第二章控制系统的数学模型-2007ppt课件

1、第二章第二章 控制系统的数学模型控制系统的数学模型n2.1 拉普拉斯变换与反变换拉普拉斯变换与反变换n2.2 传递函数传递函数n2.3 典型环节的传递函数典型环节的传递函数n2.4 动态结构图动态结构图n2.5 反馈控制系统的传递函数反馈控制系统的传递函数n2.6 信号流图与梅逊公式信号流图与梅逊公式t detfst0)(dtetftfLsFst0)()()(0001)( 1)(ttttfsesdtesFtLstst111)()( 1 00000)(ttetftsdtedteeeLsFtssttt1)(0)(022221( )sin( )( )cos( )!( )( )mmf ttF sssf

2、 ttF ssmf ttF ss若，则若，则若，则( )( ),( )1fttFs若则)()()()(22112211sFasFatfatfaL)()(asFtfeLat)()(sFetfLs)(lim)(lim0ssFtfst)(lim)(lim0ssFtfst位移定理nclearnclcnt=0:0.1:5;ny=t;nf=exp(-2*t).*y;nplot(t,f,r,LineWidth,2)nhold onn%plot(t,y)ngrid on00.511.522.533.544.5500.020.040.060.080.10.120.140.160.180.2nclearnclcn

3、w=-10:.01:10;nw1=w;na=1;nk=(a2.-w.2).2 + 4*a2*w.2;nx=(a2-w.2)./k;ny=-2*a*w./k;nk1=(a2.-w1.2).2 + 4*a2*w1.2;nx1=(a2-w1.2)./k1+1;ny1=-2*a*w1./k1;nplot(x1,y1,r,LineWidth,2)nhold onnplot(x,y)ngrid on-0.500.511.52-0.8-0.6-0.4-0.200.20.40.60.8)0()()(fssFdttdfL)0()0()()(222fsfsFsdttfdL0( )( )tF sLfds2( )(

4、)F sLf t dtsn相似性相似性1()()sLftF1()( ) ( )( )01()( ) ( )( )ftf tsFF sftf tsFF s当时，与图形相比是被压缩了；与图形相比是被放大了。当时，与图形相比是被放大了；与图形相比是被缩小了。00.20.40.60.811.21.41.61.820246810121416y=x2y=(2x)200.511.5-202-100-80-60-40-20020406080100-101-100-80-60-40-20020406080100-101-0.4-0.200.20.40.60.81-10100.511.500.511.501020

5、30405060-50050100nnnnmmmmasasasbsbsbsbsAsBsF 1111110)()()(11( ) ( )( )2jstjf tLF sF s e dsj niiinniisscsscsscsscsscsF12211)(nitsiniiiiecsscLsFLtf1111)()()()(limsFsscissiinmiiimiiisscsscsF111)()(nmiiimiiisscLsscLsFLtf111111)()()(nmitsimitsiiiecetic1111)!1(), 1()()(limnmisFsscissii), 1()()(lim)!(111mi

6、sFssdsdimcmimimssi)3() 1(2)(2sssssF3) 1(1)(43221scscscscsF43)3() 1(2) 1(lim2211sssssdsdcs21)3() 1(2) 1(lim2212ssssscs32)3() 1(2lim203ssssscs121)3() 1(2)3(lim234ssssscsttteteetf3121322143)(2222231212322()(2( )(2)s()() 1 ( )s1111 1 22111( )11nnnnnnjjtjttjtF ss sssjsjcccF ssjsjccecejjeef t )()()22()()(

7、)()1221 1sin()1cos cos()2 2tjtjttjtjtjtjteeejjeteearteej欧拉公式sin()t实有理分式反变换的一般表达式000()/2()11( )( ) ( ) ( )( )1 ( )sin( )krmnm in jijijn lltRetkrrrkrB sF sB sbsA sa ssA samnf tac ec eImt 拉氏变换在线性微分方程中的应用22222222222 2 3sin()(0)0,(0)1( ) ( )( )(0)(0)2( )(0)3 ( )(23) ( )(0)(2) (0)(0)1( ) (0)23( )?yyytyyY

8、sL y ts Y ssyysY syY ssssY sysyyssY sysssy t 习题22320221.23( ) ( ) ( )sin3 ( ) (2)9( ) 5 ( )( ) 51112. ( ),(0), ()(0) lim0 () lim03331113.( )( )( )(1)()()(tssf ttF sf tetF stsf ttF sL f tffffssssF sF sF ss ss s a s bs s 求 下 列 函 数 的 拉 氏 变 换若求求，,1)s的 反 变 换答案：223201.23( ) ( ) ( )sin3 ( ) (2)9( )5 ( )( )

