自动控制原理第三章 时域法

1、 0t0)0()0()0( ccc tAtttr0,0,0)(和和 1)(t t0)()(0tt 时时，当当)(t 。即即积积分分面面积积为为，且且定定义义：11)(000)( dttttt t0)(t 1)( tL 0,0,0)(tAttrsAtrL )(stL1)( 1 A 0,0,0)(tAtttr2sAAtL 21stL 1t0A 0,210,0)(2tAtttr3221sAAtL t)(tx032121stL 21)( 1)(23322AtdtdAtdtdtAdtdtA tASintr )(为为角角频频率率。为为振振幅幅， A22sin sAtAL0tc(t)( c0tc(t)( c

2、dt0tc(t)( c2)( cdtrtrt0tc(t)( cststptpt)(ptc)(02.0 c)(05.0 c或或pM%100)()()( cctcMppstpt0tc(t)( c)(ptc)(02.0 c)(05.0 c或或A AB B%100 BAMpsrpdtttt,pMstpMAB超调量超调量Mp =AB100%)(02.0 c)(05.0 c或或)(sC-Ts1)(sE)(sR11111)()()( TsTsTssRsCsssRttr1)(),(1)( ,11111111)(TssTsTssTssC TteTssLtc 1111)(10tc(t)1632. 01)(1 et

3、cTeTtdtdctTtt11)(00 因因为为)()(ttr TteTtctg 1)()(TtTeTttc )()(ttr )(T)(tc0t)(tc)(tr221)(ttr )1(21)(22TteTTtttc 闭环极点闭环极点（特征根）（特征根）：- -1/TtTeTtc11)( tTTeTttc1)( tTetc11)( )1(21)(22TteTTtttc )()(ttr )( 1)(ttr ttr )(221)(ttr )(sC-)2(2nnss )(sE)(sR222222)2(1)2()()()(nnnnnnnsssssssRsCs 2222)()()(nnnsssRsCs 无

4、无阻阻尼尼自自然然振振荡荡角角频频率率阻阻尼尼比比n 0222 nnss 122, 1 nns 122, 1 nns222221)()(nnnssssssC 0,cos1)( tttcn 0 njs 2, 1n n 122, 1 nns10 22, 11 nnjs122, 1 nns222222222121)(nnnnnnnnnssssssssssC 0, )11sin(11)(2122 ttgtetcntn 10 22, 11 nnjs0, )11sin(11)(2122 ttgtetcntn n 21 nd1 ns 2, 1122, 1 nns2222)(1121)(nnnnnnsssss

5、ssC )1 (1)(tetcntn 1 122, 1 nns122, 1 nnsssssCnnn1)1()1()(222 )1()1(1211)(2)1(2)1(222 ttnneetc1 122, 1 nns122, 1 nns 1 02468101200.20.40.60.811.21.41.61.820.30.30.50.5ntc( t) )10( n nncos 22, 11 nnjsn21 ndn 21 nj21 njn j 0tc(t)( cstpt)(ptc)(02.0 c)(05.0 c或或rt%100)()()( cctcMpp)10( 0, )sin(11)(2 ttet

6、cdtn rtt 1)( rtc0)sin( rdt rdt 211 tgtdr，其其中中上上升升时时间间rt)10( 增大增大 或减小或减小 ，均能减小，均能减小 ，从而加快系统的初始响应速度。从而加快系统的初始响应速度。n rt0, )sin(11)(2 ttetcdtn pt0)cos(1)sin(1)(22 pddtpdtntttetedttdcpnpnp tgttgndpd 21)(,.)2 , 1 , 0( , nntpd 21 ndptpt0)cos()sin( pddpdntt增大增大 或减小或减小 ，均能减小，均能减小 ，从而加快系统的初始响应速度。从而加快系统的初始响应速度

