Discrete Time Delay Research Articles

Global Stability and Thermal Optimal Control Strategies for Hyperthermia Treatment of Malignant Tumors

Malignant tumor (cancer) is the leading cause of death globally and the annual cost of managing cancer is trillions of dollars. Although, there are established therapies including radiotherapy, chemotherapy and phototherapy for malignant tumors, the hypoxic environment of tumors and poor perfusion act as barriers to these therapies. Hyperthermia takes advantage of oxygen deficiency and irregular perfusion in the tumor environment to destroy malignant cells. Despite successes recorded with hyperthermia, there are concerns with the post-treatment condition of patients as well as the required thermal dose to prevent harm. The investigation of the dynamics of tumor-induced immune suppression with hyperthermia treatment using mathematical analysis and optimal control theory is potentially valuable in the development of hyperthermia treatment. The role of novel tumor-derived cytokines in counterattacking immune cells is considered in this study as a mechanism accounting for the aggressiveness of malignant tumors. Since biological processes are not instantaneous, a discrete time delay is used to model biological processes involved in tumor inhibitory mechanisms by secretion, the elaboration of suppressive cells, and effector cell differentiation to produce suppressive cells. Analytical results obtained using Lyapunov’s function indicate the conditions required for global stability of the tumor-present steady-state. A thermal optimal control strategy is pursued based on optimal control theory, and the best strategy to avoid adverse outcomes is obtained. We validate the analytical results numerically and demonstrate the impact of both inadequate and excessive heat on the dynamics of interactive cell functioning.

Delay-based stabilisation and strong stabilisation of LTI systems by nonsmooth constrained optimisation

Stabilisation and strong stabilisation (i.e. the problem of stabilising by a stable controller) of linear time-invariant (LTI) multi-input multi-output systems are considered. The considered systems may be finite-dimensional or time-delay systems, which may have multiple discrete time delays. The controllers to be designed may also be either finite-dimensional or time-delay LTI controllers. In the controller design phase, a constrained nonsmooth optimisation-based design method is employed. The effectiveness of the proposed method is illustrated through various benchmark examples which are solved by using the recently developed Matlab-based software called DrStabilization . The examples also show that, in many cases, time-delay controllers may have certain advantages over conventional finite-dimensional controllers.

Stability and bifurcation analysis of a fractional‐order model of cell‐to‐cell spread of HIV‐1 with a discrete time delay

In this manuscript, fractional order is introduced onto a time‐delay differential equation model of cell‐to‐cell spread of HIV‐1. The fractional derivative of Caputo type is considered. We deal with the local stability of the resulting system and derive some necessary and sufficient conditions ensuring Hopf bifurcation to occur for this system. Explicit expressions for determining stability of critical surfaces are also given. An Adams‐type predictor–corrector technique is applied to illustrate the numerical results. The main target of this study is to describe the structure of HIV‐1 by using a fractional‐order mathematical model, and the motivation of using fractional derivatives is the ability of these operators to capture memory effects in the system.

Stability and Hopf-bifurcation analysis of four dimensional minimal neural network model with multiple time delays

In this paper, a minimal neural networks (NNs) of four dimensional (4D) model is considered with two different and discrete time delays. The time delay in information transformation follows the finite propagation speed of the signals and the finite processing time in the synapses. The linear stability and the local Hopf-bifurcation of the model are investigated by analyzing the associated characteristic equation. Stability of bifurcating periodic solutions and direction of the Hopf-bifurcation are determined using the normal form theory and center manifold theorem. Numerical simulations are performed to justify the theoretical results. Finally, the bifurcation diagrams are performed to observe the dynamical characteristics of the neuron model. Therefore, these results could be useful to understand the contribution of multiple time delays in excitable neural systems for signal processing.

Absolute stability and absolute hyperbolicity in systems with discrete time-delays

An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete time-delays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.

A Goodwin type cyclical growth model with two-time delays

In this paper, we reconsider the Goodwin 1967 growth-cycle model, where the antagonistic relationship between wages and profits is assimilated to the prey-predator conflict modeled by Volterra in 1931.Here we propose an extension of Goodwin’s basic model by adding two important elements of the business cycle theory: (i) a finite time delay between investment orders and deliveries of finished capital goods, as theorized by Kalecki (1935); (ii) a delayed reaction of real wages to the unemployment levels, as suggested by Chiarella (1990). Both these delays preserve the two-dimensionality of the original model, but it becomes a delayed differential equation system, with two discrete time delays and one-delay dependent parameters. The qualitative study of the system shows that without lags the economic meaningful equilibrium is structurally stable. Nevertheless, as soon the time delays become positive, that equilibrium loses its stability and, according to the combinations of parameters and length of the lags, either periodic or non-periodic (chaotic) fluctuations arise. Numerical simulations supporting the economic analysis show that, in the very long run, a “strange attractor” depicts the dynamic behavior of the system.

