Time-Delay Systems Research Based On Parameters Weak Coupling Linear Matrix Inequality Set

Time-delays are frequently encountered in a variety of dynamic engineering and social systems,such as economic systems,biosystems,production scheduling,chemical industry process and network control system.And they are often sources of instability and degradation in control performance in control systems.So the study of dynamic control systems with delays is important both in theory and in practice,and has thus been of great interest to a large number of researchers.Recently,various approaches have been presented to obtain stability and control conditions for time-delay systems,among which the linear matrix inequality(LMI) approach is the most popular.The first reason is that LMIs can be cast into a convex optimization problem which can be handled efficiently by resorting to the latest developed numerical algorithms. The second reason is that by the LMI approach,once the stability conditions have been established,the corresponding control problems can be solved directly,especially when the state feedback is employed.However,most of the stability and control conditions which obtained by using LMI approach are sufficient conditions,and their conservatism are very important.The conservatism of stability and control sufficient conditions can be estimated by two standards.The first one is the scope of the condition's application,that is,value range of system parameters which ensure the condition holds.The wider the range is,the less the conservatism of Condition is.The second one is the performances of systems' stability and stabilization,including admissible time-delay,time-varying rate and disturbance attenuation of system.When the value range of system parameters are the same,the larger the upper bound of time-delay is,or the faster the admissible time-varying rate is,or the smaller the lower bound of disturbance attenuation is,the less the conservatism of condition is.From this point of view,there are two corresponding levels of conservatism of condition.The priority of the first level conservatism of condition is higher than that of the second.The stability and control conditions of time-delay systems can be classified into two types:delay-dependent condition and delay-independent condition.The applicable scope of delay-dependent condition is wider than that of delay-independent condition,and the first level conservatism of delay-dependent condition is less than the one of delay-independent. But the admissible time-delay is limited,and the second level conservatism for time-delay is just the reverse.On the contrary,delay-independent condition is valid for any time-delay and the second level conservatism is lower,while the applicable scope is narrower,and the first level conservatism is higher.In addition,other stability and control performance parameters need to be considered when one discusses the second level conservatism of condition. However,there seems to be no such stability and control conditions as have wide applicable scope as well as good stability and control performances.That is,there is no stability and control sufficient conditions of which two level conservatism are both low.Time-delays in some dynamic systems are always large,for example,the time-delays of economic systems or biosystems may be days or years.So,how to obtain the stability and control sufficient conditions of which two level conservatism are both low(the scope is wide and the performances of stability and control are excellent) is very important for such systems.After studying the algebraic characteristics of delay-dependent condition and delayindependent condition,a novel approach has been developed in this dissertation.LMI approach and free-weighting matrix technology are the basis of this approach.System parameters are coupled weakly with stability and control performance parameters(time-delay, time-varying rate and disturbance attenuation) in this approach.One will obtain stability and control sufficient conditions of which two level conservatism are both low.This approach is called as Parameters Weak Coupling Linear Matrix Inequality Set(PWCLMIS) approach. This approach can be seen as shortening the gap between delay-dependent condition and delay-independent condition.The main achievements of this dissertation are as follows.(1) The concept of two levels of conservatism is developed based on the applicable scope of systems which ensure a condition holds and the stability(control) performance of the condition.Various descriptions of conservatism in other literatures are involved by this concept.The first level conservatism of condition is more important than the second one. And this is the main reason of why the delay-dependent condition is the most popular.At the same time,it is stressed that new Lyapunov-Krasovskii functional plays a very important role in reducing the first level conservatism of condition. (2) The concept and constructing approach of PWCLMIS are developped.The first level conservatism of condition which is in terms of PWCLMIS is equivalent with that in terms of other forms.And on the second level conservatism,stability and control performances parameters can be valued relatively freely,that is,very large time-delay,very large time-varying rate and very small disturbance attenuation can be obtained.So,PWCLMIS approach provides a new thought to stabilize or control large time-delay systems,for example, economic systems.(3) The problem of robust H_∞control for uncertain stochastic systems with a timevarying,delay is investigated.The delay-dependent sufficient conditions for robust stochastic stabilization and robust H_∞control are established in terms of PWCLMIS based on the Lyapunov-Krasovskii stability theory and stochastic analysis tools.The results are derived by constructing a new Lyapunov-Krasovskii functional.The large time-delay,large timevarying rate and small disturbance attenuation are achieved by employing PWCLMIS approach.(4) In some cases,the large time-delays are required strictly.The problem of large time-delays exponential stability for stochastic systems with time-varying delays,Markovian switching and nonlinearities is studied.The stability sufficient condition is presented in terms of PWCLMIS via developing a new Lyapunov-Krasovskii functional,introducing free-weighting matrices into LMIs and decomposing the algebraic equations based on the principle of Lyapunov-Krasovskii stability.The large time-delay can be obtained by exploiting this approach.(5) The problem of delay-dependent robust stabilization and robust H_∞control with large time-delay and small disturbance attenuation for uncertain singular systems is studied. The control law is designed to guarantee that the resulting closed-loop system is regular, impulse-free and stable for all admissible uncertainties.And a prescribed H_∞performance is achieved in robust H_∞control problem.This criterion is devised in terms of PWCLMIS via developing a new Lyapunov-Krasovskii functional and employing free-weighting matrices. A control law with large time-delay and small disturbance attenuation is obtained by using this criterion.At the same time,a new lemma is introduced for proving the regular and impulse-free characteristic of closed-loop singular systems.(6) The problem of robust H_∞control for uncertain singular stochastic systems with state delay and Brownian motion is investigated.The state feedback control law is designed to guarantee that the resulting closed-loop system is regular,impulse-free and mean-square asymptotically stable for all admissible uncertainties.And a prescribed H_∞performance is achieved in the robust H_∞control problem.This criterion is devised in terms of PWCLMIS. A control law with large time-delay and small disturbance attenuation is obtained.Furthermore, the effect of the Brownian motion to the regular and impulse-free characteristics of singular systems is discussed in detail for the first time.(7) A discrete and distributed time-delays dependent simultaneous approach to deterministic and uncertain stochastic high-order neural networks is presented.The result is proposed in terms of PWCLMIS by exploiting a new Lyapunov-Krasovskii functional and by making use of novel techniques for time-delay systems.The admissible time-delays of condition are large.These results improve some previous results by removing the restrictive conditions.(8) The problem of robust H_∞control for a generic linear rational expectations model of economy with uncertainties,time-varying delay and random shocks is investigated.An alternative approach is used to describe the uncertainties of the system.The result is presented in terms of PWCLMIS.Large time-delay and small disturbance attenuation are achieved without increasing the first level conservatism of result.This is the first time that LMI technology is used to analyze the problem of robust H_∞control on economic systems with a large time-delay.(9) The time-varying rates of delays are free via the PWCLMIS approach.So the conditions can be widely used for fast and slow time-varying rate systems.