Single Constant Delay Research Articles

On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation

This work is dedicated to the stability analysis of time-delay systems with a single constant delay using the Lyapunov–Krasovskii theorem. This approach has been widely used in the literature and numerous sufficient conditions of stability have been proposed and expressed as linear matrix inequalities (LMI). The main criticism of the method that is often pointed out is that these LMI conditions are only sufficient, and there is a lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel–Legendre inequality or orthogonal polynomial-based inequalities. The interest of these methods relies on their hierarchical structure with a guarantee of reduction of the level of conservatism. However, the convergence is still an open question that will be answered for the first time in this paper. The objective is to prove that the stability of a time-delay system implies the feasibility of these scalable LMI, at a sufficiently large order of the Legendre polynomials. Moreover, the proposed contribution is even able to provide an analytic estimation of this order, giving rise to a necessary and sufficient LMI for the stability of time-delay systems.

Exponential Stability of Highly Nonlinear Hybrid Differently Structured Neutral Stochastic Differential Equations with Unbounded Delays

This paper mainly studies the exponential stability of the highly nonlinear hybrid neutral stochastic differential equations (NSDEs) with multiple unbounded time-dependent delays and different structures. We prove the existence and uniqueness of the exact global solution of the new stochastic system, and then give several criteria of the exponential stability, including the q1th moment and almost surely exponential stability. Additionally, some numerical examples are given to illustrate the main results. Such systems are widely applied in physics and other fields. For example, a specific case is pantograph dynamics, in which the delay term is a proportional function. These are widely used to determine the motion of a pantograph head on an electric locomotive collecting current from an overhead trolley wire. Compared with the existing works, our results extend the single constant delay of coefficients to multiple unbounded time-dependent delays, which is more general and applicable.

Multidelay Differential Equations: A Taylor Expansion Approach

It is already well-understood that many delay differential equations with only a single constant delay exhibit a change in stability according to the value of the delay in relation to a critical delay value. Finding a formula for the critical delay is important to understanding the dynamics of delayed systems and is often simple to obtain when the system only has a single constant delay. However, if we consider a system with multiple constant delays, there is no known way to obtain such a formula that determines for what values of the delays a change in stability occurs. In this paper, we present some single-delay approximations to a multidelay system obtained via a Taylor expansion as well as formulas for their critical delays which are used to approximate where the change in stability occurs in the multidelay system. We determine when our approximations perform well and we give extra analytical and numerical attention to the two-delay and three-delay settings.

Representation of solutions to delayed differential equations with a single delay by dominant and subdominant solutions

The representation and behavior is investigated of the solutions to an equation with a single constant delay ẋ(t)=−c(t)x(t−τ), t≥t0≥0 with c a positive function provided that it has a positive solution. A scheme is suggested for determining the representation of solutions defined by arbitrary initial function if dominant and subdominant solutions are known. Applications are given to equations with constant and perturbed coefficients with open problems included for future investigation.

A Rigorous Implicit $$C^1$$ Chebyshev Integrator for Delay Equations

We present a new approach to validated numerical integration for systems of delay differential equations. We focus on the case of a single constant delay though the method generalizes to systems with multiple lags. The method provides mathematically rigorous existence results as well as error bounds for both the solution and the Frechet derivative of the solution with respect to a given past history segment. We use Chebyshev series to discretize the problem, and solve approximately using a standard numerical scheme corrected via Newton’s method. The existence/error analysis exploits a Newton–Kantorovich argument. We present examples of the rigorous time stepping procedure, and illustrate the use of the method in computer-assisted existence proofs for periodic solutions of the Mackey–Glass equation.

An Efficient Method for Computing Exact Delay-Margins of Large-Scale Power Systems

This letter proposes an efficient method for com-puting the delay-margins of large-scale power systems with single constant delay. The method computes the margin by identifying the minimal delay under which some eigenvalues are about to cross the imaginary axis from the left to the right half complex plane. Based on the Rekasius substitution, a new characterization for the crossing points is developed, followed by a method for determining the delay margins. The method only involves eigen-value decomposition on a low-order matrix whose dimension is independent of the system scale. Further leveraging the sparsity of power systems makes it computationally very attractive for practical large-scale system applications.

A Novel Delay-Dependent Asymptotic Stability Conditions for Differential and Riemann-Liouville Fractional Differential Neutral Systems with Constant Delays and Nonlinear Perturbation

The novel delay-dependent asymptotic stability of a differential and Riemann-Liouville fractional differential neutral system with constant delays and nonlinear perturbation is studied. We describe the new asymptotic stability criterion in the form of linear matrix inequalities (LMIs), using the application of zero equations, model transformation and other inequalities. Then we show the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with constant delays. Furthermore, we not only present the improved delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral system with single constant delay but also the new delay-dependent asymptotic stability criterion of a differential and Riemann-Liouville fractional differential neutral equation with constant delays. Numerical examples are exploited to represent the improvement and capability of results over another research as compared with the least upper bounds of delay and nonlinear perturbation.

