= ения входных u(t) и выходных y(t). переменных на конечном интервале времени.</p> <p>Для выявления формальных условий наблюдаемости также используем модель (4.5). Действие входного сигнала считается известным, поэтому общность реше= ния не пострадает, если предположить, что u(k) =3D 0, где k =3D 0,1, ... , n-1.= </p> <p>Допустим, что даны y(0), y(1), ... , y(n-1), тогда можно записать следую= щую систему уравнений:</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1196" type=3D"#_x0000_t75" alt=3D"" style=3D'width:159.75pt;height:1in'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge163.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i29.gif"/> </v:shape></p> <p>Используя векторную запись, получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1197" type=3D"#_x0000_t75" alt=3D"" style=3D'width:126pt;height:90pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage164.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i30.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>4.15</p> </td> </tr> </table> <p>Состояние x(0) можно получить из (4.15), если матрица наблюдаемости</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1198" type=3D"#_x0000_t75" alt=3D"" style=3D'width:79.5pt;height:96.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage165.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i31.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>4.16</p> </td> </tr> </table> <p>имеет ранг n.</p> <p>Теорема 4.4. Система (4.5) наблюдаема, если и только если ранг матрицы наблюдаемости Wn равен размерности пространства состояний.</p> <p>Аналогично формулируется и условие наблюдаемости для линейных непрерывных стационарных систем.</p> <p>Теорема 4.5. Система (4.1) наблюдаема, если и только если ранг матрицы</= p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1199" type=3D"#_x0000_t75" alt=3D"" style=3D'width:77.25pt;height:96.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage166.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i32.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>4.17</p> </td> </tr> </table> <p>равен размерности пространства состояний.</p> <p>Индексом наблюдаемости системы будет называться такое минимальное целое = число v, при котором матрица Qv, определяемая выражением</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1200" type=3D"#_x0000_t75" alt=3D"" style=3D'width:305.25pt;height:90pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge167.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i33.gif"/> </v:shape></p> <p>имеет ранг, равный n.</p> <p>Если индекс показателя качества системы равен рангу матрицы Qv (v=3Dn), = можно говорить о полной наблюдаемости динамической системы. В противном случае (v < n) речь может идти о неполной наблюдаемости системы, а индекс может использоваться для определения порядка необходимого корректирующего фильтра= .</p> <p>Свойства управляемости и наблюдаемости систем необходимо рассматривать совместно для того, чтобы задача об управлении была корректно поставлена и принципиально разрешима.</p> <h3>Контрольные вопросы к лекции № 4.</h3> <p>1. Система представлена следующей ВММ в дискретном времени</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1201" type=3D"#_x0000_t75" alt=3D"" style=3D'width:305.25pt;height:90pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge167.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i33.gif"/> </v:shape></p> <p>Определите состояние x(2)?</p> <p>2. Система представлена следующей ВММ в дискретном времени</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1202" type=3D"#_x0000_t75" alt=3D"" style=3D'width:321pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge168.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i34.gif"/> </v:shape></p> <p>Определите такую управляющую последовательность, что</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1203" type=3D"#_x0000_t75" alt=3D"" style=3D'width:261.75pt;height:41.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge169.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i35.gif"/> </v:shape></p> <p>3. Для каких систем понятия достижимости и управляемости эквивалентны?</= p> <p>4. Система представлена следующей ВММ в дискретном времени</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1204" type=3D"#_x0000_t75" alt=3D"" style=3D'width:261.75pt;height:41.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge169.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i35.gif"/> </v:shape></p> <p>Оцените достижимость, наблюдаемость и управляемость этой системы.</p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1205" type=3D"#_x0000_t75" alt=3D"" style=3D'width:74.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage170.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i36.gif"/> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1206" type=3D"#_x0000_t75" alt=3D"" style=3D'width:197.25pt;height:60pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage171.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i4/i37.gif"/> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:3'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>a) для непрерывных систем;</p> <p>b) для систем, представленных ВММ с невырожденной матрицей состояния;<= /p> </td> </tr> <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>4</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>система достижима, ненаблюдаема и управляема.</p> </td> </tr> </table> <h3>Лекция № 5.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Канонические формы уравнений состояния»<o:p>= </o:p></span></b></p> <p>Математические модели объектов управления первоначально получаются на ос= нове расчетных данных и физического поведения объекта. В этом случае переменные состояния представляют собой физические переменные объекта, и описания в пространстве состояний объективно связываются с физической реальностью.</p> <p>В некоторых случаях, однако, полезно ввести переменные состояния, которые формально определяются как линейная комбинация различных физических перемен= ных. Такое преобразование выполняется в целях получения определенных канонических форм уравнений состояния, что облегчает обнаружение некоторых свойств объек= та и системы или позволяет описать их с помощью меньшего числа параметров, а так= же установить для односвязных систем (с одним входом и одним выходом) непосредственную связь векторно-матричных моделей с моделями типа “вход - выход”.</p> <p>Рассмотрим <i>n</i>-мерный вектор <v:shape id=3D"_x0000_i1207" type=3D"#= _x0000_t75" alt=3D"" style=3D'width:34.5pt;height:15pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge005.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image001.gif"/> </v:shape>- допустимый вектор состояния некоторой системы и невырожденную матрицу <b>T</b> (n x n). Тогда вектор <b>z=3DTx</b> - также возможный вект= ор состояния рассматриваемой системы.</p> <p>Реальная система с вектором состояния <b>x</b> описывается следующими уравнениями при <b>D</b>=3D0:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1208" type=3D"#_x0000_t75" alt=3D"" style=3D'width:66pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage172.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image002.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.1)</p> </td> </tr> </table> <p>Эта же система с вектором состояния z:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1209" type=3D"#_x0000_t75" alt=3D"" style=3D'width:68.25pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage173.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image003.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.2)</p> </td> </tr> </table> <p>Подставляя выражение <b>x=3DT<sup>- 1</sup> z</b> в (5.1) , получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1210" type=3D"#_x0000_t75" alt=3D"" style=3D'width:99.75pt;height:44.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage174.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image004.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.3)</p> </td> </tr> </table> <p>Умножая первое уравнение (5.3) слева на <b>T</b>, получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1211" type=3D"#_x0000_t75" alt=3D"" style=3D'width:95.25pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage175.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image005.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.4)</p> </td> </tr> </table> <p>Сравнивая (5.4) и (5.2), легко установим, что</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1212" type=3D"#_x0000_t75" alt=3D"" style=3D'width:324.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage176.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image006.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.5)</p> </td> </tr> </table> <p>Таким образом, матрицы <b>А, В, С</b> зависят от используемого координат= ного базиса.</p> <p>Интерес представляют инварианты, полученные после преобразования.</p> <p>Теорема 5.6. Характеристическое уравнение непрерывной <v:shape id=3D"_x0= 000_i1213" type=3D"#_x0000_t75" alt=3D"" style=3D'width:81pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge177.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image007.gif"/> </v:shape>и дискретной <v:shape id=3D"_x0000_i1214" type=3D"#_x0000_t75" al= t=3D"" style=3D'width:81pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge178.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image008.gif"/> </v:shape>систем является инвариантом, если новые состояния вводятся через невырожденную матрицу <b>Т</b>.</p> <p>Для доказательства теоремы запишем характеристический полином матрицы <s= up>z</sup><b>A</b>:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1215" type=3D"#_x0000_t75" alt=3D"" style=3D'width:389.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage179.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image009.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.6)</p> </td> </tr> </table> <p>Таким образом, характеристическое уравнение матрицы состояния и ее собственное значение не зависят от базиса пространства состояний.</p> <p>Следовательно, можно утверждать, что свойства систем не изменяются при изменении базиса пространства состояний.</p> <p>Например, ранг матрицы управляемости не изменяется, так как</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1216" type=3D"#_x0000_t75" alt=3D"" style=3D'width:390pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage180.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image010.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.7)</p> </td> </tr> </table> <p>или в краткой форме</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1217" type=3D"#_x0000_t75" alt=3D"" style=3D'width:57pt;height:20.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage181.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image011.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.8)</p> </td> </tr> </table> <p>Ранг матрицы наблюдаемости также не изменится, поскольку</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1218" type=3D"#_x0000_t75" alt=3D"" style=3D'width:213pt;height:92.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage182.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image012.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.9)</p> </td> </tr> </table> <p>или</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1219" type=3D"#_x0000_t75" alt=3D"" style=3D'width:60pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage183.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image013.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.10)</p> </td> </tr> </table> <p>Из (5.8) и (5.10) могут быть получены полезные соотношения для поиска соответствующей матрицы преобразования:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1220" type=3D"#_x0000_t75" alt=3D"" style=3D'width:190.5pt;height:48.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage184.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image014.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.11)</p> </td> </tr> </table> <p>Особый интерес представляют так называемые канонические формы, названные= в силу их простоты и непосредственной связи элементов матрицы состояния с коэффициентами характеристического уравнения или для односвязных систем с коэффициентами полиномов передаточной функции.</p> <p>Каноническая форма управляемости</p> <p>Здесь и далее остановимся на рассмотрении только односвязных динамических систем.</p> <p>Предположим, что характеристическое уравнение матрицы А имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1221" type=3D"#_x0000_t75" alt=3D"" style=3D'width:246pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage185.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image015.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.12)</p> </td> </tr> </table> <p>и матрица управляемости <v:shape id=3D"_x0000_i1222" type=3D"#_x0000_t75= " alt=3D"" style=3D'width:131.25pt;height:20.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge186.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image016.gif"/> </v:shape>не вырождена. Тогда существует такое преобразование, при котором преобразованная система имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1223" type=3D"#_x0000_t75" alt=3D"" style=3D'width:187.5pt;height:117pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage187.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image017.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.13)</p> </td> </tr> </table> <p>или в компактной форме</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1224" type=3D"#_x0000_t75" alt=3D"" style=3D'width:69.75pt;height:34.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage188.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image018.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.14)</p> </td> </tr> </table> <p>Соответствующая передаточная функция системы, описываемой уравнениями состояния (5.13), имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1225" type=3D"#_x0000_t75" alt=3D"" style=3D'width:163.5pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage189.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image019.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.15)</p> </td> </tr> </table> <p>Матрица преобразования ВММ в каноническую форму управляемости может быть найдена по уравнениям (5.11). Однако этот процесс трудоемкий, поэтому можно использовать другие существующие способы ее определения.</p> <p>Первый способ основан на использовании так называемой матрицы Фробениуса [14], которая представляет собой матрицу состояния системы в не рассматрива= емой здесь канонической форме достижимости, или</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1226" type=3D"#_x0000_t75" alt=3D"" style=3D'width:210.75pt;height:91.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage190.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image020.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.16)</p> </td> </tr> </table> <p>где</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1227" type=3D"#_x0000_t75" alt=3D"" style=3D'width:129.75pt;height:20.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage191.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image021.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.17)</p> </td> </tr> </table> <p>- матрица управляемости размером n x n, а<sub>0</sub>, а<sub>1</sub>,...= , а<sub>n-1</sub> - коэффициенты характеристического уравнения (5.12), которые могут быть най= дены в результате расчета матрицы Фробениуса.</p> <p>Матрица преобразования <sup>v</sup><b>T</b> вычисляется с помощью следую= щего выражения</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1228" type=3D"#_x0000_t75" alt=3D"" style=3D'width:187.5pt;height:92.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage192.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image022.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.18)</p> </td> </tr> </table> <p>Таким образом, алгоритм преобразования ВММ к канонической форме управляемости (5.14) включает в себя следующие операции:</p> <p>1. Вычисление матрицы управляемости <b>Q</b><sub>у</sub> согласно (5.17)= .</p> <p>2. Вычисление матрицы Фробениуса <b>F</b> по формуле (5.16), транспонирование <b>F</b> для определения <sup>v</sup><b>A</b>=3D<b>F</b><s= up>T</sup>.</p> <p>3. Вычисление матрицы <sup>v</sup><b>Q</b> линейного преобразования по формуле (5.18).</p> <p>4. Вычисление матрицы выхода, определяющей выходную переменную y по ново= му вектору состояния z:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1229" type=3D"#_x0000_t75" alt=3D"" style=3D'width:51pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage193.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image023.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.19)</p> </td> </tr> </table> <p>Второй способ позволяет вычислить матрицу преобразования <sup>v</sup><b>= Q</b> рекуррентно по столбцам q<sub>i</sub></p> <p>Действительно, на основании приведенных выше рассуждений можно записать:= </p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1230" type=3D"#_x0000_t75" alt=3D"" style=3D'width:3in;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge194.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image024.gif"/> </v:shape></p> <p>или</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1231" type=3D"#_x0000_t75" alt=3D"" style=3D'width:307.5pt;height:109.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge195.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image025.gif"/> </v:shape></p> <p>Так как вектор <sup>v</sup><b>B</b> известен, то мы можем сначала вычисл= ить</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1232" type=3D"#_x0000_t75" alt=3D"" style=3D'width:68.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge196.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image026.gif"/> </v:shape></p> <p>и далее продолжить вычисления по отдельным столбцам справа налево:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1233" type=3D"#_x0000_t75" alt=3D"" style=3D'width:95.25pt;height:90pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage197.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image027.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.20)</p> </td> </tr> </table> <p>Последняя строка может служить для контроля.</p> <p>Коэффициенты <i>a<sub>i</sub></i>, <i>i</i>=3D0,1,2,...,<i>n</i>-1, матр= ицы <sup>v</sup><b>А</b> можно определить с помощью определителя <v:shape id=3D"_x0000_i1234" type= =3D"#_x0000_t75" alt=3D"" style=3D'width:60pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge198.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image028.gif"/> </v:shape>.</p> <p>Пример 5.1. Выполним переход к канонической форме управляемости для непрерывной ВММ электродвигателя постоянного тока.</p> <p>В пространстве состояния <v:shape id=3D"_x0000_i1235" type=3D"#_x0000_t7= 5" alt=3D"" style=3D'width:195.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge199.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image029.gif"/> </v:shape>получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1236" type=3D"#_x0000_t75" alt=3D"" style=3D'width:375pt;height:75.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage200.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image030.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.21)</p> </td> </tr> </table> <p>Запишем характеристическое уравнение системы</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1237" type=3D"#_x0000_t75" alt=3D"" style=3D'width:264.75pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge201.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image031.gif"/> </v:shape></p> <p>из которого следует, что матрицы состояния <v:shape id=3D"_x0000_i1238" type=3D"#_x0000_t75" alt=3D"" style=3D'width:18pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge202.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image032.gif"/> </v:shape>и управления <v:shape id=3D"_x0000_i1239" type=3D"#_x0000_t75" al= t=3D"" style=3D'width:17.25pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge203.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image033.gif"/> </v:shape>в новом координатном базисе будут иметь следующий вид</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1240" type=3D"#_x0000_t75" alt=3D"" style=3D'width:294pt;height:45.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge204.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image034.gif"/> </v:shape></p> <p>Определим матрицу преобразования <b>Q</b>.