Multiple Time-varying Delays Research Articles

Global Stability of Bidirectional Associative Memory Neural Networks With Multiple Time-Varying Delays.

This article investigates the global stability of bidirectional associative memory neural networks with discrete and distributed time-varying delays (DBAMNNs). By employing the comparison strategy and inequality techniques, global asymptotic stability (GAS) and global exponential stability (GES) of the underlying DBAMNNs are of concern in terms of p -norm ( p ≥ 2 ). Meanwhile, GES of the addressed DBAMNNs is also analyzed in terms of 1-norm. When distributed time delay is neglected, the GES of the corresponding bidirectional associative memory neural networks is presented as an M -matrix, which includes certain existing outcomes as special cases. Two examples are finally provided to substantiate the validity of theories.

Stability Analysis Method for Periodic Delay Differential Equations with Multiple Distributed and Time-Varying Delays

Dynamic stability problems leading to delay differential equations (DDEs) are found in many different fields of science and engineering. In this paper, a method for stability analysis of periodic DDEs with multiple distributed and time-varying delays is proposed, based on the well-known semidiscretization method. In order to verify the correctness of the proposed method, two typical application examples, i.e., milling process with a variable helix cutter and milling process with variable spindle speed, which can be, respectively, described by DDEs with the multidistributed and time-varying delays are considered. Then, comparisons with prior methods for stability prediction are made to verify the accuracy and efficiency of the proposed approach. As far as the milling process is concerned, the proposed method supplies a generalized algorithm to analyze the stability of the single milling systems associated with variable pith cutter, variable helix cutter, or variable spindle speed; it also can be utilized to analyze the combined systems of the aforementioned cases.

Impulsive Synchronization of Derivative Coupled Neural Networks With Cluster-Tree Topology

This article is devoted to discussing the exponential synchronization for a kind of delay derivative coupled neural networks with stochastic disturbance and multiple time-varying delays. To simulate more practical situations and widen the synchronization application fields in network science, the coupled neural networks with cluster-tree topology structure is studied by applying a novel impulsive pinning control strategy, which skillfully considered the neural networks in current cluster that directly linked to the neural networks in other clusters. Since the existence of delayed impulses, the general comparison principle for normal impulsive differential equations is efficiently extended. In view of the concept of average impulsive interval, the parameters classification discussion method and the mathematical induction method, some judgement conditions for achievement of the cluster synchronization on derivative coupled neural networks are derived. Additionally, the exponential convergence velocity of the derivative coupled neural networks is accurately estimated. Finally, numerical examples are presented to demonstrate the effectiveness of the control strategy and the theoretical results.

On stability and stabilization of T-S fuzzy systems with multiple random variables dependent time-varying delay

This paper is concerned with the problems of stability and stabilization for Takagi-Sugeno (T-S) fuzzy systems with multiple random variables dependent time-varying delay. Different from the previous works, we assume that probability distributions of delay in N intervals can be observed in advance. (N-1) Bernoulli distributed random variables are utilized to indicate which interval the time-varying delay falls into at a certain time instant. Then the original T-S fuzzy systems are transformed into a new model of T-S fuzzy systems with multiple random variables dependent time-varying delay, which includes the existed ones as its special cases. Based on the new model, an appropriate Lyapunov-Krasovskii (L-K) functional is constructed. Generalized Finsler’s lemma is introduced to avoid directly dealing with the interrelationship between these correlated random variables, and Reciprocally convex inequality is used to estimate integral terms from the infinitesimal operator of L-K functional. Less conservative stability criteria and stabilization conditions are proposed in the form of linear matrix inequalities (LMIs). Finally, two examples are given to demonstrate the effectiveness of the proposed methods.

Reliable H∞ guaranteed cost control for uncertain switched fuzzy stochastic systems with multiple time-varying delays and intermittent actuator and sensor faults

In this paper, the issue of delay-dependent passive fault-tolerant control and optimal guaranteed cost control is considered for multiple time-varying delayed switched Takagi–Sugeno (T–S) fuzzy stochastic systems against intermittent actuator and sensor faults, model uncertainties, external disturbances and measurement noise. This is the first attempt to implement H∞ guaranteed cost control for switched T–S fuzzy stochastic systems with intermittent actuator and sensor faults which allows more kinds of faults. Thus, the method has a wider application range. A dynamic output feedback controller is designed. Then, a closed-loop system is constructed. Stability results of reliable H∞ guaranteed cost control are given via piecewise fuzzy Lyapunov function in terms of linear matrix inequalities. Slack matrices are not required at all. The conservatism of this paper is relatively low. At last, the effectiveness of the addressed approach is presented by explanatory examples.