9、5112. ( ),(0),()(0)lim0 331 ()lim03tssf ttF sf tetF stsf ttF sL f tfffssfss 求下列函数的拉氏变换若求答案221122111121113.( )( )( )(1)()()(1)11111(1)111 11111111()()()()()()11 (1)tatbtF sF sF ss ss s a s bs ssLLt es ss ssLLees s a s babs bb a s a aa b s bab bb aaa bLLs sss 求，,的 反 变 换132223 1sin()23131332222tccetsjs

10、j 引例 mi(t)=I cos t1. 传递函数的概念传递函数的概念零初始条件输入信号的拉氏变换输出信号的拉氏变换传递函数 )()()()()()()()(1111011110trbtrdtdbtrdtdbtrdtdbtcatcdtdatcdtdatcdtdammmmmmnnnnnn )()(11101110sRasbsbsbsCasasasammmmnnnn )()()()()()()()(1111011110trbtrdtdbtrdtdbtrdtdbtcatcdtdatcdtdatcdtdammmmmmnnnnnn )()()()()(11101110sNsMasasasabsbsbsb

11、sRsCsGnnnnmmmm mmmmbsbsbsbsM 1110)(nnnnasasasasN 1110)()()()()()(11101110sNsMasasasabsbsbsbsRsCsGnnnnmmmm njjmiipszsK11*)()(njjmiisTsK11) 1() 1()()()()0()0()0(0rcabsGRCGKnms njjmiinnnnmmmmpszsKasasasabsbsbsbsRsCsG11*11101110)()()()()(开环增益的定义n1.K定义为开环系统开环传递函数的增益，简称为系统开环增益或开环放大倍数。也就是说，在Gk(s)中，除去积分环节之外

12、，令其s = 0代入所得到的数值，称为开环增益。 dtds微分方程传递函数 )()()()()()(1)()(11sGLsCLtksRsGsCtLsRcuidtC1,1ruidtCRidtdiL)()(1)()(sUsICssRIsLsIr)()(1sUsICsc)()() 1(2sUsURCsLCsrc11)()()(2RCsLCssUsUsGrc)()(1tkytFdttdyftF)()(222)()()()(dttdymdttdyftkytFkfsmssFsYsG21)()()(11)()()(2RCsLCssUsUsGrckfsmssFsYsG21)()()(令LCT 2RLC，121

13、)(22TssTsG12)(22TssTKsG令kmT 2fmk，kK1，1112121MdtdfdtdJMm12dtddtdMMZZi21211212dtdfdtdJM2222222fJssMssGm1)()()()(1)()()(1fJsssMssGmdtdfdtdJMm1212221/iJJJ221/ifffdtdKEmbbLmmmMdtdfdtdJM22ammiCMbaaaaaEdtdiLRiu)()()()()()()()(2233tMRdttdMLtuCdttdKCfRdttdJRfLdttdJLLaLaammbmamaama)()()(22tuKdttddttdTammmmbma

14、amKCfRJRT) 1()()()(sTsKsUssGmmam1)()()(sTKsUssGmmabmammKCfRCK比例环节比例-微分环节比例-积分-微分环节 200 000)(! 21)()()(xxxfxxxfxfxfy)()()(0000 xxxfxfxfyy0 xxx0yyy)(0 xfK 21QQdtdhAFhCQv21QhCdtdhAvFhhhh002101234567891000.511.522.533.502040608010012014016018020044.555.566.5data1data2data3110000)21()(QQhhhCdthhdAvF02010

15、hCQQv102QhhCdthdAvF102QhhCdtdhAvF02/1)()()(hCsAsQsHsGvF)(sG)(sHnnnnmmmmasasasabsbsbsbsHsG11101110)()(njjmiinjjmiisTsKpszsK1111*) 1() 1()()()()() 12() 1() 12() 1(32132112211221321321nnnnmmmmsTsTsTsssssKnllllnkknmjjjjmiimK) 1/(1Tss/1s1s122ss) 12/(122TssT)()()(tKrtcKsGtc(t)0dttrtcssG)()(1)(tc(t)0000000

16、000()( )( )1( )( )( )( )( )()( )( )( )( )( ) y tty ty sG sy tu tu tu ssty tty tu tty ty tu ttt tt )()()(11)(trtcdttdcTTSsGtc(t)0dttdrtcssG)()()(tc(t)0)()()(1)(trdttdrtcssGtc(t)0( )( )1 ( )11 ( ), ( ) ( )( )1( )C sG ssR sIf R sthen C ssssoC ttt12)(22sssG)()(2)()(222trdttdrdttrdtctc(t)022212222( )( )2