</p> <p>Первый способ. Используем выражение (5.18) , которое для нашего примера запишется следующим образом</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1241" type=3D"#_x0000_t75" alt=3D"" style=3D'width:380.25pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge205.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image035.gif"/> </v:shape></p> <p>Такой же результат получается и по второму способу. Из выражения (5.20) получаем</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1242" type=3D"#_x0000_t75" alt=3D"" style=3D'width:329.25pt;height:62.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge206.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image036.gif"/> </v:shape></p> <p>Теперь определим матрицу <sup>v</sup><b>C</b> в новом координатном базис= е</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1243" type=3D"#_x0000_t75" alt=3D"" style=3D'width:186.75pt;height:62.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge207.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image037.gif"/> </v:shape></p> <p>Таким образом, ВММ нашего объекта в канонической форме управляемости принимает следующий вид</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1244" type=3D"#_x0000_t75" alt=3D"" style=3D'width:168pt;height:90.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge208.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image038.gif"/> </v:shape></p> <h3>Каноническая форма наблюдаемости</h3> <p>Если желательно иметь матрицу в простейшей форме, то можно воспользовать= ся так называемой канонической формой наблюдаемости:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1245" type=3D"#_x0000_t75" alt=3D"" style=3D'width:198.75pt;height:117pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage209.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image039.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.22)</p> </td> </tr> </table> <p>или в компактной форме</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1246" type=3D"#_x0000_t75" alt=3D"" style=3D'width:67.5pt;height:34.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage210.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image040.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.23)</p> </td> </tr> </table> <p>Как видно из выражений (5.13) и (5.22), матрицы состояний обеих канониче= ских форм идентичны, то есть <v:shape id=3D"_x0000_i1247" type=3D"#_x0000_t75" a= lt=3D"" style=3D'width:39.75pt;height:15pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge211.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image041.gif"/> </v:shape>.</p> <p>Для канонической формы наблюдаемости матрица наблюдаемости - единичная матрица, то есть</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1248" type=3D"#_x0000_t75" alt=3D"" style=3D'width:111pt;height:90pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage212.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image042.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.24)</p> </td> </tr> </table> <p>Для нахождения матрицы <sup>N</sup><b>B</b>=3D <sup>N</sup><b>TB</b>=3D = <sup>N</sup><b>Q</b> <sup>-1</sup><b>B</b> матрица <sup>N</sup><b>Т</b> может быть вычислена согласно выражению (5.11):</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1249" type=3D"#_x0000_t75" alt=3D"" style=3D'width:93.75pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge213.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image043.gif"/> </v:shape></p> <p>Так как в нашем случае <v:shape id=3D"_x0000_i1250" type=3D"#_x0000_t75"= alt=3D"" style=3D'width:45pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge214.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image044.gif"/> </v:shape>, получаем, что матрица преобразования ВММ в каноническую форму наблюдаемости (5.22) соответствует матрице наблюдаемости исходной системы</= p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1251" type=3D"#_x0000_t75" alt=3D"" style=3D'width:96.75pt;height:90pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage215.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image045.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.25)</p> </td> </tr> </table> <p>В таком случае выражение для вычисления матрицы <sup>N</sup><b>B</b> принимает следующий вид:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1252" type=3D"#_x0000_t75" alt=3D"" style=3D'width:105pt;height:90pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage216.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image046.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(5.26)</p> </td> </tr> </table> <p>Пример 5.2. Выполним переход к канонической форме наблюдаемости для непрерывной ВММ электродвигателя постоянного тока (5.21).</p> <p>Здесь нам необходимо только определить новое содержание матрицы <sup>N</= sup><b>B</b>. Для этого воспользуемся выражением (5.26)</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1253" type=3D"#_x0000_t75" alt=3D"" style=3D'width:149.25pt;height:44.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge217.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image047.gif"/> </v:shape></p> <p>Таким образом, ВММ нашего объекта в канонической форме наблюдаемости принимает следующий вид</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1254" type=3D"#_x0000_t75" alt=3D"" style=3D'width:176.25pt;height:90.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge218.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image048.gif"/> </v:shape></p> <p>Основные свойства объектов и систем управления можно оценить в автоматизированном режиме с помощью подсистемы Анализ Компьютерного комплек= са функционального проектирования динамических систем (FuncPro 1.0). На рис. 5= .1 приведено основное окно подсистемы, в котором раскрыто меню «Анализ»</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1255" type=3D"#_x0000_t75" alt=3D"" style=3D'width:384pt;height:235.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge219.jpg" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image049.jpg"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 5.1. Основное окно подси= стемы «Анализ»</p> <h3>Контрольные вопросы к лекции № 5.</h3> <p>1. Какие свойства и характеристики системы изменятся после преобразовани= я ее векторно-матричной модели к новому координатному базису <v:shape id=3D"_x00= 00_i1256" type=3D"#_x0000_t75" alt=3D"" style=3D'width:36pt;height:12.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge220.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image050.gif"/> </v:shape>с помощью невырожденной матрицы <b>Т</b>.</p> <p>2. Допустим, что нам необходимо привести ВММ непрерывной системы с одним входом и одним выходом</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1257" type=3D"#_x0000_t75" alt=3D"" style=3D'width:65.25pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge221.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image051.gif"/> </v:shape></p> <p>к канонической форме управляемости. Предварительно определены коэффициен= ты характеристического уравнения системы и матрица преобразования <v:shape id= =3D"_x0000_i1258" type=3D"#_x0000_t75" alt=3D"" style=3D'width:39.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge222.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image052.gif"/> </v:shape>. Для вычисления какой матрицы преобразованной системы будет использована матрица <b>Q</b>.</p> <p>3. Допустим, что нам необходимо привести ВММ непрерывной системы с одним входом и одним выходом</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1259" type=3D"#_x0000_t75" alt=3D"" style=3D'width:65.25pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge221.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image051.gif"/> </v:shape></p> <p>к канонической форме наблюдаемости. Предварительно определены коэффициен= ты характеристического уравнения системы и матрица преобразования <v:shape id= =3D"_x0000_i1260" type=3D"#_x0000_t75" alt=3D"" style=3D'width:39.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge222.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image052.gif"/> </v:shape>. Для вычисления какой матрицы преобразованной системы будет использована матрица <b>Q</b>.</p> <p>4. Какова будет матрица наблюдаемости непрерывной системы</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1261" type=3D"#_x0000_t75" alt=3D"" style=3D'width:162pt;height:73.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge223.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image053.gif"/> </v:shape></p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>внутреннее содержание матриц состояния <b>А(Ф)</b>, управления по состоянию <b>В(Г)</b>, выхода <b>С</b>.</p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>матрицы выхода по состоянию <v:shape id=3D"_x0000_i1262" type=3D"#_x00= 00_t75" alt=3D"" style=3D'width:18.75pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage224.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image054.gif"= /> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:3'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>матрицы управления <v:shape id=3D"_x0000_i1263" type=3D"#_x0000_t75" a= lt=3D"" style=3D'width:18.75pt;height:15pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage225.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image055.gif"= /> </v:shape>;</p> </td> </tr> <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>4</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1264" type=3D"#_x0000_t75" alt=3D"" style=3D'width:83.25pt;height:56.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage226.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i5/image056.gif"= /> </v:shape></p> </td> </tr> </table> <h3>Лекция № 6.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Критерии качества систем управления»<o:p></o= :p></span></b></p> <p>Процесс функционального проектирования систем управления неразрывно связ= ан с решением задачи оптимизации управления, то есть задачи оптимального достиже= ния главной цели при соблюдении множества ограничений. В общем случае <i>цель управления</i> заключается в том, чтобы перевести объект из начального состояния <b>x</b>(<i>t<sub>0</sub></i>), в котором он находится в момент <= i>t<sub>0</sub></i>, в конечное состояние <b>x</b>(<i>t<sub>2</sub></i>), принадлежащее подоблас= ти <i>R<sub>1</sub></i> области допустимых состояний <i>R</i> , то есть <b>x</b>(<i>t<sub>2</sub></= i>) <v:shape id=3D"_x0000_i1265" type=3D"#_x0000_t75" alt=3D"" style=3D'width:9.75pt;he= ight:9.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge227.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i1.gif"/> </v:shape>R1. Здесь <i>R<sub>1</sub></i> <v:shape id=3D"_x0000_i1266" type= =3D"#_x0000_t75" alt=3D"" style=3D'width:9.75pt;height:9.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge227.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i1.gif"/> </v:shape><i>R</i>, что соответствует выделению в пространстве состояний области допустимых состояний R и сужению ее до некоторой области <i>R<sub>1= </sub></i>, которая для нас по каким-то причинам является желательной.</p> <p><i>Задача управления</i> заключается в том, чтобы в области допустимых управлений <i>Q</i>(<b>u</b>) найти такое управление, при котором будет достигнута цель.</p> <p>Функционалы и функции, выражающие цель управления и ограничения, называю= т <i>критериями качества</i>.</p> <p>Операция формирования критерия качества управления является наиболее ответственной на подготовительной стадии проектирования. Поэтому наиболее оптимальное ее выполнение достигается путем сочетания формализованных метод= ов и творческой деятельности проектировщика, работающего в диалоге с вычислитель= ной системой.</p> <p>Рассмотрим некоторые методические аспекты формирования критериев качества управления для решения задач предварительного синтеза системы управления.</= p> <p>Качество управления можно описать двумя способами.</p> <p><i>Первый способ</i> предусматривает или непосредственное задание динамических характеристик выходных координат системы при типовых воздейств= иях, или задание совокупности прямых и косвенных показателей качества (значение перерегулирования, времени регулирования, статической ошибки, частоты среза, полосы пропускания и т.д.).</p> <p><i>Второй способ</i> основан на введении некоторого обобщенного функцион= ала, определяемого всеми переменными системы управления <b>u</b>(<i>t</i>), <b>x= </b>(<i>t</i>), <b>y</b>(<i>t</i>).</p> <p>В теории линейных систем управления широко используются оба указанных способа.</p> <h4>Оптимальное распределение полюсов системы управления</h4> <p>При описании объекта управления с помощью векторно-матричной модели в пространстве состояний первый способ задания качества управления может быть трансформирован в оптимальное распределение на комплексной плоскости полюсов замкнутой системы. </p> <p>Для системы, описываемой векторно-матричной моделью в непрерывном времен= и</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1267" type=3D"#_x0000_t75" alt=3D"" style=3D'width:62.25pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage228.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i2.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(6.1)</p> </td> </tr> </table> <p><i>полюса системы</i> - это собственные значения матрицы А, которые обыч= но обозначаются через <v:shape id=3D"_x0000_i1268" type=3D"#_x0000_t75" alt=3D= "" style=3D'width:12.75pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge229.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i3.gif"/> </v:shape>(<b>A</b>), где <i>i</i>=3D1, 2,...,<i> n</i>. В то же время собственными значениями матрицы <b>А</b> называются корни ее характеристического уравнения</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1269" type=3D"#_x0000_t75" alt=3D"" style=3D'width:237pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage230.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i4.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(6.2)</p> </td> </tr> </table> <p class=3DMsoNormal>Для односвязных систем, которые могут быть описаны общ= ей передаточной функцией </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1270" type=3D"#_x0000_t75" alt=3D"" style=3D'width:244.5pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage231.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i5.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(6.3)</p> </td> </tr> </table> <p><i>полюса системы</i> - это корни характеристического многочлена <i>А(s)= </i>. Соответственно <i>нулями</i> системы называются корни многочлена <i>В(s)</i= >, при которых <i>Н(s)</i>=3D0.</p> <p>Расположение полюсов на комплексной плоскости во многом характеризует синтезируемую систему, определяя ее переходные и частотные характеристики, = а, следовательно, и динамические показатели качества. Так, например, устойчиво= сть системы определяется размещением полюсов в левой полуплоскости.</p> <p>Наличие нулей в замкнутой системе в определенной степени влияет на ее динамику.</p> <p>Синтезу регуляторов состояния предшествует решение задачи построения эталонной модели системы управления, которая соответствует желаемому распределению на комплексной плоскости корней характеристического уравнения= <i>A(s)</i> =3D 0 замкнутой системы. Если все составляющие вектора состояния объекта мо= гут быть измерены (имеется полная информация о векторе состояния), то обеспечен= ие заданного расположения корней не вызывает трудностей. В этом случае возника= ет вопрос о том, какое расположение корней выбрать.</p> <p>Если передаточная функция замкнутой системы не имеет нулей, то при выбор= е ее желаемого полинома <i>A(s)</i> можно руководствоваться стандартными формами (фильтрами), которые нашли достаточно широкое применение на практике. Стандартные формы определяют коэффициенты характеристического полинома (знаменателя) функции <i>Н(s)</i>, обеспечивающие в системе переходные и частотные характеристики с известными показателями качества. Если же система характеризуется наличием нулей, стандартные формы могут служить в качестве исходного материала для поиска своего <i>оптимального</i> расположения корн= ей. Как правило, в характеристическом полиноме сначала выделяются полюса для компенсации нулей, а оставшийся полином формируется из условия желаемого расположения корней.</p> <p>В табл. 6.1 и 6.2 приводятся формулы характеристического полинома и соответствующие им коэффициенты для некоторых наиболее распространенных на практике распределений:/ </p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l4 level1 lfo7;tab-stops:list 36.0pt'>распределения Бесселя (томсоновская функция), </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l4 level1 lfo7;tab-stops:list 36.0pt'>фильтра Чебышева (неравномерность передачи 0.5 дБ), </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l4 level1 lfo7;tab-stops:list 36.0pt'>фильтра Баттерворта, </= li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l4 level1 lfo7;tab-stops:list 36.0pt'>биномиального распредел= ения (Ньютона). </li> </ul> <p>Часто понятие оптимального переходного процесса связывают с минимизацией какого-либо функционала. Подобные стандартные формы получены эмпирически, и область их применения ограничивается системами невысокого порядка. К таким фильтрам относятся приведенные в табл. 6.3 распределения, минимизирующие интеграл от квадрата ошибки и оптимизирующий функционал. </p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1271" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:490.5pt;height:321.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge232.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/image006.gif"/> </v:shape></p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1272" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:482.25pt;height:369.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge233.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/image005.gif"/> </v:shape></p> <p>Качество работы системы управления характеризуется, с одной стороны, ее точностью в установившемся режиме, а c другой - переходным процессом от одн= ого установившегося состояния к другому. При исследовании переходных процессов = чаще всего полагают, что входной сигнал является единичной ступенчатой функцией.= В этом случае кривая переходного процесса называется переходной функцией и характеризуется некоторыми показателями , принимаемыми за меру качества сис= темы управления. К числу таких показателей могут быть отнесены (рис. 6.1):</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l2 level1 lfo8;tab-stops:list 36.0pt'>время нарастания tн - время, необходимое для достижения 95% конечного значения; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l2 level1 lfo8;tab-stops:list 36.0pt'>время установления (регулирования) tу - время, необходимое для попадания в некоторую окрестность конечного значения без выхода из него; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l2 level1 lfo8;tab-stops:list 36.0pt'>перерегулирование - максимальная относительная величина выброса (%) <v:shape id=3D"_x0000_= i1273" type=3D"#_x0000_t75" alt=3D"" style=3D'width:98.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image234.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i9.gif"/> </v:shape>; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l2 level1 lfo8;tab-stops:list 36.0pt'>пульсации (колебания) - число колебаний до выхода кривой в установившийся режим. </li> </ul> <p>Перерегулирование и колебания - нежелательные свойства фильтра.</p> <p class=3DMsoNormal>Типовые переходные характеристики для различных фильтр= ов при входном ступенчатом сигнале качественно представлены на рис. 6.2. </p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1274" type=3D"#_x0000_t75" alt=3D"" style=3D'width:471pt;height:300.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge235.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i10.gif"/> </v:shape></p> <p class=3DMsoNormal>Для распределений Чебышева, Баттерворта, распределения, минимизирующего оптимизирующий функционал, в характеристиках наблюдаются колебательные выбросы, которые проявляются в результате нелинейности их фазочастотных характеристик (ФЧХ). В наибольшей степени это проявляется для фильтра Чебышева (рис. 6.3, а). ФЧХ является важным параметром фильтра, обеспечивающим передачу прямоугольных и импульсных сигналов. Обеспечение максимально линейной зависимости от частоты фазового сдвига между входным и выходным сигналами помогает избежать проявления в переходных характеристиках нежелательных колебательных выбросов. Чем более нелинейна ФЧХ, тем сильнее будет искажаться выходной сигнал. То есть в идеальном случае характеристика должна иметь вид прямой фильтры аппроксимируют желаемую линейную ФЧХ. </p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1275" type=3D"#_x0000_t75" alt=3D"" style=3D'width:490.5pt;height:204pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge236.