stabil stabilization

## Syntax

                F = stabil(A, B, alfa)
                K = stabil(Sys, alfa, beta)

## Arguments

A

  square real matrix (`nx x nx`)

B

  real matrix (`nx x nu`)

alfa, beta

  real or complex vector (in conjugate pairs) or real number.

F

  real matrix (`nx x nu`)

Sys

  linear system (`syslin` list) (`m` inputs, `p` outputs).

K

  linear system (`p` inputs, `m` outputs)

## Description

`F=stabil(A,B,alfa)` returns a gain matrix `F` such that `A+B*F` is stable if pair `(A,B)` is stabilizable. Assignable poles are set to `alfa(1),alfa(2),...`. If `(A,B)` is not stabilizable a warning is given and assignable poles are set to `alfa(1),alfa(2),...`. If `alfa` is a number all eigenvalues are set to this `alfa` (default value is `alfa=-1`).

`K=stabil(Sys,alfa,beta)` returns `K`, a compensator for `Sys` such that `(A,B)`-controllable eigenvalues are set to `alfa` and `(C,A)`-observable eigenvalues are set to `beta`.

All assignable closed loop poles (which are given by the eigenvalues of `Aclosed=h_cl(Sys,K)` are set to `alfa(i)`'s and `beta(j)`'s.

## Examples

    // Gain:
    Sys=ssrand(0,2,5,list('st',2,3,3));
    A=Sys('A');B=Sys('B');F=stabil(A,B);
    spec(A) //2 controllable modes 2 unstable uncontrollable modes
    //and one stable uncontrollable mode
    spec(A+B*F) //the two controllable modes are set to -1.
    // Compensator:
    Sys=ssrand(3,2,5,list('st',2,3,3)); //3 outputs, 2 inputs, 5 states
    //2 controllables modes, 3 controllable or stabilizable modes.
    K=stabil(Sys,-2,-3);  //Compensator for Sys.
    spec(Sys('A'))
    spec(h_cl(Sys,K))   //K Stabilizes what can be stabilized.

## See also

- st_ility
- contr
- ppol