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.TH lqr 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
leqr - H-infinity LQ gain (full state)
.SH CALLING SEQUENCE
.nf
[K,X,err]=leqr(P12,Vx)
.fi
.SH PARAMETERS
.TP 10
P12
: \fVsyslin\fR list
.TP
Vx
: symmetric nonnegative matrix (should be small enough)
.TP
K,X
: two real matrices
.TP
err
: a real number (l1 norm of LHS of Riccati equation)
.SH DESCRIPTION
\fVleqr\fR computes the linear suboptimal H-infinity LQ full-state gain
for the plant \fVP12=[A,B2,C1,D12]\fR in continuous or discrete time.
.LP
\fVP12\fR is a \fVsyslin\fR list (e.g. \fVP12=syslin('c',A,B2,C1,D12)\fR).
.nf
[C1' ] [Q S]
[ ] * [C1 D12] = [ ]
[D12'] [S' R]
.fi
\fVVx\fR is related to the variance matrix of the noise \fVw\fR perturbing \fVx\fR;
(usually \fVVx=gama^-2*B1*B1'\fR).
.LP
The gain \fVK\fR is such that \fVA + B2*K\fR is stable.
.LP
\fVX\fR is the stabilizing solution of the Riccati equation.
.LP
For a continuous plant:
.nf
(A-B2*inv(R)*S')'*X+X*(A-B2*inv(R)*S')-X*(B2*inv(R)*B2'-Vx)*X+Q-S*inv(R)*S'=0
.fi
.nf
K=-inv(R)*(B2'*X+S)
.fi
.LP
For a discrete time plant:
.nf
X-(Abar'*inv((inv(X)+B2*inv(R)*B2'-Vx))*Abar+Qbar=0
.fi
.nf
K=-inv(R)*(B2'*inv(inv(X)+B2*inv(R)*B2'-Vx)*Abar+S')
.fi
with \fVAbar=A-B2*inv(R)*S'\fR and \fVQbar=Q-S*inv(R)*S'\fR
.LP
The 3-blocks matrix pencils associated with these Riccati equations are:
.nf
discrete continuous
|I -Vx 0| | A 0 B2| |I 0 0| | A Vx B2|
z|0 A' 0| - |-Q I -S| s|0 I 0| - |-Q -A' -S |
|0 B2' 0| | S' 0 R| |0 0 0| | S' -B2' R|
.fi
.LP \fVleqe\fR is the dual of \fVleqr\fR.
.SH SEE ALSO
lqr
.SH AUTHOR
F.D.
|
