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Устойчивость линейных систем автоматического управления

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на тему:

“Устойчивость линейных систем автоматического управления”

1. Общие понятия устойчивости

Устойчивость – это свойство системы возвращаться в исходное состояние
после вывода ее из состояния равновесия и прекращения действия
возмущения. Устойчивость – это одно из основных требований,
предъявляемых к системе. Если система не устойчива, то она не
работоспособна. Рассмотрим математическое понятие устойчивости.

Движение линейной системы автоматического управления описывается
линейным, неоднородным уравнением:
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при этом правая часть – входное воздействие, а левая – реакция выхода.

Решение уравнения можно записать в виде:
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 (1)

где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– представляет собой общее решение однородного уравнения и определяет
переходный процесс;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- представляет собой частное решение неоднородного уравнения и
определяет установившийся режим.

Общее решение однородного уравнения имеет вид:
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, (2)

где: Ск – постоянные интегрирования, которые зависят от начальных
условий;
picscalex100010009000003220100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c026002400c1200000026060f001a00ffffffff00001000
0000c0ffffffb7ffffff000c0000170200000b00000026060f000c004d61746854797065
0000600015000000fb02c0fe0000000000009001010000000402001054696d6573204e65
7720526f6d616e000069040000002d01000008000000320a1002570b010000006b000800
0000320a10026506010000006b0008000000320a10026c01010000006b0015000000fb02
40fe0000000000009001010000000402001054696d6573204e657720526f6d616e000069
040000002d01010004000000f001000008000000320aa0017409010000006a0008000000
320aa001710001000000700010000000fb0240fe00000000000090010100000200020010
53796d626f6c0002040000002d01000004000000f001010008000000320aa001f2090100
0000770008000000320aa001070501000000730010000000fb0240fe0000000000009001
000000020002001053796d626f6c0002040000002d01010004000000f001000008000000
320aa001b90701000000b10008000000320aa001200401000000b10008000000320aa001
c002010000003d000a00000026060f000a00ffffffff01000000000010000000fb021000
070000000000bc02000000cc0102022253797374656d00cc040000002d01000004000000
f0010100030000000000- корни характеристического уравнения:
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Рассмотрим характер решения при различных значениях корней
характеристического уравнения.

1. Если корни действительные однократные
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2. Если корни действительные кратные
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3. Если корни комплексно – сопряженные однократные
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4. Пусть корни комплексно – сопряженные кратные
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Для того чтобы система была устойчивой решение должно удовлетворять
условию
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 (3)

Это условие выполняется, если корни характеристического уравнения
системы расположены в левой полуплоскости комплексной плоскости P.

Для устойчивости линейной системы необходимо и достаточно, чтобы корни
ее характеристического уравнения располагались в левой полуплоскости
комплексной плоскости P.

Характеристическое уравнение системы можно представить в виде:

picscalex1000100090000038c0100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c022004a00d1200000026060f001a00ffffffff00001000
0000c0ffffffbcffffff600d0000dc0300000b00000026060f000c004d61746854797065
0000d00015000000fb0240fe0000000000009001000000000402001054696d6573204e65
7720526f6d616e000069040000002d01000008000000320a8002150d010000002e000800
0000320a80024a0c01000000300008000000320a8002f30901000000290008000000320a
8002450401000000280015000000fb02c0fe000000000000900100000000040200105469
6d6573204e657720526f6d616e000069040000002d01010004000000f001000008000000
320ad7037b0301000000310010000000fb02c0fe00000000000090010000000200020010
53796d626f6c0002040000002d01000004000000f001010008000000320a79028c020100
0000d50008000000320ad703de02010000003d0010000000fb0240fe0000000000009001
000000020002001053796d626f6c0002040000002d01010004000000f001000008000000
320a8002f10a010000003d0008000000320a80027006010000002d0015000000fb02c0fe
0000000000009001010000000402001054696d6573204e657720526f6d616e0000690400
00002d01000004000000f001010008000000320a3901c302010000006e0008000000320a
d7032802010000006b0008000000320af0020309010000006b0008000000320af0023401
010000006e0015000000fb0240fe0000000000009001010000000402001054696d657320
4e657720526f6d616e000069040000002d01010004000000f001000008000000320a8002
080801000000700008000000320a80021e0501000000700008000000320a800239000100
000061000a00000026060f000a00ffffffff01000000000010000000fb02100007000000
0000bc02000000cc0102022253797374656d00cc040000002d01000004000000f0010100
030000000000 (4)

