observer - observer design
Obs=observer(Sys,J) returns the observer Obs=syslin(td,A+J*C,[B+J*D,-J],eye(A)) obtained from Sys by a J output injection. (td is the time domain of Sys ). More generally, observer returns in Obs an observer for the observable part of linear system Sys : dotx=A x + Bu, y=Cx + Du represented by a syslin list. Sys has nx state variables, nu inputs and ny outputs. Obs is a linear system with matrices [Ao,Bo,Identity] , where Ao is no x no , Bo is no x (nu+ny) , Co is no x no and no=nx-m .
Input to Obs is [u,y] and output of Obs is:
xhat=estimate of x modulo unobservable subsp. (case flag='pp' ) or
xhat=estimate of x modulo unstable unobservable subsp. (case flag='st' )
case flag='st' : z=H*x can be estimated with stable observer iff H*U(:,1:m)=0 and assignable poles of the observer are set to alfa(1),alfa(2),...
case flag='pp' : z=H*x can be estimated with given error spectrum iff H*U(:,1:m)=0 all poles of the observer are assigned and set to alfa(1),alfa(2),...
If H satifies the constraint: H*U(:,1:m)=0 (ker(H) contains unobs-subsp. of Sys) one has H*U=[0,H2] and the observer for z=H*x is H2*Obs with H2=H*U(:,m+1:nx) i.e. Co, the C-matrix of the observer for H*x, is Co=H2.
In the particular case where the pair (A,C) of Sys is observable, one has m=0 and the linear system U*Obs (resp. H*U*Obs ) is an observer for x (resp. Hx ). The error spectrum is alpha(1),alpha(2),...,alpha(nx) .
nx=5;nu=1;ny=1;un=3;us=2;Sys=ssrand(ny,nu,nx,list('dt',us,us,un)); //nx=5 states, nu=1 input, ny=1 output, //un=3 unobservable states, us=2 of them unstable. [Obs,U,m]=observer(Sys); //Stable observer (default) W=U';H=W(m+1:nx,:);[A,B,C,D]=abcd(Sys); //H*U=[0,eye(no,no)]; Sys2=ss2tf(syslin('c',A,B,H)) //Transfer u-->z Idu=eye(nu,nu);Sys3=ss2tf(H*U(:,m+1:$)*Obs*[Idu;Sys]) //Transfer u-->[u;y=Sys*u]-->Obs-->xhat-->HUxhat=zhat i.e. u-->output of Obs //this transfer must equal Sys2, the u-->z transfer (H2=eye). //Assume a Kalman model //dotx = A x + B u + G w // y = C x + D u + H w + v //with Eww' = QN, Evv' = RN, Ewv' = NN //To build a Kalman observer: //1-Form BigR = [G*QN*G' G*QN*H'+G*NN; // H*QN*G'+NN*G' H*QN*H'+RN]; //the covariance matrix of the noise vector [Gw;Hw+v] //2-Build the plant P21 : dotx = A x + B1 e ; y = C2 x + D21 e //with e a unit white noise. // [W,Wt]=fullrf(BigR); //B1=W(1:size(G,1),:);D21=W(($+1-size(C,1)):$,:); //C2=C; //P21=syslin('c',A,B1,C2,D21); //3-Compute the Kalman gain //L = lqe(P21); //4- Build an observer for the plant [A,B,C,D]; //Plant = syslin('c',A,B,C,D); //Obs = observer(Plant,L); //Test example: A=-diag(1:4); B=ones(4,1); C=B'; D= 0; G=2*B; H=-3; QN=2; RN=5; NN=0; BigR = [G*QN*G' G*QN*H'+G*NN; H*QN*G'+NN*G' H*QN*H'+RN]; [W,Wt]=fullrf(BigR); B1=W(1:size(G,1),:);D21=W(($+1-size(C,1)):$,:); C2=C; P21=syslin('c',A,B1,C2,D21); L = lqe(P21); Plant = syslin('c',A,B,C,D); Obs = observer(Plant,L); spec(Obs.A)
F.D.