17、1( )11 ( ), ( )2111 ()sin()2 011( )()sin()2( ) 1( )limlimjsttjktkC sG sT sTsR sIf R sthen C sT sTssLsse dsekktjtthereC tekktTtttt n程序nclearnt=0:0.0001:.05;nm,n=size(t);nk=1000*pi;na=-100;nr=k./(a-1./ t);nw=atan(r);nfor i=1:nn if t(i)=0 n y=exp(a.*t).*sqrt(a-1./t).2+k2).*sin(k.*t+w)./(pi.*t)+3.8;n el

18、se y=exp(a.*t).*sqrt(a-1./t).2+k2).*sin(k.*t+w)./(pi.*t)+1;n endnendny1=3*103.*exp(a.*t)./(pi.*t)+1;nplot(t,y)nhold onnplot(t,y1,m)ngrid onnclc00.0050.010.0150.020.0250.030.0350.040.0450.05-20246810 x 106c(t)exp(at)结论n对微分或一阶微分环节，在单位阶跃输入信号作用下，极短时间间隔内，形成强度巨大并衰减振荡的响应c(t)。即形成了正负脉冲响应。n在实际中，这样的输入信号的能量是有限的

19、，同时系统也有多种因素制约它的强度。如放大器饱和等。n短时的较强振荡，引起系统的强烈振动，加剧相关部件的磨损和冲击。有时可能使系统崩溃。n由此可见，单位阶跃信号是衡量系统品质的重要输入信号形式。)()()(2)(222trtcdttdcTdttcdT1212)(22222TssTsssGnnn(01)tc(t)0)()()(trtcesGs11uudtdiLr)(2111iiRu)(1122cuuRidtiCuc21)()()(11sUsUsLsIr)()(12sUsICsc)()(1)(122sUsURsIc)()()(2111sIsIRsULs1Ur1R21RCs1UcU1I1I2综合点综

20、合点引出点引出点有向线段有向线段Ls1Ur1R21RCs1UcU1I1I2函数方框函数方框)()()(11 ()(sUsUsTKsUfripk)11 (sTKip1)()()(sTKsUsUsGsska1sTKss1)()()(sTKsUssGmma1sTKmmtfKssUsG)()()(tK)11 (sTKip1sTKss1sTKmmtK)(1sG)(2sG)()(21sGsG)()()()()(21sGsGsRsCsGniisGsG1)()( 2) 并联连接并联连接特点：各环节的输入信号是相同的，均为特点：各环节的输入信号是相同的，均为R(s)，输出输出C(s)为各环节的输出之和。为各环节

21、的输出之和。)()(21sGsG(s)C(s)()()()()(21sGsGsRsCsGniisGsG1)()(n为相并联的环节为相并联的环节数，包括数，包括“-”的情的情况况)C1(s)G1(s)R(s)C2(s)G2(s)R(s) )(1sG(s)C1(s)(2sGC(s)C2(s)C(s)C1(s)+C2(s)G1(s)+G2(s)R(s)结论：环节并联的结论：环节并联的等效传递函数等于等效传递函数等于所有并联环节传递所有并联环节传递函数的代数和。函数的代数和。3) 反馈连接反馈连接 )()(1)(sHsGsGR(s)C(s)(sGR(s)(sHC(s)B(s)E(s)特点：输入信号特点

22、：输入信号R(s)有与反馈信号有与反馈信号B(s)在综合点代在综合点代数相加，所得信号作为前向通道数相加，所得信号作为前向通道G(s)方框的输入信号。方框的输入信号。)()(1)()()()(sHsGsGsRsCs C(s) G (s)R(s)H(s)C(s)结论：结论：C (s)G (s)E(s)B(s)H(s)C(s)E(s)R(s)B(s)反馈通道传递函数前向通道传递函数前向通道传递函数闭环传递函数1“-”对应正反馈对应正反馈“+”对应负反对应负反馈馈3) 反馈连接反馈连接符号的移动符号的移动 )(sGR(s)(sHC(s)(a)B(s)E(s)(sGR(s)(sHC(s)(b)B(s)

23、E(s)1单位反馈单位反馈 )(1)(sGsGR(s)C(s)(b)(sGR(s)C(s)(a)E(s)(sGR(s)C(s)(a)X(s)(sGR(s)C(s)(b)X(s)(1sG C(s) R(s)G (s)X(s)= R(s)X(s)/G (s)G (s) C(s)R(s)X(s)G(s)=R(s)G (s)X(s)G (s)(sGR(s)C(s)(b)X(s)(sG(a)(sGR(s)C(s)X(s) )(sGR(s)C(s)(a)C(s)(sGR(s)C(s)(b)C(s)(sG(a)(sGR(s)C(s)R(s)(sGR(s)C(s)(b)(1sGR(s) C(s)R1(s)R2

