Function Approximation-based Sliding Mode Adaptive Control for Time-varying Uncertain Nonlinear Systems

Abstract

Dead zone characteristics exist in many physical components of control systems. They are nonlinear features particularly in direct current (DC) motor position tracking control systems, mainly caused by the uncertain time-varying nonlinear friction. They can severely limit the control performance owing to their non-smooth nonlinearities. However, dead zone characteristics usually are not easy to be known exactly and may vary with time in practical. In addition to the uncertainties in the linear part of the plant, controllers are often required to accommodate time-varying dead zone uncertainties. In general, there are two usual methods treating the systems with uncertain time-varying dead zone characteristics caused by uncertain nonlinear frictions in DC motor position control systems. The first one is to separate the unknown dead zone from the original DC motor systems and construct an adaptive dead zone inverse, and then compensate the effects of unknown dead zone characteristics (Gang & Kokotovic, 1994; Cho & Bai, 1998; Wang et al., 2004; Zhou et al., 2006). The second method is to deal with both the unknown dead zone characteristics and all the other uncertainties as one uniform uncertainty, thereupon design proper compensator (Wang et al., 2004) or adaptive controller which can counteract the effects of uncertainty(Selmic & Lewis, 2000; Tian-Ping et al., 2005). Furthermore, dead zone uncertainties' bounds remain unknown in many practical DC motor control systems. This problem can't be coped with conventional sliding mode controller (Young et al., 1999; Hung et al., 1993) and general adaptive controller (Gang & Kokotovic, 1994; Cho & Bai, 1998; Wang et al., 2004; Zhou et al., 2006; Wang et al., 2004; Selmic & Lewis, 2000; Tian Ping et al., 2005; Young et al., 1999; Hung et al., 1993). In order to deal with nonlinear systems with unknown bound time-varying uncertainties, adaptive control schemes combined with sliding mode technique have been developed (Chyau-An & Yeu-Shun, 2001; Chyau-An & Yuan-Chih, 2004; Huang & Chen, 2004; Chen & Huang, 2004). These control schemes can transform the unknown bound time-varying uncertainties into finite combinations of Fourier series as long as the uncertainties satisfy Dirichlet condition, so that they can be estimated by updating the Fourier coefficients. Since the coefficients are timeinvariant, update laws are easily obtained from the Lyapunov design to guarantee output error convergence. This chapter is devided into two parts. In the first part, for the position tracking in DC motor with unknown bound time-varying dead zone uncertainties, we’ll propose a Function