Elsevier

Automatica

Volume 40, Issue 1, January 2004, Pages 135-143
Automatica

Brief paper
Observer design for a class of MIMO nonlinear systems

https://doi.org/10.1016/j.automatica.2003.08.008Get rights and content

Abstract

A simple observer is proposed for a large class of MIMO nonlinear systems which includes many physical models. The main characteristic of the proposed observer lies in the easiness of its implementation and calibration. Indeed, the gain of this observer does not necessitate the resolution of any dynamical system and its expression is given. Moreover, its calibration is achieved through the choice of a single parameter. A simulation example is given in order to illustrate the performance of the proposed observer.

Introduction

The need to design observers for nonlinear systems is, from a control point of view, well understood by now. This paper deals with the design of observers for a special class of nonlinear systems satisfying some regularity assumptions. The general framework of this observer design is based on the works of Bornard and Hammouri (1991), Gauthier, Hammouri, and Othman (1992), Farza, Busawon, and Hammouri (1998) and Busawon, Farza, and Hammouri (1998).

In Gauthier et al. (1992) the authors have designed an observer for single output control affine systems which are observable for every input (uniformly observable). It is shown in Gauthier and Bornard (1981) (for a short proof see Gauthier et al. (1992)) that single output control affine systems which are observable for every input can be transformed locally almost everywhere into a canonical observable form. This canonical form is composed of a fixed linear dynamic component and a triangular controlled one. Using this canonical form, the authors in Gauthier et al. (1992) have designed a high-gain observer for such systems under some global Lipschitz assumptions on the controlled part. The gain of the proposed observer is issued from an algebraic Lyapunov equation. An extension of this observer synthesis to the multi-output case is given in Farza et al. (1998) and Busawon et al. (1998) for a class of systems which are observable for every input.

In the present paper, we use the general strategy of observer design adopted in Farza et al. (1998) and Busawon et al. (1998) to construct a high-gain observer for a larger class of nonlinear systems under similar regularity assumptions. This paper is organized as follows. The next section is devoted to the observer design for a class of multi-output nonlinear systems. In Section 3, an example is given in order to illustrate the performances of the proposed observer.

Section snippets

Observer design

Consider multi-output systems of the formẋ=F(s,x)x+G(u,s,x)+ε̄(t),y=C̄x,where the statex=x1x2xqRnwith xkRnk, k=1,…,q and p=n1n2⩾⋯⩾nq, k=1qnk=n; the input uU a compact subset of Rm, the output y∈Rp, s(t) is a known signal,G(u,s,x)=G1(u,s,x1)G2(u,s,x1,x2)Gq(u,s,x),F(s,x)=0F1(s,x1)0000F2(s,x1,x2)000Fq−1(s,x1,…,xq−1)00is a block matrix with each Fk, k=1,…,q−1, denoting a nk×nk+1 rectangular matrix,ε̄(t)=00ε(t),whereε=ε1εnqRnqwith each εi, i=1,…,nq being an unknown bounded

Example

As we have already mentioned, the class of systems (1) includes those considered in Farza et al. (1998) and Busawon et al. (1998). Moreover, the observers proposed in Farza et al. (1998) and Busawon et al. (1998) are of the variety proposed in this paper. As a result, simulation and real experiments, related to biochemical reactors (Farza, Busawon, & Hammouri 1998, Farza, Hammouri, Jallut, & Liéto 1999) and induction motors (Busawon et al., 1998), given in these papers, could serve here again

Conclusion

A simple observer has been synthesized for a large class of nonlinear systems. The gain of this observer does not require the resolution of any equation and is explicitly given. Moreover, its tuning is achieved through the choice of a single constant parameter. The proposed observer can be applied to many physical systems, namely biochemical reactors and induction motors. Corresponding results have been already presented in previous works. In order to put forward new structures accounted in the

M. Farza was born in Sfax, Tunisia, in 1963. He received his degrees of engineer and M.Sc. in computer sciences and applied Mathematics from ENSEEIHT Toulouse in 1988, his PhD in control sciences from INPG Grenoble in 1992. From 92 to 94 and 94 to 97, he worked at Laboratoire d'Automatique de Grenoble and at Laboratoire d'Automatique et de Génie des Procédés in Lyon (LAGEP), respectively. In September 1997, he joined the Laboratoire d'Automatique de Procédés (LAP) and the University of Caen as

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M. Farza was born in Sfax, Tunisia, in 1963. He received his degrees of engineer and M.Sc. in computer sciences and applied Mathematics from ENSEEIHT Toulouse in 1988, his PhD in control sciences from INPG Grenoble in 1992. From 92 to 94 and 94 to 97, he worked at Laboratoire d'Automatique de Grenoble and at Laboratoire d'Automatique et de Génie des Procédés in Lyon (LAGEP), respectively. In September 1997, he joined the Laboratoire d'Automatique de Procédés (LAP) and the University of Caen as an associated professor. His reasearch interests are in nonlinear control and systems and applications.

Mohammed M'SAAD was educated at Mohammadia School of Enginnering in Rabat (Morocco) where he held assistant professor position in 1978. He held a research position at the Laboratoire d'Electronique et d'Etude des Systèmes in Rabat where he prepared his doctor engineering degree in process control. In November 1982, Mohammed M'SAAD joined the Laboratoire d'Automatique de Grenoble to work on theory and applications of adaptive control. He received his Doctorat d'Etat-es-Sciences Physiques from the Institut National Polytechnique de Grenoble in April 1987. In March 1988, Mohammed M'SAAD held a research position at the Centre National de Recherche Scientifique. In September 1996, Mohammed M'SAAD held a professor position at the Ecole Nationale Supérieure d'ingénieurs de Caen where he is the head of the Laboratoire d'Automatique de Procédés. His main research areas are adaptive control theory, system identification, advanced control methodologies and applications, computer aided control engineering.

Mrs Rossignol was born in Avignon, France, in 1977. She received the Engineer's Degree from ISMRA-ENSICaen and the Masters’ Degree with specialization in measurement from University of Caen, in September 2000. From October 2000 to October 2003 she was a Ph.D. student in the Laboratoire d'Automatique des Procédés in Caen, France. During her doctoral thesis she developed methods of sensorless induction motor control.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate editor Alessandro Astolfi under the direction of editor Hassan Khalil.

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