控制系统的综合与校正

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1、%100%21epnst5 . 324214nc2412142tan221nr1)21 (21222nb2121rM)9034(%100)1sin/1 (4 . 016. 0%p)9034() 1sin1(5 . 2) 1sin1(5 . 122cst)(sR)(sC)(sG)(sH)(sE)(sGc)(sGo)(sR)(sC)(sH)(sE)(sGo)(sGc)()()(12sUsUsGc2121RRCRRT221RRR TsTssGc111)(T1T10jTsTssGc11)(TsTssGc111)(10-210-11001010510152010-210-1100101010203040

2、5060aT1T1lg20lg10mTsTssGc11)(TTc11tantan)(21)(1) 1(tanTTTm111sin21tan11mmTTlg)1lg1(lg21lg10)(mcLmTsTssGc11)(mmlg10lg10CRRT)(211212RRRbTsbTssGc11)(1Rrucu2RCbT1T1blg20mbTm1bbm11arcsinTsbTssGc11)() 1)(1() 1)(1()(sTsTsTsTsGbabac1Rrucu2R1C2C21CRTTTTbaba0tantantantan)(111111111babaTTTTbaTT11aabbdB)()(L1)

3、1)(1() 1)(1()(sTsTsTsTsGbabac)()(teKtmppcKsG)(pK)()()(dttdeTteKtmdp)1 (sTKdptippdeTKteKtm0)()()()11 (sTKipdttdeTKdeTKteKtmdptipp)()()()(0ssTsTTKip) 1)(1(21)411 (211idiTTTT14idTT)11 (sTsTKidp)11 ()(sTsTKsGidpc)411 (212idiTTTTssTsTTKip) 1)(1(21)11 ()(sTsTKsGidpccm cm lg10)( coLTm1c 0 ( ()180()() )mccm

4、c mmsin1sin111sinmc Tmc1lg10)( coLc ) 18 . 0()(ssKsG) 18 . 0(ssK1 . 0sse 45)/(5sradc KssKsKsv) 18 . 0(lim01 . 0sse-100-50050100Magnitude (dB)10-210-1100101102-180-135-90Phase (deg)Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 20 deg (at 3.43 rad/sec)Frequency (rad/sec)绘制未校正系统的伯特图，绘制未校正系统的伯特图，由该图可知

5、未校正由该图可知未校正系统的开环截止频率和相位裕量为系统的开环截止频率和相位裕量为c25. 1lg4025. 110lg204 .19)54. 38 . 0(tan901801)/(54. 3sradc计算超前网络参数计算超前网络参数和和T：方法一方法一sradcm/5 Tm1求出求出=4求出求出T=选取校正后系统的开环截止频率选取校正后系统的开环截止频率lg108 . 010lg20)( cccoL在校正后系统的开环截止频率处原系统的幅值在校正后系统的开环截止频率处原系统的幅值与校正装置的幅值大小相等、符号相反与校正装置的幅值大小相等、符号相反由超前校正网络特性由超前校正网络特性) 18 .

6、 0()(ssKsG354 . 94 .1945m7 . 3sin1sin1mm11sinmTmc1sradc/9 . 4 lg108 . 010lg20)( cccoL11 . 014 . 0)(sssGc) 18 . 0)(11 . 0() 14 . 0(10)()(0sssssGsGc9 .505 . 0tan2tan4tan90180111sys1=tf(10,0.8 1 0);sys2=1;sys3=feedback(sys1,sys2,-1);step(sys3)hold on;sys1=tf(4 10,0.08 0.9 1 0);sys2=1;sys3=feedback(sys1

7、,sys2,-1);step(sys3);hold off-80-60-40-200204060Magnitude (dB)10-1100101102-180-135-90-45045Phase (deg)Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 51.2 deg (at 4.89 rad/sec)Frequency (rad/sec)w=1.25w=2.5w=10-20 dB/dec-40 dB/dec-20 dB/dec-40 dB/dec10(0.41)50(2.5)( )( )(0.11)(0.81)(1.25)(10)cssG s

