第8章MATLAB系空间分析法

1、第8章 系统状态空间分析法8.4节和8.5节内容n系统特征方程及解n关于系统相似变换n关于系统可观性、可控性判别的n状态反馈极点配置n状态观测器8.1 系统状态方程的解n状态转移矩阵)(11asiletat若状态方程是齐次的,即有:)0()(xetxaxxatd)(buexe) t (xt0)t (a0atex1a=0 1 0;0 0 1;-6 -11 -6;x0=1;1;1;t=0:0.1:10;for i=1:length(t) x(:,i)=expm(a*t(i)*x0;endplot3(x(1,:),x(2,:),x(3,:);grid onaxx 1110 x6116100010a系

2、统的特征方程、特征值及特征向量n特征方程：|si-a|=0n特征值及特征向量：v，d=eig（a）特征向量矩阵特征值矩阵a*v = v*dex2 已知控制系统求控制系统的特征方程a=2 1 -1;1 2 -1;-1 -1 2;i=1 0 0;0 1 0;0 0 1;syms s %符号计算det(s*i-a)s=solve(det(s*i-a) %求解ans = s3-6*s2+9*s-4s = 4 1 1211121112aex2 求控制系统的特征值及特征向量v = -0.4082 0.7071 0.5774 -0.4082 -0.7071 0.5774 -0.8165 0 -0.5774d

3、 = 1.0000 0 0 0 1.0000 0 0 0 4.0000v,d=eig(a)veig=inv(v)*a*vveig = 1.0000 0 -0.0000 -0.0000 1.0000 0.0000 -0.0000 0 4.0000符号计算symbolic toolboxans = 4 1 1eigensys(a)特征值及特征向量s=determ(s*i-a) s = s3-6*s2+9*s-4矩阵行列式的值eigensys obsolete symbolic toolbox function. v,d = eigensys(a) is the same as v,d = eig(

4、sym(a)8.2 传递矩阵gcsi-a-1b+da=0 1;0 -2;b=1 0;0 1;c=1 0;0 1;d=0;syms si=1 0;0 1;g=c*inv(s*i-a)*bg = 1/s, 1/s/(s+2) 0, 1/(s+2)8.3 线性变换n状态方程的线性变换 ss2ss(sys,t)duzctyutbztatz11ducxybuaxxduzctybuzatzt111duzcyubzaz11ctctbbtatatxz ztx1ex3a=0 -2;1 -3;b=2 0;c=0 3;p=6 2;2 0;%变换矩阵变换矩阵x=pzp1=inv(p);a1=p1*a*p %z坐标系的

5、模型坐标系的模型b1=p1*bc1=c*pa1 = 0 1 -2 -3b1 = 0 1c1 = 6 0the eigenvalues of system are unchanged by the linear transformation: (线性变换不改变系统的特征值）)det()det(asiasi约当标准形ncanon(sys,model)ncanon(sys,companion)ex4利用特征值及范德蒙特矩阵求约当阵a=0 1 0;0 0 1;2 -5 4;v,d=eig(a)p=1 0 1;1 1 2;1 2 4p1=inv(p)；j=p1*a*pv = -0.5774 0.5774

6、 -0.2182 -0.5774 0.5774 -0.4364 -0.5774 0.5774 -0.8729d = 1.0000 0 0 0 1.0000 0 0 0 2.0000p = 1 0 1 1 1 2 1 2 4j = 1 1 0 0 1 0 0 0 2符号计算jo=jordan(a)jo = 2 0 0 0 1 1 0 0 18. 4 系统的可控性和可观性matlab提供函数分别计算能控性矩阵和能观测性矩阵n可控性矩阵co=ctrb(a,b)n可观测性矩阵ob=obsv(a,c)babaabbco1n21n2cacacacob可控性判定a=1 1 0;0 1 0;0 1 1;b=0

