Stability Analysis Of Time Delay System Based On Inequality Of Quadratic Function In Finite Interval

Time-delay phenomenon is common in daily life and various actual control systems.The existence of time-delay usually causes system performance deterioration or system instability.The first prerequisite for normal operation of a system is stability.Therefore,it is very important to analyze the stability of the time-delay system.To solve the problem that the linear matrix inequality(LMI)toolbox in Matlab can't solve the nonlinear matrix inequality,this paper proposes a finite interval quadratic function definition method to define the matrix inequality of quadratic function type.Through the method of drawing,N tangents are made in the specified interval of the quadratic function,and the tangents are used to replace the quadratic function,thereby converting the quadratic function matrix inequality into a linear matrix inequality.The obtained quadratic function inequality processing method is applied to the stability analysis of three types of time-delay systems.The specific research content includes the following aspects:(1)The stability of linear time-delay systems is discussed.In view of the conservativeness of the reciprocally convex inequality,this thesis introduces a suitable free matrix into the reciprocally convex inequality,extends the convex term to the second order,and proposes the quadratic reciprocally convex inequality.The proposed inequality is used to estimate the reciprocally convex term.At the same time,the problem of defining the functional derivative as a quadratic function,using the finite interval quadratic function inequality method proposed in this thesis,so as to obtain the stability in the form of linear matrix inequality condition.Finally,numerical examples are used to prove the superiority of the proposed reciprocally convex quadratic inequality and quadratic function inequality judgment method.(2)The stability of time-delay neural networks is studied.Aiming at the problem that the traditional time-delay product functionals are not comprehensive enough to consider the information,this thesis first constructs a second-order time-delay product-type functional,and considers the second-order time-delayh~2(t)into the functional,so that the functional contains more time-delay information and systems Information,thereby obtaining the second-order delay-dependent stability criterion to ensure the stability of the system.Then the finite interval quadratic function inequality definition method proposed in this thesis is used to process the second-order delay-dependent stability criterion,thereby turning it into a delay-dependent stability criterion.Finally,a numerical example verifies the validity and superiority of the second-order delay product functional and quadratic function inequality definition methods.(3)The stability of T-S fuzzy time-delay system is discussed.Based on the time-delay product functional,the reciprocally convex quadratic inequality and the method of finite interval quadratic function inequality defined proposed in this thesis,a new criterion to ensure the stability of the T-S fuzzy time-delay system is obtained.In the analysis of numerical simulation results,it is proved that the method of defining quadratic function inequality proposed in this thesis has higher definition accuracy,so that the stability criterion of T-S fuzzy time-delay system is less conservative.Finally,the application of the proposed method in the analysis of delay-dependent stability of time-delay systems is summarized,and the difficulties encountered in the related research of time-delay systems are prospected.