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.TH CONTR G "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
contr - controllability, controllable subspace
.SH CALLING SEQUENCE
.nf
[n [,U]]=contr(A,B [,tol])
[A1,B1,U,ind]=contr(A,B [,tol])
.fi
.SH PARAMETERS
.TP 6
A, B
: real matrices
.TP
tol
: may be the constant rtol or the 2 vector \fV[rtol atol]\fR
.TP 10
rtol
:tolerance used when evaluating ranks (QR factorizations).
.TP
atol
:absolute tolerance (the \fVB\fR matrix is assumed to be 0 if \fVnorm(B)<atol\fR)
.TP
n
: dimension of controllable subspace.
.TP
U
: orthogonal change of basis which puts \fV(A,B)\fR in canonical form.
.TP
A1
: block Hessenberg matrix
.TP
B1
: is \fVU'*B\fR.
.TP
ind
: vector associated with controllability indices (dimensions of subspaces \fVB,
B+A*B,...=ind(1),ind(1)+ind(2),...\fR)
.SH DESCRIPTION
\fV[n,[U]]=contr(A,B,[tol])\fR gives the controllable form of an \fV(A,B)\fR
pair.(\fVdx/dt = A x + B u\fR or \fVx(n+1) = A x(n) +b u(n)\fR).
The \fVn\fR first columns of \fVU\fR make a basis for the controllable
subspace.
.LP
If \fVV=U(:,1:n)\fR, then \fVV'*A*V\fR and \fVV'*B\fR give the controllable part
of the \fV(A,B)\fR pair.
.LP
\fV[A1,B1,U,ind]=contr(A,B)\fR returns the Hessenberg controllable
form of \fV(A,B)\fR.
.SH EXAMPLE
.nf
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))
.fi
.SH SEE ALSO
canon, cont_mat, unobs, stabil
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