Asymptotic Stability Analysis of Linear Time- Delay Systems: Delay Dependent Approach

The problem of investigation of time delay systems has been exploited over many years. Time delay is very often encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, etc. The existence of pure time lag, regardless if it is present in the control or/and the state, may cause undesirable system transient response, or even instability. During the last three decades, the problem of stability analysis of time delay systems has received considerable attention and many papers dealing with this problem have appeared (Hale & Lunel, 1993). In the literature, various stability analysis techniques have been utilized to derive stability criteria for asymptotic stability of the time delay systems by many researchers (Yan, 2001; Su, 1994; Wu & Muzukami, 1995; Xu, 1994; Oucheriah, 1995; Kim, 2001). The developed stability criteria are classified often into two categories according to their dependence on the size of the delay: delay-dependent and delay-independent stability criteria (Hale, 1997; Li & de Souza, 1997; Xu et al., 2001). It has been shown that delaydependent stability conditions that take into account the size of delays, are generally less conservative than delay-independent ones which do not include any information on the size of delays. Further, the delay-dependent stability conditions can be classified into two classes: frequency-domain (which are suitable for systems with a small number of heterogeneous delays) and time-domain approaches (for systems with a many heterogeneous delays). In the first approach, we can include the two or several variable polynomials (Kamen 1982; Hertz et al. 1984; Hale et al. 1985) or the small gain theorem based approach (Chen & Latchman 1994). In the second approach, we have the comparison principle based techniques (Lakshmikantam & Leela 1969) for functional differential equations (Niculescu et al. 1995a; Goubet-Bartholomeus et al. 1997; Richard et al. 1997) and respectively the Lyapunov stability approach with the Krasovskii and Razumikhin based methods (Hale & Lunel 1993; Kolmanovskii & Nosov 1986). The stability problem is thus reduced to one of finding solutions to Lyapunov (Su 1994) or Riccati equations (Niculescu et al., 1994), solving linear matrix inequalities (LMIs) (Boyd et al. 1994; Li & de Souza, 1995; Niculescu et al., 1995b; Gu 1997) or analyzing eigenvalue distribution of appropriate finite-dimensional matrices (Su