Multiple Time Delays Research Articles

Stability and Hopf bifurcation analysis of a fractional-order p53 multiple time delays model under PD$$^\alpha $$ control

Stability and Hopf bifurcation analysis of a fractional-order p53 multiple time delays model under PD$$^\alpha $$ control

Design of Multivariable PID Control Using Iterative Linear Programming and Decoupling

The design of multivariable process control systems is specially complicated when there are strong interactions between the different control loops, and even more with multiple time delays. This paper proposes an iterative design method of centralized proportional-integral-derivative (PID) controllers for stable linear systems. The methodology is based on the linear parameterization of equivalent loop transfer functions (ELTFs) for centralized control. These functions capture the dynamics of the other loops and, from a prior design, allow solving the design problem at each iteration with linear programming that shapes the Nyquist plot of the ELTFs in the frequency domain, which also avoids the need for approximations. Two optimizations are proposed: (I) maximizing integral gains by fulfilling linear robustness margins in each ELTF and (II) maximizing linear robustness margins by fulfilling minimum bandwidths in each loop. In both optimizations, static decoupling and decoupling at a frequency close to the bandwidth of each loop are included as constraints, which improves the decoupling performance and the procedure convergence. The effectiveness of the method is verified in three simulation examples (square and non-square) and a lab experimental process. The proposed designs achieve a similar or better response when compared to that achieved by other authors.

Stability and bifurcation analysis of fractional‐order tumor–macrophages interaction model with multi‐delays

In the realm of modeling biological systems with memory, particularly those involving intricate interactions like tumor‐immune responses, the utilization of multiple time delays and Caputo‐type fractional‐order derivatives represents a cutting‐edge approach. In this research paper, we introduce a novel fractional‐order model to investigate the dynamic interplay between tumors and macrophages, a key component of the immune system, while incorporating multiple time delays into our framework. Our proposed model comprises a system of three Caputo‐type fractional‐order differential equations, each representing distinct cell populations: tumor cells, anti‐tumor cells (specifically ‐type macrophages with pro‐inflammatory properties), and pro‐tumor cells ( ‐type macrophages with immune‐suppressive characteristics). The stability of equilibria is discussed by analyzing the characteristic equations for each case, and the existence conditions for the Hopf bifurcation are obtained according to the critical values of delay parameters. Furthermore, numerical simulations are presented in order to verify the analytical results obtained for stability and Hopf‐bifurcation with respect to the two‐time delay parameters and . The analysis shows the rich dynamics of the model according to the fractional‐order parameter and the time delay parameters.

Optimized robust control for improving frequency response of delay dependent AC microgrid with uncertainties

Converter-interface-based renewable energy incorporated into modern power systems causes deterioration in power system stability and it is proving to be a main challenge for operators using sluggish traditional generation with a growing share of converter-interface generation, and there is a need for an alternative means to deliver rapid frequency control. This paper proposes two control loops as secondary frequency control, first is an improved optimized delay-dependent frequency control to ensure robust frequency control under daily changes in system parameters that may occur in prospective, renewable-rich future AC microgrids. For a variety of operating conditions, the design process employs a mixed H2/H∞ robust control based on linear matrix inequality (LMI). In second, a speedy-acting improved power stability control loop is utilized as the proposed secondary frequency control, because demand response aggregators can be beneficial for frequency control. Robustness is evaluated against several perturbations and communication delay attacks through dynamic simulation and stability is evaluated by small-signal analysis. Different case studies subject to generation\\load loss, multiple time delay and parameter uncertainties are demonstrated on an interconnected AC microgrid system to verify the performance of the proposed control technique.

Topology property analysis and application of stable time‐delay regions for linear multiple time‐delay systems

AbstractThis study examines the relationship between the topology of stable time‐delay regions and the stability analysis of an linear multiple time delay system. To analyze the topology of stable time‐delay regions, we construct a function with a value equal to zero for the time‐delay points on the boundaries of stable time‐delay regions. The function is continuous and differentiable in the whole defining field with a global minimum of zero so that we can locate the boundaries by minimizing the value of the function. Based on the topology analysis, we proposed a performance validation approach for controllers that are designed to stabilize the system using feedback signals with time delays. The method based on the topology analysis is simple and reliable so that can deal with a linear time‐invariant system with high order and multiple time delays (more than 3). The example case study shows that the above method is reliable and enables ultra‐low latency coordination and control for the future power grid with ubiquitous power electronics.

