Fuzzy Sliding Controller Design with Adaptive Approximate Error Feedback

In this paper, a novel state error feedback sliding controller is proposed. In the controller, an optimal feedback gain is required and in this study it is assumed to be unknown. Usually, a rudimentary feedback gain is used. Besides, in order to approximate the state error feedback sliding controller with the optimal feedback gain, an adaptive fuzzy system is employed. Thus, the proposed control scheme consists of an adaptive fuzzy system and a state error feedback sliding controller with a rudimentary feedback gain. In the system framework, the rudimentary state error feedback sliding controller can be viewed as the approximate error estimator of the adaptive fuzzy system. Therefore, such an estimated error can be fed back to the learning of the fuzzy system through a modified adaptive law. With such an approximate error feedback, it is clearly evident from our simulation that the learning speed of the proposed learning scheme is faster than that of the original scheme. Also, with the proposed controller, the system stability not only is guaranteed, but also becomes more stable. time. Generally, the conventional sliding controller design has a large and inelastic control component to guarantee the system stability. It is often obtained based on the upper bounds of the system uncertainties. Since the robust controller can only use such bounds to ensure the stability if no other information is used, chattering phenomena seems unavoidable [18-20]. Please note that the integral sliding surface [29] or other sliding surfaces [30-31] are not considered in this study. An idea of resolving the above problem is to incorporate certain learning capability into the system to provide information for the robust controller. In the sliding control design for uncertain systems, adaptive approximation techniques are usually employed as learning mechanisms. In this kind of approaches, numerical [27-28, 32-34] or intelligent [12-17] approximation systems are used to estimate unknown parameters in the control law [12-17, 26-28] or to adjust the boundary layer to eliminate chattering phenomena [21-26]. In this study, an adaptive fuzzy approximation system is considered. Adaptive fuzzy systems are adaptive systems with the incorporation of linguistic fuzzy information in a form of fuzzy IF-THEN rules [5-6][8-9] and are usually employed to estimate some elements in the so-call equivalent controller. Basically, these design approaches are similar to that of indirect adaptive fuzzy control systems [6][10-11]. In this paper, a novel idea for adaptive fuzzy sliding control systems is proposed. By referring to [14], it can be found that the stable (sliding) condition, which is the negative Lyapunov energy derivative, is the key idea of the system stability. Thus, it is possible to consider the traditional stable process of approaching the sliding surface only in the design process. Of course, this design direction may violate the basis of the variable structure system design [7]. But, it may exist interesting and useful design methodologies under such a deign philosophy.