Если уравнение содержит хотя бы один положительный корень, то хотя бы
один коэффициент характеристического уравнения будет отрицательным.
Необходимое, но недостаточное условие устойчивости (при n > 2) системы –
это положительность коэффициентов характеристического уравнения.

Для нахождения корней характеристического уравнения необходимо решать
алгебраические уравнения. Аналитическое решение уравнений 3-го и 4-го
порядка громоздки, а уравнение выше 4-го порядка не имеют аналитического
решения.

В теории автоматического управления разработан ряд так называемых
критериев устойчивости, которые позволяют, не решая уравнений определять
устойчивость систем.

2. Критерий устойчивости Рауса-Гурвица

Для устойчивости линейной системы необходимо и достаточно, чтобы при а0
>0 определитель Гурвица, составленный для характеристического уравнения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, и все его диагональные миноры были положительны.

Определитель Гурвица имеет вид:
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 (5)

Диагональные миноры определяются соотношениями
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 (6)

Рассмотрим частные случаи

1. Для системы первого порядка (n = 1) характеристическое уравнение
имеет вид:
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Условие
устойчивости: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2. Для системы второго порядка (n=2) характеристическое уравнение имеет
вид:

3.
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Условие устойчивости: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4. Для системы третьего порядка (n = 3) характеристическое уравнение
имеет вид:

picscalex100010009000003ab0200000300150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02200b00201200000026060f001a00ffffffff00001000
0000c0ffffffb9ffffffc01f0000d90a00000b00000026060f000c004d61746854797065
0000900209000000fa02000010000000000000002200040000002d010000050000001402
8c039504050000001302da0a95040500000014028c03140e050000001302da0a140e1500
0000fb0240fe0000000000009001010000000402001054696d6573204e657720526f6d61
6e0000b9040000002d01010008000000320aa307901e02000000302e08000000320aa307
5c1c01000000290008000000320aa307201a01000000610008000000320aa30747180100
0000610008000000320aa307811401000000610008000000320aa307c612010000006100
08000000320aa307e61101000000280008000000320aa307051001000000610008000000
320a430a080c01000000610008000000320a430a780801000000610008000000320a430a
420501000000300008000000320aa307820c01000000300008000000320aa30762080100
0000610008000000320aa307c90401000000610008000000320a0305820c010000003000
08000000320a0305680801000000610008000000320a0305d80401000000610008000000
320aa307400001000000200008000000320a3602ab1501000000200008000000320a3602
7315010000002c0008000000320a3602781401000000300008000000320a3602d3100100
0000610008000000320a36021b0e01000000700008000000320a3602fd0b010000006100
08000000320a3602170801000000700008000000320a3602240601000000610008000000
320a36024a0201000000700008000000320a3602390001000000610015000000fb0280fe
0000000000009001010000000402001054696d6573204e657720526f6d616e0000b90400
00002d01020004000000f001010008000000320a1308131b01000000330008000000320a
13082e1901000000300008000000320a13087a1501000000320008000000320a1308ad13
01000000310008000000320a1308f81001000000330008000000320ab30afb0c01000000
330008000000320ab30a5f0901000000310008000000320a13085b090100000032000800
0000320a1308b00501000000300008000000320a73055b0901000000330008000000320a
7305bf0501000000310008000000320a1308bf0101000000330008000000320aa602c611
01000000330008000000320aa602f60c01000000320008000000320a6d01290901000000
320008000000320aa6020b0701000000310008000000320a6d0156030100000033000800
0000320aa602200101000000300010000000fb0240fe0000000000009001000000020002
001053796d626f6c0002040000002d01010004000000f001020008000000320aa307451d
010000003e0008000000320aa307e716010000002d0008000000320aa307a50e01000000
3d0008000000320aa3072603010000003d0008000000320a36022d13010000003d000800
0000320a36026c0f010000002b0008000000320a3602960a010000002b0008000000320a
3602bd04010000002b0010000000fb0240fe000000000000900101000002000200105379
6d626f6c0002040000002d01020004000000f001010008000000320aa307a90001000000
44000a00000026060f000a00ffffffff01000000000010000000fb021000070000000000
bc02000000cc0102022253797374656d00cc040000002d01010004000000f00102000300
00000000

