Section: Scientific Foundations

Subspace-based identification and detection

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 $(\lambda ,{\phi }_{\lambda })$ of the state transition matrix $F$ of a linear dynamical state-space system :

namely the $(\lambda ,{\varphi }_{\lambda })$ defined by :

The (canonical) parameter vector in that case is :

where $\Lambda$ is the vector whose elements are the eigenvalues $\lambda$, $\Phi$ is the matrix whose columns are the ${\varphi }_{\lambda }$'s, and $\mathrm{vec}$ 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  [39] . 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 $R_i\stackrel{\Delta }{=}𝐄\left(Y_k{Y}_{k-i}^T\right)$ and:

be the output covariance and Hankel matrices, respectively; and: $G\stackrel{\Delta }{=}𝐄\left(X_k{Y}_k^T\right)$. Direct computations of the $R_i$'s from the equations (9 ) lead to the well known key factorizations :

where:

are the observability and controllability matrices, respectively. The observation matrix $H$ is then found in the first block-row of the observability matrix $𝒪$. The state-transition matrix $F$ is obtained from the shift invariance property of $𝒪$. The eigenstructure $(\lambda ,{\phi }_{\lambda })$ 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:

where $\Delta \stackrel{\Delta }{=}\mathrm{diag}\left(\Lambda \right)$, and $\Lambda$ and $\Phi$ are as in (11 ). Whether a nominal parameter ${\theta }_0$ fits a given output covariance sequence ${\left(R_j\right)}_j$ is characterized by [1] :

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

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:

Residual associated with subspace identification.

Assume now that a reference ${\theta }_0$ and a new sample $Y_1,\cdots ,Y_N$ are available. For checking whether the data agree with ${\theta }_0$, the idea is to compute the empirical Hankel matrix ${\widehat{\mathscr{H}}}_{p+1,q}$:

and to define the residual vector:

Let $\theta$ be the actual parameter value for the system which generated the new data sample, and ${𝐄}_{\theta }$ be the expectation when the actual system parameter is $\theta$. From (18 ), we know that ${\zeta }_N\left({\theta }_0\right)$ has zero mean when no change occurs in $\theta$, and nonzero mean if a change occurs. Thus ${\zeta }_N\left({\theta }_0\right)$ 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] .