系统空间分析法.ppt

第8章 系统状态空间分析法,8.4节和8.5节,内容,系统特征方程及解 关于系统相似变换 关于系统可观性、可控性判别的 状态反馈极点配置 状态观测器,8.1 系统状态方程的解,状态转移矩阵,若状态方程是齐次的,即有:,EX1,a=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; end plot3(x(1,:),x(2,:),x(3,:); grid on,系统的特征方程、特征值及特征向量,特征方程：|sI-A|=0 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*D,EX2 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-4,s = 4 1 1,EX2 求控制系统的特征值及特征向量,符号计算Symbolic Toolbox,EIGENSYS Obsolete Symbolic Toolbox function. V,D = EIGENSYS(A) is the same as V,D = eig(sym(A),8.2 传递矩阵G,CsI-A-1B+D,A=0 1;0 -2;B=1 0;0 1;C=1 0;0 1;D=0; syms s I=1 0;0 1; G=C*inv(s*I-A)*B,G = 1/s, 1/s/(s+2) 0, 1/(s+2),8.3 线性变换,状态方程的线性变换 ss2ss(sys,T),EX3,A=0 -2;1 -3;B=2 0;C=0 3; P=6 2;2 0;%变换矩阵x=Pz P1=inv(P); A1=P1*A*P %z坐标系的模型 B1=P1*B C1=C*P,A1 = 0 1 -2 -3 B1 = 0 1 C1 = 6 0,The eigenvalues of system are unchanged by the linear transformation: (线性变换不改变系统的特征值）,约当标准形,canon(sys,model) canon(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 4 P1=inv(P)； J=P1*A*P,V = -0.5774 0.5774 -0.2182 -0.5774 0.5774 -0.4364 -0.5774 0.5774 -0.8729 D = 1.0000 0 0 0 1.0000 0 0 0 2.0000 P = 1 0 1 1 1 2 1 2 4 J = 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 1,8. 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 可控性矩阵CO=ctrb(A,B) 可观测性矩阵OB=obsv(A,C),可控性判定,A=1 1 0;0 1 0;0 1 1;B=0 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 disp(System is observable) elseif rOBn disp(System is unobservable) end,OB= 1 1 1 1 -2 -2 -2 -2 rOB = 1 System is unobservable,可控标准形,若S为非奇异，逆矩阵存在，设为,则，变换矩阵为P,A=-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); end P=CAM1(3,:);CAM1(3,:)*A;CAM1(3,:)*A*A; P1=inv(P); A1=P*A*P1 B1=P*B,A1 = 0 1 0 0 0 1 -2 -5 -4 B1 = 0 0 1,可观测标准形,则，变换矩阵为M=PT,若V为非奇异，逆矩阵存在，设为,8.5 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Module24，25,24.1 The Structure of State Space Feedback Control （状态反馈控制的结构）,1. State Variable Feedback Control System,n the number of state variable,If the desired location of the closed-loop poles are , the desired characteristic equation will be,We can obtained , to make the closed-loop poles to be located in desired position.,The 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）,25.1 Observer A model of the system under study （P550 Section 2）,The approach taken to solve the problem is as following : To construct a model of the system under study; Assume (subject to certain restrictions ) that the computed state variables are good approximations to the true state variables; From these computed state variables, a suitable controller for the actual system may be constructed using the techniques described in Module 24.,Where, x are assumed to be unmeasured directly.,状态观测器设计,Now, we construct a model to simulate the origin system , and assume the parameter matrix are good approximations to,But in model is different from in the origin system ,because is/are unmeasured directly.,To decrease the error , ( that is error ), we take to correct to make well approach :,Select the matrix K to make the solution of this equation on error be convergent (收敛的), then,The gain matrix K is written as:,The closed-loop poles 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.,The closed-loop system with observer,B,C,A,+,+,u,x,y,B,C,A,+,+,K,-,G,-,r,+,+,状态观测器,状态反馈,25.2 The sufficient and necessary condition of constructing a state variable observer,Observability criterion: A system A, C is state observable if and only if,参见线性控制系统工程539页POLE PLACEMENT VIA AKERMANNS FORMULA,MATLAB直接用于系统极点配置计算的函数有acker和place,A,B为系统矩阵,