Условие
устойчивости:picscalex1000100090000039a010000020015000000000005000000090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Для систем 1-го и 2-го порядка положительность коэффициентов
характеристического уравнения является необходимым и достаточным
условием устойчивости системы. Для системы 3-го порядка должно
выполняться дополнительное условие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Достоинство критерия:

1. Высокая точность, так как это алгебраический критерий.

2. Простота для систем невысокого порядка.

Недостатки критерия:

1. Необходимо иметь математическое описание системы.

2. Сложность применения для систем высокого порядка.

Рассмотрим примеры определения устойчивости по критерию Гурвица.

Пример 1. Определить устойчивость системы, если ее характеристическое
уравнение имеет
вид:picscalex10001000900000347010000020015000000000005000000090200000000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Условие устойчивости
picscalex100010009000003020100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c026002e0081200000026060f001a00ffffffff00001000
0000c0ffffffb7ffffffa0080000170200000b00000026060f000c004d61746854797065
0000600015000000fb02c0fe0000000000009001000000000402001054696d6573204e65
7720526f6d616e000081040000002d01000008000000320a100218080100000033000800
0000320a10025d0601000000300008000000320a1002bd0201000000320008000000320a
1002110101000000310015000000fb0240fe000000000000900101000000040200105469
6d6573204e657720526f6d616e000081040000002d01010004000000f001000008000000
320aa001270701000000610008000000320aa001670501000000610008000000320aa001
c20101000000610008000000320aa001390001000000610010000000fb0240fe00000000
00009001000000020002001053796d626f6c0002040000002d01000004000000f0010100
08000000320aa0010704010000003e000a00000026060f000a00ffffffff010000000000
10000000fb021000070000000000bc02000000cc0102022253797374656d00cc04000000
2d01010004000000f0010000030000000000 не выполняется, следовательно,
система не устойчива.

Пример 2. Определить устойчивость если передаточная функция разомкнутой
системы имеет вид
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.

Решение:

1. Определяем передаточную функцию замкнутой системы

picscalex107010009000003a60100000300150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02000580161200000026060f001a00ffffffff00001000
0000c0ffffffb6ffffff40160000b60400000b00000026060f000c004d61746854797065
0000200109000000fa02000010000000000000002200040000002d010000050000001402
5002fc060500000013025002aa1515000000fb0240fe0000000000009001010000000402
001054696d6573204e657720526f6d616e000066040000002d01010008000000320ac002
f915010000002e0008000000320a5c041a1501000000290008000000320a5c04ed130100
0000320008000000320a5c043c1101000000700008000000320a5c042410010000002800
08000000320a5c04eb0c01000000290008000000320a5c04da0b01000000310008000000
320a5c043e0901000000700008000000320a5c04260801000000280008000000320a5c04
4d0701000000700008000000320aa1015b0f01000000320008000000320aa101aa0c0100
0000700008000000320ac002b40401000000290008000000320ac0028003010000007000
08000000320ac002680201000000280008000000320ac0024700010000004b0015000000
fb02c0fe0000000000009001010000000402001054696d6573204e657720526f6d616e00
0066040000002d01020004000000f001010008000000320a9303940d0100000032001500
0000fb02c0fe0000000000009001010000cc0402001054696d6573204e657720526f6d61
6e00cc66040000002d01010004000000f001020008000000320a3003b20101000000e700
10000000fb0240fe0000000000009001000000020002001053796d626f6c000204000000
2d01020004000000f001010008000000320a5c048a12010000002b0008000000320a5c04
dd0e010000002b0008000000320a5c048c0a010000002b0008000000320aa101f80d0100
00002b0008000000320ac0029905010000003d000a00000026060f000a00ffffffff0100
0000000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc
040000002d01010004000000f0010200030000000000

2. Запишем характеристическое уравнение и условие устойчивости
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.

