Stability Analysis of Neural Networks With Time-Varying Delay by Constructing Novel Lyapunov Functionals

Robust non-fragile hybrid control for delayed uncertain singular impulsive jump systems based on improved impulse instant-dependent auxiliary functions method

This article researches the issue of robust non-fragile hybrid control for delayed uncertain singular impulsive jump systems (USIMJSs). The key aim is to design non-fragile hybrid state feedback controllers (including a non-fragile normal state feedback controller and a non-fragile impulsive state feedback controller), which are insensitive to the uncertainties of gains of controllers and can provide sufficient tuning margins. The non-fragile normal state feedback controller can eliminate the internal impulses and overcome the external disturbances; the non-fragile impulsive state feedback controller can suppress the interference of external unstable impulses and restrain the instantaneous jumps caused by Markovian modes switching. By introducing impulse instant-dependent auxiliary functions, the improved impulse-time-dependent Lyapunov-Krasovskii functional is constructed, which can capture the information of the impulse instants and Markovian jump modes. Novel criteria of robust admissibilization for delayed USIMJSs are acquired under linear matrix inequalities framework. Lastly, the effectiveness of the derived algorithm and designed method is confirmed by simulation examples including a direct current motor-controlled inverted pendulum device and a Quarter-Car active suspension model.

Stability Analysis for Delayed Neural Networks via a Generalized Reciprocally Convex Inequality

This article deals with the stability of neural networks (NNs) with time-varying delay. First, a generalized reciprocally convex inequality (RCI) is presented, providing a tight bound for reciprocally convex combinations. This inequality includes some existing ones as special case. Second, in order to cater for the use of the generalized RCI, a novel Lyapunov-Krasovskii functional (LKF) is constructed, which includes a generalized delay-product term. Third, based on the generalized RCI and the novel LKF, several stability criteria for the delayed NNs under study are put forward. Finally, two numerical examples are given to illustrate the effectiveness and advantages of the proposed stability criteria.

Globally Adaptive Neural Network Tracking for Uncertain Output-Feedback Systems

This article investigates the problem of global neural network (NN) tracking control for uncertain nonlinear systems in output feedback form under disturbances with unknown bounds. Compared with the existing NN control method, the differences of the proposed scheme are as follows. The designed actual controller consists of an NN controller working in the approximate domain and a robust controller working outside the approximate domain, in addition, a new smooth switching function is designed to achieve the smooth switching between the two controllers, in order to ensure the globally uniformly ultimately bounded of all closed-loop signals. The Lyapunov analysis method is used to strictly prove the global stability under the combined action of unmeasured states and system uncertainties, and the output tracking error is guaranteed to converge to an arbitrarily small neighborhood through a reasonable selection of design parameters. A numerical example and a practical example were put forward to verify the effectiveness of the control strategy.

Synchronization control of time-delay neural networks via event-triggered non-fragile cost-guaranteed control

This paper is devoted to event-triggered non-fragile cost-guaranteed synchronization control for time-delay neural networks. The switched event-triggered mechanism, which combines periodic sampling and continuous event triggering, is used in the feedback channel. A piecewise functional is first applied to fully utilize the information of the state and activation function. By employing the functional, various integral inequalities, and the free-weight matrix technique, a sufficient condition is established for exponential synchronization and cost-related performance. Then, a joint design of the needed non-fragile feedback gain and trigger matrix is derived by decoupling several nonlinear coupling terms. On the foundation of the joint design, an optimization scheme is given to acquire the minimum cost value while ensuring exponential stability of the synchronization-error system. Finally, a numerical example is used to illustrate the applicability of the present design scheme.

Impulsive strategies in nonlinear dynamical systems: A brief overview

The studies of impulsive dynamical systems have been thoroughly explored, and extensive publications have been made available. This study is mainly in the framework of continuous-time systems and aims to give an exhaustive review of several main kinds of impulsive strategies with different structures. Particularly, (i) two kinds of impulse-delay structures are discussed respectively according to the different parts where the time delay exists, and some potential effects of time delay in stability analysis are emphasized. (ii) The event-based impulsive control strategies are systematically introduced in the light of several novel event-triggered mechanisms determining the impulsive time sequences. (iii) The hybrid effects of impulses are emphatically stressed for nonlinear dynamical systems, and the constraint relationships between different impulses are revealed. (iv) The recent applications of impulses in the synchronization problem of dynamical networks are investigated. Based on the above several points, we make a detailed introduction for impulsive dynamical systems, and some significant stability results have been presented. Finally, several challenges are suggested for future works.

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.

