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Ивановский государственный энергетический университет=
p>
Кафедра Электропривод и автоматизация промышленных уста=
новок
Электронный конспект лекций
ОСНОВНЫЕ РАЗДЕЛЫ СОВРЕМЕННОЙ ТЕОРИИ АВТОМАТИЧЕСКОГО
УПРАВЛЕНИЯ
Автор: КОЛГАНОВ Алексей Руфимович, доктор технических наук,
профессор кафедры «Электропривод и автоматизация промышленных установок»
Тел. (0932) 385795, (0932) 419063
klgn@drive.ispu.ru
klgn@indi.ru
ВВЕДЕНИЕ
До пятидесятых годов ХХ века классической теорией автоматического
регулирования (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
малом» А.М. Ляпунова, но имела ярко выраженную
инженерную направленность.
Под влиянием потребностей автоматизации управления технологическими проц=
ессами
и движущимися объектами в сороковых и пятидесятых
годах прошлого века интенсивно развивалась теория автоматического управл=
ения
(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 современной. 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.
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».
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.
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 современной теории автоматического
управления. (СТАУ)
Основным характерными признаками СТАУ является описание процессов в п=
ространстве
состояний и применения для решения задач анализа и синтеза систем ме=
тодов
пространства состояний.
В курсе лекций, который перед Вами, мы представим:
- способы описания
динамических объектов управления в пространстве состояний с помощью
векторно-матричных моделей в непрерывном и дискретном времени;
- методы и алгоритмы анал=
иза
основных свойств объектов и систем управления на основе их
векторно-матричных моделей;
- методы и алгоритмы синт=
еза
систем управления с регуляторами и наблюдателями состояния.
1. Цель преподавания дисциплины
Курс имеет целью изучение современных методов теории пространства состоя=
ния
для решения задач анализа и синтеза систем автоматического управления.
Полученные в курсе сведения используются в дальнейшем при изучении
практически всех профилирующих дисциплин специальности, выполнении курсовых=
и
дипломных проектов.
2. Задачи изучения курса
Основной задачами курса являются:
- формирование навыков
построения векторно-матричных моделей объектов и систем управления в
непрерывном и дискретном времени;
- формирование устойчиво=
го
представления об использовании математического аппарата матричной алге=
бры
для анализа и синтеза систем управления;
- формирования навыков решения практических задач построения современных
систем управления с наблюдателями состояния полного и пониженно=
го
порядка;
3. Перечень дисциплин, усвоение которых необходимо для изучения курса=
h3>
а) высшая математика - дифференциальное, интегральное и операцион=
ное
исчисление, матрицы, функции от матриц, матричная алгебра, теория
дифференциальных уравнений.
б) теория автоматического управления (классическая) - принципы
построения и элементный состав систем автоматического управления,
преобразование Лапласа, передаточные функции, графы и структурные схемы.
в) теоретические основы электротехники - математическое описание
электрических цепей, переходные процессы в линейных и нелинейных цепях.
4. Общая характеристика и рекомендации по изучению материала
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.
Теоретическая часть курса дает представление о методах и алгоритмах анал=
иза
и синтеза систем управления с использованием современной теории пространства
состояния.
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.
Программное обеспечение компьютерного комплекса FuncPro 1.0, электронное
руководство по его применению, практическое пособие [6] могут быть поставлены Вам при дополнительном обращении.
Лекция № 1.
Тема:«Векторно-матричные=
модели
систем управления в непрерывном времени»
Понятие пространства состояний
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).
Рис.1.1. Скалярное (а) и векторное (б) представления
динамической системы в виде "черного ящика"
Все переменные, характеризующие систему, можно разделить на три группы.<=
/p>
1. Входные переменные или входные воздействия, генерируемые
системами, внешними по отношению к исследуемой системе. Они характеризуются
вектором входа.
r - число входов
2. Выходные переменные, характеризующие реакцию системы на указан=
ные
входные воздействия. Представляются вектором выхода
m - число выходов.
3. Промежуточные переменные, характеризующие внутреннее состояние систем=
ы, -
переменные состояния, представляются вектором
n - число переменных состояния.
Таким образом, совокупность входов можно рассматривать как один обобщенн=
ый
вход, на который воздействует вектор входа u, совокупность выходов к=
ак
вектор y, а совокупность промежуточных координат, характеризующих
состояние системы, - в виде вектора состояния x (см. рис. 1.1, б).=
p>
Состояние системы - это та минимальная информация о прошлом, кото=
рая
необходима для полного описания будущего поведения (т.е. выходов) системы, =
если
поведение ее входов известно.
Собственно система, ее входы и выходы - это три взаимосвязанных объекта,
которые в каждой конкретной ситуации определяются соответственно математиче=
ской
моделью системы, заданием множеств входных и вых=
одных
переменных.
Решение задач анализа и синтеза связано с исследованием состояний систем=
ы,
множество которых образует пространство состояний,
.
Векторно-матричные модели в непрерывном времени
В общем случае динамическая система в непреры=
вном
может быть описана парой матричных уравнений:
где F - n-мерная вектор-функция системы; Q - m=
i>-мерная
вектор-функция выхода.
Матричное уравнение (1.1) называют уравнением состояния системы. =
Его
решение, удовлетворяющее начальному условию
, дает вектор состояния системы
Матричное уравнение (1.2), определяющее выходные переменные в зависимост=
и от
x(t) и u(t), называют уравнением выхода.
В частном случае зависимости
могут быть линейными комбинациями переменных состояния xi=
и входных переменных uq. При этом динамическая система
описывается в векторно-матричной форме:
Переход к стационарным моделям позволяет оперировать с коэффициентными
матрицами, т.е. со стационарными уравнениями
где:
А - функциональная матрица размером n x n, называемая м=
атрицей
состояния системы (объекта);
В - функциональная=
матрица размером n x r, называемая матрицей управления (входа=
);
С - функциональная=
матрица размером m x n, называемая матрицей выхода по состоянию=
i>;
D - функциональная матрица размером m x r, называемая м=
атрицей
выхода по управлению.
Очень часто D=3D0, т.е. выход непосредственно не зависит от входа=
.
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).
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.
Построение ВММ реального объекта сопряжено с проблемами линеаризации
исходного математического описания и приведения его к структурированному ви=
ду -
форме Коши.
Если мы знаем физическое описание системы и можем записать уравнения,
описывающие поведения ее отдельных частей, то по=
лучить
уравнения состояния системы обычно сравнительно не трудно. Покажем эту
процедуру на нескольких примерах.
Пример 1.1. Получим уравнения состояния для простейшей
RLC-цепи, показанной на рис 1.2.
Динамическое поведение этой системы при
полностью определяется, если известны начальные значения
и входное напряжение U(t) при
. Следовательно,
можно выбрать в качестве переменных состояния, то есть
Для указанных переменных состояния можно записать дифференциальные уравн=
ения
или в векторно-матричной форме
Таким образом для рассматриваемой системы матрицы А,
В, С векторно-матричной модели будут иметь следующий вид:
Пример 1.2. На рис. 1.3. показан электродвигатель постоянн=
ого
тока независимого возбуждения, работающий при постоянном магнитном потоке (=
Ф=3Dconst).
Дифференциальные уравнения для такого объекта могут быть записаны
относительно следующих переменных состояния:
- скорости вращения ротора, тока якоря i(t), углового
перемещения ротора
. При использовании знакомых зависимостей для электродвижущей сил=
ы
и вращающего момента двигателя
получим уравнение электрической цепи
и уравнения вращающейся части
где J – приведенный момент инерции электродвигателя.
Представляя векторы состояния, входа и выхода как
получим следующую векторно-матричную модель электродвигателя
постоянного тока
То есть для рассматриваемой системы матрицы А=
,
В, С векторно-матричной модели будут иметь следующий вид:
Пример 1.3. 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.
Здесь легко выделить три функциональных элемента, соответствующие трем в=
идам
преобразования энергии:
преобразователь, осуществляющий управляемое преобразование
электрической энергии;
двигатель, выполняющий преобразование электрической энергии в механическую, - электромеханический преобразователь;=
p>
механизм, осуществляющий передачу механической энергии от вала
двигателя через редуктор к рабочему органу - платформе.
При использовании общеизвестных допущений [5] и обозначений координат и
параметров такого объекта его динамическое поведение пр=
и
МС=3D0 описывается следующей системой линейных дифференциальных уравнений:<=
/p>
Если компонентами вектора состояния выбрать
, где Uп –
напряжение преобразователя, iя - ток электродвигателя,
- скорость вращения электродвигателя, МУ - моме=
нт
упругости механизма,
- скорость вращения механизма, то элементы векторно-матричной мод=
ели
принимают следующий вид:
После подстановки реальных значений параметров объекта, которые приведен=
ы в
табл. 1.1, компоненты матриц состояния А и
управления В принимают вид (1.13).