24、(s)R3(s)R1(s)C(s)(a)R3(s)R2(s)R1(s)C(s)(b)R2(s)R3(s)C(s)(c)R2(s)R3(s)R1(s) (b)R(s)R(s)R(s)R(s)(a)R(s)R(s)R(s)R(s)n串乘并加反馈式；串乘并加反馈式；n引后支除引前复；引后支除引前复；n综前支除综后复；综前支除综后复；n相邻综引直交换；相邻综引直交换；简化系统结构图的步骤：简化系统结构图的步骤： 确定系统的一个输入量与一个输出量，确定系统的一个输入量与一个输出量，对于多个输入量或输出量，保留其中一个；对于多个输入量或输出量，保留其中一个； 移动引出点和移动引出点和/或综合点以便消除交叉

25、连接；或综合点以便消除交叉连接； 多回路无交叉连接时，应从内回路开始，多回路无交叉连接时，应从内回路开始，从里向外进行变换。从里向外进行变换。在移动引出点和在移动引出点和/或综合点时，应遵循或综合点时，应遵循以下两条原则：以下两条原则： 变换前后有关回路中各方框传递函数变换前后有关回路中各方框传递函数的乘积应保持不变；的乘积应保持不变； 变换前后有关前向通变换前后有关前向通道中各方框传递函数的乘积应保持不变。道中各方框传递函数的乘积应保持不变。n 简化步骤简化步骤n选择一入出；选择一入出；n移动消交连；移动消交连；n从里再向外；从里再向外；n n简化原则简化原则n n回前积不变；回前积不变；n

26、 例例 试化简如图所示系统结构图，求出试化简如图所示系统结构图，求出传递函数传递函数(s)=C(s)/R(s)。1GR(s)1HC(s)(a)2G2H1GR(s)1HC(s)(b)2G2H1H1GR(s)1HC(s)(c)2G2H1H1GR(s)1HC(s)(b)2G2H1H1GR(s)1HC(s)(c)2G2H1H1GR(s)21HHC(s)(d)2G1H1GR(s)21HHC(s)(d)2G1HR(s)21HHC(s)(e)11211HGGG(f)R(s)C(s)121112121GGG HGG H H12111212( )( )( )1GGC ssR sG HGG H HR(s)21HH

27、C(s)(e)12111GGG H例例 试化简如图所示系统结构图，求出试化简如图所示系统结构图，求出传递函数传递函数(s)=C(s)/R(s)。1GR(s)C(s)(a)3G4G2G1GR(s)C(s)(b)3G24/GG2G1/1 G1GR(s)C(s)(b)3G24/GG2G1/1 G1GR(s)C(s)(c)3G24/GG2G1/1 G21211GGGGR(s)C(s)(d)243/GGG 1/1 G1GR(s)C(s)(c)3G24/GG2G1/1 G4322143211)(GGGGGGGGGR(s)C(s)(e)21211GGGGR(s)C(s)(d)243/GGG 1/1 G432

28、2143211)()()()(GGGGGGGGGsRsCs)(1sG)(sH)(2sG)()(1)()()()(1)()()()()(2121sHsGsGsHsGsGsGsGsRsCsr开环传递函数前向通路传递函数1)()(1)()()()(sGsHsGsRsCsr)()()()()()()(21sHsGsHsGsGsRsB)(1sGR(s)(sHC(s)B(s)E(s)N(s)(2sG)(1sGN(s)(sHC(s)(2sG)()()(1)()()()(212sHsGsGsGsNsCsn)()()(1)()()()()(1)()()()()()(2122121sHsGsGsNsGsHsGsG

29、sRsGsGsCsCsCnr)(1sGR(s)(sHC(s)B(s)E(s)N(s)(2sG)(1sGR(s)(sHC(s)B(s)E(s)N(s)(2sGR(s)()()(21sHsGsGE(s)(2sG)(1sG(s)N(s)(sHR(s)()()(21sHsGsGE(s)(2sG)(1sG(s)N(s)(sH r(t)作用下系统的误差传递函数作用下系统的误差传递函数开环传递函数11)()()(11)()()(21sHsGsGsRsEser n(t)作用下系统的误差传递函数作用下系统的误差传递函数)()()(1)()()()()(212sHsGsGsHsGsNsEsen 系统的总误差系统的

30、总误差)()()(1)()()()()()()()()(212sHsGsGsNsHsGsRsNssRssEener1Mixed nodeinput node(source)1x2x3x4x5x6x23a32a34a45a25a44a24a12a43a1235453a1Mixed nodeinput node(source)1x2x3x4x5x6x23a32a34a45a25a44a24a12a43a1235453a1Mixed nodeinput node(source)1x2x3x4x5x6x23a32a34a45a25a44a24a12a43a1235453a1Mixed nodeinput node(source)1x2x3x4x5x6x23a32a34a45a25a44a24a12a43a1235453a543