自动控制2第二章控制系统的数学模型

1、)()()(11)()()()(11)(110110trbtrtddbtrmtdmdbtrmtdmdbtcatctddatcntdndatcntdndammnn输入输入输出输出RCicruiRu dtduCic rccuuudtdRC 111uiRur dtduCdtduCic2111 ccudtduCRu 221rcccuuudtdCRCRCRudtdCCRR )(212211222121rcccuuudtdTTTudtdTT )(3212221uru1R1C1i1ucR2C2i2输入输入输出输出1111uudtdCRur ccuudtdCRu 221rcccuuudtdCRCRudtdCC

2、RR )(2211222121uru1R1C1i1ucR2C2i2输入输入输出输出iuouLRCioiuRidtdiLu dtduCio iooouuut ddRCut ddLC 22dtdyfKyytddmydtdfKyF22 FKyydtdfytddm 22aaaaaaueiRit ddL eaCe amiCM tddJMML MaRaLaefiauJiaLM+ + - - auaaaaaaueiRit ddL eaCe amiCM tddJMML aiaeMamLaemamauCMRCtddCJRtddCJL 22aemamauCtddCJRtddCJL 22rcccuuudtdTTTu

3、dtdTT )(3212221iooouuut ddRCut ddLC 22FKyydtdfytddm 22aemamauCtddCJRtddCJL 22 0)(dtetfst)()(tfLsF 0)()(dtetfsFst)()(1sFLtf 0)(dtetfstsesdtedtettLststst11)( 1)( 1 000 s1)()()()(2121sFsFtftfL ) 0 () 0 ()()(2fsfsFstfL ) 0 (.) 0 () 0 ()()() 1(21)( nnnnnffsfssFstfL) 0()()(fssFtfL )()(2sFstfL )()()(sFstfL

4、nn )()(ssFtfL 0) 0() 0() 0() 1( nfff ssFtdtfL)()( 0|)(|)(|)(0020 tnttdttftdtftdtf nnssFtdtfL)()( )(lim)(lim0ssFtfst )(lim)(lim0ssFtfst )()(sFetfLs （6 6）延迟定理：）延迟定理：)()( sFtfeLt（7 7）位移定理：）位移定理：)(sFtfL （8 8）相似定理：）相似定理：)()()()(21021sFsFdftfLt （9 9）卷积定理：）卷积定理：)()(1sFLtf )()()(sRsCsG )0,0(,mjnibaji )()()(

5、11)()()()(11)(110110trbtrtddbtrmtdmdbtrmtdmdbtcatctddatcntdndatcntdndammnn )()()()(11101110sRbsbsbsbsCasasasammmmnnnn dtdc(t) 1)(tdt 1)( tL t0)(t 1)()( tLsR )()()()(sGsRsGsC )()()()(11sGLsCLtctg )()(tgLsG nnnnmmmmasasasabsbsbsbsRsCsG 11101110)()()(jiba ,)()()()()()()(11*00jnjimipszsKsPsQabsRsCsG izj

6、p式中：式中： 称为传递函数的零点，称为传递函数的零点，00*abK 零极点增益零极点增益 称为传递函数的极点，也称为系统的特征根。称为传递函数的极点，也称为系统的特征根。 njjmiinmsTssKsPsQabsG11)1()1()()()(jiT, 分别称为时间常数，分别称为时间常数，K称为放大系数称为放大系数显然：显然： ，,1iiz ,1jjpT jnjimipzKK 11*)2()()2()()(22112211*2121lllnljnjkkkmkimispssszssKsG ,221mmm nnn 212 式中：式中：) 12() 1() 12() 1()(221122112121

7、 lllnljnjkkkmkimiTsTsTssssKsG uFuF0iF0AiFuFuF0iF0AiFiiuuuFuF0iF0AiF)()(tKrtc KsG )(R1iuouR2_+21RuRuoi 21)()(RsURsUoi 12)()()(RRsUsUsGio )()()(trtctctddT 11)( TssG1t00.632T原点处切线斜率为原点处切线斜率为1/T21)()(ZsUZsUoi 1/)()()(21212 CsRRRZZsUsUsGio11RZ CsRRZRCsRCsRZ22222221,111 R1iuouR2_+C1111)()()( RCsCsRCssUsUs

8、GioR RC CiuouCssUCsRsUoi1)(1)( )()(trtctddT TssG1)( 1t0T21)()(ZsUZsUoi RCsZZsUsUsGio1)()()(12 Riuou_+CRZ 1CsZ12 iu 齿轮组齿轮组,iku titdttkutd00)( iu iu )()(trtddtc ssG )(1t021)()(ZsUZsUoi sRCZZsUsUsGio 12)()()(Riuou_+CRZ 2CsZ11 cu tddkkuc iu 1)( TsssG 11)()()(121212 sCRsCRCsRRZZsUsUsGio22RZ CsRZ111 R1iuo

9、u_+CR211)()()( RCsRCsCsRRsUsUsGioR RC CiuouRsUCsRsUoi)(1)( )1()( sKsG 21)()(ZsUZsUoi )1()()()(11212 CsRRRZZsUsUsGio22RZ CsRZ 1111R1iuouR2_+C)1(11)()()(2212111212 sRCCCCCsRCsCsCRZZsUsUsGiosCZ111 sCRZ221 iuou_+C1RC2) 1(11)()()( RCsRCsRsUsIsGiR RC Ciuiiui)()()(2)(222trtctctddTtctddT 121)(22 TssTsG Tn1

