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Теория автоматического управления

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Теория автоматического управления

Язык: русский
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20

1. Анализ устойчивости замкнутой системы

1.1 Анализ устойчивости системы по корням характеристического уравнения

Запишем передаточную функцию разомкнутой системы:
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. (1)

Передаточная функция замкнутой системы имеет вид:
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.

Характеристическое уравнение замкнутой системы:

picscalex1000100090000037c02000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02400420181200000026060f001a00ffffffff000010000000c0ffffff
baffffffe0170000fa0300000b00000026060f000c004d617468547970650000e0001c00
0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61
6e00d89ff377e19ff3772020f577511b6659040000002d01000008000000320add038a16
01000000307908000000320add03721302000000343708000000320add03e30f02000000
303308000000320add03830f010000002e3308000000320add03c30e0100000030330900
0000320add03f708040000003030373808000000320add039708010000002e3008000000
320add03d7070100000030300a000000320add0354010500000030303030380008000000
320add03f400010000002e3008000000320add03340001000000303008000000320a6c01
161701000000303008000000320a6c01001501000000293008000000320a6c01d8120200
0000303808000000320a6c017812010000002c3808000000320a6c01be11010000003038
08000000320a6c01b50f01000000313808000000320a6c01db0d02000000292808000000
320a6c018b0c01000000312808000000320a6c014f0c010000002c2808000000320a6c01
950b01000000302808000000320a6c018c0901000000312808000000320a6c01ae080200
0000292808000000320a6c019e0602000000303108000000320a6c013e06010000002c31
08000000320a6c01840501000000303108000000320a6c017b0301000000313108000000
320a6c011b0301000000283108000000320a6c013a000200000034381c000000fb0220ff
0000000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377
e19ff3772020f577511b6659040000002d01010004000000f001000008000000320a3103
ba0c01000000323808000000320a3103d0050100000033381c000000fb0280fe00000000
00009001000000020002001053796d626f6c0000a71a0ad5a0f11200d89ff377e19ff377
2020f577511b6659040000002d01000004000000f001010008000000320add0358150100
00003d3808000000320add034d12010000002b3808000000320add03a90d010000002d38
08000000320add03b806010000002b3808000000320a6c01e415010000003d3808000000
320a6c019f10010000002b3808000000320a6c01e30e010000002d3808000000320a6c01
760a010000002b3808000000320a6c016504010000002b3808000000320a6c01fc010100
00002b381c000000fb0280fe0000000000009001010000cc0402001054696d6573204e65
7720526f6d616e00d89ff377e19ff3772020f577511b6659040000002d01010004000000
f001000008000000320add03631101000000733808000000320add03fd0b010000007338
08000000320add031a0501000000733808000000320a6c015e1401000000733808000000
320a6c01390d01000000733808000000320a6c010c080100000073380a00000026060f00
0a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc01020222
53797374656d0059511b665900000a0021008a010000000000000000bcf3120004000000
2d01000004000000f0010100030000000000 (2)

Корни характеристического уравнения (2):

picscalex100010009000003a201000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c028004a00c1200000026060f001a00ffffffff000010000000c0ffffff
a7ffffff600c0000270400000b00000026060f000c004d617468547970650000f0001c00
0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61
6e00d89ff377e19ff3772020f577e91b665d040000002d01000008000000320ac503050b
02000000393108000000320ac503a50a010000002e3108000000320ac503250902000000
363308000000320ac5037d0502000000393308000000320ac5031d05010000002e330800
0000320ac5039d0302000000313708000000320a7f016e0602000000303508000000320a
7f010e06010000002e3509000000320a7f01ce0303000000313333651c000000fb0220ff
0000000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377
e19ff3772020f577e91b665d040000002d01010004000000f001000008000000320a2504
8f0101000000333308000000320a25045701010000002e3308000000320a2504e7000100
0000323308000000320adf01ce000100000031331c000000fb0280fe0000000000009001
010000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577
e91b665d040000002d01000004000000f001010008000000320ac503a708010000006a33
08000000320ac503400001000000733308000000320a7f0140000100000073331c000000
fb0280fe0000000000009001000000020002001053796d626f6c0000dc090aa0a0f11200
d89ff377e19ff3772020f577e91b665d040000002d01010004000000f001000008000000
320ac503390701000000b13308000000320ac5038f02010000003d3308000000320a7f01
fc02010000002d3308000000320a7f01c401010000003d330a00000026060f000a00ffff
ffff0100000000001c000000fb021000070000000000bc02000000cc0102022253797374
656d005de91b665d00000a0021008a010000000000000000bcf31200040000002d010000
04000000f0010100030000000000

Характеристическое уравнение (2) имеет два правых корня, следовательно,
данная замкнутая система неустойчива.

1.2 Анализ устойчивости системы по алгебраическому критерию

Для характеристического уравнения (2) замкнутой системы коэффициенты
ai, i=0..3,

а0=0.00008,

a1=0.0078,

a2= – 0.03,

a3=48.

Необходимым условием устойчивости системы является:

