Fuzzy Control: An Adaptive Approach Using Fuzzy Estimators and Feedback Linearization

In recent years the area of control for nonlinear systems has been the subject of many studies (Ghaebi et al., 2011; Toha & Tokhi, 2009; Yang & M., 2001). Computational advances have enabled more complex applications to provide solutions to nonlinear problems (Islam & Liu, 2011; Kaloust et al., 2004). This chapter will present an application of fuzzy estimators in the context of adaptive control theory using computational intelligence (Pan et al., 2009). The method is applicable to a class of nonlinear system governed by state equations of the form ẋ = f (x) + g(x)u, where f (x) and g(x) represent the nonlinearities of the states. Some classic control applications can be described by this specific class of nonlinear systems, for example, inverted pendulum, conic vessel, Continuously Stirred Tank Reactor (CSTR), and magnetic levitation system. The direct application of linear control techniques (e.g.: PID) may not be efficient for this class of systems. On the other hand, classical linearization methods may lead an adequate performance when the model is accurate, even though normally limited around the point in which the linearization took place (Nauck et al., 2009). However, when model uncertainties are considerable the implementation of the control law becomes difficult or unpractical. In order to deal with model uncertainties an adaptive controller is used (Han, 2005; Tong et al., 2011; 2000; Wang, 1994). The controller implements two basic ideas. First, the technique of exact feedback linearization is used to handle the nonlinearities of the system (Torres et al., 2010). Second, the control law formulation in presence of model uncertainties is made with the estimates of the nonlinear functions f (x) and g(x) (Cavalcante et al., 2008; Ying, 1998). The adaptation mechanism is used to adjust the vector of parameters θ f and θg in a singleton fuzzyfier zero-order Takagi-Sugeno-Kang (TSK) structure that provides estimates in the form f (x|θ̂ f ) and g(x|θ̂g). The fuzzy logic system is built with a product-inference rule, center average defuzzyfier, and Gaussian membership functions. One of the most important contributions of the adaptive scheme used here is the real convergence of the estimates f (x|θ̂ f ) and g(x|θ̂g) to their optimal values f (x|θf ) and g(x|θ ∗ g) while keeping the tracking error with respect to a reference signal within a compact set (Schnitman, 2001). Convergence properties are investigated using Lyapunov candidate functions. In order to illustrate the methodology, a nonlinear and open-loop unstable magnetic levitation system is used as an example. Experimental tests in a real plant were conducted to check the reliability and robustness of the proposed algorithm. Fuzzy Control: An Adaptive Approach Using Fuzzy Estimators and Feedback Linearization