Stability Analysis And Synthesis Of Neural Dynamical Systems

Neural dynamical systems are the important elements in the evolution of ar-tificial intelligence and also are the useful tools of in the research of intelligent automation.They have strong applications in various fields,including image pro-cessing,pattern recognition,memory storage,and optimization problems,to name a few.This dissertation mainly lucubrates the stability analysis and synthesis of neu-ral dynamical systems.From the view of research objects,this dissertation focuses on the recurrent neural networks(RNNs)and memristive neural networks(MNNs).MNNs,which are formed by replacing the resistors in RNNs with memristors,can be viewed as the improved version of RNNs.In many applications,the MNNs al-so display many advantages in comparison with RNNs.From the view of research topics,this dissertation mainly considers the stability analysis,stabilization,and synchronization control of neural dynamical systems.Stability is the foundation of dynamic analysis and synthesis of neural dynamical systems.Stabilization is closely related to the stability issue.For some choices of neural model parameters,if the neural networks exhibit some unstable behaviors such as oscillation or chaos,one usually has to devise a suitable controller to stabilize the systems under investiga-tion.In applications,stabilization is also usually utilized to achieve a desired goal such as better system performance or fast convergence rate.In addition,the chaotic synchronization of neural dynamical systems is applied widely in secure communi-cation and image encryption.In essence,the synchronization issue is equivalent to the stabilization of related error systems.This dissertation contains two parts in broad outline.The former is main-ly concerned with the stability of delayed neural dynamical systems by proposing the flexible terminal method and multiple integral method,respectively.The latter mainly focuses on the control synthesis of neural dynamical systems by devising sev-eral state feedback controllers,such as the sampled-data controller with time-varying switching gains and the region-partitioning-dependent intermittent controller.The main contributions of this dissertation are summarized as follows:(1)The flexible terminal method(FTM)is proposed to investigate the stability of RNNs with time-varying delays.Together with the convex combination technique,this method changes the delay interval with fixed terminals into the one with flexible terminals.Under the framework of FTM,a novel class of Lyapunov-Krasovskii functionals(LKFs)are constructed originally,in which the integral intervals associated with delayed variables are flexible.Guided by FTM,the Wirtinger-based integral inequality and free-weight matrix method are employed to develop several stability criteria of RNNs with time-varying delays,respectively.Both the theoretical and experiment analysis show that the FTM can reduce the conservatism of stability criteria effectively but not bring additional decision variables.(2)The hierarchial stability criteria are developed for delayed systems by multiple integral method.The theoretical foundation is originally provided to reveal the inverse relationship between the multiplicity of integral terms in LKFs and the conservatism of stability criteria.Thus,this foundation is the first evidence of the feasibility of multiple integral method.In addition,based on the well-known Wirtinger-based inequality,the Wirtinger-based multiple integral inequality is developed originally,which is utilized to obtain a more accurate upper bound of the derivative of LKFs with multiple integral terms.Subsequently,the delay-dependent stability criteria are derived for RNNs with time-varying delays.(3)A switching approach is proposed insightfully to explore the event-triggered control of RNNs with saturating actuators.The event-triggered mechanism is essentially a switching between the aperiodic sampling and continuous event trigger.Correspondingly,a controller with time-varying switching gains is designed,whose gains are composed of an exponentially decaying term and two gain matrices.The two gain matrices are required to be switched when the switching between the aperiodic sampling and continuous event trigger is met.By constructing a switching LKF,several sufficient conditions that ensure the local exponential stability of closed-loop systems are formulated in terms of linear matrix inequalities(LMIs).Both the exponentially decaying term and switching gains improve the feasible region of LMIs,and then they are helpful to enlarge the set of admissible initial conditions,the threshold in event-triggered mechanism,and the average waiting time.(4)A robust analysis method is originally proposed to study the stabilization of MNNs with saturating actuators.By defining a class of logical switched functions,the MNNs are converted into a tractable model,under which the connection weights of MNNs are dealt with by a robust analysis method,This way considers both the conservativeness and complexity of the stabilization criteria.Meanwhile,a saturating sampled-data controller equipped with an exponentially decaying term is designed.With the help of generalized sec-tor condition,some LMI-based conditions are formulated to ensure the local exponential stability of closed-loop systems.In addition,three optimization problems are given to design the control gain with the aims of enlarging the sampling interval,expanding the set of admissible initial conditions,and min-imizing the size of actuators,respectively.(5)The quasi-synchronization control of MNNs is investigated by taking into ac-count the switching jumps mismatch.By considering the state-dependent switch characteristic of memristor-based connection weights,a way how to use the information of switching jumps is provided.The lag quasi-synchronization of MNNs is achieved by the periodically intermittent control.The theoretical analysis shows that the error level is closely related to the switching jumps.Finally,a simple design procedure of controller is correspondingly presented to ensure that the synchronization error between the master system and the slave system converges to a predetermined level.(6)The quasi-synchronization control of delayed MNNs is explored originally by-proposing a new class of intermittent control.This method is described by three partitions of nonnegative real region and an auxiliary positive definite function.Whether the control input is imposed on the slave system or not is decided by the dynamical relationships among the three subregions and auxiliary function.From these ingredients,several succinct criteria with the associated co-design procedure are presented such that the synchronization error converges to a predetermined level.The proposed intermittent scheme is also applied to the event-triggered control,and an intermittent event-triggered mechanism is devised to investigate the quasi-synchronization of MNNs cor-respondingly.Such mechanism eliminates the events in rest time,and then it reduces the amount of samplings.