ai>0, i=0..3

Данное условие не выполняется (a2<0), следовательно, замкнутая система неустойчива.1.3 Анализ устойчивости системы по частотным критерияма) Критерий Найквиста (на комплексной плоскости)Используя передаточную функцию разомкнутой системы (1) запишем характеристическое уравнение разомкнутой системы:picscalex100010009000003a501000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02000220151200000026060f001a00ffffffff000010000000c0ffffff c6ffffffe0140000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f577481c663b040000002d01000008000000320a60013214 01000000307908000000320a60011c1201000000297908000000320a6001f40f02000000 303808000000320a6001940f010000002c3808000000320a6001da0e0100000030380800 0000320a6001d00c01000000313808000000320a6001f60a02000000292808000000320a 6001a60901000000312808000000320a60016a09010000002c2808000000320a6001b008 01000000302808000000320a6001a60601000000312808000000320a6001c80502000000 292808000000320a6001b80302000000303108000000320a60015803010000002c310800 0000320a60019e0201000000303108000000320a6001940001000000313108000000320a 600134000100000028311c000000fb0280fe000000000000900100000002000200105379 6d626f6c00006d100ae5a0f11200d89ff377e19ff3772020f577481c663b040000002d01 010004000000f001000008000000320a60010013010000003d3108000000320a6001ba0d 010000002b3108000000320a6001fe0b010000002d3108000000320a6001900701000000 2b3108000000320a60017e01010000002b311c000000fb0280fe00000000000090010100 00cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577481c 663b040000002d01000004000000f001010008000000320a60017a110100000073310800 0000320a6001540a01000000733108000000320a600126050100000073310a0000002606 0f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc0102 022253797374656d003b481c663b00000a0021008a010000000001000000bcf312000400 00002d01010004000000f0010000030000000000. (3)Найдем корни характеристического уравнения (3):picscalex1000100090000039201000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02c006a0061200000026060f001a00ffffffff000010000000c0ffffff acffffff600600006c0600000b00000026060f000c004d61746854797065000080011c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f577b91d6627040000002d01000008000000320a0606ce04 01000000357908000000320a06066e04010000002e7908000000320a0606ee0202000000 313208000000320ac0037205010000002c3208000000320ac003f8030200000031300800 0000320a7a010806010000002c3009000000320a7a01ce0303000000313030651c000000 fb0220ff0000000000009001000000cc0402001054696d6573204e657720526f6d616e00 d89ff377e19ff3772020f577b91d6627040000002d01010004000000f001000008000000 320a6606e00001000000333008000000320a2004e70001000000323008000000320ada01 ce000100000031301c000000fb0280fe0000000000009001000000020002001053796d62 6f6c0000361e0ac3a0f11200d89ff377e19ff3772020f577b91d6627040000002d010000 04000000f001010008000000320a0606e001010000003d3008000000320ac00326030100 00002d3008000000320ac003ee01010000003d3008000000320a7a01fc02010000002d30 08000000320a7a01c401010000003d301c000000fb0280fe0000000000009001010000cc 0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577b91d6627 040000002d01010004000000f001000008000000320a0606400001000000733008000000 320ac003400001000000733008000000320a7a0140000100000073300a00000026060f00 0a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc01020222 53797374656d0027b91d662700000a0021008a010000000000000000bcf3120004000000 2d01000004000000f0010100030000000000Характеристическое уравнение разомкнутой системы (3) имеет один правый корень, следовательно, разомкнутая система неустойчива.Построим годограф Найквиста. Для этого определим частотную передаточную функцию разомкнутой системы и ее действительную и мнимую части.picscalex87010009000003a804000003001c00000000000500000009020000000005000 000020101000000050000000102ffffff00050000002e0118000000050000000b0200000 000050000000c02c00820371200000026060f001a00ffffffff000010000000c0ffffffb 7ffffffe0360000770800000b00000026060f000c004d617468547970650000000208000 000fa0200001000000000000000040000002d01000005000000140201027f06050000001 3020102221c0500000014020102d21d0500000013020102a735050000001402910678010 5000000130291065f1d1c000000fb0220ff0000000000009001000000cc0402001054696 d6573204e657720526f6d616e00d89ff377e19ff3772020f5778d0f665e040000002d010 10008000000320a7807f61a01000000327908000000320a7807b41401000000337908000 000320a78072c0c01000000327908000000320a7807010901000000327908000000320a5 3052a1701000000337908000000320a5305ed0901000000327908000000320ae802722f0 1000000337908000000320ae80243240100000032791c000000fb0280fe0000000000009 001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f 5778d0f665e040000002d01020004000000f001010008000000320a2408571a010000002 97908000000320a2408db1702000000303308000000320a24087b17010000002e3308000 000320a2408bb160100000030330a000000320a2408de0f0500000030303030380008000 000320a24087e0f010000002e3008000000320a2408be0e01000000303008000000320a2 4083a0e01000000283008000000320a24088d0b01000000293008000000320a2408eb0a0 1000000313009000000320a2408e404040000003030373808000000320a2408840401000 0002e3008000000320a2408c40301000000303008000000320a240840030100000028300 8000000320aff05cd1c01000000293008000000320aff05511a020000003033080000003 20aff05f119010000002e3308000000320aff0531190100000030330a000000320aff055 4120500000030303030380008000000320aff05f411010000002e3008000000320aff053 41101000000303008000000320aff05b01001000000283008000000320aff05360f02000 000343808000000320aff05790c01000000293808000000320aff05d70b0100000031380 9000000320aff05d005040000003030373808000000320aff057005010000002e3008000 000320aff05b00401000000303008000000320aff052c0401000000283008000000320af f05b20202000000343808000000320a9403153501000000293808000000320a940399320 2000000303308000000320a94033932010000002e3308000000320a94037931010000003 0330a000000320a94039c2a0500000030303030380008000000320a94033c2a010000002 e3008000000320a94037c2901000000303008000000320a9403f82801000000283008000 000320a9403282601000000313009000000320a940326200400000030303738080000003 20a9403c61f010000002e3008000000320a9403061f01000000303008000000320a6f010 52902000000343808000000320a8d03901b01000000293808000000320a8d03601802000 000303808000000320a8d030018010000002c3808000000320a8d0346170100000030380 8000000320a8d033d1501000000313808000000320a8d036313020000002928080000003 20a8d030b1101000000312808000000320a8d03cf10010000002c2808000000320a8d031 51001000000302808000000320a8d030c0e01000000312808000000320a8d032e0d02000 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302de002501050000001302df002501050000001302df002601050000001302e00026010 50000001302e1002601050000001302e2002601050000001302e2002701050000001302e 3002701050000001302e4002701050000001302e4002801050000001302e500280105000 0001302e6002801050000001302e7002801050000001302e7002901050000001302e8002 901050000001302e9002901050000001302ea002901050000001302eb002901050000001 302ec002901050000001402f7001201040000002d010e0004000000f0010f00040000002 d010d00040000002701ffff040000002d010e0008000000fa02000001000000ff0000020 40000002d010f000500000014028d000a000500000013028d002300050000001402f7001 201040000002d010e0004000000f0010f00040000002d010d00040000002d010b0004000 000f0010e0004000000f0010d00040000002d010900040000002d010a0004000000f0010 b0005000000090200000000040000002d010100050000000902000000000700000021050 100510f86000c00050000000902000000000500000009020000000007000000210501007 70f86001a0005000000090200000000040000002d010300040000002d010100070000002 1050100280f860015000700000021050100290f860023000500000009020000000004000 0002d01000005000000090200000000040000002d0101000500000009020000000007000 00021050100500f1101a5000500000009020000000005000000090200000000070000002 1050100770f1101b10005000000090200000000040000002d010300040000002d0101000 700000021050100280f1101ac000700000021050100290f1101ba0005000000090200000 000040000002d010000050000002e010000000005000000090200000000040000002d010 30008000000fa0200000000000000000000040000002d010b0004000000f001090007000 000fc020000000000000000040000002d01090004000000f0010a0005000000090200000 000050000000102ffffff02050000000201020000000400000003010800050000000c026 400640014000000ff0600000600f60200002400010002000000000001000100020000000 10000000100020007000000fc020000ffffff0200000500000028020a000d0004000000f 0010a00050000000c021901420104000000f0010d00030000000000Рис. 1.3.1Для случая, когда разомкнутая система неустойчива критерий Найквиста звучит следующим образом: для устойчивости замкнутой системы необходимо и достаточно, чтобы годограф Найквиста охватывал особую точку (picscalex100010009000003c900000002001c000000000005000000090200000000050 00000020101000000050000000102ffffff00050000002e0118000000050000000b02000 00000050000000c02a00120021200000026060f001a00ffffffff000010000000c0fffff fc6ffffffe0010000660100000b00000026060f000c004d61746854797065000020001c0 00000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d6 16e0015abc6771eabc677c0a9c97700003000040000002d01000008000000320a60013b0 10100000031791c000000fb0280fe0000000000009001000000020002001053796d626f6 c007740000000660b0a1915abc6771eabc677c0a9c97700003000040000002d010100040 00000f001000008000000320a60013a00010000002d790a00000026060f000a00fffffff f0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656 d000000000a0021008a010000000000000000fcf31200aaafc677040000002d010000040 00000f0010100030000000000; picscalex100010009000003c900000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02000200021200000026060f001a00ffffffff000010000000c0ffffff c6ffffffc0010000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d61 6e0015abc6771eabc677c0a9c97700003000040000002d01000008000000320a6001ff00 0100000030001c000000fb0280fe0000000000009001010000000402001054696d657320 4e657720526f6d616e0015abc6771eabc677c0a9c97700003000040000002d0101000400 0000f001000008000000320a60018200010000006a000a00000026060f000a00ffffffff 0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d 000000000a0021008a010000000000000000fcf31200aaafc677040000002d0100000400 0000f0010100030000000000) в положительном направлении на угол picscalex100010009000003c900000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02c001e0011200000026060f001a00ffffffff000010000000c0ffffff c6ffffffa0010000860100000b00000026060f000c004d61746854797065000030001c00 0000fb0280fe0000000000009001010000020402001053796d626f6c0077400000009211 0a0515abc6771eabc677c0a9c97700003000040000002d01000008000000320a60019a00 0100000070791c000000fb0280fe0000000000009001010000000402001054696d657320 4e657720526f6d616e0015abc6771eabc677c0a9c97700003000040000002d0101000400 0000f001000008000000320a60012f00010000006c790a00000026060f000a00ffffffff 0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d 000000000a0021008a0100000000000000009cf31200aaafc677040000002d0100000400 0000f0010100030000000000, где l – число правых корней характеристического уравнения разомкнутой системы.Число правых корней характеристического уравнения разомкнутой системы (3) равно единице (l=1), полученный годограф не охватывает особую точку (-1, j0) на угол lр=р (годограф охватывает особую точку в направлении по часовой стрелке), следовательно, критерий Найквиста не выполняется и система неустойчива.б) Критерий Найквиста (на плоскости ЛЧХ)Построим ЛЧХ заданной системы, для этого определим расчетные выражения для L(w) и ц(w):picscalex1000100090000030303000004001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02a004e02e1200000026060f001a00ffffffff000010000000c0ffffff b8ffffffa02e0000580400000b00000026060f000c004d61746854797065000020010800 0000fa0200001000000000000000040000002d010000050000001402a703631505000000 13028b03941508000000fa0200002000000000000000040000002d010100050000001402 930394150500000013024704db15040000002d0100000500000014024704e31505000000 130244024116050000001402440241160500000013024402732e05000000140200023b15 0500000013020002932e1c000000fb0280fe0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577131b661f040000002d01 020008000000320ae303e12d01000000297908000000320ae3039b2c0100000029790800 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6002b00201000000293008000000320a600224010100000028301c000000fb0220ff0000 000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f577131b661f040000002d01030004000000f001020008000000320a37033a2d 01000000323008000000320a3703e62401000000323008000000320a37036a1d01000000 32301c000000fb0280fe0000000000009001010000cc0402001054696d6573204e657720 526f6d616e00d89ff377e19ff3772020f577131b661f040000002d01020004000000f001 030008000000320ae3039f2b01000000773008000000320ae3034b230100000077300800 0000320ae303cf1b01000000773008000000320a6002500e01000000573008000000320a 6002ea07010000004e3008000000320a6002b40101000000773008000000320a60024600 010000004c301c000000fb0220ff0000000000009001010000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f577131b661f040000002d0103000400 0000f001020008000000320ac0029a0f0100000050301c000000fb0280fe000000000000 9001000000020002001053796d626f6c0000f0170ae740f11200d89ff377e19ff3772020 f577131b661f040000002d01020004000000f001030008000000320ae303552701000000 2b3008000000320ae303d91f010000002b3008000000320ae3039d17010000002b300800 0000320a6002f110010000003d3008000000320a60027609010000003d3008000000320a 60029403010000003d300a00000026060f000a00ffffffff0100000000001c000000fb02 1000070000000000bc02000000cc0102022253797374656d001f131b661f00000a002100 8a0100000000030000005cf31200040000002d01030004000000f0010200030000000000 (7)picscalex1000100090000032403000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02600480321200000026060f001a00ffffffff000010000000c0ffffff a9ffffff40320000090400000b00000026060f000c004d617468547970650000e0000800 0000fa0200001000000000000000040000002d0100000500000014024002490805000000 13024002950b050000001402400264230500000013024002b5311c000000fb0280fe0000 000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f577ca1866f3040000002d01010008000000320aa002c7310100000029790800 0000320ad3030b2e02000000343809000000320ad303d927040000003337343408000000 320ad3037927010000002e3708000000320ad303b92601000000303708000000320aae01 233101000000293708000000320aae01af2f02000000303308000000320aae014f2f0100 00002e3308000000320aae018f2e0100000030330a000000320aae01a427050000003030 3030380008000000320aae014427010000002e3008000000320aae018426010000003030 08000000320aae01002601000000283008000000320aae017e2302000000343808000000 320aa002d42201000000283808000000320aa002301a01000000293808000000320aa002 a41801000000283809000000320aa0028c14030000006172673008000000320aa002a612 01000000297208000000320aa0021a1101000000287209000000320aa002390d03000000 6172673008000000320acc03fa0a01000000297208000000320acc036e09010000002872 08000000320aae01030b01000000297208000000320aae01770901000000287208000000 320aa002b00201000000297208000000320aa00224010100000028721c000000fb0220ff 0000000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377 e19ff3772020f577ca1866f3040000002d01020004000000f001010008000000320a2703 fc2b01000000327208000000320a0201812c0100000032721c000000fb0280fe00000000 00009001000000020002001053796d626f6c0000f5170a46a0f11200d89ff377e19ff377 2020f577ca1866f3040000002d01010004000000f001020008000000320ad303eb2c0100 00002d7208000000320ad3039f25010000002d7208000000320aae01702d010000002b72 08000000320aa002681e010000002b7208000000320aa0024c1c010000002d7208000000 320aa002141b010000003d7208000000320aa0027213010000002d7208000000320aa002 070c010000003d7208000000320aa0029403010000003d721c000000fb0280fe00000000 00009001010000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff377 2020f577ca1866f3040000002d01020004000000f001010008000000320ad303e52a0100 0000777208000000320aae016a2b01000000777208000000320aae010425010000007772 