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# fstabst

Youla's parametrization of continuous time linear dynamical systems

### Syntax

`J = fstabst(P,r)`

### Arguments

P

a continuous time linear dynamical system.

r

1x2 row vector, dimension of `P22`

J

a continuous time linear dynamical system (with same representation as `P`.

### 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 — observer based controller
• lft — linear fractional transformation
• lqg — LQG compensator
• lqg2stan — LQG to standard problem

### History

 Version Description 5.4.0 `Sl` is now checked for continuous time linear dynamical system. This modification has been introduced by this commit

### Comments

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