《现代控制理论》课程教学资源——现控实验_实验指导书

《现代控制理论》实验指导书Experiment Instructor of Modern ControlTheory张存山吴兴华齐晓妹Zhang CunShan Wu Xinghua Qi Xiaomei山东理工大学电气与电子工程学院自动化系Dept. of Automation, School of Electrical and ElectronicsEngineering, SDUT2015年4月April, 2015

《现代控制理论》实验指导书 Experiment Instructor of Modern Control Theory 张存山 吴兴华 齐晓妹 Zhang CunShan Wu Xinghua Qi Xiaomei 山东理工大学电气与电子工程学院自动化系 Dept. of Automation, School of Electrical and Electronics Engineering, SDUT 2015年4月 April，2015

目录Contents一状态空间模型的建立与状态响应的分析实验一EXPERIMENT1 Modeling with State SpaceDescription andAnalysis ofTime Response by Using MATLAB实验二 控制系统的稳定性、能控性和能观测性分析EXPERIMENT2 Analysis of Stability、Controllability and Observabi-litybyUsingMATLAB10实验三控制系统状态反馈控制律的确定EXPERIMENT3 Designing the Feedback Control Law for a 2-OrderSystem15实验四控制系统综合设计EXPERIMENT4 System Synthesis for a 3-order System by Using MATLAB

目 录 Contents 实验一 状态空间模型的建立与状态响应的分析.1 EXPERIMENT1 Modeling with State Space Description and Analysis of Time Response by Using MATLAB 实验二 控制系统的稳定性、能控性和能观测性分析.5 EXPERIMENT2 Analysis of Stability、Controllability and Observabi￾lity by Using MATLAB 实验三 控制系统状态反馈控制律的确定.10 EXPERIMENT3 Designing the Feedback Control Law for a 2-Order System 实验四 控制系统综合设计.15 EXPERIMENT4 System Synthesis for a 3-order System by Using MAT￾LAB

实验一状态空间模型的建立与状态响应分析EXPERIMENT1 Modeling with State Space Description and Analysis ofTime Response by Using MATLABI.ExperimentalPurpose1. Master the methods of obtaining the state space description2. Master the methods of state transformation.3. Master the methods of obtaining the diagonal canonical form and Jordan canonicalform by state transformation4. Master the methods of calculation of the time response of the LTI systemIl.ExperimentalContent1.1 The state space description ofthe system is「2.25-5-1.25[4 0.56242.25-4.251.25-0.25X(t) =X(t)u(t)220.25-1-0.5-1.25021.251.75-0.250.750100u(t)y(t)10oL1.1.1 Modeling with the state space descriptionThis equation can be input to the MATLAB work space by the followingstatement.>>A=[2.25.-5,-1.25.-0.5:2.25,-4.25,-1.25,-0.25:-0.25,-0.5,-1.25-1:1.25.-1.75,0.25,-0.75];>> B=[4,6;2,4;2,2;0,2];>>C=[1,0,0,0;0,1,0,0];>> D=zeros(2,2);G=ss(A,B,C,D)1

1 实验一 状态空间模型的建立与状态响应分析 EXPERIMENT1 Modeling with State Space Description and Analysis of Time Response by Using MATLAB I. Experimental Purpose 1. Master the methods of obtaining the state space description. 2. Master the methods of state transformation. 3. Master the methods of obtaining the diagonal canonical form and Jordan canonical form by state transformation 4. Master the methods of calculation of the time response of the LTI system II. Experimental Content 1.1 The state space description of the system is 2.25 5 1.25 0.5 4 6 2.25 4.25 1.25 0.25 2 4 ( ) () () 0.25 0.5 1.25 1 2 2 1.25 1.75 0.25 0.75 0 2 X t X t ut  −− −      −−− =    + − −− −       −−−   ( ) 0 1 0 0 1 0 0 0 y(t) u t       = 1.1.1 Modeling with the state space description This equation can be input to the MATLAB work space by the following statement. >>A=[2.25,-5,-1.25,-0.5;2.25,-4.25,-1.25,-0.25;-0.25,-0.5,-1.25,-1;1.25,-1.75,- 0.25,-0.75]; >> B=[4,6;2,4;2,2;0,2]; >> C=[1,0,0,0;0,1,0,0]; >> D=zeros(2,2);G=ss(A,B,C,D)

