2
Предмет: Теория Автоматического Управления
Тема: Дискретные системы автоматического управления
1. Дискретные системы автоматического управления
Дискретные системы – это системы, содержащие элементы, которые
преобразуют непрерывный сигнал в дискретный. В дискретных системах
сигналы описываются дискретными функциями времени.
Квантование – процесс преобразования непрерывного сигнала в дискретный.
В зависимости от используемого вида квантования системы можно
классифицировать:
– импульсные системы, использующие квантование по времени;
– релейные системы, использующие квантование по уровню;
– цифровые системы, использующие квантование по уровню и по времени
(комбинированное квантование).
Квантование осуществляется с помощью импульсных модуляторов, релейных
элементов, а также различного рода цифровых ключей.
Модуляция – процесс квантования по времени. В импульсных системах в
основном используются следующие виды модуляции:
– амплитудно-импульсная (АИМ)- амплитуда импульса пропорциональна
амплитуде входного сигнала (рис. 1а);
– широтно-импульсная (ШИМ)- широта импульса пропорциональна амплитуде
входного сигнала (рис. 1б);
– фазоимпульсная (ФИМ)- фаза импульса пропорциональна амплитуде входного
сигнала (рис. 1в).
а) б) в)
Рис. 1
В релейных системах управления используется импульсная манипуляция (ИМ),
в цифровых системах используются кодоимпульсная модуляция (КИМ), при
этом каждому значению амплитуды соответствует «пачка» импульсов,
представляющая код амплитуды передаваемого сигнала. Этот метод
квантования обладает хорошей помехоустойчивостью и широко используется в
цифровых системах управления.
На рис. 2 приведен пример, иллюстрирующий процесс передачи дискретных
сообщений с использованием кодоимпульсной модуляции.
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ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
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fffffffffffffffffffffffffffffffffffffffffffffffffffffffffdf3cf7bf3f9ffff
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fef3cf7bf3f9ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffff67cf7bf3f9ffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffff4fcf7bf3f9ffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffff8fcf7bf3f9ffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff9fce1b
f3f9ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffff2fcf63f3f9ffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffff2fcffff3f9ffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffe77cffef3b9ffffffffffffffffffff
fffffffffffffffffffffffffffffffffffffffffffffffffffffffffc7bcffef3b9ffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
f070c7fe0021ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
fffffffffffffffffffffffffffffff0c00000040092900aa00000000000000d00060010
0000000040000002701ffff030000000000
Рис. 2
При этом квантование по времени определяется тактовой частотой
управляющей ЭВМ, а квантование по уровню осуществляется с помощью
аналого-цифрового преобразователя (АЦП).
2. Импульсный элемент (ИЭ). Математическое описание импульсного элемента
Импульсный элемент – устройство для преобразования непрерывного сигнала
в последовательность модулированных импульсов.
Импульсный элемент может быть представлен в виде двух частей: идеального
импульсного элемента и формирователя импульсов.
Идеальный импульсный элемент (рис. 3) преобразует непрерывный
сигнал в последовательность идеальных импульсов в виде (t) –функций,
площади которых пропорциональны амплитуде передаваемого сигнала.
Рис. 3
Для выходного сигнала импульсного элемента можно записать следующее
соотношение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, (1)
где x[nT] – решетчатая функция, которая представляет собой значение
непрерывной функции в дискретные моменты времени.
