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See the recommended documentation of this function

Aide Scilab >> CACSD > contr

# contr

controllability, controllable subspace, staircase

### Calling Sequence

```n=contr(A,B [,tol])
[n,U]=contr(A,B [,tol])
[n,U,ind,V,Ac,Bc]=contr(A,B,[,tol])```

### Arguments

A, B

real matrices

tol

tolerance parameter

n

dimension of controllable subspace.

U

orthogonal change of basis which puts `(A,B)` in canonical form.

V

orthogonal matrix, change of basis in the control space.

Ac

block Hessenberg matrix `Ac=U'*A*U`

Bc

is `U'*B*V`.

ind

p integer vector associated with controllability indices (dimensions of subspaces `B, B+A*B,...=ind(1),ind(1)+ind(2),...`)

### Description

`[n,[U]]=contr(A,B,[tol])` gives the controllable form of an `(A,B)` pair.(`dx/dt = A x + B u` or `x(n+1) = A x(n) +b u(n)`). The `n` first columns of `U` make a basis for the controllable subspace.

If `V=U(:,1:n)`, then `V'*A*V` and `V'*B` give the controllable part of the `(A,B)` pair.

The pair `(Bc, Ac)` is in staircase controllable form.

```|B |sI-A      *  . . .  *      *       |
| 1|    11       .      .      .       |
|  |  A    sI-A    .    .      .       |
|  |   21      22    .  .      .       |
|  |        .     .     *      *       |
[U'BV|sI - U'AU] = |0 |     0    .     .                  |
|  |            A     sI-A     *       |
|  |             p,p-1    pp           |
|  |                                   |
|0 |         0          0   sI-A       |
|  |                            p+1,p+1|```

### Reference

Slicot library (see ab01od in SCI/modules/cacsd/src/slicot).

### Examples

```W=ssrand(2,3,5,list('co',3));  //cont. subspace has dim 3.
A=W("A");B=W("B");
[n,U]=contr(A,B);n
A1=U'*A*U;
spec(A1(n+1:\$,n+1:\$))  //uncontrollable modes
spec(A+B*rand(3,5))```

• canon — canonical controllable form
• cont_mat — controllability matrix
• unobs — unobservable subspace
• stabil — stabilization
• st_ility — stabilizability test