0a000000320aa0028d1f0500000061726374670008000000320aa0023419010000007772 08000000320aa002aa1101000000777208000000320acc03fe0901000000777208000000 320acc03780801000000507208000000320aae01070a01000000777208000000320aae01 57080100000051720a000000320aa002cc040500000061726374670008000000320aa002 b4010100000077721c000000fb0220ff0000000000009001010000cc0402001054696d65 73204e657720526f6d616e00d89ff377e19ff3772020f577ca1866f3040000002d010100 04000000f001020009000000320a0003a31604000000e7ede0ec09000000320a00033f0f 04000000f7e8f1eb1c000000fb0280fe0000000000009001010000020402001053796d62 6f6c0000961d0a36a0f11200d89ff377e19ff3772020f577ca1866f3040000002d010200 04000000f001010008000000320aa002121d0100000070e808000000320aa0021c000100 00006ae80a00000026060f000a00ffffffff0100000000001c000000fb02100007000000 0000bc02000000cc0102022253797374656d00f3ca1866f300000a0021008a0100000000 01000000bcf31200040000002d01010004000000f0010200030000000000 (8)Для построения асимптотической ЛАЧХ найдем параметры:picscalex1000100090000032804000002001c0000000000050000000902000000000500 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5d06ed03010000002e3008000000320a5d062d0301000000303008000000320ae3039716 010000003b3008000000320ae303b51301000000323008000000320ae303c20e02000000 6c6708000000320ae303380e010000002c6709000000320ae3032a0a0300000031303065 08000000320ae3033306010000002c3008000000320ae303230402000000303108000000 320ae303c303010000002e3108000000320ae303030301000000303108000000320a6c01 bf12010000003b3109000000320a6c01cf0e030000003632356508000000320a6c016f0e 010000002e3208000000320a6c01ef0c02000000333308000000320a6c01e90902000000 343808000000320a6c018708020000006c6708000000320a6c01dd060200000032300800 0000320a6c01e401020000006c6708000000320a6c013a000200000032301c000000fb02 20ff0000000000009001000000cc0402001054696d6573204e657720526f6d616e00d89f f377e19ff3772020f5770d156639040000002d01010004000000f001000008000000320a 3709e91101000000333008000000320a2b08f80d01000000313008000000320a37096c08 01000000333008000000320a3709ed0001000000333008000000320abd06741001000000 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0000000040000002d010600070000002105010066126e010c00050000000902000000000 40000002d010100050000000902000000000500000009020000000007000000210501007 7126e01170005000000090200000000040000002d010400040000002d010100070000002 105010028126e011200070000002105010029126e0120000500000009020000000004000 0002d01000005000000090200000000040000002d0101000500000009020000000008000 000210503006c6f6700e9011901050000000902000000000500000009020000000007000 000210501007712e9012c0105000000090200000000040000002d010400040000002d010 10007000000210501002812e901270107000000210501002912e90135010500000009020 0000000040000002d010000050000002e010000000005000000090200000000040000002 d010400040000002d01090004000000f0010800040000002d01070004000000f0010b000 5000000090200000000050000000102ffffff02050000000201020000000400000003010 800050000000c026400640014000000ff0600000600f6020000240001000200000000000 100010002000000010000000100020007000000fc020000ffffff0200000500000028020 8000b0004000000f0010800050000000c02f1012b0204000000f0010b00030000000000Рис. 1.3.2wср(частота среза) – частота, соответствующая пересечению ЛАЧХ с осью lgw;wкр(критическая частота) – частота, соответствующая пересечению ЛФЧХ уровня –р;Система устойчива, если выполняется условие:wср< wкрДанное условие не выполняется, следовательно, система неустойчива. Аналогичный вывод можно сделать по асимптотической ЛАЧХ и ЛФЧХ системы, построенной как сумма отдельных звеньев, входящих в систему, изображенной на рисунке (1.3.3):в) Критерий МихайловаИспользуя характеристическое уравнение замкнутой системы (2) введем функцию Михайлова:picscalex1000100090000034d01000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c020002800e1200000026060f001a00ffffffff000010000000c0ffffff c6ffffff400e0000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d61 6e0015abc6771eabc677c0a9c97700003000040000002d01000008000000320a6001cc0d 01000000297908000000320a6001400c01000000287908000000320a60017a0801000000 297908000000320a6001ee0601000000287908000000320a6001a0030100000029790800 0000320a600166010100000028791c000000fb0280fe0000000000009001010000000402 001054696d6573204e657720526f6d616e0015abc6771eabc677c0a9c977000030000400 00002d01010004000000f001000008000000320a6001d00c01000000777908000000320a 6001cc0a020000006a5908000000320a60017e0701000000775908000000320a6001c205 01000000585908000000320a60013802020000006a7708000000320a6001460001000000 44771c000000fb0280fe0000000000009001000000020002001053796d626f6c00774000 0000640c0a1115abc6771eabc677c0a9c97700003000040000002d01000004000000f001 010008000000320a60015209010000002b7708000000320a60017804010000003d770a00 000026060f000a00ffffffff0100000000001c000000fb021000070000000000bc020000 00cc0102022253797374656d000000000a0021008a010000000001000000fcf31200aaaf c677040000002d01010004000000f0010000030000000000, гдеpicscalex100010009000003ca01000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c026002e00e1200000026060f001a00ffffffff000010000000c0ffffff 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0000000000009001010000cc0402001054696d6573204e657720526f6d616e00d89ff377 e19ff3772020f577231c667f040000002d01000004000000f001010008000000320aa001 7c0a01000000772e08000000320aa001100801000000612e08000000320aa0012c050100 0000612e08000000320aa001140201000000772e08000000320aa001580001000000582e 1c000000fb0220ff0000000000009001010000cc0402001054696d6573204e657720526f 6d616e00d89ff377e19ff3772020f577231c667f040000002d01010004000000f0010000 08000000320a0002e108010000006e2e08000000320a0002fd05010000006e2e0a000000 26060f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc 0102022253797374656d007f231c667f00000a0021008a010000000000000000bcf31200 040000002d01000004000000f0010100030000000000,picscalex100010009000003e201000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02600220101200000026060f001a00ffffffff000010000000c0ffffff b7ffffffe00f0000170200000b00000026060f000c004d61746854797065000060001c00 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00009001010000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff377 2020f577e316663b040000002d01000004000000f001010008000000320aa001d60b0100 0000772e08000000320aa001780901000000612e08000000320aa001140701000000772e 08000000320aa001d20401000000612e08000000320aa001ba0101000000772e08000000 320aa001220001000000592e1c000000fb0220ff0000000000009001010000cc04020010 54696d6573204e657720526f6d616e00d89ff377e19ff3772020f577e316663b04000000 2d01010004000000f001000008000000320a0002490a010000006e2e08000000320a0002 a305010000006e2e0a00000026060f000a00ffffffff0100000000001c000000fb021000 070000000000bc02000000cc0102022253797374656d003be316663b00000a0021008a01 0000000000000000bcf31200040000002d01000004000000f0010100030000000000.Для заданной системы функция Михайлова примет вид:picscalex100010009000003e401000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c024002401f1200000026060f001a00ffffffff000010000000c0ffffff 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6d616e00d89ff377e19ff3772020f5774c1b66fc040000002d01000004000000f0010100 08000000320aa001d81c01000000773008000000320aa001941501000000773008000000 320aa0011a11010000006a3008000000320aa001b40a01000000773008000000320aa001 3802020000006a7708000000320aa00146000100000044771c000000fb0280fe00000000 00009001000000020002001053796d626f6c00009e1b0a94a0f11200d89ff377e19ff377 2020f5774c1b66fc040000002d01010004000000f001000008000000320aa001d8160100 00002d7708000000320aa0012212010000002d7708000000320aa001ad0f010000002b77 08000000320aa001ba0c010000002b7708000000320aa001bc05010000002d7708000000 320aa0018404010000003d770a00000026060f000a00ffffffff0100000000001c000000 fb021000070000000000bc02000000cc0102022253797374656d00fc4c1b66fc00000a00 21008a010000000000000000bcf31200040000002d01000004000000f001010003000000 0000picscalex1000100090000036a01000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 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73204e657720526f6d616e00d89ff377e19ff3772020f57760176636040000002d010100 04000000f001000008000000320aa001400a01000000773008000000320aa00113020100 0000773008000000320aa00158000100000058300a00000026060f000a00ffffffff0100 000000001c000000fb021000070000000000bc02000000cc0102022253797374656d0036 6017663600000a0021008a010000000000000000bcf31200040000002d01000004000000 f0010100030000000000 (9)picscalex1000100090000038301000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02400240111200000026060f001a00ffffffff000010000000c0ffffff b7ffffff00110000f70100000b00000026060f000c004d61746854797065000050001c00 0000fb0220ff0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f5777d1b6665040000002d01000008000000320af4009810 0100000033791c000000fb0280fe0000000000009001000000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f5777d1b6665040000002d0101000400 0000f00100000a000000320aa001d20b0500000030303030380008000000320aa001720b 010000002e3008000000320aa001b20a01000000303008000000320aa001d40602000000 303308000000320aa0017406010000002e3308000000320aa001b4050100000030330800 0000320aa001b60201000000293308000000320aa0012a010100000028331c000000fb02 80fe0000000000009001010000cc0402001054696d6573204e657720526f6d616e00d89f f377e19ff3772020f5777d1b6665040000002d01000004000000f001010008000000320a a001980f01000000773308000000320aa001540801000000773308000000320aa001ba01 01000000773308000000320aa00122000100000059331c000000fb0280fe000000000000 9001000000020002001053796d626f6c0000a6100a88a0f11200d89ff377e19ff3772020 f5777d1b6665040000002d01010004000000f001000008000000320aa001980901000000 2d3308000000320aa001e204010000002d3308000000320aa001a203010000003d330a00 000026060f000a00ffffffff0100000000001c000000fb021000070000000000bc020000 00cc0102022253797374656d00657d1b666500000a0021008a010000000000000000bcf3 1200040000002d01000004000000f0010100030000000000 (10)Графическое изображение функции Михайлова на комплексной плоскости при picscalex1000100090000030501000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02c00140061200000026060f001a00ffffffff000010000000c0ffffff c6ffffff00060000860100000b00000026060f000c004d61746854797065000030001c00 0000fb0280fe0000000000009001000000020002001053796d626f6c007740000000180b 0a4615abc6771eabc677c0a9c97700003000040000002d01000008000000320a6001e104 01000000a57908000000320a6001bc03010000003c7908000000320a6001420101000000 3c791c000000fb0280fe0000000000009001010000000402001054696d6573204e657720 526f6d616e0015abc6771eabc677c0a9c97700003000040000002d01010004000000f001 000008000000320a600173020100000077791c000000fb0280fe00000000000090010000 00000402001054696d6573204e657720526f6d616e0015abc6771eabc677c0a9c9770000 3000040000002d01000004000000f001010008000000320a600134000100000030790a00 000026060f000a00ffffffff0100000000001c000000fb021000070000000000bc020000 00cc0102022253797374656d000000000a0021008a010000000001000000fcf31200aaaf c677040000002d01010004000000f0010000030000000000 называется годографом Михайлова. Для устойчивости системы n-го порядка необходимо и достаточно, чтобы годограф Михайлова начинался на вещественной положительной полуоси и при увеличении частоты до ? проходил последовательно в положительном направлении n квадрантов, нигде не обращаясь в ноль.Используя выражения (9) и (10), заполним таблицу:Таблица 1.3.3w077,625-?X(w)470--?Y(w)0-39,7480-?Построим годограф Михайлова (Рис. 1.3.4):picscalex76010009000003f30b01000d001c00000000000400000003010800050000000 c02f9003901050000000b0200000000050000000902000000001c000000fb02f3ff00000 00000009001000000000400000054696d6573204e657720526f6d616e00d89ff377e19ff 3772020f577a4176678040000002d010000050000000902000000001c000000fb02f4ff0 000000000009001000000000400000054696d6573204e657720526f6d616e00d89ff377e 19ff3772020f577a4176678040000002d010100050000000902000000001c000000fb02f 4ff0000000000009001000000000400000054696d6573204e657720526f6d616e00d89ff 377e19ff3772020f577a4176678040000002d01020005000000090200000000050000000 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5000000090200000000040000002d010000050000002e010000000005000000090200000 000040000002d01040008000000fa0200000000000000000000040000002d01080004000 000f001060007000000fc020000000000000000040000002d01060004000000f00107000 5000000090200000000050000000102ffffff02050000000201020000000400000003010 800050000000c026400640014000000ff0600000600f6020000240001000200000000000 100010002000000010000000100020007000000fc020000ffffff0200000500000028020 7000a0004000000f0010700050000000c02f900390104000000f0010a00030000000000Рис. 1.3.4Полученный годограф начинается на вещественной положительной полуоси, проходит 2 квадранта в отрицательном направлении, таким образом, критерий Михайлова не выполняется, следовательно, система неустойчива.2. Построение области устойчивости в плоскости параметра КрПостроим область устойчивости, используя критерий Гурвица.Запишем характеристическое уравнение замкнутой системы в общем виде:picscalex1000100090000034901000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02600240081200000026060f001a00ffffffff000010000000c0ffffff a6ffffff00080000060200000b00000026060f000c004d61746854797065000070001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f5777a0d6667040000002d01000008000000320a80015007 01000000307908000000320a80013a0501000000297908000000320a8001080401000000 287908000000320a800110000100000031791c000000fb0280fe00000000000090010000 00020002001053796d626f6c00009f0f0a24a0f11200d89ff377e19ff3772020f5777a0d 6667040000002d01010004000000f001000008000000320a80011e06010000003d790800 0000320a8001fa00010000002b791c000000fb0280fe0000000000009001010000cc0402 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0002d01030004000000f001070008000000fa0200000000000000000000040000002d010 700040000002d0101000800000025030200b8001000c6000200040000002d01020004000 0002d01030004000000f001070004000000f001000007000000fc020000ffffff0000000 40000002d0100000400000006010100040000002d0104001700000038050100090052001 00051000f0050000f004f000f004f0010004f00110050001200510011005200100004000 0002d0102000400000006010100040000002d01030008000000fa0200000000000000000 000040000002d010700040000002d01010016000000250309005200100051000f0050000 f004f000f004f0010004f001100500012005100110052001000040000002d01020004000 0002d01030004000000f0010700040000000201010005000000090200000000040000002 e011800040000002d01050009000000320a1f004b000100000031250000040000002d010 6000c00000040092900aa000000000000002100e30002000c00040000002701ffff03000 0000000Рис. 2.1Но заметим, что исходный КР удовлетворяет этому условию, и его изменением устойчивости замкнутой системы добиться невозможно, т. к. в ХУ ЗС (2.3) а2<0, и зависит этот коэффициент от 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f004c0001000000307a0000040000002d010600040000000201010005000000090200000 000040000002e011800040000002d01050009000000320a1f008a0001000000307a00000 40000002d010600040000000201010005000000090200000000040000002e01180004000 0002d01050009000000320a1f009200010000002c7a0000040000002d010600040000000 201010005000000090200000000040000002e011800040000002d0105000f000000320a1 f0097000500000030303138326108000800080008000000040000002d010600040000002 d0100000400000006010100040000002d010400170000003805010009009e0010009d000 f009c000e009b000f009b0010009b0011009c0011009d0011009e001000040000002d010 2000400000006010100040000002d01030008000000fa020000000000000000000004000 0002d010700040000002d01010016000000250309009e0010009d000f009c000e009b000 f009b0010009b0011009c0011009d0011009e001000040000002d010200040000002d010 30004000000f00107000c00000040092900aa000000000000002100db0002000c0004000 0002701ffff030000000000Рис. 2.23. 