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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:

K=-inv(R)*(B2'*X+S)

For a discrete time plant:

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|

### See also

- lqr — LQ compensator (full state)

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