A functional analytic approach to validated numerics for eigenvalues of delay equations

This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by \begin{document}$ N \times N $\end{document} matrices for large enough \begin{document}$ N $\end{document} . We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.

Necessary conditions of exponential stability for a class of linear uncertain systems with a single constant delay

In this paper, we will investigate the necessary conditions, described by the Lyapunov matrix, for the robust exponential stability for a class of linear uncertain systems with a single constant delay and time-invariant parametric uncertainties, which are some generalizations of the existing results on uncertain linear time-delay systems. As a medium step, several pivotal properties of parameter-dependent Lyapunov matrix are proposed, which set up the relationships between fundamental matrix and Lyapunov matrix for the considered system. In addition, to calculate the parameter-dependent Lyapunov matrix, we introduce the differential equation method and the Lagrange interpolation method, respectively. Furthermore, it is noted that the proposed necessary conditions can be used to estimate the range of time delay, when the linear uncertain time-delay system is robust exponential stability. Finally, the validity of the obtained theoretical results is illustrated via numerical examples.

Dynamics of a Mathematical Hematopoietic Stem-Cell Population Model

We explore the bifurcations and dynamics of a scalar differential equation with a single constant delay which models the population of human hematopoietic stem cells in the bone marrow. One parameter continuation reveals that with a delay of just a few days, stable periodic dynamics can be generated of all periods from about one week up to one decade! The long period orbits seem to be generated by several mechanisms, one of which is a canard explosion, for which we approximate the dynamics near the slow manifold. Two-parameter continuation reveals parameter regions with even more exotic dynamics including quasi-periodic and phase-locked tori, and chaotic solutions. The panoply of dynamics that we find in the model demonstrates that instability in the stem cell dynamics could be sufficient to generate the rich behaviour seen in dynamic hematological diseases.

Asymptotic properties of delayed matrix exponential functions via Lambert function

In the case of first-order linear systems with single constant delay and with constant matrix, the application of the well-known by step method (when ordinary differential equations with delay are solved) has recently been formalized using a special type matrix, called delayed matrix exponential. This matrix function is defined on the intervals \begin{document} $(k-1)τ≤q t , \begin{document} $k=0,1,\dots$ \end{document} (where \begin{document} $τ>0$ \end{document} is a delay) as different matrix polynomials, and is continuous at nodes \begin{document} $t=kτ$ \end{document} . In the paper, the asymptotic properties of delayed matrix exponential are studied for \begin{document} $k\to∞$ \end{document} and it is, e.g., proved that the sequence of values of a delayed matrix exponential at nodes is approximately represented by a geometric progression. A constant matrix has been found such that its matrix exponential is the quotient factor that depends on the principal branch of the Lambert function. Applications of the results obtained are given as well.

Distributed Position-Based Consensus of Second-Order Multiagent Systems With Continuous/Intermittent Communication.

This paper considers the position-based consensus in a network of agents with double-integrator dynamics and directed topology. Two types of distributed observer algorithms are proposed to solve the consensus problem by utilizing continuous and intermittent position measurements, respectively, where each observer does not interact with any other observers. For the case of continuous communication between network agents, some convergence conditions are derived for reaching consensus in the network with a single constant delay or multiple time-varying delays on the basis of the eigenvalue analysis and the descriptor method. When the network agents can only obtain intermittent position data from local neighbors at discrete time instants, the consensus in the network without time delay or with nonuniform delays is investigated by using the Wirtinger's inequality and the delayed-input approach. Numerical examples are given to illustrate the theoretical analysis.

On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay

A transcendental equation $\lambda + \alpha - \beta\mathrm{e}^{-\lambda\tau} = 0$ with complex coefficients is investigated. This equation can be obtained from the characteristic equation of a linear differential equation with a single constant delay. It is known that the set of roots of this equation can be expressed by the Lambert W function. We analyze the condition on parameters for which all the roots have negative real parts by using the ``graph-like'' expression of the W function. We apply the obtained results to the stabilization of an unstable equilibrium solution by the delayed feedback control and the stability condition of the synchronous state in oscillator networks.

On the Hyers-Ulam Stability of First-Order Impulsive Delay Differential Equations

This paper proves the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of nonlinear first-order ordinary differential equation with single constant delay and finite impulses on a compact interval. Our approach uses abstract Gronwall lemma together with integral inequality of Gronwall type for piecewise continuous functions.