Stochastic Stability of Markovian Neural Networks With Generally Hybrid Transition Rates

This article studies the problem of the stability for Markovian neural networks (MNNs) with time delay. The transition rate is considered to be generally hybrid, which treats those existing ones as its special cases. The introduced generally hybrid transition rates (GHTRs) make these systems more general and practical. Apropos of the GHTRs, a double-boundary approach rather than the traditional estimation method is introduced to make full use of the error information in GHTRs. In order to fully capture system information, a parameter-type-delay-dependent-matrix (PTDDM) approach is proposed, in which the PTDDM approach removes some zero components on slack matrices in previous works. Thus, the PTDDM approach can fully link the relationship among time delay and state-related vectors. Based on these ingredients, a novel stochastic stability condition is proposed for MNNs with GHTRs. A numerical example is illustrated to demonstrate the effectiveness of the proposed approaches.

LU-Optimality Conditions in Optimization Problems With Mechanical Work Objective Functionals

In this article, we introduce interval-valued Kuhn-Tucker (KT)-pseudoinvex optimization problems governed by interval-valued path-independent curvilinear integral objective functionals. Concretely, it is proven that an interval-valued KT-pseudoinvex variational control problem is described such that every KT point is an LU-optimal solution. In addition, the main results are highlighted by two illustrative applications describing the controlled behavior of an artificial neural system.

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.

Improved Stability Criteria for Delayed Neural Networks Using a Quadratic Function Negative-Definiteness Approach

This brief is concerned with the stability of a neural network with a time-varying delay using the quadratic function negative-definiteness approach reported recently. A more general reciprocally convex combination inequality is taken to introduce some quadratic terms into the time derivative of a Lyapunov-Krasovskii (L-K) functional. As a result, the time derivative of the L-K functional is estimated by a novel quadratic function on the time-varying delay. Moreover, a simple way is introduced to calculate the coefficients of a quadratic function, which avoids tedious works by hand as done in some studies. The L-K functional approach is applied to derive a hierarchical type stability criterion for the delayed neural networks, which is of less conservatism in comparison with some existing results through two well-studied numerical examples.

H∞ State Estimation for Neural Networks With General Activation Function and Mixed Time-Varying Delays

This article deals with H_∞ state estimation of neural networks with mixed delays. In order to make full use of delay information, novel delay-product Lyapunov-Krasovskii functional (LKF) by using parameterized delay interval is first constructed. Then, generalized free-weighting-matrix integral inequality is used to estimate the derivative of LKF to reduce the conservatism. Also, a more general activation function is further applied by combining with parameterized delay interval in order to obtain a more accurate estimator model. Finally, sufficient conditions are derived to confirm that the estimation error system is asymptotically stable with a prescribed H_∞ performance. Numerical examples are simulated to show the benefits of our proposed method.

Dissipativity Analysis for Neural Networks With Time-Varying Delays via a Delay-Product-Type Lyapunov Functional Approach

This article is concerned with the problem of dissipativity and stability analysis for a class of neural networks (NNs) with time-varying delays. First, a new augmented Lyapunov-Krasovskii functional (LKF), including some delay-product-type terms, is proposed, in which the information on time-varying delay and system states is taken into full consideration. Second, by employing a generalized free-matrix-based inequality and its simplified version to estimate the derivative of the proposed LKF, some improved delay-dependent conditions are derived to ensure that the considered NNs are strictly ( Q , S , R )- γ -dissipative. Furthermore, the obtained results are applied to passivity and stability analysis of delayed NNs. Finally, two numerical examples and a real-world problem in the quadruple tank process are carried out to illustrate the effectiveness of the proposed method.

Relaxed Stability Criteria for Neural Networks With Time-Varying Delay Using Extended Secondary Delay Partitioning and Equivalent Reciprocal Convex Combination Techniques

This article investigates global asymptotic stability for neural networks (NNs) with time-varying delay, which is differentiable and uniformly bounded, and the delay derivative exists and is upper-bounded. First, we propose the extended secondary delay partitioning technique to construct the novel Lyapunov-Krasovskii functional, where both single-integral and double-integral state variables are considered, while the single-integral ones are only solved by the traditional secondary delay partitioning. Second, a novel free-weight matrix equality (FWME) is presented to resolve the reciprocal convex combination problem equivalently and directly without Schur complement, which eliminates the need of positive definite matrices, and is less conservative and restrictive compared with various improved reciprocal convex inequalities. Furthermore, by the present extended secondary delay partitioning, equivalent reciprocal convex combination technique, and Bessel-Legendre inequality, two different relaxed sufficient conditions ensuring global asymptotic stability for NNs are obtained, for time-varying delays, respectively, with unknown and known lower bounds of the delay derivative. Finally, two examples are given to illustrate the superiority and effectiveness of the presented method.

