.TH lqg2stan 1 "April 1993" "Scilab Group" "Scilab Function"
.so man1/sci.an 
.SH NAME
lqg2stan - LQG to standard problem
.SH CALLING SEQUENCE
.nf
[P,r]=lqg2stan(P22,bigQ,bigR)
.fi
.SH PARAMETERS
.TP 10
P22
: \fVsyslin\fR list (nominal plant) in state-space form
.TP
bigQ
: \fV[Q,S;S',N]\fR (symmetric) weighting matrix
.TP
bigR
: \fV[R,T;T',V]\fR (symmetric) covariance matrix
.TP
r
: \fV1\fRx\fV2\fR row vector = (number of measurements, number of inputs)  (dimension of
 the 2,2 part of \fVP\fR)
.TP
P
: \fVsyslin\fR list (augmented plant)
.SH DESCRIPTION
\fVlqg2stan\fR  returns the augmented plant for linear LQG (H2) controller 
design.
.LP
\fVP22=syslin(dom,A,B2,C2)\fR is the nominal plant; it can be in continuous 
time (\fVdom='c'\fR) or discrete time (\fVdom='d'\fR).
.nf
  . 
  x = Ax + w1 + B2u
  y = C2x + w2
.fi
for continuous time plant.
.nf
  x[n+1]= Ax[n] + w1 + B2u
      y = C2x + w2
.fi
for discrete time plant.
.LP
The (instantaneous) cost function is \fV[x' u'] bigQ [x;u]\fR.
.LP
The covariance of \fV[w1;w2]\fR is \fVE[w1;w2] [w1',w2'] = bigR\fR 
.LP
If \fV[B1;D21]\fR is a factor of \fVbigQ\fR, \fV[C1,D12]\fR
is a factor of \fVbigR\fR and \fV[A,B2,C2,D22]\fR is
a realization of P22, then \fVP\fR is a realization of
\fV[A,[B1,B2],[C1,-C2],[0,D12;D21,D22]\fR.
The (negative) feedback computed by \fVlqg\fR stabilizes \fVP22\fR,
i.e. the poles of \fVcl=P22/.K\fR are stable.
.SH EXAMPLE
.nf
ny=2;nu=3;nx=4;
P22=ssrand(ny,nu,nx);
bigQ=rand(nx+nu,nx+nu);bigQ=bigQ*bigQ';
bigR=rand(nx+ny,nx+ny);bigR=bigR*bigR';
[P,r]=lqg2stan(P22,bigQ,bigR);K=lqg(P,r);  //K=LQG-controller
spec(h_cl(P,r,K))      //Closed loop should be stable
//Same as Cl=P22/.K; spec(Cl(2))
s=poly(0,'s')
lqg2stan(1/(s+2),eye(2,2),eye(2,2))
.fi
.SH SEE ALSO
lqg, lqr, lqe, obscont, h_inf, augment, fstabst, feedback
.SH AUTHOR
F.D.