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Ajuda do Scilab >> CACSD > Linear Analysis > Stability > st_ility

# st_ility

stabilizability test

### Syntax

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

### Arguments

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 — detectability test
• contr — controllability, controllable subspace, staircase
• stabil — stabilization
• ssrand — gerador de sistema randômico
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