Stability Analysis Of Hybrid Systems With Time-varying Delays

A hybrid system includes a purely continuous system and a purely discrete system,and its structure includes flow map,jump map,flow set and jump map.It is widely used in engineering systems,such as multi-agent systems and networked control systems.Besides,time-delay is often unavoidable in engineering systems,and it may lead to deteriorate performance of a system or even make a system unstable.Therefore,time-delay systems have been paid great attention.Although a lot of work focuses on hybrid systems and time-delay systems,there are still some limitations for hybrid time-delay systems.Due to the existence of time-delay and the complexity of the structure of a hybrid system,the dynamic behavior of a hybrid time-delay system and its stability become extremely complicated.This thesis investigates the problem of stability for hybrid systems with time-varying delays.By using Lyapunov functional method,stability analysis of hybrid systems with the conditions of small delay,large delay and homogeneity is studied,and the main contents of this thesis are given as follows:Motivated by the hybrid time-delay system models and stability results for networked con-trol systems with the small delays,scheduling protocols and varying transmission intervals,this thesis aims at improving the existing results in the literature.By using Lyapunov function ap-proach,a less conservative sufficient condition is derived to guarantee the uniform global asymp-totic stability of the systems.This thesis constructs a novel Lyapunov function to verify the sta-bility,and derives useful upper bounds for the maximum allowable transmission interval and maximum allowable delay,which significantly improve the existing results.Motivated by the problem of networked control systems with logarithmic quantization and varying transmission intervals and their results for analyzing the stability,this thesis aims at establishing a hybrid system with above-mentioned factors and proposing some sufficient conditions to guarantee the uniform global asymptotic stability of the systems.By constructing a novel Lyapunov function and using the property of quantization functions,the bounds for the maximum allowable trans-mission interval and the coarsest quantization density are derived to ensure that the systems are uniformly globally exponentially stable.Based on the problem of hybrid time-delay systems for networked control systems with large delays,this thesis establishes a hybrid time-delay model with varying transmission interval-s,varying delays,packet dropouts and scheduling protocols and presents a method to guarantee its stability.Under a relaxed condition,some sufficient conditions are proposed to analyze ex-ponential stability for the model by using Lyapunov function approach.By constructing a novel Lyapunov function with varying parameters and adjusting the parameters,this thesis derives a maximum allowable update interval with different delays such that the system is exponentially stable.Motivated by stability results and methods for the continuous time-delay systems and dis-crete time-delay systems with homogeneous properties,this thesis investigates the problem of stability for hybrid time-delay systems with homogeneous properties.A hybrid time-delay sys-tem with nonstandard homogeneity is proposed,and by using Lyapunov-Krasovskii functional and Lyapunov-Razumikhin function approaches,some improved sufficient conditions are pre-sented to ensure stability for the system.A relationship of the solutions between the system with different delays and degrees is established,and the effect of different delays and different degrees on stability of the system is studied.The definition and method of input-to-state stability are proposed for a hybrid time-delay system with inputs,and the problem of input-to-state stability is studied for the system with homogeneous properties.Under a more relaxed condition,a sufficient condition is presented to guarantee its input-to-state stability by using Lyapunov-Razumikhin function approach.The characteristics of input-to-state stability for the system are analyzed with different delays and different degrees.