A nonsingular M-matrix-based global exponential stability analysis of higher-order delayed discrete-time Cohen–Grossberg neural networks

Abstract This paper focuses on the problem of global exponential stability analysis for high-order delayed discrete-time Cohen–Grossberg neural networks. Multiple time-varying delays are considered. First, a technique lemma is obtained based on the properties of nonsingular M-matrices. Second, the delay-dependent and -independent criteria under which the zero equilibrium is globally exponentially stable are derived, respectively. Last, the validity of these criteria are illustrated by a pair of numerical examples. Compared with the previous results, the merits of the proposed method are: (i) no Lyapunov–Krasovskii functional or auxiliary function is required; (ii) less computational complexity is verified; and (iii) the obtained stability criteria can easily be realized, since they are to test whether a matrix is nonsingular M-matrix.

Stability analysis of sampled-data control systems with multiple time-varying delays

This paper deals with the stability analysis of sampled-data control systems with a centralized controller and multiple actuators, where different controller-actuator delays exist. Via the input-delay approach, a sampled-data control system with multiple time-varying delays is modeled as a linear time-delay system. Though some existing criteria for time-delay systems can be applied to analyze the stability of the original system, the results are conservative since these criteria do not consider the sampling effects properly. To reduce the conservatism, a new looped-functional, which takes advantage of the hybrid characteristic of the sampled-data control system model, is proposed. Meanwhile, a new Lyapunov-Krasovskii functional that aims at fully benefiting from the Wirtinger-based inequality and utilizing the relationship information among the states associated with different delays is constructed. Based on the proposed functionals, a new stability condition for sampled-data control systems with multiple time-varying delays is presented in the form of linear matrix inequalities. Numerical examples are given to illustrate the advantages and effectiveness of the proposed method.

Controllability of fractional-order damped systems with time-varying delays in control

In this study, we focus on the controllability of fractional-order damped systems in linear and nonlinear cases with multiple time-varying delays in control. For the linear system based on the Mittag-Leffler matrix function, we define a controllability Gramian matrix, which is useful in judging whether the system is controllable or not. Furthermore, in two special cases, we present serval equivalent controllable conditions which are easy to verify. For the nonlinear system, under the controllability of its corresponding linear system, we obtain a sufficient condition on the nonlinear term to ensure that the system is controllable. Finally, two examples are given to illustrate the theory.

Observer and fault‐tolerant controller design for discrete‐time multiple state‐delayed T–S fuzzy systems

This study is concerned with observer-based fault estimation (FE) and fault-tolerant controller design for a class of discrete-time Takagi–Sugeno (T–S) fuzzy systems. There exist multiple time-varying state delays, sensor and actuator faults, local non-linear dynamics and exogenous disturbances in the systems. In comparison with the existing results, the approach suggested in this study is more flexible and feasible. By means of the n-step induction FE, a novel fuzzy adaptive descriptor observer is developed to obtain the n-step error functions. Then, an active dynamic output feedback fault-tolerant controller is designed to stabilise the closed-loop fuzzy system. Furthermore, a set of delay-dependent sufficient conditions are provided by the fuzzy Lyapunov function which utilises the form of linear matrix inequalities. The stability results from the observer and the controller in this study have less conservatism compared with the ones from the existence of observers and fault-tolerant controllers. At last, a simulation example is presented to demonstrate the advantages and effectiveness of the approach proposed in the study.

New periodic stability for a class of Nicholson's blowflies models with multiple different delays

In this research work, the existence and global exponential stability of the positive periodic solutions of Nicholson’s blowflies equation with multiple different time-varying delays are investigated. Without adopting the restriction assumption on the maximum rate of reproduction function, we establish a criterion to ensure the existence and global exponential stability of the positive periodic solutions for the addressed equation by using differential inequality techniques, the fluctuation lemma and Lyapunov functional method. The obtained result is new, which improves and complements some existing ones. Moreover, the effectiveness of the theoretical finding is illustrated by an example with numerical simulation.

