obsvf

Compute observability staircase form

Syntax

[Abar,Bbar,Cbar,T,k] = obsvf(A,B,C) obsvf(A,B,C,tol)

Description

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

$\bar{A} = T A T^{T}, \bar{B} = T B, \bar{C} = C T^{T}$

where T is unitary and the transformed system has a staircase form with the unobservable modes, if any, in the upper left corner.

$\bar{A} = [\begin{matrix} A_{n o} & A_{12} \\ 0 & A_{o} \end{matrix}], \bar{B} = [\begin{matrix} B_{n o} \\ B_{o} \end{matrix}], \bar{C} = [0 C_{o}]$

where (C_o, A_o) is observable, and the eigenvalues of A_no are the unobservable modes.

[Abar,Bbar,Cbar,T,k] = obsvf(A,B,C) decomposes the state-space system with matrices A, B, and C into the observability staircase form Abar, Bbar, and Cbar, as described above. T is the similarity transformation matrix and k is a vector of length n, where n is the number of states in A. Each entry of k represents the number of observable states factored out during each step of the transformation matrix calculation [1]. 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_o, the observable portion of Abar.

obsvf(A,B,C,tol) uses the tolerance tol when calculating the observable/unobservable subspaces. When the tolerance is not specified, it defaults to 10*n*norm(a,1)*eps.

Examples

Form the observability staircase form of

by typing

[Abar,Bbar,Cbar,T,k] = obsvf(A,B,C)
Abar =
     1     1
     4    -2
Bbar =
     1     1
     1    -1
Cbar =
     1     0
     0     1
T =
     1     0
     0     1
k =
     2     0

Algorithms

obsvf implements the Staircase Algorithm of [1] by calling ctrbf and using duality.

References

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

Version History

Introduced before R2006a