Type-Dependent Average Dwell Time Method and Its Application to Delayed Neural Networks With Large Delays

This article investigates the stability of delayed neural networks with large delays. Unlike previous studies, the original large delay is separated into several parts. Then, the delayed neural network is viewed as the switched system with one stable and multiple unstable subsystems. To effectively guarantee the stability of the considered system, the type-dependent average dwell time (ADT) is proposed to handle switches between any two sequences. Besides, multiple Lyapunov functions (MLFs) are employed to establish stability conditions. Adding more delayed state vectors increases the allowable maximum delay bound (AMDB), reducing the conservatism of stability criteria. A general form of the global exponential stability condition is put forward. Finally, a numerical example illustrates the effectiveness, and superiority of our method over the existing one.

Stability Analysis of Recurrent Neural Networks With Time-Varying Delay by Flexible Terminal Interpolation Method

This brief studies the stability problem of recurrent neural networks with time-varying delay. Based on one tunable parameter α , a flexible terminal interpolation method is proposed to change the interval with fixed terminals as 2k+1-3 ones with flexible terminals. Associated with the flexible subintervals, a novel Lyapunov-Krasovskii functional with more delay information is constructed. In order to estimate the Lyapunov-Krasovskii functional, a quadratic reciprocally convex inequality is proposed, which covers some existing ones as its special cases. Based on these ingredients, a new stability criterion is derived in the form of linear matrix inequalities. A comprehensive comparison of results is given to illustrate the newly proposed stability criterion.

Global finite-time stability of delayed quaternion-valued neural networks based on a class of extended Lyapunov-Razumikhin methods

In this paper, a class of global finite-time stability problem for quaternion-valued neural networks with time-varying delays are investigated by adopting an extended modification Lyapunov-Razumikhin (L-R) method and a new upper bounds estimation of system solution in terms of convergence rate was obtained. Firstly, a new extended method of L-R is proposed to solve the general difficulty to find a proper Lyapunov functional. Then, a new suitable controller is designed, the new conditions of inequalities global finite-time stability are obtained via combining with the former proposed L-R method in the separated real-valued system. Finally, for purpose of verifying the availability of the theorem presented, two given illustrative examples are shown.

Event-triggering load frequency control for multi-area power system based on random dynamic triggering mechanism and two-side closed functional

By taking into account sampled-data mechanism and transmission delay, the novel event-triggering load frequency control (LFC) strategy involving random dynamic triggering algorithm (RDTA) is developed for multi-area power systems in this paper. Firstly, an improved multi-area LFC model considering sampling and transmission delay (STD) simultaneously is addressed. Secondly, a modified event-triggering mechanism (ETM) with RDTA is proposed, considering parameter disturbances and a dynamic adjustment mechanism of the triggering threshold. Thirdly, a more advanced Lyapunov-Krasovskii functional (LKF) is constructed, introducing the delay-dependent matrices, more variable cross terms and the two-sided closed functional. Furthermore, two less conservative stability criteria are obtained according to the designed approach. Finally, two multi-area LFC systems are presented to verify the progressiveness of the proposed approach.

Stability and dissipativity criteria for neural networks with time-varying delays via an augmented zero equality approach

This work investigates the stability and dissipativity problems for neural networks with time-varying delay. By the construction of new augmented Lyapunov-Krasovskii functionals based on integral inequality and the use of zero equality approach, three improved results are proposed in the forms of linear matrix inequalities. And, based on the stability results, the dissipativity analysis for NNs with time-varying delays was investigated. Through some numerical examples, the superiority and effectiveness of the proposed results are shown by comparing the existing works.

Stability analysis of genetic regulatory networks via a linear parameterization approach

This paper investigates the problem of finite-time stability (FTS) for a class of delayed genetic regulatory networks with reaction-diffusion terms. In order to fully utilize the system information, a linear parameterization method is proposed. Firstly, by applying the Lagrange’s mean-value theorem, the linear parameterization method is applied to transform the nonlinear system into a linear one with time-varying bounded uncertain terms. Secondly, a new generalized convex combination lemma is proposed to dispose the relationship of bounded uncertainties with respect to their boundaries. Thirdly, sufficient conditions are established to ensure the FTS by resorting to Lyapunov Krasovskii theory, convex combination technique, Jensen’s inequality, linear matrix inequality, etc. Finally, the simulation verifications indicate the validity of the theoretical results.

