Combining Linear Algebra and Numerical Optimization for Gray-Box Affine State-Space Model Identification | IEEE Journals & Magazine | IEEE Xplore

Combining Linear Algebra and Numerical Optimization for Gray-Box Affine State-Space Model Identification


Abstract:

Accurately estimating the parameters of a gray-box linear time-invariant state-space representation is still a challenging problem especially if the number of unknowns ex...Show More

Abstract:

Accurately estimating the parameters of a gray-box linear time-invariant state-space representation is still a challenging problem especially if the number of unknowns exceeds ten. The standard nonlinear optimization-based procedures often fail because the initial guesses are not in the domain of attraction of the user-defined cost function global minimum. Herein, a new solution is developed to give access to estimated parameters which are accurate estimates of the unknown parameters directly or, at worst, good initial guesses for any standard nonlinear optimization-based method. By assuming that 1) the gray-box model to identify is minimal and structurally identifiable, 2) the parameter dependency is affine, and 3) the available datasets have been transformed into a reliable and accurate black-box state-space model, our procedure consists in determining the unique similarity transformation between the black-box and gray-box representations of the system to identify. More specifically, we introduce a technique which first extracts a system of equations linear in terms of the similarity transformation; second, it solves this system by using standard linear algebra tools; finally, completes this two-step procedure by a numerical optimization solution (dedicated now to a very small number of unknowns) if the second step does not give access to all the similarity transformation parameters effectively. In this contribution, we first describe our new procedure, prove its capabilities of giving accurate estimates under specific practical constraints, and finally demonstrates its effectiveness with simulation examples.
Published in: IEEE Transactions on Automatic Control ( Volume: 65, Issue: 8, August 2020)
Page(s): 3272 - 3285
Date of Publication: 19 September 2019

ISSN Information:


I. Introduction

In THE literature dedicated to linear time-invariant (LTI) model identification [12], [25], many reliable techniques are now available to determine black-box state-space models effectively. For instance, the subspace-based techniques [11], [26], [27] are considered as good solutions for estimating a discrete-time [17] or continuous-time (CT) [1], [8], [18] fully parameterized state-space form as well as the system order accurately. The success of the MOESP, CVA, or N4SID-like techniques probably relies on the use of tools, such as the RQ factorization or the singular value decomposition [7] which are well known for their numerical robustness as well as the nonnecessity to involve iterative gradient-based search algorithms. In order to bypass the suboptimality of the subspace-based identification methods, it can be suggested using the subspace-based state-space matrices estimates as a reliable initial guess for a prediction-error-based technique [12] (see [16], [20], [27], [28] for further details about the theoretical and numerical properties of this standard two-step approach).

References

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