Stability Analysis Of Systems With Time Delays

In practical dynamic systems, time delays are general, which often lead to instability and bad performance. Then, the stability analysis of system with time delay is the hot topic in the control field. Based on Lyapunov stability theory, many results have been obtained, which still need improvement. By constructing new Lyapunov-Krasovskii functional with the idea of delay fractioning, this paper presents new stability theorems, which become less conservative as the fraction number increases.First, we introduce the idea of delay partitioning to perform stability analysis for the delay systems with nonlinearity, such as neural networks, which have been extensively applied in engineering. We obtain the asymptotic stability condition for Hopfield neural networks and the exponential stability theorem for cellular neural networks. These results are in standard form of linear matrix inequalities and can be solved by MATLAB tools. Some examples are provided to prove the advantage of these theorems.Second, in practice, time delays are often time varying, which usually vary from nonzero lower bound. This paper first partitions the time-varying delay into constant part and time-varying part, then, apply delay fractioning to the constant part. In addition, we consider the structure parameter uncertainty and nonlinear disturbance and apply proper methods to get the stability conditions.Finally, this paper deals with some other hot-topic problems of time delay systems. For instance, we obtain new passivity theorem for neural networks with time-varying delays; present new synchronization stability of complex networks with coupling delays. Furthermore, based on the stability analysis, this paper proposes the method of state estimator design and the new idea of Smith predictor.The delay fractioning idea plays an important role in the stability analysis of time delay systems。Its advantage can be shown by many examples.