A Dynamic Event-Triggered Approach to Recursive Filtering for Complex Networks With Switching Topologies Subject to Random Sensor Failures

This article deals with the recursive filtering issue for a class of nonlinear complex networks (CNs) with switching topologies, random sensor failures and dynamic event-triggered mechanisms. A Markov chain is utilized to characterize the switching behavior of the network topology. The phenomenon of sensor failures occurs in a random way governed by a set of stochastic variables obeying certain probability distributions. In order to save communication cost, a dynamic event-triggered transmission protocol is introduced into the transmission channel from the sensors to the recursive filters. The objective of the addressed problem is to design a set of dynamic event-triggered filters for the underlying CN with a certain guaranteed upper bound (on the filtering error covariance) that is then locally minimized. By employing the induction method, an upper bound is first obtained on the filtering error covariance and subsequently minimized by properly designing the filter parameters. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed filtering scheme.

Hidden Markov Model-Based Nonfragile State Estimation of Switched Neural Network With Probabilistic Quantized Outputs

This paper focuses on the state estimator design problem for a switched neural network (SNN) with probabilistic quantized outputs, where the switching process is governed by a sojourn probability. It is assumed that both packet dropouts and signal quantization exist in communication channels. Asynchronous estimator and quantification function are described by two different hidden Markov model between the SNNs and its estimator. To deal with the small uncertain of estimators in a random way, a probabilistic nonfragile state estimator is introduced, where uncertain information is described by the interval type of gain variation. A sufficient condition on mean square stable of the estimation error system is obtained and then the desired estimator is designed. Finally, a simulation result is provided to verify the effectiveness of the proposed design method.

Reachable set bounding for neural networks with mixed delays: Reciprocally convex approach

This paper discusses the reachable set estimation problem of neural networks with mixed delays. Firstly, by means of the maximal Lyapunov-Krasovskii functional, we obtain a non-ellipsoid form of the reachable set. Further more, when calculating the derivative of the maximum Lyapunov functional, the lower bound lemma and reciprocally convex approach method are used to solve the reciprocally convex combination term, which reduce the related decision variables. Secondly, we extend the results to polytopic uncertainties neural networks and consider the case of uncertain differentiable parameters. Finally, two numerical examples and one application example are listed to show the validity of our methods.

Global Asymptotic Stability for Delayed Neural Networks Using an Integral Inequality Based on Nonorthogonal Polynomials

This brief is concerned with global asymptotic stability of a neural network with a time-varying delay. First, by introducing an auxiliary vector with some nonorthogonal polynomials, a slack-matrix-based integral inequality is established, which includes some existing one as its special case. Second, a novel Lyapunov-Krasovskii functional is constructed to suit for the use of the obtained integral inequality. As a result, a less conservative stability criterion is derived, whose effectiveness is finally demonstrated through two well-used numerical examples.

A Novel Finite-Sum Inequality-Based Method for Robust $H_\infty$ Control of Uncertain Discrete-Time Takagi-Sugeno Fuzzy Systems With Interval-Like Time-Varying Delays

This paper is concerned with the problem of robust ${H}_{\infty}$ control of an uncertain discrete-time Takagi-Sugeno fuzzy system with an interval-like time-varying delay. A novel finite-sum inequality-based method is proposed to provide a tighter estimation on the forward difference of certain Lyapunov functional, leading to a less conservative result. First, an auxiliary vector function is used to establish two finite-sum inequalities, which can produce tighter bounds for the finite-sum terms appearing in the forward difference of the Lyapunov functional. Second, a matrix-based quadratic convex approach is employed to equivalently convert the original matrix inequality including a quadratic polynomial on the time-varying delay into two boundary matrix inequalities, which delivers a less conservative bounded real lemma (BRL) for the resultant closed-loop system. Third, based on the BRL, a novel sufficient condition on the existence of suitable robust ${H}_{\infty}$ fuzzy controllers is derived. Finally, two numerical examples and a computer-simulated truck-trailer system are provided to show the effectiveness of the obtained results.