Improved delay-dependent stability criteria for linear systems with multiple time-varying delays

This paper concentrates on the stability analysis of continuous-time linear systems with multiple time-varying delays. The time-delays are assumed to be bounded and have or not lower and upper bounds on their time-derivative. Based on the Lyapunov-Krasovskii theory the proposed results formulated as linear matrix inequality conditions can verify the delay-intervals for which the system stability is preserved. The main results follow from developing new augmented Lyapunov-Krasovskii functionals that exploit the benefits of the Bessel-Legendre-based integral inequality and the reciprocally convex approach. Also, the proposed functionals allow us to take advantage of the relationships between the current and delayed states. As a result, the proposed stability conditions outperformed similar existing ones in all numerical tests accomplished in this paper.

State feedback observer-based control design for linear descriptor systems with multiple time-varying delays

Abstract In this paper, we propose an linear matrix inequality (LMI)-based design method to observer-based control problem of linear descriptor systems with multiple time-varying delays. The delay function can be continuous and bounded but not necessarily differentiable. First, by introducing a new set of improved Lyapunov–Krasovskii functionals that avoid calculating the derivative of the delay function, we obtain new delay-dependent sufficient conditions for guaranteeing the system to be regular, impulse-free and asymptotically stable. Then, based on the derived stability conditions, we design state feedback controllers and observer gains via LMIs, which can be solved numerically in standard computational algorithms. A numerical example with simulation is given to demonstrate the efficiency and validity of the proposed deign.

Multiple intermittent fault estimation and tolerant control for switched T-S fuzzy stochastic systems with multiple time-varying delays

This paper focuses on the observer-based fault estimation (FE) and active dynamic output feedback controller design for switched Takagi–Sugeno (T-S) fuzzy Itô stochastic systems against multiple time-varying delays as well as intermittent actuator faults and sensor faults. External disturbances are also considered. First, a novel descriptor sliding mode observer (SMO) is investigated to establish the error dynamics. Compared with the existing results, the proposed observer could be utilized to more cases and applied to a wider scope. Next, in light of the online FE information, a fault tolerant controller is suggested to make the closed-loop system mean-square exponentially stable. Furthermore, a novel piecewise fuzzy Lyapunov function is depicted and then the delay dependent sufficient conditions can be got by linear matrix inequalities (LMIs). The designed SMO has less conservatism than the existing ones. The reachability of the sliding mode surface in the estimation error space is assured under the adopted method. And the finite-time boundedness (FTB) issue is discussed. At last, a simulation example is acquired to validate the novelty and validity of the method proposed in the paper.

Robust H∞ dynamic output feedback control for interval type-2 T-S fuzzy multiple time-varying delays systems with external disturbance

This research focuses on the H∞ stability analysis and fuzzy dynamic output feedback control for a class of interval type-2 T-S fuzzy systems. Interval type-2 T-S fuzzy model is employed to approximate the uncertain system with multiple time-varying delays and external disturbance. Interval type-2 T-S fuzzy dynamic output feedback controller is designed by employing the interval type-2 T-S fuzzy model. Some stability criterions in terms of linear matrix inequalities (LMIs) are presented by introducing the fuzzy Lyapunov function. The closed-loop system is described as a descriptor system by employing the descriptor redundancy approach. Some less conservative stability criterions un-including the product terms between system gains and controller gains are presented and the fuzzy-basis-dependent matrix with system gains transformation is introduced. The stability criterions are extended to a finite set of LMIs constraints. Finally, simulation examples and comparison figures are presented to show the effectiveness of the proposed approach.

Fault Estimation and Tolerant Control for Discrete-Time Multiple Delayed Fuzzy Stochastic Systems With Intermittent Sensor and Actuator Faults

This article is concerned with observer-based fault estimation (FE) and tolerant controller design for a series of discrete-time Takagi-Sugeno (T-S) fuzzy stochastic systems. There exist multiple time-varying state delays, intermittent sensor and actuator faults, nonlinear dynamics, and exogenous disturbances in the systems. Compared with the results of the existence, this approach suggested in this article is more flexible and feasible. By means of the FE information, a novel fuzzy adaptive descriptor observer is developed to obtain the error dynamics. Then, an active observer-based fault-tolerant controller is designed to stabilize the closed-loop fuzzy system. Furthermore, a set of delay-dependent sufficient conditions are provided by the fuzzy Lyapunov function with the way of linear matrix inequalities (LMIs), which has less conservatism compared with the ones of the existing observers and fault-tolerant controllers. Finally, a simulation example is shown to illustrate the advantages and effectiveness of this approach depicted in this article.