Условие устойчивости выполняется, следовательно, система устойчива.

Пример 3. Для заданной системы (рис. 1) определить условие устойчивости
и критический коэффициент усиления, т.е. коэффициент усиления, при
котором система находится на границе устойчивости.

Решение:

3. Определяем передаточную функцию разомкнутой системы
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4. Определяем передаточную функцию замкнутой системы
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5. Запишем характеристическое уравнение и условие устойчивости
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4. Определим критический коэффициент усиления
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3. Критерий устойчивости Михайлова

Для оценки устойчивости систем управления кроме алгебраических
критериев, используются частотные критерии Михайлова и Найквиста.

Доказательство частотных критериев базируется на следствии из принципа
аргумента.

Допустим, задан полином
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 (7)

Если система n – го порядка содержит m неустойчивых полюсов, то угол
поворота вектора D (j) равен:
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 (8)

Формулировка критерия Михайлова:

Замкнутая система автоматического управления устойчива, если
характеристическая кривая (годограф Михайлова), начинаясь на
положительной вещественной оси в точке an, при изменении частоты 0
последовательно проходит число квадрантов равное степени
характеристического полинома.

При этом

picscalex100010009000003ef0100000300150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02c004c0121200000026060f001a00ffffffff00001000
0000c0ffffffb3ffffff80120000730400000b00000026060f000c004d61746854797065
0000000109000000fa02000010000000000000002200040000002d010000050000001402
5002a7100500000013025002e31115000000fb0240fe0000000000009001000000000402
001054696d6573204e657720526f6d616e000067040000002d01010008000000320ac002
2012010000002e0008000000320a2004dc1001000000320008000000320ac002c20f0100
0000290008000000320ac002920d01000000320008000000320ac002460a010000002800
08000000320ac0024c0801000000290008000000320ac0027c0501000000280009000000
320ac0029501030000006172670015000000fb02c0fe0000000000009001000000000402
001054696d6573204e657720526f6d616e000067040000002d01020004000000f0010100
08000000320a6204e20201000000300010000000fb0240fe000000000000900101000002
0002001053796d626f6c0002040000002d01010004000000f001020008000000320aa401
9d1001000000700008000000320ac002ef0601000000770010000000fb02c0fe00000000
00009001010000020002001053796d626f6c0002040000002d01020004000000f0010100
08000000320a62049a0601000000770008000000320a6204300401000000770015000000
fb0240fe0000000000009001010000000402001054696d6573204e657720526f6d616e00
0067040000002d01010004000000f001020008000000320ac002790e010000006d000800
0000320ac002e70a010000006e0008000000320ac0027106010000006a0008000000320a
c0022c0401000000440010000000fb0240fe000000000000900100000002000200105379
6d626f6c0002040000002d01020004000000f001010008000000320ac002320c01000000
2d0008000000320ac0024a09010000003d0008000000320ac00239000100000044001000
0000fb02c0fe0000000000009001000000020002001053796d626f6c0002040000002d01
010004000000f001020008000000320a6204f005010000002d0008000000320a62043005
01000000a30008000000320a6204890301000000a3000a00000026060f000a00ffffffff
01000000000010000000fb021000070000000000bc02000000cc0102022253797374656d
00cc040000002d01020004000000f0010100030000000000 (9)

Пример 4. Допустим, задан характеристический полином системы
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Годограф устойчивой системы имеет вид (рис. 3a).

