| Lur’e control system is a kind of typical nonlinear control system.It has a wide range of engineering background in many fields,such as aircraft control,aerospace control,hydraulic servo control and so on.The study of the Lur’e system began in the 1940s and was proposed by the famous Soviet scientist Lur’e in the study of vehicle autopilotSwitched systems are widely used in various fields of control.They al-so exist in many other fields,such as bio-ecological science,social sciences,transportation,energy environment and so on.For example,the growth and death of biological cells,traversal and landing of the aircraft,the server wait-ing for the network buffer switch,can all be modeled as switched systems.In recent years,more and more scholars pay attention to the research of switched systems.Switched Lur’e delay systems,as a class of Lur’e delay systems with switching,are widely used in real life,such as Hopfield neural networks,Lotka-Volterra ecosystems,variable structure systems and so on.Based on this,the stability problem of switched Lur’e systems with time-delay has high theoret-ical and practical significanceIn this paper,the absolute stability of switched Lur’e systems with time-delay are considered.Different research methods are adopted to select the appropriate switching signal and Lyapunov function,and the corresponding conclusions,as well as the mathematical derivation and proof are given.Be-sides,Matlab software are used to solve the algorithm,numerical simulation and so on.This paper is organized as follows.In Chapter 1,we mainly introduce the background and significance of the subject,the present development situation,the trend of this dissertation and the main results.In Chapter 2,we mainly introduce some necessary preliminaries such as definitions,properties,some related lemmas and so on.For example,some definitons and methods in stability of systems,methods for designing the switching law are introduced in this chapter.The Chapter 3,we mainly study the absolute stability of a class of linear switched Lur’e delay systems.The absolute stability of such a special Lur’e de-lay system without switching is first studied by Qinglong Han.For the research of a single system(m=1),the method of cross-term and model transforma-tion is abandoned,and a class of suitable Lyapunov functions is selected and the derivatives are bounded appropriately.The sufficient conditions for the absolute stability of a single Lur’e system with time-delay(m=1)are ob-tained.We consider the absolute stability of the switched Lur’e delay system,that is,to design appropriate switching rule between multiple subsystems and investigate the stability of the new system(m≥1).In this chapter,on the ba-sis of the predecessors,by constructing the appropriate Lyapunov-Krasovskii functional,the bound method of the Lyapunov-Krasovskii functional is further discussed,and the appropriate switching signal based on ADT is also designed such that the subsystem is still stable after switching.The results show that the method proposed in this chapter makes the Lur’e delay system has better stability performance.On the one hand,the conservative property of the ex-isting stability conclusion is reduced and its conclusion is expanded;On the other hand,the time-delay upper bound of the system is also expanded.The Chapter 4,we further study the absolute stability of the uncertain switched Lur’e system with time-delay.On the one hand,for the study of the absolute stability of a single uncertain Lur’e system with time-delay(m=1),Han Qinglong,Dong Yue,Wu Min,He Yong and Zeng Hongbing have studied and improved the absolute stability condition by different methods,and have proposed sufficient conditions for the absolute stability of a single uncertain Lur’e time-delay system.On the other hand,in the stability study of the switched delay system,the general method is to select the appropriate Lya-punov function and consider the upper bound of their derivatives firstly;And then,define them by different methods;Finally,in combination with the design of switching rule,find the conditions for the stability of the time-delay systems.It is to be noted that sometimes the Lyapunov function chosen tends to make the solution of the positive definite symmetric matrix in the stability condition less flexible and the solution process is more difficult.In this chapter,on the one hand,we extend a single uncertain Lur’e system with time-delay(m=1)to a plurality of uncertain Lur’e systems with time-delay(m≥1),and focus on the absolute stability of the switched Lur’e system with uncertainties.Also we consider the influence of the switching rules on the performance of the sys-tem,and enhance the time-delay upper bound of the system which make the system stable.On the other hand,the new Lyapunov function is constructed,so that the solution of LMIs is more flexible and the positive definite symmet-ric matrix has higher elasticity.Firstly,the time-delay interval is decomposed into n equal sub-intervals and then a suitable Lyapunov-Krasovskii functional is constructed in combination with the double integral.Secondly,by mean-s of the integral inequality and the MDADT method,the absolute stability criterion based on the LMIs technique is obtained,and the conclusion in the relevant literature is also improved.In particular,in the process of dealing with the upper bound of the derivative of the Lyapunov function,the general free weighting matrix theory is replaced by the integral inequality.Finally,a numerical example is used to simulate the simulation,which shows that the conclusion of this chapter broadens the stability of general Lur’e system with time-delay and uncertainty,and improves the upper bound of the time-delay which makes the system stable.Besides,compared with the Lyapunov func-tion selected by the general research of switched delay system,the Lyapunov function in this chapter is easier to get,and the solution flexibility is improved.In Chapter 5,the absolute stability of the switched Lur’e delay system with unstable subsystems is studied.The research on the absolute stability of the Lur’e system with time-varying delays and a single subsystem(m=1)is first studied by Qinglong Han who gives the sufficient condition of the absolute stability by selecting the Lyapunov function.In fact,there are many unstable Lur’e delay systems in real life.In this chapter,by combining the unstable subsystem with the stable subsystem,we study the absolute stability of the new system(m≥1).On the one hand,a switching rule between the subsystems is designed to make the system stable.On the other hand,different designs of the switching rules also enable the stability performance of the stabilization subsystem to be improved.First,a suitable Lyapunov function is constructed and the upper bound of the derivative of the Lyapunov function is appropriately scaled by a new lemma.The conservative nature of the condition is reduced.In particular,when a variable delay is a differentiable function that satisfies a certain condition,a better result is obtained.Then,by considering the role of the unstable subsystem and designing the appropriate switching signal,the absolute stability of the whole system can be achieved by controlling the running time ratio of the stabilizing subsystem and the unstable subsystem.Finally,through the numerical simulation,the feasibility and the superiority of the conclusion of this chapter are givenIn Chapter 6,the model of chapter 4 is extended generally,and on this basis,the absolute stability of uncertain switched Lur’e systems with time-varying delay is further studied by using different methods,and more general results are obtained,where the delay is continuous and differentiable and its lower bound is 0.Besides,the uncertain parameters involved in the system are norm-bounded.In chapter 4,when defining the derivative of Lyapunov func-tion,some useful integral items are directly ignored in the process of dealing with the integral term,so that the results are conservative to a certain extent In view of this,in this chapter,we construct a suitable Lyapunov-Kraosvskii functional,and with the help of Newton-Leibniz formula,by introducing a new free matrix,the derivative of Lyapunov-Kraosvskii functional is bounded In this process,no integral term is directly ignored.Secondly,the switching signal is designed by MDADT method,and the delay-dependent absolute sta-bility criterion based on LMI technique is obtained.Free matrix theory and MDADT method make the feasible domain of LMI solution wider,that is to say,the stability condition of the obtained solution is less conservative.Nu-merical simulation shows that the obtained results reduce the conservatism of the existing literature resultsIn Chapter 7,a conclusion of our present work is given,and we also introduce some further study ideas in the future. |
