Multiple Time-varying Delays Research Articles

The existence and uniqueness of positive periodic solutions of Nicholson-type delay systems

This paper is concerned with a class of Nicholson blowfly systems with multiple time-varying delays. By applying the method of the Lyapunov functional, some criteria are established for the existence and uniqueness of positive periodic solutions of the system. Moreover, an example is given to illustrate the main results.

Convergence and permanence of a delayed Nicholson’s Blowflies model with feedback control

In this paper, we study a generalized Nicholson’s Blowflies model with feedback control and multiple time-varying delays. Under proper conditions, we employ a novel proof to establish some criteria to guarantee the global exponential convergence and permanence of this model. Moreover, we give two examples to illustrate our main results.

Robust stability analysis of switched uncertain systems with multiple time-varying delays and actuator failures

In this paper, the robust stability issue of switched uncertain multidelay systems resulting from actuator failures is considered. Based on the average dwell time approach, a set of suitable switching signals is designed by using the total activation time ratio between the stable subsystem and the unstable one. It is first proven that the resulting closed-loop system is robustly exponentially stable for some allowable upper bound of delays if the nominal system with zero delay is exponentially stable under these switching laws. Particularly, the maximal upper bound of delays can be obtained from the linear matrix inequalities. At last, the effectiveness of the proposed method is demonstrated by a simulation example.

Exponential convergence for a class of Nicholson’s blowflies model with multiple time-varying delays

This paper is concerned with the generalized Nicholson’s blowflies model with multiple time-varying delays which is defined on the nonnegative function space. By using some differential inequalities, some new results are established to ensure that all solutions of the model converge exponentially to zero equilibrium point. Moreover, an example is given to illustrate our main results.

Global exponential stability of interval general BAM neural networks with reaction–diffusion terms and multiple time-varying delays

In this paper, we first discuss the existence and uniqueness of the equilibrium point of interval general BAM neural networks with reaction-diffusion terms and multiple time-varying delays by means of using degree theory. Then by applying the existence result of an equilibrium point and constructing a Lyapunov functional, we discuss global exponential stability for above neural networks. In the last section, we also give an example to demonstrate the validity of our global exponential stability result for above neural network.

Positive almost periodic solution for a class of Lasota–Wazewska model with multiple time-varying delays

In this paper, we study the existence and global exponential convergence of positive almost periodic solutions for the generalized Lasota–Wazewska model with multiple time-varying delays. Under proper conditions, we employ a novel proof to establish some criteria to ensure that all solutions of this model converge exponentially to a positive almost periodic solution. Moreover, we give an example to illustrate our main results.

Output-Feedback Stabilization of Discrete-Time Systems with Multiple Time-Varying Delays

Output-Feedback Stabilization of Discrete-Time Systems with Multiple Time-Varying Delays

Synchronization for general complex dynamical networks with sampled-data

In this paper, the sampled-data synchronization control problem is investigated for a class of general complex networks with time-varying coupling delays. A rather general sector-like nonlinear function is used to describe the nonlinearities existing in the network. By using the method of converting the sampling period into a bounded time-varying delay, the addressed problem is first transformed to the problem of stability analysis for a differential equation with multiple time-varying delays. Then, by constructing a Lyapunov functional and using Jensen's inequality, a sufficient condition is derived to ensure the exponential stability of the resulting delayed differential equation. Based on that, the desired sampled-data feedback controllers are designed in terms of the solution to certain linear matrix inequalities (LMIs) that can be solved effectively by using available software. Finally, a numerical simulation example is exploited to demonstrate the effectiveness of the proposed sampled-data control scheme.