При x(t) = 1(t)
picscalex100010009000003dc0100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c022004e0131200000026060f001a00ffffffff00001000
0000c0ffffffb9ffffffa0130000d90300000b00000026060f000c004d61746854797065
0000d00010000000fb02c0fe0000000000009001000000020002001053796d626f6c0002
040000002d01000008000000320a7702270b01000000e50008000000320a3c01240b0100
0000a50008000000320ad503470b010000003d0010000000fb0240fe0000000000009001
000000020002001053796d626f6c0002040000002d01010004000000f001000008000000
320a80028e0f010000002d0008000000320a80023409010000003d0008000000320a8002
d903010000003d0015000000fb02c0fe0000000000009001000000000402001054696d65
73204e657720526f6d616e000069040000002d01000004000000f001010008000000320a
d503020c01000000300008000000320ab7013206010000002a0008000000320ab701d700
010000002a0015000000fb0240fe0000000000009001000000000402001054696d657320
4e657720526f6d616e000069040000002d01010004000000f001000008000000320a8002
021301000000290008000000320a8002fd0d01000000280008000000320a800236080100
0000290008000000320a8002090701000000280008000000320a8002db02010000002900
08000000320a8002ae0101000000280008000000320a8002080001000000310015000000
fb02c0fe0000000000009001010000000402001054696d6573204e657720526f6d616e00
0069040000002d01000004000000f001010008000000320ad503960a010000006e001500
0000fb0240fe0000000000009001010000000402001054696d6573204e657720526f6d61
6e000069040000002d01010004000000f001000008000000320a8002ee10020000006e54
08000000320a8002900e01000000740008000000320a80029c0701000000740008000000
320a80024e0501000000780008000000320a8002410201000000740010000000fb0240fe
0000000000009001010000020002001053796d626f6c0002040000002d01000004000000
f001010008000000320a8002d00c0100000064000a00000026060f000a00ffffffff0100
0000000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc
040000002d01010004000000f0010000030000000000. (2)
Для любого x(t)
picscalex100010009000003e80000000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c024002600d1200000026060f001a00ffffffff00001000
0000c0ffffffd7ffffff200d0000170200000b00000026060f000c004d61746854797065
0000600015000000fb0240fe0000000000009001010000000402001054696d6573204e65
7720526f6d616e000066040000002d01000009000000320a8001860b0300000028742900
08000000320a8001490a010000002a0008000000320a8001520901000000310009000000
320a8001f805040000007828742909000000320a80018902030000002874290008000000
320a80014c01010000002a0008000000320a80014e0001000000780010000000fb0240fe
0000000000009001000000020002001053796d626f6c0002040000002d01010004000000
f001000008000000320a8001a10801000000d70008000000320a80018304010000003d00
0a00000026060f000a00ffffffff01000000000010000000fb021000070000000000bc02
000000cc0102022253797374656d00cc040000002d01000004000000f001010003000000
0000 . (3)
Это физически не реализуемо и является математической идеализацией,
вводимой для упрощения исследования дискретных систем.
Реальный импульсный элемент (рис. 4) – импульсный элемент с конечной
длительностью импульса. Он состоит из идеального импульсного элемента и
формирователя.
Формирователь преобразует идеальные импульсы в импульсы длительности – T
Рис. 4
Импульс конечной длительности можно представить в виде (рис. 5)
Рис. 5
Функция веса формирующего звена представляет собой импульс длительностью
– T, ее можно представить как сумму двух единичных функций
противоположного знака, сдвинутых на T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. (4)
Передаточная функция формирователя имеет вид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 (5)
Формирователь при = 1 называется фиксатором (или экстраполятором
нулевого порядка), при этом его передаточная функция равна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 (6)
Рассмотрим импульсный элемент при = 1 (рис. 6).
Рис. 6
Если на вход подается аналоговый сигнал, то на выходе получаем
ступенчатый сигнал. Рассмотрим схему (рис. 7), состоящую из АЦП и ЦАП:
Рис.7
Если на вход схемы поступает аналоговый сигнал, то на выходе АЦП
получаем код, значение которого соответствует амплитуде входного
сигнала, а на выходе ЦАП получаем ступенчатый сигнал.
Таким образом, для того, чтобы представить процессы в цифровых системах
необходимо использовать идеальный ИЭ и фиксатор. Импульсную систему
можно представить в виде идеального импульсного элемента и непрерывной
инерционной части, а цифровую систему в виде реального импульсного
элемента и непрерывной инерционной части. Характерная схема импульсной
системы управления приведена на рис. 8.
Рис. 8
Цифровая система автоматического управления (рис. 9) состоит из
аналого-цифрового преобразователя (АЦП), цифро-аналогового
преобразова-теля (ЦАП), цифрового автомата (ЦА) и объекта управления.
–
Рис. 9
Эту схему можно представить в виде, изображенном на рис. 10.
Ka(z)
–
Рис. 10
При этом цифровой автомат реализует алгоритм управления в реальном
масштабе времени (Ka(z) – передаточная функция алгоритма), т. е. в
течение интервала времени равного периоду дискретности –Т.
В цифровой системе квантование по уровню осуществляется с помощью АЦП, а
по времени задается цифровым автоматом. Выходной преобразователь
одновременно является экстраполятором нулевого порядка, сигнал на его
выходе в течение периода дискретности является постоянным.
Литература
1. Бесекерский В.А., Попов Е.П. “Теория систем автоматического
управления”. Профессия, 2003 г. – 752с.
2. Бронштейн И.Н., Семендяев К.Н. Справочник по математике для инженеров
и учащихся вузов. — М.: Наука,1986.
3. Основы теории автоматического управления / В.С. Булыгин, Ю.С.
Гришанин, Н.Б. Судзиловский и др.; под ред. Н.Б. Судзиловского. М.:
Машиностроение, 1985. – 512с.
4. Сборник задач по теории автоматического регулирования и управления/
Под редакцией В. А. Бесекерского. — M.: Наука, 1978.
5. Справочник по теории автоматического управления. /Под ред. А.А.
Красовского- М.: Наука, 1987. – 712с.
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