Stability and Stabilization of Time-Delay Systems

Moreover, I am grateful to Brigitte Rohrbach for her careful editing of the manuscript which helped improve the standard of my English. Finally, a special thank to my wife Franziska for her patience and invaluable support. vii Summary The stability analysis of time-delay systems forms the centre of this work. Three tools are typically used to investigate the stability of such systems: Razumikhin theory, Lyapunov-Krasovskii theory, and (for linear time-delay systems) eigenvalue considerations. However, none of these basic concepts represents applicable stability tests in terms of the system matrices. Therefore , based on the three stability concepts mentioned some suitable algebraic stability tests are developed in this work. The stability tests obtained can be categorized into four groups, depending on how much information concerning the delays is required for these tests: • Delay-independent stability criteria: The length of the delay need not be known for the application of these stability tests. The delays may be state-dependent and/or time variable. The only assumption needed is that the delays are continuous and bounded. • Stability criteria independent of constant delays: In the second group it is assumed that the delays of the system are constant; no further information on the delays is necessary. • Stability criteria independent of a delay constant: This type of stability criteria presumes that the delays are constant and that the ratios of size of the delays are known. • Delay-dependent stability criteria: This group includes exact algebraic stability criteria depending on the delay and on the system constants and stability criteria which yield an upper bound of the admissible delay. The need for delay-independent (and related) stability tests is obvious, since in practice the delays are difficult to estimate, especially those that are time variable and state dependent. While algebraic stability tests independent of delays are suitable to apply, exact algebraic stability conditions depending on the delay and the system constants are known only in some special cases. In this context a method is presented to achieve some extensions. The method permits the investigation of the stability of systems which are general enough to demonstrate the differences among the four types of stability tests. The stability of general, linear time-delay systems, however, can be checked exactly only by eigenvalue considerations. Unfortunately, the computation of viii the eigenvalues is a cumbersome task, since the corresponding transcen-dental characteristic equation contains exponential terms which induce extreme gradients. An improved version …