Retracted: Single-Input Single-Output System with Multiple Time Delay PID Control Methods for UAV Cluster Multiagent Systems

Retracted: Single-Input Single-Output System with Multiple Time Delay PID Control Methods for UAV Cluster Multiagent Systems

Global Existence and Uniqueness of Solutions of Integral Equations with Multiple Variable Delays and Integro Differential Equations: Progressive Contractions

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature.

A mathematical model for tumor-immune competitive system with multiple time delays

A mathematical model is proposed to study the complicated dynamics of tumor-immune interactions with three discrete time delays. The proposed model consists of nine coupled ordinary differential equations (ODEs) whose components are tumor cells, tumor-specific CD8+T cells, dendritic cells, macrophages, regulatory T-cells (Tregs), interleukin-10 (IL-10), IL-12, transforming growth factor-β (TGF-β) and interferon-γ (IFN-γ). By utilizing the quasi-steady-state approximation for cytokine concentrations, we obtain a system of four coupled ODEs. We introduce three discrete time delays into our deterministic system to better understand the tumor-immune dynamics. Basic properties of the system, including existence, uniqueness, positivity, boundedness, and uniform persistence are discussed. Stability analysis of both the delayed and non-delayed system has been performed and the conditions for stability and direction of Hopf bifurcation have been determined. Furthermore, we assess the length of time delay required to maintain the stability of period-1 limit cycle. Parameter estimation techniques are discussed and numerical simulations are provided to support our theoretical analysis. Notably, we observe that time delays do not significantly influence the system behavior for the existing set of parameters value.

Stability analysis of complex-valued stochastic neural networks with multi-time-delay and parameter uncertainty

This paper investigates the examination of global asymptotic stability pertaining to uncertain complex-valued stochastic neural networks (UCVSNNs) with multiple time delays. This paper employs the real-imaginary separated activation function to establish an equivalence between complex-valued neural networks (CVNNs) and two real-valued activation functions, representing the real and imaginary parts respectively. This equivalence is utilised to analyse the original UCVSNNs model in a comprehensive manner. The global asymptotic stability of the studied UCVSNNs model is ensured by constructing an appropriate Lyapunov-Krasovskii generalised function and applying the I t o ^ formula, linear matrix inequality (LMI) and other analytical techniques to derive the system stability and other sufficient conditions. In the meanwhile, we provide two numerical examples to demonstrate the reliability and benefit of the discovered results.

Analysis and simulation of vehicle following behavior with consideration of multiple time delays

With the application of intelligent transportation system, drivers can receive more traffic status information in the process of driving, but they may have multiple time delays when they are presented with different information. As a result, to further research the influence of multiple time delays on vehicle following behavior, this study comes up with a modified car following model, which takes into account multiple time delays in sensing both distance and velocity information, based on the backward looking and velocity difference model. The linear stability theory is used to explore the influence of multiple time delays on the stability of the extended model. Different nonlinear equations are derived using nonlinear theory, which include Burgers, Korteweg–de Vries, and Modified Korteweg–de Vries equations and are used to depict triangular wave, soliton wave and kink–antikink wave corresponding to stable, metastable and unstable regions respectively. In the end, numerical simulation is used to further study the improved model to depict the dynamic evolution of traffic stream. The results display that the improvement of time delay in sensing distance can induce traffic congestion, conversely, the time delay in sensing velocity offsets the negative effect of time delay in sensing distance. The theoretical results and numerical simulation results are consistent with each other.

Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis

With multiple time delays, we investigated a Caputo fractional order dynamical system involving susceptible, exposed, infected, and recovered individuals. Positivity and boundedness are also theoretically demonstrated using Laplace transform and Mittag-Leffler function. The stability of the disease-free and epidemic equilibrium points has been studied for both delayed and non-delayed model. For generating numerical solutions to the model system, we used the Adam-Bashforth-Moulton predictor-corrector technique. With the help of MATLAB (2018a), we were able to conduct graphical demonstrations and numerical simulations. The system displays Hopf bifurcation and the solutions are no longer periodic beyond a certain threshold value of the time delay parameters.

