New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems

Abstract

Stability analysis and controller design for uncertain nonlinear systems is open problem now(Vidyasagar, 1986). So far numerous design methodologies exist for the controller design of nonlinear systems(Kokotovic & Arcak, 2001). These include any of a huge number of linear design techniques(Anderson & More, 1990; Horowitz, 1991) used in conjuction with gain scheduling(Rugh & Shamma, 200); nonlinear design methodologies such as Lyapunov function approach(Vidyasagar, 1986; Kokotovic & Arcak, 2001; Cai et al., 2008; Gutman, 1979; Slotine & Li, 1991; Khalil, 1996), feedback linearization method(Hunt et al., 1987; Isidori, 1989; Slotine & Li, 1991), dynamics inversion(Slotine & Li, 1991), backstepping(Lijun & Chengkand, 2008), adaptive technique which encompass both linear adaptive(Narendra, 1994) and nonlinear adaptive control(Zheng & Wu, 2009), sliding mode control(SMC)(Utkin, 1978; Decarlo etal., 1988; Young et al., 1996; Drazenovic, 1969; Toledo & Linares, 1995; Bartolini & Ferrara, 1995; Lu & Spurgeon, 1997), and etc(Hu & Martin, 1999; Sun, 2009; Chen, 2003). The sliding mode control can provide the effective means to the problem of controlling uncertain nonlinear systems under parameter variations and external disturbances(Utkin, 1978; Decarlo et. al., 1988; Young et al., 1996). One of its essential advantages is the robustness of the controlled system to variations of parameters and external disturbances in the sliding mode on the predetermined sliding surface, s=0(Drazenovic, 1969). In the VSS, there are the two main problems, i.e., the reaching phase at the initial stage(Lee & Youn, 1994) and chattering of the input (Chern & Wu, 1992). To remove the reaching phase, the two requirements are needed, i.e., the sliding surface must be determined from an any given initial state to the origin( (0) & t 0 ( )| 0 x x s x = = = ) and the control input must satisfy the existence condition of the sliding mode on the pre-selected sliding surface for all time from the initial to the final time( 0, for 0 T s s t < ≥ $ ). In (Toledo & Linares, 1995), the sliding mode approach is applied to nonlinear output regulator schemes. The underlying concept is that of designing sliding submanifold which contains the zero tracking error sub-manifold. The convergence to a sliding manifold can be attained relying on a control strategy based on a simplex of control vectors for multi input uncertain nonlinear systems(Bartolini & Ferrara, 1995). A nonlinear optimal integral variable