1.1.2 Conversion of state space model established by 1.1.1 as the transferfunction modelThe model of transferfunction can be obtained by thefollowing statements:>>G1=tf(G)or>>[Numl,denl]=ss2tf(A,B,C,D,1)>>[Num2,den2]=ss2tf(A,B,C,D,2)1.2 The model of the system is$+5G(s) = ,$*+2s3 +3s? +4s+51.2.1 Set up the model of the transfer function of the systemTransfer function can be input to the MATLAB work space by the followingstatements.>> num=[1,5];den-[1,2,3,4,5],G=tf(num,den1.2.2 Conversion ofthe transfer function established by 1.2.1 as the state spacedescription>>GI=ss(G)or>>[A,B,C,D]=tf2ss(num,den)1.3Thestatespacedescriptionofthesystem is[1 0-1]1X=010x+0u[o02][2]y=[1 2 1]Transform the state space description into the diagonal canonical formThe transformation can be executed by the following statements.>> A=[1 0 -1;0 1 0;0 0 2];>> b=[1 0 2];2

2 1.1.2 Conversion of state space model established by 1.1.1 as the transfer function model The model of transfer function can be obtained by the following statements: >>G1=tf(G) or >> [Num1,den1]=ss2tf(A,B,C,D,1) >>[Num2,den2]=ss2tf(A,B,C,D,2) 1.2 The model of the system is 2 3 4 5 5 ( ) 4 3 2 + + + + + = s s s s s G s 1.2.1 Set up the model of the transfer function of the system Transfer function can be input to the MATLAB work space by the following statements. >> num=[1,5];den=[1,2,3,4,5],G=tf(num,den) 1.2.2 Conversion of the transfer function established by 1.2.1 as the state space description >> G1=ss(G) or >> [A,B,C,D]=tf2ss(num,den) 1.3 The state space description of the system is [1 2 1] 2 0 1 0 0 2 0 1 0 1 0 1 =           +           − = y X X u  Transform the state space description into the diagonal canonical form. The transformation can be executed by the following statements. >> A=[1 0 -1;0 1 0;0 0 2]; >> b=[1 0 2]';

>> c=[1 2 1];>>d=0;>>[V,D]=eig(A);>>[Ap,Bp,Cp,Dp]=ss2ss(A,b,c,d,inv(V))1.4 The state space description of the system is[010][0]X=001x2301y=[1 0 1]XTransform the state space description into the Jordan canonical form and thetransfor mation can be executed by the following statements.>> A=[ 0 1 0;0 0 1;2 3 0];>> b=[0 0 1]';>> c=[1 0 1];>>d=0;>>[V,J]=Jordan (A);>>[Ap,Bp,Cp,Dp]=ss2ss(A,b,c,d,inv(V))1.5ConsideraLTI systemdescribedbythefollowingstatespacedescription[。]x(0+101[ 0 ()=X(t)=-2-3y(t)=[1 2]X(t)1.5.1Determine the matrixexponiential function e4tThe matrix exponiential function can be derived by the following statements:>> A=[0 1;-2 -3];>> syms t>>eAt=expm(A*t)1.5.2 Find the zero input response of the system in the period [0, 6s]It canbecalculated bythefollowingstatements%Zeroinputresponse3

3 >> c=[1 2 1]; >> d=0; >> [V,D]=eig(A); >> [Ap,Bp,Cp,Dp]=ss2ss(A,b,c,d,inv(V)) 1.4 The state space description of the system is y X u [1 0 1] 1 0 0 2 3 0 0 0 1 0 1 0 =           +           X = X Transform the state space description into the Jordan canonical form and the transformation can be executed by the following statements. >> A=[ 0 1 0;0 0 1;2 3 0]; >> b=[0 0 1]'; >> c=[1 0 1]; >> d=0; >> [V,J]=Jordan (A); >> [Ap,Bp,Cp,Dp]=ss2ss(A,b,c,d,inv(V)) 1.5 Consider a LTI system described by the following state space description 01 0 1 ( ) ( ) ( ), (0) 23 1 2 ( ) [1 2] ( ) X t X t ut X yt Xt      =+ =         − −   =  1.5.1 Determine the matrix exponiential function t e A . The matrix exponiential function can be derived by the following statements: >> A=[0 1;-2 -3]; >> syms t >> eAt=expm(A*t) 1.5.2 Find the zero input response of the system in the period [0, 6s] It can be calculated by the following statements. %Zero input response

>> A=[0,1;-2 -3];>>B=[0;1];C=[1,2];D=[0];X0=[1;2];>> sys=ss(A,B,C,D);>>[y1,t,x1]=initial(sys,X0,0:0.02:6);>>plot(t,xl)1.5.3 When u(t) =1 (t), calculated the response ofthe system in the period [0, 6s]%The unit step response with zero initial state:>> A=[0,1;-2 -3];>> B=[0;1];C=[1,2];D=[0];>> sys=ss(A,B,C,D);>>(=[0:0.02:6];>>[y2,t,x2]=step(sys,t);>> figure(1)>> subplot(1,2,1); plot(t,x2)>> subplot(1,2,2); plot(t,y2)%full response:>> y=y1+y2;x=xl+x2;>>figure(2)>> subplot(1,2,1); plot(t,x)>> subplot(1,2,2); plot(t,y)Ill.Requirements ofthereport1.Write the relevant programs in the experimental content and compile them in thecomputer.2.All programs, results and related graphics are written on the experiment reporttogether.4

