Mixed Time Delays Research Articles

Robust stability criteria for discrete-time switched neural networks with various activation functions

This paper is concerned with the problem of robust stabilization and H∞ control for a class of discrete-time switched neural networks with various activation functions and mixed time delays. By using the average dwell time approach together with the multiple Lyapunov function technique, we propose a state feedback controller to guarantee that the switched neural networks is robustly exponentially stable with the disturbance attenuation level γ>0 about its equilibrium point for all admissible uncertainties. The obtained results are formulated in terms of linear matrix inequalities (LMIs) which can be easily verified by the MATLAB LMI control toolbox. Finally, numerical examples with simulation result are given to illustrate the effectiveness and applicability of the obtained results.

Stability analysis for a class of neutral-type neural networks with Markovian jumping parameters and mode-dependent mixed delays

This paper is concerned with the stability problem for a class of Markovian jumping neutral-type neural networks with mode-dependent mixed time-delays. The mixed time-delays are composed of discrete and distributed delays, both of which are mode-dependent. In addition, the distributed time-delays are characterized by the upper and lower bounds, both of which are mode-dependent. By constructing new Lyapunov–Krasovskii functionals, a unified framework is established to derive sufficient conditions for the concerned systems to be globally exponentially stable in mean square. A simulation example is provided to demonstrate the usefulness of the main results obtained.

Adaptive synchronization of stochastic neural networks with mixed time delays and reaction–diffusion terms

In this paper, the problem of exponential synchronization is investigated for a class of stochastic perturbed chaotic neural networks with both mixed time delays and reaction–diffusion terms. By employing Lyapunov–Krasovskii functional and stochastic analysis approaches, an adaptive controller is designed to guarantee the exponential synchronization of proposed neural networks in the mean square. In particular, the mixed time delays in this paper synchronously consist of constant delay in the leakage term (i.e., “leakage delay”), discrete time-varying delay and distributed time-varying delay which are more general than those discussed in the previous literature. Furthermore, our synchronization criteria are easily verified and do not need to solve any linear matrix inequality. Therefore, the results obtained in this paper generalize and improve those given in the previous literature. Finally, the extensive simulations are performed to show the effectiveness and feasibility of the obtained method.

Transformation from a Pure Time Delay to a Mixed Time and Phase Delay Representation in the Auditory Forebrain Pathway

Birds and mammals exploit interaural time differences (ITDs) for sound localization. Subsequent to ITD detection by brainstem neurons, ITD processing continues in parallel midbrain and forebrain pathways. In the barn owl, both ITD detection and processing in the midbrain are specialized to extract ITDs independent of frequency, which amounts to a pure time delay representation. Recent results have elucidated different mechanisms of ITD detection in mammals, which lead to a representation of small ITDs in high-frequency channels and large ITDs in low-frequency channels, resembling a phase delay representation. However, the detection mechanism does not prevent a change in ITD representation at higher processing stages. Here we analyze ITD tuning across frequency channels with pure tone and noise stimuli in neurons of the barn owl's auditory arcopallium, a nucleus at the endpoint of the forebrain pathway. To extend the analysis of ITD representation across frequency bands to a large neural population, we employed Fourier analysis for the spectral decomposition of ITD curves recorded with noise stimuli. This method was validated using physiological as well as model data. We found that low frequencies convey sensitivity to large ITDs, whereas high frequencies convey sensitivity to small ITDs. Moreover, different linear phase frequency regimes in the high-frequency and low-frequency ranges suggested an independent convergence of inputs from these frequency channels. Our results are consistent with ITD being remodeled toward a phase delay representation along the forebrain pathway. This indicates that sensory representations may undergo substantial reorganization, presumably in relation to specific behavioral output.

