Research On Passivity And Reachable Set Of Several Delayed Dynamical Systems

Time delay often occurs in various practical engineering systems,such as popula-tion,power systems,aerospace,economic systems and so on.The existence of time delay may reduce the performance of the system or even lead to instability.Therefore,the analysis of time delay on the control performance of dynamical systems is a hot topic.The study of time-delay systems has attracted more and more attention of scholars,which gets some important research results.Based on Lyapunov–Krasovskii functional theory,delay-partition method,reciprocally convex approach,free-weighting matrix technique and linear matrix inequalities?LMIs?,we investigate the passivity and reachable set esti-mation for time-delay systems in this paper.The main results of this paper are as follows:1.The passivity analysis of discrete-time stochastic delayed neural networks is re-searched.By utilizing a relaxed Lyapunov–Krasovskii functional,delay-decomposition method and reciprocally convex approach,some sufficient passivity conditions are estab-lished in the form of linear matrix inequalities.Furthermore,new criteria do not require all the symmetric matrices involved in the employed functional to be positive definite,the positive definite of the functional can be ensured by three matrix inequalities.Finally,two numerical examples are provided to demonstrate the effectiveness of the proposed results.2.The passivity analysis of discrete-time neural networks with mixed delays is in-vestigated.The system includes not only the time-varying delay,but also the distributed delay.Firstly,we introduce two zero equalities,which can reduce the conservatism.Then,by choosing a suitable Lyapunov-Krasovskii functional,based on reciprocally convex in-equality method and free-weighting matrices techniques,new criteria are derived.Finally,three numerical examples are given to illustrate the effectiveness and application in the gene regulatory network of the proposed method.3.The problem of finite-time passivity for discrete-time system with time-varying delay and nonlinear perturbations is studied.By constructing a new Lyapunov–Krasovskii functional involving variable ratios⁶-4)-1)and employing a new summation inequality named discrete Wirtinger-based inequality,finite-time passivity criteria are established.Finally,three numerical examples are given to show the effectiveness and superiority of the proposed results.4.The reachable set estimation for singular systems with time-varying delays and linear systems with mixed delays are researched.Based on a novel Lyapunov–Krasovskii functional which contains some triple integral terms,reciprocally convex approach and free-weighting matrix method,sufficient conditions are derived in terms of linear matrix inequalities.The new results reduce the conservatism of the conditions and get a more accurate set of estimation.Finally,four numerical examples are presented to illustrate the validity of the proposed method.5.The problem of reachable set bounding for discrete-time system with time-varying delay is studied.First,a relaxed Lyapunov-Krasovskii functional is established in which all the symmetric matrices are not positive definite.Then,by utilizing dis-crete Wirtinger-based inequality and reciprocally convex approach,sufficient conditions are derived to find an ellipsoid to bound the reachable sets of discrete-time system with time-varying delay when the initial state vectors are not necessary to be zero.Finally,two numerical examples are given to show the effectiveness and less conservatism of the proposed method.