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00cb1a0e66cb00000a0021008a010000000000000000bcf31200040000002d0100000400 0000f0010100030000000000, отсюдаpicscalex1000100090000039f01000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02e003a0171200000026060f001a00ffffffff000010000000c0ffffff b8ffffff60170000980300000b00000026060f000c004d617468547970650000c0000800 0000fa0200001000000000000000040000002d0100000500000014020002c80205000000 1302000245171c000000fb0280fe0000000000009001010000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f577201c664f040000002d0101000800 0000320a8c03910c01000000777908000000320a6e0139150100000077790a000000320a 6e01e00e0500000061726374670008000000320a6e01f80b0100000077720a000000320a 6e0187050500000061726374670008000000320a6e01d60202000000746708000000320a 600222000100000054671c000000fb0280fe0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577201c664f040000002d01 020004000000f001010008000000320a6e01351602000000292908000000320a6e01cb13 02000000303108000000320a6e016b13010000002e3108000000320a6e01ab1201000000 303108000000320a6e01271201000000283108000000320a6e01f40c0100000029310800 0000320a6e01720a02000000303808000000320a6e01120a010000002e3808000000320a 6e01520901000000303808000000320a6e01ce0801000000283808000000320a6e011f04 0100000028381c000000fb0280fe0000000000009001000000020002001053796d626f6c 00006f1b0a9040f11200d89ff377e19ff3772020f577201c664f040000002d0101000400 0000f001020008000000320a6e01c00d010000002d3808000000320a6e01a90401000000 2d3808000000320a60028a01010000003d380a00000026060f000a00ffffffff01000000 00001c000000fb021000070000000000bc02000000cc0102022253797374656d004f201c 664f00000a0021008a0100000000020000005cf31200040000002d01020004000000f001 0100030000000000 (13)Расчетное выражение для L(w):picscalex1000100090000030b03000004001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02a004802f1200000026060f001a00ffffffff000010000000c0ffffff b8ffffff402f0000580400000b00000026060f000c004d61746854797065000020010800 0000fa0200001000000000000000040000002d010000050000001402a703631505000000 13028b03941508000000fa0200002000000000000000040000002d010100050000001402 930394150500000013024704db15040000002d0100000500000014024704e31505000000 1302440241160500000014024402411605000000130244020f2f05000000140200023b15 05000000130200022f2f1c000000fb0280fe0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577880f66b3040000002d01 020008000000320ae3037d2e01000000297908000000320ae303372d0100000029790800 0000320ae303f12a01000000287908000000320ae303e72801000000317908000000320a e303872801000000287908000000320ae303372701000000297908000000320ae303f125 01000000297908000000320ae3036f2302000000303808000000320ae3030f2301000000 2e3808000000320ae3034f2201000000303808000000320ae303cb210100000028380800 0000320ae303c11f01000000313808000000320ae303611f01000000283808000000320a e303111e01000000293808000000320ae303cb1c01000000293808000000320ae303611a 02000000303108000000320ae303011a010000002e3108000000320ae303411901000000 303108000000320ae303bd1801000000283108000000320ae303b3160100000031310800 0000320ae303531601000000283108000000320a6e017e2102000000343808000000320a 6002d313020000006c6708000000320a6002291202000000323008000000320a6002a910 010000007c3008000000320a6002d20d010000007c3008000000320a6002580c02000000 6c6708000000320a6002ae0a02000000323008000000320a60027606020000006c670800 0000320a6002cc0402000000323008000000320a6002b00201000000293008000000320a 600224010100000028301c000000fb0220ff0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577880f66b3040000002d01 030004000000f001020008000000320a3703d62d01000000323008000000320a37039026 01000000323008000000320a37036a1d0100000032301c000000fb0280fe000000000000 9001010000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020 f577880f66b3040000002d01020004000000f001030008000000320ae303632b02000000 547708000000320ae303f52401000000777708000000320ae303cf1b0100000077770800 0000320a6002500e01000000577708000000320a6002ea07010000004e7708000000320a 6002b40101000000777708000000320a60024600010000004c771c000000fb0220ff0000 000000009001010000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f577880f66b3040000002d01030004000000f001020008000000320ac0029a0f 0100000050771c000000fb0280fe0000000000009001000000020002001053796d626f6c 0000360e0ae4a0f11200d89ff377e19ff3772020f577880f66b3040000002d0102000400 0000f001030008000000320ae303d129010000002b7708000000320ae303f12701000000 d77708000000320ae303ab20010000002b7708000000320ae303cb1e01000000d7770800 0000320ae3039d17010000002b7708000000320a6002f110010000003d7708000000320a 60027609010000003d7708000000320a60029403010000003d770a00000026060f000a00 ffffffff0100000000001c000000fb021000070000000000bc02000000cc010202225379 7374656d00b3880f66b300000a0021008a010000000003000000bcf31200040000002d01 030004000000f0010200030000000000 (14)Подставим найденное выражение Т (13) в функцию L(w) (14):picscalex1000100090000031403000004001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02a004c0341200000026060f001a00ffffffff000010000000c0ffffff b8ffffff80340000580400000b00000026060f000c004d61746854797065000020010800 0000fa0200001000000000000000040000002d010000050000001402a703060805000000 13028b03370808000000fa0200002000000000000000040000002d010100050000001402 9303370805000000130247047e08040000002d0100000500000014024704860805000000 13024402e4080500000014024402e408050000001302440248340500000014020002de07 050000001302000268341c000000fb0280fe0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577b90d6607040000002d01 020008000000320ae303b63301000000297909000000320ae30374310300000029292965 08000000320ae3030a2f02000000303108000000320ae303aa2e010000002e3108000000 320ae303ea2d01000000303108000000320ae303662d01000000283108000000320ae303 322801000000293108000000320ae303b02502000000303808000000320ae30350250100 00002e3808000000320ae303902401000000303808000000320ae3030c24010000002838 08000000320ae3035c1f01000000283808000000320ae303941d01000000283808000000 320ae3038a1b01000000313808000000320ae3032a1b01000000283808000000320ae303 da1901000000293808000000320ae303941801000000293808000000320ae30312160200 0000303808000000320ae303b215010000002e3808000000320ae303f214010000003038 08000000320ae3036e1401000000283808000000320ae303641201000000313808000000 320ae303041201000000283808000000320ae303b41001000000293808000000320ae303 6e0f01000000293808000000320ae303040d02000000303108000000320ae303a40c0100 00002e3108000000320ae303e40b01000000303108000000320ae303600b010000002831 08000000320ae303560901000000313108000000320ae303f60801000000283108000000 320a6e016c1d02000000343808000000320a60027606020000006c6708000000320a6002 cc0402000000323008000000320a6002b00201000000293008000000320a600224010100 000028301c000000fb0220ff0000000000009001000000cc0402001054696d6573204e65 7720526f6d616e00d89ff377e19ff3772020f577b90d6607040000002d01030004000000 f001020008000000320a37030f3301000000323008000000320a37033319010000003230 08000000320a37030d100100000032301c000000fb0280fe0000000000009001010000cc 0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577b90d6607 040000002d01020004000000f001030008000000320ae30378300100000077300a000000 320ae3031e2a0500000061726374670008000000320ae30336270100000077720a000000 320ae303c4200500000061726374670008000000320ae303121e02000000746708000000 320ae303981701000000776708000000320ae303720e01000000776708000000320a6002 b40101000000776708000000320a60024600010000004c671c000000fb0280fe00000000 00009001000000020002001053796d626f6c0000330f0a8ea0f11200d89ff377e19ff377 2020f577b90d6607040000002d01030004000000f001020008000000320ae303fe280100 00002d6708000000320ae303e61f010000002d6708000000320ae303741c010000002b67 08000000320ae303941a01000000d76708000000320ae3034e13010000002b6708000000 320ae3036e1101000000d76708000000320ae303400a010000002b6708000000320a6002 9403010000003d670a00000026060f000a00ffffffff0100000000001c000000fb021000 070000000000bc02000000cc0102022253797374656d0007b90d660700000a0021008a01 0000000002000000bcf31200040000002d01020004000000f0010300030000000000На критической частоте значение функции L(w), исходя из условия обеспечения минимального запаса устойчивости, должно быть равно не менее 5 дБ.picscalex100010009000003fc02000004001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02a00460321200000026060f001a00ffffffff000010000000c0ffffff b8ffffff20320000580400000b00000026060f000c004d61746854797065000020010800 0000fa0200001000000000000000040000002d010000050000001402a703ba0505000000 13028b03eb0508000000fa0200002000000000000000040000002d010100050000001402 9303eb0505000000130247043206040000002d01000005000000140247043a0605000000 130244029806050000001402440298060500000013024402fc3105000000140200029205 05000000130200021c321c000000fb0280fe0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f57762076680040000002d01 020008000000320ae3036a3101000000297909000000320ae303282f0300000029292965 08000000320ae303be2c02000000303108000000320ae3035e2c010000002e3108000000 320ae3039e2b01000000303108000000320ae3031a2b01000000283108000000320ae303 e62501000000293108000000320ae303642302000000303808000000320ae30304230100 00002e3808000000320ae303442201000000303808000000320ae303c021010000002838 08000000320ae303101d01000000283808000000320ae303481b01000000283808000000 320ae3033e1901000000313808000000320ae303de1801000000283808000000320ae303 8e1701000000293808000000320ae303481601000000293808000000320ae303c6130200 0000303808000000320ae3036613010000002e3808000000320ae303a612010000003038 08000000320ae303221201000000283808000000320ae303181001000000313808000000 320ae303b80f01000000283808000000320ae303680e01000000293808000000320ae303 220d01000000293808000000320ae303b80a02000000303108000000320ae303580a0100 00002e3108000000320ae303980901000000303108000000320ae3031409010000002831 08000000320ae3030a0701000000313108000000320ae303aa0601000000283108000000 320a6e01201b02000000343808000000320a60022a04020000006c6708000000320a6002 800202000000323008000000320a60022e000100000035301c000000fb0220ff00000000 00009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff377 2020f57762076680040000002d01030004000000f001020008000000320a3703c3300100 0000323008000000320a3703e71601000000323008000000320a3703c10d010000003230 1c000000fb0280fe0000000000009001010000cc0402001054696d6573204e657720526f 6d616e00d89ff377e19ff3772020f57762076680040000002d01020004000000f0010300 08000000320ae3032c2e0100000077300a000000320ae303d22705000000617263746700 08000000320ae303ea240100000077720a000000320ae303781e05000000617263746700 08000000320ae303c61b02000000746708000000320ae3034c1501000000776708000000 320ae303260c0100000077671c000000fb0280fe00000000000090010000000200020010 53796d626f6c000086070a69a0f11200d89ff377e19ff3772020f5776207668004000000 2d01030004000000f001020008000000320ae303b226010000002d6708000000320ae303 9a1d010000002d6708000000320ae303281a010000002b6708000000320ae30348180100 0000d76708000000320ae3030211010000002b6708000000320ae303220f01000000d767 08000000320ae303f407010000002b6708000000320a60024801010000003d670a000000 26060f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc 0102022253797374656d00806207668000000a0021008a010000000002000000bcf31200 040000002d01020004000000f0010300030000000000Из данного выражения найдем wкрwкр=308,4185, следовательно,Т=0,001198Анализируя данное значение и область устойчивости, найденную в п. 2, можно сделать вывод, что введение корректирующего звена с передаточной функцией picscalex1000100090000037c01000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02e003000c1200000026060f001a00ffffffff000010000000c0ffffff b8ffffffc00b0000980300000b00000026060f000c004d617468547970650000c0000800 0000fa0200001000000000000000040000002d0100000500000014020002e40405000000 13020002b30b1c000000fb0280fe0000000000009001010000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f5778607661d040000002d0101000800 0000320a8c03030b01000000737908000000320a6e01d70901000000737908000000320a 6002200201000000737908000000320a600216000100000057791c000000fb0280fe0000 000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f5778607661d040000002d01020004000000f001010009000000320a8c03fd07 040000003030303808000000320a8c039d07010000002e3008000000320a8c03dd060100 0000303008000000320a8c03d40401000000313008000000320a6e012909010000003130 08000000320a6e01c908010000002e3008000000320a6e01090801000000303008000000 320a6e01000601000000313008000000320a6002c20201000000293008000000320a6002 90010100000028301c000000fb0280fe0000000000009001000000020002001053796d62 6f6c0000df070a75a0f11200d89ff377e19ff3772020f5778607661d040000002d010100 04000000f001020008000000320a8c03be05010000002b3008000000320a6e01ea060100 00002b3008000000320a6002a603010000003d300a00000026060f000a00ffffffff0100 000000001c000000fb021000070000000000bc02000000cc0102022253797374656d001d 8607661d00000a0021008a010000000002000000bcf31200040000002d01020004000000 f0010100030000000000 обеспечит не только устойчивость системы, но и более чем минимальный запас устойчивости по амплитуде.4. Построение и анализ ЛЧХ системы и годографа Найквиста скорректированной системыИспользуя передаточную функцию скорректированной разомкнутой системы (12), запишем характеристическое уравнение скорректированной разомкнутой системы:picscalex100010009000003a601000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c020002a0171200000026060f001a00ffffffff000010000000c0ffffff c6ffffff60170000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f577c50e665c040000002d01000008000000320a6001ae16 01000000307908000000320a6001981401000000297908000000320a6001701202000000 303808000000320a60011012010000002c3808000000320a600156110100000030380800 0000320a60014c0f01000000313808000000320a6001720d02000000292809000000320a 6001ca09040000003030303808000000320a60016a09010000002c3008000000320a6001 b00801000000303008000000320a6001a60601000000313008000000320a6001c8050200 0000292808000000320a6001b80302000000303108000000320a60015803010000002c31 08000000320a60019e0201000000303108000000320a6001940001000000313108000000 320a600134000100000028311c000000fb0280fe00000000000090010000000200020010 53796d626f6c00004f1b0a02a0f11200d89ff377e19ff3772020f577c50e665c04000000 2d01010004000000f001000008000000320a60017c15010000003d3108000000320a6001 3610010000002b3108000000320a60017a0e010000002d3108000000320a600190070100 00002b3108000000320a60017e01010000002b311c000000fb0280fe0000000000009001 010000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577 c50e665c040000002d01000004000000f001010008000000320a6001f613010000007331 08000000320a6001d00c01000000733108000000320a600126050100000073310a000000 26060f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc 0102022253797374656d005cc50e665c00000a0021008a010000000001000000bcf31200 040000002d01010004000000f0010000030000000000 (15)Найдем корни характеристического уравнения (15):picscalex1000100090000039301000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02c006c0061200000026060f001a00ffffffff000010000000c0ffffff acffffff800600006c0600000b00000026060f000c004d61746854797065000080011c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f5771e106621040000002d01000008000000320a0606cd04 01000000357908000000320a06066d04010000002e7908000000320a0606ed0202000000 313208000000320ac0032b06010000002c3209000000320ac003f7030300000031323565 08000000320a7a010606010000002c3209000000320a7a01cd0303000000313030651c00 0000fb0220ff0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f5771e106621040000002d01010004000000f00100000800 0000320a6606e00001000000333008000000320a2004e70001000000323008000000320a da01ce000100000031301c000000fb0280fe000000000000900100000002000200105379 6d626f6c0000d0100a0ba0f11200d89ff377e19ff3772020f5771e106621040000002d01 000004000000f001010008000000320a0606e001010000003d3008000000320ac0032503 010000002d3008000000320ac003ee01010000003d3008000000320a7a01fb0201000000 2d3008000000320a7a01c401010000003d301c000000fb0280fe00000000000090010100 00cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f5771e10 6621040000002d01010004000000f001000008000000320a060640000100000073300800 0000320ac003400001000000733008000000320a7a0140000100000073300a0000002606 0f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc0102 022253797374656d00211e10662100000a0021008a010000000000000000bcf312000400 00002d01000004000000f0010100030000000000Уравнение (15) имеет один правый корень, следовательно, скорректированная разомкнутая система неустойчива.