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leqr

H-infinity LQ gain (full state)

Syntax

[K, X, err] = leqr(P12, Vx)

Arguments

P12

syslin list

Vx

symmetric nonnegative matrix (should be small enough)

K,X

two real matrices

err

a real number (l1 norm of LHS of Riccati equation)

Description

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|