Stochastic stability analysis of fuzzy hopfield neural networks with time-varying delays

The ordinary Takagi-Sugeno (TS) fuzzy models have provided an approach to represent complex nonlinear systems to a set of linear sub-models by using fuzzy sets and fuzzy reasoning. In this paper, stochastic fuzzy Hopfield neural networks with time-varying delays (SFVDHNNs) are studied. The model of SFVDHNN is first established as a modified TS fuzzy model in which the consequent parts are composed of a set of stochastic Hopfield neural networks with time-varying delays. Secondly, the global exponential stability in the mean square for SFVDHNN is studied by using the Lyapunov-Krasovskii approach. Stability criterion is derived in terms of linear matrix inequalities (LMIs), which can be effectively solved by some standard numerical packages.     

Robust H∞ Filtering for Uncertain Discrete Stochastic Systems with Time Delays

This paper considers the problem of robust H_∞ filtering for uncertain discrete-time stochastic systems with time-varying delays. The parameter uncertainties are assumed to be real time-varying norm-bounded in both the state and measurement equations. The problem to be addressed is the design of a stable filter that guarantees stochastic stability and a prescribed H_∞ performance level of the filtering error system for all admissible uncertainties and time delays. A sufficient condition for the existence of such filters is obtained in terms of a linear matrix inequality (LMI). For the case when this LMI is feasible, an explicit expression for a desired filter is given. An illustrative example is also provided to demonstrate the effectiveness and applicability of the proposed method.     

H ∞ Filtering for Stochastic Systems with Markovian Switching and Partly Unknown Transition Probabilities

This paper considers the H _∞ filtering problem for stochastic systems. The systems under consideration involve Markovian switching, mode-dependent delays, Itô-type stochastic disturbance, distributed time-varying delays and partly unknown transition probabilities. Our aim is to design a full-order filter such that the corresponding filtering error system is stochastically stable and satisfies a prescribed H _∞ disturbance attenuation level. By using a new Lyapunov–Krasovskii functional, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the proposed main results.     

New Criteria of Robust $$H_\infty $$ Stability for Fuzzy Mixed-Delay Systems with Nonlinear Noise Disturbances

This paper investigates new criteria of the robust \(H_\infty \) stability for a class of uncertain stochastic fuzzy mixed-delay systems with nonlinear noise disturbances by employing an improved free-weighting matrix approach. The fuzzy system is based on the Takagi–Sugeno model that is often used to represent the complex nonlinear systems in terms of fuzzy sets and fuzzy reasoning. To reflect more realistic dynamical behaviors of the system, both the parameter uncertainties and stochastic disturbances are considered, the stochastic disturbances are given in the form of a Brownian motion. The mixed delays comprise both discrete and distributed time-varying delays. In terms of a stochastic fuzzy Lyapunov functional, a sufficient criterion is proposed to investigate dynamical behaviors of the system in the mean-square sense with an \(H_\infty \) performance index.     

Exponential H∞ filtering for time-varying delay systems: Markovian approach

This paper discusses the H_∞ filtering problem for a class of deterministic systems with time-varying delays, where the stochastic property of time-varying delays described by Markovian approach is taken into consideration in filter design. Firstly, the delay interval is separated into several subintervals, which can be described by Markov process. Then, a new H_∞ filtering method for deterministic system with time-varying delay is given, whose filter can switch with time delay in terms of Markov process. Sufficient conditions for the existence of H_∞ filter are obtained as linear matrix inequalities, where the mode transition rates are known exactly or inexactly. Finally, numerical examples are used to demonstrate the utility of the given methods.     

Delay-Dependent H∞ Control and Filtering for Uncertain Markovian Jump Systems With Time-Varying Delays

This paper deals with the problems of delay-dependent robust H_infin control and filtering for Markovian jump linear systems with norm-bounded parameter uncertainties and time-varying delays. In terms of linear matrix inequalities, improved delay-dependent stochastic stability and bounded real lemma (BRL) for Markovian delay systems are obtained by introducing some slack matrix variables. The conservatism caused by either model transformation or bounding techniques is reduced. Based on the proposed BRL, sufficient conditions for the solvability of the robust H_infin control and H_infin filtering problems are proposed, respectively. Dynamic output feedback controllers and full-order filters, which guarantee the resulting closed-loop system and the error system, respectively, to be stochastically stable and satisfy a prescribed H_infin performance level for all delays no larger than a given upper bound, are constructed. Numerical examples are provided to demonstrate the reduced conservatism of the proposed results in this paper.     

Delay-Dependent Exponential Stability for Uncertain Stochastic Hopfield Neural Networks With Time-Varying Delays

This paper provides new delay-dependent conditions that guarantee the robust exponential stability of stochastic Hopfield type neural networks with time-varying delays and parameter uncertainties. Both the cases of the time-varying delays which are differentiable and may not be differentiable are considered. The stability conditions are derived by using the recently developed free-weighting matrices technique and expressed in terms of linear matrix inequalities. Numerical examples are provided to demonstrate the effectiveness of the proposed stability criteria. It is shown that the proposed stability results are less conservative than some previous ones in the literature.      

