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# dt_ility

detectability test

### Syntax

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

### Arguments

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")```