Presence Of Time Delay Research Articles

Variable-step [formula omitted] method combined with time two-grid algorithm for multi-singularity problems arising from two-dimensional nonlinear delay fractional equations

In this paper, we focus on the numerical simulation for two-dimensional nonlinear fractional sub-diffusion equations in the presence of time delay. Firstly, we investigate the existence, uniqueness and regularity of the solution for such problems. The theoretical result implies that the solution at τ+ is smoother than that at 0+, where τ is a constant time delay, and this is an improvement for the work (Tan et al., 2022). Secondly, a high-order difference scheme based on L1 method is constructed. For the sake of repairing the convergence order in temporal direction and improving the computational efficiency, an efficient time two-grid algorithm based on nonuniform meshes is first developed. The convergence order of the two-grid scheme reaches O(NF−min{rα,2−α}+NC−min{2rα,4−2α}+h12+h22), where NF and NC represent the number of the fine and coarse grids respectively, while h1 and h2 are the space-step sizes. Furthermore, stability and convergence analysis of the proposed scheme are carefully verified by energy method. Finally, numerical experiments are carried out to show the validity of theoretical statements.

Stability analysis of an atherosclerotic plaque formation model with time delay

Atherosclerosis is a chronic inflammatory disease that poses a serious threat to human health. It starts with the buildup of plaque in the artery wall, which results from the accumulation of pro‐inflammatory factors and other substances. In this paper, we propose a mathematical model of early atherosclerosis with a free boundary and time delay. The time delay represents the time required for macrophages to transit to foam cells through cholesterol accumulation. We obtain an explicit solution and analyze the stability of the model and the effect of the time delay on plaque size. We show that in the form of perturbation (where represents the mode of angle), when or 1, the steady‐state solution is linearly stable; when , there exists a critical parameter such that the steady‐state solution is linearly stable for and unstable for . Moreover, we find that smaller plaque are associated with the presence of time delay.

Stability of twisted states in power-law-coupled Kuramoto oscillators on a circle with and without time delay.

Other than the fully synchronized state, a twisted state can also be an equilibrium solution in the Kuramoto model and its variations. In the present work, we explore the stability of the twisted state in Kuramoto oscillators put on a rim of a planar circle in the two-dimensional space in the presence of power-law decaying interaction strength (∼r^{-α} with the distance r) and time delays due to a finite speed of information transfer. For example, our model can phenomenologically mimic a large sports stadium where many people try to sing or clap their hands in unison; the sound intensity decays with the distance and there can exist a time delay proportional to the distance due to the finiteness of sound speed. We first consider the case without the time delay effect and numerically find that stable twisted states emerge when the exponent α exceeds a critical value of α_{c}≈2. In other words, for α<α_{c}, the fully synchronized state, not the twisted state, is the only stable fixed point of the dynamics. In our analytic approach, we also derive an equation for α_{c} and discuss its solutions. In the presence of time delay, we find that it is possible that the synchronized state becomes unstable while twisted states are stable.

Delay-induced nutrient recycling in plankton system: Application to Sundarban mangrove wetland

The paper discusses the nutrient–plankton system with effect of time delay in nutrient recycling and toxin determined function response (TDFR). The designed model system explores the delay-induced system dynamics. We present the local stability analysis of interior equilibrium points in absence as well as in presence of time delay. Further, the direction of Hopf bifurcation is obtained. We perform the numerical computation and observe that time delay in nutrient recycling can generate the periodic solution in a stable nutrient–plankton system. Some other essential parameters, such as input concentration of nutrients and natural removal rate of nutrients, also regulate the dynamical system. The system shows Hopf and double-Hopf bifurcation in the presence of time delay. Our study shows that the delay in the nutrient recycling causes instability transition phenomenon. The delay-induced nutrient recycling and different input concentrations of nutrients can regulate the estuarine system. Finally, the stability switching is observed for delayed system.

