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Scilab help >> CACSD > lqg2stan

# lqg2stan

LQG to standard problem

### Calling Sequence

`[P,r]=lqg2stan(P22,bigQ,bigR)`

### Arguments

P22

`syslin` list (nominal plant) in state-space form

bigQ

`[Q,S;S',N]` (symmetric) weighting matrix

bigR

`[R,T;T',V]` (symmetric) covariance matrix

r

`1`x`2` row vector = (number of measurements, number of inputs) (dimension of the 2,2 part of `P`)

P

`syslin` list (augmented plant)

### Description

`lqg2stan` returns the augmented plant for linear LQG (H2) controller design.

`P22=syslin(dom,A,B2,C2)` is the nominal plant; it can be in continuous time (`dom='c'`) or discrete time (`dom='d'`).

```.
x = Ax + w1 + B2u
y = C2x + w2```

for continuous time plant.

```x[n+1]= Ax[n] + w1 + B2u
y = C2x + w2```

for discrete time plant.

The (instantaneous) cost function is `[x' u'] bigQ [x;u]`.

The covariance of `[w1;w2]` is `E[w1;w2] [w1',w2'] = bigR`

If `[B1;D21]` is a factor of `bigQ`, `[C1,D12]` is a factor of `bigR` and `[A,B2,C2,D22]` is a realization of P22, then `P` is a realization of `[A,[B1,B2],[C1,-C2],[0,D12;D21,D22]`. The (negative) feedback computed by `lqg` stabilizes `P22`, i.e. the poles of `cl=P22/.K` are stable.

### Examples

```ny=2;nu=3;nx=4;
P22=ssrand(ny,nu,nx);
bigQ=rand(nx+nu,nx+nu);bigQ=bigQ*bigQ';
bigR=rand(nx+ny,nx+ny);bigR=bigR*bigR';
[P,r]=lqg2stan(P22,bigQ,bigR);K=lqg(P,r);  //K=LQG-controller
spec(h_cl(P,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))```

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

F.D.
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