На рис. 1.5. приведено окно редактирования векторно-матричной модели (1.=
13)
в среде Компьютерного комплекса функционального проектирования динамичес=
ких
систем.
Контрольные вопросы к лекции № 1.
1. Какие переменные при построении математического описания системы прин=
ято
называть
a) входными переменными;
b) выходными переменными;
c) переменными состояния?
2. Математическое описание объекта с одним входом и одним выходом
представлено структурной схемой, содержащей q элементов, представленных
передаточной функцией общего вида
Как в этом случае можно определить размерность пространства состояния
для описания этого объекта?
3. Математическое описание объекта с двумя входами
и одним выходом y(t) представлено следующим уравнением в
операторной форме
Какова в этом случае будет размерность пространства состояния n для опис=
ания
этого объекта?
4. Выберите из приведенных ниже записей возможные формы представления уравнения
состояния для непрерывных систем.
5. Объект управления имеет r – входов, m - выходов, его
математическое описание в непрерывном времени содержит n дифференциальных
уравнений первого порядка. Какова в этом случае будет размерность матрицы
состояния?
6. Сформируйте векторно-матричную модель фильтра, электрическая схема
которого представлена на рис. 1.6.
Здесь следует учесть, что
- объект имеет один вход =
- U1
один выход - iH ;
- все параметры электриче=
ской
схемы R1, R2, L, C1, C2=
, RH
известны и являются постоянными;
- могут быть использованы
следующие обозначения
7.При составлении математического описания динамических процессов в упру=
гом
электромеханическом объекте, влючающем в себя электродвигатель постоянного =
тока
независимого возбуждения (Ф=3Dconst) и ме=
ханизм,
модель которого представляется двухмассовой системой (см. пример 1.3), могут
быть использованы следующие переменные:
- iя
- ток электродвигателя,
-
- скорость вращения электродвигателя,
- Му –
упругий момент механизма,
-
- скорость вращения механизма,
-
- угол поворота ротора электродвигателя,
- l – линейное
перемещение механизма.
Какие из этих переменных, и в какой последовательности включены в состав
вектора состояния
приведенной ниже векторно-матричной модели?
ОТВЕТЫ
№ задания
|
Ответ
|
1.
|
a) переменные, характеризующие реакцию системы на входные воздействия;=
b) переменные, генерируемые системами, внешними по отношению к исследу=
емой
системе;
c) промежуточные переменные, характеризующие внутреннее состояние сист=
емы.
|
2.
|
|
3.
|
n=3D2
|
4.
|
a, e, f, g
|
5.
|
(n x n): n – строк и n – столбцов
|
6.
|
|
7.
|
|
Лекция № 2.
Тема: «Векторно-матричные модели систем управления в
дискретном времени»
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.
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.
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
, необходимо определить зависимости между ними.
Построение дискретного эквивалента непрерывной системы называется ква=
нтованием
непрерывной системы.
В дальнейшем будем считать, что непрерывная система описывается уравнени=
ями
Наиболее распространенная ситуация в цифровом управлении состоит в том, =
что
АЦП сохраняет уровень аналогового сигнала постоянным до тех пор, пока не
потребуется новое преобразование
Так как управляющий сигнал прерывист, необходимо установить его поведени=
е в
точках разрыва. Допустим, что сигнал непрерывен справа и представляется
дискретным процессом
Определим связь между переменными системы в моменты квантования. При
заданном состоянии в момент квантования tk состояние в
некоторый момент t можно получить, решив систему (2.1).
Первому уравнению системы (2.1) соответствует однородное уравнение
решение которого равно
Обозначим
тогда решение уравнения можно представить как
Допустим, что решение неоднородного уравнения имеет вид
Дифференцируя (2.4) по t, получим
Сравнение первого уравнения системы (2.1) и уравнения (2.5) позволяет
получить следующее соотношение:
откуда
Подставим (2.7) в (2.4), тогда
Здесь учтено, что
Вектор С2 определяется по
начальным условиям. При t=3Dtk
Таким образом,
На основании решения (2.22) определим состояние и выход системы в следую=
щий
момент квантования tk+1 :
где
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.
Для периодического квантования с периодом Т, tk=3DkT, =
модель
(2.10) сводится к стационарной системе, которую будем рассматривать в
дальнейшем. Для простоты записи примем Т=3D1, тогда
где
Если матрица А не особая, то возможна =
запись
где I- единичная матрица.
Таким образом, для получения ВММ непрерывной системы в дискретном времени
требуется вычислить матричную экспоненту и проинтегрировать ее.
Вначале рассмотрим простейший случай. Используя разложение матричной
экспоненты в ряд, получим следующие выражения:
При i=3D1 выражения (2.15) принимают вид
По существу такая аппроксимация совпадает с простейшим методом решения д=
ифференциальных
уравнений по Эйлеру.
Модель (2.16) будем называть дискретной моделью по Эйлеру.
Более точное нахождение матриц Ф и =
Г
может быть осуществлено различными способами, в том числе такими, как:
- разложение матричной
экспоненты в ряд;
- применение преобразован=
ия
Лапласа
- использование теоремы
Гамильтона - Кэли.
Применение указанных способов проиллюстрируем примерами.
Разложение матричной экспоненты в ряд
Пример 2.1. Получим ВММ в дискретном времени для интеграто=
ра
второго порядка, структурная схема которого приведена на рис. 2.2. Динамиче=
ский
процесс для такого объекта при
описывается дифференциальным уравнением
Вводя
как состояния системы, получим векторно-матричную запись уравнения
(2.18)
Определим матрицы Ф и Г разложе=
нием
матричной экспоненты в ряд.
Так как А2=3D0, приведенный=
ряд
сходится точно.
Таким образом, векторно-матричная модель двойного интегратора в дискретн=
ом
времени имеет вид
Использование преобразования Лапласа.
Пример 2.2. Для непрерывной модели электродвигателя
постоянного тока будем считать, что Lя=3D0, Mс=3D0.
Тогда уравнения электродвигателя будут иметь вид
В нормализованной форме записи значений параметров электродвигателя, то =
есть
при
в пространстве состояния
получим следующую запись ВММ в непрерывном времени.
Найдем изображение матричной экспоненты по Лапласу:
Вычисляя обратное преобразование элементов полученной матрицы, получим
матрицу
Для нахождения матрицы Г используем выражение (2.32):
Таким образом, векторно-матричная модель рассматриваемой системы в
дискретном времени имеет вид
Использование теоремы Гамильтона - Кэли.
Для систем с реальными параметрами чаще всего применяют теорему
Гамильтона - Кэли, согласно которой любую функцию квадратной матрицы =
p>
можно вычислить через характеристический полином
где
- корни характеристического уравнения матрицы=
А
Пример 2.3. Применяя указанную теорему, построим ВММ в
дискретном времени для непрерывной системы
Первоначально определим корни характеристического уравнения
решение которого дает следующие значения:
.
Так как порядок матрицы равен 2, имеем
Коэффициенты
определяем из системы уравнений
решение которой имеет вид
Следовательно,
Контрольные вопросы к лекции № 2.
1. Запишите дискретную модель по Эйлеру для непрерывного объекта
представленного следующей векторно-матричной моделью в непрерывном времени<=
/p>
2. Для непрерывного объекта, представленного на рис. 2.3 в виде структур=
ной
схемы, определите внутреннее содержание матриц Ф=
,
Г, С векторно-матричной модели в дискретном времени с периодом
квантования Т.
3. Непрерывный объект в дискретном времени с периодом квантования Т=
i>
представлен векторно-матричной моделью следующего вида
Запишите уравнение состояния этого объекта в непреры=
вном
времении.
ОТВЕТЫ
Лекция № 3.
Тема: «Вычислительные алгоритмы формирования
векторно-матричных моделей в дискретном времени»
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.
Алгоритм 1. Вычисление матричной экспоненты с помощью
степенного ряда при заданном значении периода квантования Т.
Получение относительно точного решения с помощью приведенного в предыдущ=
ей
лекции степенного ряда
сопряжено с необходимостью вычисления высоких степеней матрицы А. Однако с помощью алгоритма, построенного на
основании теоремы Кели-Гамильтона матричная экспонента может быть вычислена=
с
помощью (n-1) степеней матрицы А.
Согласно этому алгоритму вычисление матриц Ф<=
/span>
и Г производится в несколько этапов (шагов).
Шаг 1: вычисляют первые (n - 1) степеней матрицы A.=
Шаг 2: вычисляют коэффициенты характеристического уравнения
по следующим формулам:
где Tk=3Dtr(Ak) - след ма=
трицы Ak.
Шаг 3: согласно теореме Кели-Гамильтона n - ая степень мат=
рицы
вычисляется через коэффициенты характеристического уравнения как
a (n + m)-я степень матрицы A находится с помощью последов=
ательного
умножения этого соотношения на матрицу A.