Robustness, performance analysis and practical improvements of modified uncertainty and disturbance estimator based control

Modified uncertainty and disturbance estimator (MUDE) based control shows great control performance and robustness against unexpected dynamics in the presence of time delay. However, there are still some problems that have not been analyzed theoretically but actually hinder its further applications, such as noise amplification and control signal fluctuations under delay mismatch. On the other hand, the trade-off between robustness and control performance is still a critical issue in MUDE based controller design. However, some doubts arise when extending the single-loop sensitivity function analysis method to MUDE (multi-loop) control systems. To this end, this paper comprehensively analyzes the performance of MUDE, including control performance and robustness, and proposes some practical improvements to address its issues (denoted as IUDE). An ambiguity in analysis of sensitivity functions for multi-loop control systems is first clarified. Then, based on the two-degree-of-freedom equivalent control structure and sensitivity functions, the control performance and robustness of the MUDE based control system are analyzed in detail, including noise sensitivity and control signal response. Finally, some practical improvements are proposed and a quantitative tuning rule is also derived to facilitate its application. Simulation results show that the proposed control method effectively mitigates the problems of noise amplification and control signal fluctuation while almost not degrading the control performance. Water-tank control experiments have also been performed to validate the efficacy of the proposed method, which depicts a promising prospect in control practice.

Novel twofold control for delayed industrial processes with integrating and inverse response characteristics

Non-self- regulating nature of integrating plant causes their regulation a great challenge to researchers. Presence of time delay and positive zero in such processes makes their control more demanding. A novel control technique is developed for such processes to cope with the difficult situations and achieve enhanced performance and robustness. Suggested method is twofold: first the Inner loop controller’s parameters are adjusted to stabilize the process using maximum sensitivity concept and then the outer loop control block is designed by specifying desired setpoint tracking response in the frame of direct synthesis approach involving single tunable parameter. The tunable parameter’s values are obtained so that the robust stability condition is satisfied along with the justification of robust performance constraint in the presence of unstructured uncertainties. The effectiveness of the proposed method is verified on different plants and its advantages over some existing methods are also highlighted. Suggested approach yields improved performance with less control effort and enhanced robustness.

Time Delay Effect in Suppressing Drill-String Stick-Slip Vibration

The role of time delay in the control processes aimed at mitigating stick-slip vibration in drill-strings, is of paramount importance. Its influence on the controlled system's settling time cannot be understated; indeed, it can lead to significant consequences such as controller instability. In this study, a numerical analysis was conducted, initially a sliding mode control was employed to suppress stick-slip vibration in drill-string. Then we explore a spectrum of time delay values alongside varying reference angular velocities. The results of this investigation suggest a critical finding: the controller's susceptibility to limit cycles is notably heightened in the presence of time delay and its experiences a decline when faced with fluctuating reference angular velocities. Despite this degradation, it is noteworthy that the system does not always reach a state of complete instability. These insights are invaluable, offering essential considerations for the development of robust control strategies. Addressing the challenges posed by time delay-induced issues in suppressing stick-slip vibration during drilling operations requires a nuanced approach, taking into account the intricacies highlighted in this study.

Stability switches and chaos induced by delay in a reaction-diffusion nutrient-plankton model

In this paper, we investigate a reaction-diffusion model incorporating dynamic variables for nutrient, phytoplankton, and zooplankton. Moreover, we account for the impact of time delay in the growth of phytoplankton following nutrient uptake. Our theoretical analysis reveals that the time delay can trigger the emergence of persistent oscillations in the model via a Hopf bifurcation. We also analytically track the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Our simulation results demonstrate stability switches occurring for the positive equilibrium with an increasing time lag. Furthermore, the model exhibits homogeneous periodic-2 and 3 solutions, as well as chaotic behaviour. These findings highlight that the presence of time delay in the phytoplankton growth can bring forth dynamical complexity to the nutrient-plankton system of aquatic habitats.

