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# lqg2stan

LQG to standard problem

### Syntax

[P_aug,r]=lqg2stan(P,Qxu,Qwv)

### Arguments

- P22
State space representation of the nominal plant (

`nu`

inputs,`ny`

outputs,`nx`

states).- Qxu
`[Q,S;S',N]`

symmetric`nx+nu`

by`nx+nu`

weighting matrix.- Qwv
`[R,T;T',V]`

symmetric`nx+ny`

by`nx+ny`

covariance matrix.- r
Row vector

`[ny nu]`

.- P_aug
Augmented plant state space representation (see: syslin)

### Description

`lqg2stan`

returns the augmented plant for linear LQG (H2) controller
design problem defined by:

The nominal plant

`P22`

:described byThe (instantaneous) cost function .

The noises covariance matrix

Up to Scilab-5.5.2 lqg2stan returns wrong inverted values
(see bug 13751)
to obtain the good result one had to use This bug is fixed since Scilab-6.0.0, old codes must be modified accordingly. |

### Algorithm

If `[B1;D21]`

is a factor of
`Qxu`

, `[C1,D12]`

is a
factor of `Qwv`

(see: fullrf) then
`P_aug=syslin(P.dt,P.A,[B1,P.B],[C1;-P.C],[0,D12;D21,P.D])`

### Examples

ny=2;nu=3;nx=4; P22=ssrand(ny,nu,nx); Qxu=rand(nx+nu,nx+nu);Qxu=Qxu*Qxu'; Qwv=rand(nx+ny,nx+ny);Qwv=Qwv*Qwv'; [P_aug,r]=lqg2stan(P,Qxu,Qwv); K=lqg(P_aug,r); //K=LQG-controller spec(h_cl(P_aug,r,K)) //Closed loop should be stable //Same as Cl=P22/.K; spec(Cl('A')) s=poly(0,'s') lqg2stan(1/(s+2),eye(2,2),eye(2,2))

### See also

- lqg — LQG compensator
- lqr — LQ compensator (full state)
- lqe — linear quadratic estimator (Kalman Filter)
- obscont — observer based controller
- h_inf — Continuous time H-infinity (central) controller
- augment — augmented plant
- fstabst — Youla's parametrization of continuous time linear dynamical systems
- feedback — feedback operation

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