Delay-Dependent Synchronization Criterion for Lur'e Systems with Delay Feedback Control

In recent years, the research on chaotic synchronization has received considerable attention [Pecora & Carroll, 1991; Curran & Chua, 1997; Curran et al., 1997; Suykens & Vandewalle, 1997; Chen & Dong, 1998; Suykens et al., 1999; Chen & Liu, 2000; Yalcin et al., 2001; Cao et al., 2005]. An overview of synchronization methods has been presented in [Chen & Dong, 1998]. A number of master-slave synchronization schemes for Lur’e systems have been proposed [Curran & Chua, 1997; Curran et al., 1997; Suykens & Vandewalle, 1997; Suykens et al., 1999]. Chen and Liu [2000] first handled propagation delay in master-slave synchronization schemes and introduced the possibility of applying synchronization to optical communication. For the two remote chaotic systems, the existence of a time-delay may destroy synchronization. Yalcin et al. [2001] studied master-slave schemes with identical Lur’e systems and supposed that the output of a master system was received at the slave systems with delay, which is assumed to be a known value. Recently, Cao et al. [2005] added the linear feedback term into the delay feedback controller to assist the time-delay feedback control term to stabilize the error systems. However, it is clear that the time-delay feedback control term becomes useless if the nondelay feedback term is available and used because the latter can stabilize the error system directly. For the delay-dependent global asymptotic stability criterion presented in [Yalcin et al., 2001], the first-order model transformation for the error system was employed. However, there exist some additional eigenvalues in the transformed system when the first-order model transformation [Fridman & Shaked, 2003] was employed. Thus, it may be not equivalent to the original one, see e.g. [Gu & Niculescu, 2000]. Although Park [1999] and Moon et al. [2001]’s inequalities and descriptor model transformation [Fridman & Shaked, 2003] can overcome the conservatism of the first-order model