Delay-dependent Lurie-Postnikov type Lyapunov-Krasovskii functionals for stability analysis of discrete-time delayed neural networks.

This paper addresses the influence of time-varying delay and nonlinear activation functions with sector restrictions on the stability of discrete-time neural networks. Compared to previous works that mainly focuses on the influence of delay information, this paper devotes to activation nonlinear functions information to help compensate the analysis technique based on Lyapunov-Krasovskii functional (LKF). A class of delay-dependent Lurie-Postnikov type integral terms involving sector constraints of nonlinear activation function is proposed to complement the LKF construction. The less conservative criteria for the stability analysis of discrete-time delayed networks is given by using improved LKF. Numerical examples show that conservatism can be reduced by the delay-dependent integral terms involving nonlinear activation functions.

A Degree-Dependent Polynomial-Based Reciprocally Convex Matrix Inequality and Its Application to Stability Analysis of Delayed Neural Networks.

In this article, several improved stability criteria for time-varying delayed neural networks (DNNs) are proposed. A degree-dependent polynomial-based reciprocally convex matrix inequality (RCMI) is proposed for obtaining less conservative stability criteria. Unlike previous RCMIs, the matrix inequality in this article produces a polynomial of any degree in the time-varying delay, which helps to reduce conservatism. In addition, to reduce the computational complexity caused by dealing with the negative definite of the high-degree terms, an improved lemma is presented. Applying the above matrix inequalities and improved negative definiteness condition helps to generate a more relaxed stability criterion for analyzing time-varying DNNs. Two examples are provided to illustrate this statement.

Discrete-State Decomposition Technique of Dissipativity Analysis for Discrete-Time Singular Systems With Time-Varying Delays.

This article is concerned with the problem of dissipativity for discrete-time singular systems with time-varying delays. First, the discrete-state decomposition technique is proposed after performing the restricted equivalent transformation for singular systems. To reduce the use of decision variables, the state-decomposed Lyapunov function is established based on the decomposed state vectors. Second, to obtain the condition with less conservatism, the two zero-value equations, especially concerning difference subsystems and algebraic ones, the discrete Wirtinger-based inequality and the extended reciprocally convex inequality are employed to bound the forward difference of the Lyapunov function. Then, the less conservative dissipativity criteria with lower computational complexity are obtained. Finally, simulation results are provided to demonstrate the superiority of the proposed technique.

Hierarchical Passivity Criterion for Delayed Neural Networks via A General Delay-Product-Type Lyapunov-Krasovskii Functional.

This article is concerned with passivity analysis of neural networks with a time-varying delay. Several techniques in the domain are improved to establish the new passivity criterion with less conservatism. First, a Lyapunov-Krasovskii functional (LKF) is constructed with two general delay-product-type terms which contain any chosen degree of polynomials in time-varying delay. Second, a general convexity lemma without conservatism is developed to address the positive-definiteness of the LKF and the negative-definiteness of its time-derivative. Then, with these improved results, a hierarchical passivity criterion of less conservatism is obtained for neural networks with a time-varying delay, whose size and conservatism vary with the maximal degree of the time-varying delay polynomial in the LKF. It is shown that the conservatism of the passivity criterion does not always reduce as the degree of the time-varying delay polynomial increases. Finally, a numerical example is given to illustrate the proposed criterion and benchmark against the existing results.

Exponential Synchronization of Memristor-Based Competitive Neural Networks With Reaction-Diffusions and Infinite Distributed Delays.

Taking into account the infinite distributed delays and reaction-diffusions, this article investigates the global exponential synchronization problem of a class of memristor-based competitive neural networks (MCNNs) with different time scales. Based on the Lyapunov-Krasovskii functional and inequality approach, an adaptive control approach is proposed to ensure the exponential synchronization of the addressed drive-response networks. The closed-loop system is a discontinuous and delayed partial differential system in a cascade form, involving the spatial diffusion, the infinite distributed delays, the parametric adaptive law, the state-dependent switching parameters, and the variable structure controllers. By combining the theories of nonsmooth analysis, partial differential equation (PDE) and adaptive control, we present a new analytical method for rigorously deriving the synchronization of the states of the complex system. The derived m-norm (m ≥ 2)-based synchronization criteria are easily verified and the theoretical results are easily extended to memristor-based neural networks (NNs) without different time scales and reaction-diffusions. Finally, numerical simulations are presented to verify the effectiveness of the theoretical results.

