A Differential Algebraic Approach - ScienceDirect

7 downloads 0 Views 466KB Size Report
Class (BSC) in the Diop-Fliess' Observability sense. A differential algebraic approach is proposed for the estimation of the state of a class of bilinear system.
Mathl. Comput.

08957177(94)00190-1

Modelling Vol. 20, No. 12, pp. 125-132, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 08957177194 $9.50 + 0.00

Observers for a Multi-Input Multi-Output Bilinear System Class: A Differential Algebraic Approach R. Universidad

MARTINEZ-GUERRA* Aut6noma

Departamento Apartado

Postal

Metropolitana-Iztapalapa de Matem&icas

D.F., 09340, Mexico rafaQxanum.uam.mx 55-534,

MBxico,

J. DE LI?ON-MORALES Universidad Aut6noma de Nuevo Le6n Facultad de Ingenieria Mechnica y El&trica Apartado Postal 148-F San Nicol& de 10s Garza, Nuevo Le6n, CP. 66451, Mexico (Received

and accepted

July

1994)

Abstract-h

exponential observer is constructed for a multi-output observable Bilinear System Class (BSC) in the Diop-Fliess’ Observability sense. A differential algebraic approach is proposed for the estimation of the state of a class of bilinear system. A result on Multi-output Translated Fliess’ Generalized Observability Canonical Form (MTGOCF) is given. Keywords-Differential primitive element, Multi-output Translated ability Canonical Form, Exponential observer, Bilinear System Class.

Flies& Generalized Observ-

1. INTRODUCTION In recent years, a variety of approaches have been used on the study of the synthesis of estimation and control algorithms. The control techniques assume a complete knowledge of the state vector at any time. This is not always the case since techniques for on-line measurements of some process are almost always indirect. Due to this restriction, it is necessary to design and implement estimators. An important problem in system theory is the state measurement which is solved by introducing a state estimator. A system whose task is to give an estimation of the state is called an observer. An estimation of the state for linear systems is given by the Luenberger’s observer, while for bilinear systems, and in general for nonlinear ones, the construction of observers is more difficult. Many authors have worked on the development of the state estimators for a restricted class of nonlinear systems such as bilinear systems either by the differential geometric viewpoint (theory mainly due to Hermann, Krener, Isidori, Van der Schaft, Nijmeijer, Gauthier [l], Bornard [2], Hammouri [3,4], Zeitz, Williamson, Levine, Marino, etc.), or by the algebraic one (introduced by Fliess in 1986). The latter approach is based on differential algebra and has, among other features, the ability to define observability [5,6], and consequently, gives an estimation of the state through *Author to whom correspondence should be addressed. The authors are indebted to the reviewers for their helpful comments.

125

R. MARTINEZ-GUERRA AND

126

J. DE LEON-MORALES

the design of observers [7-91 for systems represented by an arbitrary set of algebredifferential equations. This paper is addressed in the algebraic direction. We propose the construction of an exponential nonlinear observer for a multi-input multi-output observable bilinear system class (in the Diop-Fliess’ Observability sense). An application to a chemical reactor model is given.

2. STATEMENT

OF THE PROBLEM

We start with some basic definitions which are considered in [5]. DEFINITION

2.1.

A Dynamics is a finitely generated differential algebraic extension G/k(u).

This

means that any element of G satisfies an algebraic differential equation with coefficients which are rational functions over k in the components of u and a finite number of their time derivatives. 2.2. Let a subset {u, y} of G in a dynamics G/k(u). An element in G is said to be observable with respect to {u, y} if it is algebraic over k(u, y). Therefore, a state 3 is said to be observable if and only if it is observable with respect to {u, y}. DEFINITION

DEFINITION

2.3.

A Dynamics G/k( u ) with output y is said to be observable if and only if any

state is so. Here, the concept of observability means that the differential field extension G/k(u, y) is algebraic; that is to say, that the whole differential information is contained in k(u, y). Let us now consider the following Bilinear System Class (BSC):

(1) where u = (ui, . . . , u,) E IV, x = (~1,. . . ,x,) E ET, y = (~1,. . . , yP) E JRp,Ai,O 5 i 5 m, B, C and D are real matrices of appropriate size. Suppose that BSC (1) is observable in the Diop-Fliess’ Observability sense. Then let us pose the following problem. Is it possible to synthesize an exponential observer for the system (l)?

3. MULTI-OUTPUT FLIESS’ LOCAL GENERALIZED OBSERVABILITY CANONICAL FORM According to the theorem of the differential primitive element, there exist k~elernze;tsyk rd_‘l”,” , qk, 0 5 qk 5 n, an integer such that ypk) is algebraically dependent on yk, yk , yk , . . . , yk ’ 21,u(l), uc2), . . . , i.e., yk(‘k) = -c,,

yk,. . . ,y;*-l),u,u(l)r..

. ,d7))

(2)

Let ~=~,171+~,771+r/2+1,...,~1+r/2+~~~+rlp-l+lr c l

’ 7