Delay induced interaction of humoral- and cell-mediated immune responses with cancer.

A mathematical model for the quantitative analysis of the interaction between cancer cells and cell-mediated immune system with two discrete-time delays, considering the role of antibodies, is studied in this paper. The model is analyzed both analytically and numerically to understand the dynamics of interaction delay and proliferation enhancement effect delay in the eradication of cancer. Hopf bifurcation occurs when one-time delay crosses the critical value in the absence of the other. It is also observed that as we increase any of the delays, it will increase the cancer burden. The analysis shows that a cancer-free state cannot be obtained solely by the killing rate of cytotoxic T-lymphocytes, but humoral response in combination with cell-mediated immune responses plays an important role to eradicate cancer.

α− dissipative filtering for Singular Markov jump system with time delays

This paper considers the [Formula: see text] dissipative filtering problem for a class of Singular Markov jump systems (SMJSs) with distributed time delays and discrete time delays. First, using Lyapunov’s stability theory and combining delay partitioning technique, integral partitioning technique, and free weight matrix method, the sufficient conditions for stochastic admissibility and [Formula: see text] dissipation of system are studied. Then, a filtering design method based on linear matrix inequalities (LMIs) is given to make the filtering error system stochastically admissible and [Formula: see text] dissipative. Finally, numerical simulations verify the effectiveness of the resulting method.

Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays

<p style='text-indent:20px;'>Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays are investigated. First, stability analysis of the model is given, and then the conditions on parameters at which the system has a Hopf bifurcation are determined. Second, bifurcation analysis is given by taking one of delay parameters as a bifurcation parameter while fixing the other in its stability interval to show the existence of Hopf bifurcations. The normal form theory and the center manifold reduction for functional differential equations have been utilized to determine some properties of the Hopf bifurcation including the direction and stability of bifurcating periodic solution. Finally, numerical simulations are performed to support theoretical results. Analytical results together with numerics present that time delay has a crucial role on the dynamical behavior of Chlorine Dioxide-Iodine-Malonic Acid (CIMA) reaction governed by a system of nonlinear differential equations. Delay causes oscillations in the reaction system. These results are compatible with those observed experimentally.</p>

DrStabilization: A Matlab-based Software to Design Structured Time-delay Controllers

A new Matlab-based software called DrStabilization is introduced. The main purpose of the software is to stabilize multi-input multi-output linear time-invariant systems which may have an arbitrary number of discrete time-delays. The stabilization problem is formulated as the minimization of spectral abscissa function over the controller parameters. By modifying this problem, it is also possible to consider strong stability, i.e., stability achieved by a stable controller. The software allows designing structured dynamic output feedback controllers. By structuring the controller matrices, one can design decentralized controllers and/or well-known servocompansators such as PI/PID/PIR. Furthermore, the software also allows designing time-delay controllers in which case an arbitrary number of artificial time-delays can be introduced both to the I/O channels and the dynamics of the controller itself. The utilization of the software is illustrated through an example.

PERFORMANCE COMPARISON OF PID CONTROLLER AND FUZZY LOGIC CONTROLLER FOR WATER LEVEL CONTROL WITH APPLYING TIME DELAY

Liquid level control is a system often found in most industrial facilities. Usually, embedded control systems are used, as these systems play an increasingly important role in control engineering. Time delay is one of the most important parameters which affects the control systems performance. This study aims to design a water level control system in MATLAB/ Simulink environment using the proportional integral derivative (PID) controller and Fuzzy Logic (FL) controller and then to investigate the effect of the time delay on their performance. The two control systems were tested and compared in terms of performance, with and without applying a time delay. The study highlights the application of two types of delays (distributed time delay and discrete-time delay) to the output of the built-in control system and discusses the results. The results showed that the FL controller is better than the PID controller in terms of stability and performance. Although the rise time of both controllers was almost the same, the FL controller showed higher performance in terms of overshoot and settling time. While the overshoot for the PID controller was 9.33 %, it did not appear when using the FL controller. Moreover, the settling time of the PID controller was 10 seconds, while the FL controller took only 8.6 seconds to achieve stabilization. As a result, the FL controller has demonstrated the ability to eliminate the overshoot problem that appears when using PID controllers and stabilize faster.