Design of Delay-Dependent Exponential Estimator for T–S Fuzzy Neural Networks with Mixed Time-Varying Interval Delays Using Hybrid Taguchi-Genetic Algorithm

This paper considers the design of state estimator for Takagi---Sugeno (T---S) fuzzy neural networks with mixed time-varying interval delays. The mixed time-delays that consist of both the discrete time-varying and distributed time-delays with a given range are presented. The activation functions are assumed to be globally Lipschitz continuous. By using the Lyapunov-Krasovskii method, a linear matrix inequality (LMI) approach is developed to construct sufficient conditions for the existence of admissible state estimator such that the error-state system is exponentially globally stable. To avoid complex mathematical derivations and conservative results, a new hybrid Taguchi-genetic algorithm method is integrated with a LMI method to seek the estimator gains that satisfy the Lyapunov-Krasovskii functional stability inequalities. The proposed new approach is straightforward and well adapted to the computer implementation. Therefore, the computational complexity can be reduced remarkably and facilitate the design task of the estimator for T---S fuzzy neural networks with time-varying interval delays. Two illustrative examples are exploited in order to illustrate the effectiveness of the proposed state estimator.

Linear matrix inequality approach for synchronization control of fuzzy cellular neural networks with mixed time delays

Fuzzy cellular neural networks (FCNNs) are special kinds of cellular neural networks (CNNs). Each cell in an FCNN contains fuzzy operating abilities. The entire network is governed by cellular computing laws. The design of FCNNs is based on fuzzy local rules. In this paper, a linear matrix inequality (LMI) approach for synchronization control of FCNNs with mixed delays is investigated. Mixed delays include discrete time-varying delays and unbounded distributed delays. A dynamic control scheme is proposed to achieve the synchronization between a drive network and a response network. By constructing the Lyapunov—Krasovskii functional which contains a triple-integral term and the free-weighting matrices method an improved delay-dependent stability criterion is derived in terms of LMIs. The controller can be easily obtained by solving the derived LMIs. A numerical example and its simulations are presented to illustrate the effectiveness of the proposed method.

Global exponential stability of impulsive fuzzy Cohen–Grossberg neural networks with mixed delays and reaction–diffusion terms

This paper is concerned with the problem of exponential stability for a class of impulsive fuzzy Cohen–Grossberg neural networks with mixed time delays and reaction–diffusion. The mixed delays include time-varying delays and continuously distributed delays. Based on the Lyapunov method, Poincaré Integral Inequality, and the linear matrix inequality (LMI) approach, we found some new sufficient conditions ensuring the global exponential stability of equilibrium point for impulsive fuzzy Cohen–Grossberg neural networks with mixed time delays and reaction–diffusion terms. These global exponential stability conditions depend on the reaction–diffusion terms and time delays. The results presented in this paper are less conservative than the existing sufficient stability conditions. Finally, some examples are given to show the effectiveness and superiority of the theoretical results.

State Estimation for Discrete-Time Neural Networks with Markov-Mode-Dependent Lower and Upper Bounds on the Distributed Delays

This paper is concerned with the state estimation problem for a new class of discrete-time neural networks with Markovian jumping parameters and mixed time-delays. The parameters of the neural networks under consideration switch over time subject to a Markov chain. The networks involve both the discrete-time-varying delay and the mode-dependent distributed time-delay characterized by the upper and lower boundaries dependent on the Markov chain. By constructing novel Lyapunov-Krasovskii functionals, sufficient conditions are firstly established to guarantee the exponential stability in mean square for the addressed discrete-time neural networks with Markovian jumping parameters and mixed time-delays. Then, the state estimation problem is coped with for the same neural network where the goal is to design a desired state estimator such that the estimation error approaches zero exponentially in mean square. The derived conditions for both the stability and the existence of desired estimators are expressed in the form of matrix inequalities that can be solved by the semi-definite programme method. A numerical simulation example is exploited to demonstrate the usefulness of the main results obtained.