H ∞ Guaranteed Cost Filtering for Uncertain Discrete-Time Switched Systems With Multiple Time-Varying Delays

This paper deals with the problem of H∞ guaranteed cost filtering for uncertain discrete-time switched systems with multiple time-varying delays. The switched system under consideration is subject to time-varying norm-bounded parameter uncertainties in all the system matrices. The aim is to design a filter, which guarantees the asymptotical stability of the error system with prescribed disturbance attenuation for all admissible uncertainties and the cost function value is not more than a specified upper bound. By resorting to a switched Lyapunov function, some delay-dependent sufficient conditions are presented in terms of linear matrix inequalities. A numerical example is provided to demonstrate the effectiveness of the proposed algorithms.

Positive almost periodic solution for a class of Nicholson’s blowflies model with multiple time-varying delays

In this paper, we study the existence and exponential convergence of positive almost periodic solutions for the generalized Nicholson’s blowflies model with multiple time-varying delays. Under proper conditions, we establish some criteria to ensure that the solutions of this model converge locally exponentially to a positive almost periodic solution. Moreover, we give some examples to illustrate our main results.

Approximation-Based Adaptive Tracking Control of Pure-Feedback Nonlinear Systems With Multiple Unknown Time-Varying Delays

This paper presents adaptive neural tracking control for a class of non-affine pure-feedback systems with multiple unknown state time-varying delays. To overcome the design difficulty from non-affine structure of pure-feedback system, mean value theorem is exploited to deduce affine appearance of state variables x(i) as virtual controls α(i), and of the actual control u. The separation technique is introduced to decompose unknown functions of all time-varying delayed states into a series of continuous functions of each delayed state. The novel Lyapunov-Krasovskii functionals are employed to compensate for the unknown functions of current delayed state, which is effectively free from any restriction on unknown time-delay functions and overcomes the circular construction of controller caused by the neural approximation of a function of u and [Formula: see text] . Novel continuous functions are introduced to overcome the design difficulty deduced from the use of one adaptive parameter. To achieve uniformly ultimate boundedness of all the signals in the closed-loop system and tracking performance, control gains are effectively modified as a dynamic form with a class of even function, which makes stability analysis be carried out at the present of multiple time-varying delays. Simulation studies are provided to demonstrate the effectiveness of the proposed scheme.

Adaptive robust control of uncertain dynamical systems with multiple time-varying delays

The problem of adaptive robust stabilisation is considered for a class of dynamical systems with multiple time-varying delayed state perturbations, time-varying uncertain parameters, and external disturbances. It is assumed that the upper bounds of the delayed state perturbations, uncertainties and external disturbances are unknown, and that the time-varying delays are any non-negative continuous and bounded functions. In particular, it is not required that the derivatives of time-varying delays have to be less than one. For such a class of uncertain time-delay systems, a new method is presented whereby a class of memoryless continuous adaptive robust state feedback controllers is proposed. By employing a quasi-Lyapunov function, it is shown that the solutions of uncertain time-delay systems can be guaranteed to be uniformly exponentially convergent towards a ball which can be as small as desired. In addition, since the proposed adaptive robust state feedback controllers are completely independent of time delays, the results obtained in the study may be also applicable to a class of dynamical systems with uncertain time delays. Finally, a numerical example is given to demonstrate the validity of the results.

Leader-following consensus of second-order agents with multiple time-varying delays

In this paper, a leader-following consensus problem of second-order multi-agent systems with fixed and switching topologies as well as non-uniform time-varying delays is considered. For the case of fixed topology, a necessary and sufficient condition is obtained. For the case of switching topology, a sufficient condition is obtained under the assumption that the total period over which the leader is globally reachable is sufficiently large. We not only prove that a consensus is reachable asymptotically but also give an estimation of the convergence rate. An example with simulation is presented to illustrate the theoretical results.

Tracking for nonlinear plants with multiple unknown time‐varying state delays using sliding mode with adaptation

AbstractIn this paper, we develop a sliding mode model reference adaptive control (MRAC) scheme for a class of nonlinear dynamic systems with multiple time‐varying state delays, which is robust with respect to unknown plant delays, to a nonlinear perturbation, and to an external disturbance with unknown bounds. An appropriate Lyapunov–Krasovskii‐type functional is introduced to design the adaptation algorithms, and to prove stability. Copyright © 2009 John Wiley & Sons, Ltd.