7、。闭环极点离实轴越远，峰值时间越小。闭环极点离实轴越远，峰值时间越小。n pt 最大超调量最大超调量pM%100%100)()()(21 ecctcMpp)(tc21 ndpt221211)sin(11)( eetcp0, )sin(11)(2 ttetcdtn 21sin)sin( n 21 nj21 njn j 211)( etcp最大超调量只和阻尼比最大超调量只和阻尼比 有关，有关， 越大，越大， 越小。越小。 pM00.10.20.30.40.50.60.70.80.910102030405060708090100 pM%10021 eMp 调节时间调节时间st%)sin(12 ted

8、tn211 tne0, )sin(11)(2 ttetcdtn 10)(tctnst %)1ln(2 %12 snte10)(tctstst 211 tne 112 4912. 3)02. 0ln( 3996. 2)05. 0ln( nst %)1ln(2 时当时当52,3,4nnst8 . 00 调节时间近似与调节时间近似与 成反比。成反比。n 闭环极点离虚轴越远，闭环极点离虚轴越远， 越小。越小。st 1 st0 10 )10( %10021 eMppM 。，一一定定时时，当当， snnnstt )3(4或8 . 04 . 0 707. 021 )(sR )1( TssK)(sCsTK25

9、. 016 ， pM%16 pMn st2222221)(nnnssTKsTsTKKsTsKs TTKnn122 25. 025. 0162121)/(825. 016KTsradTKn 0.5， 解得%,16%10021eMp45 . 025. 0414122 TK%44%10021 eMp )%5( ,5.1825.033)%2( ,2825.044误误差差带带取取误误差差带带取取sstnns 2222)()(nnnsssRsC tetcdtnn sin1)(2 ttcnn sin)( tnntetc 2)()(12)()1()1(222ttnnneetc 1)( sR21)(ssR )2

10、sin(12)(2 tettcdntnn)21(22)(tettcntnnn tntnnnneettc )1(222)1(222221212121212122)( )sin(1)(tttcnn )2()(2)1()()(222222nnnnnnsszzsssssRsC 10 1 zssR1)( )111sin(1)2(1)(221222 aatgteaaatcntnnza )111sin(1)2(1)(221222 aatgteaaatcntn0, )sin(11)(2 ttetcdtn )111sin(1)2(1)(221222 aatgteaaatcntn0, )sin(11)(2 tte

11、tcdtn )111sin(1)2(1)(221222 aatgteaaatcntn)2(2nnss )(sR)(sC 1 s 222222222)2( 2) 1(2) 1()2() 1(1)2() 1()()()(nnnnnnnnnnnnddssssssssssssssRsCs )(sCd2nd 2222)(nnnsss 222)2( 2) 1()(nnnndssss TKsTKsTKKsKTsKsKTssKsKTssKs 1)1()1(1)1()(22)1( TssKs )(sR)(sC5 .01 TKsTKsTKs 1)(2 TKTKn 12112n1则： KTTKn21 加加微微分分负

12、负反反馈馈前前 KTKTKn2111 n K 1%16%100%21111 eP25. 0 5 . 01 0625. 01611, 21 KK 求得：)%2()( 185 . 04411误误差差带带对对应应stns nnnnmmmmasasasabsbsbsbs 11101110.)(nnnsspszsksnjnknknkkjmiig 21112212,)2()()()(12 2112221021)(1)()(nknknkkknkknkkknjjjssCsBpsAsAsssC teCteBeAAtcknktnkkknktnkknjtpjnkknkkj2121101sin1cos)(221 nk

13、kjp ,kkjCBA,kkjCBA,)(2()()(222pssszssnnn z p n dj dj 55 nnpz 以以及及)2()(222nnnsspzs pzpssszsssssCtcnnnsst )(2()(1lim)(lim)(lim22200 简简化化前前pzpsszssssCtcnnnsst )2(1lim)(lim)(lim22200 简简化化后后0)(lim tgt)()(.)()(11101110sDsBasasasabsbsbsbsRsCnnnnmmmm )()()()(110jjjjKikjijsjspsasB )sincos()(11tBtAeectcjjrjjj

14、tkitpiji kirjjjjjjjiijsjsspscsRsDsBsC11)()()()()()( j 0, 00110aasasa 10,aa0202112, 1212024, 0aaaaasasasa 210,aaa01110 nnnnasasasa01110 nnnnasasasa1321321321531420gdddcccbbbaaaaaa04321ssssssnnnnnaaaaaaaaabaaaaaaaaabaaaaaaaaab 1321321321531420gdddcccbbbaaaaaa0432

