observer

Arguments

Sys

syslin list (linear system)

J

nx x ny constant matrix (output injection matrix)

flag

character strings ('pp' or 'st' (default))

alfa

location of closed-loop poles (optional parameter, default=-1)

Obs

linear system (syslin list), the observer

U

orthogonal matrix (see dt_ility)

m

integer (dimension of unstable unobservable (st) or unobservable (pp) subspace)

Description

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).

Examples

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)

See Also

• dt_ility — detectability test
• unobs — unobservable subspace
• stabil — stabilization

Authors

F.D.