Stochastic stability analysis of piecewise homogeneous Markovian jump neural networks with mixed time-delays

In this paper, the problem of stochastic stability analysis is considered for piecewise homogeneous Markovian jump neural networks with both discrete and distributed delays by use of linear matrix inequality (LMI) method. Based on a Lyapunov functional that accounts for the mixed time-delays, a delay-dependent stability condition is given, which is formulated by LMIs and thus can be easily checked. Some special cases are also investigated. Finally, a numerical example is given to show the validness of the proposed result.

Delay-dependent robust stabilization and H∞ control for neural networks with various activation functions

This paper considers the problem of robust stabilization for a class of uncertain neural networks with various activation functions and mixed time delays. The aim is to derive a H∞ control law to ensure the robust stability of the closed-loop system about its equilibrium with parameter uncertainties. By employing the Lyapunov stability theory and the matrix inequality technique, a new set of sufficient conditions is presented for the existence of the H∞ control problem. The stability criteria are derived in terms of linear matrix inequalities (LMIs) which can be solved easily by the Matlab LMI toolbox. In addition to the requirement of global robust stabilization, for a prescribed H∞ performance level the stabilizing controller gain matrices for all delays to satisfy the upper bound of the time-varying delay are required to be obtained. Numerical examples are presented to illustrate the effectiveness of the proposed method.

Passivity analysis of Markov jump neural networks with mixed time-delays and piecewise-constant transition rates

In this paper, passivity analysis is considered for Markov jump neural networks with both mixed time-delays and time-varying transition rates. The mixed time-delays consist of both discrete and distributed delays. The time-varying character of transition rates is assumed to be piecewise-constant. By use of the linear matrix inequality (LMI) method and a Lyapunov functional that accounts for the mixed time-delays, a delay-dependent passivity condition is derived, which can be easily checked. The result presented depends upon not only discrete delay but also distributed delay. A numerical example is proposed to show the effectiveness of the proposed method.

Stability analysis of Markovian jump stochastic BAM neural networks with impulse control and mixed time delays.

This paper discusses the issue of stability analysis for a class of impulsive stochastic bidirectional associative memory neural networks with both Markovian jump parameters and mixed time delays. The jumping parameters are modeled as a continuous-time discrete-state Markov chain. Based on a novel Lyapunov-Krasovskii functional, the generalized Itô's formula, mathematical induction, and stochastic analysis theory, a linear matrix inequality approach is developed to derive some novel sufficient conditions that guarantee the exponential stability in the mean square of the equilibrium point. At the same time, we also investigate the robustly exponential stability in the mean square of the corresponding system with unknown parameters. It should be mentioned that our stability results are delay-dependent, which depend on not only the upper bounds of time delays but also their lower bounds. Moreover, the derivatives of time delays are not necessarily zero or smaller than one since several free matrices are introduced in our results. Consequently, the results obtained in this paper are not only less conservative but also generalize and improve many earlier results. Finally, two numerical examples and their simulations are provided to show the effectiveness of the theoretical results.

H<sub>∞</sub> Dynamic Output Feedback Control for a Class of Discrete-Time Stochastic Systems with Sector Nonlinearities and Mixed Time-Delays

This paper deals with the dynamic output feedback control problem for a class of discrete-time stochastic systems involving sector nonlinearities and mixed time-delays. Based on the control theory and Lyapunov-Krasovskii functional method, an effective approach is proposed to design a dynamic output feedback controller such that the resultant closed-loop system is stochastically stable with a prescribed disturbance attention in the mean square sense and has a performance level. A numerical example demonstrates the effectiveness of the method.

Robust Exponential Stability of Stochastically Nonlinear Jump Systems with Mixed Time Delays

In this paper, the problem of robust exponential stability is investigated for a class of stochastically nonlinear jump systems with mixed time delays. By applying the Lyapunov–Krasovskii functional and stochastic analysis theory as well as matrix inequality technique, some novel sufficient conditions are derived to ensure the exponential stability of the trivial solution in the mean square. Time delays proposed in this paper comprise both time-varying and distributed delays. Moreover, the derivatives of time-varying delays are not necessarily less than 1. The results obtained in this paper extend and improve those given in the literature. Finally, two numerical examples and their simulations are provided to show the effectiveness of the obtained results.