More general results of aperiodically intermittent synchronization for stochastic Markovian switching complex networks with multi-links and time-varying coupling structure

The paper is concerned with the pth moment exponential synchronization for stochastic Markovian switching complex networks with multi-links via aperiodically intermittent control. Different from the current literature on multi-links systems, the networks we consider have time-varying topology structure. New differential inequalities are established in which the Markovian switching and multiple time-varying delays are taken into account and the constraint on parameter that determines exponential convergence rate is reduced. Based on Lyapunov method, graph theory and differential inequality techniques, some sufficient conditions are obtained to achieve exponential synchronization. As a result, the developed synchronization criteria have more accurate exponential convergence rate than that in most existing literature which focuses on delayed systems with intermittent control. Finally, an application to modified Chua’s circuits and corresponding numerical simulations are presented to demonstrate the effectiveness of the proposed theoretical results.

Delay-Dependent $H_\infty$ Guaranteed Cost Control for Uncertain Switched T–S Fuzzy Systems With Multiple Interval Time-Varying Delays

This article deals with delay-dependent H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> guaranteed cost control for uncertain switched Takagi-Sugeno (T-S) fuzzy systems with multiple interval time-varying delays. It is the first time that the fault tolerant control (FTC) of nonlinear systems subject to measurement noise, external disturbances, nonlinear functions, and intermittent faults (IFs) in sensors and actuators is investigated. A passive dynamic full-order output feedback controller is first addressed for the system to guarantee the exponential stability. Compared with the existing results, the restrictions on the multiple time-varying delays are relaxed in the derivation of this work. The suggested controller has a wider application. Moreover, as the piecewise fuzzy Lyapunov function is introduced with some slack matrices, the delay-dependent sufficient conditions are provided in the way of a cluster of linear matrix inequalities (LMIs) and then less conservatism is attained by our design contrast to the existing fault tolerant controllers. At last, two simulation examples and some comparisons are achieved to illustrate the novelty and feasibility of the method gathered in the article.

New results on global exponential stability for a periodic Nicholson\u2019s blowflies model involving time-varying delays

This paper investigates a periodic Nicholson’s blowflies equation with multiple time-varying delays. By using differential inequality techniques and the fluctuation lemma, we establish a criterion to ensure the global exponential stability on the positive solutions of the addressed equation, which improves and complements some existing ones. The effectiveness of the obtained result is illustrated by some numerical simulations.

Global Attractivity for Lasota–Wazewska-Type System with Patch Structure and Multiple Time-Varying Delays

This paper aims to study the asymptotic behavior of Lasota–Wazewska-type system with patch structure and multiple time-varying delays. Based on the fluctuation lemma and some differential inequality techniques, we prove that the positive equilibrium is a global attractor of the addressed system with small time delay. Finally, we provide an example to illustrate the feasibility of the theoretical results.

Positivity and Stability Analysis for Fractional-Order Delayed Systems: A T-S Fuzzy Model Approach

This article investigates positivity, external positivity, and asymptotic stability for a large class of incommensurate fractional-order nonlinear systems (FONSs) with bounded multiple time-varying delays by virtue of the T-S fuzzy method. The Laplace transformation technique is used to obtain the solutions of T-S fuzzy FONSs. A sufficient and necessary condition is derived for characterizing (internal) positivity, and certain criteria are also provided to guarantee external positivity of FONSs with or without time-varying delays. It is indicated that the positivity of the considered systems is determined purely by system matrices rather than the magnitudes of time-varying delays. Moreover, a sufficient and necessary condition for asymptotic stability of positive FONSs is obtained, and a state-feedback controller is also designed to guarantee that state variables not only converge to the origin asymptotically but also remain nonnegative, where the control gain matrix is obtained by solving an linear programming (LP) problem. Three numerical simulation examples are given to expound validity and feasibility of the theoretical results.