Energy-to-Peak State Estimation for Static Neural Networks With Interval Time-Varying Delays

This paper is concerned with energy-to-peak state estimation on static neural networks (SNNs) with interval time-varying delays. The objective is to design suitable delay-dependent state estimators such that the peak value of the estimation error state can be minimized for all disturbances with bounded energy. Note that the Lyapunov-Krasovskii functional (LKF) method plus proper integral inequalities provides a powerful tool in stability analysis and state estimation of delayed NNs. The main contribution of this paper lies in three points: 1) the relationship between two integral inequalities based on orthogonal and nonorthogonal polynomial sequences is disclosed. It is proven that the second-order Bessel-Legendre inequality (BLI), which is based on an orthogonal polynomial sequence, outperforms the second-order integral inequality recently established based on a nonorthogonal polynomial sequence; 2) the LKF method together with the second-order BLI is employed to derive some novel sufficient conditions such that the resulting estimation error system is globally asymptotically stable with desirable energy-to-peak performance, in which two types of time-varying delays are considered, allowing its derivative information is partly known or totally unknown; and 3) a linear-matrix-inequality-based approach is presented to design energy-to-peak state estimators for SNNs with two types of time-varying delays, whose efficiency is demonstrated via two widely studied numerical examples.

Synchronization of Coupled Reaction-Diffusion Neural Networks With Directed Topology via an Adaptive Approach

This paper investigates the synchronization issue of coupled reaction-diffusion neural networks with directed topology via an adaptive approach. Due to the complexity of the network structure and the presence of space variables, it is difficult to design proper adaptive strategies on coupling weights to accomplish the synchronous goal. Under the assumptions of two kinds of special network structures, that is, directed spanning path and directed spanning tree, some novel edge-based adaptive laws, which utilized the local information of node dynamics fully are designed on the coupling weights for reaching synchronization. By constructing appropriate energy function, and utilizing some analytical techniques, several sufficient conditions are given. Finally, some simulation examples are given to verify the effectiveness of the obtained theoretical results.

Hierarchical Type Stability Criteria for Delayed Neural Networks via Canonical Bessel-Legendre Inequalities

This paper is concerned with global asymptotic stability of delayed neural networks. Notice that a Bessel-Legendre inequality plays a key role in deriving less conservative stability criteria for delayed neural networks. However, this inequality is in the form of Legendre polynomials and the integral interval is fixed on . As a result, the application scope of the Bessel-Legendre inequality is limited. This paper aims to develop the Bessel-Legendre inequality method so that less conservative stability criteria are expected. First, by introducing a canonical orthogonal polynomial sequel, a canonical Bessel-Legendre inequality and its affine version are established, which are not explicitly in the form of Legendre polynomials. Moreover, the integral interval is shifted to a general one . Second, by introducing a proper augmented Lyapunov-Krasovskii functional, which is tailored for the canonical Bessel-Legendre inequality, some sufficient conditions on global asymptotic stability are formulated for neural networks with constant delays and neural networks with time-varying delays, respectively. These conditions are proven to have a hierarchical feature: the higher level of hierarchy, the less conservatism of the stability criterion. Finally, three numerical examples are given to illustrate the efficiency of the proposed stability criteria.

Stabilization of Neural-Network-Based Control Systems via Event-Triggered Control With Nonperiodic Sampled Data

This paper focuses on a problem of event-triggered stabilization for a class of nonuniformly sampled neural-network-based control systems (NNBCSs). First, a new event-triggered data transmission mechanism is designed based on the nonperiodic sampled data. Different from the previous works, the proposed triggering scheme enables the NNBCSs design to enjoy the advantages of both nonuniform and event-triggered sampling schemes. Second, under the nonperiodic event-triggered data transmission scheme, the nonperiodic sampled-data three-layer fully connected feedforward neural-network (TLFCFFNN)-based event-triggered controller is constructed, and the resulting closed-loop TLFCFFNN-based event-triggered control system is modeled as a state delay system based on time-delay system modeling approach. Then, the stability criteria for the closed-loop system is formulated using Lyapunov-Krasovskii functional approach. Third, the sufficient conditions for the codesign of the TLFCFFNN-based controller and triggering parameters are given in terms of solvability of matrix inequalities to guarantee the asymptotical stability of the closed-loop system and an upper bound on the given cost function while reducing the updates of the controller. Finally, three numerical examples are provided to illustrate the effectiveness and benefits of the proposed results.

