Scilab Function
Last update : 00/00/0000

dt_ility - detectability test

### Calling Sequence

  `[k, [n [,U [,Sld ] ] ]]=dt_ility(Sl [,tol])`

### Parameters

- `Sl`: linear system (`syslin` list)
- `n`: dimension of unobservable subspace
- `k`: dimension of unstable, unobservable subspace (`k<=n`).
- `U`: orthogonal matrix
- `Sld`: linear system (`syslin` list)
- `tol`: threshold for controllability test.

### Description

Detectability test for `sl`, a linear system in state-space representation. `U` is a basis whose `k` first columns span the unstable, unobservable subspace of `Sl` (intersection of unobservable subspace of `(A,C)` and unstable subspace of `A`). Detectability means `k=0`.

`Sld = (U'*A*U,U'*B,C*U,D)` displays the "detectable part" of `Sl=(A,B,C,D)`, i.e.

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

    C*U = [0,0,*]

with `(A33,C3)` observable (dimension `nx-n`), `A22` stable (dimension `n-k`) and `A11` unstable (dimension `k`).

### Examples

    A=[2,1,1;0,-2,1;0,0,3];
    C=[0,0,1];
    X=rand(3,3);A=inv(X)*A*X;C=C*X;
    W=syslin('c',A,[],C);
    [k,n,U,W1]=dt_ility(W);
    W1("A")
    W1("C")

### See Also

`contr`, `st_ility`, `unobs`, `stabil`,