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i11.gif"/> </v:shape></p> <p class=3DMsoNormal>Отсутствие перерегулирования в переходных характеристи= ках для распределения Бесселя и биномиального распределения показывают, насколь= ко хорошо эти фильтры аппроксимируют желаемую линейную ФЧХ. Однако ФЧХ не описывает полностью свойства передачи фильтра. Другим важным фактором оценки фильтра является его амплитудно-частотная характеристика (АЧХ) (рис. 6.3, б= ). Идеальный фильтр (рис. 6.4) характеризуется следующими показателями: </p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l1 level1 lfo9;tab-stops:list 36.0pt'>нулевыми потерями и пульсациями в полосе пропускания; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l1 level1 lfo9;tab-stops:list 36.0pt'>нулевой шириной в переходной области (бесконечная крутизна на частоте среза); </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l1 level1 lfo9;tab-stops:list 36.0pt'>бесконечным затуханием в полосе пропускания. </li> </ul> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1276" type=3D"#_x0000_t75" alt=3D"" style=3D'width:324pt;height:207pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge237.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i12.gif"/> </v:shape></p> <p>Как видно из рис. 6.3, наиболее полно сформулированным требованиям отвеч= ает фильтр Баттерворта, имеющий максимально плоскую АЧХ в полосе пропускания и достаточно хорошую крутизну затухания. Распределение Чебышева используется в том случае, когда более важным параметром выступает крутизна нарастания затухания. Высокую скорость нарастания затухания обеспечивает отсутствие гладкой характеристики в полосе пропускания. Наихудшие показатели по АЧХ у фильтра Бесселя. Следует заметить, что при улучшении АЧХ фазочастотная характеристика ухудшается и наоборот. Поэтому между ними важно найти компро= мисс.</p> <p>При синтезе системы управления среди частотных и временных характеристик предпочтение отдается последним, т.к. переходные кривые позволяют более наглядно представить поведение системы с учетом всестороннего влияния линей= ных и нелинейных внешних факторов. Сравнительная таблица примерных показателей качества переходных процессов для стандартных распределений (табл. 6.4) , а также рис. 6.2 дают наглядное представление о преимуществах или недостатках каждого из рассматриваемых здесь фильтров. Как видно, наилучшими показателя= ми во временной области обладает фильтр Бесселя. Широкое распространение получ= или также распределение Баттерворта и распределение, минимизирующее оптимизирую= щий функционал. В любом случае к выбору расположения корней следует подходить и= сходя из конкретных целей и задач и с учетом свойств объекта проектирования.</p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1277" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:481.5pt;height:422.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge238.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/image008.gif"/> </v:shape></p> <h3>Контрольные вопросы к лекции № 6.</h3> <p>1. Объект управления первоначально представлен передаточной функций</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1278" type=3D"#_x0000_t75" alt=3D"" style=3D'width:207.75pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge239.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i6/i13.gif"/> </v:shape></p> <p>Какое число корней Вы должны расположить на комплексной плоскости для задания желаемого динамического качества проектируемой системы управления?<= /p> <p>2. Технологические требования к проектируемой системе управления не допускают перерегулирования входного сигнала y(t) более 1% при ступенчатом входном воздействии u(t). Какое стандартное распределение полюсов следует выбрать в качестве критерия качества управления?</p> <p>3. Какой из рассмотренных здесь фильтров имеет амплитудно-частотную характеристику максимально приближающуюся к АЧХ идеального фильтра?</p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>n=3D4</p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>a) распределение Бесселя;</p> <p>b) биномиальное распределение.</p> </td> </tr> <tr style=3D'mso-yfti-irow:3;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Фильтр Баттерворта.</p> </td> </tr> </table> <h3>Лекция № 7.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Критерии качества систем управления» (продол= жение)<o:p></o:p></span></b></p> <p><b>Выбор желаемого характеристического полинома с помощью среднегеометрического корня</b></p> <p>Выражение характеристического полинома в нормированном виде может быть записано через значение среднегеометрического корня <v:shape id=3D"_x0000_i= 1279" type=3D"#_x0000_t75" alt=3D"" style=3D'width:17.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge240.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image001.gif"/> </v:shape>, характеризующего быстроту протекания переходного процесса: </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1280" type=3D"#_x0000_t75" alt=3D"" style=3D'width:281.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage241.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image002.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.1)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1281" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:39pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge242.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image003.gif"/> </v:shape>коэффициенты полинома из таблиц стандартных распределений. На рис. 7.1 и 7.2 показано влияние изменения значения на форму и качество временных= и частотных характеристик (на примере фильтра Баттерворта). </p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1282" type=3D"#_x0000_t75" alt=3D"" style=3D'width:438.75pt;height:213.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge243.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image004.gif"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 7.1. Зависимость переход= ных характеристик от значения среднегеометрического корня <v:shape id=3D"_x0000= _i1283" type=3D"#_x0000_t75" alt=3D"" style=3D'width:54.75pt;height:16.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge244.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image005.gif"/> </v:shape>(а) и от порядка системы n =3D 2...7 (б) на примере фильтра Батте= рворта</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1284" type=3D"#_x0000_t75" alt=3D"" style=3D'width:408.75pt;height:177.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge245.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image006.gif"/> </v:shape></p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'>Рис. 7.2. З= ависимость частотных характеристик АЧХ (а) и ФЧХ (б) от среднегеометрического корня <v= :shape id=3D"_x0000_i1285" type=3D"#_x0000_t75" alt=3D"" style=3D'width:17.25pt;h= eight:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge240.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image001.gif"/> </v:shape>=3D 2...8 на примере фильтра Баттерворта</p> <p>Рост <v:shape id=3D"_x0000_i1286" type=3D"#_x0000_t75" alt=3D"" style=3D= 'width:17.25pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge240.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image001.gif"/> </v:shape>способствует улучшению динамики системы за счет уменьшения времени нарастания и времени установления. Но к вопросу повышения значения среднегеометрического корня следует подходить осторожно, учитывая изменение частотных свойств системы. Необходимо найти распределение корней, позволяющ= ее системе использовать максимально возможный диапазон частот и амплитуд задаю= щего сигнала в области линейной зоны.</p> <p>Степень характеристического полинома задает порядок синтезируемой систем= ы. Для фильтров высокого порядка реальные АЧХ значительно улучшаются (рис. 7.2= ). В то же время высокий порядок связан с усложнением схемной реализации и, как следствие этого, с повышением стоимости. Кроме того, с ростом порядка систе= мы несколько ухудшаются ее некоторые динамические показатели. Функции высокого порядка имеют более значительный затухающий процесс.</p> <p>Таким образом, синтез должен предусматривает решение задачи построения системы управления минимального порядка, удовлетворяющей заданным требовани= ям. Правильный выбор порядка системы и расположения корней (полюсов) на комплек= сной плоскости является важным моментом на начальном этапе проектирования и оказывает влияние на последующую работу на этапах анализа и синтеза систем управления.</p> <p>На рис. 7.3 показано сравнительное расположение корней для рассматриваем= ых здесь распределений 4-го порядка. При этом следует напомнить, что удаленнос= ть корня в левой полуплоскости от мнимой оси характеризует степень устойчивости системы. Для биномиального распределения характерно расположение всех корней характеристического полинома в комплексной плоскости на прямой, параллельной действительной оси, со значением модуля, соответствующего заданной степени устойчивости. Переходный процесс имеет апериодический характер.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1287" type=3D"#_x0000_t75" alt=3D"" style=3D'width:427.5pt;height:287.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge246.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image007.gif"/> </v:shape></p> <p>Для других фильтров корни характеристического полинома располагаются на полуокружности в левой полуплоскости. Возможное пересечение полуокружности с действительной осью характеризует единственный действительный корень полино= ма, остальные корни - комплексно-сопряженные, равномерно распределенные на полу= окружности.</p> <p>По мере роста порядка системы вещественная часть комплексного корня уменьшается по модулю, а мнимая возрастает. А это, в свою очередь, вызывает уменьшение степени устойчивости замкнутой системы и увеличение ее колебательности. Переходные процессы для таких фильтров по сравнению с биномиальным распределением корней будут более колебательными.</p> <p>Желаемое распределение полюсов проектируемой системы управления достаточ= но просто выбрать в среде компьютерного комплекса FuncPro. На рис. 7.5 представлено окно формирования желаемого распределения полюсов, в котором д= ля системы 5-го порядка выбран фильтр Баттерворта, и приведены динамические характеристики эталонной модели для различных значений среднегеометрического корня.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1288" type=3D"#_x0000_t75" alt=3D"" style=3D'width:417.75pt;height:293.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge247.jpg" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image008.jpg"/> </v:shape></p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'>Рис. 7.5. О= кно формирования оптимального распределения полюсов проектируемой системы управления</p> <p>Обобщенный функционал качества управления</p> <p>Для детерминированных процессов с непрерывным временем, описываемых в пространстве состояний, функционал качества в общем случае имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1289" type=3D"#_x0000_t75" alt=3D"" style=3D'width:207pt;height:48.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage248.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image009.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.2)</p> </td> </tr> </table> <p>Здесь V3 - заданная с точностью до матрицы параметров <b>Q</b>k скалярная функция конечного состояния - терминальная функция; L - скалярная функция, действующая из пространств состояния, управления; <b>Q</b> - матрица параме= тров зависимости L от x; <b>R</b> - матрица параметров зависимости L от u.</p> <p>Функционал (7.2) можно назвать классическим. Он использовался в классиче= ских задачах вариационного исчисления:</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l0 level1 lfo10;tab-stops:list 36.0pt'>в задаче Больца, где V= 3<v:shape id=3D"_x0000_i1290" type=3D"#_x0000_t75" alt=3D"" style=3D'width:11.2= 5pt; height:12pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image249.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image010.g= if"/> </v:shape>0, L<v:shape id=3D"_x0000_i1291" type=3D"#_x0000_t75" alt=3D= "" style=3D'width:11.25pt;height:12pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image249.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image010.g= if"/> </v:shape>0; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l0 level1 lfo10;tab-stops:list 36.0pt'>в задаче Лагранжа, где V3=3D0, L<v:shape id=3D"_x0000_i1292" type=3D"#_x0000_t75" alt=3D"" st= yle=3D'width:11.25pt; height:12pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image249.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image010.g= if"/> </v:shape>0; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l0 level1 lfo10;tab-stops:list 36.0pt'>в задаче Майера, где V= 3<v:shape id=3D"_x0000_i1293" type=3D"#_x0000_t75" alt=3D"" style=3D'width:11.2= 5pt; height:12pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.file= s/image249.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image010.g= if"/> </v:shape>0, L=3D0. </li> </ul> <p>Опустим векторы параметров как аргументы функций в (7.2). Тогда функцион= ал Больца запишется в общей форме как</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1294" type=3D"#_x0000_t75" alt=3D"" style=3D'width:162pt;height:48.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage250.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image011.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.3)</p> </td> </tr> </table> <p>На практике применяют частные формы классического функционала (7.3). Некоторые из этих форм рассмотрим ниже.</p> <p>Классический функционал с аддитивной функцией затрат на управление</p> <p>Функция L может быть представлена в виде суммы функций:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1295" type=3D"#_x0000_t75" alt=3D"" style=3D'width:180pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage251.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image012.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.4)</p> </td> </tr> </table> <p>Функция <v:shape id=3D"_x0000_i1296" type=3D"#_x0000_t75" alt=3D"" style= =3D'width:42.75pt; height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge252.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image013.gif"/> </v:shape>, как правило, имеет смысл тех или иных затрат на управление. Поэ= тому функционал</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1297" type=3D"#_x0000_t75" alt=3D"" style=3D'width:213pt;height:45.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage253.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image014.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.5)</p> </td> </tr> </table> <p>называют классическим функционалом с аддитивной функцией затрат на управление.</p> <p>Классический квадратичный функционал (Летова- Калмана) </p> <p>Пусть все три функции V3, Q3, U3 выражаются квадратичными формами:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1298" type=3D"#_x0000_t75" alt=3D"" style=3D'width:315pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage254.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image015.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.6)</p> </td> </tr> </table> <p>где <b>Q</b><sub>k</sub>, <b>Q</b> - положительно полуопределенные квадратные n x n- матрицы; <b>R</b> - положительно определенная квадратная = r x r - матрица; <b>Q</b> и <b>R</b> могут быть нестационарными матрицами.</p> <p>Напомним, что некоторую матрицу <b>F</b>(n x n) называют положительно полуопределенной (неотрицательно определенной), если она симметрична, то ес= ть F=3DF<sup>T</sup>, и при любом <i>n</i>-мерном векторе x <v:shape id=3D"_x0= 000_i1299" type=3D"#_x0000_t75" alt=3D"" style=3D'width:11.25pt;height:12pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge249.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image010.gif"/> </v:shape>0 выполняется неравенство x<sup>T</sup>Fx <v:shape id=3D"_x0000_i= 1300" type=3D"#_x0000_t75" alt=3D"" style=3D'width:9.75pt;height:12pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge255.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image016.gif"/> </v:shape>0.</p> <p>Положительно определенная матрица F в тех же условиях обладает свойством xTFx > 0. </p> <p>Соответствующий функционал</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1301" type=3D"#_x0000_t75" alt=3D"" style=3D'width:286.5pt;height:45.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage256.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image017.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.7)</p> </td> </tr> </table> <p>является классическим квадратичным функционалом.</p> <p>Проанализируем смысловое содержание функционала (7.7).</p> <p>Первое слагаемое в его составе характеризует ошибку управления в конечный момент времени tk и используется с целью обеспечить малость этой ошибки.</p> <p>Второе слагаемое оценивает отклонения реальных переменных состояния от желаемых и представляет собой своеобразный “штраф” за большие ошибки при лю= бом t0 <v:shape id=3D"_x0000_i1302" type=3D"#_x0000_t75" alt=3D"" style=3D'widt= h:10.5pt; height:10.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge257.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image018.gif"/> </v:shape>t <v:shape id=3D"_x0000_i1303" type=3D"#_x0000_t75" alt=3D"" styl= e=3D'width:10.5pt; height:10.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge257.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image018.gif"/> </v:shape>tk.</p> <p>Последнее слагаемое, будучи всегда положительным, оценивает стоимость управления. Физически оно характеризует затрагиваемую энергию на управление= .</p> <p>Основное затруднение формирования квадратичного критерия качества связан= о с выбором элементов весовых матриц <b>Q</b> и <b>R</b>.</p> <p>Предварительный выбор значений элементов весовых матриц <b>Q</b> и <b>R<= /b>. может быть осуществлен с помощью следующих рекомендаций.</p> <p>1. Обычно матрицы <b>Q</b> и <b>R</b> назначаются постоянными и диагональными, т. е. матрица <b>Q</b> - содержит n ненулевых элементов qii, i=3D1, 2,..., n, а матрица <b>R</b> - r элементов rjj, j=3D1,2,...,n.</p> <p>2. Принимаем, что максимально допустимые отклонения переменных состояния x(t) в любой момент времени вносят в функционал качества одинаковый вклад. Распространяя аналогичные рассуждения и на отклонения сигналов управления u= (t), можно записать</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1304" type=3D"#_x0000_t75" alt=3D"" style=3D'width:175.5pt;height:41.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage258.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image019.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.8)</p> </td> </tr> </table> <p>Здесь хimax - максимально допустимое отклонение i-й переменной состояния (i=3D2, 3, ..., n), определяемое техническим заданием; uj max - максимально допустимое отклонение j-го сигнала управления (j=3D2,3, ..., r) согласно техническому заданию.</p> <p>Общий вклад максимально допустимых отклонений переменных состояния должен приблизительно соответствовать общему вкладу максимально допустимых отклоне= ний сигналов управления.</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1305" type=3D"#_x0000_t75" alt=3D"" style=3D'width:108pt;height:34.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage259.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image020.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(7.9)</p> </td> </tr> </table> <p>3. Произвольно выбираем значение r11 (например, r11=3D1), и по формулам = (7.8), (7.9) вычисляем значения остальных коэффициентов.</p> <p>Полученные значения весовых коэффициентов следует рассматривать как начальные оценки. Если отклонения по всем переменным достигают своего макси= мума не одновременно, формулы (7.8), (7.9) неверно отражают требования, предъявляемые к системе. Поэтому окончательный выбор весовых коэффициентов целесообразно производить после нескольких пробных процедур синтеза и моделирования системы управления.</p> <h3>Контрольные вопросы к лекции № 7.