A DELAY NONAUTONOMOUS PREDATOR–PREY MODEL FOR THE EFFECTS OF FEAR, REFUGE AND HUNTING COOPERATION

Fear of predation may assert privilege to prey species by restricting their exposure to potential predators, meanwhile it can also impose costs by constraining the exploration of optimal resources. A predator–prey model with the effect of fear, refuge, and hunting cooperation has been investigated in this paper. The system’s equilibria are obtained and their local stability behavior is discussed. The existence of Hopf-bifurcation is analytically shown by taking refuge as a bifurcation parameter. There are many ecological factors which are not instantaneous processes, and so, to make the system more realistic, we incorporate three discrete time delays: in the effect of fear, refuge and hunting cooperation, and analyze the delayed system for stability and bifurcation. Moreover, for environmental fluctuations, we further modify the delayed system by incorporating seasonality in the fear, refuge and cooperation. We have analyzed the seasonally forced delayed system for the existence of a positive periodic solution. In the support of analytical results, some numerical simulations are carried out. Sensitivity analysis is used to identify parameters having crucial impacts on the ecological balance of predator–prey interactions. We find that the rate of predation, fear, and hunting cooperation destabilizes the system, whereas prey refuge stabilizes the system. Time delay in the cooperation behavior generates irregular oscillations whereas delay in refuge stabilizes an otherwise unstable system. Seasonal variations in the level of fear and refuge generate higher periodic solutions and bursting patterns, respectively, which can be replaced by simple 1-periodic solution if the cooperation and fear are also allowed to vary with time in the former and latter situations. Higher periodicity and bursting patterns are also observed due to synergistic effects of delay and seasonality. Our results indicate that the combined effects of fear, refuge and hunting cooperation play a major role in maintaining a healthy ecological environment.

Time-Delayed Bioreactor Model of Phenol and Cresol Mixture Degradation with Interaction Kinetics

This paper is devoted to a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The biomass specific growth rate is presented as sum kinetics with interaction parameters (SKIP). A discrete time delay is introduced and incorporated into the biomass growth response. These two aspects—the mutual influence of the two substrates and the natural biological time delay in the biomass growth rate—are new in the scientific literature concerning bioreactor (chemostat) models. The equilibrium points of the model are determined and their local asymptotic stability as well as the occurrence of local Hopf bifurcations are studied in dependence on the delay parameter. The existence and uniqueness of positive solutions are established, and the global stabilizability of the model dynamics is proved for certain values of the delay. Numerical simulations illustrate the global behavior of the model solutions as well as the transient oscillations as a result of the Hopf bifurcation. The performed theoretical analysis and computer simulations can be successfully used to better understand the biodegradation dynamics of the chemical compounds in the bioreactor and to predict and control the system behavior in real life conditions.

Learning delay dynamics for multivariate stochastic processes, with application to the prediction of the growth rate of COVID-19 cases in the United States

Delay differential equations form the underpinning of many complex dynamical systems. The forward problem of solving random differential equations with delay has received increasing attention in recent years. Motivated by the challenge to predict the COVID-19 caseload trajectories for individual states in the U.S., we target here the inverse problem. Given a sample of observed random trajectories obeying an unknown random differential equation model with delay, we use a functional data analysis framework to learn the model parameters that govern the underlying dynamics from the data. We show the existence and uniqueness of the analytical solutions of the population delay random differential equation model when one has discrete time delays in the functional concurrent regression model and also for a second scenario where one has a delay continuum or distributed delay. The latter involves a functional linear regression model with history index. The derivative of the process of interest is modeled using the process itself as predictor and also other functional predictors with predictor-specific delayed impacts. This dynamics learning approach is shown to be well suited to model the growth rate of COVID-19 for the states that are part of the U.S., by pooling information from the individual states, using the case process and concurrently observed economic and mobility data as predictors.

