Stability Analysis And Networked Synchronization Control Of Recurrent Neural Networks With Time Delays

Neural networks(NNs)are a kind of nonlinear dynamic systems.They can exhibit a great of complex dynamic behaviors when the parameters are properly selected.Since a lot of successful applications have been witnessed in many fields, such as signal processing,pattern recognition,dynamic image processing,secure communication,global optimization,and so on,the study of dynamic behaviors of NNs have attracted considerable attention in recent years.NNs are adaptive and self-organizing systems and a lot of their engineering applications are strongly relative to their stability.On the other hand,time delays inevitably exist in NNs and are frequently important sources of oscillation or even chaos.Consequently,the studies of stability,including asymptotical stability,exponential stability,absolute stability,etc.,of NNs with time delays are of profound theoretical and practical significance.Also,a lot of research results on them have emerged in recent years. Meanwhile,many efforts have also been performed on the synchronization control of delayed NNs,especially on the synchronization of chaotic NNs.This thesis contains two parts in broad outline.In the former,it mainly focuses on the studies of the stability problems for recurrent neural networks(RNNs)with various time delays, including constant time delays,interval time delays and time delays with known probability distribution.By utilizing many useful tools,such as Lyapunov stability theory,S-procedure,linear matrix inequalities(LMIs)and so on,the corresponding stability criteria are achieved.In the latter,the networked synchronization control problems for RNNs with time delays are considered.As far as the research subjects are concerned,we develop two new models in this thesis.(1)Establish a probability-distribution-dependent model for a class of delayed RNNs and provide corresponding stability criteria.In the existing references, the time delays in NNs are deterministic and the achieved stability criteria can be divided into two categories:delay-independent and delay-dependent.Generally speaking,the delay-dependent criteria can lead to less conservative results than these by the delay-independent criteria,especially when the time delays are small enough.However,these stability criteria were derived based only on the information of variation range of the time delays.When it comes to the case that some values of the time delays are very large but the probabilities of the delays taking such large values are very small,the existing methods may lead to more conservative results. Actually,the time delays in some NNs are often existent in a stochastic fashion and their probabilistic characteristics can often be measured by statistical methods. Considering these probability distributions,a new modeling method is introduced by translating the distribution probabilities of the time delays into parameter matrices of the transferred systems.Consequently,by combining the Lyapunov method and LMI technique,some delay-distribution-dependent stability criteria of the delayed RNNs are achieved.(2)Introduce a new model for networked synchronization control of delayed RNNs and provide the corresponding control design method.In real applications of many synchronization control systems,the signal transmissions sometimes have to relay on the common communication networks.Two or more NNs which are either chaotic or periodic sharing a common dynamic behavior by coupling or external forcing and transmitting signals mutually through common communication networks fall into the category of networked synchronization control of NNs. Since the communication will suffer from network-induced time delays,high rate of frame losses,high rate of bit errors,environment disturbance,and so on,the existing synchronization control schemes can not work well in the networked synchronization control systems.On this issue,a new mathematic model of networked synchronization control is introduced,in which the network performances are indexed and the stochastic fluctuations are described in term of a Brownian motion.By using Lyapunov method and some well known inequalities,the exponential mean-square networked synchronization control problems of delayed RNNs are concerned about and the corresponding control design approaches are provided.As far as the analysis methods are concerned,this thesis contains the following: (1)Develop some LMI-based lemmas.These lemmas are important complements of the existing rule sets of LMI theory.Also,they play a great role in improving the existing stability criteria of delayed RNNs.(2)Introduce a new kind of delay-centralpoint method which is based on the piecewise analysis method.By introducing the center point of the time delay variation interval,developing appropriative Lyapunov functions and employing the piecewise analysis method,some sufficient conditions are achieved which ensure the stability of a class of RNNs with interval time delays.(3) By developing a new Lyapunov-Krasovskii functional and employing the improved LMI technique,the robust global exponential stability problem of a class of stochastic RNNs is investigated.More specifically,the main contents of this thesis are as follows:(1)Robust global asymptotical stability of RNNs with interval time delaysFirstly,a special case is considered,where the lower bound of the time delay interval is assumed to be zero.By constructing an appropriative Lyapunov-Krasovskii functional,this thesis studies the robust global asymptotical stability problem of a class of RNNs with slow time-varying delays and fast time-varying delays,respectively. It is well known that the performance of a gained criterion depends not only on the construction of Lyapunov functional,but also on the approach to estimate the upper bound of derivative of the Lyapunov functional.However,in the existing methods,some convexities in certain matrix equation are often ignored when estimating the upper bound of derivative of the Lyapunov functional.In this thesis, the convexity in the matrix inequality is picked up,and a new lemma is developed which is a supplement of the existing LMIs rule sets.Furthermore,this thesis improves the existing stability analysis approach of combining the Lyapunov method and free-weighting matrix technique.On the other hand,the variation intervals of time delay are often assumed to be started from zero in the existing references.Actually,the lower bounds of time delay intervals are not zero in many important systems.This thesis analyzes the global asymptotical stability problem of continuous-time RNNs and discretetime RNNs with interval time delays,respectively.The delay-central-point method which is latest reported in the control system is reconsidered and a delay-centralpoint approach which is based on the piecewise analysis method is introduced to investigate the stability of RNNs with interval time delays.The existing robust global asymptotical stability criteria for continuous-time and for discrete-time RNNs, respectively,are both improved in this thesis.(2)Robust global exponential stability of stochastic RNNs with time-varying delaysIn real systems,the connection weights of the neurons depend on certain resistance and capacitance values which include uncertainties.On the other hand, the information storage and neurotransmission frequently suffer from the stochastic fluctuations.Therefore,when designing a neural network,both the parameter uncertainties and the stochastic fluctuations should be involved.By constructing a novel Lyapunov-Krasovskii functional and reconsidering the existing free-weighting matrix technique,some sufficient conditions of robust global exponential stability for a class of stochastic RNNs with norm-bounded parameter uncertainties are achieved,which can provide less conservatism than currently available stability criteria.(3)Delay-distribution-dependent stability criteria for a class of RNNsThe existing stability criteria are often dependent on the variation range of time delays in term of the deterministic time delays.The stability problems for a class of RNNs in which the probability distribution of time delays can be obtained by statistical methods are studied.A new modeling method is firstly introduced. By utilizing a stochastic variable which satisfies Bernoulli random binary distribution, the probability distribution of the time delay is translated into the transformed NNs'parameter matrices.In terms of different kinds of RNNs,appropriative Lyapunov-Krasovskii functions are developed and distinguished analysis techniques are performed.Some delay-distribution-dependent stability criteria are provided for discrete-time RNNs,discrete-time stochastic RNNs with norm-bounded parameter uncertainties and continuous-time RNNs,respectively.Also,some comparisons are performed with the existing results.In addition,this modeling method can describe the time delays more exactly and can be extended to many other dynamic systems and control systems.(4)Networked synchronization control of RNNs with time delaysThe synchronization control methods of a class of RNNs with time delays are studied,in which the reference signals are transmitted through common communication networks.The networked synchronization models are firstly given in this thesis and the communication networks'Quality of Services(QoS)are indexed.The elementary principles of networked synchronization Control of stochastic RNNs are introduced.By utilizing the Lyapunov method and exploiting LMIs technique,the problem of mean-square exponential synchronization control of stochastic delayed RNNs is concerned about and the gained results are verified on the TrueTime platform. Then,according to the probability distribution of the indexed QoS of networks, the mathematic model of the network is improved and some new results based on the probability distribution of the communication networks'QoS are achieved.