Delay-Dependent Global Exponential Stability for Delayed Recurrent Neural Networks

This paper deals with the global exponential stability for delayed recurrent neural networks (DRNNs). By constructing an augmented Lyapunov-Krasovskii functional and adopting the reciprocally convex combination approach and Wirtinger-based integral inequality, delay-dependent global exponential stability criteria are derived in terms of linear matrix inequalities. Meanwhile, a general and effective method on global exponential stability analysis for DRNNs is given through a lemma, where the exponential convergence rate can be estimated. With this lemma, some global asymptotic stability criteria of DRNNs acquired in previous studies can be generalized to global exponential stability ones. Finally, a frequently utilized numerical example is carried out to illustrate the effectiveness and merits of the proposed theoretical results.

Stability of Recurrent Neural Networks With Time-Varying Delay via Flexible Terminal Method

This brief is concerned with the stability criteria for recurrent neural networks with time-varying delay. First, based on convex combination technique, a delay interval with fixed terminals is changed into the one with flexible terminals, which is called flexible terminal method (FTM). Second, based on the FTM, a novel Lyapunov-Krasovskii functional is constructed, in which the integral interval associated with delayed variables is not fixed. Thus, the FTM can achieve the same effect as that of delay-partitioning method, while their implementary ways are different. Guided by FTM, Wirtinger-based integral inequality and free-weight matrix method are employed to develop several stability criteria, respectively. Finally, the feasibility and the effectiveness of the proposed results are tested by two numerical examples.

Neuronal State Estimation for Neural Networks With Two Additive Time-Varying Delay Components

This paper is concerned with the state estimation for neural networks with two additive time-varying delay components. Three cases of these two time-varying delays are fully considered: 1) both delays are differentiable uniformly bounded with delay-derivative bounded by some constants; 2) one delay is continuous uniformly bounded while the other is differentiable uniformly bounded with delay-derivative bounded by certain constants; and 3) both delays are continuous uniformly bounded. First, an extended reciprocally convex inequality is introduced to bound reciprocally convex combinations appearing in the derivative of some Lyapunov-Krasovskii functional. Second, sufficient conditions are derived based on the extended inequality for three cases of time-varying delays, respectively. Third, a linear-matrix-inequality-based approach with two tuning parameters is proposed to design desired Luenberger estimators such that the error system is globally asymptotically stable. This approach is then applied to state estimation on neural networks with a single interval time-varying delay. Finally, two numerical examples are given to illustrate the effectiveness of the proposed method.

Stabilization of systems with interval time-varying delay based on delay decomposing approach

The paper considers the stabilization for systems with interval time-varying delay. By decomposing the delay interval into multiple equidistant subintervals and considering the triple integral terms, a novel Lyapunov-krasovskii functional(LKF) is defined. Then extended integral inequality and convex combination approach are used to estimate the derivative of the constructed functional, and as a result, the new stability criterion with less conservatism and decision variables is obtained. On this basis, the state feedback controller is designed, by using linearization method, the existence condition of controller is obtained in terms of linear matrix inequalities(LMIs), and the specific form of controller is also given, moreover, by selecting the appropriate parameter value, the stabilization time of the system can be reduced. Numerical examples are given to illustrate the effectiveness of the proposed method.

Complete stability of delayed recurrent neural networks with Gaussian activation functions

This paper addresses the complete stability of delayed recurrent neural networks with Gaussian activation functions. By means of the geometrical properties of Gaussian function and algebraic properties of nonsingular M-matrix, some sufficient conditions are obtained to ensure that for an n-neuron neural network, there are exactly 3^k equilibrium points with 0≤k≤n, among which 2^k and 3^k-2^k equilibrium points are locally exponentially stable and unstable, respectively. Moreover, it concludes that all the states converge to one of the equilibrium points; i.e., the neural networks are completely stable. The derived conditions herein can be easily tested. Finally, a numerical example is given to illustrate the theoretical results.