</h3> <p class=3DMsoNormal><v:shape id=3D"_x0000_i1306" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:24pt;height:24pt'/>для увеличения быстродействия эталонной модели, построенной на основе стандартного фильтра Баттерворта? </p> <p>2. Поясните физический смысл составляющих классического квадратичного функционала</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1307" type=3D"#_x0000_t75" alt=3D"" style=3D'width:278.25pt;height:39pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge260.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image021.gif"/> </v:shape></p> <p>3. Какой формы и какого размера должны быть матрицы весовых коэффициенто= в <b>Q</b> и <b>R</b>., используемые для вычисления классического квадратичного функционала?</p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Первое слагаемое - ошибка управления в конечный момент времени tk и исполь-зуется с целью минимизации этой ошибки.</p> <p>Второе слагаемое - отклонение реальных переменных состояния от желаемы= х и представляет собой своеобразный “штраф” за большие ошибки при любом t0 <v= :shape id=3D"_x0000_i1308" type=3D"#_x0000_t75" alt=3D"" style=3D'width:10.5pt;= height:10.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage257.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image018.gif"= /> </v:shape>t <v:shape id=3D"_x0000_i1309" type=3D"#_x0000_t75" alt=3D"" st= yle=3D'width:10.5pt; height:10.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage257.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i7/image018.gif"= /> </v:shape>tk.</p> <p>Третье слагаемое оценивает стоимость управления, физически характеризу= ет за-трагиваемую энергию на управление.</p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>a) оценка управляемости объекта;</p> <p>b) приведение векторно-матричной модели объекта управления к каноничес= кой форме управляемости;</p> </td> </tr> <tr style=3D'mso-yfti-irow:3;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Матрицы <b>Q</b> и <b>R</b> назначаются постоянными и диагональными, т= . е. матрица <b>Q</b> (n x n) - содержит n ненулевых элементов qii, i=3D1, 2,.= .., n, а матрица <b>R</b>(r x r) - r элементов rjj, j=3D1,2,...,r.</p> </td> </tr> </table> <h3>Лекция № 8.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Принципы детерминированного синтеза систем управления»<o:p></o:p></span></b></p> <p>Понятие “синтез” означает нахождение такой структуры и параметров системы управления, при которых выходные переменные объекта управления (ОУ) отвечают задан-ным требованиям или критериям качества. Решение проблем автоматизации процедуры син-теза систем управления (СУ) сопряжено с построением универсал= ьных алгоритмов, обеспе-чивающих поиск оптимальных структуры и параметров СУ, не зависимо от сложности ОУ.</p> <p>Очевидно, что универсальность алгоритмов и их инвариантность к сложности= ОУ может быть достигнута только при использовании высокоформализованных упроще= нных математических моделей ОУ, которыми являются векторно-матричные модели.</p> <p>Если синтез осуществляется в пространстве состояний, то при известном состоянии x(t1) в момент времени t1 должна существовать возможность нахожде= ния такого управления, которое будет удовлетворять требованиям, налагаемым на в= ыход объекта. Эти требования формируются с помощью критериев качества управления, которые рассмотрены в предыду-щем разделе.</p> <p>В большинстве практических задачах синтеза СУ возникает противоречивая ситуа-ция: объект управления обладает нелинейными свойствами, устройства управления предпо-лагается реализовать с помощью средств цифровой (микропроцессорной) техники, а наибо-лее простыми, доступными и эффективными являются алгоритмы и программы синтеза СУ с помощью линейных непрерывных моделей.</p> <p>В том случае, когда ВММ удовлетворяет требованиям адекватности описания нели-нейного ОУ и требованиям представления дискретного управляющего устрой= ства примене-ние указанных алгоритмов вполне оправдано и наиболее рационально. Однако и при невы-полнении этих требований выполнение процедуры синтеза СУ целесообразно начать с ис-пользования ВММ с последующим их усложнением до требуемого вида. Таким образом, не-смотря на техническую сложность современ= ных ЭМС, нелинейные свойства ОУ и дискрет-ный характер управления, алгоритмы синтеза и анализа СУ, основанные на матричных опе-рациях с векторно-матричн= ыми моделями, не потеряли свою актуальность.</p> <p>Этап функционального проектирования систем управления предусматривает выпол-нение следующих проектных операций: </p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l10 level1 lfo11;tab-stops:list 36.0pt'>синтез СУ с регулятор= ом состояния; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l10 level1 lfo11;tab-stops:list 36.0pt'>синтез оптимального управления; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l10 level1 lfo11;tab-stops:list 36.0pt'>синтез наблюдателей состояния; </li> </ul> <h3>Синтез системы управления с регулятором состояния</h3> <p>В линейном случае мы всегда выражаем вектор входа через линейную комбина= цию компонент вектора состояния, т.е. в непрерывном времени с помощью уравнения= </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1310" type=3D"#_x0000_t75" alt=3D"" style=3D'width:104.25pt;height:16.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage261.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image001.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.1)</p> </td> </tr> </table> <p>а в дискретном случае с помощью уравнения</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1311" type=3D"#_x0000_t75" alt=3D"" style=3D'width:110.25pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage262.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image002.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.2)</p> </td> </tr> </table> <p>где r(t), r(k) - задающая переменная.</p> <p>Следует отметить, что уравнения (8.1), (8.2) имеют вид уравнений выхода векторно-матричной модели в пространстве состояний и, таким образом, управляющее устройство, определяемое уравнениями (8.1), (8.2), является статическим. Требуемый астатизм обес-печивается дополнительным включением в= СУ элементов, обладающих интегрирующими свойствами.</p> <p>Для решения задачи синтеза в такой постановке, необходимо измерение всех компо-нент вектора состояния объекта x(t). Если состояние объекта неизмеряе= мо, его надо оценить. В детерминированных системах это осуществляется с помощью наблюдателя. Сначала оце-нивается вектор состояния <v:shape id=3D"_x0000_i1= 312" type=3D"#_x0000_t75" alt=3D"" style=3D'width:9.75pt;height:12.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge263.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image003.gif"/> </v:shape>(t), а затем рассчитывается вектор входа объекта</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1313" type=3D"#_x0000_t75" alt=3D"" style=3D'width:102.75pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage264.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image004.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.3)</p> </td> </tr> </table> <p>Одним из фундаментальных методов проектирования детерминированных систем управления в пространстве состояний является метод расположения полюсов.</p> <p>Как было отмечено в разделе. 3, в линейных системах качество управления и динами-ческие показатели системы можно задать с помощью корней характеристического уравнения или полинома замкнутой системы.</p> <p>Общая постановка задачи. Для стационарного непрерывного управляемого объекта, уравнение динамики которого имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1314" type=3D"#_x0000_t75" alt=3D"" style=3D'width:63pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage265.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image005.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.4)</p> </td> </tr> </table> <p>и управляющего устройства, описываемого уравнением</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1315" type=3D"#_x0000_t75" alt=3D"" style=3D'width:61.5pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage266.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image006.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.5)</p> </td> </tr> </table> <p>необходимо определить матрицу К, такую, чтобы замкнутая система, получае= мая подстанов-кой (8.5) в (8.4),</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1316" type=3D"#_x0000_t75" alt=3D"" style=3D'width:156pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage267.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image007.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.6)</p> </td> </tr> </table> <p>имела желаемый характеристический полином</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1317" type=3D"#_x0000_t75" alt=3D"" style=3D'width:227.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage268.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image008.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.7)</p> </td> </tr> </table> <p>Общая схема системы управления будет иметь вид, представленный в среде компью-терного комплекса FuncPro 1.0 на рис 8.1.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1318" type=3D"#_x0000_t75" alt=3D"" style=3D'width:431.25pt;height:301.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge269.jpg" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image009.jpg"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 8.1. Схема системы управ= ления с регулятором состояния</p> <p>Методические аспекты выбора желаемого полинома (8.7) подробно изложены в лек-ции 6.. Здесь же рассмотрим ряд алгоритмов вычисления матрицы коэффицие= нтов обратных связей К.</p> <p>Алгоритм 1. Рассмотрим управляемый объект с одним входом и одним выходом. Подстановкой</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1319" type=3D"#_x0000_t75" alt=3D"" style=3D'width:39pt;height:15pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage270.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image010.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.8)</p> </td> </tr> </table> <p>преобразуем его к канонической форме управляемости</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1320" type=3D"#_x0000_t75" alt=3D"" style=3D'width:180.75pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage271.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image011.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.9)</p> </td> </tr> </table> <p>Для линейных стационарных систем характер свободного движения не изменит= ся, ес-ли выбрать вместо (8.5) регулятор вида</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1321" type=3D"#_x0000_t75" alt=3D"" style=3D'width:51.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage272.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image012.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.10)</p> </td> </tr> </table> <p>или</p> <p align=3Dcenter style=3D'text-align:center'><v:shape id=3D"_x0000_i1322" = type=3D"#_x0000_t75" alt=3D"" style=3D'width:48pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge273.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image013.gif"/> </v:shape>, где <v:shape id=3D"_x0000_i1323" type=3D"#_x0000_t75" alt=3D"" = style=3D'width:85.5pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge274.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image014.gif"/> </v:shape>.</p> <p>Подставляя (8.10) в (8.9), получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1324" type=3D"#_x0000_t75" alt=3D"" style=3D'width:263.25pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage275.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image015.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.11)</p> </td> </tr> </table> <p>В этом случае характеристический полином имеет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1325" type=3D"#_x0000_t75" alt=3D"" style=3D'width:290.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage276.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image016.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.12)</p> </td> </tr> </table> <p>Сопоставляя полиномы (8.12) и (8.7), получаем соотношения для вычисления коэф-фициентов матрицы регулятора kRT:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1326" type=3D"#_x0000_t75" alt=3D"" style=3D'width:155.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage277.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image017.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.13)</p> </td> </tr> </table> <p>Связь между управляющей переменной u и вектором состояния х определяется со-гласно выражению (8.10):</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1327" type=3D"#_x0000_t75" alt=3D"" style=3D'width:129.75pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage278.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image018.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.14)</p> </td> </tr> </table> <p>то есть</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1328" type=3D"#_x0000_t75" alt=3D"" style=3D'width:50.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage279.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image019.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.15)</p> </td> </tr> </table> <p>Из этого следует, что для вычисления коэффициентов обратных связей нужно опреде-лить еще и матрицу преобразования <v:shape id=3D"_x0000_i1329" type= =3D"#_x0000_t75" alt=3D"" style=3D'width:44.25pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge280.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image020.gif"/> </v:shape></p> <p>Таким образом, алгоритм 1 синтеза регулятора состояния может быть сформулирован следующим образом:</p> <p>1. На основании требований технического задания к динамическим характеристикам проектируемой СУ формируется критерий качества управления в форме характеристического полинома (8.13), т.е. определяются коэффициенты d= 0, d1, d2, ..., dn-1.</p> <p>2. Векторно-матричная модель объекта преобразуется к канонической форме управляемо-сти. В процессе преобразования определяются коэффициенты характеристического урав-нения a0 , a1 , a2 , ..., an-1 , прямая и обратная матрицы преобразования <v:shape id=3D"_x0000_i1330" type=3D"#_x0000_t75" al= t=3D"" style=3D'width:65.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge281.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image021.gif"/> </v:shape></p> <p>3. На основании выражения (8.13) вычисляются коэффициенты матрицы <v:sha= pe id=3D"_x0000_i1331" type=3D"#_x0000_t75" alt=3D"" style=3D'width:15.75pt;h= eight:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge282.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image022.gif"/> </v:shape>. </p> <p>4. По формуле (8.15) определяется матрица реальных коэффициентов обратных связей К по полному вектору состояния.</p> <p>Алгоритм 2. Общность постановки задачи не нарушится, если считать, что оптималь-ные динамические характеристики проектируемой системы задаются в в= иде эталонной моде-ли</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1332" type=3D"#_x0000_t75" alt=3D"" style=3D'width:70.5pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage283.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image023.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.16)</p> </td> </tr> </table> <p>имеющей тот же порядок, что и модель объекта (8.4).</p> <p>Тогда коэффициенты регулятора состояния однозначно определятся путем реш= ения уравнения</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1333" type=3D"#_x0000_t75" alt=3D"" style=3D'width:113.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage284.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image024.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.17)</p> </td> </tr> </table> <p>Процесс синтеза сводится к двум задачам:</p> <p>1) определение эталонной модели (8.16);</p> <p>2) решение уравнения (8.17).</p> <p>Наиболее просто желаемое качество управления можно задать с помощью эталонной модели, представленной в канонической форме управляемости </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1334" type=3D"#_x0000_t75" alt=3D"" style=3D'width:221.25pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage285.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image025.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.18)</p> </td> </tr> </table> <p>Коэффициенты аэ0, аэ1, ... , аэ n-1 в этом случае соответствуют коэффициентам d0, d1, ..., dn-1 желаемого характеристического полинома (8.7= ).</p> <p>Для согласования объекта и эталонной модели введем линейное преобразован= ие</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1335" type=3D"#_x0000_t75" alt=3D"" style=3D'width:70.5pt;height:16.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage286.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image026.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.19)</p> </td> </tr> </table> <p>После преобразования (8.9) система “объект + регулятор” (8.6) относитель= но нового базиса имеет вид</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1336" type=3D"#_x0000_t75" alt=3D"" style=3D'width:223.5pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge287.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image027.gif"/> </v:shape></p> <p>Из условия равенства собственных движений синтезируемой системы и эталон= ной модели следует, что <v:shape id=3D"_x0000_i1337" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:70.5pt;height:16.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge286.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image026.gif"/> </v:shape>.</p> <p>Откуда получается выражение для вычисления значений матрицы обратных свя= зей К регулятора состояния:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1338" type=3D"#_x0000_t75" alt=3D"" style=3D'width:92.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage288.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image029.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.20)</p> </td> </tr> </table> <p>Таким образом, для синтеза регулятора состояния в этом случае необходимо= :</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l7 level1 lfo12;tab-stops:list 36.0pt'>задать векторно-матрич= ную модель объекта управления; </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l7 level1 lfo12;tab-stops:list 36.0pt'>сформировать эталонную модель в канонической форме управляемости (8.18); </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l7 level1 lfo12;tab-stops:list 36.0pt'>вычислить матрицы преобразования ОУ к канонической форме управляемости Q и Q - 1. </li> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l7 level1 lfo12;tab-stops:list 36.0pt'>из матричного выражения (8.20) вычислить К. </li> </ul> <p>Алгоритм 3. Если матрица входов В ВММ объекта управления имеет больше од= ного ненулевого элемента, то применение алгоритма 2 вызывает определенные затруднения. Для их устранения используем следующую методику.</p> <p>Представим эталонную матрицу как сумму двух матриц:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1339" type=3D"#_x0000_t75" alt=3D"" style=3D'width:83.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage289.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image030.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.21)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1340" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:131.25pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge290.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image031.gif"/> </v:shape>- коэффициентные матрицы объекта управления в канониче-ской форме управляемости; Кk - матрица коэффициентов обратных связей в канонической фо= рме.</p> <p>Подставляя (8.21) в (8.20):</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1341" type=3D"#_x0000_t75" alt=3D"" style=3D'width:274.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge291.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image032.gif"/> </v:shape></p> <p>получим </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1342" type=3D"#_x0000_t75" alt=3D"" style=3D'width:58.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage292.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image033.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.22)</p> </td> </tr> </table> <p>Так как <v:shape id=3D"_x0000_i1343" type=3D"#_x0000_t75" alt=3D"" style= =3D'width:135pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge293.