8、 G sssss ssc()1 01 2 .5() ()( 0 .81)(1 .2 5 )()1 2 .51()() (1 .2 5 )(1 .2 5 )()1 2 .5() (1) 5ccasGsGssssssaasGs GsasssaTasssas 0校 正 后 系 统 特 征 方 程 为 ：要 求 系 统 性 能 指 标 为 ：4 52c* / 9 0 4 .41 10 .1 60 .4 (1)0 .1 6s in21 .5 (1)2 .5 (1)21 .4 2 8 04 0 .5 0 3 9 5 .5 5 9 22 .84 .8 0 2 crrrrsnsr a dsMMKKMMttsj

9、 取 故则估 计 期 望 主 导 极 点 位 置 为 ：令 校 正 后 闭 环 系 统 第 三 个 极 点 为 (2 .84 .8 )(2 .84 .8 )() ( 2 )spsjsjsp *则 期 望 系 统 特 征 方 程 ：32232(1.25)()12.5 () ( 1) (1.25)1.25(10 )12.5(2.84.8)(2.84.8)() (2) (5.602330.9052)() (5.6023)(5.6023as ssassasasasjsjspssspspsp *30.9052)30.9052 4.3523 1.25 12.5 4.230.9052 12.5 33.350

10、 7.308 3.06642.95spapapapap *令则有解得， 59 3.0664(2.3832)( )7.30838.33(2.3832)( )( )(1.25)(7.308)ccsG sssG s G ss ss0246800.511.5Step ResponseTime (sec)Amplitude0()180 ( ()180 + () )ccc c cbT )10151(1bLclg20)(0 20) 15 . 0)(12 . 0(lim)(lim000KsssKsssGKssv) 15 . 0)(12 . 0(20)(0ssssG) 15 . 0)(12 . 0()(0sss

11、KsG-150-100-50050100Magnitude (dB)10-210-1100101102103-270-225-180-135-90Phase (deg)Bode DiagramGm = -9.12 dB (at 3.16 rad/sec) , Pm = -24.2 deg (at 5.1 rad/sec)Frequency (rad/sec)w=2w=505 . 02 . 020lg20ccc6 .30)85. 55 . 0(tan)85. 52 . 0(tan9018011)/(85. 5sradc180)5 . 0(tan)2 . 0(tan90)(11ggg0)(Im0j

12、G)/(16. 3sradg)(125 . 02 . 020lg20dBhggg1331235180)(cc )/(16. 1sradc 43)5 . 0(tan)2 . 0(tan11ccbclg2020lg20cbT 511132.7413 . 411)(ssTsbTssGc) 132.74)(15 . 0)(12 . 0() 13 . 4(20)()(0ssssssGsGc35)16. 132.74(tan)16. 13 . 4(tan)16. 15 . 0(tan)16. 12 . 0(tan901801111)/(91. 2sradg )(2 .14)()(lg200dBjGjGhg

13、gc 180)32.74(tan)3 . 4(tan)5 . 0(tan)2 . 0(tan90180)(1111ggggg)( 3 .115 . 032.743 . 420lg20dBhgggg cc gg ,hh -100-50050100Magnitude (dB)10-310-210-1100101102-270-225-180-135-90-450Phase (deg)Bode DiagramGm = 14.2 dB (at 2.91 rad/sec) , Pm = 39.1 deg (at 1.03 rad/sec)Frequency (rad/sec)w=0.0135w=0.23

14、2w=2w=5-20 dB/dec-40 dB/dec-20 dB/dec-40 dB/dec-60 dB/dec20(4.31)11.576(0.232)( )( )(0.21)(0.51)(74.321)(2)(5)(0.01346)cssG s Gssssss ssssys1=tf(86 20,conv(0.1 0.7 1,74.32 1 0);sys2=1;sys3=feedback(sys1,sys2,-1);step(sys3)figure;sys1=tf(20,0.1 0.7 1 0);sys2=1;sys3=feedback(sys1,sys2,-1);step(sys3) 1

15、5 . 0)(12 . 0(20sss) 132.74)(15 . 0)(12 . 0() 13 . 4(20sssss) 1)(1() 1)(1()(sTsTsTsTsGbabac0lg20)(lg200 cbcTL cc ) 1125. 0)(1()(0sssKsG020lg20cc6 .16125. 0tantan9018011cc-150-100-50050100Magnitude (dB)10-210-1100101102103-270-225-180-135-90Phase (deg)Bode DiagramGm = -6.94 dB (at 2.83 rad/sec) , Pm