A Deep Time Delay Filter for Cooperative Adaptive Cruise Control

Cooperative adaptive cruise control (CACC) is a smart transportation solution to alleviate traffic congestion and enhance road safety. The performance of CACC systems can be remarkably affected by communication time delays, and traditional control methods often compromise control performance by adjusting control gains to maintain system stability. In this paper, we present a study on the stability of a CACC system in the presence of time delays and highlight the trade-off between control performance and tuning controller gains to address increasing delays. We propose a novel approach incorporating a neural network module called the deep time delay filter (DTDF) to overcome this limitation. The DTDF leverages the assumption that time delays primarily originate from the communication layer of the CACC network, which can be subject to adversarial delays of varying magnitudes. By considering time-delayed versions of the car states and predicting the present (un-delayed) states, the DTDF compensates for the effects of communication delays. The proposed approach combines classical control techniques with machine learning, offering a hybrid control system that excels in explainability and robustness to unknown parameters. We conduct comprehensive experiments using various deep-learning architectures to train and evaluate the DTDF models. Our experiments utilize a robot platform consisting of MATLAB, Simulink, the Optitrack motion capture system, and the Qbot2e robots. Through these experiments, we demonstrate that when appropriately trained, our system can effectively mitigate the adverse effects of constant time delays and outperforms a traditional CACC baseline in control performance. This experimental comparison, to the best of the author’s knowledge, is the first of its kind in the context of a hybrid machine learning CACC system. We thoroughly explore initial conditions and range policy parameters to evaluate our system under various experimental scenarios. By providing detailed insights and experimental results, we aim to contribute to the advancement of CACC research and highlight the potential of hybrid machine learning approaches in improving the performance and reliability of CACC systems.

Reference management-based resilient control for delayed periodic piecewise polynomial systems under proportional integral observer

This article examines the proportional integral observer (PIO)-based resilient tracking control problem for periodic piecewise polynomial systems (PPPSs) in the presence of time delay and disturbances. To be precise, the PPPSs are formulated by dividing the fundamental period of periodic systems into numerous subintervals, each of which is defined using matrix polynomial functions. Further, the PIO strategy is employed to estimate the states of the undertaken system with high precision wherein the integral term in PIO enhances the system's design flexibility and also increases its robustness. Taking advantage of these characteristics, a PIO-based tracking control is configured with gain perturbations to track the desired reference states. Moreover, to attenuate the implications made by disturbances in considered system design, H ∞ performance is employed. Furthermore, in order to establish adequate conditions in the form of linear matrix inequalities, the time-varying polynomial Lyapunov–Krasovskii functional is implemented. Then, the time-varying gain matrices of the control scheme and PIO configuration are computed by solving the obtained criteria through the MATLAB platform. Concludingly, to evidence for the discoveries' potential and usefulness, a numerical example is offered.

Stability of Impaired Humoral Immunity HIV-1 Models with Active and Latent Cellular Infections

This research aims to formulate and analyze two mathematical models describing the within-host dynamics of human immunodeficiency virus type-1 (HIV-1) in case of impaired humoral immunity. These models consist of five compartments, including healthy CD4+ T cells, (HIV-1)-latently infected cells, (HIV-1)-actively infected cells, HIV-1 particles, and B-cells. We make the assumption that healthy cells can become infected when exposed to: (i) HIV-1 particles resulting from viral infection (VI), (ii) (HIV-1)-latently infected cells due to latent cellular infection (CI), and (iii) (HIV-1)-actively infected cells due to active CI. In the second model, we introduce distributed time-delays. For each of these systems, we demonstrate the non-negativity and boundedness of the solutions, calculate the basic reproductive number, identify all possible equilibrium states, and establish the global asymptotic stability of these equilibria. We employ the Lyapunov method in combination with LaSalle’s invariance principle to investigate the global stability of these equilibrium points. Theoretical findings are subsequently validated through numerical simulations. Additionally, we explore the impact of B-cell impairment, time-delays, and CI on HIV-1 dynamics. Our results indicate that weakened immunity significantly contributes to disease progression. Furthermore, the presence of time-delays can markedly decrease the basic reproductive number, thereby suppressing HIV-1 replication. Conversely, the existence of latent CI spread increases the basic reproductive number, intensifying the progression of HIV-1. Consequently, neglecting latent CI spread in the HIV-1 dynamics model can lead to an underestimation of the basic reproductive number, potentially resulting in inaccurate or insufficient drug therapies for eradicating HIV-1 from the body. These findings offer valuable insights that can enhance the understanding of HIV-1 dynamics within a host.

Dynamical integrity estimation in time delayed systems: A rapid iterative algorithm

The robustness of dynamical systems against external perturbations is crucial in engineering; however, it is often overlooked for the lack of methods for rapidly computing it. This paper proposes a novel algorithm for estimating the robustness of systems subject to time delay. More precisely, the algorithm iteratively estimates the so-called local integrity measure (LIM), that is, the radius of the largest hypersphere centered at the fixed point and located within its basin of attraction. Since time-delayed systems are infinite dimensional, initial conditions are restricted to a constrained type of initial functions. The semi-discretization method is used for rapidly simulating the dynamics of the systems, while trajectories are classified as converging or diverging using a subdivision of the reduced phase space into cells. The algorithm was tested on four different mechanical systems, and in all cases it very quickly provided an accurate estimation of the LIM. Moreover, it enabled the study of LIM trends in a multi-dimensional parameter space, which would have been unfeasible with alternative methods. This breakthrough in computational efficiency has important implications for engineering design, allowing for careful consideration of dynamical integrity and enhancing the safety and reliability of engineered systems, especially in the presence of time delays.

