Specifications

obsvf
16-162
16obsvf
Purpose 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 Iftheobservability matrixof(A,C) hasrank , wherenis the sizeofA,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 isobservable, and the eigenvaluesof are the unobservable
modes.
[Abar,Bbar,Cbar,T,k] = obsvf(A,B,C) decomposes the state-space system
with matrices
A, B,andC 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,wheren is the number of states in
A.Eachentryofk
representsthenumberofobservable 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,andsum(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
A =
1 1
4 -2
B =
rn
≤
ATAT
T
,= BTB,= CCT
T
=
T
A
A
no
A
12
0 A
o
,=
B
B
no
B
o
,= C
0 C
o
=
C
o
A
o
,()
A
no
A
o