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image034.gif"/> </v:shape>то согласно (8.21) элементы матрицы Кk можно опреде-лить из уравн= ения</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1344" type=3D"#_x0000_t75" alt=3D"" style=3D'width:275.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage294.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image035.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(8.23)</p> </td> </tr> </table> <p>Итак, алгоритм 3 формируется как последовательность следующих операций:<= /p> <p>1. Вычисление матриц преобразования ОУ к канонической форме управляемост= и Q и Q-1.</p> <p>2. Решение уравнения (8.23) относительно Кk.</p> <p>3. Вычисление реальных коэффициентов обратных связей (матрицы К) по форм= уле (8.22).</p> <p>Сравнительно высокая сложность приведенных выше алгоритмов делает практически неосуществимыми проектные операции синтеза регулятора состояния= без ЭВМ.</p> <p>Компьютерная реализация представленных выше алгоритмов синтеза регулятора со-стояния при построенной ВММ ОУ затруднений не вызывает, так как все вычисления по-строены на основе типовых операций обработки матриц и матричн= ой алгебры.</p> <p>Эффективность проектных решений, полученных в результате вычислительных экс-периментов, выполняемых с помощью компьютерных средств реализации указа= нных алго-ритмов, может быть повышена за счет включения пользователя в заключительный этап уточнения параметров регулятора состояния (РС).</p> <p>В практических ситуациях часто нет необходимости использовать в дальнейш= их опе-рациях точные значения расчетных параметров РС. Это объясняется возможн= ыми погрешно-стями и трудностями реализации. Поэтому вполне оправдано уже на начальных стадиях про-ектирования "поиграть" параметрами РС и дополнительных устройств, а в отдельных случа-ях даже сократить число обрат= ных связей СУ.</p> <p>Пример 8.1. Для упругого электромеханического объекта, который подробно пред-ставлен в лекции 1, синтезируем регулятор состояния.</p> <p>При построении ВММ откажемся от учета инерционности преобразователя и вв= едем следующие компоненты вектора состояния <v:shape id=3D"_x0000_i1345" type=3D= "#_x0000_t75" alt=3D"" style=3D'width:102.75pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge295.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image036.gif"/> </v:shape>, где <v:shape id=3D"_x0000_i1346" type=3D"#_x0000_t75" alt=3D"" = style=3D'width:24pt; height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge296.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image037.gif"/> </v:shape>- скорость враще-ния платформы, F- усилие, под действие которого вращается платформа, <v:shape id=3D"_x0000_i1347" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:23.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge297.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image038.gif"/> </v:shape>- скорость вращения электродвигателя, i - ток электродвигателя.</= p> <p align=3Dright style=3D'text-align:right'>Таблица 8.1. Параметры электромеханического объекта</p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1348" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:481.5pt;height:93pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge298.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image039.gif"/> </v:shape></p> <p>Векторно-матричная модель объекта с учетом численных значений параметров пре-образователя, электродвигателя и механизма, приведенных в табл. 8.1., принимает следую-щий вид:</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1349" type=3D"#_x0000_t75" alt=3D"" style=3D'width:236.25pt;height:92.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge299.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image040.gif"/> </v:shape></p> <p>Желаемые динамические показатели управления были определены с помощью инст-рументальных средств формирования критерия качества путем некоторой модификации стандартного распределения Баттерворта для среднегеометрического корня <v:shape id=3D"_x0000_i1350" type=3D"#_x0000_t75" alt=3D"" style=3D'w= idth:53.25pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge300.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image041.gif"/> </v:shape>. В ре-зультате был выбран желаемый характеристический полином</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1351" type=3D"#_x0000_t75" alt=3D"" style=3D'width:276pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge301.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image042.gif"/> </v:shape></p> <p>которому соответствует эталонная модель, представленная в канонической ф= орме</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1352" type=3D"#_x0000_t75" alt=3D"" style=3D'width:306.75pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge302.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image043.gif"/> </v:shape></p> <p>Вычислительные эксперименты, выполненные с помощью компьютерного комплек= са функционального проектирования СУ, позволили получить расчетные параметры регулятора состояния в виде значений коэффициентов обратных связей по векто= ру состояния, которые приведены во второй строке табл. 8.2.</p> <p align=3Dright style=3D'text-align:right'>Таблица 8.2. Параметры регулято= ра состояния</p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1353" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:481.5pt;height:83.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge303.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image044.gif"/> </v:shape></p> <p>Анализ полученных результатов показывает, что для обеспечения заданного качества динамических характеристик СУ электромеханическим объектом необход= имо введение сла-бой положительной обратной связи по току электродвигателя, что нежелательно. В этой свя-зи было принято решение в дальнейшем отказаться от реализации обратной связи по току, а значения других коэффициентов округлит= ь в допустимых пределах. Выбранные значения параметров дальнейшей реализации регулятора состояния приведены в третьей строке табл. 8.2.</p> <p>Обоснованность принятого решения доказывают приведенные на рис. 8.2 предвари-тельные результаты вычислительных экспериментов с моделями синтезированных систем управления. Суммарное среднеквадратичное отклонения выходной координаты не превыша-ет здесь 1.5%.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1354" type=3D"#_x0000_t75" alt=3D"" style=3D'width:387pt;height:302.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge304.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image045.gif"/> </v:shape></p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1355" type=3D"#_x0000_t75" alt=3D"" style=3D'width:386.25pt;height:307.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge305.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image046.gif"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'><v:shape id=3D"_x0000_i1356" = type=3D"#_x0000_t75" alt=3D"" style=3D'width:67.5pt;height:3.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge306.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image047.gif"/> </v:shape>- расчетные параметры регулятора состояния</p> <p align=3Dcenter style=3D'text-align:center'><v:shape id=3D"_x0000_i1357" = type=3D"#_x0000_t75" alt=3D"" style=3D'width:67.5pt;height:3.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge307.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image048.gif"/> </v:shape>- параметры реализации</p> <p align=3Dcenter style=3D'text-align:center'>Рис. 8.2. Сравнительные динам= ические характеристики СУ</p> <h3>Контрольные вопросы к лекции № 8.</h3> <p>1. Для объекта, описанного ВММ, матрицы которой имеют размеры А(7x7), В(7x1), С(1x7), синтезирован регулятор состояния при допущении, что все переменные со-стояния измеряемы. Каков будет размер матрицы состояния систе= мы «объект – регу-лятор»?</p> <p>2. Какие операции должны быть обязательно выполнены при реализации любог= о из ал-горитмов синтеза регулятора состояния?</p> <p>3. Каковы будут значения параметров регулятора состояния, синтезированно= го для объ-екта? Векторно-матричная модель объекта и эталонная модель системы управления приведены ниже.</p> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 6.45pt 0cm = 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"50%" style=3D'width:50.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1. ВММ объекта управления</= p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2. Эталонная модель СУ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"50%" style=3D'width:50.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1358" type=3D"#_x0000_t75" alt=3D"" style=3D'width:177.75pt;height:75.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage308.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image049.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>4</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1359" type=3D"#_x0000_t75" alt=3D"" style=3D'width:157.5pt;height:55.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage309.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image050.gif"= /> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:2;mso-yfti-lastrow:yes'> <td style=3D'padding:0cm 0cm 0cm 0cm'> <p class=3DMsoNormal><o:p> </o:p></p> </td> <td style=3D'border:none;padding:0cm 0cm 0cm 0cm'> <p class=3DMsoNormal><span style=3D'font-size:10.0pt'><o:p> </o:p></= span></p> </td> </tr> </table> <p>4. Каковы будут значения параметров регулятора состояния, синтезированно= го для объ-екта? Векторно-матричная модель объекта и эталонная модель системы управления приведены ниже.</p> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 6.45pt 0cm = 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"50%" style=3D'width:50.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>5. ВММ объекта управления</= p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>6. Эталонная модель СУ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"50%" style=3D'width:50.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>7</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1360" type=3D"#_x0000_t75" alt=3D"" style=3D'width:177.75pt;height:75.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage308.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image049.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>8</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1361" type=3D"#_x0000_t75" alt=3D"" style=3D'width:157.5pt;height:55.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage309.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image050.gif"= /> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:2;mso-yfti-lastrow:yes'> <td style=3D'padding:0cm 0cm 0cm 0cm'> <p class=3DMsoNormal><o:p> </o:p></p> </td> <td style=3D'border:none;padding:0cm 0cm 0cm 0cm'> <p class=3DMsoNormal><span style=3D'font-size:10.0pt'><o:p> </o:p></= span></p> </td> </tr> </table> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>(7x7)</p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>a) оценка управляемости объекта;</p> <p>b) приведение векторно-матричной модели объекта управления к каноничес= кой форме управляемости;</p> </td> </tr> <tr style=3D'mso-yfti-irow:3'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p><v:shape id=3D"_x0000_i1362" type=3D"#_x0000_t75" alt=3D"" style=3D'wi= dth:92.25pt; height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage310.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i8/image053.gif"= /> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>4</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Не смогу рассчитать</p> </td> </tr> </table> <h3>Лекция № 9.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Синтез оптимального управления»<o:p></o:p></= span></b></p> <p>В традиционной постановке задача синтеза оптимального управления в пространстве состояний предусматривает определение вектора управляющих сигн= алов u0(t) на основании минимизации некоторого критерия качества и формулируется следующим образом.</p> <p>Для объекта управления, который описывается векторными дифференциальным и ал-гебраическими уравнениями</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1363" type=3D"#_x0000_t75" alt=3D"" style=3D'width:126.75pt;height:60.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage311.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image001.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(9.1)</p> </td> </tr> </table> <p>необходимо найти закон управления u0(t), при котором достигается минимум квадратичного функционала качества </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1364" type=3D"#_x0000_t75" alt=3D"" style=3D'width:276pt;height:39pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage312.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image002.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(9.2)</p> </td> </tr> </table> <p>который подробно представлен в лекции 7.</p> <p>Общая математическая постановка указанной задачи приводит к уравнению Беллма-на, которое имеет следующий вид:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1365" type=3D"#_x0000_t75" alt=3D"" style=3D'width:286.5pt;height:37.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage313.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image003.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(9.3)</p> </td> </tr> </table> <p>Вывод уравнения Беллмана, характеристики входящих в него переменных и функций приведены в приложении 1.</p> <p>Решение уравнения (9.3) для объекта управления, который описывается векторно-матричной моделью (9.1), позволяет определить закон оптимального управления в виде</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1366" type=3D"#_x0000_t75" alt=3D"" style=3D'width:206.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage314.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image004.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(9.4)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1367" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:117pt; height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge315.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image005.gif"/> </v:shape>, P(t) - решение матричного дифференциального уравнения Рик-кати<= /p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1368" type=3D"#_x0000_t75" alt=3D"" style=3D'width:281.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage316.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image006.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(9.5)</p> </td> </tr> </table> <p>c граничным условием <v:shape id=3D"_x0000_i1369" type=3D"#_x0000_t75" a= lt=3D"" style=3D'width:47.25pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge317.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image007.gif"/> </v:shape>.</p> <p>Вывод уравнения Риккати приведен в приложении 2.</p> <p>В соответствии с вышеизложенным алгоритм синтеза оптимального уравнения пред-ставляет собой следующую последовательность действий:</p> <p>1) построение векторно-матричной модели ОУ (9.1);</p> <p>2) выбор элементов весовых матриц F, Q(t), R(t) в (9.2), при которых переходные процессы в системе управления удовлетворяют заданным требованиям= ;</p> <p>3) решение матричного дифференциального уравнения Риккати (9.5);</p> <p>4) анализ динамических характеристик в оптимальной системе управления и оценка ее качества.</p> <p>Основные трудности возникают здесь при решении матричного дифференциальн= ого уравнения Риккати. Интегрирование этого уравнения удобно выполнять в обратн= ом времени <v:shape id=3D"_x0000_i1370" type=3D"#_x0000_t75" alt=3D"" style=3D= 'width:41.25pt; height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge318.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image021.gif"/> </v:shape>. В этом случае задача сводится к задаче Коши с начальными услови= ями <v:shape id=3D"_x0000_i1371" type=3D"#_x0000_t75" alt=3D"" style=3D'width:45pt;heig= ht:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge319.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image008.gif"/> </v:shape>. Ввиду симметричности матрицы P(t) уравнение (9.5) равносильно системе n(n+1)/2 обыкно-венных нелинейных дифференциальных уравнений первого порядка с переменными во вре-мени коэффициентами.</p> <p>Для стационарных систем, в которых A, B, Q, R - коэффициентные матрицы и= <v:shape id=3D"_x0000_i1372" type=3D"#_x0000_t75" alt=3D"" style=3D'width:36.75pt;h= eight:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge320.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image009.gif"/> </v:shape>, матричное дифференциальное уравнение Риккати вырождается в алгебраическое</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1373" type=3D"#_x0000_t75" alt=3D"" style=3D'width:157.5pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage321.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image010.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(9.6)</p> </td> </tr> </table> <p>решением которого является симметричная положительно определенная матриц= а Р.</p> <p>Решение уравнения (9.28) для стационарных систем при <v:shape id=3D"_x00= 00_i1374" type=3D"#_x0000_t75" alt=3D"" style=3D'width:29.25pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge322.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image011.gif"/> </v:shape>и <v:shape id=3D"_x0000_i1375" type=3D"#_x0000_t75" alt=3D"" styl= e=3D'width:36.75pt; height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge320.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image009.gif"/> </v:shape>имеет предел </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1376" type=3D"#_x0000_t75" alt=3D"" style=3D'width:61.5pt;height:21.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage323.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image012.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(9.7)</p> </td> </tr> </table> <p>Поэтому матрицу Р можно вычислить как предельное значение решения уравне= ния (9.5) при достаточно большом Т.</p> <p>По аналогии с (9.4) оптимальное управление определится из выражения</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1377" type=3D"#_x0000_t75" alt=3D"" style=3D'width:96.75pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage324.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image013.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(9.8)</p> </td> </tr> </table> <p>Достоверность представленных алгоритмов подтвердим практическим примером= . </p> <p>Пример 9.1. Для электромеханического объекта с упругой передачей механического движения от вала электродвигателя к валу рабочего механизма, численные значения пара-метров которого приведены в табл. 9.1., выполним си= нтез оптимального управления (9.4) и безынерционного регулятора состояния.</p> <p align=3Dright style=3D'text-align:right'>Таблица 9.1. Параметры электромеханического объекта</p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1378" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:481.5pt;height:92.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge325.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image014.gif"/> </v:shape></p> <p>Результатом серии вычислительных экспериментов явились:</p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l5 level1 lfo13;tab-stops:list 36.0pt'>внутреннее содержание весовых матриц Q, и R </li> </ul> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1379" type=3D"#_x0000_t75" alt=3D"" style=3D'width:247.5pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge326.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image015.gif"/> </v:shape></p> <ul type=3Ddisc> <li class=3DMsoNormal style=3D'mso-margin-top-alt:auto;mso-margin-bottom-a= lt:auto; mso-list:l3 level1 lfo14;tab-stops:list 36.0pt'>временные характеристи= ки </li> </ul> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1380" type=3D"#_x0000_t75" alt=3D"" style=3D'width:125.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge327.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image016.gif"/> </v:shape></p> <p>полученные в результате решения уравнения Риккати (9.5) в обратном време= ни, которые приведены на рис. 9.1</p> <p>Для постановки имитационных экспериментов используем приведенные в табл.= 9.2 постоянные расчетные значения коэффициентов обратных связей, соответствующие t=3D0, и значения реализации.</p> <p align=3Dright style=3D'text-align:right'>Таблица 9.2. Значения коэффицие= нтов обратных связей</p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1381" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:481.5pt;height:83.