для m=3D0,1,2...; где
а остальные коэффициенты определяются из рекуррентных соотношений:
Шаг 4: для любого заданного T функцию eAT
можно записать как
Таким образом, функцию eAT мо=
жно
определить с любой заданной точностью с помощью уже вычисленных матриц A=
2,
A3,..., A n-1 и коэффициентов qij<=
/sub>
без вычисления и суммирования степеней матрицы A выше (n-1).<=
/p>
Интегрирование матричной экспоненты при разложении в ряд заменяется
взвешенной суммой
которая вычисляется аналогично матричной эксп=
оненте
Применение соотношений (3.8-3.10) возможно только в том случае, если зар=
анее
определена величина периода квантования.
К преимуществам данного метода можно отнести простоту алгоритма построен=
ия,
быструю сходимость и малое время расчета при малых значениях Т. А то
обстоятельство, что алгоритм не требует вычисления высоких степеней матрицы,
позволяет избежать зацикливания исполняющей программы, если заданная точность не 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.
Методы вычисления матричной экспоненты разложением в степенной ряд имеют
существенный недостаток - изменение периода квантования требует повторения =
всех
этапов алгоритма.
Кроме того, сходимость алгоритма и точность решения для конкретной модели
определятся значением периода квантования Т. Определение критического
значения Тmax 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.
Алгоритм 2. Вычисление матричной экспоненты при неизвес=
тном
значении периода квантования Т.
Особую актуальность представляют символьно-численные алгоритмы
формирования дискретной ВММ, когда величина периода квантования заранее
неизвестна.
Для этих целей могут быть использованы представление eA=
sup>T
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
где
- различные собственные значения матрицы A (то есть
), mk - кратность собственного значения
как нуля минимального многочлена матрицы A, Zkl
- компоненты матрицы A, определяемые через значения приведенной
присоединенной матрицы.
Если характеристический многочлен матрицы А=
span>
имеет все различные нули, то mk=3D1 для k=3D1,2,
… ,n и минимальный многочлен совпадает с характеристическим. При этом основ=
ная
формула теоремы Сильвестра приводится к следующему виду:
Если первоначально считать период квантования неизвестным, то будет
целесообразным представить матрицы Ф и Г
формируемой ВММ трехмерными Ф(n x n x n), Г<=
/b>(n
x m x n).
В этом случае матрицы дискретной модели записывается как произведение
некоторых матричных коэффициентов, умноженных на собственные моды
, то есть
Таким образом, трехмерные матрицы дискретных моделей содержат n
квадратных коэффициентных матриц Fi или Gi
и при выбранном значении периода квантования Т численное значение
каждого элемента матриц вычисляется по формулам
Причем моды комплексных собственных значений записываются через синус и
косинус, например, для
значения корней характеристического полинома определяются численн=
ым
методом по алгоритму, в основу которого положен метод нахождения комплексных
корней многочлена Берстоу [12].
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.
Для подтверждения преимуществ символьно-численного алгоритма рассмотрим
варианты формирования дискретной модели непрерывного объекта 3-го порядка,
представляющего собой линеаризованную модель сериесного электродвигателя.=
p>
В непрерывном времени объект описывается векторно-матричной моделью вида=
:
Компонентами вектора состояния здесь являются: магнитный поток - Ф, ток - 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'>
.
В результате вычислительных экспериментов сформировано выражение для
вычисления матричной экспоненты
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.
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.
Заканчивая рассмотрение векторно-матричных моделей, целесообразно
сформулировать следующие выводы.
1. Векторно-матричные модели являются высокоформализованным средством
математического описания систем управления, общая структура которого не зав=
исит
от сложности объекта.
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.
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>
4. Проектные операции построения ВММ в дискретном времени практически не
осуществимы без применения ЭВМ.
Контрольные вопросы к лекции № 3.
1. Для построения дискретной модели непрерывного объекта 5-го порядка
используется алгоритм вычисления матричной экспоненты с помощью степенного =
ряда
25 степени (i=3D0, 1, …, 25). Какую максимальную степень (k)м=
атрицы А (Ak) при этом необх=
одимо
вычислить?
2. Укажите основной недостаток алгоритма вычисления матричной экспоненты=
с
помощью степенного ряда.
3. Назовите главное достоинство символьно-численного алгоритма формирова=
ния
дискретной векторно-матричной модели.
4. При каких условиях для вычисления матрицы состояния Ф
дискретной ВММ с помощью символьно-численного алгоритма дважды используются
одинаковые матричные коэффициенты, т.е. Fk=3D F=
k+1.
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
k=3D4
|
2
|
Изменение периода T квантования требует повторения всех этапов
алгоритма
|
3
|
Наиболее трудоемкие вычислительные операции по определению постоянных
матричных коэффициентов выполняются один раз для любого количества
исследуемых значений периода квантования
|
4
|
В том случае, если имеются комплексные собственные значения матрицы А комплексно, т.е.
|
Лекция № 4.
Тема: «Анализ свойств объектов и систем управления»=
Здесь мы рассмотрим вопросы анализа специфических свойств объектов и сис=
тем
управления, представленных векторно-матричными моделями в непрерывном време=
ни
моделями типа "вход - выход"
где
а также векторно-матричными моделями в дискретном времени
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).
Общее решение уравнений состояния в непрерывном времени было выполнено п=
ри
рассмотрении алгоритмов вычисления матриц состояния (Ф) и управления=
(Г)
векторно-матричной модели непрерывного объекта в дискретном времени.
Решение уравнения состояния в дискретном времени.
Дискретная стационарная система может быть описана разностными уравнения=
ми
(4.3), если значение периода квантования для простоты записи предварительно
принято равным Т=3D1.
Предположим, что известны начальный вектор x(k0=
) и
входные сигналы: u(k0), u(k0+1), u(k0+2),... .
Систему уравнений (4.3) можно решить просто, выполнив следующие итерации=
:
или для любого значения k имеем
Полученное решение (4.4) состоит из двух частей: одна зависит от начальн=
ых
условий, другая является взвешенной суммой входных сигналов.
Достижимость и управляемость
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 достижимость, управляемость и наблюдаемость. Нал=
ичие
этих свойств у объектов управления позволяет рассчитывать оптимальное
управление с помощью простых математических операций.
Сформулируем определения понятий достижимости и управляемости.
Определение 4.1. Состояние x(t1) линейной систе=
мы
достижимо, если существует момент времени t0 < t1
и такой вход, который переводит начальное состояние системы x(t0)=3D0
в желаемое состояние x(t1), при условии, что интервал ( t0 - t0) конечен=
.
Определение 4.2. Состояние x(t1) линейной систе=
мы
управляемо, если существует момент времени t2 > t1
и такой вход, который переводит состояние системы x(t1) в
состояние x(t2)=3D0 (начало координат), при условии, что
интервал (t2 - t1) конечен.
Для непрерывных систем вида (4.1) каждое достижимое состояние =
управляемо.
Поэтому при анализе непрерывных систем говорят только об управляемости.
Для исследования достижимости используем векторно-матричную модель объек=
та
управления (ОУ) в дискретном времени при Т=3D1
Теорема 4.1. Состояние системы x(n) достижимо, если=
и
только если ранг матрицы достижимости
равен размерности пространства состояний n.
Предположим, что задано начальное состояние x(0). Тогда состояние в моме=
нт времени
n ( n - порядок системы) определяе=
тся
соотношением
где
Если матрица WD имеет ранг n , то можно найти <=
i>n
уравнений, решением которых является такой управляющий сигнал, что из
начального состояния x(0) система перейдет в желаемое конечное состояние x(=
n).
Условия управляемости могут быть получены также из выражения (4.6). Из
определения 4.2 следует, что
. Тогда
Для того чтобы система была управляемой, т. е. могла=
быть
переведена из состояния x(0)
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
Векторы должны быть линейно независимы, так как в противном случае
состояние x(n)=3D 0 не может быть достигнуто. Из этого следует:
Теорема 4.2. Состояние x(0)
0 системы (4.5) управляемо, если и только если ранг матрицы
равен размерности пространства состояний n.
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 det (Ф
)
0
Если матрица Ф не вырождена, условия достижим=
ости и
управляемости эквивалентны.
В непрерывных системах требование достижимости совпадает с требованием
управляемости. Поэтому здесь используют только понятие управляемости, замен=
яя
его в большинстве случаев понятием достижимости.
Теорема 4.3. Состояние непрерывной системы управляемо, если и только если
ранг матрицы
равен размерности пространства состояний.
Дополнительно можно ввести понятие индекса управляемости системы (4.1).
Индексом управляемости системы называется такое минимальное целое число y, =
при
котором матрица Qy ,определяемая выражением
, имеет ранг, равный размерности пространства состояний n. В общем
случае
.
