VIII ERMAC-R3 8 o Encontro Regional de Matem´ atica Aplicada e Computacional 20 a 22-Novembro-2008 Universidade Federal do Rio Grande do Norte - Natal/RN Adaptive fuzzy sliding mode control and its application to underwater robotic vehicles Wallace Moreira Bessa CEFET/RJ, Centro Federal de Educa¸c˜ ao Tecnol´ogica Celso Suckow da Fonseca Av. Maracan˜a 229, 20271-110, Rio de Janeiro, RJ, Brasil E-mail: wmbessa@cefet-rj.br Max Suell Dutra COPPE/UFRJ, Universidade Federal do Rio de Janeiro Caixa Postal 68503, 21945-970, Rio de Janeiro, RJ, Brasil E-mail: max@mecanica.ufrj.br Edwin Kreuzer TechnischeUniversit¨atHamburg-Harburg Eissendorfer Strasse 42, D-21071, Hamburg, Deutschland E-mail: kreuzer@tuhh.de Abstract This work presents a discussion about the convergence properties of a variable structure controller for uncer- tain single-input-single-output nonlinear systems. The adopted approach is based on the sliding mode control strategy and enhanced by an adaptive fuzzy algorithm to cope with modeling inaccuracies and external distur- bances that can arise. The convergence of the tracking error vector is analytically proven using Lyapunov’s di- rect method and Barbalat’s lemma. An application of this adaptive fuzzy sliding mode controller to an un- derwater robotic vehicle is introduced to illustrate the controller design method. Numerical results are also presented in order to demonstrate the control system performance. Palavras-chave Adaptive algorithms, Fuzzy logic, Nonlinear control, Underwater robotic vehicles, Sliding modes. Introduction Sliding mode control, due to its robustness against modeling imprecisions and external disturbances, has been successfully employed to nonlinear control prob- lems. But a known drawback of conventional sliding mode controllers is the chattering effect. To overcome the undesired effects of the control chattering, Slotine [16] proposed the adoption of a thin boundary layer neighboring the switching surface, by replacing the sign function by a saturation function. This substitution can minimize or, when desired, even completely elimi- nate chattering, but turns perfect tracking into a track- ing with guaranteed precision problem, which actually means that a steady-state error will always remain. In order to enhance the tracking performance inside the boundary layer, some adaptive strategy should be used for uncertainty/disturbance compensation. Due to the possibility to express human experience in an algorithmic manner, fuzzy logic has been largely employed in the last decades to both control and iden- tification of dynamical systems. In spite of the simplic- ity of this heuristic approach, in some situations a more rigorous mathematical treatment of the problem is re- quired. Recently, much effort [1, 5, 6, 9, 14, 17, 19] has been made to combine fuzzy logic with sliding mode methodology. In this work, an adaptive fuzzy sliding mode con- troller (AFSMC) is proposed to deal with impre- cise single-input-single-output nonlinear systems. The adopted controller is primarily based on the sliding mode control methodology, but a stable adaptive fuzzy inference system is embedded in the boundary layer to cope with structured (or parametric) uncertainties, unstructured uncertainties (or unmodeled dynamics) and external disturbances. Using Lyapunov’s second method (also called Lyapunov’s direct method) and Barbalat’s lemma, the convergence properties of the tracking error vector is analytically proven. Based on the proposed control scheme, a depth regulator is in- troduced for remotely operated underwater vehicles to illustrate the controller design method. Numeri- cal results shows that, when compared with a conven- tional sliding mode controller, the AFSMC shows an improved performance. Adaptive fuzzy sliding mode con- troller Consider a class of nth-order nonlinear systems: x (n) = f (x)+ b(x)u + d (1)