Robust Stability Analysis Of Systems With Time Delays And Uncertainties

Time delays widely occur in many practical systems, and may cause undesirable dynamic behaviors such as oscillation and instability. On the other hand, uncertainties are unavoidable due to modeling errors, measurement errors, and linearization approxi-mations, external perturbations, environmental noises and incomplete information of the system. The characteristics of dynamic systems are significantly affected by the pres-ence of the uncertainty, even to the extend of instability in extreme situation. In view of this, robust stability analysis of dynamic systems with time delays and uncertainties has received considerable attention by many researchers, a great deal of results have been reported in the literature. This dissertation focuses on the robust stability for several delayed dynamical systems with uncertainties such as neutral systems, stochastic Hop-field neural networks, discrete-time systems and bidirectional associative memory (BAM) neural networks based on Lyapunov stability theory, stochastic theory and linear matrix inequality (LMI) technique. The main contributions of this dissertation are as follows:(1)The problem of robust stability for the uncertain neutral system with interval time varying discrete delay and the neutral system with nonlinear perturbations is investigated. Based on Lyapunov-Krasovskii functional method and LMI technique, several new criteria are derived. The developed free weight matrices technique and two integral equalities are employed to overcome the conservative estimation when dealing with the cross terms. Numerical examples are given to demonstrate the effectiveness and the improvement of the proposed method.(2) Impacted on the surrounding environment noise, time delays tend to be ran-dom. It is worth noting that only the deterministic time delay case was concerned and the stability criteria were derived based only on the information of variation range or bounds of the time delay in the existing references. To this end, a new type of discrete system with random delay is proposed. The problem of robustly globally exponentially stable in the mean square sense for the proposed system is investigated. By defining a Lyapunov-Krasovskii functional by utilizing some new finite sum equalities for the bound-ing of cross term, a delay-distribution-dependent criterion is obtained. Different from the existing ones, the proposed criterion depends on not only the size of the delay but also the probability distribution of it. Numerical examples suggest that the results are effective.(3)The problem of robustly stable in the mean square for a class of stochastic Hop-field neural networks with mixed time-varying delays and parameter uncertainties is in-vestigated. The time-varying discrete delay is assumed to belong to an interval. By defining a new Lyapunov-Krasovskii functional and employing the stochastic stability theory, several delay-range-dependent stability criteria are established in terms of linear matrix inequalities (LMIs), which describe the effect of the stochastic noise on the system stability. Two bounding integral equalities are utilized to reduce the conservativeness. Then, the problem of robustly exponentially stable in the mean square for delayed un-certain Hopfield neural networks with Markovian jumping parameters is studied. Based on Lyapunov-Krasovskii functional method and the integral equality, some delay-range-dependent exponential stability criteria are presented in terms of LMIs. The decay rate can be any finite positive value without any other constraints, which is more general than the conventional assumptions in some relative lectures.(4)The stochastic stability problem for a class of Markovian jumping continuous time BAM neural networks with mode-dependent time delays and a class of Markovian jumping discrete time BAM neural networks with time delays are studied. Stochastic perturbation in two forms and generalized excitation function which is more general then Lipschitz conditions are considered in the former. Based on Lyapunov method and stochastic analysis techniques, mode-dependent criteria that ensure the mean-square stability of the system are deduced. The criteria are established in terms of LMIs. In view of the discrete model, we utilize the Lyapunov stability theorem and two finite sum equalities and obtain some stability conditions in terms of LMIs. Simulation results show the effectiveness of the proposed stability conditions.