Пример 5. Допустим, задан характеристический полином системы

picscalex1000100090000031f0200000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02e005a0151200000026060f001a00ffffffff00001000
0000c0ffffffb3ffffff60150000930500000b00000026060f000c004d61746854797065
0000400115000000fb0240fe0000000000009001000000000402001054696d6573204e65
7720526f6d616e000067040000002d01000008000000320a1e050715010000002e000800
0000320a1e050c0f01000000290008000000320a1e05bf0601000000280008000000320a
1e05680401000000290008000000320a1e05970101000000280008000000320a0b02cd11
010000002c0008000000320a0b025e0301000000290008000000320a0b02970101000000
280015000000fb02c0fe0000000000009001000000000402001054696d6573204e657720
526f6d616e000067040000002d01010004000000f001000008000000320a8e054f140100
0000310008000000320a5504260e01000000320008000000320a8e05fb0b010000003000
08000000320a8e055b0801000000320008000000320a7b02ee1001000000320008000000
320a7b02510c01000000310008000000320a4201c80801000000320008000000320a7b02
b20601000000300015000000fb0240fe0000000000009001010000000402001054696d65
73204e657720526f6d616e000067040000002d01000004000000f001010008000000320a
1e05771301000000610008000000320a1e05c511010000006a0008000000320a1e05050b
01000000610008000000320a1e05600701000000610008000000320a1e058c0201000000
6a0008000000320a1e05470001000000440008000000320a0b02f30f0100000061000800
0000320a0b023a0d01000000700008000000320a0b02790b01000000610008000000320a
0b02b40701000000700008000000320a0b02bc0501000000610008000000320a0b027002
01000000700008000000320a0b02470001000000440010000000fb0240fe000000000000
9001010000020002001053796d626f6c0002040000002d01010004000000f00100000800
0000320a1e05431201000000770008000000320a1e05a20c01000000770008000000320a
1e050a0301000000770010000000fb0240fe000000000000900100000002000200105379
6d626f6c0002040000002d01000004000000f001010008000000320a1e050a1001000000
2b0008000000320a1e05a509010000002d0008000000320a1e056605010000003d000800
0000320a0b028c0e010000002b0008000000320a0b02120a010000002b0008000000320a
0b025c04010000003d000a00000026060f000a00ffffffff01000000000010000000fb02
1000070000000000bc02000000cc0102022253797374656d00cc040000002d0101000400
0000f0010000030000000000

Годограф устойчивой системы имеет вид (рис. 3б).

Пример 6. Допустим, задан характеристический полином системы
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Годограф устойчивой системы имеет вид (рис. 3в).

Рис. 3

Пример. Для заданной системы (рис. 4) определить условие устойчивости,
частоту собственных колебаний системы и критический коэффициент
усиления, т.е. коэффициент усиления, при котором система находится на
границе устойчивости.

Определить устойчивость при T1 = T2 = 1 c и kv = 1 c-1.

Решение:

1. Определяем передаточную функцию разомкнутой системы
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гдеpicscalex950100090000030a01000002001500000000000500000009020000000004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2. Определяем передаточную функцию замкнутой системы
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3. Запишем характеристическое уравнение
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4. Определим частоту собственных колебаний системы и критический
коэффициент усиления из условия границы устойчивости
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Откуда частота собственных колебаний системы равна:
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Критический коэффициент усиления равен:
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5. Определим устойчивость при T1 = T2 = 1 c и kv = 1 c-1.