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge328.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image017.gif"/> </v:shape></p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1382" type=3D"#_x0000_t75" alt=3D"" style=3D'width:6in;height:370.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge329.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image018.gif"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 9.1. Динамические характ= еристики K0(t)</p> <p>Сравнительные динамические характеристики (см. рис. 9.2), систем управле= ния, в ко-торых параметры регулятора соответствуют значениям реализации коэффициентов обратных связей (табл. 9.2) и значениям регулятора состояния, синтезированного при использовании в качестве критерия качества биномиально= го распределения корней (<v:shape id=3D"_x0000_i1383" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:57.75pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge330.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image019.gif"/> </v:shape>), подтвер-ждают корректность алгоритмического и программного обеспечения синтеза оптимального управления.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1384" type=3D"#_x0000_t75" alt=3D"" style=3D'width:427.5pt;height:367.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge331.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image020.gif"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 9.2. Сравнительные динам= ические характеристики систем управления с регулятором состояния</p> <h3>Контрольные вопросы к лекции № 9.</h3> <p>1. Укажите основные особенности численного решения матричного дифференциального уравнения Риккати?</p> <p>2. При каких условиях матричное дифференциальное уравнение Риккати вырождается в алгебраическое?</p> <p>3. Какие значения должны принимать коэффициенты обратных связей для «точ= ной» реализации оптимального управления?</p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Интегрирование уравнения Риккати, как правило, выполняется в обратном времени <v:shape id=3D"_x0000_i1385" type=3D"#_x0000_t75" alt=3D"" style= =3D'width:41.25pt; height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage318.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image021.gif"= /> </v:shape>.</p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Для стационарных систем, в которых A, B, Q, R - коэффициентные матрицы= и <v:shape id=3D"_x0000_i1386" type=3D"#_x0000_t75" alt=3D"" style=3D'width:36.75pt= ;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage320.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image009.gif"= /> </v:shape>, матричное дифференциальное уравнение Риккати вырождается в алгебраическое <v:shape id=3D"_x0000_i1387" type=3D"#_x0000_t75" alt=3D""= style=3D'width:157.5pt; height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage321.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image010.gif"= /> </v:shape></p> </td> </tr> <tr style=3D'mso-yfti-irow:3;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Значения коэффициентов обратных связей должны непрерывно изменяться во времени в соответствии с результатом решения уравнения Риккати <v:shape i= d=3D"_x0000_i1388" type=3D"#_x0000_t75" alt=3D"" style=3D'width:125.25pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage327.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i9/image016.gif"= /> </v:shape>.</p> </td> </tr> </table> <h3>Лекция № 10.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Наблюдатель состояния полного порядка»<o:p><= /o:p></span></b></p> <p>При решении практических задач управления методами теории пространства состоя-ний мы часто встречаемся со случаями, когда часть переменных вектора состояния оказыва-ются неизмеримыми. Если имеется математическая модель системы, то можно попытаться вычислить состояние системы по наблюдаемым вхо= дам и выходам.</p> <p>Восстановление вектора состояния x(t) называется его оценкой, а устройст= во, обеспе-чивающее получение оценки по измерениям управления u (t) и вектора выхода y(t) на конеч-ном интервале времени, - наблюдателем.</p> <p>Пусть стационарный объект описывается традиционной системой уравнений</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1389" type=3D"#_x0000_t75" alt=3D"" style=3D'width:65.25pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage332.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image001.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.1)</p> </td> </tr> </table> <p>Предположим, что матрицы A, B, C известны, тогда вектор x можно аппроксимиро-вать состоянием <v:shape id=3D"_x0000_i1390" type=3D"#_x0000_t= 75" alt=3D"" style=3D'width:11.25pt;height:13.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge333.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image003.gif"/> </v:shape>модели</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1391" type=3D"#_x0000_t75" alt=3D"" style=3D'width:62.25pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage334.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image004.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.2)</p> </td> </tr> </table> <p>которая имеет тот же вход, что и объект (10.1).</p> <p>Если модель (10.2) является идеальной аппроксимацией системы (10.1), в т= ом смысле, что их параметры и начальные условия идентичны, то состояния x и <v= :shape id=3D"_x0000_i1392" type=3D"#_x0000_t75" alt=3D"" style=3D'width:11.25pt;h= eight:13.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge333.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image003.gif"/> </v:shape>также совпадают. Если начальные условия для систем (10.1) и (10.2) различны, то <v:shape id=3D"_x0000_i1393" type=3D"#_x0000_t75" alt=3D"" sty= le=3D'width:11.25pt; height:13.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge333.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image003.gif"/> </v:shape>сходится к x только то-гда, когда система (10.1) асимптотически устойчива.</p> <p>При восстановлении (10.2) не используется измеряемый выход. Качество восстанов-ления улучшается, если ввести в модель разность измеренного выход= а и его оценки <v:shape id=3D"_x0000_i1394" type=3D"#_x0000_t75" alt=3D"" style= =3D'width:36pt; height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge335.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image006.gif"/> </v:shape>в виде обратной связи:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1395" type=3D"#_x0000_t75" alt=3D"" style=3D'width:130.5pt;height:26.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage336.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image007.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.3)</p> </td> </tr> </table> <p>Здесь L - некоторая матрица, обеспечивающая требуемый вид переходных процессов оценки вектора состояния.</p> <p>Введем ошибку восстановления</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1396" type=3D"#_x0000_t75" alt=3D"" style=3D'width:45pt;height:12.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage337.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image008.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.4)</p> </td> </tr> </table> <p>Вычитая (10.3) из дифференциального уравнения (10.1), получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1397" type=3D"#_x0000_t75" alt=3D"" style=3D'width:343.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage338.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image009.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.5)</p> </td> </tr> </table> <p>Очевидно, для того чтобы ошибка восстановления стремилась к нулю, необхо= димо выбрать матрицу L так, чтобы система (10.5) была асимптотически устойчива.<= /p> <p>Таким образом, вводя обратную связь в модель восстановления, можно нивелировать ошибку, даже если система (10.1) неустойчива.</p> <p>Мы видим, что модель (10.3) восстанавливает все составляющие вектора состояния, поэтому она называется наблюдателем состояния полного порядка.</= p> <p>Общая схема системы управления с наблюдателем состояния полного порядка будет иметь вид, представленный в среде компьютерного комплекса FuncPro 1.0= на рис 10.1.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1398" type=3D"#_x0000_t75" alt=3D"" style=3D'width:431.25pt;height:301.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge339.jpg" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image010.jpg"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 10.1. Схема системы упра= вления с наблюдателем полного порядка</p> <p>При синтезе наблюдателей обычно используют два принципа: принцип разделимости и принцип дуальности.</p> <p>Принцип разделимости. Допустим, что объект описывается уравнениями (10.1= ), регу-лятор состояния -</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1399" type=3D"#_x0000_t75" alt=3D"" style=3D'width:55.5pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage340.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image011.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.6)</p> </td> </tr> </table> <p>а наблюдатель полного порядка - уравнением (10.3).</p> <p>Тогда уравнения состояния всей системы имеют вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1400" type=3D"#_x0000_t75" alt=3D"" style=3D'width:195pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage341.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image012.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.7)</p> </td> </tr> </table> <p>Если ввести ошибку (10.4), то с помощью линейного преобразования</p> <table class=3DMsoNormalTable border=3D0 cellpadding=3D0 style=3D'mso-cells= pacing:1.5pt'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:.75pt 6.45pt .75pt .75pt'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1401" type=3D"#_x0000_t75" alt=3D"" style=3D'width:88.5pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage342.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image013.gif= "/> </v:shape></p> </td> <td style=3D'padding:.75pt 6.45pt .75pt .75pt'> <p align=3Dright style=3D'text-align:right'>(10.8)</p> </td> </tr> </table> <p>получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1402" type=3D"#_x0000_t75" alt=3D"" style=3D'width:183pt;height:73.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage343.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image014.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.9)</p> </td> </tr> </table> <p>Из (10.9) очевидно, что характеристическое уравнение всей системы имеет = вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1403" type=3D"#_x0000_t75" alt=3D"" style=3D'width:194.25pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage344.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image015.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.10)</p> </td> </tr> </table> <p>Откуда следует, что характеристические полиномы замкнутой системы управл= ения и наблю-дателя независимы и, следовательно, мы можем назначать полюсы наблюдателю независимо от полюсов системы управления.</p> <p>Процесс, описываемый уравнениями (10.9), состоит из двух взаимно независ= имых процессов: из процесса оценки вектора <v:shape id=3D"_x0000_i1404" type=3D"= #_x0000_t75" alt=3D"" style=3D'width:11.25pt;height:13.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge333.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image003.gif"/> </v:shape>и из собственно процесса управления.</p> <p>Эту возможность разделения процессов в наблюдателе и регуляторе принято называть принципом разделимости.</p> <p>Принцип дуальности (приводится без доказательства). Условием наблюдаемос= ти сис-темы (10.1) одновременно является полная управляемость сопряженной сист= емы</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1405" type=3D"#_x0000_t75" alt=3D"" style=3D'width:75.75pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage345.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image017.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.11)</p> </td> </tr> </table> <p>С учетом принципов дуальности и разделимости синтез наблюдателя состояния вы-полняется независимо от синтеза регулятора состояния по алгоритмам, изложенным в 4.1.</p> <p>Действительно, для системы (10.3) регулятор состояния будет иметь вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1406" type=3D"#_x0000_t75" alt=3D"" style=3D'width:48pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage346.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image018.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(10.12)</p> </td> </tr> </table> <p>Так как <v:shape id=3D"_x0000_i1407" type=3D"#_x0000_t75" alt=3D"" style= =3D'width:152.25pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge347.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image019.gif"/> </v:shape>элементы матрицы L определяются по алгоритмам вычисления матрицы К для регулятора состояния.</p> <p>В этой связи, для автоматизированного выполнения проектных операций синт= еза на-блюдателей состояния полного порядка могут быть использованы алгоритмы и компью-терные средства синтеза регуляторов состояния.</p> <p>Многочисленные вычислительные эксперименты, выполненные в целях построен= ия систем управления с наблюдателями полного порядка и регуляторами состояния, подтвер-ждают корректность приведенного выше положения.</p> <p>На рис. 10.2 приведены сравнительные динамические характеристики упругого элек-тромеханического объекта, системы управления которого построены как при непосредствен-ном изменении координат, так и при использовании наблюдателя (10.3). Причем, при поста-новке имитационных экспериментов с моделью ЭМС с наблюдателем были изменены от-дельные параметры объекта в пределах <v:shape id=3D"_x0000_i1408" type=3D"#_x0000_t75" alt=3D"" style=3D'width:57pt;heig= ht:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge348.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image020.gif"/> </v:shape>, учтены люфты в механических переда-чах и инерционность преобразователя.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1409" type=3D"#_x0000_t75" alt=3D"" style=3D'width:397.5pt;height:364.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge349.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image021.gif"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 10.2. Сравнительные дина= мические характеристики ЭМС при непосредственном измерении координат и использовании наблюдателя полного порядка</p> <h3>Контрольные вопросы к лекции №10</h3> <p>1. Для достижения каких целей и из каких условий выбирается матрица L наблюдателя полного порядка?</p> <p>2. Назовите два основных принципа синтеза наблюдателя. </p> <p>3. В какой последовательности следует выполнять проектные операции синте= за регуля-тора и наблюдателя состояния?</p> <p>4. Вы решили воспользоваться методом расположения полюсов для синтеза наблюдате-ля полного порядка. С какой векторно-матричной моделью должна выполняться эта операция, если ВММ объекта управления имеет вид</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1410" type=3D"#_x0000_t75" alt=3D"" style=3D'width:162pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge350.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image022.gif"/> </v:shape></p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Матрица L обеспечивает требуемый вид переходных процессов оценки векто= ра состояния и выбирается из условий достижения асимптотической устойчивости замкнутой системы объект – наблюдатель.</p> </td> </tr> <tr style=3D'mso-yfti-irow:2'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Принцип разделимости и принцип дуальности.</p> </td> </tr> <tr style=3D'mso-yfti-irow:3'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>3</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Проектные операции могут быть выполнены в любой последовательности.</p> </td> </tr> <tr style=3D'mso-yfti-irow:4;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>4</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1411" type=3D"#_x0000_t75" alt=3D"" style=3D'width:141pt;height:74.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage351.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i10/image023.gif= "/> </v:shape></p> </td> </tr> </table> <h3>Лекция № 11.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Оценка вектора состояния при случайных возму= щениях и наличии помех»<o:p></o:p></span></b></p> <p>При решении практических задач очень часто возникает ситуация, когда необходимые для управления координаты вектора состояния объекта не измеряют= ся или измеряются с существенными случайными ошибками, а движение объекта управления подвержено случайным воздействиям. В таких ситуациях управление определяется на основе результатов оценивания состояния системы, которое им= еет статистическую связь с данными наблюдений.</p> <p>Рассмотрим алгоритм линейной оценки вектора состояния системы с минималь= ной дисперсией.</p> <p>Пусть динамическая система описывается векторным дифференциальным уравнением.</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1412" type=3D"#_x0000_t75" alt=3D"" style=3D'width:177.75pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage352.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image001.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.1)</p> </td> </tr> </table> <p>Здесь, кроме представленных ранее матриц и векторов, w(t)- k-мерный вект= ор случайных воздействий; Т- матрица размером n x k.</p> <p>Вектор измеряемых выходных координат, который доступен наблюдению, определяется соотношением</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1413" type=3D"#_x0000_t75" alt=3D"" style=3D'width:106.5pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage353.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image002.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.2)</p> </td> </tr> </table> <p>где v(t) - m-мерный вектор случайных помех, сопровождающих измерения.</p> <p>Предполагается, что система (11.1), (11.2) при w(t) <v:shape id=3D"_x000= 0_i1414" type=3D"#_x0000_t75" alt=3D"" style=3D'width:9.75pt;height:9pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge354.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image003.gif"/> </v:shape>0 и v(t) <v:shape id=3D"_x0000_i1415" type=3D"#_x0000_t75" alt=3D= "" style=3D'width:9.75pt;height:9pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge354.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image003.gif"/> </v:shape>0 наблюдаема. Будем считать w(t) и v(t) гауссовскими случайными процессами типа белого шума с нулевыми математическими ожиданиями</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1416" type=3D"#_x0000_t75" alt=3D"" style=3D'width:139.5pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge355.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image004.gif"/> </v:shape></p> <p>и корреляционными матрицами</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1417" type=3D"#_x0000_t75" alt=3D"" style=3D'width:235.5pt;height:37.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage356.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image005.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.3)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1418" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:23.25pt; height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge357.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image006.gif"/> </v:shape>- дельта-функция; Q(t)- симметричная неотрицательно определенная = k x k - матрица интенсивности белого шума w(t);</p> <p>R(t)- симметричная положительно определенная m x m - матрица интенсивнос= ти белого шума v(t).</p> <p>Допустим, что начальное состояние системы х(t0)- n-мерный случайный вект= ор с известным математическим ожиданием</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1419" type=3D"#_x0000_t75" alt=3D"" style=3D'width:65.25pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge358.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image007.gif"/> </v:shape></p> <p>и корреляционной матрицей</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1420" type=3D"#_x0000_t75" alt=3D"" style=3D'width:231pt;height:37.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage359.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image008.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.4)</p> </td> </tr> </table> <p>Кроме того, будем считать, что х(t0), w(t), v(t) взаимно не коррелирован= ны.