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

$\begin{matrix} \bar{A} = T A T^{T}, & \bar{B} = T B, & \bar{C} = C T^{T} \end{matrix}$

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.

$\begin{matrix} \bar{A} = [\begin{matrix} A_{u c} & 0 \\ A_{21} & A_{c} \end{matrix}], & \bar{B} = [\begin{array}{l} 0 \\ B_{c} \end{array}], & \bar{C} = [C_{n c} C_{c}] \end{matrix}$

where (A_c, B_c) is controllable, all eigenvalues of A_uc are uncontrollable, and $C_{c} {(s I - A_{c})}^{- 1} B_{c} = C {(s I - A)}^{- 1} B$ .

[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

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