Gain-Constrained Extended Kalman Filtering with Stochastic Nonlinearities and Randomly Occurring Measurement Delays

In this paper, the gain-constrained extended Kalman filtering problem is studied for discrete time-varying nonlinear system with stochastic nonlinearities and randomly occurring measurement delays. Both deterministic and stochastic nonlinearities are simultaneously present in the model, where the stochastic nonlinearities are described by first moment and can encompass several classes of well-studied stochastic nonlinear functions. A diagonal matrix composed of mutually independent Bernoulli random variables is introduced to reflect the phenomenon of randomly occurring measurement delays caused by unfavorable network conditions. The aim of the addressed filtering problem is to design a finite-horizon recursive filter such that, for all stochastic nonlinearities, randomly occurring measurement delays and gain constraint, the upper bound of the cost function involving filtering error is minimized at each sampling time. It is shown that the filter gain is obtained by solving matrix equations. A numerical simulation example is provided to illustrate the effectiveness of the proposed algorithm.     

Robust Delay-Dependent Stability Criteria for Dynamic Systems with Nonlinear Perturbations and Leakage Delay

This paper studies the global asymptotic stability for uncertain systems with mixed delays. The mixed delays include constant delay in the leakage term (i.e., leakage delay) and time-varying delays. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties. Based on a appropriate Lyapunov–Krasovskii functional with triple integral terms, some integral inequalities and convex combination technique, a novel delay-dependent stability criterion is derived in terms of linear matrix inequalities (LMIs). Finally, three numerical examples are included to show the superiority of the proposed method.     

Non-Fragile H_{\infty } Filter Design for Uncertain Stochastic Nonlinear Time-Delay Markovian Jump Systems

This paper investigates the problem of non-fragile and mode-dependent \(H_{\infty }\) filter design for a class of nonlinear stochastic Markovian jump systems with mode-dependent time-varying delays and norm-bounded parameter uncertainties. The non-fragile and mode-dependent filter to be designed is assumed to include multiplicative gain variations which result from inaccuracies in filter implementation, and the desired filter ensures the filtering error system is not only exponentially mean-square stable, but also satisfies a prescribed \(H_{\infty }\) -norm level for all admissible uncertainties. A stochastic Lyapunov–Krasovskii function is proposed to reflect the information of Markovian jump modes and the mode-dependent time-varying delays, and a set of strict linear matrix inequalities are utilized to derive sufficient conditions that guarantee the desired filter can be constructed. A numerical example and a vertical take-off and landing (VTOL) helicopter system are utilized to illustrate the effectiveness and usefulness of the main results obtained.     

LMI-based criterion for robust stability of 2-D discrete systems with interval time-varying delays employing quantisation / overflow nonlinearities

This paper is concerned with the problem of global asymptotic stability of a class of nonlinear uncertain two-dimensional (2-D) discrete systems described by the Fornasini-Marchesini second local state-space model with time-varying state delays. The class of systems under investigation involves norm bounded parameter uncertainties, interval-like time-varying delays and various combinations of quantisation and overflow nonlinearities. A linear matrix inequality-based delay-dependent criterion for the global asymptotic stability of such systems is proposed. An example is given to illustrate the effectiveness of the proposed method.     

Delay-Probability-Distribution-Dependent Stability of Uncertain Stochastic Genetic Regulatory Networks with Time-Varying Delays

This paper investigates the delay-probability-distribution-dependent stability problem of uncertain stochastic genetic regulatory networks (SGRNs) with time-varying delays. The information of the probability distribution of the time-delay is considered and transformed into parameter matrices of the transferred SGRNs model. Based on the Lyapunov–Krasovskii functional and a stochastic analysis approach, a delay-probability-distribution-dependent sufficient condition is obtained in the linear matrix inequality (LMI) form such that delayed SGRNs are robustly globally asymptotically stable in the mean square for all admissible uncertainties. Three numerical examples are given to illustrate the effectiveness of our theoretical results.     

Mean Square Exponential Stability for Uncertain Delayed Stochastic Neural Networks with Markovian Jump Parameters

This paper is concerned with the problem of delay-dependent mean square exponential stability for a class of delayed stochastic Hopfield neural networks with Markovian jump parameters. The delays here are time-varying delays. Based on a new Lyapunov–Krasovskii functional, delay-dependent stability conditions are derived by means of linear matrix inequalities (LMIs). It is shown that the proposed results can contain some existing stability conditions as a special case. Finally, three numerical examples are given to illustrate the effectiveness of the proposed method, and the simulations show that our results are less conservative than the existing ones.     

