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

$$\overline{A}=TA{T}^T\text{, }\overline{B}=TB,\overline{C}=C{T}^T$$

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

$$\overline{A}=\left[\begin{array}{cc}{A}_{no}& A_{12}\\ 0& A_o\end{array}\right],\overline{B}=\left[\begin{array}{c}{B}_{no}\\ B_o\end{array}\right],\overline{C}=\left[0C_o\right]$$

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

A =
     1     1
     4    -2

B =
     1    -1
     1    -1

C =
     1     0
     0     1

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