第2讲自动控制系统的数学模型ppt课件

1、 数学模型: 数学表达式 微分方程 差分方程 状态方程 等等第一节 控制系统微分方程的编写一.线性元件的微分方程)()()()(1)(22ttdtdRCdtLCdtdCiidtCdtdiLtuututuduuRiuuuuiooooiiOLR+-+-uo(t)ui(t) RLC电路i(t)TTRL21RCT2,时间常数令mFkxmkxdtdxfFdtdxfkxFdxmtd22Fkxdtdxfdxmtd22 弹簧-质量-阻尼器系统MJkftddtdMdJ1122MdtddJkftd1122 机械转动系统机械转动系统 +-eaia负载LaRaMc+-（1电枢回路ueRiiLaaaaaadtd（2反电

2、势Ceea（3电磁力矩iCamM icMamdc（4轴上动力学方程McMdtdJ)(22MMTKKTtdTTccamummcdtdUadtddRLTaaa，CCRTmeamJ，CKeu1，JTKmm 电枢控制的他励直流电动机电枢控制的他励直流电动机yyy0y0ABx0 xx0)()()()()(000000 xyxxxdtdyyxdtdyfxfyxfyxxxx 功率功率放大放大运放运放I运放运放II反馈联接反馈联接测速发电机测速发电机减速器减速器负载负载uru1u2Mcuf+-+-转速自动控制系统原理图转速自动控制系统原理图uaX运算放大器I运算放大器II功率放大器电动机测速电机+-urueu

3、1u2uaMcufuKuuKuefr111)()(1122uuKudtduKua23KuffMKKuuTcmrrmdtddtdKdtd)()(0MKKKTcmmUrdtdUrKdtdt)()1 ()(00KKKKKu321KKKKKK054321MKuKTcmaumdtd （忽略电枢电阻，电抗）i(t)1111)()()()()(0)()()(TsRCssssssRCsttUdttdRCuuuuuuuioiooioo初始条件下在11)()()(,11)(TssGssTssGuuio则令：)(suiG(s)(suo+uo(t)ui(t)-)()()()()()(01110111txdtxdtxt

4、ydtydtybtdbtdbatdatdammnmmmnnnnnn y(t) - 输出量 x(t) - 输入量asasabsbsbnnnnmmmmsXsYsG011011)()()( ui(t)R1R2C2uO(t)CRTCRSTSUUGSCRSCRZZUUSCRZRZsssssioio212221221222211,1)()()(1)()(1比例比例积分控制器积分控制器PIPI控制器控制器ui(t)C1R2uO(t)R1CRRRCRRRZZUUGRZSCRSCRZKsKSSSSio11121112122211111,) 1() 1()()()(11比例比例微分控制器微分控制器PD控制器控制器

5、ui(t)C1R2uO(t)R1C2(a)yiy0 xf1f2k1k2(b)ab 只适应线性定常系统；只适应线性定常系统； 由系统元件的参数、结构决定，是系统的动态数学模型；由系统元件的参数、结构决定，是系统的动态数学模型； n m ,n n m ,n 为系统的阶数为系统的阶数 单输入单输入单输出单输出 传递函数写法：传递函数写法： 三、关于传递函数的几点说明012201asasabsb) s (G有理分式有理分式: :)as)(as ()bs () s (G010零极点形式零极点形式: :) 1sT)(1sT(1s) s (G21时间常数形式：时间常数形式：常数ksXsYsGtkxty)()

6、()(),()(TssksXsYsGdttxkty1)()()(,)()(11)()()()()()(TssXsYsGtxtydttdyTT:时间常数)(tx)(ty)(tx1632. 00T1斜率Ttety1)(Ttetyttx1)()( 1)(时，当tTetYTsssTssXsGsY11)(111111)()()(过程：)(tx)(ty)(tx)(ty0)()()(22)(22ttxtydttdyTdttydT，,12221)(TSSTsG2222)(,1nSnSnsGTn则令的相点是一对共轭复数时)(,10sG)(2222)(sXnnsnsYsnnsnsYssX12222)(,1)(则当

