leqr — H-infinity LQ gain (full state)
[K,X,err]=leqr(P12,Vx)
syslin list
symmetric nonnegative matrix (should be small enough)
two real matrices
a real number (l1 norm of LHS of Riccati equation)
leqr computes the linear suboptimal H-infinity LQ full-state gain
for the plant P12=[A,B2,C1,D12] in continuous or discrete time.
P12 is a syslin list (e.g. P12=syslin('c',A,B2,C1,D12)).
[C1' ] [Q S] [ ] * [C1 D12] = [ ] [D12'] [S' R]
Vx is related to the variance matrix of the noise w perturbing x;
(usually Vx=gama^-2*B1*B1').
The gain K is such that A + B2*K is stable.
X is the stabilizing solution of the Riccati equation.
For a continuous plant:
(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
K=-inv(R)*(B2'*X+S)
For a discrete time plant:
X-(Abar'*inv((inv(X)+B2*inv(R)*B2'-Vx))*Abar+Qbar=0
K=-inv(R)*(B2'*inv(inv(X)+B2*inv(R)*B2'-Vx)*Abar+S')
with Abar=A-B2*inv(R)*S' and Qbar=Q-S*inv(R)*S'
The 3-blocks matrix pencils associated with these Riccati equations are:
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|