7、 1;1 0;0 1;n=length(a)co=ctrb(a,b);rco=rank(co);if rco=n disp(system is controllable)elseif rcon disp(system is uncontrollable)end n = 3 co = 0 1 1 1 2 1 1 0 1 0 1 0 0 1 1 1 2 1 rco = 2 system is uncontrollable可观测性判定a=-3 1;1 -3;b=1 1;1 1;c=1 1;1 1;d=0;n=length(a);ob=obsv(a,c);rob=rank(ob)if rob=n di

8、sp(system is observable)elseif robn disp(system is unobservable)endob= 1 1 1 1 -2 -2 -2 -2rob = 1system is unobservable可控标准形babaabbs1n2若s为非奇异，逆矩阵存在，设为tnt2t11ssscs则，变换矩阵为p1ntntntnasasspa=-2 2 -1;0 -2 0;1 -4 0;b=0 1 1;n=length(a);cam=ctrb(a,b);if det(cam)=0 cam1=inv(cam);endp=cam1(3,:);cam1(3,:)*a;cam

9、1(3,:)*a*a;p1=inv(p);a1=p*a*p1b1=p*ba1 = 0 1 0 0 0 1 -2 -5 -4b1 = 0 0 1可观测标准形则，变换矩阵为m=ptt1nttttcacacv)(若v为非奇异，逆矩阵存在，设为1nttntt2t11avavvv)(cmcamma18.5 系统状态反馈与状态观测器利用反馈结构，研究在什么条件下能实现闭环系统极点的任意配置，以达到预期要求。状态反馈与状态观测器原理参见线性控制系统工程module24，2524.1 the structure of state space feedback control（状态反馈控制的结构）u0 asi1

10、. state variable feedback control system0)(bgasi,21nggggn the number of state variableif the desired location of the closed-loop poles are , the desired characteristic equation will be n,210)()(21nsssthe eigenvalues of the closed-loop system will be given by),(0)(21nggggbgasi0)()()(21nsssbgasiletwew

11、e can obtained , to make the closed-loop poles to be located in desired position.), 2, 1(nigithe principle of designing a state space controller 2. the sufficient and necessary condition of state feedback for closed-loop placement: (状态反馈实现极点配置的充要条件）the state variables of system are all controllable.

12、25.1 observer a model of the system under study（p550 section 2）the approach taken to solve the problem is as following :(1) to construct a model of the system under study;(2) assume (subject to certain restrictions ) that the computed state variables are good approximations to the true state variabl

13、es;(3) from these computed state variables, a suitable controller for the actual system may be constructed using the techniques described in module 24. uwhere, x are assumed to be unmeasured directly.状态观测器设计uubaxxcynow, we construct a model to simulate the origin system , and assume the parameter ma

14、trix are good approximations tocba,cba,cba,cba,cxybuaxxcbamodelxcyubxaxcbamodeltoequals,but in model is different from in the origin system ,because is/are unmeasured directly. xxcba,0 x is called the estimated value of .xxto decrease the error , ( that is error ), we take to correct to make well ap

15、proach : xyyxxyyxxu)2()() 1 (xcyxckkcxubxayykubxaxcxybuaxx)(: )2() 1 (;xxkcaxkckcxxaaxxxccbbaawhenekcaethenxxename)(,select the matrix k to make the solution of this equation on error be convergent (收敛的收敛的), then, xxore , 0)(:xxkcaxxobserverthe gain matrix k is written as: nkkkk21the closed-loop pol

16、es of this model (observer) can be selected by selecting the gain matrix k , so that the state variables will be same as in the end.hence, we can use as the state variables in the state variable feedback system.xxxq the closed-loop system with observerbc asi+uxybc asi+x y k-g-r+状态观测器状态反馈25.2 the sufficient and necessary condition of constructing a state variable observerthe state variables of system are all observable.observability criterion:a system a, c is state observable if and only ifncacacacrankn12参见线性控制系统工程539页pole placement via akerm