Rich dynamics of a delay-induced stage-structure prey–predator model with cooperative behaviour in both species and the impact of prey refuge

The prey–predator interaction between organisms living in an ecosystem is greatly affected by the cooperation among the species as well as immature prey refuge in presence of time delay. This study has developed and analysed a stage-structured prey–predator model that includes refuge behaviour for prey and two types of cooperative behaviours, namely cooperative behaviour among prey as well as cooperative behaviour among predators, in addition to gestational delay. The presence of Saddle–node, Transcritical, and Hopf-bifurcations is explained by analytically and numerically. We examine the model parameter’s sensitivity to determine the species’ favourable and unfavourable impacts and to identify the most significant parameters. The dynamics of delay and non-delay systems are compared, and we recognise that a consistent increase in time delay causes chaotic oscillations with period-doubling fluctuations which can be regulated by the refuge for immature prey. However, we demonstrate that the immature prey refuge destabilises the system in absence of time delay while it acts as a control parameter in presence of delay. Finally, we wrap up the paper by highlighting the key biological implications of the numerical and analytical findings.

Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread

This paper presents and analyzes two mathematical models for the human immunodeficiency virus type-1 (HIV-1) infection with Cytotoxic T Lymphocyte cell (CTL) immune impairment. These models describe the interactions between healthy CD4+T cells, latently and actively infected cells, HIV-1 particles, and CTLs. The healthy CD4+T cells might be infected when they make contact with: (i) HIV-1 particles due to virus-to-cell (VTC) contact; (ii) latently infected cells due to latent cell-to-cell (CTC) contact; and (iii) actively infected cells due to active CTC contact. Distributed time delays are considered in the second model. We show the nonnegativity and boundedness of the solutions of the systems. Further, we derive basic reproduction numbers ℜ0 and ℜ˜0, that determine the existence and stability of equilibria of our proposed systems. We establish the global asymptotic stability of all equilibria by using the Lyapunov method together with LaSalle’s invariance principle. We confirm the theoretical results by numerical simulations. The effect of immune impairment, time delay and CTC transmission on the HIV-1 dynamics are discussed. It is found that weak immunity contributes significantly to the development of the disease. Further, we have established that the presence of time delay can significantly decrease the basic reproduction number and then suppress the HIV-1 replication. On the other hand, the presence of latent CTC spread increases the basic reproduction number and then enhances the viral progression. Thus, neglecting the latent CTC spread in the HIV-1 infection model will lead to an underestimation of the basic reproduction number. Consequently, the designed drug therapies will not be accurate or sufficient to eradicate the viruses from the body. These findings may help to improve the understanding of the dynamics of HIV-1 within a host.

A second-order difference scheme for the nonlinear time-fractional diffusion-wave equation with generalized memory kernel in the presence of time delay

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α+1 where 0<α<1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis.

Quantized-state-based decentralized neural network control of a class of uncertain interconnected nonlinear systems with input and interaction time delays

This paper presents a quantized-state-based decentralized control strategy for uncertain interconnected nonlinear systems with time-varying delays under a networked control environment. Full state variables quantized via a uniform-hysteretic quantizer are available only for a decentralized adaptive control design. Unlike the existing decentralized adaptive recursive control designs, this study is focused on establishing a quantized-state-based decentralized learning mechanism for neural networks using discontinuously quantized states. Moreover, the stability of a decentralized neural network tracking system in the presence of time delays is analyzed. For each subsystem, a neural-network-based local adaptive tracker using a delay compensator is designed based on the local quantized state feedback. Technical lemmas pertaining to quantization errors and adaptive laws are presented to ensure the boundedness of all closed-loop signals and the convergence of local tracking errors to an adjustable neighborhood of the origin. Finally, illustrative simulations clarify and verify the decentralization strategy of the developed state-quantized adaptive tracking system. The control performance is evaluated using the root mean square control errors, where the values are 0.0293 and 0.0134 for each subsystem in Example 1 and 0.0467 and 0.0384 for each subsystem in Example 2.

