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AN UNKNOWN INPUT OBSERVER DESIGN FOR LINEAR TIME-DELAY SYSTEMS Anas FATTOUH, Olivier SENAME and Jean-Michel DION

Laboratoire d’Automatique de Grenoble INPG - CNRS UMR 5528 ENSIEG-BP 46 38402 Saint Martin d’H`eres Cedex, FRANCE Tel. +33.4.76.82.62.32 - FAX: +33.4.76.82.63.88 {Anas.Fattouh, Olivier.Sename, Jean-Michel.Dion}@inpg.fr

ABSTRACT This paper presents an algebraic method to design a dynamic gain unknown input observer for linear systems with commensurate delays in state, input and output variables. Necessary and sufficient conditions for the existence of such an observer are provided as well as a constructive procedure. An example is given in order to illustrate the proposed method.

der to construct an observer for linear systems with time-delay : • Infinite dimensional approach [2]. • Algebraic and polynomial approaches: ring theory [3, 4, 5] and finite spectrum assignment theory [6, 7]. • Frequency domain approach [8, 9].

Keywords: Time-delay systems; Observers; State space.

1 Introduction It is often desirable to use a state feedback to control a system. One of the major difficulties in implementing a state-feedback control law is that all of the state variables of the system are required for the controller synthesis. However, in the most practical situations, this condition is rarely satisfied and an observer has to be build up in order to estimate the state variables from the output and the input measurements. For linear time-invariant delay-free systems, several schemes based on various assumptions have been developed in order to construct observers [1]. This problem is a more complicated issue in systems with any kind of delay than it is in a delay-free systems. Indeed, a functional state vector has to be considered instead of an instantaneous state vector in delay-free systems. Different usual approaches can be distinguished in or-

• Lyapunov functional approach based on Lyapunov-Krasovskii theory [10] or LyapunovRazumikhin theory [11]. One of the key points in designing an observer is to take into account the nonlinearities and parameter changes of the system under consideration. These different types of uncertainties as well as unknown external excitations can be conveniently represented as an unknown input. This approach can be used either in control systems or in diagnostic systems. The problem of unknown input observers for linear time-delay systems has only been considered by Sename [12]. In that work the usual method to design an unknown input observer for linear delay-free systems has been generalized to the case of time-delay systems through a polynomial approach. However, only some necessary conditions for the existence of such an observer are given and no design procedure has been proposed. In the other hand, it is well known that, for time-delay systems over a ring, a static state feedback is not such a powerful tool for the control of those systems [13]. This fact is also true for the construction of observers

for systems over a ring since the problem of detectability is dual to this of stabilizability. This paper can be considered as an extension of the work of Sename [12]. A dynamic gain Luenberger-type observer is considered while a static one is considered in [12]. This change allows us to give necessary and sufficient conditions for the existence of such an observer as well as a constructive procedure. This paper is organized as follows. Section 2 presents an example to show the interest of using a dynamic gain. Section 3 is devoted to the problem statement. In section 4 necessary and sufficient conditions for the existence of the observer as well as a constructive procedure are given. In section 5 an example illustrates the proposed method. Finally this paper concludes with Section 6. Notations : < is the field of real numbers N is the set of non-negative integers. ∇ is the delay operator defined by ∇f (t) := f (t − h), 0≤h∈<