State feedback fuzzy-model-based control for Markovian jump nonlinear systems

Abstract

This paper deals with the fuzzy-model-based control design for a class of Markovian jump nonlinear systems. A fuzzy system modeling is proposed to represent the dynamics of this class of systems. The structure of the fuzzy system is composed of two levels, a crisp level which describes the Markovian jumps and a fuzzy level which describes the system nonlinearities. A sufficient condition on the existence of a stochastically stabilizing controller using a Lyapunov function approach is presented. The fuzzy-model-based control design is formulated in terms of a set of linear matrix inequalities. Simulation results for a single-machine infinite-bus power system which is modeled as a Markovian jump nonlinear system in the infinite-bus voltage are presented to illustrate the applicability of the technique.<br>Neste artigo, apresentam-se projetos de controladores fuzzy para uma classe de sistemas não-lineares com saltos Markovianos. Uma modelagem fuzzy é apresentada para representar esta classe de sistemas na vizinhança de pontos de operação escolhidos. A estrutura do sistema fuzzy é composta de dois níveis, um para descrição dos saltos Markovianos e outro para descrição das não-linearidades no estado do sistema. Uma condição suficiente para a estabilização estocástica do sistema fuzzy considerado é derivada usando uma função de Lyapunov acoplada. O projeto de controle fuzzy é então formulado a partir de um conjunto de desigualdades matriciais lineares. Resultados de simulações em um sistema de potência máquina-barramento infinito modelado como um sistema não-linear com saltos Markovianos na tensão do barramento infinito são apresentados para ilustrar a aplicabilidade da técnica

Stability analysis and synthesis of Markovian jump nonlinear systems with incomplete transition descriptions via fuzzy control

“…Hence (12) is equivalent to (11). Therefore, in the presence of unknown elements in the transition rates matrix, one can readily conclude that the system is stable if and only if (10) and (11) hold for ∈ and ∈ , respectively.…”

“…By Schur complement, we can obtain that (26) is equivalent to (27). In a similar way, if ∈ , (18) and (19) can be worked out from (11). (10)-(11) will be satisfied in Theorem 1 such that the closed-loop system (8) is stochastically stable.…”

“…Proof: Consider the system (1) with the control input (7) and replace 0 , by 0 , + , , in (10) and (11), respectively. Then if ∈ , performing a congruence transformation to (10) by −1 , , we can obtain…”

State feedback fuzzy-model-based control for discrete-time markovian jump nonlinear systems with time-varying delays

“…In fact, it can be shown that T-S fuzzy systems are universal approximators [13]. Recently, a fuzzy-model-based control technique for a class of Markovian jump nonlinear system was introduced in [14] and [15]. In this approach, the class of systems is represented by a fuzzy system model with two levels of structure: A crisp level which describes the jumps of the Markov process and a fuzzy level which describes the system nonlinearities.…”

“…In this paper, we include in the fuzzy control design the approximation error between the fuzzy model and the nonlinear system and the two-level structured fuzzy model as in [14][15]. Actually, the effect of the approximation error can deteriorate the stability and control performance of nonlinear control systems [16].…”