Some Studies On The Stability Of Time-Delay Systems

The stability of time-delay systems has received considerable attention in the last few decades. The theoretical and practical importance of this topic has been well recognized. However, the stability of time-delay systems is not fully investigated due to the complexity. Roughly speaking, there are two types of approaches to study the stability: namely, time domain ones and the eigenvalue-based ones. In this dissertation, some studies of the author based on these two approaches are introduced.In Chapter 2, the researches based on simple Lyapunov functional methods are introduced. There are three parts in the chapter. 1) A new method for the stability analysis, an integral equality method, is proposed. First, an integral equality is presented in this part. This equality is better than the integral inequality in the stability analysis of time-delay systems. In addition, the selecting rules of the free terms are given. The integral equality and the selecting rules constitute a systematic method to study the stability. This new method is less conservative than the one based on the integral inequality. 2) The stability of neutral systems with distributed delays is studied. By constructing a modified Lyapunov-Krasovskii functional, a new and less conservative stability criterion is obtained. 3) The stabilization of networked control system, in which the plant has state and input delays, is addressed.In Chapter 3, the studies based on discretized Lyapunov functional method are presented. Four contributions are introduced therein. 1) A new complete Lyapunov-Krasovskii functional for neutral systems is proposed. Based on this new complete functional, the discretized functional method is applicable to a wider class of robust analysis problem. 2) The stability of neutral systems with mixed delays is studied. A new discretized Lyapunov functional method is proposed. By the method, the results are less conservative than the existing ones and are very close to the analytical results. 3) A new exponential estimate method, based on discretized Lyapunov functional method, is proposed. The estimate results by the method are better than the ones obtained by the methods based simple Lyapunov functionals. 4) The stability of the systems with time-varying delays is discussed. The proposed method does not introduce the input-output method. As a consequence, the method is easier than the existing ones. In addition, the method is less conservative than the previous methods.Chapter 2 and Chapter 3 are in the time domain framework. A main motivation for this framework is to further reduce the conservatism. The time domain methods can be used to effectively study the robustness and design controller. However, the results by the methods are inevitably conservative. In addition, the obtained stability conditions are sufficient though not necessary. Considering this, two new non-time-domain methods are proposed.In Chapter 4, a new method, mixed algebra and frequency method is proposed. It is known that the algebra method cannot lead to delay interval which guarantees the stability while the frequency method cannot be utilized to study the system stability for given delay. As to the time-domain methods, the results are inevitably conservative. One can hardly apply these methods to address the stability of the systems with incommensurate multiple delays. The mixed algebra and frequency method can overcome the respective disadvantages of the algebra method and the frequency method. By this new method, one can accurately find the delay interval for system stability.In Chapter 5, a new methodology is presented to study the complete stability of time-delay systems. First, we study the crossing directions of the imaginary roots and some important properties are obtained. According to these properties we can directly determine the crossing directions via the structure information. More specifically, here the structure information means which imaginary roots belong to one factored quasi-polynomial. The crossing direction of an imaginary root can be directly determined by the magnitude order of the imaginary roots of the factored quasi-polynomial where it belongs. Moreover, we can confirm that a time invariant delayed system will be ultimately unstable as delay increases if there exists at least one imaginary root. To obtain the desired structure information, a frequency sweeping method is presented. By performing the frequency sweeping method, the complete stability regions can be depicted. The method is easier than the existing ones.