fstabst

Scilab Function
Last update : 12/01/2004

fstabst - Youla's parametrization

### Calling Sequence

  `[J]=fstabst(P,r)`

### Parameters

- `P`: `syslin` list (linear system)
- `r`: 1x2 row vector, dimension of `P22`
- `J`: `syslin` list (linear system in state-space representation)

### Description

Parameterization of all stabilizing feedbacks.

`P` is partitioned as follows:

    P=[ P11 P12;
        P21 P22]

(in state-space or transfer form: automatic conversion in state-space is done for the computations)

`r` = size of `P22` subsystem, (2,2) block of `P`

              J =[ J11 J12;
                   J21 J22]

`K` is a stabilizing controller for `P` (i.e. `P22`) iff `K=lft(J,r,Q)` with `Q` stable.

The central part of `J` , `J11` is the lqg regulator for `P`

This `J` is such that defining `T` as the 2-port `lft` of `P` and `J` : `[T,rt]=lft(P,r,J,r)` one has that `T12` is inner and `T21` is co-inner.

### 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);
    J=fstabst(P,r);
    Q=ssrand(nu,ny,1);Q('A')=-1;  //Stable Q
    K=lft(J,r,Q);
    A=h_cl(P,r,K); spec(A)

### See Also

`obscont`, `lft`, `lqg`, `lqg2stan`,