10、2222)(nnnsssG mIeRn21nj21njc(t)t101011)()()(2 RCsLCssUsUsGioiuouiuouLRCiiooouuut ddRCut ddLC 22)()( trtcsesG )(se R RC C)(sUi)(sUo)(sIRsUsUsIoi)()()( )(1)(sICssUo 111)()()(RsUsUsIi )(1)(22sIsCsUo uiu1R1C1i i1 1uoR2C2i i2 2 )()(1)(2111sIsIsCsU 212)()()(RsUsUsIo 111)()()(RsUsUsIi )(1)(22sIsCsUo )()(1)

11、(2111sIsIsCsU 212)()()(RsUsUsIo uiu1R1C1i i1 1uoR2C2i i2 2 dtiiCiRur)(121111 dtiCiRdtiiC22222111)(1dtduCio22 dtiiCiRur)(121111sCsIsIsIRsUr12111)()()()( )(1)(1)(111121sIsCsCRsIsCsUr )(sUr)(1sI1111 sCRsC)(2sIsC11)(1)()()(2222121sIsCsIRsCsIsI dtiCiRdtiiC22222111)(1)()()()(2211221sICCsCRsIsI )(1sI21121C

12、CsCR )(2sI)(2sI)()(22ssUCsIo sC21)(2sI)(sUodtduCio22 )(sUr)(1sI1111 sCRsC)(2sIsC1121121CCsCR )(2sI)(2sIsC21)(sUo变换前后各变量间的变换前后各变量间的数学关系保持不变。数学关系保持不变。)()(1sGsGini )(1sG)(3sG)(sR)(sC)(2sG)(1sU)(2sU)()()(321sGsGsG)(sR)(sC)()()(321sGsGsG )(sR)(sC)(1sG)(3sG)(sR)(sC)(2sG)(1sU)(3sU)(2sU)()(1sGsGini )()(1)(s

13、HsGsG )(sR)(sC)()()(sEsGsC )(sH)(sR)(sC)(sG)(sE)(sB)()()(sCsHsB )()()(sBsRsE )()()()()(sCsHsRsGsC )()(1)()()()(sHsGsGsRsCs )(1sX)(sG)(2sX)(sY )(1sX)(sN)(sG)(2sX)(sY )(1sX)(sG)(sG)(2sX)(sY ),()()()(21sGsXsXsY ),()()()()(21sNsXsGsXsY :又)()(sGsN )(sG)(1sX)(2sX )(sY)(sG)(sN )(sY)(1sX)(2sX)(1sG),()()()(2

14、1sXsGsXsY ),()()()()()(21sGsNsXsGsXsY )(1)(sGsN )(1)(sGsN )()()()(11sXsNsGsX )(sG)(1sX)(1sX)(sY)(sG)(1sX)(sY)(sN)(1sX)(1sG)(sG)(1sX)(sY)(sY)(1sX)(sG)(sN)(sY)(sY)(sG),()()(1sYsGsX )()(sGsN )()()(1sYsNsX )(1sX)(2sX)(3sX)(sY )(1sX)(3sX)(2sX)(sY)(sG)(sX)(sY)(1sX)(2sX)(sG)(sX)(sY)(2sX)(1sX)(sG )(2sX)(3sX

15、)(sX)(sG )(2sX)(3sX)(sXH1H3G1G4G2G3H3H11G)(sR- -+)(sC2G3G4G1H2H3H+- - -+1G)(sR)(sC2G3G4G1H2H32HG+- - -+- -+1G)(sR)(sC2G3G4G1H2H32HG+- - -+- -+1432124333243211)()()(HGGGGHGGHGGGGGGsRsCs )(1sG)(2sG)(sH)(sR)(sN- - -)(sC)(sE)(sR)(sC)(sN)(sR)(sN)(sC)(1sG)(2sG)(sH)(sR)(sCR)(sE)(sBHGGGGsRsCsRR21211)()()( )

16、(1)(2121sRHGGGGsCR 0)( sNHGGsGsNsCsNN2121)()()()()(1)(212sNHGGGsCN )(1sG)(2sG)(sH)(sC)(sB)(sN)(sE)(1sG)(2sG)(sC)(sN )(sH输出：输出：)()()()()(sNssRssCN )(),(ssN HGGGGsRsCsR21211)()()( HGGsGsNsCsNN2121)()()()( 1G)(sR- -+)(sC2G3G4GH+- - -+)(sNHGGGGGGGHGHGGGGGGGHGGGGHGGGGs321312123213121223122311)1(1)1()( 1G

17、)(sR- -+)(sC2G3G4GH+- - -+)(sN1G- -)(sC4G+- -+)(sNHGGG2231 1G- -)(sC4G+- -+)(sNHGGG2231 - -)(sC4G+)(sN)1(2231HGGGG )1(2231HGGGG - - -)(sC4G+)(sN)1(2231HGGGG )1(2231HGGGG - -HGGGHGGGGGHGHGGGGGGGGGGHGGGHGGGGGsN32123121243214214312232234111)1(11 1)1()( HGGGHGGGGGHGHGGGGGGGGGGsN321231212432142143111)( H

18、GGGGGGGHGHGGGGGGGs3213121232131211)( )()()()()(sNssRssCN 分分母母多多项项式式。具具有有完完全全相相同同的的和和显显然然，传传递递函函数数)()(ssN 信号流图与结构图的转换信号流图与结构图的转换X1X2X3X4a1a2a3X1X2X4a1a3a2a4abX1X2X1X2aba1传传输输从从源源节节点点至至任任何何节节点点的的T nkkkPT11条条前前向向通通道道的的传传输输；第第kPk kjijiiLLLLLL1特特征征式式的的传传输输的的乘乘积积之之和和；所所有有三三个个互互不不接接触触回回环环的的传传输输的的乘乘积积之之和和；所所有有两两两两互互不不接接触触回回环