Если индекс управляемости системы равен рангу матрицы Qy (y=
=3Dn), в
этом случае речь может идти о полной управляемости динамической системы. В
противном случае (y < n) система характеризуется неполной управляемостью=
.
Рассмотрим два характерных случая:
а) В - матрица-столбец размером n x 1.
Матрица управляемости Qy в этом случае - квадратная. Вычисляется
определитель. Если определитель det(Qy)
0, матрица имеет ранг, равный порядку системы. В этом случае сис=
тема
полностью управляема;
б) В - матрица
размером n x m , то есть система имеет m каналов управления. Матрица
управляемости Qy 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.
Пример 4.1. Определим управляемость системы
a)
матрица управляемости
система управляема.
б)
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 Qу. 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.
В случае представления объекта управления моделью типа “вход - выход” (4=
.2)
условием его управляемости является отсутствие общих корней полиномов А(s) и B(s), то есть система (4.2) управляема, если и
только если алгебраические уравнения
<=
/o:p>
не имеют общих корней.
Данное условие может быть проверено как непосредственным вычислением кор=
ней
полиномов, так и косвенным путем.
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
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.
Результант формируется следующим образом:
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=
0 по строке записываются коэффициенты в строке - нулев=
ые.
Общее число коэффициентов в строке - (n+m).
б) следующие n строк результанта формируются аналогично с использованием
коэффициентов полинома числителя bk (k=3D0,1,...,m).
Пример. 4.2. Определим управляемость системы, представленной передаточной
функцией
<=
/o:p>
|
, det(R)=3D0, система не управляема!
|
Прямой расчет корней числителя и знаменателя дает аналогичные результаты,
приведенные в табл. 4.1
Таблица 4.1. Результаты расчета=
корней
полиномов числителя и знаменателя передаточной функции
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)
Наблюдаемость
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).
Таким образом, возникает вопрос: можно ли определить вектор состояния по
измеряемому вектору выхода и вектору входа?
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.
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.=
Допустим, что даны y(0), y(1), ... , y(n-1), тогда можно записать следую=
щую
систему уравнений:
Используя векторную запись, получим
Состояние x(0) можно получить из (4.15), если матрица наблюдаемости
имеет ранг n.
Теорема 4.4. Система (4.5) наблюдаема, если и только если ранг матрицы
наблюдаемости Wn равен размерности пространства состояний.
Аналогично формулируется и условие наблюдаемости для линейных непрерывных
стационарных систем.
Теорема 4.5. Система (4.1) наблюдаема, если и только если ранг матрицы=
p>
равен размерности пространства состояний.
Индексом наблюдаемости системы будет называться такое минимальное целое =
число
v, при котором матрица Qv, определяемая выражением
имеет ранг, равный n.
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=
.
Свойства управляемости и наблюдаемости систем необходимо рассматривать
совместно для того, чтобы задача об управлении была корректно поставлена и
принципиально разрешима.
Контрольные вопросы к лекции № 4.
1. Система представлена следующей ВММ в дискретном времени
Определите состояние x(2)?
2. Система представлена следующей ВММ в дискретном времени
Определите такую управляющую последовательность, что
3. Для каких систем понятия достижимости и управляемости эквивалентны?=
p>
4. Система представлена следующей ВММ в дискретном времени
Оцените достижимость, наблюдаемость и управляемость этой системы.
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
|
2
|
|
3
|
a) для непрерывных систем;
b) для систем, представленных ВММ с невырожденной матрицей состояния;<=
/p>
|
4
|
система достижима, ненаблюдаема и управляема.
|
Лекция № 5.
Тема: «Канонические формы уравнений состояния»=
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.
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”.
Рассмотрим n-мерный вектор
- допустимый вектор состояния некоторой системы и невырожденную
матрицу T (n x n). Тогда вектор z=3DTx - также возможный вект=
ор
состояния рассматриваемой системы.
Реальная система с вектором состояния x описывается следующими
уравнениями при D=3D0:
Эта же система с вектором состояния z:
Подставляя выражение x=3DT- 1 z в (5.1) , получим
Умножая первое уравнение (5.3) слева на T, получим
Сравнивая (5.4) и (5.2), легко установим, что
Таким образом, матрицы А, В, С зависят от используемого координат=
ного
базиса.
Интерес представляют инварианты, полученные после преобразования.
Теорема 5.6. Характеристическое уравнение непрерывной
и дискретной
систем является инвариантом, если новые состояния вводятся через
невырожденную матрицу Т.
Для доказательства теоремы запишем характеристический полином матрицы zA:
Таким образом, характеристическое уравнение матрицы состояния и ее
собственное значение не зависят от базиса пространства состояний.
Следовательно, можно утверждать, что свойства систем не изменяются при
изменении базиса пространства состояний.
Например, ранг матрицы управляемости не изменяется, так как
или в краткой форме
Ранг матрицы наблюдаемости также не изменится, поскольку
или
Из (5.8) и (5.10) могут быть получены полезные соотношения для поиска
соответствующей матрицы преобразования:
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.
Каноническая форма управляемости
Здесь и далее остановимся на рассмотрении только односвязных динамических
систем.
Предположим, что характеристическое уравнение матрицы А имеет вид
и матрица управляемости
не вырождена. Тогда существует такое преобразование, при котором
преобразованная система имеет вид
или в компактной форме
Соответствующая передаточная функция системы, описываемой уравнениями
состояния (5.13), имеет вид
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.
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
где
- матрица управляемости размером n x n, а0, а1,...=
, аn-1
- коэффициенты характеристического уравнения (5.12), которые могут быть най=
дены
в результате расчета матрицы Фробениуса.
Матрица преобразования vT вычисляется с помощью следую=
щего
выражения
Таким образом, алгоритм преобразования ВММ к канонической форме
управляемости (5.14) включает в себя следующие операции:
1. Вычисление матрицы управляемости Qу согласно (5.17)=
.
2. Вычисление матрицы Фробениуса F по формуле (5.16),
транспонирование F для определения vA=3DFT.
3. Вычисление матрицы vQ линейного преобразования по
формуле (5.18).
4. Вычисление матрицы выхода, определяющей выходную переменную y по ново=
му
вектору состояния z:
Второй способ позволяет вычислить матрицу преобразования v=
Q
рекуррентно по столбцам qi
Действительно, на основании приведенных выше рассуждений можно записать:=
или
Так как вектор vB известен, то мы можем сначала вычисл=
ить
и далее продолжить вычисления по отдельным столбцам справа налево:
Последняя строка может служить для контроля.
Коэффициенты ai, i=3D0,1,2,...,n-1, матр=
ицы vА
можно определить с помощью определителя
.
Пример 5.1. Выполним переход к канонической форме управляемости для
непрерывной ВММ электродвигателя постоянного тока.
В пространстве состояния
получим
Запишем характеристическое уравнение системы
из которого следует, что матрицы состояния
и управления
в новом координатном базисе будут иметь следующий вид
Определим матрицу преобразования Q.
Первый способ. Используем выражение (5.18) , которое для нашего примера
запишется следующим образом
Такой же результат получается и по второму способу. Из выражения (5.20)
получаем
Теперь определим матрицу vC в новом координатном базис=
е
Таким образом, ВММ нашего объекта в канонической форме управляемости
принимает следующий вид
Каноническая форма наблюдаемости
Если желательно иметь матрицу в простейшей форме, то можно воспользовать=
ся
так называемой канонической формой наблюдаемости:
или в компактной форме
Как видно из выражений (5.13) и (5.22), матрицы состояний обеих канониче=
ских
форм идентичны, то есть
.
Для канонической формы наблюдаемости матрица наблюдаемости - единичная
матрица, то есть
Для нахождения матрицы NB=3D NTB=3D =
NQ
-1B матрица NТ может быть вычислена
согласно выражению (5.11):
Так как в нашем случае
, получаем, что матрица преобразования ВММ в каноническую форму
наблюдаемости (5.22) соответствует матрице наблюдаемости исходной системы=
p>
В таком случае выражение для вычисления матрицы NB
принимает следующий вид:
Пример 5.2. Выполним переход к канонической форме наблюдаемости для
непрерывной ВММ электродвигателя постоянного тока (5.21).
Здесь нам необходимо только определить новое содержание матрицы N=
sup>B.
Для этого воспользуемся выражением (5.26)
Таким образом, ВММ нашего объекта в канонической форме наблюдаемости
принимает следующий вид
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»
Рис. 5.1. Основное окно подси=
стемы
«Анализ»
Контрольные вопросы к лекции № 5.
1. Какие свойства и характеристики системы изменятся после преобразовани=
я ее
векторно-матричной модели к новому координатному базису
с помощью невырожденной матрицы Т.
2. Допустим, что нам необходимо привести ВММ непрерывной системы с одним
входом и одним выходом
к канонической форме управляемости. Предварительно определены коэффициен=
ты
характеристического уравнения системы и матрица преобразования
. Для вычисления какой матрицы преобразованной системы будет
использована матрица Q.
