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ABSTRACT Observer-based indirect adaptive fuzzy sliding mode control with state variable filters for unknown nonlinear dynamical systems

Recently, adaptive fuzzy control system designs have been extensively discussed in the literature [1,3,6,17]. The fundamental idea of adaptive fuzzy control is as follows: based on the universal approximation theorem [17], one first constructs a fuzzy model to describe the input/output behavior of the controlled system. After that a controller is designed based on the fuzzy model, the adaptive laws are derived to adjust the parameters of the fuzzy models.

In an effort to improve the robustness of the adaptive fuzzy control system, many works have been published on the design of adaptive fuzzy sliding mode controller (AFSMC) [7,8,21,18,23,24], which integrates the sliding mode controller (SMC) [2,14–16,19] design technique into the adaptive fuzzy control to improve the stability and the robustness of the control system. Conventionally, AFSMC design is based on the assumption that the system states are available for measurement, so the adaptive laws of AFSMC are formulated as functions of the tracking error vector of the system [7,8,21,18,23,24]. However, in practice, not all the states of the controlled system are available for measurement. It implies that as one could not obtain all elements ofthe tracking error vector, the conventional adaptive laws would be difficult to realize. In order to treat this problem, several studies apply an observer to estimate the tracking error vector [9,10,12,13,20,22] and use the SPR-Lyapunov design approach [15] to design an adaptive scheme [9,10,13,20,22].

Unlike these works, this paper proposes an indirect AFSMC with state variable filters to tackle the above-mentioned problem. For a given unknown nonlinear dynamic system in which not all the states are available for measurement, we first construct the fuzzy models to describe the input/output behavior of the nonlinear dynamic system. Then, an observer is employed to estimate the tracking error vector. Based on the observer, a fuzzy sliding model controller is developed for guaranteeing the tracking performance.

Subsequently, by passing the observation error vector to a set of state variable filters [4,5], we obtain a filtered observation error vector. Finally, based on the filtered observation error vector, we propose the adaptive laws to adjust the free parameters of the fuzzy models. The stability of the overall control system is analyzed based on the Lyapunov method. Simulation results illustrate the design procedures and demonstrate the tracking performance of the proposed controller.

This paper is organized as follows: Section 2 presents the fuzzy sliding mode controller (FSMC) design based on the observer. With the aid of the state variable filters, the adaptive laws for adjusting the free parameters of the proposed control strategy are presented in Section 3. In Section 4, simulations ofan inverted pendulum system are given to confirm the validity of the proposed control scheme. Section 5 gives the conclusions.