4 >> A=[0,1;-2 -3]; >>B=[0;1];C=[1,2];D=[0];X0=[1;2]; >> sys=ss(A,B,C,D); >> [y1,t,x1]=initial(sys,X0,0:0.02:6); >> plot(t,x1) 1.5.3 When u(t) = 1 (t), calculated the response of the system in the period [0, 6s] %The unit step response with zero initial state: >> A=[0,1;-2 -3]; >> B=[0;1];C=[1,2];D=[0]; >> sys=ss(A,B,C,D); >> t=[0:0.02:6]; >> [y2,t,x2]=step(sys,t); >> figure(1) >> subplot(1,2,1); plot(t,x2) >> subplot(1,2,2); plot(t,y2) %full response: >> y=y1+y2;x=x1+x2; >> figure(2) >> subplot(1,2,1); plot(t,x) >> subplot(1,2,2); plot(t,y) III. Requirements of the report 1.Write the relevant programs in the experimental content and compile them in the computer. 2.All programs, results and related graphics are written on the experiment report together

实验二控制系统的稳定性、能控性和能观测性分析Experiment II Analysis of Stability, Controllability and Observabilityby Using MATLABI.Experimental Purpose1. Master the methods of analyzing the stability of the LTI system by the Lyapunovmethod.2. Master the methods of determining the controllability and observability of acontrol system.3. Master the methods of transforming the state space description into controllablecanonical form and observable canonical form4.Master the method of structure decomposition ofa control system5.Masterthe method ofminimal realization of theLTI systemII.ExperimentalContent2.1 Consider the state space description ofa LTI system is4.5-3.50.51X =2-4.54.5-0.5A[-1 1.50.5-2.5y=[1 o 1]xDetermine the stability by the Lyapunov 2 nd Methodclose allclear all% the state space model of a systemA=[1 -3.5 4.5;2 -4.5 4.5;-1 1.5 -2.5];%Q=IQ=eye(3,3);%Finding the sol ution matrix P of Lyapunov equation A'P +PA=-Q5

5 实验二 控制系统的稳定性、能控性和能观测性分析 Experiment II Analysis of Stability, Controllability and Observability by Using MATLAB I. Experimental Purpose 1. Master the methods of analyzing the stability of the LTI system by the Lyapunov method. 2. Master the methods of determining the controllability and observability of a control system. 3. Master the methods of transforming the state space description into controllable canonical form and observable canonical form. 4. Master the method of structure decomposition of a control system. 5. Master the method of minimal realization of the LTI system. II. Experimental Content 2.1 Consider the state space description of a LTI system is y [ ]X X X u 1 0 1 0.5 0.5 0.5 1 1.5 2.5 2 4.5 4.5 1 3.5 4.5 =           − − − +           − − − −  = Determine the stability by the Lyapunov 2nd Method. close all clear all % the state space model of a system A=[1 -3.5 4.5;2 -4.5 4.5;-1 1.5 -2.5]; %Q=I Q=eye(3,3); % Finding the solution matrix P of Lyapunov equation T A P PA Q + =−

P=lyap(A,Q)% Display the principle minor determinant of the matrix Pand determine thestability of the system.flag=0;n=length(A);for i=l:ndet(P(1:i,1:i)if(detP(1:i,1:i)> A=[1 2 0;3 -1 1;0 2 0];>>B=[2;1;1];6

6 P=lyap(A,Q) % Display the principle minor determinant of the matrix P and determine the stability of the system. flag=0; n=length(A); for i=1:n det(P(1:i,1:i)) if((detP(1:i,1:i))> A=[1 2 0;3 -1 1;0 2 0]; >>B=[2;1;1];

>>Qc=ctrb(A,B)>>nc=rank(Qc)2.2.2 Transform the state space description into the controllable canonical form:>>A=[1 2 0;3 -1 1;0 2 0];>>B=[2;1;1];>>C=[0 1 1];D=[0];>>cp=poly(A);>>P=[A 2 *B,A *B,B]*[1,0,0;cp(2),1,0; cp(3), cp(2),1]>>[Ac,Bc,Cc,Dc]=ss2ss(A,B,C,D,inv(P))2.3 Consider the state space description ofa LTI system is102X=22110-21[y=[0 1 1]2.3.1 Determine the observable matrix and find whether the system is observableObservability matrix and its rank can be calculated as the following statements>>A=[1 0 2 ;2 1 1;1 0 -2];B=[1;2;1]>>C=[0 1 1]; D=[0];>>Qo=obsv(A,C)>>no=rank(Qo)2.3.2 Transform the state space description into the observable canonical formTransform the state space description into the observable canonical form>> A=[1 0 2 ;2 1 1;1 0 -2];B=[1;2;1];>> C=[0 1 1]; D=[0];>> cp=poly(A);>> T-[1, cp(2), cp(3);0,1,cp(2);0,0,1]*[C*A^2;C*A;C];>>[Ao,Bo,Co,Do]=ss2ss(A,B,C,D,T)7

