User`s guide

ctrbf
11-45
11ctrbf
Purpose Compute the controllability staircase form
Syntax [Abar,Bbar,Cbar,T,k] = ctrbf(A,B,C)
[Abar,Bbar,Cbar,T,k] = ctrbf(A,B,C,tol)
Description If the controllability matrix of has rank , where n is the size of
, then there exists a similarity transformation such that
where isunitary, and thetransformed systemhas a staircaseform, inw hich
the uncontrollable modes, if there are any, are in the upper left corner.
where is controllable, all eigenvalues of are uncontrollable, and
[Abar,Bbar,Cbar,T,k] = ctrbf(A,B,C) decomposes the state-space system
represented by
A, B,andCinto the controllability staircase form, Abar, Bbar,
and
Cbar, described above. T is the similarity transformation matrix and k is a
vector of length n,wherenis the order of the system represented by
A.Each
entryof
k represents the number ofcontrol lablestatesfac toredout duringeach
stepof thetransformationmatrixcalculation. Thenumber ofnonzero elements
in
k indicates how many ite rations were necessary to calculate T, and sum(k) is
the number of states in , the controllable portion of
Abar.
ctrbf(A,B,C,tol) uses the tolerance tol when calculating the controllable/
uncontrollable subspaces. When the tolerance is not specified, it defaults to
10*n*norm(A,1)*eps.
AB
,()
rn
≤
A
ATAT
T
,= BTB,= CCT
T
=
T
A
A
uc
0
A
21
A
c
,=
B
0
B
c
,= C
C
nc
C
c
=
A
c
B
c
,()
A
uc
C
c
sI A
c
–()
1
–
B
c
CsI A–()
1
–
B.=
A
c