| In recent years,descriptor systems,also known as singular systems,generalized state-space systems and implicit systems,appear in a range of practical systems,including robots,networks and economic systems.Descriptor systems have a wide range of far-reaching practical significance and promising applications,and thus have attracted the attention of many scholars at home and abroad.Among the directions extended from descriptor systems,nonlinear descriptor time-delay systems are the hot spot gradually discovered by scholars in recent years.For the study of uniformly asymptotic stability of nonlinear descriptor time-delay systems,the regularity and the impulse-free property of the systems should be guaranteed first,then prove stability,and then the uniformly asymptotic stability should be considered.The criterion of uniformly asymptotic stability is related to the selection of the Lyapunov-Krasovskii functional(abbreviated as L-K functional)and the estimation of the integral terms produced by the L-K functional after derivation.Compared with Jensen inequality and Wirtinger inequality,Bessel-Legendre inequality(abbreviated as B-L inequality)is less conservative.Inspired by this,the B-L inequality can be applied to nonlinear descriptor time-delay systems in order to obtain uniformly asymptotic stability criterion with less conservatism.Based on the Lyapunov’s second method of stability and constrained equivalent transformation of descriptor systems,a new L-K functional is established by the B-L inequality to solve the uniformly asymptotic stability problem of nonlinear descriptor time-delay systems.The uniformly asymptotic stability conditions are given in the form of strict linear matrix inequalities(abbreviated as LMI),which is better than some currently known research results in terms of conservatism.The main work is as follows:(1)Firstly,the problem of uniformly asymptotic stability of nonlinear descriptor time-varying delay systems is addressed.By constructing a new augmented L-K functional,and then,the derivative of the L-K functional,using the third order B-L inequalities and double third order B-L inequalities to enlarge the derivative of some integral terms,a new uniformly asymptotic stability criterion is obtained in terms of LMI for nonlinear descriptor time-varying delay systems.Notably,B-L inequality is closer to the truth value of the amplified integral term than Wirtinger inequality,auxiliary function inequality and other inequalities.So the criterion of the uniformly asymptotic stability is less conservative.Finally,a numerical example is given to verify the feasibility and effectiveness of the proposed method,indicating that the proposed method is less conservative than other one in the literature.(2)Furthermore,the problem of uniformly asymptotic stability of nonlinear descriptor systems with distributed time delay is studied.Under the assumption that the nonlinear descriptor system with distributed time delay is regular and impulse-free,the augmented and with triple integral L-K functional are constructed.Then,some of integral terms produced by L-K functional after derivation are dealt with by the third order B-L inequalities and double third order B-L inequalities.Consequently,the uniformly asymptotic stability criterion for nonlinear descriptor systems with distributed time delay is obtained in terms of LMI.Different from(1),due to the addition of distributed time delay in nonlinear descriptor systems,there are few results on this kind of system models at present,so no similar papers can be found to compare the results.Finally,a numerical example is provided to demonstrate feasibility and validity of the proposed method. |
