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.TH onserver 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
observer - observer design
.SH CALLING SEQUENCE
.nf
Obs=observer(Sys,J)
[Obs,U,m]=observer(Sys [,flag,alfa])
.fi
.SH PARAMETERS
.TP 10
Sys
: \fVsyslin\fR list (linear system)
.TP
J
: nx x ny constant matrix (output injection matrix)
.TP
flag
: character strings (\fV'pp'\fR or \fV'st'\fR (default))
.TP
alfa
: location of closed-loop poles (optional parameter, default=-1)
.TP
Obs
: linear system (\fVsyslin\fR list), the observer
.TP
U
: orthogonal matrix (see \fVdt_ility\fR)
.TP
m
: integer (dimension of unstable unobservable (\fVst\fR) or unobservable
(\fVpp\fR) subspace)
.SH DESCRIPTION
\fVObs=observer(Sys,J)\fR returns the observer
\fVObs=syslin(td,A+J*C,[B+J*D,-J],eye(A))\fR obtained from \fVSys\fR
by a \fVJ\fR output injection. (td is the time domain of \fVSys\fR).
More generally, \fVobserver\fR returns in \fVObs\fR an observer for
the observable part of linear system
\fVSys\fR: \fVdotx=A x + Bu, y=Cx + Du\fR represented by a \fVsyslin\fR list.
\fVSys\fR has \fVnx\fR state variables, \fVnu\fR inputs and \fVny\fR outputs.
\fVObs\fR is a linear system with matrices \fV[Ao,Bo,Identity]\fR,
where \fVAo\fR is \fVno x no\fR, \fVBo\fR is \fVno x (nu+ny)\fR, \fV Co\fR is
\fVno x no\fR and \fVno=nx-m\fR.
.LP
Input to \fVObs\fR is \fV[u,y]\fR and output of \fVObs\fR is:
.LP
xhat=estimate of x modulo unobservable subsp. (case \fVflag='pp'\fR)
or
.LP
xhat=estimate of x modulo unstable unobservable subsp. (case \fVflag='st'\fR)
.LP
case \fVflag='st'\fR:
z=H*x can be estimated with stable observer iff \fVH*U(:,1:m)=0\fR
and assignable poles of the observer are set to \fValfa(1),alfa(2),...\fR
.LP
case \fVflag='pp'\fR:
z=H*x can be estimated with given error spectrum iff \fVH*U(:,1:m)=0\fR
all poles of the observer are assigned and set to \fValfa(1),alfa(2),...\fR
.LP
If H satifies the constraint: \fVH*U(:,1:m)=0\fR (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.
.LP
In the particular case where the pair \fV(A,C)\fR of \fVSys\fR is
observable, one has \fVm=0\fR and the linear system \fVU*Obs\fR (resp.
\fVH*U*Obs\fR) is an observer for \fVx\fR (resp. \fVHx\fR).
The error spectrum is \fValpha(1),alpha(2),...,alpha(nx)\fR.
.SH EXAMPLE
.nf
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).
.fi
.SH SEE ALSO
dt_ility, unobs, stabil
.SH AUTHOR
F.D.
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