Построим годограф Найквиста. Для этого определим частотную передаточную функцию скорректированной разомкнутой системы и ее действительную и мнимую части.picscalex750100090000039a04000003001c00000000000500000009020000000005000 000020101000000050000000102ffffff00050000002e0118000000050000000b0200000 000050000000c02c008c03e1200000026060f001a00ffffffff000010000000c0ffffffb 7ffffff803e0000770800000b00000026060f000c004d617468547970650000000208000 000fa0200001000000000000000040000002d01000005000000140201027f06050000001 30201029e1e05000000140201024e2005000000130201023f3d050000001402910678010 500000013029106fe201c000000fb0220ff0000000000009001000000cc0402001054696 d6573204e657720526f6d616e00d89ff377e19ff3772020f577de1c664a040000002d010 10008000000320a78076f1f01000000327908000000320a7807301e01000000337908000 000320a7807d80c01000000327908000000320a7807920b01000000327908000000320a5 3054e1f01000000337908000000320a53052f0d01000000327908000000320ae8024f330 1000000327908000000320ae8028a2a0100000033791c000000fb0280fe0000000000009 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0000320ad3038417010000002d3808000000320ad3037110010000002b3808000000320a d3035a06010000002b3808000000320aae01f512010000002d3808000000320aae01670a 010000002d3808000000320aa002be03010000003d380a00000026060f000a00ffffffff 0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d 00305e1f663000000a0021008a010000000001000000bcf31200040000002d0101000400 0000f0010200030000000000 (17)Используя выражения (16) и (17), заполним таблицу:Таблица 4.1w0-328,8237?P-480-0,4850Q0-00Построим годограф Найквиста (Рис. 4.1):picscalex79010009000003e80d000010001c00000000000400000003010800050000000 c02db004001050000000b0200000000050000000902000000001c000000fb02f3ff00000 00000009001000000000400000054696d6573204e657720526f6d616e00d89ff377e19ff 3772020f57737136685040000002d010000050000000902000000001c000000fb02f4ff0 000000000009001000000000400000054696d6573204e657720526f6d616e00d89ff377e 19ff3772020f57737136685040000002d010100050000000902000000001c000000fb02f 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9125e00230005000000090200000000040000002d0100000500000009020000000004000 0002d0101000500000009020000000007000000210501005012c900a3000500000009020 00000000500000009020000000007000000210501007712c900af0005000000090200000 000040000002d010400040000002d01010007000000210501002812c900aa00070000002 10501002912c900b80005000000090200000000040000002d010000050000002e0100000 00005000000090200000000040000002d01040008000000fa02000000000000000000000 40000002d010b0004000000f001090007000000fc020000000000000000040000002d010 90004000000f0010a0005000000090200000000050000000102ffffff020500000002010 20000000400000003010800050000000c026400640014000000ff0600000600f60200002 40001000200000000000100010002000000010000000100020007000000fc020000fffff f0200000500000028020a000d0004000000f0010a00050000000c02d900350104000000f 0010d00030000000000Рис. 4.1Для случая, когда разомкнутая система неустойчива критерий Найквиста звучит следующим образом: для устойчивости замкнутой системы необходимо и достаточно, чтобы годограф Найквиста охватывал особую точку (picscalex100010009000003c900000002001c000000000005000000090200000000050 00000020101000000050000000102ffffff00050000002e0118000000050000000b02000 00000050000000c02a00120021200000026060f001a00ffffffff000010000000c0fffff fc6ffffffe0010000660100000b00000026060f000c004d61746854797065000020001c0 00000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d6 16e0015abc6771eabc677c0a9c97700003000040000002d01000008000000320a60013b0 10100000031791c000000fb0280fe0000000000009001000000020002001053796d626f6 c007740000000660b0a1915abc6771eabc677c0a9c97700003000040000002d010100040 00000f001000008000000320a60013a00010000002d790a00000026060f000a00fffffff f0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656 d000000000a0021008a010000000000000000fcf31200aaafc677040000002d010000040 00000f0010100030000000000; picscalex100010009000003c900000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02000200021200000026060f001a00ffffffff000010000000c0ffffff c6ffffffc0010000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d61 6e0015abc6771eabc677c0a9c97700003000040000002d01000008000000320a6001ff00 0100000030001c000000fb0280fe0000000000009001010000000402001054696d657320 4e657720526f6d616e0015abc6771eabc677c0a9c97700003000040000002d0101000400 0000f001000008000000320a60018200010000006a000a00000026060f000a00ffffffff 0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d 000000000a0021008a010000000000000000fcf31200aaafc677040000002d0100000400 0000f0010100030000000000) в положительном направлении на угол picscalex100010009000003c900000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02c001e0011200000026060f001a00ffffffff000010000000c0ffffff c6ffffffa0010000860100000b00000026060f000c004d61746854797065000030001c00 0000fb0280fe0000000000009001010000020402001053796d626f6c0077400000009211 0a0515abc6771eabc677c0a9c97700003000040000002d01000008000000320a60019a00 0100000070791c000000fb0280fe0000000000009001010000000402001054696d657320 4e657720526f6d616e0015abc6771eabc677c0a9c97700003000040000002d0101000400 0000f001000008000000320a60012f00010000006c790a00000026060f000a00ffffffff 0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d 000000000a0021008a0100000000000000009cf31200aaafc677040000002d0100000400 0000f0010100030000000000, где l – число правых корней характеристического уравнения разомкнутой системы.Число правых корней характеристического уравнения разомкнутой системы равно единице (l=1), полученный годограф охватывает особую точку (-1, j0) на угол lр=р, следовательно, критерий Найквиста выполняется и система устойчива.Построим ЛЧХ 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100010002000000010000000100020007000000fc020000ffffff0200000500000028020 8000b0004000000f0010800050000000c0229022b0204000000f0010b00030000000000Рис. 4.2wср(частота среза) – частота, соответствующая пересечению ЛАЧХ с осью lgw;wкр(критическая частота) – частота, соответствующая пересечению ЛФЧХ уровня –р;Система устойчива, если выполняется условие:wср< wкрДанное условие выполняется, следовательно, система устойчива. Запас устойчивости по амплитуде: Lзап= 5,8 дБЗапас устойчивости по фазе: цзап=0,2 радАналогичный вывод можно сделать по асимптотической ЛАЧХ и ЛФЧХ системы, построенной как сумма отдельных звеньев, входящих в систему.5. Анализ качества системы в переходном режимеОпределим прямые показатели качества, для этого построим переходную характеристику:picscalex100010009000003db01000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c024004c00b1200000026060f001a00ffffffff000010000000c0ffffff b8ffffff800b0000f80300000b00000026060f000c004d617468547970650000e0000800 0000fa0200001000000000000000040000002d0100000500000014022002dc0905000000 13022002ac0a1c000000fb0280fe0000000000009001000000020002001053796d626f6c 000019180a87a0f11200d89ff377e19ff3772020f577fc16669d040000002d0101000800 0000320ada03c40a01000000fe7908000000320ac202c40a01000000fd7908000000320a aa01c40a01000000fc7908000000320ada03fe0501000000ee7908000000320ac202fe05 01000000ed7908000000320aaa01fe0501000000ec7908000000320a8002e60201000000 3d791c000000fb0220ff0000000000009001000000020002001053796d626f6c0000fb17 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500000013027e00a1010500000013027e00a2010500000014027e00a9010500000013027 e00a9010500000013027e00aa010500000013027e00ab010500000013027e00ac0105000 00013027e00ad010500000013027e00ae010500000013027e00af010500000013027e00b 0010500000013027e00b1010500000013027e00b2010500000013027e00b301050000001 3027e00b4010500000013027e00b5010500000013027e00b6010500000013027e00b7010 500000013027e00b8010500000013027e00b9010500000013027e00ba010500000014023 300be01040000002d010a0004000000f0010b00040000002d010900040000002701ffff0 40000002d01070004000000f0010a0004000000f0010900040000002d010500040000002 d01060004000000f001070005000000090200000000040000002d0101000500000009020 0000000040000002d01000005000000090200000000040000002d0101000500000009020 0000000040000002d010000050000002e010000000005000000090200000000040000002 d01030008000000fa0200000000000000000000040000002d01070004000000f00105000 7000000fc020000000000000000040000002d01050004000000f00106000500000009020 0000000050000000102ffffff02050000000201020000000400000003010800050000000 c026400640014000000ff0600000600f6020000240001000200000000000100010002000 000010000000100020007000000fc020000ffffff0200000500000028020600090004000 000f0010600050000000c021201d70104000000f0010900030000000000Рис. 5.1По рисунку (5.1) определим: hmax=0.3; hуст=0.17; h(0)=0, время регулирования на уровне 0.05 (hуст-h(0)).Коридор: [0.95 (hуст-h(0)); 1.05 (hуст-h(0))].Коридор: [0.1615; 0.1785].Время регулирования: tрег= 0,15 с.Перерегулирование равно:picscalex98010009000003c301000003001c00000000000500000009020000000005000 000020101000000050000000102ffffff00050000002e0118000000050000000b0200000 000050000000c02c004c00d1200000026060f001a00ffffffff000010000000c0ffffffa 7ffffff800d0000670400000b00000026060f000c004d617468547970650000100108000 000fa0200001000000000000000040000002d0100000500000014024002f602050000001 302400237091c000000fb0280fe0000000000009001000000000402001054696d6573204 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100000068f108000000320a7f01d6060100000068f108000000320a7f011003010000006 8f11c000000fb0280fe0000000000009001010000020402001053796d626f6c007740000 000d5040ab25bd4f47764d4f4770100000000003000040000002d01020004000000f0010 10008000000320aa0021c000100000073f10a00000026060f000a00ffffffff010000000 0001c000000fb021000070000000000bc02000000cc0102022253797374656d000000000 a000000040000000000010000000100000000003000040000002d01010004000000f0010 200030000000000 (5.3)picscalex900100090000039401000003001c00000000000500000009020000000005000 000020101000000050000000102ffffff00050000002e0118000000050000000b0200000 000050000000c02e003c0121200000026060f001a00ffffffff000010000000c0ffffffb 9ffffff80120000990300000b00000026060f000c004d617468547970650000c00008000 000fa0200001000000000000000040000002d0100000500000014020002c602050000001 3020002db081c000000fb0280fe0000000000009001000000cc0402001054696d6573204 e657720526f6d616e00d89ff377e19ff3772020f5774c1d6609040000002d01010008000 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Используя передаточную функцию скорректированной замкнутой системы (21), запишем частотную передаточную функцию скорректированной замкнутой системы:picscalex710100090000037604000003001c00000000000500000009020000000005000 000020101000000050000000102ffffff00050000002e0118000000050000000b0200000 000050000000c02c00880411200000026060f001a00ffffffff000010000000c0ffffffb affffff404100007a0800000b00000026060f000c004d617468547970650000000208000 000fa0200001000000000000000040000002d010000050000001402fe01cb05050000001 302fe01f120050000001402fe01ab22050000001302fe01e43f05000000140291067d010 5000000130291067e201c000000fb0220ff0000000000009001000000cc0402001054696 d6573204e657720526f6d616e00d89ff377e19ff3772020f577501c662f040000002d010 10008000000320a7b07c71f01000000327908000000320a7b07841e01000000337908000 000320a7b07190d01000000327908000000320a7b07cf0b01000000327908000000320a5 305bd1e01000000337908000000320a53058c0c01000000327908000000320ae802b03e0 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500000013021a018a010500000013021a018b010500000013021a018c010500000013021 a018d010500000013021a018e010500000013021a018f010500000013021a01900105000 00013021a0191010500000013021a0192010500000013021b0192010500000013021b019 3010500000013021b0194010500000013021b0195010500000013021b019601050000001 3021b0197010500000014022d009701040000002d010e0004000000f0010f00040000002 d010d00040000002701ffff040000002d010e0008000000fa02000001000000ff0000020 40000002d010f00050000001402a5000a00050000001302a50023000500000014022d009 701040000002d010e0004000000f0010f00040000002d010d00040000002d010b0004000 000f0010e0004000000f0010d00040000002d010900040000002d010a0004000000f0010 b0005000000090200000000040000002d010100050000000902000000000700000021050 1004e009e000d00050000000902000000000500000009020000000007000000210501007 7009e001b0005000000090200000000040000002d010700040000002d010100070000002 105010028009e001600070000002105010029009e0024000500000009020000000004000 0002d01000005000000090200000000040000002d0101000500000009020000000007000 0002105010077004101e6000500000009020000000005000000090200000000040000002 d010000050000002e010000000005000000090200000000040000002d01070008000000f a0200000000000000000000040000002d010b0004000000f001090007000000fc0200000 00000000000040000002d01090004000000f0010a0005000000090200000000050000000 102ffffff02050000000201020000000400000003010800050000000c026400640014000 000ff0600000600f60200002400010002000000000001000100020000000100000001000 20007000000fc020000ffffff0200000500000028020a000d0004000000f0010a0005000 0000c024901b10104000000f0010d00030000000000Рис. 5.2По рисунку (5.2) определим: picscalex1000100090000033301000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02400240081200000026060f001a00ffffffff000010000000c0ffffff a6ffffff00080000e60100000b00000026060f000c004d61746854797065000060001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f57721186602040000002d01000009000000320a8001bf05 030000003030376508000000320a80015f05010000002e3008000000320a80019f040100 000031301c000000fb0220ff0000000000009001000000cc0402001054696d6573204e65 7720526f6d616e00d89ff377e19ff3772020f57721186602040000002d01010004000000 f001000009000000320ae0017201030000006d6178651c000000fb0280fe000000000000 9001000000020002001053796d626f6c0000b91d0ae3a0f11200d89ff377e19ff3772020 f57721186602040000002d01000004000000f001010008000000320a8001910301000000 3d611c000000fb0280fe0000000000009001010000cc0402001054696d6573204e657720 526f6d616e00d89ff377e19ff3772020f57721186602040000002d01010004000000f001 000008000000320a80014c00010000004e610a00000026060f000a00ffffffff01000000 00001c000000fb021000070000000000bc02000000cc0102022253797374656d00022118 660200000a0021008a010000000000000000bcf31200040000002d01000004000000f001 0100030000000000; picscalex1000100090000031d01000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c020002c0071200000026060f001a00ffffffff000010000000c0ffffff c6ffffff80070000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f577a01d6670040000002d01000008000000320a6001f205 02000000313708000000320a60019205010000002e3708000000320a6001d20401000000 303708000000320a6001bc0201000000293708000000320a6001fc010100000030370800 0000320a600178010100000028371c000000fb0280fe0000000000009001000000020002 001053796d626f6c0000ea1c0ae1a0f11200d89ff377e19ff3772020f577a01d66700400 00002d01010004000000f001000008000000320a6001a003010000003d371c000000fb02 80fe0000000000009001010000cc0402001054696d6573204e657720526f6d616e00d89f f377e19ff3772020f577a01d6670040000002d01000004000000f001010008000000320a 60014c00010000004e370a00000026060f000a00ffffffff0100000000001c000000fb02 1000070000000000bc02000000cc0102022253797374656d0070a01d667000000a002100 8a010000000001000000bcf31200040000002d01010004000000f0010000030000000000 .