Finite Time Delays Research Articles

Network Synchronization Revisited: Time Delays in Mutually Coupled Oscillators

Coordinated and efficient operation in large, complex systems requires the synchronization of the rhythms of spatially distributed components. Such systems are the basis for critical infrastructure such as satellite navigation, mobile communications, and services like the precision time protocol and Universal Coordinated Time. Different concepts for the synchronization of oscillator networks have been proposed, in particular mutual synchronization without and hierarchical synchronization from a reference clock. Established network synchronization models in electrical engineering address the role of inevitable cross-coupling time delays for network synchronization. Mutual synchronization has been studied using linear approximations of the coupling functions of these models. We review previous work and present a general model in which we study synchronization taking into account nonlinearities and finite time delays. As a result, dynamical phenomena in networks of coupled electronic oscillators induced by time delays, such as the multistability and stabilization of synchronized states can be predicted and observed. We study the linear stability of nonlinear states and predict for which system parameters synchronized states can be stable. We use these results to discuss the implementation of mutual synchronization for complex system architectures. A key finding is that mutual synchronization can result in stable in- and anti-phase synchronized states in the presence of large time delays. We provide the condition for which such synchronized states are guaranteed to be stable.

Asynchronous Impulsive Protocols With Asymmetric Feedback Saturation on Leader-Based Formation Control of Multiagent Systems.

Saturation phenomena often exist due to limited system resources, and impulsive protocols can lead to a reduction in communication cost. From these issues, this article investigates a leader-based formation control problem of multiagent systems via asynchronous impulsive protocols with saturated feedback. General linear system models with and without finite time-varying time delays under asymmetric saturated feedback control are concurrently considered. The asynchronous impulsive protocols only permit communication at impulsive instants and each agent has its own communication instants independently. Moreover, to improve system performance, an offset only containing desired formation information is introduced. Finally, because the feedbacks are saturated, admissible regions are proved to exist, which are also estimated by a mean of optimization. Numerical simulations are presented to demonstrate the validity of the proposed schemes.

Fiscal policy delays and the classical growth cycle

This paper deals with the impact of fiscal policy delays on the national income adjustment process. Here we reconsidered the pioneering work by Wolfstetter, who introduced the public sector in the well-known Goodwin’s classical growth cycle model, where the conflict between capital and labor on the distribution of income is formalized. Unlike Wolfstetter, we take into account two finite time delays characterizing the public economic activity. The former delay concerns the structure of the tax system and the government tax revenues; the latter pertains the political process governing the public purchase decisions and the actual expenditures. The result is a system of delayed differential equations (DDEs). Choosing delay terms as bifurcation parameters, we proved the existence of Hopf bifurcations. Therefore, we studied the stability and the direction of the bifurcating periodic solutions by using the first Lyapunov coefficient. Some numerical simulations carried out to support theoretical results show that, in the basic model, which coincides with the one by Wolfstetter. The effectiveness of policies (pro-cyclical and counter-cyclical) are strictly dependent on the length of the lags and on their particular combinations. As the basic model lacks an investment function, because investments passively equals the saving, we add that function taking into account the profit expectations. Furthermore, we assumed that the size of the public expenditure decreases quickly with the rise of the employment rate. These new hypotheses are such that to yield an extended model, where, unlike the basic model, we proved that, without lags, a pro-cyclical policy does not assure the stabilization of the economy if the government adopts weak reduction of the public expenditure. In this case, regular cycles around the equilibrium arise. When the lags are positive, the government might stabilize the system only by a low discretional expenditure, if the policy is counter-cyclical, and by low reduction of its expenditure, if the policy is pro-cyclical. This on condition that some particular pairs of the two delays subsist.

Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations

In this paper, we analyze an alcoholism model which involves the impact of Twitter via Liouville–Caputo and Atangana–Baleanu–Caputo fractional derivatives with constant- and variable-order. Two fractional mathematical models are considered, with and without delay. Special solutions using an iterative scheme via Laplace and Sumudu transform were obtained. We studied the uniqueness and existence of the solutions employing the fixed point postulate. The generalized model with variable-order was solved numerically via the Adams method and the Adams–Bashforth–Moulton scheme. Stability and convergence of the numerical solutions were presented in details. Numerical examples of the approximate solutions are provided to show that the numerical methods are computationally efficient. Therefore, by including both the fractional derivatives and finite time delays in the alcoholism model studied, we believe that we have established a more complete and more realistic indicator of alcoholism model and affect the spread of the drinking.

Existence–uniqueness and stability of reaction–diffusion stochastic Hopfield neural networks with S-type distributed time delays

Abstract The aim of this paper is to develop some theories of mild solutions to reaction–diffusion stochastic Hopfield neural networks with S-type distributed time delays driven by infinite dimensional Wiener processes. First, we transform the networks with infinite time delays defined on ( − ∞ , 0 ] into those with finite time delays defined on [ − τ , 0 ] by truncation method. Then, the existence–uniqueness theorem of the networks with finite time delays is established in terms of the generalized Halanay inequality. Next, the existence of equilibrium is demonstrated with homotopy invariance theorem and topological degree theory. And the stability is considered using M-function. Finally, with the help of approximation method, the well-posedness and stability of the original networks are obtained. An interesting example shows the effectiveness of our results.

