Fractional-order Memristor-based Neural Networks Research Articles

Stabilization analysis of incommensurate fractional-order memristor-based neural networks via delay-dependent distributed controller

This paper studies the stabilization analysis problem of a class of incommensurate fractional-order memristor-based neural networks with multiple time-varying delays (IFOMNNs-MTDs). In previously published studies of IFOMNNs-MTDs, the controller is usually designed as a general delay-independent feedback controller. Due to the simple form of the feedback controller, less system information is used and less adjustable parameters are considered, which limits the flexibility and control effect of controller. Specially, the use of delay information is insufficient in the existing results, which will make the controller insensitive to the influence of delay factors and affect the control performance. Thereby, the accurate analysis of the dynamic characteristics of IFOMNNs-MTDs by designing appropriate controller needs further study. This paper aims to propose a new delay-dependent controller to improve the stabilization analysis of IFOMNNs-MTDs. Firstly, a delay-dependent distributed controller with distributed control gain and time-delay state summation term is proposed, which accords with the transmission law of activation function weights and is helpful to consider the historical information (i.e., time delay factors) of system. Thereby, the flexibility and control effect of controller are improved by adding control parameters and improving the use of system information. Secondly, a new less-conservatism stabilization criterion of IFOMNNs-MTDs is established by using the designed controller. Moreover, the controller gain can be searched in a large range by using the established criterion. Finally, a numerical simulation is provided to verify the validity of the obtained results.

Function matrix projection synchronization for the multi-time delayed fractional order memristor-based neural networks with parameter uncertainty

Fractional Order Memristor-Based Neural Networks (FOMBNNs) has the strong sensitivity to initial values and shows more complex paths, so its Projective Synchronization (PS) and applications have been widely used in the fields of security communication. Our main work intends to extend the scaling factor of PS to a function matrix depending on time t and proposes a new synchronization type for the first time, i.e., Function Matrix Projective Synchronization (FMPS) for FOMBNNs, whose scaling factor is highly variable over time and difficult to predict. However, the FOMBNNs is a state dependent discontinuous system and it is easy to produce complex nonlinearity, which makes the study of the FMPS becomes a challenge. Therefore, our work will commit to solving this problem and realizing the FMPS for multi-time delayed FOMBNNs with parameter uncertainty. Firstly, the error functions and FMPS are defined, which can be degenerated to matrix PS, modified PS, PS, anti-synchronization and complete synchronization. Then, for the multi-time delayed FOMBNNs with parameter uncertainty, the active controller is designed and the sufficient condition for realizing the FMPS is proved by using a Lyapunov functional and some Lemmas of fractional calculus. Finally, the FMPS of four numerical examples are given and trajectories of their synchronization errors approach to 0, which illustrate the efficiency of the proposed synchronization analysis. This research will provide a general method for studying the FMPS of other dynamical systems.

Asymptotic Stabilization Control of Fractional-Order Memristor-Based Neural Networks System via Combining Vector Lyapunov Function With M-Matrix

This article examines a new measure of combining the vector Lyapunov function with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula> -matrix for settling the asymptotic stabilization control of fractional-order memristor-based neural networks system (FOMBNNS) has large delays in various dimensional forms. Some new stability and stabilization criteria are deduced. First, the vector Lyapunov function and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula> -matrix are imported for investigating stabilization control for the above system. Then, we solve the problem for a special type of situation that the activation functions no longer consider Lipschitz parameters via the new method. Finally, four numerical examples from different kinds of situations are simulated for expounding the validity of the novel asymptotic stability and stabilization criteria. Compared with the methods mentioned in the current references, the proposed asymptotic stability and stabilization criteria in this article have strong generality and universality. They can be applied not only to the most common feedback control, accordingly, the feedback control law based on which they are designed but also to all fractional-order parameters from 0 to 1. In addition, the new method has lower conservativeness and fewer constraints. Moreover, the new stability and stabilization criteria can also overcome the difficulty in dealing with the above system owning large delays.

Stability of Memristor-based Fractional-order Neural Networks with Mixed Time-delay and Impulsive

Stability of Memristor-based Fractional-order Neural Networks with Mixed Time-delay and Impulsive

Asymptotic and Finite-Time Synchronization of Fractional-Order Memristor-Based Inertial Neural Networks with Time-Varying Delay

This paper emphasized on studying the asymptotic synchronization and finite synchronization of fractional-order memristor-based inertial neural networks with time-varying latency. The fractional-order memristor-based inertial neural network model is offered as a more general and flexible alternative to the integer-order inertial neural network. By utilizing the properties of fractional calculus, two lemmas on asymptotic stability and finite-time stability are provided. Based on the two lemmas and the constructed Lyapunov functionals, some updated and valid criteria have been developed to achieve asymptotic and finite-time synchronization of the addressed systems. Finally, the effectiveness of the proposed method is demonstrated by a number of examples and simulations.

Bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks under signed digraph via adaptive strategy

In this work, an adaptive controller, formulated as linear feedback controls plus nonlinear parts, is synthesized to achieve bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks (FMNNs), in which follower FMNNs are linearly coupled under a signed digraph. The salient features of this research lie in two aspects: (1) the assumption on time-varying delays is very weak, since it neither requires boundedness of delays nor restricts the differentiation of time delay functions; (2) the adaptive controller contains no delay term, and it is feasible for both bounded and unbounded activation functions. As the preparatory work for stability analysis of the controlled synchronization error system, the ready-made results on delayed incommensurate fractional-order linear positive systems, especially stability condition and comparison principle, are perfected by relaxing premises of time-varying delays. Besides, a group of differential inclusion inequalities are established as a powerful aid in scaling the bipartite synchronization error system. More relevantly, an algebraic synchronization criterion, formulated in terms of coupling strength, inner coupling matrix, Laplacian matrix and FMNN system parameters, is proved with the benefit of vector Lyapunov function and positive system theory. With the controller utilized, linear feedback controls can be exerted merely on partial neurons of some selected FMNNs, which is exemplified by numerical simulations.

Lag quasi-synchronization of incommensurate fractional-order memristor-based neural networks with nonidentical characteristics via quantized control: A vector fractional Halanay inequality approach

This work realizes lag quasi-synchronization of incommensurate fractional-order memristor-based neural networks (FMNNs) with nonidentical characteristics via quantized control. The motivations behind this research work are threefold: (1) quantized controllers, which generate discrete control signals, can be more easily realized in computers than non-quantized controllers, and can consume smaller communication capacity; (2) incommensurate orders in a single FMNN and nonidentical characteristics in drive-response FMNNs are inescapable due to the differences among the circuit elements used to implement FMNNs; (3) convergence analysis of delayed incommensurate fractional-order nonlinear systems, which is the basis for the derivation of synchronization criterion, has not been handled perfectly. As an effective tool for convergence analysis of delayed incommensurate fractional-order nonlinear systems, especially for estimation of ultimate state bound, a vector fractional Halanay inequality is established at first. Then, a quantized synchronization controller, in which the dead-zone is introduced into some logarithmic quantizers to avoid chattering phenomenon, is designed. By means of vector Lyapunov function together with the newly derived vector fractional Halanay inequality, the synchronization criterion is proved theoretically. Lastly, numerical simulations supplementarily illustrate the correctness of the synchronization criterion. In contrast with the hypotheses in the relevant literature, the hypotheses in this paper are weaker.

Improved Results on Finite-Time Passivity and Synchronization Problem for Fractional-Order Memristor-Based Competitive Neural Networks: Interval Matrix Approach

This research paper deals with the passivity and synchronization problem of fractional-order memristor-based competitive neural networks (FOMBCNNs) for the first time. Since the FOMBCNNs’ parameters are state-dependent, FOMBCNNs may exhibit unexpected parameter mismatch when different initial conditions are chosen. Therefore, the conventional robust control scheme cannot guarantee the synchronization of FOMBCNNs. Under the framework of the Filippov solution, the drive and response FOMBCNNs are first transformed into systems with interval parameters. Then, the new sufficient criteria are obtained by linear matrix inequalities (LMIs) to ensure the passivity in finite-time criteria for FOMBCNNs with mismatched switching jumps. Further, a feedback control law is designed to ensure the finite-time synchronization of FOMBCNNs. Finally, three numerical cases are given to illustrate the usefulness of our passivity and synchronization results.

Pinning multisynchronization of delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations

This paper concerns the multisynchronization issue for delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations. First, the coexistence of multiple equilibrium states for isolated subnetwork is analyzed. By means of state-space decomposition, fractional-order Halanay inequality and Caputo derivative properties, the novel algebraic sufficient conditions are derived to ensure that the addressed networks with arbitrary activation functions have multiple locally stable almost periodic orbits or equilibrium points. Then, based on the obtained multistability results, a pinning control strategy is designed to realize the multisynchronization of the N coupled networks. By the aid of graph theory, depth first search method and pinning control law, some sufficient conditions are formulated such that the considered neural networks can possess multiple synchronization manifolds. Finally, the multistability and multisynchronization performance of the considered neural networks with different activation functions are illustrated by numerical examples.