15、1ssssssnnnnn 11231121311bababbbbaac 11351131512bababbbbaac 11471141713bababbbbaac 1321321321531420gdddcccbbbaaaaaa04321ssssssnnnnn 0322130 asasasa0123ssss003130213120aaaaaaaaaa 3210,aaaa03021 aaaa054322345 sssss012345ssssss0050093205905 . 15 . 0532411 - -1 3 0（ 2） 1 0 0（ ） 329 0122234 ssss 22 0 0010

16、02202)( 002211101234sssss 0161620128223456 ssssss3456ssss000001612201612216208108624 ss01243 ss033 ss0123ssss831830310161620128223456 ssssss0)4)(2(22 ss08624 ss2,24,32, 1jsjs 设系统特征方程为：设系统特征方程为：s4+5s3+7s2+5s+6=0劳劳 斯斯 表表s0s1s2s3s451756116601 劳斯表何时会出现零行劳斯表何时会出现零行?2 出现零行怎么办出现零行怎么办?3 如何求对称的根如何求对称的根? 由零行的

17、上一行构成由零行的上一行构成辅助方程辅助方程: 有大小相等符号相反的有大小相等符号相反的特征根时会出现零行特征根时会出现零行s2+1=0对其求导得零行系数对其求导得零行系数: 2s1211继续计算劳斯表继续计算劳斯表1第一列全大于零第一列全大于零,所以系统稳定所以系统稳定错啦错啦!求解辅助方程得求解辅助方程得: s1,2=j由综合除法可得另两由综合除法可得另两个根为个根为s3,4= -2,-30.1110 nnnnasasasa00 a1ananaaaaaaaaaaaaaaa000000000042053164207531 nn 043223140 asasasasa4203142031000

18、000aaaaaaaaaa 0, 030212031211 aaaaaaaaa0, 000431420313 aaaaaaa05432234 ssss56514253101234 sssss)40( , 0 iaiKsssKsssKsssKs 158)5)(3(1)5)(3()(23015823 Ksss)5)(3( sssK )(sR)(sCKKKssss812081510123 0 K1200012008120 KKKK有120 LK015823 Ksssaas aazs aa068523 sss022, 06) 1( 8) 1( 5) 1(2323 zzzzzz即022110123zzz

19、z1z1z0222 z12, 1jz 辅辅助助方方程程的的解解为为1 zssTsTsTTsTCsGmmme01. 0,048. 0) 1(1)(2 0,58 . 401. 0048. 0 TTTm) 1(1) 1(1)(2 sTCsTTsTCsGmemme) 1(12 sTTsTCmme1K)(sR)(sC sK2)1()(221 sTTsTsCKKsGmmeemmeCKKssTTsTCKKs/)(212321 emmeCKKssTTsTCKKs/)(212321 1001/21 TCKKeemeemmeCKKssTCKKCKKssTTsTCKKs/)(21221212321 120,01.

20、0,048. 0,121 KKTTCme50,01. 0,048. 0,121KKTTCme niinssssssssese1212121)(1)1()1)(1(11)(11111 )()()(0tctct )(0tc)(tc)(limttss ss - -)(s )(0sC)(1sG)(2sG)(sH)(sR)(sN-+)(sC)(sE)()()(tbtrte )(tr)(tb)(limteetss sse- -)(s )(0sC)(1sG)(2sG)(sH)(sR)(sN-+)(sC)(sE)(sB)(sE)()(0tctr )(trsssse - -)(s )(0sC)(1sG)(2sG

21、)(sH)(sR)(sN-+)(sC)(sE)(sB)(sE- -)(s )(0sC)(1sG)(2sG)(sH)(sR)(sN-+)(sC)(sE)(sB)(sE)()(0tctr )(trsssse )()()()()()()()()(0ssHsCsHsCsHsBsRsE )()()(0sCsHsR 0)( sE- -)(s )(0sC)(1sG)(2sG)(sH)(sR)(sN-+)(sC)(sE)(sB)(sE)(sE)(s )(sR)(sN)(sC)(2sG)(1sG)(sE)(sH)(sB)(11)()()(11)()()(21sGsHsGsGsRsEskE )(sH)(sR)(s

