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# lqg2stan

LQG to standard problem

### Syntax

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

### Arguments

P

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 `P`: described by • The (instantaneous) cost function .

• The noises covariance matrix  ### 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;
P = 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 = P/.K; spec(Cl('A'))

lqg2stan(1/(%s+2), eye(2,2), eye(2,2))```

• 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

### History

 Version Description 6.0 It is no longer necessary to enter `-P` to get `P_aug` instead of `-P_aug` (bug 13751 fixed).