obsvf (Control System Toolbox)

Compute the observability staircase form

Syntax

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

Description

If the observability matrix of (A,C) has rank

, where n is the size of A, then there exists a similarity transformation such that

where

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

where

is observable, and the eigenvalues of

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

, 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.

Example

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

Algorithm

obsvf is an M-file that implements the Staircase Algorithm of [1] by calling ctrbf and using duality.