New Stability Conditions for Nonlinear Systems Described by Multiple Model Approach

Abstract

This paper studies the global asymptotic stability and the tracking control problem of an uncertain non stationary continuous system described by the multiple model approach. It is based on the construction of a basis of models containing four extreme models and possibility of addition of an average model. Once the basis of models is generated, an operation of fusion of these different models is made to the level of the elementary control law and the partial output using the geometric method. New sufficient conditions for the stability are derived via Lyapunov technique. The matrices of feedback gains and tracking gains are determined while solving systems of LMI constraints (Linear Matrix Inequalities). The case of an unstable continuous nonlinear model of electrical circuit operating in pseudo-periodic system is considered to illustrate the proposed approach. Keyword: Electrical circuit INTRODUCTIONThe multiple model approach is proving very interesting whenever we are confronted with complex systems and/or nonlinear. It is to represent the system studied by a family of simpler and easier to manipulate mathematical models [1], [2], [3]. This approach has been recently developed in several science and engineering domains, with typical applications in the electrical and mechanical engineering areas, with application to modelling, control and/or fault detection. It was introduced as an efficient and powerful method to cope with modelling and control difficulties when complex non linear and/or uncertain processes are concerned. The multiple model approach assumes that it is possible to replace a unique non linear representation by a combination of simpler models thus building a so-called model-base. Usually, each model of this base describes the considered process at a specific operating point [4]. The interaction between the different models of the base through normalized activation functions allows the modelling of the global nonlinear and complex system. The stability of these models is, most of the time, studied using the quadratic Lyapunov approach [5] The stability conditions based on the use of the quadratic Lyapunov function are conservative as a single common symmetric positive definite matrix verifying all Lyapunov inequalities is required [13]. It is also rejected by certain systems such as the saturated systems, the piecewise linear systems, etc. Some works show the contribution of the polyquadratic and the piecewise quadratic Lyapunov functions, [15], [16], [17].The study proposed in this paper focuses on a class of uncertain systems and complex continuing bounded parameters [18], [19]. The global model can be obtained either by using the switching operation or fusion. In this study, we are interested in the fusion using the geometric method [19].

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