Olivier Prot

Identification of Parameterized Gray-Box State-Space Systems: From a Black-Box Linear Time-Invariant Representation to a Structured One

While determining the order as well as the matrices of a black-box linear state-space model is now an easy problem to solve, it is well-known that the estimated (fully parameterized) state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. By assuming that the system has been identified consistently in a fully parameterized form, the question addressed in this paper then is how to recover the physical parameters from this initially estimated black-box form. Two solutions to solve such a parameterization problem are more precisely introduced. First, a solution based on a null-space-based reformulation of a set of equations arising from the aforementioned similarity transformation problem is considered. Second, an algorithm dedicated to nonsmooth optimization is presented to transform the initial fully parameterized model into the structured state-space parameterization of the system to be identified. A specific constraint on the similarity transformation between both system representations is added to avoid singularity. By assuming that the physical state-space form is identifiable and the initial fully parameterized model is consistent, it is proved that the global solutions of these two optimization problems are unique. The proposed algorithms are presented, along with an example of a physical system.

A Null-Space-based Technique for the Estimation of Linear-Time Invariant Structured State-Space Representations*

Abstract Estimating the order as well as the matrices of a linear state-space model is now an easy problem to solve. However, it is well-known that the state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. The question addressed in this paper then is, how to recover the physical parameters once the system has been identified in a fully-parameterized form. The novelty of our approach is on transforming the bilinear equations arising from the similarity transformation equations as a null-space problem. We show that the null-space of a certain matrix contains the physical parameters. Extracting the physical parameters then requires the solution of a non-convex optimization problem in a reduced dimensional space. By assuming that the physical state-space form is identifiable and the initial fully-parameterized model is consistent, the solution of this optimization problem is unique. The proposed algorithm is presented, along with an example of a physical system.

Linear fractional LPV model identification from local experiments: An H ∞ -based optimization technique

In this paper, a new identification technique is introduced to estimate a linear fractional representation of a linear parameter-varying (LPV) system from local experiments by using a dedicated non-smooth optimization procedure. More precisely, the developed approach consists in estimating the parameters of an LPV state-space model from local fully-parameterized identified state-space models through the non-smooth optimization of a specific H∞-based criterion. The method presented in this paper results directly in an LPV model whose parametric matrices can be rational functions of the scheduling variables without any interpolation step (required usually by the local approach) and without writing the local fully-parameterized LTI state-space models with respect to a coherent basis. A numerical example is used to illustrate the performance of the suggested technique.

Analytical Modelling and Grey-box Identification of a Flexible Arm using a Linear Parameter-varying Model

Abstract In this paper, a methodology is investigated for the determination of a dynamical model of robotic manipulators. Rather than building a model either from the law of Physics or from experimental data independently, a combination of an analytical and an experimental approach is performed in order to identify a linear parameter-varying (LPV) model of the system. A local approach dedicated to LPV models is developed. As a sample, the case of a 2-DOF flexible manipulator is addressed. The identification procedure is evaluated from simulated data. This study shows that an LPV model with polynomial variation of the state equation matrices with respect to one scheduling parameter provides good results in terms of output fit.

Initialization of Gradient-based Optimization Algorithms for the Identification of Structured State-Space Models

Abstract Estimating consistent parameters of a structured (grey-box) state-space representation requires a reliable initialization when the vector of parameters is computed by using a gradient-based algorithm. Assuming that a reliable initial fully-parameterized state-space model is available, an algorithm dedicated to non-smooth optimization is introduced in this paper in order to transform this initial model into the structured state-space parameterization of the system to be identified. A specific constraint on the similarity transformation between both system representations is added to avoid singularity. Numerical examples highlight the performance of the developed approach.

H∞-norm-based optimization for the identification of gray-box LTI state-space model parameters

In this paper, the challenging problem of determining the unknown parameters of an identifiable LTI state-space representation of a stable system is addressed by resorting to a specific H∞-norm-based optimization algorithm. More specifically, by assuming the availability of a reliable fully-parameterized representation of the system to identify, the algorithm developed herein consists in restructuring this initial black-box representation of the system dynamics via the optimization of a dedicated maximum eigenvalue-based criterion. This study shows that this H∞-norm-based approach can be seen as a good solution for the identification of gray-box LTI state-space representations and, by extension, as an interesting alternative or a reliable initialization step for the standard output-error techniques.

New developments for matrix fraction descriptions

This article aims at giving a new answer for the challenging problem of the parametrisation of multi-input multi-output matrix fraction descriptions. In order to reach this goal, new parametrisations of matrix fraction descriptions, called fully-parametrised left matrix fraction descriptions (F-LMFD), are first introduced. Their structural properties as well as their suitability for multi-input multi-output model description are more precisely analysed. As any over-parametrised model description, the F-LMFD cannot describe a transfer function uniquely. The structure of the space of equivalent F-LMFD is then investigated through the determination of its basis. The study carried out in this article is the prelude to a computational improvement of the identification of matrix fraction descriptions with gradient-based optimisation methods.

Comparison of a gradient-based algorithm and a proximity control algorithm for gray-box LTI model identification

In this paper, the challenging problem of determining the unknown parameters of an identifiable LTI statespace representation of a stable system is addressed by resorting to a specific H∞-norm-based optimization algorithm. More specifically, by assuming the availability of a reliable fully-parameterized representation of the system to identify, the algorithm developed herein consists in restructuring this initial black-box representation of the system dynamics via the optimization of a dedicated maximum eigenvalue-based criterion. This study shows that this H∞-norm-based approach can be seen as a good solution for the estimation of gray-box LTI state-space forms and, by extension, as an interesting alternative or a reliable initialization step for the standard output-error techniques.