Global stabilization of fractional-order memristor-based neural networks with incommensurate orders and multiple time-varying delays: a positive-system-based approach

This paper addresses global stabilization of fractional-order memristor-based neural networks (FMNNs) with incommensurate orders and multiple time-varying delays (MTDs), where the time delay functions are not necessarily bounded. First, without assuming that time delay functions are bounded, the asymptotical stability condition is given for fractional-order linear positive system with incommensurate orders and MTDs. Then, comparison principle for such a system is established. By virtue of two kinds of vector Lyapunov functions (absolute-value-function-based and square-function-based vector Lyapunov functions), stability condition of fractional-order linear positive system and comparison principle, two stabilization criteria are derived and the equivalence between them is illustrated. In comparison with the reported criterion, the criteria derived in this paper are less conservative, since they allow controller parameters to satisfy weaker algebraic conditions. Lastly, numerical examples are displayed to validate the availability of the controller and correctness of the stabilization criteria.

Pinning synchronization of fractional-order memristor-based neural networks with multiple time-varying delays via static or dynamic coupling

This paper investigates global asymptotical synchronization between fractional-order memristor-based neural networks (FMNNs) with multiple time-varying delays (MTDs) by pinning control. Two classes of coupling manners, static manner and dynamic manner, are introduced into the pinning controller respectively. For the case of static coupling, to make the controller exclude fraction, 1-norm Lyapunov function and fractional Halanay inequality in MTDs case are utilized for synthesis of controller and convergence analysis of synchronization error. For the case of dynamic coupling, a fractional differential inequality is proved and discussed in an elaborate way, and then global asymptotical synchronization is analyzed by means of Lyapunov-like function and the newly-proved inequality. Lastly, numerical simulations are carried out to show the practicability of the pinning controllers and the feasibility of the obtained synchronization criteria.

New criteria for finite-time stability of fractional order memristor-based neural networks with time delays

In this paper, the finite-time stability (FTS) problems for a class of fractional order memristor-based neural network (FOMNN) with time delays are investigated. Based on the method of steps, set-valued mapping, the theory of differential inclusion and fractional order Gronwall inequality, a new delay dependent criterion for the FTS of FOMNN with time delays and the fractional order 0<μ<1 is derived, which is less conservative than the existing criteria. Furthermore, a sufficient condition to ensure the FTS of FOMNN with the fractional-order 1<μ<2 is proposed. Finally, two examples are given to illustrate the validity of the proposed results.

New Results on Synchronization of Fractional-Order Memristor‐Based Neural Networks via State Feedback Control

This paper addresses the synchronization issue for the drive-response fractional-order memristor‐based neural networks (FOMNNs) via state feedback control. To achieve the synchronization for considered drive-response FOMNNs, two feedback controllers are introduced. Then, by adopting nonsmooth analysis, fractional Lyapunov’s direct method, Young inequality, and fractional-order differential inclusions, several algebraic sufficient criteria are obtained for guaranteeing the synchronization of the drive-response FOMNNs. Lastly, for illustrating the effectiveness of the obtained theoretical results, an example is given.

New criterion for finite-time synchronization of fractional order memristor-based neural networks with time delay

A new fractional order Gronwall inequality with time delay is developed in this paper. Based on this inequality, a new criterion for finite-time synchronization of fractional order memristor-based neural networks (FMNNs) with time delay is derived. In addition, two numerical examples are exhibited to illustrate the effectiveness of the obtained results.

Synchronization for commensurate Riemann-Liouville fractional-order memristor-based neural networks with unknown parameters

This paper deals with the synchronization for commensurate Riemann-Liouville fractional-order memristor-based neural networks with unknown parameters through Lyapunov direct method. Two inequalities on Riemann-Liouville fractional derivative of quadratic function are given, which play a critical role in the theoretical analysis of Riemann-Liouville fractional differential system. Combining the adaptive controller with parameter update laws, synchronization and parameter identification of fractional-order memristor-based neural networks with unknown parameters can be achieved simultaneously. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results.

