Subspace-based methods for eigenstructure identification and diagnosis
The
problem is to identify, or diagnose changes in, the eigenstructure
of a linear system of the form
X(n) = F
X(n-1) + V(n)
Y(n) = H X(n-1) +
W(n)
where V is the input excitation
noise, W is the measurement noise, X is the state, and
Y
is the measurement. Only Y is observed. The eigenstructure consists
of the collection of eigenvalues of F, and associated eigenvectors
premultiplied
by
the observation matrix H.
Eigenstructure
identification and diagnosis is, for instance, of interest in
vibration
mechanics , where this eigenstructre represents the modal information
about the structure, and output only identification and monitoring is considered,
i.e., the input excitation is natural and not measured.
Subspace algorithms for eigenstructure identification[BAB2000]
use the covariance Hankel matrix H of the observation
as a starting point:
Hij = cov( Y(n),
YT(n-i-j) )
and uses the fact that H and the observability matrix
of the pair (H,F) possess identical left kernel space (hence
the name of the technique). Since the left kernel of pair (H,F)
characterize the latter up to a change of basis in the state space, it
is enough to compute this left kernel, and this can be performed by using
the Hankel matrix H. More precisely, an abstract presentation
of the method can be the following :
-
estimate H based on available output samples ;
-
compute a matrix S of maximal rank, satisfying STH=
0 ;
-
find a pair (H,F) such that its observability matrix O(H,F)
satisfies
also STO(H,F) = 0.
Subspace algorithms for eigenstructure monitoring
and diagnosis [BAB2000]
are derived from the former one as follows :
-
a nominal pair (H,F) is assumed, with its associated observability
matrix O(H,F) ;
-
compute a matrix S of maximal rank, satisfying STO(H,F)
= 0 ;
-
estimate H based on current available output samples
;
-
confront H and S by computing STH,
and check whether or not STH = 0
holds.
Since the Hankel matrix H contains empirical covariance
estimates, it is contaminated by noise and therefore the quantity STH
is a random variable and checking whether or not STH
= 0 holds must be regarded as a statistical testing problem.
We handle this by showing that, for a long sample, the quantity STH
is approximately Gaussian, with a known covariance (reflecting the uncertainty
due to excitation and measurement noises), and finally it remains to test
if this Gaussian quantity is zero mean or not.
Important properties of subspace algorithms
are
listed now :
-
[BF85][BAB2000]
The above algorithms for eigenstructure identification and monitoring are
still valid if the input excitation noise V is nonstationary,
i.e., has a time-varying covariance matrix; equivalently, it amounts
to considering observations Y with fixed eigenstructure and
time-varying
zeroes.
-
[MBaBeG01]
[BaBeGHMvdA2001]
[MBaBeG02a]
[MBaBeG02b]
There exists a variation of the above subspace identification algorithm,
which can be used with moving sensors; this means that a subset of the
sensors are kept fixed (the so-called "references") and the other are moved
for the successive records. This measurement technique is frequently used
in vibration mechanics, it allows to reconstruct, with fewer sensors, a
large number of eigenvector coordinates.