picscalex100010009000003540000000000120000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02600240011200000026060f001a00ffffffff00001000
0000c0ffffffb7ffffff00010000170200000b00000026060f000c004d61746854797065
000060000a00000026060f000a00ffffffff010000000000030000000000picscalex107
010009000003be0200000200150000000000050000000902000000000400000002010100
050000000102ffffff00040000002e01180005000000310201000000050000000b020000
0000050000000c02200640311200000026060f001a00ffffffff000010000000c0ffffff
b8ffffff00310000d80500000b00000026060f000c004d61746854797065000050011500
0000fb0240fe0000000000009001010000000402001054696d6573204e657720526f6d61
6e0000a0040000002d01000008000000320a7605251802000000292e08000000320a7605
e81201000000310008000000320a76051d1201000000280008000000320a760557100100
00006a0008000000320a7605b30d01000000290008000000320a7605170a010000003200
08000000320a7605880701000000310008000000320a7605bd0601000000280008000000
320a7605a0040200000020200c000000320a760540000a00000020202020202020202020
08000000320a37021a2f01000000290008000000320a3702d12c01000000540008000000
320a3702302b01000000540008000000320a3702712601000000310008000000320a3702
a62501000000280008000000320a3702e023010000006a0008000000320a370263200200
0000295d08000000320a37021a1e01000000540008000000320a3702921a010000005400
08000000320a3702ce1901000000280008000000320a3702c914010000006b0008000000
320a3702d413010000005b0008000000320a37028b1101000000290008000000320a3702
2b0f01000000280008000000320a37028f0d020000006a5908000000320a3702eb0a0100
0000290008000000320a37028b0801000000280008000000320a37024307010000005800
08000000320a3702d70401000000290008000000320a3702b602010000006a0008000000
320a3702820101000000280008000000320a3702470001000000440015000000fb0280fe
0000000000009001010000000402001054696d6573204e657720526f6d616e0000a00400
00002d01010004000000f001000008000000320aad04d61601000000320008000000320a
ad04640c01000000320008000000320aa702cb2d01000000320008000000320aa702182c
01000000310008000000320a6e015f2a01000000320008000000320aa702141f01000000
320008000000320aa7027a1b01000000310008000000320a6e01da180100000032001000
0000fb0240fe0000000000009001010000020002001053796d626f6c0002040000002d01
000004000000f001010008000000320a7605541501000000770008000000320a7605d510
01000000770008000000320a7605e20a01000000770008000000320a3702dd2801000000
770008000000320a37025e2401000000770008000000320a370258170100000077000800
0000320a3702e80f01000000770008000000320a3702480901000000770008000000320a
3702340301000000770010000000fb0240fe000000000000900100000002000200105379
6d626f6c0002040000002d01010004000000f001000008000000320a7605171401000000
2d0008000000320a76059c0e010000002b0008000000320a7605b708010000002d000800
0000320a76057905010000003d0008000000320a37020330010000003d0008000000320a
3702a027010000002d0008000000320a37022522010000002b0008000000320a3702cf1c
010000002b0008000000320a37021b16010000002d0008000000320a3702741201000000
3d0008000000320a3702d40b010000002b0008000000320a3702c005010000003d000a00
000026060f000a00ffffffff01000000000010000000fb021000070000000000bc020000
00cc0102022253797374656d00cc040000002d01000004000000f0010100030000000000

5. Строим характеристическую
кривуюpicscalex100010009000003e20000000200150000000000050000000902000000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(рис. 5) по
данным, приведенным в таблице 1.

Таблица 1
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()10-
1-

Y()0picscalex100010009000003ed000000040015000000000005000000090200000000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-

В соответствии с критерием Михайлова, рассматриваемая система является
устойчивой.

4. Частотный критерий устойчивости Найквиста

Частотный критерий устойчивости Найквиста позволяет по виду частотной
характеристики разомкнутой системы судить об устойчивости замкнутой
системы, т.е. он применим для замкнутых систем.

Рассмотрим функцию, которая связывает характеристики разомкнутых и
замкнутых систем
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 (6)

где D(p) – характеристический полином замкнутой системы;

A(p) – характеристический полином разомкнутой системы.

При этом степени полиномов A(p) и D(p) одинаковы исходя из условия
физической реализуемости системы.

В соответствии со следствием из принципа аргумента
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 (7)

Рассмотрим разные случаи.

Система, устойчивая в разомкнутом состоянии.

Так как разомкнутая система устойчива, то она не содержит корней в
правой полуплоскости (т.е. m = 0), для того чтобы и замкнутая система
была устойчива, должно выполняться условие:
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 (8)

Графически это обозначает, что годограф вектора W (j) не охватывает
начала координат, а вектора K (j) – точку с координатами (-1, j0), как
показано на рис. 6. Точка с координатами (-1, j0) называется
критической.

Рис. 6.

Система, неустойчивая в разомкнутом состоянии.

Так как разомкнутая система неустойчива, то она содержит m корней в
правой полуплоскости, для того, чтобы замкнутая система была устойчивой,
должно выполняться условие
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 (9)

Графически это обозначает, что годограф вектора K (j) охватывает точку с
координатами (-1, j0) m/2 – раз.