Enhancing the settling time estimation of fixed-time stability and applying it to the predefined-time synchronization of delayed memristive neural networks with external unknown disturbance

This paper concentrates on the global predefined-time synchronization of delayed memristive neural networks with external unknown disturbance via an observer-based active control. First, a global predefined-time stability theorem based on a non-negative piecewise Lyapunov function is proposed, which can obtain more accurate upper bound of the settling time estimation. Subsequently, considering the delayed memristive neural networks with disturbance, a disturbance-observer is designed to approximate the external unknown disturbance in the response system with a Hurwitz theorem and then to eliminate the influence of the unknown disturbance. With the help of global predefined-time stability theorem, the predefined-time synchronization is achieved between two delayed memristive neural networks via an active control Lyapunov function design. Finally, two numerical simulations are performed, and the results are given to show the correctness and feasibility of the predefined-time stability theorem.

Stubborn State Estimation for Delayed Neural Networks Using Saturating Output Errors

This paper is concerned with the stubborn state estimation of delayed neural networks that subject to a general class of disturbances in measurements, including outliers and impulsive disturbances as its special cases. This class of disturbances may be unbounded, irregular, and assorted; therefore, they can hardly be suppressed by existing identification-based estimation approaches. In this paper, a stubborn state estimator is constructed by intentionally devising a saturation scheme on the injection of output estimation error. The embedded saturation can effectively resist the influences from these measurement disturbances by saturating them. Moreover, the saturation threshold in the designed scheme is not constant but governed by a dynamic equation with parameters to be designed. Benefiting from this adaptiveness, the estimator obtains more freedom in dealing with various disturbances. By combining a novel Lyapunov functional, the generalized sector condition and two latest integral inequalities, a delay-dependent criterion is derived in a less conservative way to check whether the estimation error system with this dynamic saturation is globally stable. A sufficient condition with two tuning scalars is further provided to codesign the gain of the state estimator and the evolution law of the saturation threshold. Finally, two numerical examples are used to illustrate the stubbornness of this state estimator in the presence of measurement outliers or impulsive disturbances.

New H∞ state estimation criteria of delayed static neural networks via the Lyapunov-Krasovskii functional with negative definite terms

In the estimation problem for delayed static neural networks (SNNs), constructing a proper Lyapunov-Krasovskii functional (LKF) is crucial for deriving less conservative estimation criteria. In this paper, a delay-product-type LKF with negative definite terms is proposed. Based on the third-order Bessel-Legendre (B-L) integral inequality and mixed convex combination approaches, a less conservative estimator design criterion is derived. Furthermore, the desired estimator gain matrices and the H_∞ performance index are obtained by solving a set of linear matrix inequalities (LMIs). Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Passivity Analysis for Quaternion-Valued Memristor-Based Neural Networks With Time-Varying Delay

This paper is concerned with the problem of global exponential passivity for quaternion-valued memristor-based neural networks (QVMNNs) with time-varying delay. The QVMNNs can be seen as a switched system due to the memristor parameters are switching according to the states of the network. This is the first time that the global exponential passivity of QVMNNs with time-varying delay is investigated. By means of a nondecomposition method and structuring novel Lyapunov functional in form of quaternion self-conjugate matrices, the delay-dependent passivity criteria are derived in the forms of quaternion-valued linear matrix inequalities (LMIs) as well as complex-valued LMIs. Furthermore, the asymptotical stability criteria can be obtained from the proposed passivity criteria. Finally, a numerical example is presented to illustrate the effectiveness of the theoretical results.

New Results on Stability Analysis for Delayed Markovian Generalized Neural Networks With Partly Unknown Transition Rates

The stability of delayed Markovian generalized neural networks is studied where the transition rates of the modes are partly unknown. The partly unknown transition rates generalize the traditional works that are with all known transition rates. Then, a Lyapunov-Krasovskii functional (LKF) with a delay-product-type (DPT) term is constructed. The DPT term is not only simple but also fully utilizes the information of time delay. Based on the new DPT LKF, stability criteria are presented, which are with lower computational complexity and less conservative. In the end, the validity and superiorities of the analytical results are verified by several examples.

Stability Analysis for Delayed Neural Networks via Improved Auxiliary Polynomial-Based Functions

This brief is concerned with stability analysis for delayed neural networks (DNNs). By establishing polynomials and introducing slack variables reasonably, some improved delay-product type of auxiliary polynomial-based functions (APFs) is developed to exploit additional degrees of freedom and more information on extra states. Then, by constructing Lyapunov-Krasovskii functional using APFs and integrals of quadratic forms with high order scalar functions, a novel stability criterion is derived for DNNs, in which the benefits of the improved inequalities are fully integrated and the information on delay and its derivative is well reflected. By virtue of the advantages of APFs, more desirable performance is achieved through the proposed approach, which is demonstrated by the numerical examples.

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.