Hybrid Adjusting Variables-Dependent Event-Based Finite-Time State Estimation for Two-Time-Scale Markov Jump Complex Networks.

This article investigates the problem of dynamic event-triggered finite-time H∞ state estimation for a class of discrete-time nonlinear two-time-scale Markov jump complex networks. A hybrid adjusting variables-dependent dynamic event-triggered mechanism (DETM) is proposed to regulate the releases of measurement outputs of a node to a remote state estimator. Such a DETM contains both an additive dynamically adjusting variable (DAV) and a multiplicative adaptively adjusting variable. The aim is to design a DETM-based mode-dependent state estimator, which guarantees that the resultant error dynamics is stochastically finite-time bounded with H∞ performance. By constructing a mode-dependent Lyapunov function with multiple DAVs and a singular perturbation parameter associated with time scales, a matrix-inequalities-based sufficient condition is derived, the feasible solutions of which facilitate the design of the parameters of the state estimator. The validity of the designed state estimator and the superiority of the devised DETM are verified by two examples. It is verified that the devised DETM is capable of saving network resources and simultaneously improving the estimation performance.

Mixed-Delay-Based Augmented Functional for Sampled-Data Synchronization of Delayed Neural Networks With Communication Delay.

The synchronization control for delayed neural networks (DNNs) via a sampled-data controller considering communication delay is studied by input delay approach. Although few scholars have put forward the coexistence of transmission delay and communication delay in this problem, no report has clarified the interaction between transmission delay and communication delay. Also, the time-squared terms are underutilized. Thus, a novel augmented Lyapunov functional, which consists of a mixed-delay-based augmented part and a time-squared two-sided looped part, is proposed to fill this gap. In the mixed-delay-based augmented part, not only the information of transmission delay and communication delay themselves, but also the interaction between those two delays is considered. Time-dependent quadratic terms as well as the sampling integral states are introduced in the two-sided looped part, so that more characteristic information of the sampling pattern is encompassed and the relationship of the states at the sampling instant is enhanced. Then, this novel augmented functional is applied to the synchronization control of DNNs. A less conservative synchronization criterion is obtained in the form of linear matrix inequalities. A numerical example illustrates the validity and superiority of the presented synchronization criterion.

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.

Networked Output-Feedback MPC: A Bounded Dynamic Variable and Time-Varying Threshold-Dependent Event-Based Approach.

The event-triggered model predictive control (MPC) problem is addressed for polytopic uncertain systems. A new dynamic event-triggered mechanism (DETM) with a bounded dynamic variable and a time-varying threshold is proposed to manage measurement data packet releases. The dynamic output-feedback MPC issue is detailed as a "min-max" optimization problem (OP) with an objective function over an infinite horizon, where the hard constraint on the predictive control is required. By applying a Lyapunov-like function containing the bounded dynamic variable, an auxiliary OP constrained by several matrix inequalities is proposed, and the design methods of the output-feedback gains are provided if this auxiliary OP is feasible. The designed MPC controller ensures that the closed-loop system is input-to-state practically stable. Two examples including an event-triggered DC motor are given to illustrate the validity of the developed MPC algorithm. Simulation results verify that the proposed DETM has advantages over some existing triggering mechanisms in decreasing the consumption of resources while meeting the required performance.

Stability Analysis of Discrete-Time Neural Networks With a Time-Varying Delay: Extended Free-Weighting Matrices Zero Equation Approach.

This research investigates the stability of discrete-time neural networks (DNNs) with a time-varying delay by using the Lyapunov-Krasovskii functional (LKF) method. Recent researches acquired some less conservatism stability criteria for time-varying delayed systems via some augmented LKFs. However, the forward difference of such LKFs resulted in high-degree time-varying delay-dependent polynomials. This research aims to develop some augmented state-related vectors and the corresponding extended free-weighting matrices zero equations to avoid the appearance of such high-degree polynomials and help to provide more freedom for the estimation results. Besides, an augmented delay-product-type LKF is also established for ameliorating the stability conditions of the time-varying delayed DNNs. Then, based on the above methods and Jensen's summation inequality, the auxiliary-function-based summation inequality, and the reciprocally convex matrix inequality, some less conservatism stability criteria for time-varying delayed DNNs are formulated. The validity of the proposed time-varying delay-dependent stability criteria is illustrated by two numerical examples.