Free-form and hybrid lens models for SDSS J1004+4112: substructure and central image time delay constraints

ABSTRACT SDSS J1004+4112 is a well-studied gravitational lens with a recently measured time delay between its first and fourth arriving quasar images. Using this new constraint, we present updated free-form lens reconstructions using the lens inversion method grale, which only uses multiple image and time delay data as inputs. In addition, we obtain hybrid lens reconstructions by including a model of the brightest cluster galaxy as a Sersic lens. For both reconstructions, we use two sets of images as input: one with all identified images, and the other a revised set leaving out images that have been potentially misidentified. We also develop a source position optimization Markov chain Monte Carlo (MCMC) routine, performed on completed grale runs, that allows each model to better match observed image positions and time delays. All the reconstructions produce similar mass distributions, with the hybrid models finding a steeper profile in the centre. Similarly, all the mass distributions are fitted by the Navarro–Frenk–White profile, finding results consistent with previous parametric reconstructions and those derived from Chandra X-ray observations. We identify an ∼5 × 1011 M⊙ substructure apparently unaffiliated with any cluster member galaxy and present in all our models, and study its reality. Using our free-form and hybrid models, we predict a central quasar image time delay of ∼2980 ± 270 and ∼3280 ± 215 d, respectively. A potential future measurement of this time delay will, while being an observational challenge, further constrain the steepness of the central density profile.

Fractional order PD control of the Hopf bifurcation of HBV viral systems with multiple time delays

This paper will explore a fractional-order hepatitis B virus (HBV) infection model that takes into account cell-cell and virus-cell transmission and multiple treatment modalities. The desired control strategy is realized by means of a fractional order PD controller. Firstly, we calculated the basic regeneration number and equilibrium point of the HBV model. Afterwards, for the uncontrolled HBV virus system, the adequate conditions for both stability and Hopf bifurcation are systematically investigated via choosing the appropriate time delay as a parameter for bifurcation. Subsequently, under fractional order PD controller, the effect of a proposed controller on system stability and Hopf bifurcation is studied. The desired dynamic characteristics can be obtained afterwards. Finally, numerical simulations show that all three treatments significantly reduce R0. The onset of oscillations can be delayed by decreasing the order of the fractional order. There are three control pathways for fractional order PD control, and the generation of bifurcation can also be delayed by changing the gain parameter. Using the above methods, the diffusion of HBV virus particles in the body can be effectively controlled. The conclusions drawn in this paper are extremely novel and have potential theoretical value for the future treatment of hepatitis B illness.

Adaptive interference mitigation space‐time array reconfiguration by joint selection of antenna and delay tap

AbstractSpace‐time adaptive processing (STAP) array has found extensive use in the global navigation satellite system for interference suppression. Nevertheless, the conventional space‐time array necessitates multiple antennas and time delay taps, resulting in a heavy burden on both hardware and computation resources. Thus, lowering the cost and complexity of STAP becomes a considerable problem demanding prompt solutions. This work designs a space‐time array reconfiguration scheme based on the joint selection strategy of the antenna and delay tap to address this problem. The reconstructed array can induce a high signal to interference plus noise power ratio (SINR) when using fewer antennas and delay taps. The space‐time correlation coefficient (STCC) is first presented to manifest the influence of reconfiguration on output SINR. Then, we formulate the space‐time array reconfiguration mathematical model, aiming to minimise STCC. Two reconfiguration methods are provided to solve the optimal array configuration of antennas and delay taps that can reach the minimum STCC and maximum SINR. The optimal antenna‐delay tap pairs are finally selected to comprise the reconstructed array. Substantial simulation experiments verify that the presented scheme and strategy are effective and reliable in space‐time array reconfiguration.