3. Допустим, что нам необходимо привести ВММ непрерывной системы с одним
входом и одним выходом
к канонической форме наблюдаемости. Предварительно определены коэффициен=
ты
характеристического уравнения системы и матрица преобразования
. Для вычисления какой матрицы преобразованной системы будет
использована матрица Q.
4. Какова будет матрица наблюдаемости непрерывной системы
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
внутреннее содержание матриц состояния А(Ф), управления по
состоянию В(Г), выхода С.
|
2
|
матрицы выхода по состоянию
|
3
|
матрицы управления
;
|
4
|
|
Лекция № 6.
Тема: «Критерии качества систем управления»
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 цель
управления заключается в том, чтобы перевести объект из начального
состояния x(t0), в котором он находится в момент <=
i>t0,
в конечное состояние x(t2), принадлежащее подоблас=
ти R1
области допустимых состояний R , то есть x(t2=
i>)
R1. Здесь R1
R, что соответствует выделению в пространстве состояний
области допустимых состояний R и сужению ее до некоторой области R1=
,
которая для нас по каким-то причинам является желательной.
Задача управления заключается в том, чтобы в области допустимых
управлений Q(u) найти такое управление, при котором будет
достигнута цель.
Функционалы и функции, выражающие цель управления и ограничения, называю=
т критериями
качества.
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.
Рассмотрим некоторые методические аспекты формирования критериев качества
управления для решения задач предварительного синтеза системы управления.=
p>
Качество управления можно описать двумя способами.
Первый способ 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.).
Второй способ основан на введении некоторого обобщенного функцион=
ала,
определяемого всеми переменными системы управления u(t), x=
(t),
y(t).
В теории линейных систем управления широко используются оба указанных
способа.
Оптимальное распределение полюсов системы управления
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.
Для системы, описываемой векторно-матричной моделью в непрерывном времен=
и
полюса системы - это собственные значения матрицы А, которые обыч=
но
обозначаются через
(A), где i=3D1, 2,..., n. В то же время
собственными значениями матрицы А называются корни ее
характеристического уравнения
Для односвязных систем, которые могут быть описаны общ=
ей
передаточной функцией
полюса системы - это корни характеристического многочлена А(s)=
.
Соответственно нулями системы называются корни многочлена В(s),
при которых Н(s)=3D0.
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.
Наличие нулей в замкнутой системе в определенной степени влияет на ее
динамику.
Синтезу регуляторов состояния предшествует решение задачи построения
эталонной модели системы управления, которая соответствует желаемому
распределению на комплексной плоскости корней характеристического уравнения=
A(s)
=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.
Если передаточная функция замкнутой системы не имеет нулей, то при выбор=
е ее
желаемого полинома A(s) можно руководствоваться стандартными формами
(фильтрами), которые нашли достаточно широкое применение на практике.
Стандартные формы определяют коэффициенты характеристического полинома
(знаменателя) функции Н(s), 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 оптимального расположения корн=
ей.
Как правило, в характеристическом полиноме сначала выделяются полюса для
компенсации нулей, а оставшийся полином формируется из условия желаемого
расположения корней.
В табл. 6.1 и 6.2 приводятся формулы характеристического полинома и
соответствующие им коэффициенты для некоторых наиболее распространенных на
практике распределений:/
- распределения Бесселя
(томсоновская функция),
- фильтра Чебышева
(неравномерность передачи 0.5 дБ),
- фильтра Баттерворта, =
li>
- биномиального распредел=
ения
(Ньютона).
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.
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):
- время нарастания tн -
время, необходимое для достижения 95% конечного значения;
- время установления
(регулирования) tу - время, необходимое для попадания в некоторую
окрестность конечного значения без выхода из него;
- перерегулирование -
максимальная относительная величина выброса (%)
;
- пульсации (колебания) -
число колебаний до выхода кривой в установившийся режим.
Перерегулирование и колебания - нежелательные свойства фильтра.
Типовые переходные характеристики для различных фильтр=
ов при
входном ступенчатом сигнале качественно представлены на рис. 6.2.
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.
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:
- нулевыми потерями и
пульсациями в полосе пропускания;
- нулевой шириной в
переходной области (бесконечная крутизна на частоте среза);
- бесконечным затуханием в
полосе пропускания.
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.
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.
Контрольные вопросы к лекции № 6.
1. Объект управления первоначально представлен передаточной функций
Какое число корней Вы должны расположить на комплексной плоскости для
задания желаемого динамического качества проектируемой системы управления?<=
/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?
3. Какой из рассмотренных здесь фильтров имеет амплитудно-частотную
характеристику максимально приближающуюся к АЧХ идеального фильтра?
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
n=3D4
|
2
|
a) распределение Бесселя;
b) биномиальное распределение.
|
3
|
Фильтр Баттерворта.
|
Лекция № 7.
Тема: «Критерии качества систем управления» (продол=
жение)
Выбор желаемого характеристического полинома с помощью
среднегеометрического корня
Выражение характеристического полинома в нормированном виде может быть
записано через значение среднегеометрического корня
, характеризующего быстроту протекания переходного процесса:
где
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).
Рис. 7.1. Зависимость переход=
ных
характеристик от значения среднегеометрического корня
(а) и от порядка системы n =3D 2...7 (б) на примере фильтра Батте=
рворта
Рис. 7.2. З=
ависимость
частотных характеристик АЧХ (а) и ФЧХ (б) от среднегеометрического корня
=3D 2...8 на примере фильтра Баттерворта
Рост
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.
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.
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.
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.
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.
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.
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.
Рис. 7.5. О=
кно
формирования оптимального распределения полюсов проектируемой системы
управления
Обобщенный функционал качества управления
Для детерминированных процессов с непрерывным временем, описываемых в
пространстве состояний, функционал качества в общем случае имеет вид
Здесь V3 - заданная с точностью до матрицы параметров Qk скалярная
функция конечного состояния - терминальная функция; L - скалярная функция,
действующая из пространств состояния, управления; Q - матрица параме=
тров
зависимости L от x; R - матрица параметров зависимости L от u.
Функционал (7.2) можно назвать классическим. Он использовался в классиче=
ских
задачах вариационного исчисления:
- в задаче Больца, где V=
3
0, L
0;
- в задаче Лагранжа, где
V3=3D0, L
0;
- в задаче Майера, где V=
3
0, L=3D0.
Опустим векторы параметров как аргументы функций в (7.2). Тогда функцион=
ал
Больца запишется в общей форме как
На практике применяют частные формы классического функционала (7.3).
Некоторые из этих форм рассмотрим ниже.
Классический функционал с аддитивной функцией затрат на управление
Функция L может быть представлена в виде суммы функций:
Функция
, как правило, имеет смысл тех или иных затрат на управление. Поэ=
тому
функционал
называют классическим функционалом с аддитивной функцией затрат на
управление.
Классический квадратичный функционал (Летова- Калмана)
Пусть все три функции V3, Q3, U3 выражаются квадратичными формами:
где Qk, Q - положительно полуопределенные
квадратные n x n- матрицы; R - положительно определенная квадратная =
r x
r - матрица; Q и R могут быть нестационарными матрицами.
Напомним, что некоторую матрицу F(n x n) называют положительно
полуопределенной (неотрицательно определенной), если она симметрична, то ес=
ть
F=3DFT, и при любом n-мерном векторе x
0 выполняется неравенство xTFx
0.
Положительно определенная матрица F в тех же условиях обладает свойством
xTFx > 0.
Соответствующий функционал
является классическим квадратичным функционалом.
Проанализируем смысловое содержание функционала (7.7).
Первое слагаемое в его составе характеризует ошибку управления в конечный
момент времени tk и используется с целью обеспечить малость этой ошибки.
Второе слагаемое оценивает отклонения реальных переменных состояния от
желаемых и представляет собой своеобразный “штраф” за большие ошибки при лю=
бом
t0
t
tk.
Последнее слагаемое, будучи всегда положительным, оценивает стоимость
управления. Физически оно характеризует затрагиваемую энергию на управление=
.
Основное затруднение формирования квадратичного критерия качества связан=
о с
выбором элементов весовых матриц Q и R.
Предварительный выбор значений элементов весовых матриц Q и R<=
/b>.
может быть осуществлен с помощью следующих рекомендаций.
1. Обычно матрицы Q и R назначаются постоянными и
диагональными, т. е. матрица Q - содержит n ненулевых элементов qii,
i=3D1, 2,..., n, а матрица R - r элементов rjj, j=3D1,2,...,n.
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
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.
Общий вклад максимально допустимых отклонений переменных состояния должен
приблизительно соответствовать общему вкладу максимально допустимых отклоне=
ний
сигналов управления.