16、= -13.9 deg (at 4.15 rad/sec)Frequency (rad/sec) w = 1 w = 8-20 dB/dec-40 dB/dec-60 dB/dec0lg20)(lg200 cbcTLsradtsc/11. 2) 1sin1(5 . 2) 1sin1(5 . 122 sradc/2 . 2 0lg2020lg20lg20 ccc1 .92 .22020 c)11. 01)(1 . 91 ()1)(1 ()(sssssGaac)11. 01)(1 . 91)(125. 01 ()1 (20)()(0ssssssGsGaac(12.33 )(1)0.99914(0.

17、4292)(1)( )(121.2 )(10.11 )(0.04717)(9.091)cssssG ssssssrada/43. 0 5011. 0tan1 . 9tan125. 0tan90tan1801111caccac020(12.33 )( )( )(1 0.125 )(121.2 )(1 0.11 )159.8624(0.4292)(1) (8)(1)(0.04717)(9.091)csG s G ssssssss ssss 21.512 . 211. 0tan2 . 22 .21tan2 . 2125. 0tan902 . 233. 2tan1801111)(75. 3) 1sin

18、1(5 . 2) 1sin1(5 . 122stcs -150-100-50050100Magnitude (dB)10-310-210-1100101102103-270-180-90090Phase (deg)Bode DiagramGm = 17 dB (at 8.14 rad/sec) , Pm = 51.9 deg (at 2.11 rad/sec)Frequency (rad/sec)w = 0.047w = 0.43 w = 1 w = 8 w = 9.1 -20 dB/dec-40 dB/dec-20 dB/dec-40 dB/dec-60 dB/dec)()()(0jGjGj

19、Gc)(/ )()(0jGjGjGc)()()(0LLLc23/l11sin1llsin1rM122lc123llc) 1007. 0)(19 . 0()(0sssKsG09 . 01000lg20cc2 .11007. 0tan9 . 0tan9018011cc-150-100-50050100150Magnitude (dB)10-210-1100101102103104-270-225-180-135-90Phase (deg)Bode DiagramGm = -16.8 dB (at 12.6 rad/sec) , Pm = -11 deg (at 32.9 rad/sec)Frequ

20、ency (rad/sec)-20 dB/dec-40 dB/dec-60 dB/dec w = 1.111 w = 142.910001000( )(0.91)(0.0071)(1.111)(142.9)G sssss sssradc/40 ) 1(4 . 016. 0rpM35. 1rM) 1(5 . 2) 1(5 . 122 rrcsMMtsradc/56.35 7 . 611rrMMlsradlc/4 .10122sradllc/70123srad /102srad /1433) 1005. 0() 11 . 0() 15 . 2() 19 . 0()(sssssGc 6 .49400

21、05. 0tan40007. 0tan405 . 2tan90401 . 0tan1801111) 1005. 0)(1007. 0)(15 . 2() 11 . 0(1000)()(0ssssssGsGc)(213. 0) 1sin1(5 . 2) 1sin1(5 . 122stcs %5 .28%100)1sin1(4 . 016. 0% p-150-100-50050100150Magnitude (dB)10-210-1100101102103104-270-180-90090Phase (deg)Bode DiagramGm = 17.6 dB (at 159 rad/sec) ,

22、Pm = 49.9 deg (at 39.1 rad/sec)Frequency (rad/sec) -20 dB/dec-60 dB/dec w = 1.111 w = 142.9 w = 200 w = 0.4 w = 10-40 dB/dec-20 dB/dec-40 dB/dec-60 dB/decL-L =L -40 dB/dec7.2 (s+1.111) (s+10)- (s+0.4) (s+200)(1sG)(2sG)(3sG)(sGc)()(1)()()()(2321sGsGsGsGsGsGc1)()()(1)()(1)()()()()(2312321jGjGsGsGsGjGj