On Designing Time-Varying Delay Feedback Controllers for Master–Slave Synchronization of Lur'e Systems

This paper is concerned with the problem of designing time-varying delay feedback controllers for master-slave synchronization of Lur'e systems. Two cases of time-varying delays are fully considered; one is the time-varying delay being continuous uniformly bounded while the other is the time-varying delay being differentiable uniformly bounded with the derivative of the delay bounded by a constant. Based on Lyapunov-Krasovskii functional approach, some delay-dependent synchronization criteria are first obtained and formulated in the form of linear matrix inequalities (LMIs). The relationship between synchronization criteria for the two cases of time-varying delays is built. Then, sufficient conditions on the existence of a time-varying delay feedback controller are derived by employing these newly-obtained synchronization criteria. The controller gains can be achieved by solving a set of LMIs. Finally, Chua's circuit is used to illustrate the effectiveness of the design method.     

LMI Conditions for Robust Stability Analysis of?Stochastic Hopfield Neural Networks with Interval Time-Varying Delays and Linear Fractional Uncertainties

In this paper, the delay-dependent robust stability for a class of stochastic neural networks with linear fractional uncertainties is studied. The time-varying delay is assumed to belong to an interval, which means that the lower and upper bounds of interval time-varying delays are available. Based on the Lyapunov–Krasovskii functional, stochastic stability theory and some inequality techniques, delay-interval dependent stability criteria are obtained in terms of linear matrix inequalities (LMIs). In order to derive less conservative results, few free-weighting matrices are introduced. Three numerical examples are presented to show the effectiveness and improvement of the proposed method.     

Anti-Synchronization of Markovian Jumping Stochastic Chaotic Neural Networks with Mixed Time Delays

The anti-synchronization control is investigated for a class of uncertain stochastic chaotic neural networks with both Markovian jump parameters and mixed delays. The mixed delays consists of discrete and distributed time-varying delays. First, by combining the Lyapunov method and a generalized Halanay-type inequality for stochastic differential equations, a delay-dependent criterion is established to guarantee the state variables of the discussed stochastic chaotic neural networks to be globally exponential anti-synchronized. Next, by utilizing a novel lemma and the Jensen integral inequality, a delay-dependent criterion is proposed to achieve the globally stochastic robust anti-synchronization. With some parameters being fixed in advance, the proposed conditions are all expressed in terms of linear matrix inequalities, which can be solved numerically by employing the standard Matlab LMI toolbox package. Finally, two examples are proposed to demonstrate the effectiveness and usefulness of the proposed results.     

Sliding Mode Control for Linear Systems with Time-Varying Input and State Delays

This paper deals with a sliding mode control for linear systems with input and state time-varying delays. Two different time delays are considered for the sliding mode control of a continuous linear system. Delay-dependent sufficient conditions are given for the existence of sliding manifolds in terms of linear matrix inequalities. A reaching motion controller is also proposed for such systems by means of both reaching law and inequality approach. An inverted pendulum system is utilized to illustrate the effectiveness of the developed techniques.     

Stability and Constrained Control of a Class of Discrete-Time Fuzzy Positive Systems with Time-Varying Delays

This paper deals with the stability of nonlinear discrete-time positive systems with time-varying delays represented by the Takagi–Sugeno (T–S) fuzzy model. The time-varying delays in the systems can be unbounded. Sufficient conditions of stability which are not relevant to the magnitude of delays are derived by a solution trajectory. Based on the stability results, the problems of controller design via the parallel distributed compensation (PDC) scheme are solved. The control is under the positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. Constrained control is also considered, further requiring that the state trajectory of the closed-loop system be bounded by a prescribed boundary if the initial condition is bounded by the same boundary. The stability results and control laws are formulated as linear matrix inequalities (LMIs) and linear programs (LPs). A numerical example and a real plant are studied to demonstrate the application of the proposed methods.     

Stability Analysis of Stochastic Fuzzy Neural Networks with Time-Varying Delays and Reaction–Diffusion Terms

This paper investigates the problem of the global exponential stability for a class of stochastic Takagi–Sugeno fuzzy neural networks (STSFNNs) with time-varying delays and reaction–diffusion terms. Based on the piecewise Lyapunov–Krasovskii functional, Poincaré integral inequality, and Itô differential formula, we obtain some sufficient conditions ensuring the global exponential stability of an equilibrium point for STSFNNs with time-varying delays and reaction–diffusion terms. These sufficient conditions depend on the reaction–diffusion terms and time delays. Finally, some examples are given to show the effectiveness and superiority of the proposed approach.     

Passivity Analysis of Neural Networks With Time-Varying Delays

This brief considers the problem of passivity analysis for neural networks with both time-varying delays and norm-bounded parameter uncertainties. For two types of time-varying delays, delay-dependent passivity conditions are derived, respectively. These conditions are expressed in terms of linear matrix inequalities. An example is provided to show the less conservatism of the proposed conditions.