.TH onserver 1 "April 1993" "Scilab Group" "Scilab Function"
.so man1/sci.an 
.SH NAME
observer - observer design
.SH CALLING SEQUENCE
.nf
[Obs,U,m]=observer(Sys,flag,alfa)
.fi
.SH PARAMETERS
.TP 10
Sys
: \fVsyslin\fR list (linear system)
.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
.SH DESCRIPTION
Observer 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 'pp')
or
.LP
xhat=estimate of x modulo unstable unobservable subsp. (case 'st')
.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 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.
.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);ss2tf(Obs*sysdiag(Idu,Sys)*[Idu;Idu])  
// Transfer u-->[u;u]-->w=[u;y=Sys*u]-->Obs*w  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.