3. Произвольно выбираем значение r11 (например, r11=3D1), и по формулам =
(7.8),
(7.9) вычисляем значения остальных коэффициентов.
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.
Контрольные вопросы к лекции № 7.
для увеличения быстродействия эталонной
модели, построенной на основе стандартного фильтра Баттерворта?
2. Поясните физический смысл составляющих классического квадратичного
функционала
3. Какой формы и какого размера должны быть матрицы весовых коэффициенто=
в Q
и R., используемые для вычисления классического квадратичного
функционала?
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
Первое слагаемое - ошибка управления в конечный момент времени tk и
исполь-зуется с целью минимизации этой ошибки.
Второе слагаемое - отклонение реальных переменных состояния от желаемы=
х и
представляет собой своеобразный “штраф” за большие ошибки при любом t0
t
tk.
Третье слагаемое оценивает стоимость управления, физически характеризу=
ет
за-трагиваемую энергию на управление.
|
2
|
a) оценка управляемости объекта;
b) приведение векторно-матричной модели объекта управления к каноничес=
кой
форме управляемости;
|
3
|
Матрицы Q и R назначаются постоянными и диагональными, т=
. е.
матрица Q (n x n) - содержит n ненулевых элементов qii, i=3D1, 2,.=
.., n,
а матрица R(r x r) - r элементов rjj, j=3D1,2,...,r.
|
Лекция № 8.
Тема: «Принципы детерминированного синтеза систем
управления»
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.
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.
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.
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.
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.
Этап функционального проектирования систем управления предусматривает
выпол-нение следующих проектных операций:
- синтез СУ с регулятор=
ом
состояния;
- синтез оптимального
управления;
- синтез наблюдателей
состояния;
Синтез системы управления с регулятором состояния
В линейном случае мы всегда выражаем вектор входа через линейную комбина=
цию
компонент вектора состояния, т.е. в непрерывном времени с помощью уравнения=
а в дискретном случае с помощью уравнения
где r(t), r(k) - задающая переменная.
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.
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
(t), а затем рассчитывается вектор входа объекта
Одним из фундаментальных методов проектирования детерминированных систем
управления в пространстве состояний является метод расположения полюсов.
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.
Общая постановка задачи. Для стационарного непрерывного управляемого
объекта, уравнение динамики которого имеет вид
и управляющего устройства, описываемого уравнением
необходимо определить матрицу К, такую, чтобы замкнутая система, получае=
мая
подстанов-кой (8.5) в (8.4),
имела желаемый характеристический полином
Общая схема системы управления будет иметь вид, представленный в среде
компью-терного комплекса FuncPro 1.0 на рис 8.1.
Рис. 8.1. Схема системы управ=
ления с
регулятором состояния
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.
Алгоритм 1. Рассмотрим управляемый объект с одним входом и одним выходом.
Подстановкой
преобразуем его к канонической форме управляемости
Для линейных стационарных систем характер свободного движения не изменит=
ся,
ес-ли выбрать вместо (8.5) регулятор вида
или
, где
.
Подставляя (8.10) в (8.9), получим
В этом случае характеристический полином имеет вид
Сопоставляя полиномы (8.12) и (8.7), получаем соотношения для вычисления
коэф-фициентов матрицы регулятора kRT:
Связь между управляющей переменной u и вектором состояния х определяется
со-гласно выражению (8.10):
то есть
Из этого следует, что для вычисления коэффициентов обратных связей нужно
опреде-лить еще и матрицу преобразования
Таким образом, алгоритм 1 синтеза регулятора состояния может быть
сформулирован следующим образом:
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.
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
3. На основании выражения (8.13) вычисляются коэффициенты матрицы
.
4. По формуле (8.15) определяется матрица реальных коэффициентов обратных
связей К по полному вектору состояния.
Алгоритм 2. Общность постановки задачи не нарушится, если считать, что
оптималь-ные динамические характеристики проектируемой системы задаются в в=
иде эталонной
моде-ли
имеющей тот же порядок, что и модель объекта (8.4).
Тогда коэффициенты регулятора состояния однозначно определятся путем реш=
ения
уравнения
Процесс синтеза сводится к двум задачам:
1) определение эталонной модели (8.16);
2) решение уравнения (8.17).
Наиболее просто желаемое качество управления можно задать с помощью
эталонной модели, представленной в канонической форме управляемости
Коэффициенты аэ0, аэ1, ... , аэ n-1 в этом случае соответствуют
коэффициентам d0, d1, ..., dn-1 желаемого характеристического полинома (8.7=
).
Для согласования объекта и эталонной модели введем линейное преобразован=
ие
После преобразования (8.9) система “объект + регулятор” (8.6) относитель=
но
нового базиса имеет вид
Из условия равенства собственных движений синтезируемой системы и эталон=
ной
модели следует, что
.
Откуда получается выражение для вычисления значений матрицы обратных свя=
зей
К регулятора состояния:
Таким образом, для синтеза регулятора состояния в этом случае необходимо=
:
- задать векторно-матрич=
ную
модель объекта управления;
- сформировать эталонную
модель в канонической форме управляемости (8.18);
- вычислить матрицы
преобразования ОУ к канонической форме управляемости Q и Q - 1.
- из матричного выражения
(8.20) вычислить К.
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.
Представим эталонную матрицу как сумму двух матриц:
где
- коэффициентные матрицы объекта управления в канониче-ской форме
управляемости; Кk - матрица коэффициентов обратных связей в канонической фо=
рме.
Подставляя (8.21) в (8.20):
получим
Так как
то согласно (8.21) элементы матрицы Кk можно опреде-лить из уравн=
ения
Итак, алгоритм 3 формируется как последовательность следующих операций:<=
/p>
1. Вычисление матриц преобразования ОУ к канонической форме управляемост=
и Q
и Q-1.
2. Решение уравнения (8.23) относительно Кk.
3. Вычисление реальных коэффициентов обратных связей (матрицы К) по форм=
уле
(8.22).
Сравнительно высокая сложность приведенных выше алгоритмов делает
практически неосуществимыми проектные операции синтеза регулятора состояния=
без
ЭВМ.
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.
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).
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.
Пример 8.1. Для упругого электромеханического объекта, который подробно
пред-ставлен в лекции 1, синтезируем регулятор состояния.
При построении ВММ откажемся от учета инерционности преобразователя и вв=
едем
следующие компоненты вектора состояния
, где
- скорость враще-ния платформы, F- усилие, под действие которого
вращается платформа,
- скорость вращения электродвигателя, i - ток электродвигателя.=
p>
Таблица 8.1. Параметры
электромеханического объекта
Векторно-матричная модель объекта с учетом численных значений параметров
пре-образователя, электродвигателя и механизма, приведенных в табл. 8.1.,
принимает следую-щий вид:
Желаемые динамические показатели управления были определены с помощью
инст-рументальных средств формирования критерия качества путем некоторой
модификации стандартного распределения Баттерворта для среднегеометрического
корня
. В ре-зультате был выбран желаемый характеристический полином
которому соответствует эталонная модель, представленная в канонической ф=
орме
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.
Таблица 8.2. Параметры регулято=
ра
состояния
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.
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%.
- расчетные параметры регулятора состояния
- параметры реализации
Рис. 8.2. Сравнительные динам=
ические
характеристики СУ
Контрольные вопросы к лекции № 8.
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»?
2. Какие операции должны быть обязательно выполнены при реализации любог=
о из
ал-горитмов синтеза регулятора состояния?
3. Каковы будут значения параметров регулятора состояния, синтезированно=
го
для объ-екта? Векторно-матричная модель объекта и эталонная модель системы
управления приведены ниже.
1. ВММ объекта управления=
p>
|
2. Эталонная модель СУ
|
3
|
4
|
|
=
span>
|
4. Каковы будут значения параметров регулятора состояния, синтезированно=
го
для объ-екта? Векторно-матричная модель объекта и эталонная модель системы
управления приведены ниже.
5. ВММ объекта управления=
p>
|
6. Эталонная модель СУ
|
7
|
8
|
|
=
span>
|
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
(7x7)
|
2
|
a) оценка управляемости объекта;
b) приведение векторно-матричной модели объекта управления к каноничес=
кой
форме управляемости;
|
3
|
|
4
|
Не смогу рассчитать
|
Лекция № 9.
Тема: «Синтез оптимального управления»=
span>
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.
Для объекта управления, который описывается векторными дифференциальным и
ал-гебраическими уравнениями
необходимо найти закон управления u0(t), при котором достигается минимум
квадратичного функционала качества
который подробно представлен в лекции 7.
Общая математическая постановка указанной задачи приводит к уравнению
Беллма-на, которое имеет следующий вид:
Вывод уравнения Беллмана, характеристики входящих в него переменных и
функций приведены в приложении 1.
