contr controllability, controllable subspace, staircase

## Syntax

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

## See also

- canon
- cont_mat
- unobs
- stabil
- st_ility