7 >>Qc=ctrb(A,B) >>nc=rank(Qc) 2.2.2 Transform the state space description into the controllable canonical form: >> A=[1 2 0;3 -1 1;0 2 0]; >>B=[2;1;1]; >>C=[0 1 1];D=[0]; >>cp=poly(A); >>P=[A^2*B,A*B,B]*[1,0,0;cp(2),1,0; cp(3), cp(2),1]; >>[Ac,Bc,Cc,Dc]=ss2ss(A,B,C,D,inv(P)) 2.3 Consider the state space description of a LTI system is  [ ]         =           +           − = 0 1 1 1 2 1 1 0 2 2 1 1 1 0 2 y X X u  2.3.1 Determine the observable matrix and find whether the system is observable Observability matrix and its rank can be calculated as the following statements >> A=[1 0 2 ;2 1 1;1 0 -2];B=[1;2;1] >>C=[0 1 1]; D=[0]; >>Qo=obsv(A,C) >>no=rank(Qo) 2.3.2 Transform the state space description into the observable canonical form: Transform the state space description into the observable canonical form: >> A=[1 0 2 ;2 1 1;1 0 -2];B=[1;2;1]; >> C=[0 1 1]; D=[0]; >> cp=poly(A); >> T=[1, cp(2), cp(3);0,1,cp(2);0,0,1]*[C*A^2;C*A;C]; >>[Ao,Bo,Co,Do]=ss2ss(A,B,C,D,T)

2.4 Consider the state space descriptionofa LTI system is[-10 1010-1000X=00000000y=[1 00ox2.4.1Takethe controllablecanonical decompositionforthe systemWith the following statement:>>A=[-1010 0;0 -10 0 0;0 01 0;0 0 01];>>B=[1;1;0;0];>> C-[1 0 0 0];>>[A1,B1,C1,T,k]=ctrbf(A,B,C)2.4.2TaketheobservablecanonicaldecompositionforthesystemWith the following statement:>>A=[-10 1 0 0;0 -10 0 0;0 0 1 0;0 0 0 1];>>B=[1;1;0;0];>> C-[1 0 0 0];>>[A1,B1,C1,T,k]=obsvf(A,B,C)2.4.3DetermineaminimalrealizationofthesystemAchieve with the following statement>>A=[-10100;0-1000;0010;0001];>> B=[1;1;0;0];>> C-[1 0 0 0]; D=[0];>>[Am,Bm,Cm,Dm]=minreal(A,B,C,D);>>[z1,p1,k1]=ss2zp(A,B,C,D);>>sys1=zpk(zl,p1,k1)>>[22,p2,k2]=ss2zp(Am,Bm,Cm,Dm)>> sys2=zpk(Z2,p2,k2)8

8 2.4 Consider the state space description of a LTI system is y [ ]X X X u 1 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 10 0 0 10 1 0 0 =             +             − −  = 2.4.1 Take the controllable canonical decomposition for the system. With the following statement: >> A=[-10 1 0 0;0 -10 0 0;0 0 1 0;0 0 0 1]; >> B=[1;1;0;0]; >> C=[1 0 0 0]; >> [A1,B1,C1,T,k]=ctrbf(A,B,C) 2.4.2 Take the observable canonical decomposition for the system. With the following statement: >> A=[-10 1 0 0;0 -10 0 0;0 0 1 0;0 0 0 1]; >> B=[1;1;0;0]; >> C=[1 0 0 0]; >> [A1,B1,C1,T,k]=obsvf(A,B,C) 2.4.3 Determine a minimal realization of the system. Achieve with the following statement: >> A=[-10 1 0 0;0 -10 0 0;0 0 1 0;0 0 0 1]; >> B=[1;1;0;0]; >> C=[1 0 0 0]; D=[0]; >> [Am,Bm,Cm,Dm]=minreal(A,B,C,D); >>[z1,p1,k1]=ss2zp(A,B,C,D); >>sys1=zpk(z1,p1,k1) >>[z2,p2,k2]=ss2zp(Am,Bm,Cm,Dm) >> sys2=zpk(z2,p2,k2)