On nonparametric estimation of nonlinear dynamic systems by the Fourier series estimate

doi:10.1016/0165-1684(96)00067-9

Signal Processing

Volume 52, Issue 3, August 1996, Pages 299-321

https://doi.org/10.1016/0165-1684(96)00067-9 Get rights and content

Abstract

The paper deals with estimation of block oriented systems by the Fourier series estimate. Three types of systems are considered: cascade of nonlinear memoryless systems, Hammerstein system and Wiener system. The optimal model of memoryless cascade system is derived and estimated by the Fourier series regression estimate. Nonlinear dynamical systems of Hammerstein and Wiener type with arbitrary nonlinearities are estimated by means of nonparametric techniques. Convergence and the rates of estimation procedures are investigated. Convergence results are verified in computer experiments.

Zusammenfassung

Dieser Artikel behandelt die Schätzung blockorientierter Systeme mit Hilfe des Fourierreihen-Schätzers. Drei Typen von Systemen werden betrachtet, nämlich die Serienschaltung nichtlinearer gedächtnisloser Systeme, das Hammerstein-System und das Wiener-System. Das optimale Modell eines gedächtnislosen Serienschaltungs-Systems wird abgeleitet und mit Hilfe des Fourierreihen-Regressionsschätzers geschätzt. Nichtlineare dynamische Systeme des Hammerstein- und Wiener-Typs mit beliebigen Nichtlinearitäten werden mittels nichtparametrischer Methoden geschätzt. Die Konvergenz und die Geschwindigkeit von Schätzprozeduren werden untersucht. Die Konvergenzergebnisse werden durch Computer-Experimente bestätigt.

Résumé

Cet article parle de l'estimation de systèmes orientés par blocs à l'aide de l'estimée de la série de Fourier. Trois types de systèmes sont considérés: une cascade de systèmes nonlinéaires sans mémoire, le système de Hammerstein et le système de Wiener. Le modèle optimal de système en cascade sans mémoire est dérivé et évalué, à l'aide de l'estimée de la régression de la série de Fourier. Les systèmes dynamiques nonlinéaires de types Hammerstein et Wiener avec des nonlinéarites arbitraires sont estimés au moyen de techniques non paramétriques. La convergence et les taux des différentes procédures d'estimation sont étudiés. Les résultats de convergence sont vérifiés par des simulations informatiques.