Показатель колебательности M есть отношение максимальной ординаты амплитудно-частотной характеристики замкнутой системы к начальной ординате:picscalex1000100090000035801000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c022004c0061200000026060f001a00ffffffff000010000000c0ffffff bdffffff80060000dd0300000b00000026060f000c004d617468547970650000e0000800 0000fa0200001000000000000000040000002d0100000500000014020002400305000000 130200026d061c000000fb0280fe0000000000009001000000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f57773186628040000002d0101000800 0000320a8b03db0501000000297908000000320a8b031b0501000000307908000000320a 8b0397040100000028791c000000fb0220ff0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f57773186628040000002d01 020004000000f001010009000000320ac9019804030000006d6178651c000000fb0280fe 0000000000009001010000cc0402001054696d6573204e657720526f6d616e00d89ff377 e19ff3772020f57773186628040000002d01010004000000f001020008000000320a8b03 6c03010000004e6108000000320a69017903010000004e6108000000320a600240000100 00004d611c000000fb0280fe0000000000009001000000020002001053796d626f6c0000 dc1a0ab7a0f11200d89ff377e19ff3772020f57773186628040000002d01020004000000 f001010008000000320a60021102010000003d610a00000026060f000a00ffffffff0100 000000001c000000fb021000070000000000bc02000000cc0102022253797374656d0028 7318662800000a0021008a010000000001000000bcf31200040000002d01010004000000 f0010200030000000000picscalex1000100090000030601000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02c001a0071200000026060f001a00ffffffff000010000000c0ffffff c6ffffff60070000860100000b00000026060f000c004d61746854797065000030001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f5777516664b040000002d01000009000000320a60016c04 040000003932333508000000320a60010c04010000002e3208000000320a60014c030100 000035321c000000fb0280fe0000000000009001000000020002001053796d626f6c0000 381e0ad140f11200d89ff377e19ff3772020f5777516664b040000002d01010004000000 f001000008000000320a60012002010000003d321c000000fb0280fe0000000000009001 010000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577 7516664b040000002d01000004000000f001010008000000320a60014000010000004d32 0a00000026060f000a00ffffffff0100000000001c000000fb021000070000000000bc02 000000cc0102022253797374656d004b7516664b00000a0021008a010000000001000000 5cf31200040000002d01010004000000f0010000030000000000Определим запасы устойчивости системы.Найдем критическую частоту – частоту, на которой значение ц(w) равняется –р.picscalex1000100090000036302000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02800400201200000026060f001a00ffffffff000010000000c0ffffff a9ffffffc01f0000290400000b00000026060f000c004d617468547970650000f0000800 0000fa0200001000000000000000040000002d0100000500000014024002e40b05000000 13024002271f1c000000fb0280fe0000000000009001000000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f5771d17661f040000002d0101000800 0000320aa002391f01000000297908000000320ad303591b01000000297908000000320a d303b71a0100000031790a000000320ad3032f130600000030303038353608000000320a d303cf12010000002e3008000000320ad3030f1201000000303008000000320ad303b310 01000000283008000000320ad303390f02000000343808000000320aae01951e01000000 293809000000320aae01951b040000003036393208000000320aae01351b010000002e36 08000000320aae01751a0100000030360b000000320aae01441108000000303030303030 363408000000320aae01e410010000002e3008000000320aae0124100100000030300800 0000320aae01a00f01000000283008000000320aae011e0d02000000343808000000320a a002540b01000000283808000000320aa002b00201000000293808000000320aa0022401 0100000028381c000000fb0220ff0000000000009001000000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f5771d17661f040000002d0102000400 0000f001010008000000320a2703d21801000000323808000000320a0201671801000000 32381c000000fb0280fe0000000000009001000000020002001053796d626f6c00005a16 0ab3b4e11200d89ff377e19ff3772020f5771d17661f040000002d01010004000000f001 020008000000320ad303c119010000002d3808000000320ad3033d11010000002d380800 0000320aae015619010000002b3808000000320aae01fe0b010000002d3808000000320a a002e806010000002b3808000000320aa002cc04010000002d3808000000320aa0029403 010000003d381c000000fb0280fe0000000000009001010000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f5771d17661f040000002d0102000400 0000f001010008000000320ad303bb1701000000773808000000320aae01501701000000 773808000000320aae01a40e0100000077380a000000320aa0020d080500000061726374 673008000000320aa002b4010100000077721c000000fb0280fe00000000000090010100 00020402001053796d626f6c00005a160ab4b4e11200d89ff377e19ff3772020f5771d17 661f040000002d01010004000000f001020008000000320aa00292050100000070720800 0000320aa0021c00010000006a720a00000026060f000a00ffffffff0100000000001c00 0000fb021000070000000000bc02000000cc0102022253797374656d001f1d17661f0000 0a0021008a010000000002000000d0e31200040000002d01020004000000f00101000300 00000000 (23)picscalex1000100090000035302000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c028004201f1200000026060f001a00ffffffff000010000000c0ffffff a9ffffffe01e0000290400000b00000026060f000c004d617468547970650000f0000800 0000fa0200001000000000000000040000002d0100000500000014024002000b05000000 13024002431e1c000000fb0280fe0000000000009001000000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f577401c66c9040000002d0101000800 0000320aa002551e01000000297908000000320ad303751a01000000297908000000320a d303d3190100000031790a000000320ad3034b120600000030303038353608000000320a d303eb11010000002e3008000000320ad3032b1101000000303008000000320ad303cf0f 01000000283008000000320ad303550e02000000343808000000320aae01b11d01000000 293809000000320aae01b11a040000003036393208000000320aae01511a010000002e36 08000000320aae0191190100000030360b000000320aae01601008000000303030303030 363408000000320aae010010010000002e3008000000320aae01400f0100000030300800 0000320aae01bc0e01000000283008000000320aae013a0c02000000343808000000320a a002700a0100000028381c000000fb0220ff0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577401c66c9040000002d01 020004000000f001010008000000320a2703ee1701000000323808000000320a02018317 0100000032381c000000fb0280fe0000000000009001000000020002001053796d626f6c 00006d1e0a79a0f11200d89ff377e19ff3772020f577401c66c9040000002d0101000400 0000f001020008000000320ad303dd18010000002d3808000000320ad303591001000000 2d3808000000320aae017218010000002b3808000000320aae011a0b010000002d380800 0000320aa0020406010000002b3808000000320aa002e803010000002d3808000000320a a002b002010000003d3808000000320aa0023a00010000002d381c000000fb0280fe0000 000000009001010000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f577401c66c9040000002d01020004000000f001010008000000320ad303d716 01000000773808000000320aae016c1601000000773808000000320aae01c00d01000000 77380a000000320aa0022907050000006172637467301c000000fb0280fe000000000000 9001010000020402001053796d626f6c00006d1e0a7aa0f11200d89ff377e19ff3772020 f577401c66c9040000002d01010004000000f001020008000000320aa002ae0401000000 707208000000320aa00242010100000070720a00000026060f000a00ffffffff01000000 00001c000000fb021000070000000000bc02000000cc0102022253797374656d00c9401c 66c900000a0021008a010000000002000000bcf31200040000002d01020004000000f001 0100030000000000wкр=328,824Рассчитаем запас по амплитуде:picscalex1000100090000038802000004001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02a004e0231200000026060f001a00ffffffff000010000000c0ffffff b8ffffffa0230000580400000b00000026060f000c004d61746854797065000020010800 0000fa0200001000000000000000040000002d010000050000001402a703040805000000 13028b03350808000000fa0200002000000000000000040000002d010100050000001402 9303350805000000130247047c08040000002d0100000500000014024704840805000000 13024402e2080500000014024402e208050000001302440269230500000014020002dc07 050000001302000289231c000000fb0280fe0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577680a6671040000002d01 020008000000320ae303d72201000000297908000000320ae30391210100000029790800 0000320ae3030f1f02000000303808000000320ae303af1e010000002e3808000000320a e303ef1d01000000303808000000320ae3036b1d01000000283808000000320ae303621b 01000000313808000000320ae303841a02000000292808000000320ae3033e1901000000 292809000000320ae3033c15040000003030303808000000320ae303dc14010000002e30 08000000320ae3031c1401000000303008000000320ae303981301000000283008000000 320ae3038f1101000000313008000000320ae303b11002000000292808000000320ae303 6b0f01000000292808000000320ae303010d02000000303108000000320ae303a10c0100 00002e3108000000320ae303e10b01000000303108000000320ae3035d0b010000002831 08000000320ae303540901000000313108000000320ae303f40801000000283108000000 320a6e01fb1402000000343808000000320a60027506020000006c6708000000320a6002 cb0402000000323008000000320a6002af0201000000293008000000320a600223010100 000028301c000000fb0220ff0000000000009001000000cc0402001054696d6573204e65 7720526f6d616e00d89ff377e19ff3772020f577680a6671040000002d01030004000000 f001020008000000320a3703302201000000323008000000320a3703dd19010000003230 08000000320a37030a100100000032301c000000fb0280fe0000000000009001010000cc 0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577680a6671 040000002d01020004000000f001030008000000320ae303952001000000773008000000 320ae303421801000000773008000000320ae3036f0e01000000773008000000320a6002 b30101000000773008000000320a60024600010000004c301c000000fb0280fe00000000 00009001000000020002001053796d626f6c0000f31f0a8aa0f11200d89ff377e19ff377 2020f577680a6671040000002d01030004000000f001020008000000320ae3034c1c0100 00002b3008000000320ae3037912010000002b3008000000320ae3033e0a010000002b30 08000000320a60029303010000003d300a00000026060f000a00ffffffff010000000000 1c000000fb021000070000000000bc02000000cc0102022253797374656d0071680a6671 00000a0021008a010000000002000000bcf31200040000002d01020004000000f0010300 030000000000 (24)picscalex1000100090000033801000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c020002200d1200000026060f001a00ffffffff000010000000c0ffffff c6ffffffe00c0000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f577b71f66e4040000002d01000009000000320a6001900a 030000003739376508000000320a6001300a010000002e3908000000320a600170090100 0000353908000000320a6001820601000000293909000000320a60014204030000003832 346508000000320a6001e203010000002e3209000000320a6001a2010300000033323865 08000000320a600124010100000028321c000000fb0280fe000000000000900100000002 0002001053796d626f6c0000821d0a74a0f11200d89ff377e19ff3772020f577b71f66e4 040000002d01010004000000f001000008000000320a60019e08010000002d3208000000 320a60016607010000003d321c000000fb0280fe0000000000009001010000cc04020010 54696d6573204e657720526f6d616e00d89ff377e19ff3772020f577b71f66e404000000 2d01000004000000f001010008000000320a60014600010000004c320a00000026060f00 0a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc01020222 53797374656d00e4b71f66e400000a0021008a010000000001000000bcf3120004000000 2d01010004000000f0010000030000000000Запас по амплитуде: Lзап= 5,797 дБНайдем частоту среза – частоту, на которой значение L(w) равняется 0, используя выражение (24):picscalex1000100090000037002000004001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02a004a0211200000026060f001a00ffffffff000010000000c0ffffff b8ffffff60210000580400000b00000026060f000c004d61746854797065000020010800 0000fa0200001000000000000000040000002d010000050000001402a703740305000000 13028b03a50308000000fa0200002000000000000000040000002d010100050000001402 9303a5030500000013024704ec03040000002d0100000500000014024704f40305000000 130244025204050000001402440252040500000013024402dc1e05000000140200024c03 0500000013020002fc1e1c000000fb0280fe0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577fe17666f040000002d01 020008000000320a6002a02001000000307908000000320ae3034a1e0100000029790800 0000320ae303041d01000000297908000000320ae303821a02000000303808000000320a e303221a010000002e3808000000320ae303621901000000303808000000320ae303de18 01000000283808000000320ae303d41601000000313808000000320ae303f61502000000 292808000000320ae303b01401000000292809000000320ae303ae100400000030303038 08000000320ae3034e10010000002e3008000000320ae3038e0f01000000303008000000 320ae3030a0f01000000283008000000320ae303000d01000000313008000000320ae303 220c02000000292808000000320ae303dc0a01000000292808000000320ae30372080200 0000303108000000320ae3031208010000002e3108000000320ae3035207010000003031 08000000320ae303ce0601000000283108000000320ae303c40401000000313108000000 320ae303640401000000283108000000320a6e016d1002000000343808000000320a6002 e401020000006c6708000000320a60023a000200000032301c000000fb0220ff00000000 00009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff377 2020f577fe17666f040000002d01030004000000f001020008000000320a3703a31d0100 0000323008000000320a37034f1501000000323008000000320a37037b0b010000003230 1c000000fb0280fe0000000000009001000000020002001053796d626f6c0000191d0a0a 40f11200d89ff377e19ff3772020f577fe17666f040000002d01020004000000f0010300 08000000320a60026e1f010000003d3008000000320ae303be17010000002b3008000000 320ae303ea0d010000002b3008000000320ae303ae05010000002b301c000000fb0280fe 0000000000009001010000cc0402001054696d6573204e657720526f6d616e00d89ff377 e19ff3772020f577fe17666f040000002d01030004000000f001020008000000320ae303 081c01000000773008000000320ae303b41301000000773008000000320ae303e0090100 000077300a00000026060f000a00ffffffff0100000000001c000000fb02100007000000 0000bc02000000cc0102022253797374656d006ffe17666f00000a0021008a0100000000 020000005cf31200040000002d01020004000000f0010300030000000000wср=232,624Рассчитаем запас по фазе, используя выражение (23):picscalex1000100090000034a01000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02000240091200000026060f001a00ffffffff000010000000c0ffffff c6ffffff00090000c60100000b00000026060f000c004d61746854797065000050001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f577e01f66d3040000002d01000009000000320a6001be06 030000003136386508000000320a60015e06010000002e3608000000320a60019e050100 0000303608000000320a6001b00201000000293608000000320a60012401010000002836 1c000000fb0280fe0000000000009001000000020002001053796d626f6c0000eb1e0ac4 a0f11200d89ff377e19ff3772020f577e01f66d3040000002d01010004000000f0010000 08000000320a6001cc04010000002d3608000000320a60019403010000003d361c000000 fb0280fe0000000000009001010000cc0402001054696d6573204e657720526f6d616e00 d89ff377e19ff3772020f577e01f66d3040000002d01000004000000f001010008000000 320a6001b4010100000077361c000000fb0280fe00000000000090010100000204020010 53796d626f6c0000eb1e0ac5a0f11200d89ff377e19ff3772020f577e01f66d304000000 2d01010004000000f001000008000000320a60011c00010000006a360a00000026060f00 0a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc01020222 53797374656d00d3e01f66d300000a0021008a010000000000000000bcf3120004000000 2d01000004000000f0010100030000000000Запас по фазе: цзап=0,168 рад.6. Анализ качества системы в установившемся режимеУстановившаяся ошибка системы равна:picscalex1000100090000034c01000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c026002600c1200000026060f001a00ffffffff000010000000c0ffffff a6ffffff200c0000060200000b00000026060f000c004d61746854797065000070001c00 0000fb0220ff0000000000009001010000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f577931c6699040000002d01000009000000320ae0013a0a 04000000f3f1f26609000000320ae0012a0504000000f3f1f25809000000320ae0011601 03000000f3f1f2581c000000fb0280fe0000000000009001010000020402001053796d62 6f6c0000a71c0acca0f11200d89ff377e19ff3772020f577931c6699040000002d010100 04000000f001000008000000320a80014c090100000065f108000000320a80013c040100 000065f108000000320a800128000100000065f11c000000fb0280fe0000000000009001 000000020002001053796d626f6c0000521f0a64a0f11200d89ff377e19ff3772020f577 931c6699040000002d01000004000000f001010008000000320a80013908010000002bf1 08000000320a80011603010000003df11c000000fb0220ff0000000000009001000000cc 0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577931c6699 040000002d01010004000000f001000008000000320ae0014a070100000030f10a000000 26060f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc 0102022253797374656d0099931c669900000a0021008a010000000000000000bcf31200 040000002d01000004000000f0010100030000000000 (25)еустХо=С0Х0(t)+ С1Х'0(t)+…еуст f =С0F0(t)+ С1F'0(t)+…Так как в заданном случае задающее и возмущающее воздействия – константы, необходимо найти лишь первые коэффициенты функций ошибок.