st_ility

Scilab Function
Last update : 12/01/2004

st_ility - stabilizability test

### Calling Sequence

  `[ns, [nc, [,U [,Slo] ]]]=st_ility(Sl [,tol])`

### Parameters

- `Sl` : `syslin` list (linear system)
- `ns` : integer (dimension of stabilizable subspace)
- `nc` : integer (dimension of controllable subspace `nc <= ns` )
- `U` : basis such that its `ns` (resp. `nc` ) first components span the stabilizable (resp. controllable) subspace
- `Slo` : a linear system (`syslin` list)
- `tol` : threshold for controllability detection (see contr)

### Description

`Slo=( U'*A*U, U'*B, C*U, D, U'*x0 )` (`syslin` list) displays the stabilizable form of `Sl` . Stabilizability means `ns=nx` (dim. of `A` matrix).

             [*,*,*]            [*]
    U'*A*U = [0,*,*]     U'*B = [0]
             [0,0,*]            [0]

where `(A11,B1)` (dim(A11)= `nc` ) is controllable and `A22` (dim(A22)=`ns-nc` ) is stable. "Stable" means real part of eigenvalues negative for a continuous linear system, and magnitude of eigenvalues lower than one for a discrete-time system (as defined by `syslin` ).

### Examples

    A=diag([0.9,-2,3]);B=[0;0;1];Sl=syslin('c',A,B,[]);
    [ns,nc,U]=st_ility(Sl);
    U'*A*U
    U'*B
    [ns,nc,U]=st_ility(syslin('d',A,B,[]));
    U'*A*U
    U'*B

### See Also

`dt_ility` , `contr` , `stabil` , `ssrand` ,

### Author

S. Steer INRIA 1988