Comments on "A Multichannel IOS Small Gain Theorem for Systems With Multiple Time-Varying Communication Delays

The small-gain condition presented by Polushin may be replaced by a strictly weaker one to obtain essentially the same result. The necessary minor modifications of the proof are given. Using essentially the same arguments, a global version of the result is also presented.

Delay-dependent robust stabilization for uncertain singular systems with multiple time-varying state delays

In this note, the problem of delay-dependent robust stabilization for singular systems with multiple time-varying state delays has been investigated, and the problem is solved via state feedback controller in terms of a linear matrix inequality technique. Numerical examples are given to show the validity of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Delay-dependent approach to robust H ∞ filtering for discrete-time singular systems with multiple time-varying delays and polytopic uncertainties

A delay-dependent approach to robust H∞ filtering is proposed for discrete-time singular systems with multiple time-varying delays and polytopic uncertainties. The delays are interval time-varying delays and uncertainties are a convex compact set of polytopic types. The aim of the designed filter is to guarantee regular, causal, asymptotic stability with H∞ norm bound of filtering error singular systems. By establishing a finite sum inequality based on quadratic terms, a new delay-dependent bounded lemma (BL) for singular systems with interval time-varying delays is proposed. Based on the result, the existence condition of the robust H∞ filter and the filter design method are presented in terms of linear matrix inequality (LMI). Finally, two examples are also given to illustrate the effectiveness of the proposed methodology.

Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple time-varying delays

In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the stability analysis for stochastic cellular neural networks with multiple time-varying delays using linear matrix inequality (LMI) theory. A novel LMI-based stability criterion is derived to guarantee the asymptotic stability of stochastic cellular neural networks with multiple time-varying delays which are represented by T–S fuzzy models. In order to derive delay-dependent stability conditions, free-weighting matrices method has been introduced, which may develop less-conservative results. In fact, these techniques lead to generalized and less-conservative stability condition that guarantee the wide stability region. Our results can be specialized to several cases including those studied extensively in the literature. Finally, numerical examples are given to demonstrate the effectiveness and conservativeness of our results.

Robust integral sliding mode control for uncertain systems with multiple time delays

This paper proposes a robust integral sliding mode (RISM) manifold and the corresponding stabilization control law for uncertain systems with multiple time-varying time delays based on the techniques of linear matrix inequalities (LMI). The sufficient condition for the existence of the RISM manifold is given in terms of LMI, and then, the sliding mode control (SMC) law that can keep the system state on the RISM manifold from the initial time moment is developed. The efficiency and feasibility of the results are illustrated by a numerical example.

Robust $H_{\infty}$ Filtering for a Class of Nonlinear Networked Systems With Multiple Stochastic Communication Delays and Packet Dropouts

In this paper, the robust H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> filtering problem is studied for a class of uncertain nonlinear networked systems with both multiple stochastic time-varying communication delays and multiple packet dropouts. A sequence of random variables, all of which are mutually independent but obey Bernoulli distribution, are introduced to account for the randomly occurred communication delays. The packet dropout phenomenon occurs in a random way and the occurrence probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in the interval. The discrete-time system under consideration is also subject to parameter uncertainties, state-dependent stochastic disturbances and sector-bounded nonlinearities. We aim to design a linear full-order filter such that the estimation error converges to zero exponentially in the mean square while the disturbance rejection attenuation is constrained to a give level by means of the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance index. Intensive stochastic analysis is carried out to obtain sufficient conditions for ensuring the exponential stability as well as prescribed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance for the overall filtering error dynamics, in the presence of random delays, random dropouts, nonlinearities, and the parameter uncertainties. These conditions are characterized in terms of the feasibility of a set of linear matrix inequalities (LMIs), and then the explicit expression is given for the desired filter parameters. Simulation results are employed to demonstrate the effectiveness of the proposed filter design technique in this paper.