Fractional Bloch equation with delay

In this paper we investigate a fractional generalization of the Bloch equation that includes both fractional derivatives and time delays. The appearance of the fractional derivative on the left side of the Bloch equation encodes a degree of system memory in the dynamic model for magnetization. The introduction of a time delay on the right side of the equation balances the equation by also adding a degree of system memory on the right side of the equation. The analysis of this system shows different stability behavior for the T 1 and the T 2 relaxation processes. The T 1 decay is stable for the range of delays tested (1–100 μs), while the T 2 relaxation in this model exhibited a critical delay (typically 6 μs) above which the system was unstable. Delays are expected to appear in NMR systems, in both the system model and in the signal excitation and detection processes. Therefore, by including both the fractional derivative and finite time delays in the Bloch equation, we believe that we have established a more complete and more realistic model for NMR resonance and relaxation.

Quantum Feedback Networks: Hamiltonian Formulation

A quantum network is an open system consisting of several component Markovian input-output subsystems interconnected by boson field channels carrying quantum stochastic signals. Generalizing the work of Chebotarev and Gregoratti, we formulate the model description by prescribing a candidate Hamiltonian for the network including details of the component systems, the field channels, their interconnections, interactions and any time delays arising from the geometry of the network. (We show that the candidate is a symmetric operator and proceed modulo the proof of self- adjointness.) The model is non-Markovian for finite time delays, but in the limit where these delays vanish we recover a Markov model and thereby deduce the rules for introducing feedback into arbitrary quantum networks. The type of feedback considered includes that mediated by the use of beam splitters. We are therefore able to give a system-theoretic approach to introducing connections between quantum mechanical state-based input-output systems, and give a unifying treatment using non-commutative fractional linear, or Möbius, transformations.

BOUNDARY LAYER USING DITHERING IN SLIDING MODE CONTROL

Practical issues in the implementation of sliding mode control systems, such as finite switching frequency, time delays and unmodelled dynamics, determine the state to belong to a neighborhood of the sliding manifold (the boundary layer) instead of lying on it. Dithering is here proposed as a technique for the regulation of the boundary layer when the actuator has a natural on-off behaviour. It is proved that dithering maintains the state of the system close to the ideal sliding manifold and that state error and practical stability depend on the dither amplitude, period and shape. Simulations on DC/DC power converters show that dither can play an important role for stability and robustness by using sliding mode control.

OPTIMUM MIXING TANK HEAD REGULATION

The regulation of a connected tank system which is used for mixing dyes, in aqueous solution, is considered. Novel modelling, in matrix impedance (P̄,Q̄,R̄) form, from which the state space and transfer function, admittance, realisations of the system are generated, is proposed. Independent level control of the mixing tanks is investigated. Steady-state non-interaction of the closed-loop system is required, using pneumatically operated valves, each of which has finite time delays. Optimum operating conditions are specified. Minimum closed-loop control effort and almost critically damped response transients are achieved.

Dynamics of Controlled Mechanical Systems with Delayed Feedback

5R15. Dynamics of Controlled Mechanical Systems with Delayed Feedback. - H Hu and Z Wang (Inst of Vib Eng Res, Nanjing Univ of Aeronaut and Astronaut, Nanjing, 210016, PR China). Springer-Verlag, Berlin. 2002. 294 pp. ISBN 3-540-43733-9. $89.95.Reviewed by DB Schaechter (Lockheed Martin, Bldg 201, Org L9-24, 3251 Hanover St, Palo Alto CA 94304-1191).Hu and Wang have compiled an engineering monograph that can be used as a reference for the subject title. Their text is suited for upper division control engineers, graduate students, and control engineering professionals, and presupposes knowledge of control systems, differential and difference equations, analysis, frequency response methods, and linear and nonlinear stability criteria. The text is divided into eight chapters dealing with modeling, fundamentals, stability analyses, periodic motions, chaotic systems, and control of delayed dynamic systems. It consists of good quality figures, a decent index, no problem sets, no appendices, several pages of references, and interesting highly analytical (theorem-proof) technical material solidified occasionally with seemingly contrived examples. Readability suffers slightly in places from lack of grammatical editing. In the text, the authors present a formalized treatment of systems (both linear and nonlinear) containing delays (both single and multiple) and address both the stability and performance aspects of such systems. They present a refreshingly systematic and analytical approach to the subject matter, much more than just the e−sT unity gain, linear phase impact of a constant time delay and its manifestations in linear systems. These techniques include, linear stability theory, Lyapunov stability, approximation techniques for short time delays, stability regions, delay independent stability, generalized Sturm stability, perturbation methods, and periodic behavior. A noticeable omission, perhaps by choice, is treatment at any length of Laplace transforms and how they might be applied to analyzing and designing controllers for such systems. There is a wealth of relevant and detailed information pertaining to systems which encompass time delays, however, the sequence of chapters seems to be somewhat non-intuitive, at times, with occasional meandering away from the chapter topic. For example, Chapter 1, which deals with the modeling of delay dynamic systems, rapidly delves in unwarranted detail into genetic algorithm approaches for characterizing and identifying such systems. This chapter, in turn, is followed by an excellent chapter entitled Fundamentals of Delay Differential Equations, with content that one might expect pedagogically to serve as introductory material for the entire text. The remainder of the text then penetrates the core material of the book, how one characterizes, recognizes, analyzes, approximates, and controls systems in which single or multiple finite time delays play a significant role in the overall system response. All in all, Dynamics of Controlled Mechanical Systems with Delayed Feedback would add depth and breadth to any engineering library and would prove beneficial to control engineers who would like to probe more deeply into some of the more subtle ramifications of the presence of time delays in mechanical systems.