Формулировка критерия Найквиста: Замкнутая система автоматического
управления устойчива, если амплитудно-фазовая частотная характеристика
разомкнутой, неустойчивой системы, имеющей m корней в правой
полуплоскости, охватывает точку с координатами (–1, j0) m/2-раз.

Иногда по графику трудно определить охватывает ли АФХ критическую точку.
В этом случае можно использовать правило переходов. Переходами
называются точки пересечения АФХ отрезка оси (-.. – 1). Знак перехода
определяется по следующему правилу: если фаза убывает – переход
отрицательный.

Формулировка критерия Найквиста: Замкнутая система автома-тического
управления устойчива, если разность положительных и отрицательных
переходов равна m/2, где m – количество корней в правой полуплоскости
разомкнутой неустойчивой системы, т.е.

(10)

Пример 8. Для заданной системы (рис. 7) определить условие устойчивости
и критический коэффициент усиления.

Определить устойчивость при T1 = T2 = 1 c и kv = 1 c-1.

Решение:

1. Определяем передаточную функцию разомкнутой системы
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2. Строим АФХ разомкнутой системы
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При T1 = T2 = 1 c и kv = 1 c-1 АФХ разомкнутой системы имеет вид
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Расчетные данные приведены в таблице 2, а график АФХ на рис. 8.

Таблица 2

01P()-2-1/20Q()-

00

Как видно из рисунка (8) и таблицы 2, АФХ разомкнутой системы не
охватывает критическую точку, следовательно, замкнутая система, при
заданной структуре и параметрах, устойчива.

Определим критический коэффициент усиления из условия:
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picscalex100010009000003690200000300150000000000050000000902000000000400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5. Определение областей устойчивости

Устойчивость систем зависит от структуры и параметров системы. При
расчете систем автоматического управления возникает задача опреде-ления
диапазона изменения варьируемых параметров системы, при кото-рых она
устойчива.

Область устойчивости – это совокупность значений параметров системы, при
которых она устойчива.

Коэффициенты характеристического уравнения являются функциями от
параметров системы, и они определяют расположение корней в комплексной
плоскости, при изменении параметров корни перемещаются в комплексной
плоскости и система может стать не устойчивой.

Для определения областей устойчивости можно использовать различные
методы, наиболее часто используют метод D – разбиения. D-разбиение может
быть выполнено по одному и более параметрам.

Рассмотрим алгоритм определения областей устойчивости с помощью метода D
– разбиения по одному параметру на конкретных примерах.

Пример 9. Определить область возможных значений параметра «к», при
которых заданная система (рис. 9) устойчива

Порядок определения областей устойчивости

1. Определяем передаточную функцию замкнутой системы
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picscalex100010009000003870300000200360100000000050000000201010000000500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2. Определяем характеристический полином

3. Разрешим уравнение относительно параметра – к
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4. Строим кривую D – разбиения (см. таблицу 3 и рис. 10)

Таблица 3

0122X()0124Y()0-104

Так как параметр является вещественной положительной величиной, то
областью устойчивости являются значения параметра – к, расположенные на
вещественной положительной оси, т.е.] 0, 2 [, что может быть провере-но
по критерию Гурвица.

Литература

1. Бронштейн И.Н., Семендяев К.Н. Справочник по математике для инженеров
и учащихся вузов. – М.: Наука, 1986.

2. Брюханов В.Н. и др. Теория автоматического управления. – М: Высшая
школа, 2000.

3. Егупов Н.Д., Пупков К.А., Баркин А.И. Синтез регуляторов систем
автоматического управления. МГТУ им. Н.Э. Баумана, 2004.

4. Ким Д.П., Дмитриева Н.Д. Сборник задач по теории автоматического
управления. Линейные системы. ФИЗМАТЛИТ, 2007. – 168 с.

5. Лукас В.А. Теория автоматического управления. – М.: Недра, 1990. –
416 с.

6. Сборник задач по теории автоматического регулирования и управления/
Под редакцией В.А. Бесекерского. – M.: Наука, 1978.

7. Справочник по теории автоматического управления. /Под ред.
А.А. Красовского – М.: Наука, 198 – 712 с.

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