</p> <p>Найти необходимо линейную несмещенную оценку вектора х(t0), построенную = на основе наблюдения <v:shape id=3D"_x0000_i1421" type=3D"#_x0000_t75" alt=3D"= " style=3D'width:24.75pt; height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge360.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image009.gif"/> </v:shape>, <v:shape id=3D"_x0000_i1422" type=3D"#_x0000_t75" alt=3D"" styl= e=3D'width:52.5pt; height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge361.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image010.gif"/> </v:shape>.</p> <p>Несмещенная оценка <v:shape id=3D"_x0000_i1423" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:22.5pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge362.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image011.gif"/> </v:shape>предполагает равенство ее математического ожидания математическому ожиданию истинной величины <v:shape id=3D"_x0000_i1424" type=3D"#_x0000_t75= " alt=3D"" style=3D'width:22.5pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge363.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image012.gif"/> </v:shape>.</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1425" type=3D"#_x0000_t75" alt=3D"" style=3D'width:90.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage364.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image013.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.5)</p> </td> </tr> </table> <p>Предположим, что <v:shape id=3D"_x0000_i1426" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:21pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge365.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image014.gif"/> </v:shape>получается на выходе фильтра</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1427" type=3D"#_x0000_t75" alt=3D"" style=3D'width:175.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage366.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image015.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.6)</p> </td> </tr> </table> <p>Чтобы процесс на выходе фильтра был несмещенной оценкой, должно выполнят= ься равенство (11.5). Вычислим математическое ожидание обеих частей уравнения (11.6):</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1428" type=3D"#_x0000_t75" alt=3D"" style=3D'width:238.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage367.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image016.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.7)</p> </td> </tr> </table> <p>но из (11.2) следует, что</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1429" type=3D"#_x0000_t75" alt=3D"" style=3D'width:126pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage368.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image017.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.8)</p> </td> </tr> </table> <p>На основании (11.6) - (11.8) получим дифференциальное уравнение для сред= него значения вектора состояния системы:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1430" type=3D"#_x0000_t75" alt=3D"" style=3D'width:180.75pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage369.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image018.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.9)</p> </td> </tr> </table> <p>Вычисляя математическое ожидание от (11.13), получим еще одно уравнение = для среднего значения вектора состояния</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1431" type=3D"#_x0000_t75" alt=3D"" style=3D'width:123pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage370.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image019.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.10)</p> </td> </tr> </table> <p>Сравнивая (11.10) и (11.9), определим первое условие несмещенности оценки вектора состояния:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1432" type=3D"#_x0000_t75" alt=3D"" style=3D'width:112.5pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage371.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image020.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.11)</p> </td> </tr> </table> <p>Второе условие состоит в том, чтобы уравнения (11.21) и (11.22) решались= при одном и том же начальном условии</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1433" type=3D"#_x0000_t75" alt=3D"" style=3D'width:153pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage372.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image021.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.12)</p> </td> </tr> </table> <p>При выполнении этих условий уравнение фильтра примет вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1434" type=3D"#_x0000_t75" alt=3D"" style=3D'width:232.5pt;height:37.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage373.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image022.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.13)</p> </td> </tr> </table> <p>Остается определить матрицу коэффициентов усиления фильтра L(t), которая обеспечила бы оптимальную оценку в том смысле, что составляющие ошибки оценивания должны иметь минимальную дисперсию.</p> <p>Согласно принципу дуальности и с учетом того, что дисперсия случайной функции представляет собой среднее значение квадрата разности между случайн= ой функцией и ее средним значением, можно утверждать, что матрица коэффициентов усиления оптимального фильтра определяется выражением</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1435" type=3D"#_x0000_t75" alt=3D"" style=3D'width:131.25pt;height:21.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage374.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image023.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.14)</p> </td> </tr> </table> <p>которое получается по алгоритму синтеза оптимального управления для сопряженной системы</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1436" type=3D"#_x0000_t75" alt=3D"" style=3D'width:75.75pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge345.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image024.gif"/> </v:shape></p> <p>Здесь P(t) - корреляционная матрица ошибок оценивания, является решением матричного дифференциального уравнения Риккати</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1437" type=3D"#_x0000_t75" alt=3D"" style=3D'width:272.25pt;height:37.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage375.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image025.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.27)</p> </td> </tr> </table> <p>Наблюдатель, построенный по приведенному алгоритму, называют оптимальным фильтром Калмана – Бьюси.</p> <p>Для стационарной системы, в которой A, B, C, T, Q, R- коэффициентные матрицы, уравнение фильтра Калмана - Бьюси принимает вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1438" type=3D"#_x0000_t75" alt=3D"" style=3D'width:187.5pt;height:18.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage376.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image026.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.28)</p> </td> </tr> </table> <p>Матрица коэффициентов усиления фильтра постоянна и определяется выражени= ем</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1439" type=3D"#_x0000_t75" alt=3D"" style=3D'width:61.5pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage377.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image027.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.29)</p> </td> </tr> </table> <p>где Р- положительно определенная матрица, являющаяся решением алгебраического матричного уравнения Риккати</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1440" type=3D"#_x0000_t75" alt=3D"" style=3D'width:183.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage378.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image028.gif= "/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(11.30)</p> </td> </tr> </table> <p>Многочисленные вычислительные эксперименты, выполненные в целях анализа систем управления с наблюдателями, построенными по принципу фильтра Калмана= - Бьюси при учете помех и случайных возмущений, подтверждают корректность приведенных алгоритмов и компьютерных средств их реализации. На рис. 11.1 приведены сравнительные динамические характеристики упругого электромеханического объекта, системы управления которого построены как при непосредственном изменении координат, так и при использовании наблюдателя (11.25). При постановке экспериментов с моделью ЭМС с наблюдателем случайные возмущения и помехи имитированы с помощью гармонических сигналов, а регулят= оры и наблюдатели представлены в дискретном виде с помощью разностных уравнений= .</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1441" type=3D"#_x0000_t75" alt=3D"" style=3D'width:419.25pt;height:296.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge379.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i11/image029.gif"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 11.1. Сравнительные дина= мические характеристики ЭМС при непосредственном измерении координат и использовании наблюдателя, построенного по принципу фильтра Калмана - Бьюси.</p> <h3>Контрольные вопросы к лекции №11</h3> <p>1. Какой из рассмотренных ранее алгоритмов используется для вычисления матрицы коэффициентов усиления оптимального фильтра?</p> <p>2. При каких условиях матрица коэффициентов усиления фильтра Калмана-Бью= си постоянна во времени?</p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Алгоритм синтеза оптимального управления для сопряженной системы.</p> </td> </tr> <tr style=3D'mso-yfti-irow:2;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Для стационарной системы.</p> </td> </tr> </table> <h3>Лекция № 12.</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Тема: «Наблюдатель состояния пониженного порядка»<o= :p></o:p></span></b></p> <p>Для получения рациональной оценки координат вектора состояния при отсутс= твии шумов в измерениях Люенбергером был предложен метод, позволяющий восстанавливать только требуемые переменные вектора состояния системы.</p> <p>Рассмотрим линейную наблюдаемую стационарную систему</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1442" type=3D"#_x0000_t75" alt=3D"" style=3D'width:102pt;height:52.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage380.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i1.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.1)</p> </td> </tr> </table> <p>в которой <b>y</b>(<i>t</i>) - <i>m</i>- мерный вектор выходных координа= т, причем <i>m < n</i> и <i>rang</i> <b>C</b>=3D<i>m</i>, т.е. имеется m ли= нейно независимых уравнений для определения m переменных вектора состояния по век= тору выхода y(t). Следовательно, порядок наблюдателя может быть снижен до (n - m= ).</p> <p>Предположим, что оценка вектора состояния системы может быть выполнена с помощью фильтра <i>(n - m)</i>-го порядка</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1443" type=3D"#_x0000_t75" alt=3D"" style=3D'width:148.5pt;height:36pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage381.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i2.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.2)</p> </td> </tr> </table> <p>где <b>z</b>(<i>t</i>) - (<i>n - m</i>)- мерный вектор состояния; <b>F</= b>, <b>G1</b>, <b>G2</b> - матрицы с размерами <i>(n - m)</i>x<i>(n - m)</i>, <i>(n - m)</= i>x <i>m</i> , <i>(n - m)</i>x <i>z</i> соответственно. </p> <p>Определим условия, которым должны удовлетворять матрицы <b>F</b>, <b>G1<= /b>, <b>G2</b> фильтра (12.2). Допустим, что параметры фильтра можно подобрать такие, чтобы </p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1444" type=3D"#_x0000_t75" alt=3D"" style=3D'width:59.25pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage382.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i3.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.3)</p> </td> </tr> </table> <p>Умножим обе части уравнения (12.2) слева на матрицу <b>Т</b>, тогда с уч= етом (12.1) получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1445" type=3D"#_x0000_t75" alt=3D"" style=3D'width:122.25pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage383.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i4.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.4)</p> </td> </tr> </table> <p class=3DMsoNormal><span style=3D'display:none;mso-hide:all'><o:p> <= /o:p></span></p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1446" type=3D"#_x0000_t75" alt=3D"" style=3D'width:177pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage384.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i5.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.5)</p> </td> </tr> </table> <p>Из равенств (12.4), (12.5) следует</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1447" type=3D"#_x0000_t75" alt=3D"" style=3D'width:151.5pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage385.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i6.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.6)</p> </td> </tr> </table> <p>Оценку вектора состояния будем искать в виде</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1448" type=3D"#_x0000_t75" alt=3D"" style=3D'width:91.5pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage386.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i7.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.7)</p> </td> </tr> </table> <p>где <b>Н</b> и <b>G</b> -матрицы с размерами <i>n</i> x<i>(n - m)</i> и = <i>(n</i> x <i>m)</i> соответственно. </p> <p>Потребуем выполнение условия <v:shape id=3D"_x0000_i1449" type=3D"#_x000= 0_t75" alt=3D"" style=3D'width:27.75pt;height:12.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge387.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i8.gif"/> </v:shape>, т.е.</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1450" type=3D"#_x0000_t75" alt=3D"" style=3D'width:153pt;height:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage388.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i9.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.8)</p> </td> </tr> </table> <p>откуда следует, что</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1451" type=3D"#_x0000_t75" alt=3D"" style=3D'width:64.5pt;height:13.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage389.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i10.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.9)</p> </td> </tr> </table> <p>где <b>I</b> - единичная матрица.</p> <p>Элементы пяти матриц: <b>F</b>, <b>G1</b>, <b>G2</b>, <b>G</b>, <b>H</b>, связанные уравнениями (12.6), (12.9), могут выбираться до некоторой степени произвольно.</p> <p>Векторно-матричная модель системы управления с наблюдателем Люенбергера = при <v:shape id=3D"_x0000_i1452" type=3D"#_x0000_t75" alt=3D"" style=3D'width:78pt;heig= ht:15.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge390.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i11.gif"/> </v:shape>имеет следующий вид:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1453" type=3D"#_x0000_t75" alt=3D"" style=3D'width:270.75pt;height:33.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage391.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i12.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.10)</p> </td> </tr> </table> <p>или</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1454" type=3D"#_x0000_t75" alt=3D"" style=3D'width:256.5pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage392.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i13.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(12.11)</p> </td> </tr> </table> <p>Наблюдатель Люенбергера не изменяет полюсы замкнутой системы управления,= а лишь добавляет к ним свои собственные.</p> <p>Таким образом, синтез наблюдателя пониженного порядка может выполняться = по следующему алгоритму.</p> <p>1. Проверить наблюдаемость исходной системы с определением индекса наблюдаемости <i>m</i>.</p> <p>2. Выполнить анализ матрицы состояния <b>А</b> объекта управления с определением корней характеристического уравнения.</p> <p>3. Выбрать матрицу <b>F</b> таким образом, чтобы обеспечить требуемое вр= емя переходного процесса в наблюдателе.</p> <p>4. Произвольно задать матрицу <b>G1</b>, соблюдая при этом выполнение условия управляемости фильтра (12.2) по вектору <b>y</b>(<i>t</i>), т.е. необходимо, чтобы</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1455" type=3D"#_x0000_t75" alt=3D"" style=3D'width:175.5pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage393.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i14.gif"/> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'> </p> </td> </tr> </table> <p>где <i>p=3Dn - m</i> .</p> <p>5. Решить матричное уравнение <v:shape id=3D"_x0000_i1456" type=3D"#_x00= 00_t75" alt=3D"" style=3D'width:81pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge394.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i15.gif"/> </v:shape>относительно <b>Т</b>.</p> <p>6. Вычислить матрицу <v:shape id=3D"_x0000_i1457" type=3D"#_x0000_t75" a= lt=3D"" style=3D'width:47.25pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge395.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i16.gif"/> </v:shape>.</p> <p>7. Вычислить матрицы <b>Н</b> и <b>G</b> из уравнения (12.9).</p> <p>Общая схема системы управления с наблюдателем пониженного порядка будет иметь вид, представленный в среде компьютерного комплекса FuncPro 1.0 на рис 12.1.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1458" type=3D"#_x0000_t75" alt=3D"" style=3D'width:426.75pt;height:297.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge396.jpg" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i17.jpg"/> </v:shape></p> <p align=3Dcenter style=3D'text-align:center'>Рис. 12.1. Схема системы упра= вления с наблюдателем пониженного порядка</p> <p>Компьютерная реализация указанного алгоритма позволяет конструировать наблюдатели координат, измерение которых затруднено или практически невозмо= жно. Так, например, для построения системы управления электромеханического объек= та с упругой передачей механического движения от вала электродвигателя к валу рабочего механизма требуется измерение упругого момента. Проще и эффективнее здесь использовать вычислительное устройство, построенное как наблюдатель первого порядка, а координаты тока, скоростей электродвигателя и механизма регистрировать с помощью датчиков.</p> <p align=3Dright style=3D'text-align:right'>Таблица 12.1. Параметры электромеханического объекта</p> <p align=3Dright style=3D'text-align:right'><v:shape id=3D"_x0000_i1459" ty= pe=3D"#_x0000_t75" alt=3D"" style=3D'width:481.5pt;height:92.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge397.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/image018.gif"/> </v:shape></p> <p>Выполнение отмеченной процедуры для электромеханического объекта с параметрами, приведенными в табл. 12.1., приводит следующим результатам.</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1460" type=3D"#_x0000_t75" alt=3D"" style=3D'width:180.75pt;height:89.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge398.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i18.gif"/> </v:shape></p> <p align=3Dright style=3D'text-align:right'> </p> <p>Корректность полученных результатов подтверждают динамические характерис= тики системы с регулятором состояния и наблюдателем упругого момента, приведенны= е на рис. 12.2.</p> <p>Регулятор состояния здесь синтезирован при использовании в качестве крит= ерия качества биномиального распределения корней (<v:shape id=3D"_x0000_i1461" t= ype=3D"#_x0000_t75" alt=3D"" style=3D'width:15.75pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge033.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i33.gif"/> </v:shape>=3D8.5 с-1).</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1462" type=3D"#_x0000_t75" alt=3D"" style=3D'width:420pt;height:387pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge399.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i19.gif"/> </v:shape></p> <p align=3Dright style=3D'text-align:right'> </p> <p align=3Dcenter style=3D'text-align:center'>Рис. 12.2. Динамические характеристики ЭМС при непосредственном измерении всех координат и при оцен= ке упругого момента наблюдателем пониженного порядка.</p> <h3>Контрольные вопросы к лекции №12</h3> <p>1. Какая матрица <v:shape id=3D"_x0000_i1463" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:150pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge400.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i20.gif"/> </v:shape>определяет требуемое время оценки неизмеряемых компонент вектора состояния системы с помощью наблюдателя Люенбергера?