Запишем передаточную функцию замкнутой системы по ошибке по задающему воздействию:picscalex100010009000003b202000004001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c022006001f1200000026060f001a00ffffffff000010000000c0ffffff b8ffffffc01e0000d80500000b00000026060f000c004d617468547970650000e0010800 0000fa0200000800000000000000040000002d010000050000001402f403760905000000 1302f4038f1e08000000fa0200001000000000000000040000002d010100050000001402 000271070500000013020002af1e1c000000fb0280fe0000000000009001000000cc0402 001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577dd1e66f20400 00002d01020008000000320a8005fd1d01000000297908000000320a8005d51b02000000 303808000000320a8005751b010000002e3808000000320a8005b51a0100000030380800 0000320a8005ac1801000000313808000000320a8005d21602000000292809000000320a 80052a13040000003030303808000000320a8005ca12010000002e3008000000320a8005 0a1201000000303008000000320a8005011001000000313008000000320a8005230f0200 0000292808000000320a8005130d02000000303108000000320a8005b30c010000002e31 08000000320a8005f30b01000000303108000000320a8005ea0901000000313108000000 320a80058a0901000000283108000000320a62034b1302000000343808000000320a5404 610701000000313808000000320a6e01b01201000000313808000000320a60024f050100 0000293808000000320a60021d040100000028381c000000fb0260ff0000000000009001 000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577 dd1e66f2040000002d01030004000000f001020008000000320a72038103010000003038 1c000000fb0280fe0000000000009001010000cc0402001054696d6573204e657720526f 6d616e00d89ff377e19ff3772020f577dd1e66f2040000002d01020004000000f0010300 08000000320a80055b1d01000000733808000000320a8005301601000000733808000000 320a8005810e01000000733808000000320a6002ad040100000073381c000000fb02c0fd 0000000000009001010000cc0402001054696d6573204e657720526f6d616e00d89ff377 e19ff3772020f577dd1e66f2040000002d01030004000000f001020008000000320a9202 130001000000d4381c000000fb0220ff0000000000009001010000cc0402001054696d65 73204e657720526f6d616e00d89ff377e19ff3772020f577dd1e66f2040000002d010200 04000000f001030008000000320a3a03dd020100000058381c000000fb0280fe00000000 00009001000000020002001053796d626f6c0000751e0a6da0f11200d89ff377e19ff377 2020f577dd1e66f2040000002d01030004000000f001020008000000320a800596190100 00002b3808000000320a8005da17010000002d3808000000320a8005eb10010000002b38 08000000320a8005d40a010000002b3808000000320a54044b08010000002b3808000000 320a60023306010000003d381c000000fb0280fe00000000000090010100000204020010 53796d626f6c000001070a93a0f11200d89ff377e19ff3772020f577dd1e66f204000000 2d01020004000000f001030008000000320a2203bc010100000065380a00000026060f00 0a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc01020222 53797374656d00f2dd1e66f200000a0021008a010000000003000000bcf3120004000000 2d01030004000000f0010200030000000000picscalex1000100090000035d02000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c020004c00f1200000026060f001a00ffffffff000010000000c0ffffff b8ffffff800f0000b80300000b00000026060f000c004d617468547970650000d0000800 0000fa0200001000000000000000040000002d0100000500000014026c002a0905000000 130294032a090500000014020002ca0d0500000013020002780f1c000000fb0280fe0000 000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f577b41f6608040000002d01010008000000320a8c03e40d0200000034370800 0000320a6e01410e01000000313708000000320a6002920801000000293708000000320a 600260070100000028371c000000fb0220ff0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577b41f6608040000002d01 020004000000f001010008000000320aa6037d0a01000000303708000000320ac0023e01 0100000030371c000000fb0260ff0000000000009001000000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f577b41f6608040000002d0101000400 0000f001020008000000320a7203c4060100000030371c000000fb0280fe000000000000 9001000000020002001053796d626f6c0000dd160a8ca0f11200d89ff377e19ff3772020 f577b41f6608040000002d01020004000000f001010008000000320a6002bc0c01000000 2d3708000000320a6002840b010000003d3708000000320a60024502010000003d371c00 0000fb0220ff0000000000009001000000020002001053796d626f6c00007c1e0a76a0f1 1200d89ff377e19ff3772020f577b41f6608040000002d01010004000000f00102000800 0000320aa603f909010000003d371c000000fb0220ff0000000000009001010000cc0402 001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f577b41f66080400 00002d01020004000000f001010008000000320aa6036a0901000000533708000000320a 3a0320060100000058371c000000fb0280fe0000000000009001010000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577b41f6608040000002d01 010004000000f001020008000000320a6002f00701000000733708000000320a60022e00 0100000043371c000000fb02c0fd0000000000009001010000cc0402001054696d657320 4e657720526f6d616e00d89ff377e19ff3772020f577b41f6608040000002d0102000400 0000f001010008000000320a9202560301000000d4371c000000fb0280fe000000000000 9001010000020402001053796d626f6c00007c1e0a78a0f11200d89ff377e19ff3772020 f577b41f6608040000002d01010004000000f001020008000000320a2203ff0401000000 65370a00000026060f000a00ffffffff0100000000001c000000fb021000070000000000 bc02000000cc0102022253797374656d0008b41f660800000a0021008a01000000000200 0000bcf31200040000002d01020004000000f0010100030000000000Установившаяся ошибка системы по задающему воздействию:picscalex100010009000003ae01000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02e003800d1200000026060f001a00ffffffff000010000000c0ffffff b8ffffff400d0000980300000b00000026060f000c004d617468547970650000c0000800 0000fa0200001000000000000000040000002d01000005000000140200027c0505000000 130200023c070500000014020002740b0500000013020002220d1c000000fb0280fe0000 000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f5775c1e666b040000002d01010008000000320a8c038e0b0200000034370800 0000320a6e01ee0b01000000353708000000320a6002140801000000353708000000320a 8c039f0502000000343708000000320a6e018c060100000031371c000000fb0220ff0000 000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f5775c1e666b040000002d01020004000000f001010008000000320ac0023703 0100000030371c000000fb0280fe0000000000009001000000020002001053796d626f6c 000073180a31a0f11200d89ff377e19ff3772020f5775c1e666b040000002d0101000400 0000f001020008000000320a6002660a010000002d3708000000320a60022e0901000000 3d3708000000320a6002840701000000d73708000000320a6e019605010000002d370800 0000320a60023e04010000003d371c000000fb0220ff0000000000009001010000cc0402 001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f5775c1e666b0400 00002d01020004000000f001010009000000320ac002170104000000f3f1f2581c000000 fb0280fe0000000000009001010000020402001053796d626f6c000073180a32a0f11200 d89ff377e19ff3772020f5775c1e666b040000002d01010004000000f001020008000000 320a600228000100000065f10a00000026060f000a00ffffffff0100000000001c000000 fb021000070000000000bc02000000cc0102022253797374656d006b5c1e666b00000a00 21008a010000000002000000bcf31200040000002d01020004000000f001010003000000 0000Запишем передаточную функцию замкнутой системы по ошибке по возмущению:picscalex820100090000032404000004001c00000000000500000009020000000005000 000020101000000050000000102ffffff00050000002e0118000000050000000b0200000 000050000000c02000820381200000026060f001a00ffffffff000010000000c0ffffffa 6ffffffe0370000a60700000b00000026060f000c004d617468547970650000e00108000 000fa0200000800000000000000040000002d0100000500000014021202560f050000001 3021202c515050000001402d405f308050000001302d4050c1e08000000fa02000010000 00000000000040000002d010100050000001402e003ee06050000001302e0032c1e05000 0001402e003dc1f050000001302e003dc371c000000fb0280fe000000000000900100000 0cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f57771166 68f040000002d01020008000000320a6c054a3701000000297908000000320a6c0522350 2000000303808000000320a6c05c234010000002e3808000000320a6c050234010000003 03808000000320a6c05f93101000000313808000000320a6c051f3002000000292809000 000320a6c05772c040000003030303808000000320a6c05172c010000002e30080000003 20a6c05572b01000000303008000000320a6c054e2901000000313008000000320a6c057 02802000000292808000000320a6c05602602000000303108000000320a6c05002601000 0002e3108000000320a6c05402501000000303108000000320a6c0537230100000031310 8000000320a6c05d72201000000283108000000320a6c05f61f020000003438080000003 20a4e03ef3201000000293809000000320a4e03472f040000003030303808000000320a4 e03e72e010000002e3008000000320a4e03272e01000000303008000000320a4e031e2c0 1000000313008000000320a4e03402b02000000292808000000320a4e033029020000003 03108000000320a4e03d028010000002e3108000000320a4e03102801000000303108000 000320a4e03072601000000313108000000320a4e03a72501000000283108000000320a4 e03272402000000313208000000320a60077a1d01000000293208000000320a6007521b0 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70000000000bc02000000cc0102022253797374656d00a36f1866a300000a0021008a010 0000000010000005cf31200040000002d01010004000000f0010200030000000000Установившаяся ошибка системы по задающему воздействию:picscalex1000100090000037201000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02e003600b1200000026060f001a00ffffffff000010000000c0ffffff b8ffffff200b0000980300000b00000026060f000c004d617468547970650000c0000800 0000fa0200001000000000000000040000002d0100000500000014020002c40405000000 13020002720605000000140200026c0905000000130200021a0b1c000000fb0280fe0000 000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f577fe17662f040000002d01010008000000320a8c0386090200000034370800 0000320a6e01740902000000313208000000320a60022c0701000000313208000000320a 8c03de0402000000343708000000320a6e01cc040200000031321c000000fb0280fe0000 000000009001000000020002001053796d626f6c0000b71c0a49b4e11200d89ff377e19f f3772020f577fe17662f040000002d01020004000000f001010008000000320a60022e08 010000003d3208000000320a6002ba0601000000d73208000000320a6002860301000000 3d321c000000fb0220ff0000000000009001010000cc0402001054696d6573204e657720 526f6d616e00d89ff377e19ff3772020f577fe17662f040000002d01010004000000f001 020009000000320ac002160104000000f3f1f2661c000000fb0280fe0000000000009001 010000020402001053796d626f6c0000b71c0a4ab4e11200d89ff377e19ff3772020f577 fe17662f040000002d01020004000000f001010008000000320a600228000100000065f1 0a00000026060f000a00ffffffff0100000000001c000000fb021000070000000000bc02 000000cc0102022253797374656d002ffe17662f00000a0021008a010000000001000000 d0e31200040000002d01010004000000f0010200030000000000Рассчитаем установившуюся ошибку системы, используя выражение (25):picscalex1000100090000038401000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02e003e00c1200000026060f001a00ffffffff000010000000c0ffffff b8ffffffa00c0000980300000b00000026060f000c004d617468547970650000c0000800 0000fa0200001000000000000000040000002d0100000500000014020002550405000000 13020002030605000000140200028307050000001302000231090500000014020002e10a 05000000130200028f0c1c000000fb0280fe0000000000009001000000cc040200105469 6d6573204e657720526f6d616e00d89ff377e19ff3772020f577e71d662f040000002d01 010008000000320a8c03fb0a02000000343708000000320a6e01580b0100000037370800 0000320a8c039d0702000000343708000000320a6e01fd0701000000353708000000320a 8c036f0402000000343708000000320a6e015d040200000031321c000000fb0280fe0000 000000009001000000020002001053796d626f6c0000b31f0a6ca0f11200d89ff377e19f f3772020f577e71d662f040000002d01020004000000f001010008000000320a6002a309 010000003d3208000000320a60025d06010000002d3208000000320a6002170301000000 3d321c000000fb0220ff0000000000009001010000cc0402001054696d6573204e657720 526f6d616e00d89ff377e19ff3772020f577e71d662f040000002d01010004000000f001 020009000000320ac002170103000000f3f1f2651c000000fb0280fe0000000000009001 010000020402001053796d626f6c0000b31f0a6da0f11200d89ff377e19ff3772020f577 e71d662f040000002d01020004000000f001010008000000320a600228000100000065f1 0a00000026060f000a00ffffffff0100000000001c000000fb021000070000000000bc02 000000cc0102022253797374656d002fe71d662f00000a0021008a010000000001000000 bcf31200040000002d01010004000000f0010200030000000000Приведем размерность установившейся ошибки к размерности входного сигнала:picscalex1000100090000033901000002001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02400220071200000026060f001a00ffffffff000010000000c0ffffff a6ffffffe0060000e60100000b00000026060f000c004d61746854797065000060001c00 0000fb0280fe0000000000009001000000cc0402001054696d6573204e657720526f6d61 6e00d89ff377e19ff3772020f5771a1866fe040000002d01000008000000320a80011806 01000000367908000000320a8001020401000000297908000000320a8001d00201000000 28791c000000fb0280fe0000000000009001000000020002001053796d626f6c0000b919 0ab240f11200d89ff377e19ff3772020f5771a1866fe040000002d01010004000000f001 000008000000320a8001e604010000003d791c000000fb0280fe00000000000090010100 00cc0402001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f5771a18 66fe040000002d01000004000000f001010008000000320a800160030100000073790800 0000320a800116000100000057791c000000fb0220ff0000000000009001010000cc0402 001054696d6573204e657720526f6d616e00d89ff377e19ff3772020f5771a1866fe0400 00002d01010004000000f001000008000000320ae001530102000000ced10a0000002606 0f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc0102 022253797374656d00fe1a1866fe00000a0021008a0100000000000000005cf312000400 00002d01000004000000f0010100030000000000;picscalex1000100090000037301000003001c0000000000050000000902000000000500 0000020101000000050000000102ffffff00050000002e0118000000050000000b020000 0000050000000c02e003800c1200000026060f001a00ffffffff000010000000c0ffffff b8ffffff400c0000980300000b00000026060f000c004d617468547970650000c0000800 0000fa0200001000000000000000040000002d0100000500000014020002550405000000 1302000203060500000014020002cf090500000013020002370c1c000000fb0280fe0000 000000009001000000cc0402001054696d6573204e657720526f6d616e00d89ff377e19f f3772020f577710b6623040000002d01010009000000320a8c03e9090300000032383265 08000000320a6e01a30a01000000373808000000320a6002710701000000363808000000 320a8c036f0402000000343708000000320a6e01cc040100000037371c000000fb0280fe 0000000000009001000000020002001053796d626f6c00001d110a5c40f11200d89ff377 e19ff3772020f577710b6623040000002d01020004000000f001010008000000320a6002 9108010000003d3708000000320a60025d0601000000b83708000000320a600217030100 00003d371c000000fb0220ff0000000000009001010000cc0402001054696d6573204e65 7720526f6d616e00d89ff377e19ff3772020f577710b6623040000002d01010004000000 f001020009000000320ac002170103000000f3f1f2651c000000fb0280fe000000000000 9001010000020402001053796d626f6c00001d110a5d40f11200d89ff377e19ff3772020 f577710b6623040000002d01020004000000f001010008000000320a6002280001000000 65f10a00000026060f000a00ffffffff0100000000001c000000fb021000070000000000 bc02000000cc0102022253797374656d0023710b662300000a0021008a01000000000100 00005cf31200040000002d01010004000000f0010200030000000000Система является статической как относительно возмущения, так и относительно задающего воздействия, установившаяся ошибка системы равна 7/282.