Stability Analysis for Delayed Neural Networks via a Novel Negative-Definiteness Determination Method.

The stability of neural networks with a time-varying delay is studied in this article. First, a relaxed Lyapunov-Krasovskii functional (LKF) is presented, in which the positive-definiteness requirement of the augmented quadratic term and the delay-product-type terms are set free, and two double integral states are augmented into the single integral terms at the same time. Second, a new negative-definiteness determination method is put forward for quadratic functions by utilizing Taylor's formula and the interval-decomposition approach. This method encompasses the previous negative-definiteness determination approaches and has less conservatism. Finally, the proposed LKF and the negative-definiteness determination method are applied to the stability analysis of neural networks with a time-varying delay, whose advantages are shown by two numerical examples.

Event-Triggered Fault Detection Filter Design for Discrete-Time Memristive Neural Networks With Time Delays.

In this article, the fault detection (FD) filter design problem is addressed for discrete-time memristive neural networks with time delays. When constructing the system model, an event-triggered communication mechanism is investigated to reduce the communication burden and a fault weighting matrix function is adopted to improve the accuracy of the FD filter. Then, based on the Lyapunov functional theory, an augmented Lyapunov functional is constructed. By utilizing the summation inequality approach and the improved reciprocally convex combination method, an FD filter that guarantees the asymptotic stability and the prescribed H∞ performance level of the residual system is designed. Finally, numerical simulations are provided to illustrate the effectiveness of the presented results.

Robust Delay-Dependent Load Frequency Control of Wind Power System Based on a Novel Reconstructed Model.

This article presents a novel reconstructed model for the delayed load frequency control (LFC) schemes considering wind power, which aims to improve the computational efficiency for PID controllers while retaining their dynamic performance. Via fully exploiting system states influenced by time delays directly, this novel reconstructed method is proposed with a controller isolated. Hence, when the PID controllers are unknown, the stability criterion based on this model can resolve controller gains with less time consumed. For given PID gains, this model can be employed to establish criteria for stability analysis, which can realize the tradeoff between the calculation accuracy and efficiency. The case study is first based on a two-area traditional LFC system to validate the merits of a novel reconstructed model, including accurately estimating the influence of time delay on system frequency stability with increased computational capability. Then, under traditional and deregulated environments, case studies are carried out on the two-area and three-area schemes, respectively. Through the novel reconstructed model, the efficiency of obtaining controller parameters is highly improved while their robustness against the random wind power, tie-line power changes, inertial reductions, and time delays remains almost unchanged.

Stability Analysis of Continuous-Time Switched Neural Networks With Time-Varying Delay Based on Admissible Edge-Dependent Average Dwell Time.

This article investigates the stability of the switched neural networks (SNNs) with a time-varying delay. To effectively guarantee the stability of the considered system with unstable subsystems and reduce conservatism of the stability criteria, admissible edge-dependent average dwell time (AED-ADT) is first utilized to restrict switching signals for the continuous-time SNNs, and multiple Lyapunov-Kravosikii functionals (LKFs) combining relaxed integral inequalities are employed to develop two novel less-conservative stability conditions. Finally, the numeral examples clearly indicate that the proposed criteria can reduce conservatism and ensure the stability of continuous-time SNNs.

Reachable Set Estimation for Discrete-Time Markovian Jump Neural Networks With Generally Incomplete Transition Probabilities.

This paper is concerned with the problem of reachable set estimation for discrete-time Markovian jump neural networks with generally incomplete transition probabilities (TPs). This kind of TP may be exactly known, merely known with lower and upper bounds, or unknown. The aim of this paper is to derive a precise reachable set description for the considered system via the Lyapunov-Krasovskii functional (LKF) approach. By constructing an augmented LKF, using an equivalent transformation method to deal with the unknown TPs and utilizing the extended reciprocally convex matrix inequality, and the free matrix weighting approach to estimate the forward difference of the constructed LKF, several sufficient conditions that guarantee the existence of an ellipsoidal reachable set are established. Finally, a numerical example with simulation results is given to demonstrate the effectiveness and superiority of the proposed results.

