Study On Stability And Control For Dynamical Systems With Time-delays

Time delay is commonly encountered in real systems, and its existence is frequently a source of instability and poor performance. The problem of stability analysis and controller synthesis for time-delay systems is an important topic in automatic control. The existing stability results for time-delay systems can be classified into two types: delay-independent stability and delay-dependent stability. In the first case, the stability condition that the system parameters satisfy is applicable for any bounded delay. In the second case, the stability condition that the system parameters satisfy only guarantee the system to be stable up to some maximum value for the time-delay. In applications, system delays are frequently limited, and their bounds are known. So, delay-independent stability analysis and controller synthesis may be conservative. In this paper, by combing M-matrix approach or descriptor system approach with inequality techniques, we study the problems of delay-dependent stability and delay-dependent controller design for several classes of time-delay systems. For the M-matrix approach, the idea is to transform the problem of estimating the solutions of the time-delay system into the problem of estimating one element of the solution by using M-matrix theory. Then using the inequality techniques, we can obtain delay-dependent conditions for stability of the time-delay system of concern. For the descriptor system approach, the idea is to transform the original system into a descriptor system with time-delay and construct an appropriate Lyapunov functional for the descriptor system. Then, applying the new Lyapunov functional and using inequality on product of two vectors, we can derive sufficient conditions for stability of the time-delay system of concern in terms of delay-dependent linear matrix inequalities. Finally based on the delay-dependent stability result, we can design the controllers to be considered. Although the purpose of this thesis is to study the problems of delay-dependent analysis and control for time-delay systems, we derive new results concerning delay-independent analysis and control. The main work and research results of this thesis lie in the following.1. By applying M-matrix theory and using inequality techniques, we study the problem of absolute stability for Lur'e indirect control systems with variable delays. We extend the existing delay-independent stability results for Lur'e indirect control systems with constant delays to the systems with variable delays. Moreover, we first obtain delay-dependent stability results for the above systems. In addition, we successfully apply this idea to study the problem of delay-dependent exponential stability for neural networks with variable delays, and derive delay-dependent criteria for exponential stability based on M-matrix.2. By using the descriptor system transformation, we construct a new type of Lyapunov functional to study the absolute stability for Lur'e direct control systems with delays. Based on linear matrix inequalities, we obtain sufficient conditions for delay-dependent absolute stability for the systems with constant or continuous bounded delays. We also derive delay-independent sufficient conditions for absolute stability for the systems with constant delays. Numerical examples illustrate the effectiveness of our methods. 3. By combining descriptor system approach with inequality techniques, we study the problem of exponential stability for neural networks with variable delays. Based on linear matrix inequalities, we obtain sufficient conditions for delay-dependent exponential stability for neural networks with constant or continuous bounded delays. We also derive delay-independent sufficient conditions for exponential stability for neural networks with constant delays in terms of linear matrix inequalities. Numerical examples are presented which show that our results are less conservative that the existing stability criteria.4. By extending the descriptor approach to the It-type stochastic system with delays,...