22、B)(sE)(1sG)(sC)(2sG)(sR)(sN)(sC)(2sG)(1sG)(sE)(sH)(sB)(1sG)(2sG)(sH)(sC)(sB)(sN)(sE1)()()(1)()()()()(212sHsGsGsHsGsNsEsNE )()()()()(sNssRssENEE )()()(1)()()()()()(1)(21221sHsGsGsNsHsGsHsGsGsR )(sR)(sN)(sC)(2sG)(1sG)(sE)(sH)(sB)()()(1)()()()(212sHsGsGsGsNsCsNN )(11)(21sRHGGsE )(1sG)(2sG)(sH)(sR)(sN-+

23、)(sC)(sE)(lim)(lim0ssEteestss ) 12)(1() 15 . 0(ssssK)(sR)(sC0)5 . 01(3223 KsKss60 K)15 . 0()12)(1()12)(1()(11)()()( sKsssssssGsRsEskE21)(ssR 21)15 . 0()12)(1()12)(1()(ssKsssssssE KssKsssssssssEessss11)15 . 0()12)(1()12)(1(lim)(lim200 61 sse1 . 0 sse)(lim)(lim0ssEteestss 22)( ssR)(11)(11)(21sRGsRHGGs

24、Ek )(1)(lim)(lim)(lim00sGssRssEteeksstssr ssre)()12()1()12()1()(01211212121sGsKsTsTsTssssKsGnllllnjjmkkkkmiik K )(0sGnnnmmmG 212102,2,1)0( KssRssGsKssRsGssRevvsvsksssr )(lim)(1)(lim)(1)(lim10000)(sGkKssRssGsKssRsGssRevvsvsksssr )(lim)(1)(lim)(1)(lim10000012)(lim0sGKksp ,)(lim00KsKGKsp ，时当1 ssR1)(pks

25、ksssrKsGsGssRe 11)(lim11)(1)(lim00)(lim)(lim000sGsKsGKsksp ，时当0 Kessr 11,)(lim00 sGsKKsp 0 ssrepKpKssepKKeKsKGKssrsp 11,)(lim000，时当 0,)(lim100 ssrspesGsKK ，时当1 0 KeKsKGKssrsp 11,)(lim000，时当 1 0 21)(ssRvksksssrKsGssGssRe1)(lim1)(1)(lim00 ，时当0 ，时当1 ，时当2 )(lim0ssGKksv )(lim)(lim)(lim010000sGsKsGssKssGK

26、ssksv , 0)(lim00 ssKGKsv ssre,)(lim00KsKGKsv Kessr1 ,)(lim010 sGsKKvsv0 ssre ssrsvessKGK,0)(lim000，时当 KeKsKGKssrsv1,)(lim100 ，时当 0,)(lim2010 ssrvsvesGsKK，时当 vKssevKvKKeKsKGKssrsv1,)(lim00 型型系系统统，21)(ssR31)(ssRaksksssrKsGssGssRe1)(lim1)(1)(lim200 ssrsaesKGsK,0)(lim0020，时当 KeKsKGKssrsa1,)(lim200 ，时当 0

27、,)(lim3020 ssrvsaesGsKK，时当 )(lim20sGsKksa )(lim)(lim)(lim02002020sGsKsGsKssGsKssksa ssrsaesKGsK,0)(lim1010，时当 KeKsKGKssrsa1,)(lim200 ，时当 0,)(lim300 ssrsaesGsKK，时当 aKsseaKaK ssrsaesKGsK,0)(lim0020，时当 ssrsaesKGsK,0)(lim1010，时当 KeKsKGKssrsa1,)(lim00 型系统，型系统，31)(ssR时时，当当2)(2CtBtAtr avpssrKCKBKAe 1有有pKR 1vKRaKR)()(11)()()(sHsGsRsE