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Over the past two decades, various methods for identification of Hammerstein systems have been studied extensively. These can be roughly classified into several categories, such as the iterative method (Liu & Bai, 2007; Narendra & Gallman, 1966; Rangan, Wolodkin, & Polla, 1995; Stoica, 1981; Voros, 1997), the over-parameterization method (Chang & Luus, 1971; Ding, Chen, & Iwai, 2007a; Hsia, 1976), the blind approach (Bai & Fu, 2002), the subspace method (Verhaegen & Westwick, 1996), the least squares method (Ding & Chen, 2005; Goethals, Pelckmans, Suykens, & Moor, 2005), the parametric instrumental variables method (Laurain, Gilson, & Garnier, 2009; Stoica & Soderstrom, 1981), the stochastic method (Bilings & Fakhouri, 1978; Greblicki, 1996; Pawlak, 1991) and the non-parametric identification method (Bai, 2003; Greblicki & Pawlak, 1987; Krzyak, 1993, 1996). Recently, a decomposition-based Newton iterative identification approach for a Hammerstein nonlinear FIR system with ARMA noise was presented by Ding, Deng, and Liu (2014).
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Bounded error identification of Hammerstein systems through sparse polynomial optimization
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In this paper we present a procedure for the evaluation of bounds on the parameters of Hammerstein systems, from output measurements affected by bounded errors. The identification problem is formulated in terms of polynomial optimization, and relaxation techniques, based on linear matrix inequalities, are proposed to evaluate parameter bounds by means of convex optimization. The structured sparsity of the formulated identification problem is exploited to reduce the computational complexity of the convex relaxed problem. Analysis of convergence properties and computational complexity is reported.
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Therefore, once different orders of output spectrum are achieved, many existing algorithms (e.g., Least Square) can be employed to estimate model parameters involved in each order output spectrum, and finally the nonlinear system model could be obtained. Although frequency domain identification of block-oriented nonlinear systems has been discussed in the literature, existing methods usually assume known structure and invertibility of nonlinearity, or require the linear part to be a rational transfer function, or involve many repeated experiments with different magnitudes and frequencies restrictively using sinusoidal input signals [32,35–38,40]. In this section, to estimate different orders of output spectrum, no priori knowledge is required for model structure and orders.
Block-oriented nonlinear models including Wiener models, Hammerstein models and Wiener–Hammerstein models, etc. have been extensively applied in practice for system identification, signal processing and control. In this study, analytical frequency response functions including generalized frequency response functions (GFRFs) and nonlinear output spectrum of block-oriented nonlinear systems are developed, which can demonstrate clearly the relationship between frequency response functions and model parameters, and also the dependence of frequency response functions on the linear part of the model. The nonlinear part of these models can be a more general multivariate polynomial function. These fundamental results provide a significant insight into the analysis and design of block-oriented nonlinear systems. Effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems. Compared with some existing frequency domain identification methods, the new estimation algorithms do not necessarily require model structure information, not need the invertibility of the nonlinearity and not restrict to harmonic inputs. Simulation examples are given to illustrate these new results.
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Hammerstein system identification from measurements affected by bounded noise is considered in the paper. First, we show that computation of tight parameter bounds requires the solution to nonconvex optimization problems where the number of decision variables increases with the length of the experimental data sequence. Then, in order to reduce the computational burden of the identification problem, we propose a procedure to relax the previously formulated problem to a set of polynomial optimization problems where the number of variables does not depend on the size of the measurements sequence. Advantages of the presented approach with respect to previously published results are discussed.
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We apply nonparametric techniques to identify nonlinear dynamic block-oriented systems of Hammerstein type. Hammerstein system consists of a memoryless nonlinear system followed by a dynamic, linear system. We introduce identification algorithms based on input-output observations for both systems and study their convergence and the rates. The performance of identification algorithms is validated in simulation studies. We apply Hammerstein system identification algorithms to identification of nonlinearities in a flexible robot manipulator.
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Nonlinear dynamic systems of Hammerstein type are identified from input and output measurements. Identification algorithms for a memoryless nonlinear part and for a linear dynamic part are proposed. Convergence and rates of convergence of the algorithms are investigated. The class of nonlinearities considered in the paper is very large and cannot be parameterized therefore nonparametric approach is used. The performance of identification algorithms is studied in simulation experiments.

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¹: This research was supported by NSERC grant OGP000270, FCAR grant 92-RE-0147, Alexander von Humboldt Fellowship, and by the Vinberg Scholarship, while the author spent sabbatical leave at Computer Science Department at Technion-Israel Institute of Technology. The author gratefully acknowledges the support of the Frontier Research Program RIKEN, Artificial Brain Systems Laboratory, Wakoshi, Japan.

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PaperOn nonparametric estimation of nonlinear dynamic systems by the Fourier series estimate

Abstract

Zusammenfassung

Résumé

Automatica

J. Franklin Institute

Automatica

Identification of nonlinear systems — A survey

Non-linear system identification using the Hammerstein model

Internat. J. Syst. Sci.

Conditions necessaires et suffisantes de convergence pour une classe d'estimateurs de la densite

C. R. Acad. Sci. Paris

A noniterative method for identification using Hammerstein system

IEEE Trans. Automat. Control

Analysis and parameter estimation of nonlinear systems with Hammerstein model using Taylor series approach

IEEE Trans. Circuits Systems

Identification of complex-cell intensive nonlinearities in a cascade model of cat visual cortex

Biological Cybernetics

Use of Hammerstein models in the identification of nonlinear systems

Amer. Inst. Chem. Eng. J.

Nonlinear neuronal mode analysis of action potential encoding in the cockroach tactile spine neuron

Biological Cybernetics

An iterative method for identification of nonlinear systems using a Uryson model

IEEE Trans. Automat. Control.

Nonparametric identification of Wiener systems

IEEE Trans. Inform. Theory

Non-parametric identification of a memoryless system with cascade structure

Internat. J. Syst. Science

Fourier and Hermite series estimates of regression functions

Ann. Inst. Statist. Math.

Identification of discrete Hammerstein systems using kernel regression estimates

IEEE Trans. Automat. Control

Identifiability of large-scale interconnected linear zero-memory systems

Internat. J. Systems Sci.

Paper
On nonparametric estimation of nonlinear dynamic systems by the Fourier series estimate