Parameter identification of a class of Hammerstein plants

doi:10.1016/S0005-1098(01)00010-3

Automatica

Volume 37, Issue 5, May 2001, Pages 749-756

https://doi.org/10.1016/S0005-1098(01)00010-3 Get rights and content

Abstract

We consider parameter identification of plants that can be described by a piecewise affine nonlinear element in series with linear dynamics. The identification scheme is based on a suitable plant parametrization obtained by an orthogonal expansion of the nonlinearity. Then, a persistently exciting (PE) input sequence is generated to ensure the convergence of the estimated model to the true one.

Introduction

In many situations Hammerstein model may be a good approximation for nonlinear plants. They are composed of a static nonlinear gain and linear dynamics. Hammerstein-model-based identification has been dealt with using both parametric and nonparametric methods. In the parametric approach, the nonlinear element is usually modeled by a finite-order polynomial with unknown coefficients (but known order). Based on this assumption, parameter estimation can be dealt with using correlation techniques, recursive algorithms or combinations of these, see e.g. (Billings & Fakhouri, 1979; Stoica & Söderström, 1982). In the nonparametric methods the nonlinearity is just supposed to be a continuous function (e.g. Lang 1993, Lang 1997) or a Lebesgue/Borel measurable function (e.g. Greblicki & Pawlak, 1991; Greblicki, 1989; Pawlak, 1991; Krzyzak, 1989). Plant identification is then based on two approximations: (i) the dynamics are represented by a truncated version of its impulse response, (ii) the nonlinear element is represented by a truncated series approximation involving Fourier series (Greblicki, 1989; Krzyzak, 1989), Laguerre, Legendre and Hermite polynomials (Greblicki & Pawlak, 1994; Pawlak, 1991), block-pulse functions (Kung & Shinh, 1986), polygonal approximation (Lang, 1993). The convergence of the parameter estimates has been analyzed, using stochastic tools, both for parametric and nonparametric methods. In Stoica and Söderström (1982) it is shown that consistency can be achieved, with a parametric instrumental variable method, using as input a strictly persistently exciting sequence or a white noise. Specific random inputs have been used in nonparametric methods to ensure consistency and other properties e.g. convergence in the mean integrated square error (MISE) of the nonlinear element estimates (e.g. Pawlak, 1991), uniformly and MISE (Lang, 1993).

In this paper we are considering the identification problem in the case where the nonlinear element can, as in Lang (1993), be modeled by a piecewise affine function. This approximation, together with the effect of external disturbances, is accounted for through a modeling error sequence in the global plant model. A linear-in-the-parameters plant representation, is then obtained through the expansion of the nonlinear element over a specific set of N piecewise affine functions. The greater N the smaller the modeling error (so the better the quality of the identified model). Parameter estimation is then performed by resorting to a recursive gradient algorithm augmented by a parameter projection. The last feature removes the integral behavior of the standard gradient algorithm and makes the resulting algorithm robust, with respect to the modeling error. Persistent excitation (PE) of the involved observation vector, is shown to be a sufficient condition for the parameter estimates to converge to their true values. Such a property heavily depends on the richness of the plant input. It is worth noticing that, for nonlinear plants, it is generally not clear how rich the input should be for the PE to be guaranteed. For the considered class of nonlinear static elements, a specific input is designed and shown to ensure the PE property. Consequently, parameter estimates converge in the mean close to their true values. The smaller the modeling error the closer the estimates. In the ideal case (no modeling error) the convergence becomes exponential. PE and parameter convergence are achieved here using deterministic tools while these are arrived to using fundamentally stochastic tools in Stoica and Söderström (1982). The present paper is organized as follows: the plant to be identified is described in Section 2, the identification scheme is presented in Section 3 and analyzed in Section 4. Finally, a simulation comes to illustrate the effectiveness of the proposed identification method.

Section snippets

Identification problem formulation

Class of plants: We are interested in plants that can be described by a Hammerstein model defined by $y(t)= B(q^{−1}) A(q^{−1}) u(t)+z(t), u(t)=F(v(t)),$ $A(q^{−1})=1+a_{1} q^{−1} +⋯+a_{na} q^{−na},$ $B(q^{−1})=b_{1} q^{−1} +⋯+b_{nb} q^{−nb},$ where $v(t), y(t)$ are the plant input and output, respectively; u(t) is a nonavailable internal sequence; z(t) is a bounded modeling error. The function F is supposed to be continuous, piecewise affine within some given finite interval $[v_{min}, v_{max}]$ including zero (see an example in Fig. 1). These properties can

Adaptive identification scheme design

The identification scheme is designed in three steps: (i) a plant model parametrization is obtained; (ii) the latter is based upon to estimate a plant model; and (iii) an exciting input sequence is generated.

