Research Of Stability And Control Of 2-D Discrete Systems With Time-delay Based On Finite Sum Ineqyalities

With the development of modern industry and national economy,people need to deal with a growing number of multivariable system and multidimensional signals,such as rolling mill system,image processing,digital filtering and so on.Most of them are characterized by two-dimensional(2-D)discrete system model.Therefore,in recent years,the theory of2-D system has been paid much attention by many scholars.In practical engineering,the existence of time delay is unavoidable,which usually leads to the system is instability or reduces the performance.Therefore,many scholars have paid more and more attention to the stability analysis,control and filtering problems of 2-D discrete systems with time delay.Among them,delay-dependent stability and control has become the main research direction of 2-D discrete system with time-varying delay.Aiming at such systems,most of them use the inequalities method to deal with the Lyapunov-Krasovskii functions,so as to get the stability criteria and guarantee the stability of the system.Therefore,it is very important to obtain the more rigorous inequalities to ensure that the system gets less conservatism.In this paper,we propose new finite sum inequalities for 2-D discrete systems with time-varying delay,and use it to deal with the Lyapunov-Krasovskii functions and analyze the stability and control problems of the system.First of all,this paper introduces the research background and the research status of 2-D discrete system,and briefly introduces and analyzes the system model,the basic theory of stability and the state feedback control.Secondly,according to finite sum inequalities of one dimensional(1-D)discrete systems,the new finite sum inequalities forms of 2-D discrete systems are derived.The difference of Lyapunov-Krasovskii functions are processed by combining it with the reciprocally convex inequality,and the stability criterion is derived in terms of linear matrix inequalities.Not only the system conservatism is smaller,but also the upper bound of time delay is improved.Considering the state feedback of the system,the state feedback controller is designed to guarantee the stability of the closed-loop system,and the state feedback control gain of the system is obtained,which optimizes the performance of the system.Finally,we study the 2-D Markov jump system with time-varying delay described by the Roesser model.When constructing Lyapunov-Krasovskii functions,we consider the upper bound and the lower bound of the time delay.The stability criterion is obtained by using the finite sum inequalities of 2-D discrete systems to processing the difference of functions.And the maximum allowable upper bound of time delay is given.Compared with the existing result,the effectiveness and advantage of employing the proposed results are illustrated via numerical examples.