Active disturbance rejection for time-varying state-delay systems based on equivalent-input-disturbance approach.

Exogenous disturbances largely affect the control performance of systems with time delays. This study considers a control problem of rejecting a disturbance in a PI control system for a time-varying state-delay plant. The equivalent-input-disturbance (EID) approach is integrated in a PI control system. An EID estimator estimates the overall effects of a time-varying delay and a disturbance. An EID estimate is combined into a PI control law to improve control performance. A less-conservative stability condition of the control system is derived using a Lyapunov-Krasovskii functional together with the Jensen's integral inequality and the reciprocally convex combination lemma. Parameters of the controllers in the system are calculated using the condition. Engine idle speed control is used to verify the effectiveness of this approach. Compared with the generalized extended-state observer and the sliding-mode control methods, our method reduced the tracking error to about one third and one sixth, respectively. This demonstrates the validity and superiority of our method.

Stochastic Finite-Time H∞ State Estimation for Discrete-Time Semi-Markovian Jump Neural Networks With Time-Varying Delays.

In this article, the finite-time H∞ state estimation problem is addressed for a class of discrete-time neural networks with semi-Markovian jump parameters and time-varying delays. The focus is mainly on the design of a state estimator such that the constructed error system is stochastically finite-time bounded with a prescribed H∞ performance level via finite-time Lyapunov stability theory. By constructing a delay-product-type Lyapunov functional, in which the information of time-varying delays and characteristics of activation functions are fully taken into account, and using the Jensen summation inequality, the free weighting matrix approach, and the extended reciprocally convex matrix inequality, some sufficient conditions are established in terms of linear matrix inequalities to ensure the existence of the state estimator. Finally, numerical examples with simulation results are provided to illustrate the effectiveness of our proposed results.

Stability and Stabilization of T-S Fuzzy Systems With Time-Varying Delays via Delay-Product-Type Functional Method.

This paper is concerned with the stability and stabilization problems of T-S fuzzy systems with time-varying delays. The purpose is to develop a new state-feedback controller design method with less conservatism. First, a novel Lyapunov-Krasovskii functional is constructed by combining delay-product-type functional method together with the state vector augmentation. By utilizing Wirtinger-based integral inequality and an extended reciprocally convex matrix inequality, a less conservative delay-dependent stability condition is developed. Then, the corresponding controller design method for the closed-loop delayed fuzzy system is derived based on parallel distributed compensation scheme. Finally, two classic numerical examples are given to show the effectiveness and merits of the proposed approaches.

Extended Dissipativity Analysis for Markovian Jump Neural Networks With Time-Varying Delay via Delay-Product-Type Functionals.

This paper investigates the problem of extended dissipativity for Markovian jump neural networks (MJNNs) with a time-varying delay. The objective is to derive less conservative extended dissipativity criteria for delayed MJNNs. Toward this aim, an appropriate Lyapunov-Krasovskii functional (LKF) with some improved delay-product-type terms is first constructed. Then, by employing the extended reciprocally convex matrix inequality (ERCMI) and the Wirtinger-based integral inequality to estimate the derivative of the constructed LKF, a delay-dependent extended dissipativity condition is derived for the delayed MJNNs. An improved extended dissipativity criterion is also given via the allowable delay sets method. Based on the above-mentioned results, the extended dissipativity condition of delayed NNs without Markovian jump parameters is directly derived. Finally, three numerical examples are employed to illustrate the advantages of the proposed method.

Stability Analysis of Discrete-Time Neural Networks With Time-Varying Delay via an Extended Reciprocally Convex Matrix Inequality.

This paper is concerned with the stability analysis of discrete-time neural networks with a time-varying delay. Assessment of the effect of time delays on system stability requires suitable delay-dependent stability criteria. This paper aims to develop new stability criteria for reduction of conservatism without much increase of computational burden. An extended reciprocally convex matrix inequality is developed to replace the popular reciprocally convex combination lemma (RCCL). It has potential to reduce the conservatism of the RCCL-based criteria without introducing any extra decision variable due to its advantage of reduced estimation gap using the same decision variables. Moreover, a delay-product-type term is introduced for the first time into the Lyapunov function candidate such that a delay-variation-dependent stability criterion with the bounds of delay change rate is established. Finally, the advantages of the proposed criteria are demonstrated through two numerical examples.