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 rankr≤n, wherenis the size ofA, then there exists a similarity transformation such that

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

whereTis unitary, and the transformed system has astaircaseform, 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 ofA_ucare 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 byA,B, andCinto the controllability staircase form,Abar,Bbar, andCbar, described above.Tis the similarity transformation matrix andkis a vector of lengthn, wherenis the order of the system represented byA. Each entry ofkrepresents the number of controllable states factored out during each step of the transformation matrix calculation. The number of nonzero elements inkindicates how many iterations were necessary to calculateT, andsum(k)is the number of states inA_c, the controllable portion ofAbar.

ctrbf(A,B,C,tol)uses the tolerancetolwhen calculating the controllable/uncontrollable subspaces. When the tolerance is not specified, it defaults to10*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 systemAbarshows an uncontrollable mode located at -3 and a controllable mode located at 2.

Algorithms

ctrbfimplements the Staircase Algorithm of(1].

References

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

版本历史

Introduced before R2006a