Identification scheme analysis

First, let us introduce the following partitions: $Φ (t)=[φ_{0} (t)^{T} φ_{1} (t)^{T} ⋯φ_{ς−1} (t)^{T} φ_{ς+1} (t)^{T} ⋯φ_{N} (t)^{T}]^{T},$ $θ ̂ (t)=[θ ̂_{0} (t)^{T} θ ̂_{1} (t)^{T} ⋯ θ ̂_{ς−1} (t)^{T} θ ̂_{ς+1} (t)^{T} ⋯ θ ̂_{N} (t)^{T}]^{T},$ where $φ_{0} (t)^{T} =[−y(t−1) ⋯ −y(t−n)F_{ς} (v(t−1)) ⋯ F_{ς} (v(t−n))],$ $φ_{j} (t)^{T} =[F_{j} (v(t−1)) ⋯ F_{j} (v(t−n))](j=1,…ς−1,ς+1,…,N),$ $θ ̂_{0} (t)=[a ̂_{1} (t) ⋯ a ̂_{n−1} (t) b ̂_{1} (t) ⋯ b ̂_{n} (t)]^{T},$ $θ ̂_{j} (t)=[μ ̂_{1,j} (t) ⋯ μ ̂_{n,j} (t)]^{T} .$ Let us also introduce the following notations: $φ_{0} ′(t)=[−x(t−1)⋯−x(t−n)F_{ς} (v(t−1))⋯F_{ς} (v(t−n))]^{T} with x(t)= B(q^{−1}) A(q^{−1}) u(t),$ $Φ ′(t)=[φ_{0} ′(t)^{T} φ_{1} (t)^{T} ⋯φ_{ς−1} (t)^{T} φ_{ς+1} (t)^{T} ⋯φ_{N} (t)^{T}]^{T} .$ Comparing (4.2a) with

Conclusions

This paper has focused on nonlinear system identification based on the Hammerstein model (2.1), where the nonlinear element $F(.)$ is approximated by an affine function. The first step in designing the identification scheme, consisted in using the specific expansion , , , to get the plant representation , , , , which is linear in the unknown parameters. The latter have then estimated using the modified gradient algorithm , . Convergence of the parameter estimates to their true values has been

Fouad Giri was born in Sefrou, Morocco, in 1957. He obtained an Electrical Engineering degree from the Ecole Mohammadia d'Ingénieurs, Rabat-Morocco, in 1982, and a Ph.D. degree in Automatic Control and Signal Processing, from the Institut National Polytechnique de Grenoble, France, in 1988. He has spent long-term visits at the Laboratoire d'Automatique de Grenoble, and the University of Southern California and the Ruhr University, Bochum-Germany.

Since 1982, he has been successively Assistant

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Since 1982, he has been successively Assistant Professor and Professor in the Ecole Mohammadia d'Ingénieurs he is currently with the Laboratoire d'Automatique des Procédés, ISMRA. His research interests are in system identification, adaptive control, nonlinear control, applications of advanced control to production plants and hydraulic and electromechanical systems. He has published over 60 journal/conference papers. He is a co-author of the book, in French, feedback systems in control and regulation, volume 1: representations, analysis and performances (Eyrolles, 1993), volume 2: synthesis, applications and instrumentation (Eyrolles, 1994).

Fatima-Zahra Chaoui was born in Oujda, Morocco. She obtained an Electrical Engineering degree from the Ecole Nationale d'Electricité et de Mécanique, Casablanca-Morocco, in 1993, and in currently completing a Ph.D. degree in Automatic Control and Production systems, at the Institut National Polytechnique de Grenoble, France. She is an Assistant Professor of automatic control in the Ecole Normale d'Enseignement Technique. Her research interest are in system identification and adaptive control of linear and nonlinear systems.

Youssef Rochdi was born in Marrakech, in 1969. From 1991 to 1995, he received his undergraduate education at the Ecole Normale Supérieure de l'Enseignement Technique (ENSET) in Rabat (Morocco). In 1997, he received his Agrégation degree in Electrical Engineering, with first class honors, from the ENSET, where he is now Professor-Agrégé. He is currently completing his Ph.D. in Automatic Control. His field of interest includes identification and control of nonlinear systems.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. Vicino under the direction of Editor Torsten Söderström.

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Brief PaperParameter identification of a class of Hammerstein plants☆

Abstract

Introduction

Section snippets

Identification problem formulation

Adaptive identification scheme design

Identification scheme analysis

Conclusions

Signal Processing

Nonlinear system identification using the Hammerstein model

International Journal of Systems Sciences

Stability of adaptive controllers

On the robustness of discrete-time indirect linear adaptive controllers

Automatica

Adaptive filtering, prediction and control

Non parametric orthogonal series identification of Hammerstein systems

International Journal of Systems Sciences

Brief Paper
Parameter identification of a class of Hammerstein plants☆