7、)1gtt1sin(1e1) t (y212n2tny(t)y(t)x(t)txxnj21jS平面sT1TsTs) s (G1s2sK) s (G1sK) s (GKS) s (G) t (xdt) t (dx2td) t (xdK) t (y) t (xdt) t (dxK) t (ydt) t (dxK) t (y1222221sT1实现微分环节纯微分环节一阶微分环节二阶微分环节)t (x) t (ye)s (X)s (Y)s (Gsx(t)tty(t)G(s)C(s)R(s)R(s)R(s)-B(s)B(s)+- 步骤: 典型环节表示系统中的具体元件 按信号传递顺序连接各典型环节,从而构

8、成整个系统结构图K1K2(s+1)K3-Km/（TmS+1）Ku/（TmS+1）运算放大器运算放大器运算放大器运算放大器功率放大器功率放大器电动机电动机测速电动机测速电动机Ur+-UfUeU1U1U2U2UaMcUaUf+KfK1K2(s+1)K3Ku/（Tms+1）Ur(s)Ue(s)U1(s)U2(s)Ua(s)-Mc(s)Uf(s)(s)+-+Kf-Km/（Tms+1）) s (G) s (G21) s (Xc) s (Xr) s (G1) s (G2) s (Xr) s (Xc) s (X1) s (G) s (G) s (X) s (X) s (X) s (X) s (X) s (X

9、) s (G) s (G) s (X) s (X),s (G) s (X) s (X),s (G) s (X) s (X211cr1rcrc21c1r1则：又) s (G1) s (G2) s (Xr) s (X1) s (X2+) s (Xc) s (G) s (G21) s (Xr) s (Xc)()()()()()()()()()()(),()()(),()()(2121212111ssssssssGsXcssssssssGGXXXXXXXGXXGXXrrccr则：又 并联) s (G) s (H) s (XrB(s)+-E(s) s (Yc) s (H) s (G1) s (G) s

10、(Xr) s (Yc) s (Y) s (H) s (B)s (B) s (X)s (G) s (Ycrc) s (X) s (G) s (Y) s (H) s (G)s (Y) s (H) s (X)s (G) s (Yrccrc)()(1)()()(sHsGsGssXYrc 反响+-) s (GYX1X2) s (G) s (GX1X2Y)s (X) s (X)s (G) s (Y21) s (X) s (G) s (X) s (G) s (Y21) s (GX1X2Y+-) s (GYX1X2) s (G1XX) s (G) s (Y21X) s (G1X)s (G) s (Y21 综合

11、点后移 综合点前移+-+-G(s)YX1X1G(s) s (G1YX1G(s)YYG(s)YX1G(s)Y 分支点后移 分支点前移) s (G1) s (G4) s (G2) s (G3) s (H) s (C) s (R+-) s (G1) s (G4) s (G2) s (G3) s (H) s (C) s (R+-+) s (G2) s (G1) s (G4) s (G2) s (G3) s (H) s (G2) s (C) s (R+-+GGG421HGG1G323) s (C) s (RHGG1GGGGG3243321) s (C) s (R) s (G1) s (G4) s (G2

12、) s (G3) s (H) s (C) s (R+-) s (G1) s (G4) s (G2) s (G3) s (H) s (C) s (R+-+) s (G2) s (G1) s (G4) s (G2) s (G3) s (H) s (G2) s (C) s (R+-+GGG421HGG1G323) s (C) s (RHGG1GGGGG3243321) s (C) s (RG1G3G2G4H2H3H1) s (R) s (C+-G1G3G2G4H2H3H1) s (R) s (C+-G14G1GG43G2H3H1) s (R) s (C+-GH42HGGGGHGGHGG1GGGG)

13、s (R) s (C143213432324321) s (R+-R11R12) s (C1C2s1C1s)()()()()(21sHsGsGsRsB表示)(sGkR(s)G2(S)+)(1sGH(S)N(S)+B(S)E(S)C(S)()()(1)()()()(2121SHSGSGSGSGSRSC)()()(1)()()(212sHsGsGsGsNsC)()()(1)()()()()(1)()()()(2122121sHsGsGsNsGsHsGsGsRsGsGsCXYGXYG。NREPQCG1G2-H1n1kkkP*1)s(GfedcbaLLLLLL1n : 前向通路数P k : 第k条前向通讯的增益 : 各回路的回路增益之和: 两两互不接触回路增益之和: 所有不接触回路中,每次取其 三个回路 增益之和. :第K条前向通路的余子式,即把与 该通路相接触的回路的回路增益值为0aLcbLLuedLLLk)(sR)(1sG)(2sG)(3sG)(4sG321