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МИНИСТЕРСТВО ВЫСШЕГО И СРЕДНЕГО

СПЕЦИАЛЬНОГО ОБРАЗОВАНИЯ РФ

НОВОСИБИРСКИЙ ГОСУДАРСТВЕННЫЙ ТЕХНИЧЕСКИЙ УНИВЕРСИТЕТ

Расчетно-графическая работа №1

По курсу “Теория автоматического управления”

Студент: Стариков Д.А.

Группа: АС-513

Преподаватель: кандидат технических наук, доцент

Кошкин Юрий Николаевич

К защите: 1 декабря 1997г

Оценка:_________________________

Подпись преподавателя: __________

Новосибирск, 1997 г.

Вариант 25V

Вид воздействия: V(p)

Виды передаточных функций:

Параметры схемы:

Показатели качества управления:

1. Найти передаточные функции системы в разомкнутом и замкнутом
состоянии по управляющему V(p) и возмущающему F(p) воздействиям,
характеристическое уравнение и матрицы А,В и С.

Для записи характеристического уравнения приравняем знаменатель
передаточной функции замкнутой системы к нулю.

Переходим к записи дифференциального уравнения, описывающему поведение
исследуемой системы в динамике

Используя переменные состояния в виде:

можно перейти к дифференциальным уравнениям состояния в форме Коши:

Из этого определяем матрицы А,В,С :

2. Определение устойчивости исследуемой системы двумя критериями.

2.1 Частотный критерий Найквиста в логарифмическом масштабе.

Запишем передаточную функцию разомкнутой системы:

Данная система состоит из 3 типовых звеньев:

Расчетная таблица для ЛАХ и ЛФХ:

Из графиков ЛАХ и ЛФК видно, что точка пересечения ЛАХ с осью абсцисс
лежит правее точки, где фазовый сдвиг достигает значения равного –180.

Значит система неустойчива.

2.2 Критерий Гурвица

Приравниваем знаменатель передаточной функции замкнутой системы к нулю и
записываем характеристическое уравнение:

Составляем определитель Гурвица:

Для того, чтобы линейная динамическая система была устойчива,
необходимо и достаточно, чтобы все диагональные миноры определителя
Гурвица и сам определитель имели знаки, одинаковые со знаком первого
коэффициента характеристического уравнения, т.е. были положительными:

3. Определяем значение критического коэффициента усиления разомкнутой
системы, при котором САУ будет находиться на границе устойчивости, с
помощью критерия Гурвица

Выпишем знаменатель ПФ в замкнутом состоянии и приравняем его к нулю,
получим характеристическое уравнение:

Для определения критического коэффициента приравняем к нулю (n – 1)
диагональный минор в определители Гурвица для данного
характеристического уравнения и получим выражение:

4. Исследовать влияние одного из параметров системы на устойчивость
системы (метод Д-разбиения).

Исследуем влияние параметра T1 на устойчивость системы методом
Д-разбиения.

Для получения кривой Д-разбиения решим характеристическое уравнение
(знаменатель ПФ в замкнутом состоянии) относительно T1.

Задаваясь частотой –( ( ( ( +( строим кривую Д-разбиения и штрихуем
левую сторону кривой при движении по ней с увеличением частоты от –( до
+(.

В 1 области К правых корней

Из 1 во 3 (К+1) правых корней

Из 3 во 2 (К+2) правых корней

Из 2 в 3 (К+1) правых корней

Из 3 в 1 К правых корней

Из 1 в 4 (К-1) правых корней

Далее проводим анализ полученных полуплоскостей с точки зрения выделения
полуплоскости, претендующей на устойчивость, т.е. такой, которая будет
содержать наименьшее число правых корней.

Таким образом, полуплоскость 4 – полуплоскость претендент на
устойчивость. Проверим по критерию Гурвица устойчивость для того
значения параметра, который находиться внутри полуплоскости –
претендента, т.е. в отрезке лежащем на вещественной оси от 19 до +(.

Расчетная таблица:

Возьмем T1=25

Тогда, характеристическое уравнение будет:

Составляем определитель Гурвица:

Все определители больше нуля значит, система устойчива при 19(T1((.

5.Синтез корректирующего устройства, обеспечивающее требуемые показатели
качества в установившемся и переходном режимах.

Синтезируем корректирующее устройство для заданной системы, т.к.
согласно п.2 она неустойчива. По заданным показателям качества строим
желаемую ЛАХ разомкнутой системы.

Определяем (по графику для определения коэффициента K0 по допустимому
перерегулированию):

Проводим асимптоту с наклоном -20 дб/дек через частоту среза до
пересечения с заданной ЛАХ. В высокочастотной области проводим асимптоту
с наклоном –80 дб/дек и получаем желаемую ЛАХ.

Вычитание ЛАХ исходной системы из ЛАХ желаемой системы получаем ЛАХ
корректирующего устройства. По полученной ЛАХ подбираем корректирующее
устройство, его передаточная функция имеет вид:

Строим структурную схему скорректированной системы:

Записываем ПФ скорректированной системы в разомкнутом и замкнутом
состояниях:

где L4(w) – ЛАХ и F4(w) – ЛФК скорректированной системы.

Запас устойчивости по фазе (=15(

По построенным ЛФХ и ЛАХ видно, что скорректированная система устойчива
(критерий Найквиста).

Для проверки показателей качества скорректированной системы строим ВЧХ
замкнутой системы:

Трапеции будут выглядить так:

Получили четыре трапеции, теперь определим параметры для каждой из
трапеций.

Wd1 12

Wd2 18

Wd3 19

Wd4 23

Wp1 18

Wp2 19

Wp3 23

Wp4 30

P1 -1,8

P2 12,2

P3 -9,07

P4 -1,6

X1 0,666

X2 0,9

X3 0,82

X4 0,766

h1 h2 h3 h4 x1( ) x2( ) x3( ) x4( ) t1 t2 t3 t4

0 0 0 0 0 0 0 0 0 0 0 0 0

0,5 0,269 0,304 0,286 0,277 -0,4842 3,7088 -2,145 -0,443 0,02778 0,026
0,022 0,0167

1 0,515 0,593 0,5545 0,536 -0,927 7,2346 -4,1588 -0,858 0,05556 0,053
0,043 0,0333

1,5 0,732 0,832 0,785 0,76 -1,3176 10,1504 -5,8875 -1,216 0,08333 0,079
0,065 0,05

2 0,909 1,003 0,965 0,94 -1,6362 12,2366 -7,2375 -1,504 0,11111 0,105
0,087 0,0667

2,5 1,04 1,12 1,087 1,069 -1,872 13,664 -8,1525 -1,71 0,13889 0,132
0,109 0,0833

3 1,127 1,176 1,159 1,144 -2,0286 14,3472 -8,6925 -1,83 0,16667 0,158
0,13 0,1

3,5 1,168 1,175 1,1725 1,168 -2,1024 14,335 -8,7938 -1,869 0,19444 0,184
0,152 0,1167

4 1,169 1,131 1,1525 1,163 -2,1042 13,7982 -8,6438 -1,861 0,22222 0,211
0,174 0,1333

4,5 1,148 1,071 1,105 1,129 -2,0664 13,0662 -8,2875 -1,806 0,25 0,237
0,196 0,15

5 1,108 1,001 1,045 1,071 -1,9944 12,2122 -7,8375 -1,714 0,27778 0,263
0,217 0,1667

5,5 1,06 0,951 0,9865 1,018 -1,908 11,6022 -7,3988 -1,629 0,30556 0,289
0,239 0,1833

6 1,043 0,92 0,9415 0,958 -1,8774 11,224 -7,0613 -1,533 0,33333 0,316
0,261 0,2

6,5 0,956 0,903 0,915 0,938 -1,7208 11,0166 -6,8625 -1,501 0,36111 0,342
0,283 0,2167

7 0,951 0,915 0,9095 0,919 -1,7118 11,163 -6,8213 -1,47 0,38889 0,368
0,304 0,2333

7,5 0,936 0,946 0,9235 0,913 -1,6848 11,5412 -6,9263 -1,461 0,41667
0,395 0,326 0,25

8 0,945 0,986 0,9495 0,938 -1,701 12,0292 -7,1213 -1,501 0,44444 0,421
0,348 0,2667

10 1,016 1,062 1,054 1,038 -1,8288 12,9564 -7,905 -1,661 0,55556 0,526
0,435 0,3333

12 1,036 0,96 1,0075 1,027 -1,8648 11,712 -7,5563 -1,643 0,66667 0,632
0,522 0,4

14 0,997 0,976 0,963 0,976 -1,794???????????????????????????

Методом трапеций строим график переходного процесса.

Переходной процесс:

=0.5с, что удовлетворяет заданным требованиям.

Литература

Теория автоматического управления / Под ред. А.А.Воронова. – М. : Высшая
школа. -1977.-Ч.I.-304с.

Бесекерский В.А., Попов Е.П. Теория автоматического регулирования. – М.
: Наука, 1974.

Егоров К.В. Основы теории автоматического управления. – М. : “Энергия”,
1967

PAGE

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