Admissibility and dissipativity analysis for discrete-time singular systems with mixed time-varying delays

The problems of admissibility and dissipativity are studied for discrete-time singular systems with mixed time-varying delays. The mixed time delays consist of both discrete and distributed delays. By taking advantage of the delay partitioning technique, a criterion that warrants the admissibility of the considered systems is established. Based on the criterion, a dissipativity condition is also given. All the results presented depend upon not only discrete delay and distributed delay, but also depend upon the number of delay partitions. Some numerical examples are included to demonstrate the improvement and effectiveness of the proposed methods.

Dynamical Behaviors of Impulsive Stochastic Reaction‐Diffusion Neural Networks with Mixed Time Delays

We discuss the dynamical behaviors of impulsive stochastic reaction‐diffusion neural networks (ISRDNNs) with mixed time delays. By using a well‐known L‐operator differential inequality with mixed time delays and combining with the Lyapunov‐Krasovkii functional approach, as well as linear matrix inequality (LMI) technique, some novel sufficient conditions are derived to ensure the existence, uniqueness, and global exponential stability of the periodic solutions for ISRDNNs with mixed time delays in the mean square sense. The obtained sufficient conditions depend on the reaction‐diffusion terms. The results of this paper are new and improve some of the previously known results. The proposed model is quite general since many factors such as noise perturbations, impulsive phenomena, and mixed time delays are considered. Finally, two numerical examples are provided to verify the usefulness of the obtained results.

Stability in Switched Cohen‐Grossberg Neural Networks with Mixed Time Delays and Non‐Lipschitz Activation Functions

The stability for the switched Cohen‐Grossberg neural networks with mixed time delays and α‐inverse Hölder activation functions is investigated under the switching rule with the average dwell time property. By applying multiple Lyapunov‐Krasovskii functional approach and linear matrix inequality (LMI) technique, a delay‐dependent sufficient criterion is achieved to ensure such switched neural networks to be globally exponentially stable in terms of LMIs, and the exponential decay estimation is explicitly developed for the states too. Two illustrative examples are given to demonstrate the validity of the theoretical results.

Generalized Projective Synchronization between Two Different Neural Networks with Mixed Time Delays

The generalized projective synchronization (GPS) between two different neural networks with nonlinear coupling and mixed time delays is considered. Several kinds of nonlinear feedback controllers are designed to achieve GPS between two different such neural networks. Some results for GPS of these neural networks are proved theoretically by using the Lyapunov stability theory and the LaSalle invariance principle. Moreover, by comparison, we determine an optimal nonlinear controller from several ones and provide an adaptive update law for it. Computer simulations are provided to show the effectiveness and feasibility of the proposed methods.

Stability Analysis for Stochastic Markovian JumpReaction‐Diffusion Neural Networks with Partially Known TransitionProbabilities and Mixed Time Delays

The stability problem is proposed for a new class of stochastic Markovian jump reaction‐diffusion neural networks with partial information on transition probability and mixed time delays. The new stability conditions are established in terms of linear matrix inequalities (LMIs). To reduce the conservatism of the stability conditions, an improved Lyapunov‐Krasovskii functional and free‐connection weighting matrices are introduced. The obtained results are dependent on delays and the measure of the space AND, therefore, have less conservativeness than delay‐independent and space‐independent ones. An example is given to show the effectiveness of the obtained results.

Stability analysis for discrete-time Markovian jump neural networks with mixed time-delays

The problem of delay-dependent stability analysis is investigated for discrete-time Markovian jump neural networks with mixed time-delays (both discrete and infinity-distributed time delays). The Markov chain in the underlying neural networks is finite piecewise homogeneous. A delay-dependent condition is derived for the addressed neural networks to be globally asymptotically stable. As an extension, we further consider the stability analysis problem for the same type of neural networks but with partially known transition probabilities. Two numerical examples are given to demonstrate the usefulness of the derived methods.