Role of multiple time delays on a stage-structured predator–prey system in a toxic environment

In this paper, a stage-structured predator–prey model is considered in a toxic environment, emphasizing on the affect of the toxic substances on growth of the species in presence of time delays. The positivity and boundedness of solutions have been discussed. The analytical and numerical simulations of the proposed model show that the coexistence equilibrium point is stable for low and moderate values of the delay parameter associated with the predator’s maturity. Even though there is an interval of the maturation time in which the system exhibits oscillatory behaviour, a longer maturity duration may lead to the extinction of the predator species. The impact of toxicity on the proposed system is also investigated. The stability of the system repeatedly switches as the time delay, related to the changes of toxicity incurred in the adult predator, varies, and finally becomes unstable for a significant value of it. Besides that, the delay related to the prey’s toxicity has also led some exciting results.

Coordinated Distributed Predictive Control for Voltage Regulation of DC Microgrids With Communication Delays and Data Loss

This paper is concerned with the voltage tracking problem of DC microgrids subject to communication delays and packet losses, for which existing work commonly adopts passive fault-tolerant approaches. To accurately compensate for the above communication constraints experienced by DC microgrids, a coupled discrete microgrid model is developed on the basis of the physical laws of actual DC microgrids. Unlike the practice of passively tolerating communication delays, through the physical model established, this paper suggests a consensus-based proportional-integral predictive control strategy, which can actively compensate for communication delays and consecutive packet dropouts encountered by DC microgrids. Under the observer-based distributed networked predictive control framework, each distributed generation subsystem in the microgrid exchanges its own measurements over the network and integrates information from controllers of other units to achieve output voltage consensus in the presence of time delays and packet losses. Furthermore, to demonstrate the generality of the proposed method, the sufficient and necessary conditions for the DC microgrid system to accomplish voltage tracking are given. These conditions are rendered in the form of matrix eigenvalues associated with the physical connections and communication couplings between distributed generation units. Besides, the stability and convergence of the closed-loop microgrid system are given in the form of linear matrix inequality based on the Lyapunov function. Finally, the performance of the proposed control scheme is evaluated in terms of its convergence, robustness to load variations, and plug-and-play functionality through the built photovoltaic cell-based (with battery banks) DC microgrid hardware system.

Dynamics of a delay-induced prey–predator system with interaction between immature prey and predators

In biological pest control systems, several pests (including insects, mites, weeds, etc.) are controlled by biocontrol agents that rely primarily on predation. Following this biocontrol management ecology, we have created a three-tier prey–predator model with prey phase structure and predator gestation delay. Several studies have demonstrated that predators with Holling type-II functional responses sometimes consume immature prey. A study of the well-posedness and local bifurcation (such as saddle-node and transcritical) near the trivial and planer equilibrium points is carried out. Without any time lag, the prey development coefficient has a stabilizing impact, while increasing attack rate accelerates instability. Energy transformation rate and handling time are shown to cause multiple stability switches in the system. Numerical results demonstrate time delay is the key destabilizer that destroys stability. Our model can replicate more realistic events by including time-dependent factors and exploring the dynamic behavior of nonautonomous systems. In the presence of time delay, sufficient conditions of permanence and global attractivity of the nonautonomous system are derived. Finally, MATLAB simulations are performed to validate the analytical findings.

Decentralized Event-Triggered Cooperative Localization in Multirobot Systems Under Random Delays: With/Without Timestamps Mechanism

This article addresses the problem of the decentralized event-triggered cooperative localization (DECL) for a group of mobile robots in the presence of time delays. First, we introduce a DECL algorithm for multirobot systems under time delays. We consider two different scenarios: timestamped and non-timestamped that leads to different DECL algorithms. Then, we provide the stochastic boundedness of filtering error considering bounded random delays. We show that if the delay due to multirobot communication is sufficiently small, then by choosing proper event-triggering parameters, the filtering error and covariance remain bounded while reducing the transfer of information. Finally, simulation and experimental results using a team of four e-puck2 mobile robots demonstrate the effectiveness of the proposed algorithm.