</p> <p>2. Из каких условий выбирается матрица <b>G</b><sub>1</sub> фильтра <v:s= hape id=3D"_x0000_i1464" type=3D"#_x0000_t75" alt=3D"" style=3D'width:150pt;hei= ght:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge400.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i20.gif"/> </v:shape>необходимого для синтеза наблюдателя Люенбергера?</p> <h3>ОТВЕТЫ</h3> <table class=3DMsoNormalTable border=3D1 cellspacing=3D1 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:.7pt;mso-padding-alt:0cm 0cm 0cm 0cm= '> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>№ задания</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Ответ</p> </td> </tr> <tr style=3D'mso-yfti-irow:1'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>1</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>Матрицы F</p> </td> </tr> <tr style=3D'mso-yfti-irow:2;mso-yfti-lastrow:yes'> <td width=3D"14%" style=3D'width:14.94%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'>2</p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p>Матрицу <b>G</b><sub>1</sub> можно задавать произвольно, соблюдая при = этом условия управляе-мости фильтра (12.2) по вектору <b>y</b>(<i>t</i>), т.е. необходимо, чтобы</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1465" type=3D"#_x0000_t75" alt=3D"" style=3D'width:175.5pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage393.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/i12/i14.gif"/> </v:shape></p> <p>где <i>p=3Dn - m<o:p></o:p></i></p> </td> </tr> </table> <h3>Приложение 1 к лекции 9</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Вывод уравнения Беллмана<o:p></o:p></span></b></p> <p>Пусть управляемый объект описывается векторным дифференциальным уравнени= ем общего вида</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1466" type=3D"#_x0000_t75" alt=3D"" style=3D'width:100.5pt;height:16.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage401.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image001.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.1)</p> </td> </tr> </table> <p>а критерий оптимальности также имеет общий вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1467" type=3D"#_x0000_t75" alt=3D"" style=3D'width:106.5pt;height:39pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage402.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image002.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.2)</p> </td> </tr> </table> <p>Необходимо в классе допустимых управлений найти управление u<sub>0</sub>= (t), при котором функционал I достигает минимального значения, т.е.</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1468" type=3D"#_x0000_t75" alt=3D"" style=3D'width:159.75pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage403.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image003.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.3)</p> </td> </tr> </table> <p>а объект переводится за время T- t0 из заданного начального состояния x(= t0) в произвольное конечное, принадлежащее пространству состояний.</p> <p>В основу дальнейших рассуждений положен принцип оптимальности, утверждаю= щий, что любой оставшийся конечный участок оптимальной траектории сам по себе является также оптимальной траекторией.</p> <p>Предположим, что уже найдены оптимальное управление u<sub>0</sub>(t) и соответствующая ему траектория движения объекта x(t). Выберем на оптимальной траектории две точки, соответствующие моментам времени t и t+<v:shape id=3D= "_x0000_i1469" type=3D"#_x0000_t75" alt=3D"" style=3D'width:15pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge404.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image004.gif"/> </v:shape> , где <v:shape id=3D"_x0000_i1470" type=3D"#_x0000_t75" alt=3D""= style=3D'width:15pt; height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge404.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image004.gif"/> </v:shape>- малая величина. Согласно принципу оптимальности участки оптимал= ьной траектории от точек t и t+<v:shape id=3D"_x0000_i1471" type=3D"#_x0000_t75"= alt=3D"" style=3D'width:15pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge404.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image004.gif"/> </v:shape> до конечной точки Т являются оптимальными. Обозначим минимальное значение функционала (П.1.2), соответствующее этим участкам:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1472" type=3D"#_x0000_t75" alt=3D"" style=3D'width:233.25pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage405.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image005.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.4)</p> </td> </tr> </table> <p class=3DMsoNormal><span style=3D'display:none;mso-hide:all'><o:p> <= /o:p></span></p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1473" type=3D"#_x0000_t75" alt=3D"" style=3D'width:256.5pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage406.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image006.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.5)</p> </td> </tr> </table> <p>На основании выражений (П.1.4), (П.1.5) можно записать</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1474" type=3D"#_x0000_t75" alt=3D"" style=3D'width:270.75pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage407.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image007.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.6)</p> </td> </tr> </table> <p>Учитывая малость <v:shape id=3D"_x0000_i1475" type=3D"#_x0000_t75" alt= =3D"" style=3D'width:15pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge404.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image004.gif"/> </v:shape>, запишем</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1476" type=3D"#_x0000_t75" alt=3D"" style=3D'width:215.25pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage408.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image008.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.7)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1477" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:26.25pt; height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge409.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image009.gif"/> </v:shape>- малая с большим порядком малости, чем <v:shape id=3D"_x0000_i14= 78" type=3D"#_x0000_t75" alt=3D"" style=3D'width:15pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge404.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image004.gif"/> </v:shape>.</p> <p>Функцию х(t+ ) разложим в ряд Тейлора в окрестности точки t, представив = его в виде</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1479" type=3D"#_x0000_t75" alt=3D"" style=3D'width:148.5pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage410.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image010.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.8)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1480" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:27.75pt; height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge411.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image011.gif"/> </v:shape>- совокупность последующих членов ряда.</p> <p>Предполагая дифференцируемость функции V по своим аргументами и учитывая (П.1.8), функцию V(x(t+ ), t+ ) разложим в ряд Тейлора в окрестности точки (x(t), t):</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1481" type=3D"#_x0000_t75" alt=3D"" style=3D'width:273.75pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage412.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image012.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.9)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1482" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:148.5pt; height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge413.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image013.gif"/> </v:shape>- вектор-строка частных производных в точке (x(t),t); <v:shape id= =3D"_x0000_i1483" type=3D"#_x0000_t75" alt=3D"" style=3D'width:76.5pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge414.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image014.gif"/> </v:shape>- приращение вектора x(t); <v:shape id=3D"_x0000_i1484" type=3D"#= _x0000_t75" alt=3D"" style=3D'width:27pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge415.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image015.gif"/> </v:shape>- совокупность последующих членов ряда Тейлора.</p> <p>Подставим (П.1.7), (П.1.9) в (П.1.6):</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1485" type=3D"#_x0000_t75" alt=3D"" style=3D'width:429.75pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage416.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image016.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.10)</p> </td> </tr> </table> <p>где <v:shape id=3D"_x0000_i1486" type=3D"#_x0000_t75" alt=3D"" style=3D'= width:27.75pt; height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge417.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image017.gif"/> </v:shape>- все члены с порядком малости, большим, чем у <v:shape id=3D"_x0= 000_i1487" type=3D"#_x0000_t75" alt=3D"" style=3D'width:15pt;height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge404.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image004.gif"/> </v:shape>.</p> <p>Так как величина V(x(t),t) не зависит от управления, вынесем ее из-под символа минимума и взаимно уничтожим с левой частью выражения. Оставшиеся ч= лены разделим на <v:shape id=3D"_x0000_i1488" type=3D"#_x0000_t75" alt=3D"" styl= e=3D'width:15pt; height:14.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge404.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image004.gif"/> </v:shape>. Производную <v:shape id=3D"_x0000_i1489" type=3D"#_x0000_t75" a= lt=3D"" style=3D'width:33pt;height:16.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge418.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image018.gif"/> </v:shape>, не зависящую от u(t), вынесем за скобки, а производную <v:shape id=3D"_x0000_i1490" type=3D"#_x0000_t75" alt=3D"" style=3D'width:28.5pt;he= ight:16.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge419.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image019.gif"/> </v:shape>заменим согласно (П.1.1) функцией f. Тогда с учетом того, что <v:= shape id=3D"_x0000_i1491" type=3D"#_x0000_t75" alt=3D"" style=3D'width:66pt;heig= ht:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge420.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image020.gif"/> </v:shape>, получим уравнение Беллмана</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"86%" style=3D'width:86.64%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1492" type=3D"#_x0000_t75" alt=3D"" style=3D'width:276.75pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage421.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p1/image021.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.1.11)</p> </td> </tr> </table> <h3>Приложение 2 к лекции 9</h3> <p align=3Dcenter style=3D'text-align:center;mso-outline-level:4'><b><span style=3D'color:#996600'>Вывод уравнения Риккати<o:p></o:p></span></b></p> <p>Решим уравнение Беллмана (9.3) для объекта управления (9.1) и функционала качества (9.2).</p> <p>Наличие неинтегрального члена в составе функционала (9.2) не влияет на структуру уравнения Беллмана, если принять</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1493" type=3D"#_x0000_t75" alt=3D"" style=3D'width:264pt;height:38.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage422.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image001.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.1)</p> </td> </tr> </table> <p>где <i>L</i> - подынтегральная функция из (9.2).</p> <p>Тогда уравнение Беллмана принимает вид</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1494" type=3D"#_x0000_t75" alt=3D"" style=3D'width:449.25pt;height:31.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage423.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image002.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.2)</p> </td> </tr> </table> <p>Оптимальное уравнение будем искать из условия</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1495" type=3D"#_x0000_t75" alt=3D"" style=3D'width:62.25pt;height:33pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge424.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image003.gif"/> </v:shape></p> <p>Выполнив дифференцирование, получим</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1496" type=3D"#_x0000_t75" alt=3D"" style=3D'width:151.5pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage425.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image004.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.3)</p> </td> </tr> </table> <p>Из (П.2.3) получаем оптимальное уравнение</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1497" type=3D"#_x0000_t75" alt=3D"" style=3D'width:174.75pt;height:36.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage426.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image005.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.4)</p> </td> </tr> </table> <p>Чтобы использовать выражение (П.2.4), необходимо найти функцию <b>V</b>(= <b>x</b>(<i>t</i>),<i>t</i>). Для этого подставим (П.2.4) в (П.2.2):</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1498" type=3D"#_x0000_t75" alt=3D"" style=3D'width:436.5pt;height:36.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage427.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image006.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.5)</p> </td> </tr> </table> <p>Решение (П.2.5) при <i>t=3DT</i> должно удовлетворять условию</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1499" type=3D"#_x0000_t75" alt=3D"" style=3D'width:132.75pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage428.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image007.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.6)</p> </td> </tr> </table> <p>Общее решение будем искать в виде квадратной формы переменных состояния<= /p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1500" type=3D"#_x0000_t75" alt=3D"" style=3D'width:133.5pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage429.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image008.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.7)</p> </td> </tr> </table> <p>где <b>P</b>(<i>t</i>)=3D<b>P</b><sup>T</sup>(<i>t</i>) - неизвестная симметричная нестационарная <i>n</i> x <i>n</i>-матрица.</p> <p>Подставим в (П.2.5) выражение (П.2.7) и </p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1501" type=3D"#_x0000_t75" alt=3D"" style=3D'width:447.75pt;height:63.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge430.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image009.gif"/> </v:shape></p> <p>Последнее слагаемое представим суммой</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1502" type=3D"#_x0000_t75" alt=3D"" style=3D'width:306.75pt;height:30.75pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge431.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image010.gif"/> </v:shape></p> <p>Тогда полученное соотношение преобразуется к виду</p> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape id= =3D"_x0000_i1503" type=3D"#_x0000_t75" alt=3D"" style=3D'width:367.5pt;height:31.5pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge432.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image011.gif"/> </v:shape></p> <p>При любых состояниях <b>x</b>(<i>t</i>)<v:shape id=3D"_x0000_i1504" type= =3D"#_x0000_t75" alt=3D"" style=3D'width:11.25pt;height:11.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/ima= ge146.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/i39.gif"/> </v:shape> 0 предыдущее соотношение выполняется, если</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1505" type=3D"#_x0000_t75" alt=3D"" style=3D'width:292.5pt;height:18pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage433.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image012.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.8)</p> </td> </tr> </table> <p>В результате получили нелинейное дифференциальное уравнение, известное п= од названием <i>матричного уравнения Риккати</i>.</p> <p>Очевидно, что его решение следует искать при граничном условии</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"90%" style=3D'width:90.0%;padding:0cm 6.45pt 0cm 0cm'> <p align=3Dcenter style=3D'text-align:center'><b>P</b>(<i>T</i>)=3D<b>F</= b></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.9)</p> </td> </tr> </table> <p>Для стационарных систем уравнение (П.2.8) становится алгебраическим:</p> <table class=3DMsoNormalTable border=3D0 cellspacing=3D0 cellpadding=3D0 wi= dth=3D"100%" style=3D'width:100.0%;mso-cellspacing:0cm;mso-padding-alt:0cm 0cm 0cm 0cm'> <tr style=3D'mso-yfti-irow:0;mso-yfti-firstrow:yes;mso-yfti-lastrow:yes'> <td width=3D"86%" style=3D'width:86.64%;padding:0cm 6.45pt 0cm 0cm'> <p class=3DMsoNormal align=3Dcenter style=3D'text-align:center'><v:shape = id=3D"_x0000_i1506" type=3D"#_x0000_t75" alt=3D"" style=3D'width:154.5pt;height:17.25pt'> <v:imagedata src=3D"Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/i= mage434.png" o:href=3D"http://www.ispu.ru/library/lessons/kolganov2/p2/image013.gif"= /> </v:shape></p> </td> <td style=3D'padding:0cm 6.45pt 0cm 0cm'> <p align=3Dright style=3D'text-align:right'>(П.2.10)</p> </td> </tr> </table> <h3>Рекомендуемая литература</h3> <p>1. Андреев Ю.Н. Управление конечномерными линейными объектами. - М.: Нау= ка, 1976.- 423 с.</p> <p>2. Воронов А.А. Введение в динамику сложных управляемых систем. М.: Наук= а. Гл. ред. физ.-мат. лит., 1985.- 352 с.</p> <p>3. Воронов А.А. Устойчивость, управляемость, наблюдаемость. - М.: Наука, 1979 - 336 с.</p> <p>4. Директор С., Рорер Р. Введение в теорию систем:/ Пер. с англ.- М.: Ми= р, 1974.- 464 с.</p> <p>5. Колганов А.Р., Буренин С.В. Алгоритмы и программы функционального проектирования систем управления электромеханическими объектами: Учеб. посо= бие/ Иван. гос. энерг. ун-т. - Иваново, 1997. - 140 с.</p> <p>6. Колганов А.Р., Комаров А.Б. Компьютерный комплекс функционального проектирования систем управления динамическими объектами: Практ. пособие/ И= ван. гос. энерг. ун-т. - Иваново, 2001. - 60 с.</p> <p>7. Колганов А.Р., Комаров А.Б. Компьютерный комплекс функционального проектирования электромеханических систем:/ Свидетельство об официальной регистрации программы для ЭВМ № 2001610471 от 19 апреля 2001 г. – М.: РОСПАТЕНТ, 2001.</p> <p>8. Кузовков Н.Т. Модальное управление и наблюдающие устройства М.: Машин= остроение , 1976. - 184 с. </p> <p>9. Летов А.М. Аналитическое конструирование регуляторов// Автоматика и телемеханика. - 1960. № 4-6. </p> <p>10. Мошиц Г., Хорн П. Проектирование активных фильтров: Пер. с англ.- М.: Мир, 1984.- 320 с.</p> <p>11. Острем К., Виттенмарк Б. Системы управления с ЭВМ: Пер. с англ.- М.: Мир, 1987.- 480 с.</p> <p>12. Сигорский В.П. Математический аппарат инженера. - Киев.: Технiка, 19= 77.- 768 с.</p> <p>13. Справочник по теории автоматического управления / Под ред. А.А. Красовского.- М.: Наука. Гл.ред. физ.-мат-лит., 1987.- 712 с.</p> <p>14. Стрейц В. Метод пространства состояний в теории дискретных линейных систем управления: Пер. с англ.- М.: Наука, 1985.- 298 с.</p> <p>15. Тарарыкин С.В., Тютиков В.В. Системное проектирование линейных регуляторов состояния: Учеб. пособие/ Иван. гос. энерг. ун-т. - Иваново, 19= 97. - 92 с.</p> <p>16. Ключев В.И. Теория электропривода. - М.: Энергоатомиздат, 1985. - 56= 0 с.</p> <p class=3DMsoNormal><o:p> </o:p></p> </div> </body> </html> ------=_NextPart_01C5168D.19EF4C40 Content-Location: file:///C:/9EA24656/Osnovn_razdel_teorii_avtomat_upravl-Kolganov.files/image001.png Content-Transfer-Encoding: base64 Content-Type: image/png iVBORw0KGgoAAAANSUhEUgAAAiIAAAC+CAMAAAA/UZQJAAADAFBMVEUEBAQEBCQEBE8EBG8EBHoE GgQEGiQEJC8EJDoEJE8ELwQELy8EOgQEOkQPBAQPBFoPBG8PLxoaDwQaDyQaDy8aD0QaGgQaGiQk DwQkGhokGi8kGk8kGlokJCQkmg8kpA8vJAQvjw8vmhovpA86jxo6mho6miQ6mjo6pBo6pCQ6xARE eiREhCREhC9EjyREmhpEmiREmi9Emm9EryREr09EuhpEukRExBpEzy9E2hpE5BpPhDpPjy9PjzpP pFpPpGRPry9Pr09PujpPxCRPxC9PxDpPxERPzxpPzyRPzy9Pz0RP5BpP5CRP5C9P5DpahERajzpa mjpamm9apDpapG9apHpary9ar09ar29ar3paujpaunpaxC9axDpazyRazy9azzpa2i9a2mRa5CRa 5C9a5DpkmkRkpDpkr0Rkr29kujpkxCRkxDpk2jpk2kRk5CRk5C9k5ERk7zpvmk9vr09vuiRvuk9v xERvxE9vxG9vzy9vz09v2jpv7zp6mmR6pFp6r296xER6xFp6zwR6z1p62i962k965E+Ez0SEz2SE 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