Optimal Sliding Mode Control for a Class of Uncertain Nonlinear Systems Based on Feedback Linearization

Abstract

Optimal control is one of the most important branches in modern control theory, and linear quadratic regulator (LQR) has been well used and developed in linear control systems. However, there would be several problems in employing LQR to uncertain nonlinear systems. The optimal LQR problem for nonlinear systems often leads to solving a nonlinear two-point boundary-value (TPBV) problem (Tang et al. 2008; Pang et al. 2009) and an analytical solution generally does not exist except some simplest cases (Tang & Gao, 2005). Additionally, the optimal controller design is usually based on the precise mathematical models. While if the controlled system is subject to some uncertainties, such as parameter variations, unmodeled dynamics and external disturbances, the performance criterion which is optimized based on the nominal system would deviate from the optimal value, even the system becomes unstable (Gao & Hung, 1993 ; Pang & Wang, 2009). The main control strategies to deal with the optimal control problems of nonlinear systems are as follows. First, obtain approximate solution of optimal control problems by iteration or recursion, such as successive approximate approach (Tang, 2005), SDRE (Shamma & Cloutier, 2001), ASRE (Cimen & Banks, 2004). These methods could have direct results but usually complex and difficult to be realized. Second, transform the nonlinear system into a linear one by the approximate linearization (i.e. Jacobian linearization), then optimal control can be realized easily based on the transformed system. But the main problem of this method is that the transformation is only applicable to those systems with less nonlinearity and operating in a very small neighborhood of equilibrium points. Third, transform the nonlinear system into a linear one by the exact linearization technique (Mokhtari et al. 2006; Pang & Chen, 2009). This differs entirely from approximate linearization in that the approximate linearization is often done simply by neglecting any term of order higher than 1 in the dynamics while exact linearization is achieved by exact state transformations and feedback. As a precise and robust algorithm, the sliding mode control (SMC) (Yang & Ozguner, 1997; Choi et al. 1993; Choi et al. 1994) has attracted a great deal of attention to the uncertain nonlinear system control problems. Its outstanding advantage is that the sliding motion exhibits complete robustness to system uncertainties. In this chapter, combining LQR and SMC, the design of global robust optimal sliding mode controller (GROSMC) is concerned. Firstly, the GROSMC is designed for a class of uncertain linear systems. And then, a class of