observer

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)