Stability Research For Time-delay Systems Using LOI/LMI Method

Time delays arise in the transport of energy,mass,information and such,and are omnipresent in natural and engineered systems.The presence of time delay often leads to the performance degradation and even instability of the system.Therefore,the study of time delay systems has always been a hot topic in control theory and research.With the development of the construction method of Lyapunov Krasovskii functional and linear inequality technology,Lyapunov Krasovskii method and LMI technique have been the main tools to the study of time-delay systems in recent two decades.Recently,the development of Sum Of Square technology and the newly proposed PIETOOLS(Partial Integral Equation Tools)bring the new sight into the research of time-delay systems.Based on linear matrix inequality and linear operator inequality technique,this paper investigates the stability problems of linear and nonlinear time-delay systems.The main works are summarized as follows.(1)The stability problem of continuous linear systems with multiple delays is studied.Considering the fundamental-state concept,we represent the linear multi-delay systems into a fundamental-state-space equation and propose the corresponding DPS formation.A complete quadratic Lyapunov-Krasovskii functional in the form of inner product of linear partial integral operator is constructed and a novel stability criterion in the forms of linear operator inequality is given,which is solved by MATLAB PIETOOLS.The numerical example shows the correctiveness and effectiveness of our method.(2)The H_? optimal observer synthesis for linear multi-time-delay systems with sensor noise is studied.The designed PDE-form observer can correct both the current state and the history of state of the system.A complete quadratic Lyapunov-Krasovskii functional based on Partial Integral operator is constructed,and a delay-dependent optimal estimation condition is obtained in the form of linear operator inequality.The method to solve the observer gain operator which can satisfies the H_?robust performance is provided.Using the MATLAB PIETOOLS solves the optimal H_?norm bound.The numerical examples show that the bound value is very close to the theoretical analysis bound value obtained by Padétool,even upto the fourth decimal point.(3)The stability analysis of discrete time-delay systems is studied.We propose a less conservative summation inequality,which relaxes the Wirtinger inequality.By means of a new summation inequality and the directly constructed Lyapunov-Krasovskii functional,new delay-dependent stability criteria in the form of linear matrix inequalities are obtained for discrete-time linear delay system and discrete-time recurrent neural network system respectively,which can be solved by MATLAB LMITOOLS.Less conservativeness of the criteria over some reported results is shown by numerical example.(4)The stability analysis and sample-date-based control of T-S fuzzy time-delay system are studied.The state feedback controller is constructed containing the current and history of state of the system.Based on the concept of loop function,a new two-side loop function is constructed.The stability performance is derived from the derivative change condition of sum of this loop functional and Lyapunov-Krasovskii functional.Based on the improved integral inequality technique,new delay-dependent sampled-data control strategy and stability criterion are obtained in the form of linear matrix inequality,which can be solved by MATLAB LMITOOLS.The numerical example shows the correctiveness and advantage of our method.