Решение уравнения (9.3) для объекта управления, который описывается
векторно-матричной моделью (9.1), позволяет определить закон оптимального
управления в виде
где
, P(t) - решение матричного дифференциального уравнения Рик-кати<=
/p>
c граничным условием
.
Вывод уравнения Риккати приведен в приложении 2.
В соответствии с вышеизложенным алгоритм синтеза оптимального уравнения
пред-ставляет собой следующую последовательность действий:
1) построение векторно-матричной модели ОУ (9.1);
2) выбор элементов весовых матриц F, Q(t), R(t) в (9.2), при которых
переходные процессы в системе управления удовлетворяют заданным требованиям=
;
3) решение матричного дифференциального уравнения Риккати (9.5);
4) анализ динамических характеристик в оптимальной системе управления и
оценка ее качества.
Основные трудности возникают здесь при решении матричного дифференциальн=
ого
уравнения Риккати. Интегрирование этого уравнения удобно выполнять в обратн=
ом
времени
. В этом случае задача сводится к задаче Коши с начальными услови=
ями
. 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.
Для стационарных систем, в которых A, B, Q, R - коэффициентные матрицы и=
, матричное дифференциальное уравнение Риккати вырождается в
алгебраическое
решением которого является симметричная положительно определенная матриц=
а Р.
Решение уравнения (9.28) для стационарных систем при
и
имеет предел
Поэтому матрицу Р можно вычислить как предельное значение решения уравне=
ния
(9.5) при достаточно большом Т.
По аналогии с (9.4) оптимальное управление определится из выражения
Достоверность представленных алгоритмов подтвердим практическим примером=
.
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.
Таблица 9.1. Параметры
электромеханического объекта
Результатом серии вычислительных экспериментов явились:
- внутреннее содержание
весовых матриц Q, и R
- временные характеристи=
ки
полученные в результате решения уравнения Риккати (9.5) в обратном време=
ни,
которые приведены на рис. 9.1
Для постановки имитационных экспериментов используем приведенные в табл.=
9.2
постоянные расчетные значения коэффициентов обратных связей, соответствующие
t=3D0, и значения реализации.
Таблица 9.2. Значения коэффицие=
нтов
обратных связей
Рис. 9.1. Динамические характ=
еристики
K0(t)
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 (
), подтвер-ждают корректность алгоритмического и программного
обеспечения синтеза оптимального управления.
Рис. 9.2. Сравнительные динам=
ические
характеристики систем управления с регулятором состояния
Контрольные вопросы к лекции № 9.
1. Укажите основные особенности численного решения матричного
дифференциального уравнения Риккати?
2. При каких условиях матричное дифференциальное уравнение Риккати
вырождается в алгебраическое?
3. Какие значения должны принимать коэффициенты обратных связей для «точ=
ной»
реализации оптимального управления?
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
Интегрирование уравнения Риккати, как правило, выполняется в обратном
времени
.
|
2
|
Для стационарных систем, в которых A, B, Q, R - коэффициентные матрицы=
и
, матричное дифференциальное уравнение Риккати вырождается в
алгебраическое
|
3
|
Значения коэффициентов обратных связей должны непрерывно изменяться во
времени в соответствии с результатом решения уравнения Риккати
.
|
Лекция № 10.
Тема: «Наблюдатель состояния полного порядка»<=
/o: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.
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.
Пусть стационарный объект описывается традиционной системой уравнений
Предположим, что матрицы A, B, C известны, тогда вектор x можно
аппроксимиро-вать состоянием
модели
которая имеет тот же вход, что и объект (10.1).
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
также совпадают. Если начальные условия для систем (10.1) и (10.2)
различны, то
сходится к x только то-гда, когда система (10.1) асимптотически
устойчива.
При восстановлении (10.2) не используется измеряемый выход. Качество
восстанов-ления улучшается, если ввести в модель разность измеренного выход=
а и
его оценки
в виде обратной связи:
Здесь L - некоторая матрица, обеспечивающая требуемый вид переходных
процессов оценки вектора состояния.
Введем ошибку восстановления
Вычитая (10.3) из дифференциального уравнения (10.1), получим
Очевидно, для того чтобы ошибка восстановления стремилась к нулю, необхо=
димо
выбрать матрицу L так, чтобы система (10.5) была асимптотически устойчива.<=
/p>
Таким образом, вводя обратную связь в модель восстановления, можно
нивелировать ошибку, даже если система (10.1) неустойчива.
Мы видим, что модель (10.3) восстанавливает все составляющие вектора
состояния, поэтому она называется наблюдателем состояния полного порядка.=
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.
Рис. 10.1. Схема системы упра=
вления с
наблюдателем полного порядка
При синтезе наблюдателей обычно используют два принципа: принцип
разделимости и принцип дуальности.
Принцип разделимости. Допустим, что объект описывается уравнениями (10.1=
),
регу-лятор состояния -
а наблюдатель полного порядка - уравнением (10.3).
Тогда уравнения состояния всей системы имеют вид
Если ввести ошибку (10.4), то с помощью линейного преобразования
получим
Из (10.9) очевидно, что характеристическое уравнение всей системы имеет =
вид
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.
Процесс, описываемый уравнениями (10.9), состоит из двух взаимно независ=
имых
процессов: из процесса оценки вектора
и из собственно процесса управления.
Эту возможность разделения процессов в наблюдателе и регуляторе принято
называть принципом разделимости.
Принцип дуальности (приводится без доказательства). Условием наблюдаемос=
ти
сис-темы (10.1) одновременно является полная управляемость сопряженной сист=
емы
С учетом принципов дуальности и разделимости синтез наблюдателя состояния
вы-полняется независимо от синтеза регулятора состояния по алгоритмам,
изложенным в 4.1.
Действительно, для системы (10.3) регулятор состояния будет иметь вид
Так как
элементы матрицы L определяются по алгоритмам вычисления матрицы К
для регулятора состояния.
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.
Многочисленные вычислительные эксперименты, выполненные в целях построен=
ия
систем управления с наблюдателями полного порядка и регуляторами состояния,
подтвер-ждают корректность приведенного выше положения.
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
, учтены люфты в механических переда-чах и инерционность
преобразователя.
Рис. 10.2. Сравнительные дина=
мические
характеристики ЭМС при непосредственном измерении координат и использовании
наблюдателя полного порядка
Контрольные вопросы к лекции №10
1. Для достижения каких целей и из каких условий выбирается матрица L
наблюдателя полного порядка?
2. Назовите два основных принципа синтеза наблюдателя.
3. В какой последовательности следует выполнять проектные операции синте=
за
регуля-тора и наблюдателя состояния?
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
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
Матрица L обеспечивает требуемый вид переходных процессов оценки векто=
ра
состояния и выбирается из условий достижения асимптотической устойчивости
замкнутой системы объект – наблюдатель.
|
2
|
Принцип разделимости и принцип дуальности.
|
3
|
Проектные операции могут быть выполнены в любой последовательности.
|
4
|
|
Лекция № 11.
Тема: «Оценка вектора состояния при случайных возму=
щениях
и наличии помех»
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.
Рассмотрим алгоритм линейной оценки вектора состояния системы с минималь=
ной
дисперсией.
Пусть динамическая система описывается векторным дифференциальным
уравнением.
Здесь, кроме представленных ранее матриц и векторов, w(t)- k-мерный вект=
ор
случайных воздействий; Т- матрица размером n x k.
Вектор измеряемых выходных координат, который доступен наблюдению,
определяется соотношением
где v(t) - m-мерный вектор случайных помех, сопровождающих измерения.
Предполагается, что система (11.1), (11.2) при w(t)
0 и v(t)
0 наблюдаема. Будем считать w(t) и v(t) гауссовскими случайными
процессами типа белого шума с нулевыми математическими ожиданиями
и корреляционными матрицами
где
- дельта-функция; Q(t)- симметричная неотрицательно определенная =
k x
k - матрица интенсивности белого шума w(t);
R(t)- симметричная положительно определенная m x m - матрица интенсивнос=
ти
белого шума v(t).
Допустим, что начальное состояние системы х(t0)- n-мерный случайный вект=
ор с
известным математическим ожиданием
и корреляционной матрицей
Кроме того, будем считать, что х(t0), w(t), v(t) взаимно не коррелирован=
ны.
Найти необходимо линейную несмещенную оценку вектора х(t0), построенную =
на
основе наблюдения
,
.
Несмещенная оценка
предполагает равенство ее математического ожидания математическому
ожиданию истинной величины
.