Global Hopf bifurcation for differential equations with multiple threshold‐type state‐dependent delays

Using the ‐equivariant degree, we develop a global Hopf bifurcation theory for system of differential equations with multiple threshold‐type state‐dependent delays whose prototype is the human respiratory system with multiple blood transport time delays. To establish a theoretic framework for modeling practices of periodic breathing, we further investigate the periodic oscillations of carbon dioxide concentrations in the lung, brain, and tissue compartments and conduct a local and global Hopf bifurcation analysis for the model when varying the commensurate scale of the multiple delays in a transformed system. Such a global Hopf bifurcation will indicate the onset and persistence of the periodic oscillations.

The Optimization of Control Parameters: Finite-Time Bipartite Synchronization of Memristive Neural Networks With Multiple Time Delays via Saturation Function.

This article studies the memristive neural networks with multiple time delays (MNNsMTDs). The topology of networks is signed, which contains both cooperative and competitive relationships. Two controllers without time delays are designed to achieve finite-time bipartite synchronization (FTBS) and practical FTBS (PFTBS) of MNNsMTDs. A novel controller with a saturation function rather than a sign function is proposed to avoid chattering. Along with the Lyapunov function method, some mathematical techniques, and scaling inequalities, some sufficient conditions for FTBS and PFTBS of MNNsMTDs are attained. Besides, this article also concerns fixed-time bipartite synchronization (FXBS) and practical FXBS (PFXBS) of MNNsMTDs. An optimization model is designed to obtain some optimal control parameters. An algorithm based on particle swarm optimization (PSO) is provided to solve this model. Some numerical examples are included to demonstrate the correctness and applicability of the approaches.

Bifurcation−Driven Tipping in A Novel Bicyclic Crossed Neural Network with Multiple Time Delays

Bifurcation−Driven Tipping in A Novel Bicyclic Crossed Neural Network with Multiple Time Delays

A two-degree-of-freedom controller for a high-precision air temperature control system with multiple disturbances

The key equipment to prevent precision machinery such as lithography from being influenced by air temperature fluctuation is a high-precision air temperature control (ATC) system. The ATC system developed for a lithography immersion testing platform suffers from the difficulties of multiple disturbances, large inertia and time delay. However, seldom have control methods can solve these problems systematically. A two-degree-of-freedom (2DOF) controller is proposed in this paper for this system, where model predictive control (MPC) is developed to overcome large inertia and time delay in the main control loop, and the measurable disturbance from the inlet air is compensated; based on the loop shaping approach, the controller in the disturbance rejection control loop is designed to attenuate the load disturbance; a Kalman filter is designed to reduce the influence of noise having no stable frequency characteristic. The experimental results show that the MPC controller and Kalman filter can effectively avoid the system oscillation, and the disturbance rejection control loop can significantly attenuate the load disturbance in the frequency range from 0.01 to 0.02 rad/s. Therefore, the 1σ precision is increased by 3.3 mK to 8.7 mK by the 2DOF controller when the system is influenced by the load disturbance with different fluctuation amplitudes.

Model-Based Distributed Asynchronous Sampling for Multiagent Consensus With Event-Detecting Delays and Network Delays

In this paper, the practical average consensus problem of linear multi-agent systems with model-based distributed asynchronous sampling is investigated. Both internal event-detecting delays and external network delays are allowed in systems. Data-sampling instants are generated asynchronously by local event-triggering conditions. Particularly, data-sampling instants and event-triggering instants of each agent are also asynchronous due to the existence of event-detecting delays. Prediction models are utilized in detecting events and updating control inputs. Artificial interpolation instants are introduced in each sampling interval to address theoretical challenges brought by internal and external delays. To deal with this kind of complex asynchronous setup with multiple time delays, based on Barbalat's Lemma, a new procedure for deriving conditions for average consensus and a theoretical analysis approach are presented.

A norm stability condition of neutral-type Cohen-Grossberg neural networks with multiple time delays

By constructing an appropriate Lyapunov functional, this paper obtains a novel delay-independent stability criterion of neutral-type Cohen-Grossberg neural networks possessing multiple time delays. Although this type of system cannot be represented as vector–matrix form due to the presence of multiple delays, our stability conclusion is fully defined by the infinite norm of parametric matrices and the network parameters first time. Due to the feasibility and simplicity of method proposed, our stability conclusion reducing the computational complexity while also reducing the conservatism compared with several onetime literature. Two concrete neural network models are applied to confirm the effectiveness and superiority of our conclusion.