Compute controllability staircase form
[Abar,Bbar,Cbar,T,k] = ctrbf(A,B,C)
If the controllability matrix of (A, B) has rank r ≤ n, where n is the size of A, then there exists a similarity transformation such that
where T is unitary, and the transformed system has a staircase form, in which the uncontrollable modes, if there are any, are in the upper left corner.
where (Ac, Bc) is controllable, all eigenvalues of Auc are uncontrollable, and .
[Abar,Bbar,Cbar,T,k] = ctrbf(A,B,C) decomposes
the state-space system represented by
C into the controllability staircase form,
Cbar, described above.
the similarity transformation matrix and
k is a
vector of length n, where n is
the order of the system represented by
k represents the number of controllable
states factored out during each step of the transformation matrix
calculation. The number of nonzero elements in
how many iterations were necessary to calculate
sum(k) is the number of states in Ac,
the controllable portion of
ctrbf(A,B,C,tol) uses the tolerance
calculating the controllable/uncontrollable subspaces. When the tolerance
is not specified, it defaults to
Compute the controllability staircase form for
A = 1 1 4 -2 B = 1 -1 1 -1 C = 1 0 0 1
and locate the uncontrollable mode.
[Abar,Bbar,Cbar,T,k]=ctrbf(A,B,C) Abar = -3.0000 0 -3.0000 2.0000 Bbar = 0.0000 0.0000 1.4142 -1.4142 Cbar = -0.7071 0.7071 0.7071 0.7071 T = -0.7071 0.7071 0.7071 0.7071 k = 1 0
The decomposed system
Abar shows an uncontrollable
mode located at -3 and a controllable mode located at 2.
ctrbf implements the Staircase Algorithm
 Rosenbrock, M.M., State-Space and Multivariable Theory, John Wiley, 1970.