LMI-based criterion for global Mittag-Leffler lag quasi-synchronization of fractional-order memristor-based neural networks via linear feedback pinning control

This paper addresses global Mittag-Leffler lag quasi-synchronization (GMLQS) of fractional-order memristor-based neural networks (FMNNs). Firstly, for a general class of fractional-order nonlinear systems with interval uncertainties, a kind of linear feedback pinning controller, which works by feeding back partial state errors to the partial controlled variables, is designed to achieve global Mittag-Leffler ultimate boundedness (GMUB). Correspondingly, a GMUB criterion in the form of linear matrix inequality (LMI) is derived with the aid of a Lyapunov function and a newly established fractional-order differential inequality. Then, the linear feedback pinning controller is employed for GMUB of the synchronization error system, which is equivalent to GMLQS of FMNNs. Since there exists the propagation delay between drive-response FMNNs and the synchronization error system can not be obtained straightforwardly, an identical equation about Caputo’s fractional derivatives is established to overcome this difficulty. Moreover, the detailed pinning control scheme design procedures for a prescribed GMLQS performance are presented, where the performance involves both ultimate error bound and transient behavior. In the control scheme, an auxiliary LMI and an optimization objective function are introduced, which can significantly reduce control cost. Finally, a numerical example is presented to show the feasibility of the control scheme. The results obtained in this paper have improved the existing criterion on quasi-synchronization of FMNNs, and will also provide a novel insight into pinning control of fractional-order nonlinear systems.

Finite-time synchronization for fractional-order memristor-based neural networks with discontinuous activations and multiple delays

This paper addresses the finite-time synchronization problem for fractional-order memristor-based neural networks (FMNNs) with discontinuous activations, in which multiple delays are considered. Firstly, on the basis of set-valued mapping as well as differential inclusions theory, the synchronization issue of drive-response systems is considered as the stabilization of the error system. Then, the state feedback controllers, which contain both discontinuous part and time-delayed part, are designed to analyze the finite-time synchronization of the concerned network model. Making use of the stability theorem of fractional-order systems with multiple time delays, some fractional derivative inequalities and comparison theorem, several sufficient criteria are established for confirming that the synchronization error of the concerned system can reach zero within a limited time. Additionally, the settling time can be optimized by adjusting controller parameter. Finally, the effectiveness of synchronization strategies is validated through the simulation results.

Fixed-Time Synchronization of Delayed Fractional-Order Memristor-Based Fuzzy Cellular Neural Networks

In this article, we discussed the fixed-time synchronization of fractional-order memristor-based fuzzy cellular neural networks (FMFCNNs) with time-varying delays. Based on the differential inclusion theory and discontinuous state feedback control technique, some useful novel criteria of the fixed-time synchronization for FMFCNNs are derived by constructing appropriate Lyapunov function. The results in our paper improve and generalize current literature research results significantly. Finally, the effectiveness of our control schemes for the fixed-time synchronization is demonstrated through numerical simulations.

Global Stabilization of Fractional-Order Memristor-Based Neural Networks With Time Delay.

This paper addresses the global stabilization of fractional-order memristor-based neural networks (FMNNs) with time delay. The voltage threshold type memristor model is considered, and the FMNNs are represented by fractional-order differential equations with discontinuous right-hand sides. Then, the problem is addressed based on fractional-order differential inclusions and set-valued maps, together with the aid of Lyapunov functions and the comparison principle. Two types of control laws (delayed state feedback control and coupling state feedback control) are designed. Accordingly, two types of stabilization criteria [algebraic form and linear matrix inequality (LMI) form] are established. There are two groups of adjustable parameters included in the delayed state feedback control, which can be selected flexibly to achieve the desired global asymptotic stabilization or global Mittag-Leffler stabilization. Since the existing LMI-based stability analysis techniques for fractional-order systems are not applicable to delayed fractional-order nonlinear systems, a fractional-order differential inequality is established to overcome this difficulty. Based on the coupling state feedback control, some LMI stabilization criteria are developed for the first time with the help of the newly established fractional-order differential inequality. The obtained LMI results provide new insights into the research of delayed fractional-order nonlinear systems. Finally, three numerical examples are presented to illustrate the effectiveness of the proposed theoretical results.

Uniformly stable and attractive of fractional-order memristor-based neural networks with multiple delays

Memristive neural networks (MNN) have been wildly studied. Nevertheless, fractional -order memristive neural networks (FMNN) still remain a wide-open dilemma. This paper addresses the problem of FMNN systems with multiple delays and grew several related theories through the study. First, the existence of the system equilibrium point is investigated based on contraction mapping principle, and a new sufficient criterion is obtained. Second, the delay-free uniform stability of the system is discussed by employing differential inclusion theory. Third, a novel asymptotic stability criterion is proposed which is less conservative. Finally, one descriptive numerical example and simulation results emphasize the accuracy and reliability of the proposed results.