Scilab Function
Last update : 12/01/2004

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])`

### Parameters

- `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*V`
- `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 SCIDIR/routines/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))

### See Also

`canon`, `cont_mat`, `unobs`, `stabil`, `st_ility`,