Stability Analysis Of Singular Time-delay Systems

In practical control systems, there often exist time-delay phenomena, which are alwaysregarded as the cause that reduce system performance and even lead to system instability. Therefore,it has great theoretical and practical significances to research time-delay systems. Since the singularsystem was first proposed by Rosenbrock in the1970s, many scholars have done much research onit. In just a few decades, singular systems have already been an indispensable branch of moderncontrol theory. In fact, time-delay extensive exists in the singular systems, it has become a hotspotin the field of control. The most important thing in system research is the stability analysis, becausethe stability is the basis for other research.This paper, first, analyzes and summarizes the results of previous studies, finding: Lessconservative delay-dependent stability criteria can be obtained by using delay decompositionapproach to study the stability of regular systems. However, nowadays, delay decompositionapproach is usually equisection method. The same length of the delay subinterval would bring theconversation for the results and the study objects are regular systems, while the studies on singularsystems are rare. A less conservative result was obtained by utilizing the equal delay decompositionapproach proposed by professor Han, to analyze singular time-delay systems with constant delay.According to this, if an unequal delay decomposition approach is applied to the analysis of stabilityof linear singular time-delay systems, an even less conservative result will be obtained.Hence, this paper studies the above problems on the basis of singular systems with constantdelay, time-varying delay and interval time-varying delay, respectively. According to the differentdelay characteristics of three kinds of time-delay singular systems，the unequal delay decompositionapproach is used to divide delay into two parts. Then the new Lyapunov-Krasovskii functionals areproposed. Meanwhile, effective items are considered when analyzing the derivative ofLyapunov-Krasovskii functionals. Finally, the less conservative stability criteria are obtained.In the first part, the stability analysis of singular systems with constant delay is considered.The regularity, impulse-free and asymptotic stability of singular systems with constant delay isstudied. By using the unequal delay decomposition approach, the constant delay h is divided intoh1andh2. A less conservative delay-dependent stability criterion is obtained by proposing a newLyapunov-Krasovskii functional and combining Jensen inequality to handle the integral terms. Atthe end of this chapter, a simulation example shows that by this method, a less conservative result isobtained compared with the equal delay decomposition approach. In the second part, the stability analysis of singular systems with time-varying delay isconsidered. The regularity, impulse-free and asymptotic stability of singular systems with timevarying-delay is studied. Based on the unequal delay decomposition approach, which is mentionedin the first part, the time varying-delay h (t)is divided into h1(t)and h2(t), then a newLyapunov-Krasovskii functional is proposed. Meanwhile, effective items are considered whenanalyzing the derivative of Lyapunov-Krasovskii functional, which takes the full use of the relevantinformation of time-delay. As a result, a less conservative stability criterion is obtained. At the endof this chapter, two simulation examples are showed to prove the criterion is effective.In the third part, the stability analysis of singular systems with interval time-varying delay isconsidered. It focus on the regularity, impulse-free and stability of varying time-delay singularsystems when the lower-bound of h(t)is hm≠0. On the basis of part two, interval time-varyingdelay h (t)is divided into h1(t)and h2(t), and lower-bound of h1(t)ish1m, which h1m≠0.Then a new Lyapunov-Krasovskii functional is proposed, containing the information ofhm andh1m.In addition, when dealing with the Lyapunov-Krasovskii functional,effective items are kept toobtain the less conservative stability criteria. At last, two simulation examples illustrate theeffectiveness of this conclusion.In the end, this paper summarizes all the have-done studies and forecasts the following worksthat need to be done for the deeper research.