Предположим, что
получается на выходе фильтра
Чтобы процесс на выходе фильтра был несмещенной оценкой, должно выполнят=
ься
равенство (11.5). Вычислим математическое ожидание обеих частей уравнения
(11.6):
но из (11.2) следует, что
На основании (11.6) - (11.8) получим дифференциальное уравнение для сред=
него
значения вектора состояния системы:
Вычисляя математическое ожидание от (11.13), получим еще одно уравнение =
для
среднего значения вектора состояния
Сравнивая (11.10) и (11.9), определим первое условие несмещенности оценки
вектора состояния:
Второе условие состоит в том, чтобы уравнения (11.21) и (11.22) решались=
при
одном и том же начальном условии
При выполнении этих условий уравнение фильтра примет вид
Остается определить матрицу коэффициентов усиления фильтра L(t), которая
обеспечила бы оптимальную оценку в том смысле, что составляющие ошибки
оценивания должны иметь минимальную дисперсию.
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
которое получается по алгоритму синтеза оптимального управления для
сопряженной системы
Здесь P(t) - корреляционная матрица ошибок оценивания, является решением
матричного дифференциального уравнения Риккати
Наблюдатель, построенный по приведенному алгоритму, называют оптимальным
фильтром Калмана – Бьюси.
Для стационарной системы, в которой A, B, C, T, Q, R- коэффициентные
матрицы, уравнение фильтра Калмана - Бьюси принимает вид
Матрица коэффициентов усиления фильтра постоянна и определяется выражени=
ем
где Р- положительно определенная матрица, являющаяся решением
алгебраического матричного уравнения Риккати
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=
.
Рис. 11.1. Сравнительные дина=
мические
характеристики ЭМС при непосредственном измерении координат и использовании
наблюдателя, построенного по принципу фильтра Калмана - Бьюси.
Контрольные вопросы к лекции №11
1. Какой из рассмотренных ранее алгоритмов используется для вычисления
матрицы коэффициентов усиления оптимального фильтра?
2. При каких условиях матрица коэффициентов усиления фильтра Калмана-Бью=
си
постоянна во времени?
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
Алгоритм синтеза оптимального управления для сопряженной системы.
|
2
|
Для стационарной системы.
|
Лекция № 12.
Тема: «Наблюдатель состояния пониженного порядка»
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.
Рассмотрим линейную наблюдаемую стационарную систему
в которой y(t) - m- мерный вектор выходных координа=
т,
причем m < n и rang C=3Dm, 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=
).
Предположим, что оценка вектора состояния системы может быть выполнена с
помощью фильтра (n - m)-го порядка
где z(t) - (n - m)- мерный вектор состояния; F=
b>, G1,
G2 - матрицы с размерами (n - m)x(n - m), (n - m)=
i>x m
, (n - m)x z соответственно.
Определим условия, которым должны удовлетворять матрицы F, G1<=
/b>,
G2 фильтра (12.2). Допустим, что параметры фильтра можно подобрать
такие, чтобы
Умножим обе части уравнения (12.2) слева на матрицу Т, тогда с уч=
етом
(12.1) получим
<=
/o:p>
Из равенств (12.4), (12.5) следует
Оценку вектора состояния будем искать в виде
где Н и G -матрицы с размерами n x(n - m) и =
(n
x m) соответственно.
Потребуем выполнение условия
, т.е.
откуда следует, что
где I - единичная матрица.
Элементы пяти матриц: F, G1, G2, G, H,
связанные уравнениями (12.6), (12.9), могут выбираться до некоторой степени
произвольно.
Векторно-матричная модель системы управления с наблюдателем Люенбергера =
при
имеет следующий вид:
или
Наблюдатель Люенбергера не изменяет полюсы замкнутой системы управления,=
а
лишь добавляет к ним свои собственные.
Таким образом, синтез наблюдателя пониженного порядка может выполняться =
по
следующему алгоритму.
1. Проверить наблюдаемость исходной системы с определением индекса
наблюдаемости m.
2. Выполнить анализ матрицы состояния А объекта управления с
определением корней характеристического уравнения.
3. Выбрать матрицу F таким образом, чтобы обеспечить требуемое вр=
емя
переходного процесса в наблюдателе.
4. Произвольно задать матрицу G1, соблюдая при этом выполнение
условия управляемости фильтра (12.2) по вектору y(t), т.е.
необходимо, чтобы
где p=3Dn - m .
5. Решить матричное уравнение
относительно Т.
6. Вычислить матрицу
.
7. Вычислить матрицы Н и G из уравнения (12.9).
Общая схема системы управления с наблюдателем пониженного порядка будет
иметь вид, представленный в среде компьютерного комплекса FuncPro 1.0 на рис
12.1.
Рис. 12.1. Схема системы упра=
вления с
наблюдателем пониженного порядка
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.
Таблица 12.1. Параметры
электромеханического объекта
Выполнение отмеченной процедуры для электромеханического объекта с
параметрами, приведенными в табл. 12.1., приводит следующим результатам.
Корректность полученных результатов подтверждают динамические характерис=
тики
системы с регулятором состояния и наблюдателем упругого момента, приведенны=
е на
рис. 12.2.
Регулятор состояния здесь синтезирован при использовании в качестве крит=
ерия
качества биномиального распределения корней (
=3D8.5 с-1).
Рис. 12.2. Динамические
характеристики ЭМС при непосредственном измерении всех координат и при оцен=
ке
упругого момента наблюдателем пониженного порядка.
Контрольные вопросы к лекции №12
1. Какая матрица
определяет требуемое время оценки неизмеряемых компонент вектора
состояния системы с помощью наблюдателя Люенбергера?
2. Из каких условий выбирается матрица G1 фильтра
необходимого для синтеза наблюдателя Люенбергера?
ОТВЕТЫ
№ задания
|
Ответ
|
1
|
Матрицы F
|
2
|
Матрицу G1 можно задавать произвольно, соблюдая при =
этом
условия управляе-мости фильтра (12.2) по вектору y(t), т.е.
необходимо, чтобы
где p=3Dn - m
|
Приложение 1 к лекции 9
Вывод уравнения Беллмана
Пусть управляемый объект описывается векторным дифференциальным уравнени=
ем
общего вида
а критерий оптимальности также имеет общий вид
Необходимо в классе допустимых управлений найти управление u0=
(t),
при котором функционал I достигает минимального значения, т.е.
а объект переводится за время T- t0 из заданного начального состояния x(=
t0)
в произвольное конечное, принадлежащее пространству состояний.
В основу дальнейших рассуждений положен принцип оптимальности, утверждаю=
щий,
что любой оставшийся конечный участок оптимальной траектории сам по себе
является также оптимальной траекторией.
Предположим, что уже найдены оптимальное управление u0(t) и
соответствующая ему траектория движения объекта x(t). Выберем на оптимальной
траектории две точки, соответствующие моментам времени t и t+
, где
- малая величина. Согласно принципу оптимальности участки оптимал=
ьной
траектории от точек t и t+
до конечной точки Т являются оптимальными. Обозначим минимальное
значение функционала (П.1.2), соответствующее этим участкам:
<=
/o:p>
На основании выражений (П.1.4), (П.1.5) можно записать
Учитывая малость
, запишем
где
- малая с большим порядком малости, чем
.
Функцию х(t+ ) разложим в ряд Тейлора в окрестности точки t, представив =
его
в виде
где
- совокупность последующих членов ряда.
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):
где
- вектор-строка частных производных в точке (x(t),t);
- приращение вектора x(t);
- совокупность последующих членов ряда Тейлора.
Подставим (П.1.7), (П.1.9) в (П.1.6):
где
- все члены с порядком малости, большим, чем у
.
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
. Производную
, не зависящую от u(t), вынесем за скобки, а производную
заменим согласно (П.1.1) функцией f. Тогда с учетом того, что
, получим уравнение Беллмана
Приложение 2 к лекции 9
Вывод уравнения Риккати
Решим уравнение Беллмана (9.3) для объекта управления (9.1) и функционала
качества (9.2).
Наличие неинтегрального члена в составе функционала (9.2) не влияет на
структуру уравнения Беллмана, если принять
где L - подынтегральная функция из (9.2).
Тогда уравнение Беллмана принимает вид
Оптимальное уравнение будем искать из условия
Выполнив дифференцирование, получим
Из (П.2.3) получаем оптимальное уравнение
Чтобы использовать выражение (П.2.4), необходимо найти функцию V(=
x(t),t).
Для этого подставим (П.2.4) в (П.2.2):
Решение (П.2.5) при t=3DT должно удовлетворять условию
Общее решение будем искать в виде квадратной формы переменных состояния<=
/p>
где P(t)=3DPT(t) - неизвестная
симметричная нестационарная n x n-матрица.
Подставим в (П.2.5) выражение (П.2.7) и
Последнее слагаемое представим суммой
Тогда полученное соотношение преобразуется к виду
При любых состояниях x(t)
0 предыдущее соотношение выполняется, если
В результате получили нелинейное дифференциальное уравнение, известное п=
од
названием матричного уравнения Риккати.
Очевидно, что его решение следует искать при граничном условии
Для стационарных систем уравнение (П.2.8) становится алгебраическим:
Рекомендуемая литература
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////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
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////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////
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