Team-i4s:Subspace-based identification and detection

Inria / Raweb 2009
Presentation of the Team i4s

Section: Scientific Foundations

Subspace-based identification and detection

See module 6.5 .

For reasons closely related to the vibrations monitoring applications described in module 4.2 , we have been investigating subspace-based methods, for both the identification and the monitoring of the eigenstructure $Im34 ${(\#955 ,\#966 _\#955 )}$$ of the state transition matrix F of a linear dynamical state-space system :

$Im35 ${\mfenced o={ \mtable{...},}$$

(9)

namely the $Im36 ${(\#955 ,\#981 _\#955 )}$$ defined by :

$Im37 ${det~{(F-\#955 ~I)}=0,~~{(F-\#955 ~I)}~\#966 _\#955 =0,~~\#981 _\#955 \mover =\#916 H~\#966 _\#955 }$$

(10)

The (canonical) parameter vector in that case is :

$Im38 ${\#952 \mover =\#916 \mfenced o=( c=) \mtable{...}}$$

(11)

where $\upper_lambda$ is the vector whose elements are the eigenvalues $\lambda$ , $\upper_phi$ is the matrix whose columns are the $Im39 $\#981 _\#955 $$ 's, and vec is the column stacking operator.

Subspace-based methods is the generic name for linear systems identification algorithms based on either time domain measurements or output covariance matrices, in which different subspaces of Gaussian random vectors play a key role [56] . A contribution of ours, minor but extremely fruitful, has been to write the output-only covariance-driven subspace identification method under a form that involves a parameter estimating function, from which we define a residual adapted to vibration monitoring [1] . This is explained next.

Covariance-driven subspace identification.

Let $Im40 ${R_i\mover =\#916 \#119812 \mfenced o=( c=) Y_k~Y_{k-i}^T}$$ and:

$Im41 ${\#8459 _{p+1,q}\mover =\#916 \mfenced o=( c=) \mtable{...}\mover =\#916 Hank\mfenced o=( c=) R_i}$$

(12)

be the output covariance and Hankel matrices, respectively; and: $Im42 ${G\mover =\#916 \#119812 \mfenced o=( c=) X_kY_k^T}$$ . Direct computations of the R_i 's from the equations (9 ) lead to the well known key factorizations :

$Im43 $\mtable{...}$$

(13)

where:

$Im44 ${\#119978 _{p+1}{(H,F)}\mover =\#916 \mfenced o=( c=) \mtable{...}~~~\mtext and~~~\#119966 _q{(F,G)}~\mover =\#916 ~{(G~FG~\#8943 ~F^{q-1}G)}}$$

(14)

are the observability and controllability matrices, respectively. The observation matrix H is then found in the first block-row of the observability matrix $Im45 $\#119978 $$ . The state-transition matrix F is obtained from the shift invariance property of $Im45 $\#119978 $$ . The eigenstructure $Im34 ${(\#955 ,\#966 _\#955 )}$$ then results from (10 ).

Since the actual model order is generally not known, this procedure is run with increasing model orders.

Model parameter characterization.

Choosing the eigenvectors of matrix F as a basis for the state space of model (9 ) yields the following representation of the observability matrix:

$Im46 ${\#119978 _{p+1}{(\#952 )}=\mfenced o=( c=) \mtable{...}}$$

(15)

where $Im47 ${\#916 \mover =\#916 diag{(\#923 )}}$$ , and $\upper_lambda$ and $\upper_phi$ are as in (11 ). Whether a nominal parameter $\theta$ ₀ fits a given output covariance sequence (R_j)_j is characterized by [1] :

$Im48 ${\#119978 _{p+1}{(\#952 _0)}~~\mtext and~~\#8459 _{p+1,q}~~\mtext have~\mtext the~\mtext same~\mtext left~\mtext kernel~\mtext space.}$$

(16)

This property can be checked as follows. From the nominal $\theta$ ₀ , compute $Im49 ${\#119978 _{p+1}{(\#952 _0)}}$$ using (15 ), and perform e.g. a singular value decomposition (SVD) of $Im49 ${\#119978 _{p+1}{(\#952 _0)}}$$ for extracting a matrix U such that:

$Im50 ${U^T~U=I_s~~\mtext and~~U^T~\#119978 _{p+1}{(\#952 _0)}=0}$$

(17)

Matrix U is not unique (two such matrices relate through a post-multiplication with an orthonormal matrix), but can be regarded as a function of $\theta$ ₀ . Then the characterization writes:

$Im51 ${U{(\#952 _0)}^T~\#8459 _{p+1,q}=0}$$

(18)

Residual associated with subspace identification.

Assume now that a reference $\theta$ ₀ and a new sample $Im52 ${Y_1,\#8943 ,Y_N}$$ are available. For checking whether the data agree with $\theta$ ₀ , the idea is to compute the empirical Hankel matrix $Im53 $\mover \#8459 ^_{p+1,q}$$ :

$Im54 ${\mover \#8459 ^_{p+1,q}\mover =\#916 Hank\mfenced o=( c=) \mover R^_i,~~~\mover R^_i\mover =\#916 1/{(N-i)}~\munderover \#8721 {k=i+1}NY_k~Y_{k-i}^T}$$

(19)

and to define the residual vector:

$Im55 ${\#950 _N{(\#952 _0)}\mover =\#916 \sqrt N~vec\mfenced o=( c=) U{(\#952 _0)}^T~\mover \#8459 ^_{p+1,q}}$$

(20)

Let $\theta$ be the actual parameter value for the system which generated the new data sample, and $Im56 $\#119812 _\#952 $$ be the expectation when the actual system parameter is $\theta$ . From (18 ), we know that $\zeta$ _N( $\theta$ ₀) has zero mean when no change occurs in $\theta$ , and nonzero mean if a change occurs. Thus $\zeta$ _N( $\theta$ ₀) plays the role of a residual.

It is our experience that this residual has highly interesting properties, both for damage detection [1] and localization [3] , and for flutter monitoring [8] .

Other uses of the key factorizations.

Factorization ( 3.5.1 ) is the key for a characterization of the canonical parameter vector $\theta$ in (11 ), and for deriving the residual. Factorization (13 ) is also the key for :

Proving consistency and robustness results [6] ;
Designing an extension of covariance-driven subspace identification algorithm adapted to the presence and fusion of non-simultaneously recorded multiple sensors setups [7] ;
Proving the consistency and robustness of this extension [9] ;
Designing various forms of input-output covariance-driven subspace identification algorithms adapted to the presence of both known inputs and unknown excitations [10] .

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