1-4244-0023-6/06/$20.00 ©2006 IEEE CIS 2006 Decoupling Adaptive Fuzzy Sliding-Mode Control with Rule Reduction for Nonlinear System Lon-Chen Hung, Hung-Yuan Chung Department of Electrical Engineering National Central University Jhongli City, Taoyuan County 320, Taiwan (R.O.C.) hychung@ee.ncu.edu.tw Abstract—In this paper, adaptive fuzzy sliding-mode controller design approach with decoupling method is proposed. The decoupling method provides a simple way to achieve asymptotic stability for a class of fourth-order nonlinear system. The adaptive fuzzy sliding-mode control system is comprised of fuzzy controller and a compensation controller. The compensation controller is designed to compensate for the difference between the ideal computational controller and the fuzzy controller. Using this approach, the response of system will converge faster than that of previous reports. The simulation results for a ball-beam system presented to demonstrate the effectiveness and robustness of the method. Keywords—fuzzy, mobile, robot I. INTRODUCTION Fuzzy logic control, as one of the most useful approaches for utilizing expert knowledge, has had extensive research in the past decade [3]. Fuzzy logic control is generally applicable to plants that are mathematically poorly modeled and where experienced operators are available for providing qualitative guiding. Although achieving much practical success, fuzzy control has not been viewed as a rigorous science, due to a lack of formal synthesis techniques which guarantee the very basic requirements of global stability and acceptable performance [13]. In stability analysis [5,16], it is commonly assumed that the mathematical model of the plants are known, this assumption contradicts the very basic premise of fuzzy control systems, i.e., to control processes that are poorly modeled from a mathematical view. Based on fuzzy systems which are capable of approximating, with arbitrary accuracy, any real continuous function on a compact set, a globally stable adaptive controller is firstly synthesized from a collection of fuzzy IF-THEN rules [2]. The fuzzy system, used to approximate an optimal controller, is adjusted by an adaptive law based on a Lyapunov function synthesis approach. In recent years, there have been attempts to design the fuzzy based on the sliding-mode control law [1,4,6-12,14,15]. They have shown that the boundary layer can reached in finite time and the ultimate boundedness of states is obtained asymptotically even though there exist some disturbance of dynamic uncertainties of the system. Palm showed that the analogy between a simple and sliding-mode controller with a boundary layer [14]. Hwang et al. proposed a fuzzy sliding- mode controller and opened a way of designing for higher order nonlinear system [8]. The sliding-mode control provides a good performance in tracking of some nonlinear systems. Nevertheless, a notorious characteristic of sliding-mode control approach is the discontinuity around the switching hyper-plane, that means some of the state variable are vibrant. One of the methods to cope with the problem is to utilize a feed-forward compensator to offset unpredictable affect of system uncertainties. In most studies, the fuzzy controller of second-order systems is designed on a phase plane built by error e and change of error e i that are produced from the states x and x i . For example, in a cart-pole system only the pole subsystem is considered ignoring the cart subsystem and it is thus impossible to achieve a good control around the set point (distance=0). In this study, a decoupling fuzzy controller design is proposed. This controller guarantees some properties, such as the robust performance and stability properties. Further, a class of fourth-order nonlinear systems is investigated. A decoupling adaptive fuzzy sliding-mode control design scheme is presented through width of consequence adaptation for a class of fourth-order nonlinear systems. Each subsystem, which is decoupling into two second-order systems, is said to have main and sub-control purpose. Two sliding surfaces are constructed through the state variables of the decoupling subsystem. We define main and sub-target condition for these sliding surfaces and introduce an intermediate variable from the sub-sliding surface condition. The proposed adaptation law, which results from the direct adaptive approach, is used to appropriately determine the width of the unknown system variables. And the membership functions in the THEN part will vary with the width adaptation of consequence that one tuning factor is characterized to adapt the control rules. A tuning methodology is derived via the rules regulation. The online tuning algorithm is derived in the Lyapunov sense; thus, the stability of the control system can be guaranteed. To illustrate the effectiveness of the proposed design method, a comparison between a decouped fuzzy sliding-mode control [12] and the proposed control is made. The rest of the paper is divided into five sections. In Section 2, the systems are described. In Section 3, the adaptive 97