Integral Sliding-Based Robust Control

In this chapter we will study the robust performance control based-on integral sliding-mode for system with nonlinearities and perturbations that consist of external disturbances and model uncertainties of great possibility time-varying manner. Sliding-mode control is one of robust control methodologies that deal with both linear and nonlinear systems, known for over four decades (El-Ghezawi et al., 1983; Utkin & Shi, 1996) and being used extensively from switching power electronics (Tan et al., 2005) to automobile industry (Hebden et al., 2003), even satellite control (Goeree & Fasse, 2000; Liu et al., 2005). The basic idea of sliding-mode control is to drive the sliding surface s from s = 0 to s = 0 and stay there for all future time, if proper sliding-mode control is established. Depending on the design of sliding surface, however, s = 0 does not necessarily guarantee system state being the problem of control to equilibrium. For example, sliding-mode control drives a sliding surface, where s = Mx − Mx0, to s = 0. This then implies that the system state reaches the initial state, that is, x = x0 for some constant matrix M and initial state, which is not equal to zero. Considering linear sliding surface s = Mx, one of the superior advantages that sliding-mode has is that s = 0 implies the equilibrium of system state, i.e., x = 0. Another sliding surface design, the integral sliding surface, in particular, for this chapter, has one important advantage that is the improvement of the problem of reaching phase, which is the initial period of time that the system has not yet reached the sliding surface and thus is sensitive to any uncertainties or disturbances that jeopardize the system. Integral sliding surface design solves the problem in that the system trajectories start in the sliding surface from the first time instant (Fridman et al., 2005; Poznyak et al., 2004). The function of integral sliding-mode control is now to maintain the system’s motion on the integral sliding surface despite model uncertainties and external disturbances, although the system state equilibrium has not yet been reached. In general, an inherent and invariant property, more importantly an advantage, that all sliding-mode control has is the ability to completely nullify the so-called matched-type uncertainties and nonlinearities, defined in the range space of input matrix (El-Ghezawi et al., 1983). But, in the presence of unmatched-type nonlinearities and uncertainties, the conventional sliding-mode control (Utkin et al., 1999) can not be formulated and thus is unable to control the system. Therefore, the existence of unmatched-type uncertainties has the great possibility to endanger the sliding dynamics, which identify the systemmotion on the sliding surface after matched-type uncertainties are nullified. Hence, another control action simultaneously stabilizes the sliding dynamics must be developed. 8