I4S - 2011 - Annual activity report

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I4S - 2011

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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 $(λ, φ_{λ})$ of the state transition matrix $F$ of a linear dynamical state-space system :

\{\begin{matrix} X_{k + 1} & = & F X_{k} + V_{k + 1} \\ Y_{k} & = & H X_{k} \end{matrix},

(9)

namely the $(λ, ϕ_{λ})$ defined by :

det (F - λ I) = 0, (F - λ I) φ_{λ} = 0, ϕ_{λ} \overset{Δ}{=} H φ_{λ}

(10)

The (canonical) parameter vector in that case is :

θ \overset{Δ}{=} (\begin{matrix} Λ \\ vec Φ \end{matrix})

(11)

where $Λ$ is the vector whose elements are the eigenvalues $λ$ , $Φ$ is the matrix whose columns are the $ϕ_{λ}$ '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 [37] . 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 $R_{i} \overset{Δ}{=} 𝐄 (Y_{k} Y_{k - i}^{T})$ and:

ℋ_{p + 1, q} \overset{Δ}{=} (\begin{matrix} R_{0} & R_{1} & ⋮ & R_{q - 1} \\ R_{1} & R_{2} & ⋮ & R_{q} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ R_{p} & R_{p + 1} & ⋮ & R_{p + q - 1} \end{matrix}) \overset{Δ}{=} Hank (R_{i})

(12)

be the output covariance and Hankel matrices, respectively; and: $G \overset{Δ}{=} 𝐄 (X_{k} Y_{k}^{T})$ . Direct computations of the $R_{i}$ 's from the equations (9 ) lead to the well known key factorizations :

\begin{matrix} R_{i} & = & H F^{i} G \\ ℋ_{p + 1, q} & = & 𝒪_{p + 1} (H, F) 𝒞_{q} (F, G) \end{matrix}

(13)

where:

𝒪_{p + 1} (H, F) \overset{Δ}{=} (\begin{matrix} H \\ H F \\ ⋮ \\ H F^{p} \end{matrix}) and 𝒞_{q} (F, G) \overset{Δ}{=} (G F G \dots 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 $𝒪$ . The state-transition matrix $F$ is obtained from the shift invariance property of $𝒪$ . The eigenstructure $(λ, φ_{λ})$ 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:

𝒪_{p + 1} (θ) = (\begin{matrix} Φ \\ Φ Δ \\ ⋮ \\ Φ Δ^{p} \end{matrix})

(15)

where $Δ \overset{Δ}{=} diag (Λ)$ , and $Λ$ and $Φ$ are as in (11 ). Whether a nominal parameter $θ_{0}$ fits a given output covariance sequence ${(R_{j})}_{j}$ is characterized by [1] :

𝒪_{p + 1} (θ_{0}) and ℋ_{p + 1, q} have the same left kernel space.

(16)

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

U^{T} U = I_{s} and U^{T} 𝒪_{p + 1} (θ_{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 $θ_{0}$ . Then the characterization writes:

U {(θ_{0})}^{T} ℋ_{p + 1, q} = 0

(18)

Residual associated with subspace identification.

Assume now that a reference $θ_{0}$ and a new sample $Y_{1}, \dots, Y_{N}$ are available. For checking whether the data agree with $θ_{0}$ , the idea is to compute the empirical Hankel matrix ${\hat{ℋ}}_{p + 1, q}$ :

{\hat{ℋ}}_{p + 1, q} \overset{Δ}{=} Hank ({\hat{R}}_{i}), {\hat{R}}_{i} \overset{Δ}{=} 1 / (N - i) \sum_{k = i + 1}^{N} Y_{k} Y_{k - i}^{T}

(19)

and to define the residual vector:

ζ_{N} (θ_{0}) \overset{Δ}{=} \sqrt{N} vec (U {(θ_{0})}^{T} {\hat{ℋ}}_{p + 1, q})

(20)

Let $θ$ be the actual parameter value for the system which generated the new data sample, and $𝐄_{θ}$ be the expectation when the actual system parameter is $θ$ . From (18 ), we know that $ζ_{N} (θ_{0})$ has zero mean when no change occurs in $θ$ , and nonzero mean if a change occurs. Thus $ζ_{N} (θ_{0})$ 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 $θ$ 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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