Controller Synthesis for a Class of State-Dependent Switched Nonlinear Systems

In the last decade, switched systems have gained much attention since a large class of practical systems can be modeled as switched systems (see e.g., (Dayawansa & Martin, 1999)) and there exist systems that cannot be asymptotically stabilized by a single continuous feedback control law (see e.g., (Brockett, 1983)). For stability analysis of switched systems, many interesting results have been proposed in the literature, see e.g., (Narendra & Balakrishnan, 1994; Johansson & Rantzer, 1998; Branicky, 1998; Ye et al., 1998; Dayawansa & Martin, 1999; Liberzon et al., 1999; Skafidas et al., 1999; Mancilla-Aguilar, 2000; and Chatterjee & Liberzon, 2007). Another topic is the derivation of stabilizing switching rules for switched systems, see et al., (Peleties & DeCarlo, 1991; Liberzon & Morse, 1999; and Pettersson, 2003). For feedback controller synthesis of switched control systems (with input signals), most of the proposed results consider the linear subsystems case, see e.g., (Daafouz et al., 2002; Sun & Ge, 2003; Petterson, 2004; Hespanha & Morse, 2004; and Seatzu et al., 2006). Only a few results have been proposed for feedback controller synthesis of switched nonlinear control systems, see e.g., (Sun & Zhao, 2001; El-Farra et al., 2005; Wu, 2008; and Wu, 2009). In (El-Farra et al., 2005), an integrated synthesis of feedback controllers together with switching laws has been proposed. In (Sun & Zhao, 2001), a common control Lyapunov function (CCLF) approach has been introduced also for constructing feedback control laws together with switching signals for switched nonlinear control systems. The concept of CCLF is motivated by the control Lyapunov function approach (see, e.g., (Artstein, 1983) and (Sontag, 1983 and 1989)) for designing stabilizing feedback laws for (non-switched) nonlinear systems. In (Wu, 2008), for switched nonlinear control systems under arbitrarily switching, conditions for the existence of CCLFs has been derived and a globally uniformly asymptotically stabilizing feedback law has been proposed. However, no systematical approaches have been provided for constructing CCLFs. Moreover, the obtained feedback law is complicated. In (Wu, 2009), for switched nonlinear control systems, which arbitrarily switching between a set of subsystems in strict feedback form, the backstepping approach (see e.g., (Krstic et al., 1995) and (Sepulchre et al., 1997)) has been employed to construct CCLFs, and a simpler stabilizing feedback law has been proposed. However, till now, few results have been reported in the literature about stabilizing feedback controllers design for state-dependent switched nonlinear control systems. The 3