Control System Toolbox™

ctrbf - Compute controllability staircase form

Syntax

[Abar,Bbar,Cbar,T,k] = ctrbf(A,B,C)
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 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 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, and C into the controllability staircase form, Abar, Bbar, and Cbar, described above. T is the similarity transformation matrix and k is a vector of length n, where n is the order of the system represented by A. Each entry of k represents the number of controllable states factored out during each step of the transformation matrix calculation. The number of nonzero elements in k indicates how many iterations 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.

Example

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.

Algorithm

ctrbf is an M-file that implements the Staircase Algorithm of [1].

References

[1] Rosenbrock, M.M., State-Space and Multivariable Theory, John Wiley, 1970.

See Also

ctrb, minreal

ctrb

ctrlpref