International Conference on Electromechanical Engineering (ICEE'2012) Skikda, Algeria, 20-22 November 2012 Direct Adaptive Fuzzy Control for a Class of SISO Nonlinear Systems with Unknown Control Direction Salim Labiod, Hamid Boubertakh, Nabil Oucief LAJ, Faculté des Sciences et de la Technologie, Université de Jijel BP.98, 18000, Jijel, Algeria labiod_salim@yahoo.fr ; boubert_hamid@yahoo.com ; ouciefnabil@yahoo.fr Abstract—In this paper, a direct adaptive control scheme using fuzzy systems is presented for a class of SISO uncertain nonlinear systems with unknown control direction. Within this scheme, a fuzzy system is employed to generate directly the control input signal without dynamic system estimation, and the Nussbaum- type function is used to deal with the unknown control direction. The stability analysis of the closed-loop system is performed using a Lyapunov approach. Simulation results are provided to verify the effectiveness of the proposed design. Index Terms—Fuzzy Control, Adaptive control, Nonlinear systems, Nussbaum function. I. INTRODUCTION Over the past two decades, fuzzy logic control has found extensive applications for plants that are complex and ill- defined [1]. In most of these applications, the rule base of the fuzzy controller is constructed from expert knowledge. However, it is sometimes difficult to build the rule base of some plants, or the need may arise to tune the controller parameters if the plant dynamics change. In the hope to overcome this problem, based on the universal approximation theorem and on-line learning ability of fuzzy systems, several stable adaptive fuzzy control schemes have been developed to incorporate the expert knowledge systematically [1]. The stability study in such schemes is performed by using the Lyapunov design approach. Conceptually, there are two distinct approaches that have been formulated in the design of a fuzzy adaptive control system: direct and indirect schemes. In the direct scheme, the fuzzy system is used to approximate an unknown ideal controller [1-3]. On the other hand, the indirect scheme uses fuzzy systems to estimate the plant dynamics and then synthesizes a control law based on these estimates [1, 2, 4]. However, in the aforementioned papers, the control direction is assumed known a priori, i.e., the sign of the control gain is assumed known for the designer. Without this assumption, adaptive controllers design becomes much more difficult, because in this case, one cannot decide the direction along which the control operates and/or the direction of the search of controller parameters. In the adaptive control literature, the unknown control direction problem is mainly solved by using the Nussbaum-type function for controller design for both linear and nonlinear systems [5-9]. In this paper we present a direct adaptive fuzzy control scheme for a class of uncertain nonlinear systems. In the design, a fuzzy system is used to approximate an unknown ideal controller, and a Nussbaum gain function is introduced to solve the problem of unknown control direction. It is proved that the proposed adaptive fuzzy control approach can guarantee that all the signals of the closed-loop system are bounded and that the tracking error converges asymptotically to zero. This paper is organized as follows. The problem formulation and fuzzy systems description are given in section 2. The proposed direct adaptive control scheme is presented in section 3 with its adaptive law and stability analysis. In section 4, the proposed adaptive control algorithm is used to control a simple nonlinear system. II. PROBLEM FORMULATION AND FUZZY SYSTEMS Consider the class of single-input single-output (SISO) nonlinear systems modeled by     1 1 , 1, , 1 i i n x x i n x f g u y x             x x    (1) or, equivalently       n y f g u   x x (2) where   1 , , T n n x x   x   , is the state vector of the system which is assumed available for measurement, u   is the scalar control input, y   is the scalar system output,   f x and   g x are unknown smooth nonlinear functions. In respect of the dynamic system (1), the following assumption will be made: Assumption 1: The control gain   g x and its sign are unknown with   0 g g g    x , where g and g are positive constants. The objective is to design an adaptive fuzzy controller for system (1) such that the system output  y t follows a desired