Admissibility Analysis Of Singular Systems With Mixed Time Delays

As a kind of dynamic systems,singular systems have more general form and wider application than regular ones.Due to singularity of the state derivative coefficient matrix,the research of singular systems is more complicated and there are unique properties of singular systems.Delays are widely found in many engineering systems,such as chemical engineering,biological research,hydraulic systems,etc.Because of the complex and special dynamic characteristics,time-delay systems become the focus in control engineering research areas.Delay is often a source of instability and poor performance of the dynamical systems.Thus,the stability conditions of time-delay systems have attracted the attention of scholars.In this paper,a class of singular systems with mixed time delays consisting of discrete and distributed delays is studied.Firstly,two types of augmented Lyapunov-Krasovskii functionals(abbreviated as LKF)are constructed.Then,the integral inequality technique is used to deal with the integral term generated by the derivative of LKF.Finally,the admissibility conditions of time-delay singular systems are obtained,which are less conservative than other ones in the literature.The work of this paper is described as follows:(1)For singular systems with mixed time delays,a new type of LKF is constructed,and the Bessel-Legendre inequality(abbreviated as B-L inequality)is used to deal with the integral term generated by derivation on LKF.In terms of linear matrix inequality(abbreviated as LMI),an admissibility condition with less conservatism of singular systems with mixed time delays is given.Firstly,an augmented type LKF is establised by means of expanding dimension of quadratic function state vector added to the integral term.It is noted that the new LKF includes more time-delay information compared with the existing ones in the literature.Then,the B-L inequality is used to deal with the single integral term and double integral term generated by derivation on the LKF.Then,a new admissibility condition of singular systems with mixed time delays is obtained in terms of LMI.Finally,the numerical examples are shown to demonstrate that the proposed methods are effective.Compared with the results in the literature,the larger delay upper bound is obtaied by our results.It is indicated that the proposed method is less conservative and the admissibility conditions of singular systems with mixed delays is improved.(2)For singular systems with mixed time delays,a new augmented LKF is constructed in order to obtain an admissibility condition with less conservatism.Firstly,the dimention of state vectors of the single integral term and the double integral term are expanded by adding the new state vector,the derivative of the state vector and the single integral term.The purpose is that more time-delay information is inclued in the new LKF,which is helpful to reducing the conservativeness of the result.Then,the B-L inequality is used to deal with the single integral term and double integral term generated by derivation on the LKF.Then,a new admissibility condition of singular systems with mixed time delays is obtained again in terms of LMI.Compared with the results in the literature,the new one is less conservative,which is verified by the numerical example.