Stability Of Time Delay Systems Based On KYP Lemma And Augmented State LK Functionals

The analysis and synthesis of time-delay systems is a research hotspot in the field of control theory.Delay difference systems,which are a special class of time-delay systems,play a vital role in the stability analysis of certain time-delay systems.For instance,in the study of neutral-type time delay systems,the stability of delay-difference systems included in the derivative part is a critical technical assumption to ensure the stability of the overall system.Neutral-type time-delay systems are another important type of time-delay systems that find widespread practical applications in various engineering fields.The distinguishing characteristic of neutral-type time-delay systems is that they contain both the derivative term of the current state and that of the past state.Compared with general time-delay systems,neutral-type time-delay systems can more accurately and profoundly reflect the changing laws of phenomena,and most general time-delay systems can be regarded as special cases of neutral-type time-delay systems.However,the stability analysis of neutral-type time-delay systems is more challenging than that of general time-delay systems due to the presence of state delay derivatives.Therefore,studying neutral-type time-delay systems holds significant theoretical value.Stability analysis of time-delay systems is generally classified into two categories:delay-independent stability and delay-dependent stability.The former holds for any delay value,while the latter depends on the value of delays.For delay-difference systems and neutral-type time-delay systems,delay-independent stability can be further divided into strong stability and weak stability.In the case of weakly stable of delay-difference systems and neutral-type time-delay systems,although they are asymptotically stable,they may lose stability due to any small disturbance in the delay.Therefore,strong stability is typically of greater interest for these two types of systems,implying that sufficiently small delay perturbations do not change the stability of the system.Based on the above analysis,this dissertation focuses on studying the strong stability analysis and delay-dependent stability analysis of delay difference systems and neutraltype time-delay systems,and the obtained results are applied in robust stability analysis and controller design.The specific research results are as follows:1.The strong stability problem of linear continuous-time delay-difference systems with multiple time delays is studied.By using Kalman-Yakubovich-Popov(KYP)lemma,a strong stability condition in terms of linear matrix inequalities(LMI)is proposed.A time-domain interpretation of the proposed LMI-based condition is given in terms of a Lyapunov-Krasovskii(LK)functional,which allows us to reveal relations with an existing result.The LMI condition can easily be reformulated in a form where the dependence on the coefficients of the delay-difference system is linear,which is instrumental to establishing a sufficient LMI condition for robust strong stability of delay-difference systems with norm-bounded uncertainty.A necessary and sufficient condition for robust strong stability is also given,in the form of a structured singular value characterization.Based on the fact that integral delay systems belong to delay difference systems in a certain sense,the proposed method in this chapter is applied in the stability analysis of integral delay systems,and delay-dependent stability criteria are established for integral delay systems with multiple delays.2.The strong stability analysis problem of linear neutral-type time-delay systems with commensurate delays is studied.Different from the existing method with the considered original system being transformed into a high dimensional neutral-type system with a single delay,our proposed method directly deals with the original system.First,by using KYP lemma,a strong stability condition in terms of LMIs is established.It is shown that the proposed condition possesses a lower computational burden than the existing results.Then,a time-domain interpretation of the proposed condition is given in terms of a quadratic integral Lyapunov functional.Finally,based on the fact that the established condition involves matrices that are linear functions of the coefficients of the neutral-type time-delay systems,the proposed condition is further used to solve the robust strong stability analysis problem of neutral-type time-delay systems with norm-bounded uncertainty.3.The stability analysis problem of linear neutral-type time-delay systems subject to two different delays in both the state variables and the retarded derivatives of state variables is studied.By choosing a suitable state vector indexed by an integer k,a new augmented LK functional is constructed,and a delay-dependent stability criterion based on LMIs is developed accordingly.It is shown that the proposed condition is less conservative than the existing methods due to the introduction of the delay-product-type integral terms in the LK functional.The resulting stability criterion is then applied to the robust stability analysis of neutral-type time-delay systems with norm-bounded uncertainty,and a robust delay-dependent stability condition is obtained.Moreover,a delay-independent stability criterion is developed based on the proposed LK functional,and its frequency-domain interpretation is also given.4.The stability and stabilization problems of linear stochastic neutral-type timedelay systems with two delays are studied.Firstly,a less restrictive constraint is imposed to ensure the mean square exponential stability of the difference(D)operator.Such a constraint is necessary and sufficient for the stability of the difference operator in the deterministic setting.Then,for a specified augmented state vector,a new LK functional is constructed.Different from existing LK functionals,the proposed one not only contains all possible integrals but also avoids using absolute values of certain integral terms.By using the constructed LK functional,a novel delay-dependent stability criterion in terms of LMIs is derived.The proposed criterion is less conservative than the existing ones since it requires a less restrictive constraint on the difference operator and involves more decision variables.Based on the derived stability criterion,a method for designing state feedback controller is established.