23、GsGsGsGccc1)()()()()()()()(23212jGjGsGsGsGsGsGsGcc)()(1)()()()(2321sGsGsGsGsGsGc1)()()(1)()(1)()()()()(2312321jGjGsGsGsGjGjGsGsGsGccc1)()()()()()()()(23212jGjGsGsGsGsGsGsGcc) 1007. 0)(19 . 0(19 . 01007. 0)(3210sssKsKsKsKsG1007. 01sK)(sGc19 . 02sKsK3-150-100-50050100150Magnitude (dB)10-210-1100101102

24、103104-270-225-180-135-90Phase (deg)Bode DiagramGm = -16.8 dB (at 12.6 rad/sec) , Pm = -11 deg (at 32.9 rad/sec)Frequency (rad/sec)-20 dB/dec-40 dB/dec-60 dB/dec w = 1.111 w = 142.910001000( )(0.91)(0.0071)(1.111)(142.9)G sssss ss09 . 01000lg20cc2 .11007. 0tan9 . 0tan9018011ccsradc/10 ) 1(4 . 016. 0

25、rpM225. 1rM) 1(5 . 2) 1(5 . 122 rrcsMMtsradc/68. 9 89. 911rrMMlsradlc/84. 1122 sradllc/2 .18123 ) 14 . 0)(19 . 0(40)()(2ssssGsGc5 . 2lg40105 . 2lg201000lg201114 . 0/40)(2ssKsGc19 . 0)(22sKsG1007. 01sK)(sGc19 . 02sKsK311007. 0Ks3Ks)()()()(3212sGsGsGsG)()(/)()(31sGsGsGsGcc)()()()(/)()()()()()(2313212s

26、GsGsGsGsGsGsGsGsGsGccc-150-100-50050100150Magnitude (dB)10-310-210-1100101102103104-270-180-90090Phase (deg)Bode DiagramGm = 27.7 dB (at 122 rad/sec) , Pm = 66.9 deg (at 10.2 rad/sec)Frequency (rad/sec) w = 0.025 w = 1.111 w = 2.5 w = 108.9 w = 142.9-20 dB/dec-40 dB/dec-20 dB/dec-40 dB/dec-60 dB/dec

27、G2Gc 67)1036. 01/(103 .41tan10007. 0tan90104 . 0tan1802111) 13 .4136. 0)(1007. 0() 14 . 0(1000)()(1)()()()(22321ssssssGsGsGsGsGsGc)(68. 0) 1sin1(5 . 2) 1sin1(5 . 122stcs %6 .19%100)1sin1(4 . 016. 0% pg13=tf(1000,0.007 1 0);g2=tf(1,0.9 1);gcl=g13*g2/(1+g13*g2);step(gcl,0.5)hold ong1c=tf(1000*0.4 1,0.

28、007 1 0);g2c=tf(1,0.36 41.3 1);gclc=g1c*g2c/(1+g1c*g2c);step(gclc,0.5)hold off180cmjniijpszs1111)2(4)(sssG5 . 0%2nst/4sn432212jjsnnd-80-60-40-2002040Magnitude (dB)10-1100101102-180-135-90Phase (deg)Bode DiagramGm = Inf dB (at Inf rad/sec) , Pm = 51.8 deg (at 1.57 rad/sec)Frequency (rad/sec) w = 0.5-

29、20 dB/dec -40 dB/dec)()(cdcdpszs30)90120(180)2(4(180) 12(dssssk)()()(cccpszssG9 . 2cz4 . 5cp4 . 5s9 . 2s) s (Gc)ps ()zs (K) s (G) s (G) s (Gcccc)4 . 5)(2()9 . 2(4 . 59 . 2)2(4)()(ssssKssKsssGsGcccKK414 . 529 . 2ddddssssK6 .186 . 38 . 44 . 31 . 4K65. 44/ KKc)4 . 5)(2()9 . 2(6 .18)()(sssssGsGc1005)4 .

