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 (A, B) has rank r ≤ n, where n is the size of A, then there exists a similarity transformation such that

$$\begin{array}{ccc}\overline{A}=TA{T}^T,& \overline{B}=TB,& \overline{C}=C{T}^T\end{array}$$

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.

$$\begin{array}{ccc}\overline{A}=\left[\begin{array}{cc}{A}_{uc}& 0\\ A_{21}& A_c\end{array}\right],& \overline{B}=\left[\begin{array}{l}0\\ B_c\end{array}\right],& \overline{C}=\left[C_{nc}{C}_c\right]\end{array}$$

where (A_c, B_c) is controllable, all eigenvalues of A_uc are uncontrollable, and $$C_c{(sI-A_c)}^{-1}{B}_c=C{(sI-A)}^{-1}B$$.

[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 A_c, 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.

Examples

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.

Algorithms

ctrbf implements the Staircase Algorithm of [1].

References

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

Version History

Introduced before R2006a