Stability Analysis And Control Design For Several Classes Of Delayed Dynamical Systems

Time delay(TD) frequently occurs in a large amount of practical engineering applications such as aerospace, metallurgy, petrochemical, communication, electricity, biology, population, economy, and so on. Delayed dynamical system has attracted a good deal of attention of many scholars at home and abroad due to a wide range of its application. Meanwhile, Time delay is one of many key factors of instability, oscillator, or even poor performance for these dynamical systems. Whereupon, stability analysis and control design for delayed dynamical systems have been a very hot research topic. How to understand and control the performance of delayed dynamical systems is an important research subject. Based on Lyapunov-Krasovskii functional(LKF) theory, delay-partition approach(DPA), inequality processing techniques, linear matrix inequalities(LMIs) etc,we investigate the stability analysis and control design for neutral-type delayed neural networks(NNs), neural networks with distributed delay, neural networks with mixed delays and delayed Lurie systems(LSs). The main research results of this paper are as follows:1. The stability analysis of neutral-type delayed NNs. By constructing a novel LKF that contains triple and quadruple-integral terms, improved delay-dependent stability results are formulated. Our derivation makes full use of the idea of second-order convex combination and the property of quadratic convex function, which play a key role in reducing further the conservatism of conditions. Finally, four numerical examples are presented to illustrate the effectiveness and advantages of the theoretical results.2. The stability analysis of NNs with distributed time delay. The novelty of this paper lies in the consideration of a new integral inequality proved to be less conservative than the celebrated Jensen’s inequality and takes fully the relationship between the terms in the Leibniz-Newton formula in the within the framework of LMIs into account.By employing a general DPA, an appropriate LKF is constructed. Based on the new integral inequality approach and DPA, modified delay-dependent stability criteria are derived.Finally, four numerical examples are given to demonstrate the effectiveness and the advantage of the proposed method.3. The stability analysis of NNs with mixed time delays. By making full use of a novel a multiple integral inequality proved to be better than the celebrated Jensen’s inequality, new stability results are established. By constructing an appropriate LKF including the multiple integral terms, improved delay-dependent stability conditions are obtained. Furthermore, less conservative stability results are derived by dividing the distributed delay into multiple nonuniformly subinterval. Finally, three numerical examples are presented to illustrate the effectiveness and advantages of the theoretical results4. H∞controller design for delayed NNs. The main goal of this work is to design a desired and effective H∞control law to ensure the asymptotical stability of the closed-loop system with a given disturbance attenuation level γ > 0. By constructing an augmented LKF and introducing a general and complete delay-partition approach, a novel delay-dependent stability criterion is obtained in terms of LMIs. Moreover, by getting the utmost out of a modified Wirtinger’s integral inequality, an improved sufficient condition is achieved for the existence of the H∞control problem. Finally, two numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed methods.5. The stability analysis of uncertain neutral-type LSs with mixed time delays. The system not only has time-varying uncertainties and sector-bounded nonlinearity, but also discrete and distributed delays. By constructing an appropriate LKF and employing effective mathematical techniques, less conservative delay-dependent stability results are established. Finally, three numerical examples are presented to illustrate lesser conservatism and the advantage of the proposed main results.6. CLSs synchronization with time-varying-delay feedback control. A novel integral inequality is developed by employing two adjustable parameters, which encompasses the celebrated Wirtinger’s integral inequality and Jensen’s inequality as two special cases. By constructing an augmented LKF taking fully the information time-varying-delay range into account, less conservative delay-dependent synchronization criteria are established in the form of LIMs. Besides, the desired controller gain can be achieved by introducing new nonlinear function conditions. Finally, two numerical example of typical Chua’s circuit is presented to show the improvements over the existing criteria and the effectiveness of the design approach.7. Delayed CLSs synchronization with sampled-data feedback control. This paper proposes a novel approach to study the problem of master-slave synchronization for CLSs with sampled-data feedback control. Specifically, first, it is assumed that the sampling intervals are randomly variable but bounded. By getting the utmost out of the usable information on the actual sampling pattern and the nonlinear part condition, a newly augmented LKF is constructed via a more general delay-partition approach. Second, in order to obtain less conservative synchronization criteria, a novel integral inequality is developed by the mean of the new adjustable parameters. Third, a longer sampling period is achieved by using a double integral form of Wirtinger-based integral inequality(WBII).Finally, three numerical examples with simulations of Chua’s circuit are given to demonstrate the effectiveness and merits of the proposed method.8. CLSs synchronization with stochastic sampled-data control. This study investigates the problem of designing stochastic sampled-data controller for master-slave synchronization of CLSs via a novel approach. Specially, first, we assume that the occurrence probabilities of the sampling intervals are fixed constants and satisfy a Bernoulli distribution. In order to take full advantage of the sawtooth structure characteristics of the sampling input delay, we construct a newly augmented LKF based on the extended Wirtinger inequality. Second, by using a novel free-matrix-based integral inequality(FMBII) including well-known integral inequalities as special cases, an exponentially mean-square synchronization criterion is proposed for analyzing the corresponding synchronization error system. Third, based on the above method, the desired feedback gain matrix can be designed successfully. Finally, three numerical simulation examples of Chua’s circuit and neural network are given to illustrate the effectiveness and superiorities of the proposed method.