30、 5)(2()9 . 2(6 .18lim)()(limsssssssGssGKscsv322jsd)4 . 5)(2()9 . 2(6 .188 .104 . 7)4 . 5)(2()9 . 2(6 .181)()(123ssssssssssssHsG4 . 7)322()322(3sjjsi4 . 33s9 . 2cz-6-5-4-3-2-10-15-10-5051015Root LocusReal AxisImaginary Axis)2/2/()Im()Re(tgsspddc)2/2/()Im()Re(tgsszddcds B11( )11cccsszbTsbTG sbbTsspsT1

31、1cczbp1058. 033. 0jsd5 . 0167. 0sn1053. 0)2)(1(06. 1limsssssKsv15 sKv110vvKKb8zs0cd1 . 0cz01. 0cp110cczpb01. 01 . 0101)( sssGcccKsssG01. 01 . 0101)()2)(1)(01. 0() 1 . 0( )2)(1(06. 101. 01 . 010)()(sssssKsssssKsGsGcc10/06. 1cKK 7cdcpsz51. 028. 0jsdstns4 .1428. 0/4)/(4%)2(5 . 0156. 05 . 0/28. 0sn98. 0

32、1 . 02101. 0dddddsssssK25. 998. 006. 11006. 110KKc) 15 . 0)(1)(1100() 110(9 . 4)2)(1)(01. 0() 1 . 0(98. 0)()(sssssssssssGsGc109 . 4)()(limssGssGKcsv31. 23p137. 04p1 . 0z51. 028. 0jsd根轨迹法做滞后校正根轨迹法做滞后校正*01( ) 0.1(2)(5)(0.21)(0.51)5 %40% 6 vvvskkG skks sssssksts原系统和期望性能指标期望性能指标取*0.1550vkkk ， 作原系统根轨迹Roo

33、t LocusReal AxisImaginary Axis-10-50-10-50510System: gsGain: 1.18Pole: 0.0123 + 3.18iDamping: -0.00388Overshoot (%): 101Frequency (rad/sec): 3.18System: gsGain: 0.0677Pole: -0.88 + 2.37e-008iDamping: 1Overshoot (%): 0Frequency (rad/sec): 0.88221,2|ln|0.28 0.5, 5.24 ln 1.20.61.039snsttsj 由性能指标得取则主导极点

34、位置：则期望特征方程为：*22432(1.21.44)()(1.2)(1.21)(1.441.2 )1.44sssrsqsrsrsrq sq 验证表明：主导极点靠近原系统根轨迹，但对应点的增益不满足静态指标要求。*0*()( )50()1( )( ) (2)(5)()50 5 510cccccccccccccvcckszG sspkszzG s G spbzs sssppbkkkbkbb校正后开环传递函数其中为保证则校正后系统的特征方程为：432(7)(710)(1050 )50 5.8 7 1.2 1010 50 1.44 1.2 050 1.44 0101ccccccccspspspb s

35、pprprqpbrqpq *令 =得05.8701.211010501.441.2050001.440cpbrq*34*00.11 0.24, 5.9051, 3.6496 0.7007, 5.2093, 0.24()0.24(0.46)( )( )0.1150()12(0.46)( )( )(2)(5)()(2)(5)(0.11)cccccccccccpbrqsskbkszsG sG sspskszsG s G ss sssps sss解得：，故Root LocusReal AxisImaginary Axis-14-12-10-8-6-4-2024-10-8-6-4-20246810Sys

36、tem: gGain: 4.34Pole: 0.00826 + 2.73iDamping: -0.00302Overshoot (%): 101Frequency (rad/sec): 2.7305101520253000.20.40.60.811.21.41.61.8Step ResponseTime (sec)Amplitude%7 4 .4 % 2 0 .7 9 %4 8 .0 5 % 8 .9 3 6 0 .1 9 9 2scscssvssvctstsee0246810121416182002468101214161820)(1)(1sGsGN)(1sG)(2sG)(sGN)(1)(0

37、sGsGr)(0sG)(sGr)()()(1sRsCs )(1)()(111sssG)(1)()()(11ssRsEse)(1)()(sGsGs)(0sG)(sGr)(0sG)(sGr01( )0G s)(0sG)(sGr) 1()(1110sasasasKsGnnnnv)(1)(1)()()()(001sGsGsGsRsCsrsKasasasKssGsGsGsGsssGvnnnnvrr)1 ()()1 ()()(1)(1)()(1)()(112112100111) 1( ssK1114. 01456. 0)(sssGcsradc/56. 4 22.49)()(1)()(1)(2121sGsGsGsGsre) 1( ssK)(sGr1114. 01456. 0ss1)(122TssssGr