**State Estimation via Observers with Unknown Inputs: Application to a Particular Class of Uncertain Takagi-Sugeno Systems**

**Wafa JAMEL ^{1}, Atef KHEDHER^{2}, Nasreddine BOUGUILA^{1}, Kamel Ben OTHMAN^{2}**

**ATSI, National Engineering School of Monastir**

^{1 }Rue Ibn El Jazzar, 5019 Monastir-Tunisia

(wafa_jamel, nasreddinebouguila)@yahoo.fr

**LARA Automatique, National Engineering School of Tunis**

^{2 }BP 37, le Belvédère, 1002 Tunis

(khedher_atef, kamelbenothman)@yahoo.fr

**Abstract**: This paper deals with the design of a multiple observer allowing estimating the state vector of a nonlinear system described by a Takagi-Sugeno multiple model subject to modelling and input uncertainties which are considered as unknown inputs. The main contribution of the paper is the conception of a multiple observer based on the elimination of these unknown inputs. Convergence conditions are established in order to guarantee the convergence of the state estimation error. These conditions are expressed in Linear Matrix Inequality (LMI) formulation. An example of simulation is given to illustrate the proposed method.

**Keywords**: Multiple model approach, Takagi-Sugeno models, multiple observer, unknown inputs, state estimate, modelling and input uncertainties.

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**CITE THIS PAPER AS**:

Wafa JAMEL, Atef KHEDHER, Nasreddine BOUGUILA*, *Kamel Ben OTHMAN, **State Estimation via Observers with Unknown Inputs: Application to a Particular Class of Uncertain Takagi-Sugeno Systems**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 19 (3), pp. 219-228, 2010.

**1. Introduction**

State estimation plays a significant role in the context of monitoring and/or of diagnosis of systems. It is an analytical source of redundancy used to generate failure symptoms of the system by making a comparison between the real behaviour signals of the system and the estimated signals. A non desired variation between these signals indicates the possible presence of faults affecting the system. These faults indicators are named residues. Their generation is based on the use of the state observers.

A state observer is a dynamical system allowing the state reconstruction from the system model and the measurements of its inputs and outputs [20]. In fact, the observer controlled by the same inputs applied to the system is able to provide the same output signals provided that the model employed reproduces with precision the behaviour of the system to be supervised. The observer design can be delicate according to the type and the complexity of the considered model.

Two types of models are distinguished according to the linear or non linear character of the system. Linear models have simple structures. They are the base of several applications and research works.

In the case of linear systems, observers can be designed for uncertain systems with time-delay perturbations [8] and unknown input systems [7]. Several researches were achieved concerning the state estimation in the presence of both known and unknown inputs [26], [29], [7]. These works can be gathered into two categories. The first one supposes an a priori knowledge of information on these non measurable inputs; in particular, Johnson [17] proposes a polynomial approach and Meditch [21] suggests approximating the unknown inputs by the response of a known dynamic system. The second category proceeds either by estimation of the unknown inputs [18], [19], or by their complete elimination from the system equations [11].

However, in the majority of real cases the nonlinear nature of the process cannot be neglected. The assumption of linearity is checked only locally around an operating point. Real physical processes present complex behaviours with nonlinear laws. As, it is delicate to synthesize an observer for a nonlinear system, the multiple model approach constitutes a tool which is largely used in the modelling of nonlinear systems [22], [6].

The principle of the multiple model approach is based on the reduction of the system complexity by the decomposition of its operating space in a finite number of operating zones. Each zone is characterized by a local model (named also, sub-model). Each sub-model is a simple and linear system around an operating point. The relative contribution of each sub-model is quantified with the help of a weighting function. The global behaviour of the nonlinear system is obtained by the sum of the local models balanced by the weighting functions.

Various studies dealing with the presence of unknown inputs acting on the nonlinear system were published [3], [4], [18], [19]. The problem of state estimation of nonlinear systems submitted to uncertainties has received considerable attention [13], [1], [24], [28]. In practice, there are many situations where some of the system inputs are inaccessible. The recourse to the use of an unknown input observer is then necessary in order to be able to estimate the state of the considered system. For state estimation, the suggested technique consists in associating to each local model a local unknown input observer. The multiple observer or global observer is the sum of the local observers weighted by the weighting functions associated to the local models [25].

In this paper, the problem of state estimation of an uncertain Takagi-Sugeno multiple model is addressed. The purpose of this work is to extend the principle of the design of observers with unknown inputs to uncertain system case. Only model’s and input uncertainties are considered in this paper.

Others works dealing with uncertain systems choose to estimate the sate using different kinds of observers, such as sliding mode observer [1], Proportional and Proportional Integral observer [23]. The main contribution in this paper is the development of an unknown input multiple observer for uncertain nonlinear systems modelled by Takagi-Sugeno models. The convergence conditions of the sate estimation error are expressed in terms of linear matrix inequalities (LMI).

The paper is organized as follows. Section 2 recalls the multiple model approach. In section 3, the multiple observer of a system with unknown inputs is presented.

Section 4 presents the main results concerning the synthesis of a multiple observer to estimate the state of nonlinear system submitted to modelling and inputs uncertainties. Finally, in section 5, a numerical example is given to show the validity of the proposed methodology.

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