Extreme stability and almost periodicity in continuous and discrete neuronal models with finite delays

We consider the dynamical characteristics of a continuous-time isolated Hopfield-type neuron subjected to an almost periodic external stimulus. The model neuron is assumed to be dissipative having finite time delays in the process of encoding the external input stimulus and recalling the encoded pattern associated with the external stimulus. By using non-autonomous Halanay-type inequalities we obtain sufficient conditions for the hetero-associative stable encoding of temporally non-uniform stimuli. A brief study of a discrete-time model derived from the continuous-time system is given. It is shown that the discrete-time model preserves the stability conditions of the continuous-time system.

Process system regulation with minimum control effort

The regulation of process systems which may be described by linear multi-variable Laplace-transformed models having finite time delays is considered. Conventional frequency response methods are employed to assess relatively stability. A feedback strategy which will minimize the control effort required to confine output changes following disturbance inputs is employed. Typical process system models are investigated using output feedback. Computer-simulated response characteristics are presented.

Time-delay effects in feedback quantum-nondemolition measurements

The effect of time delay on the feedback quantum-nondemolition measurement is analyzed by the use of nonexclusive probability densities. For zero time delay our results are similar to those of others, but for finite time delays new results are predicted.

Almost periodic solutions for delay-differential equations with infinite delays

One of the most widely methods for establishing the existence of almost periodic (a.p. for short) solutions of systems or ordinary differential equations is based on a general result due to Amerio [ 1 J, a generalization of an earlier result for linear a.p. systems due to Favard. In essence, this basic method consists of showing that if a solution is in a compact set in the statespace for all t and has a certain separation property with respect to other such solutions, then if this same property holds for all systems in the socalled hull of the given system, this solution will be a.p. This result in fact relates to a necessary as well as sufficient stability condition for a fairly general dynamical system to have an a.p. solution; cf. [2, Chap. 51. In fact Miller [3] used the idea of embedding the a.p. system in a more general dynamical system and was thus able to obtain a local stability condition for the existence of a.p. solutions without using the separation conditions due to Favard and Amerio. Most stability conditions for the existence of a.p. solutions known prior to Miller’s result were in terms of global stability conditions. For a fairly complete and comprehensive discussion of a.p. solutions for a.p. systems, the books by Fink [4] and Yoshizawa [5] are recommended. For a more general discussion of the idea of embedding a.p. systems in dynamical systems, cf. Sell [6]. For ordinary differential equations with fixed finite time delays, much of the method based on Amerio’s separation result can and has been adapted to obtain similar stability conditions for the existence of a.p. solutions; cf. [7], for example. Miller in [3] shows that local stability conditions can also be used for such delay-differential equations. Results for delay equations with infinite time delays based on Amerio’s separation condition have also been obtained; cf., for example, Hino [B]. However, for such systems which do not involve substantially fading memory, not much seems to have been done; one of the main difficulties in such infinite delay systems arises from the fact that bounded solutions may no longer remain in compact subsets of the state

Stabilizability of production-inventory systems incorporating unfilled-order backlogs with retarded control policies

In this paper it is shown that the method of D-partition, in conjunction with a decoupling state transformation technique, can be used to investigate the stabilizability by linear state feedback of a class of production-inventory systems incorporating unfilled-order backlogs in which the control policies are retarded by finite time-delays. The theory is illustrated by the presentation of typical transient and steady-state response characteristics of controlled production-inventory systems of this class.

Stabilizability of production-inventory systems with retarded control policies

In this paper it is shown that the method of D-partition can be used to investigate the stabilizability by linear state feedback of a class of continuous-time production-inventory systems in which the control policies are retarded by finite time-delays. The theory is illustrated by the presentation of typical transient and steady-state response characteristics of controlled production-inventory systems of this class.