Scilab Function

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
```