Consensus networks with time-delays over finite fields

ABSTRACTIn this paper, we investigate the consensus problem in networks with time-delays over finite fields. The delays are categorised into three cases: single constant delay, multiple constant delays, and time-varying bounded delays. For all cases, some sufficient and necessary conditions for consensus are derived. Furthermore, assuming that the communication graph is strongly connected, some of the obtained necessary conditions reveal that the conditions for consensus with time-delays over finite fields depend not only on the diagonal entries but also on the off-diagonal entries, something that is intrinsically distinct from the case over real numbers (where having at least one nonzero diagonal entry is a sufficient and necessary condition to guarantee consensus). In addition, it is shown that delayed networks cannot achieve consensus when the interaction graph is a tree if the corresponding delay-free networks cannot reach consensus, which is consistent with the result over real numbers. As for average consensus, we show that it can never be achieved for delayed networks over finite fields, although it indeed can be reached under several conditions for delay-free networks over finite fields. Finally, networks with time-varying delays are discussed and one sufficient condition for consensus is presented by graph-theoretic method.

Controllability of semilinear systems with fixed delay in control

In this paper, different sufficient conditions for exact controllability of semilinear systems with a single constant point delay in control are established in infinite dimensional space. The existence and uniqueness of mild solution is also proved under suitable assumptions. In particular, local Lipschitz continuity of a nonlinear function is used. To illustrate the developed theory some examples are given.

Discussion on “Estimation of linear positive systems with unknown time-varying delays”

Delay systems have attracted a considerable amount of interest δðtnÞ for t≥t, which in practice is unrealistic. Conceptually, this in the control theory community in recent years, due to important methodological and practical issues involved in their analysis and design. New and emergent application areas, such as control of networked systems, are inherently characterized by the presence of time delays [7]. Stability analysis, control design, as well as estimation problems for both linear and nonlinear systems with delays in the input, output or state equations have been studied in an impressive number of papers [8]. Due to the inherent complexity of time-delay systems, it is well known that there is still much to be done even in the relatively simple context of linear timeinvariant systems. An important issue in this area is to achieve results that are general enough with respect to the nature of the delay. The trend of research results with respect to the kind of delay which has moved from the case of a single constant delay to multiple constant delays, time varying continuous delays, timevarying bounded (and non-necessarily continuous) delays. It is likely that the case of unknown delays is the next important step forward, and for this reason, the paper [2] of Rami, Jordan and Schonlein is particularly relevant, since it is one of the first attempts to provide a methodological framework in which this difficult problem can be casted and solved. I will briefly discuss three issues, namely (i) the importance of the contribution, (ii) its limitations and (iii) the points that in my opinion can be further developed. Why unknown delays are important. Consider the problem of controlling a linear system affected by time varying delays in the actuator or in the measurement process. It is reasonable to assume that in practice the delay function δðtÞ on the output is bounded and known at time t, since this can be achieved through the timestamps associated to the measurement. If the delay is due to the communication on a shared network, the delay function may be not continuous, but it is still known and bounded. Interestingly, however, when the delay affects the input the knowledge of δðtÞ at time t is not sufficient to generate a feedback with a delay dependent gain, since the controller would not know which delay affects the input signal generated at time t. This delay is δðtnÞ, with t 1⁄4 tn−δðtnÞ, and therefore t≥t (notice that t might also be not unique). Thus, in the case of delayed input it is necessary to know

On the Computation of the Third Order Terms of the Series Defining the Center Manifold for a Scalar Delay Differential Equation

When computing the third order terms of the series of powers of the function whose graph is the center manifold, at an equilibrium point of a scalar delay differential equation with a single constant delay r > 0, some problems occur at the term \({w_{2,1}z^2\overline{z}.}\) More precisely, in order to determine the values at 0, respectively −r of the function w 2,1(.), an algebraic system of equations must be solved. We show that the two equations are dependent, hence the system has an infinity of solutions. Then we show how we can overcome this lack of uniqueness and provide a formula for w 2,1(0).

Model approximation of multiple delay transfer function models using multiple-point step response fitting

Compared with model approximation for general SISO transfer function models with or without a single constant delay, the approximation problem for SISO transfer function models with multiple delays has received much less attention. In this paper, we attack this problem and thus present a multiple-point step response fitting based approximation method to derive the reduced models. A simple frequency-domain weighted recursive least squares (RLS) algorithm is proposed to determine a set of parameters so that the reduced models can approximate the original models by minimizing the defined frequency-domain squared-error between the step responses of the original and the reduced model. Numerical examples have demonstrated that the proposed approximation approach can not only introduce less dynamic approximation error, but also yield zero steady-state error.

H ∞ filtering for discrete-time systems with time-varying delay

The problem of H ∞ filtering for discrete-time systems with time-varying delay in measurement is investigated in this paper. First, under the assumption that the time-varying delay is of a known upper bound, the delayed measurement is re-described as the one with multiple state delays. Then the proposed H ∞ filtering problem is transformed into one for systems with multiple measurement channels that contain the same state information as the original measurement and each channel has a single constant delay. Finally, based on the reorganized innovation analysis approach in Krein space, a necessary and sufficient condition for the existence of an H ∞ filter which guarantees a prescribed attenuation level is derived. The solution to the H ∞ filtering is given in terms of the solutions to Riccati and matrix difference equations.