Time-Delay Synchronization and Anti-Synchronization of Variable-Order Fractional Discrete-Time Chen–Rossler Chaotic Systems Using Variable-Order Fractional Discrete-Time PID Control

In this research paper, we solve the problem of synchronization and anti-synchronization of chaotic systems described by discrete and time-delayed variable fractional-order differential equations. To guarantee the synchronization and anti-synchronization, we use the well-known PID (Proportional-Integral-Derivative) control theory and the Lyapunov–Krasovskii stability theory for discrete systems of a variable fractional order. We illustrate the results obtained through simulation with examples, in which it can be seen that our results are satisfactory, thus achieving synchronization and anti-synchronization of chaotic systems of a variable fractional order with discrete time delay.

ANALYSIS OF BOGDANOV–TAKENS BIFURCATION OF CODIMENSION 2 IN A GAUSE-TYPE MODEL WITH CONSTANT HARVESTING OF BOTH SPECIES AND DELAY EFFECT

In this paper, we investigate the dynamics of a system in which both prey and predator are harvested with constant rate. Our main objective is to find the effects of harvesting on equilibria, stability, and bifurcations in the system, which may be useful for biological management. The existence and stability of equilibrium points of the model are further investigated. A thorough qualitative analysis has been carried out based on bifurcation theory in dynamical systems and to validate our analytical findings, a large scale numerical simulation has been performed by using plausible values of parameters involved. It is shown that the model can exhibit Hopf bifurcation. The first Lyapunov coefficient is calculated to determine the direction of limit cycle of Hopf bifurcation. Also, it has been proven analytically that the system exhibits Bogdanov–Takens bifurcation of codimension 2. Moreover, discrete-time delay effect has been included due to gestation of the predator species on the same system and observed Hopf bifurcation with respect to the delay parameter. This study renders important tools for investigations of the dynamics of biotic organisms for the management and control of over harvesting. Some phase plane analysis has been carried out to support our analytical results.

Petri-Net Based Multi-Objective Optimization in Multi-UAV Aided Large-Scale Wireless Power and Information Transfer Networks

Power consumption in wireless sensor networks is high, and the lifetime of a battery has become a bottleneck, restricting network performance. Wireless power transfer with a ground mobile charger is vulnerable to interference from the terrain and other factors, and hence it is difficult to deploy in practice. Accordingly, a novel paradigm is adopted where a multi-UAV (unmanned aerial vehicle) with batteries can transfer power and information to SDs (sensor devices) in a large-scale sensor network. However, there are discrete events, continuous process, time delay, and decisions in such a complicated system. From the perspective of a hybrid system, a hybrid colored cyber Petri net system is proposed here to depict and analyze this problem. Furthermore, the energy utilization rate and information collection time delay are conflict with each other; therefore, UAV-aided wireless power and information transfer is formulated as a multi-objective optimization problem. For this reason, the MAC-NSGA II (multiple ant colony-nondominated sorting genetic algorithm II) is proposed in this work. Firstly, the optimal trajectory of multiple UAVs was obtained, and on this basis, the above two objectives were optimized simultaneously. Large-scale simulation results show that the proposed algorithm is superior to NSGA II and MOEA/D in terms of energy efficiency and information collection delay.

A new robust discrete-time sliding mode control design for systems with time-varying delays on state and input and unknown unmatched parameter uncertainties

This paper proposes a new robust Discrete-Time Sliding Mode Control (dt-smc) scheme for linear discrete-time delay system. The delays, affecting the state and the input, are assumed to be known, time-varying and bounded. The considered system is also subject to time-varying Unknown But norm-Bounded (ubb) unmatched uncertainties, that act on the state and the delayed state vectors. The designed controller is based on a novel sliding surface. Using a new Lyapunov functional candidate and by introducing slack variables, a less conservative delay-dependent sufficient condition ensuring the asymptotic stability of the closed loop system is derived in terms of Linear Matrix Inequalities (lmis). With respect to that condition a new dt-smc law is synthesized. The proposed control algorithm is applied to an Autonomous Underwater Vehicle (auv). The simulation results demonstrate the effectiveness of the proposed controller.

Exploring the dynamics of a tumor-immune interplay with time delay

With the effect of discrete time delay in deliberation, we propose and analyze a conceptual mathematical model for the tumor-immune interaction. The proposed model is delineated by a system of three coupled non-linear ordinary differential equations (ODEs), namely tumor cells, effector cells and cytokine Interleukin-2 (IL-2). Though simple, the model can have complicated dynamical behaviors. We have discussed the qualitative properties of the mathematical model including existence and positivity of the solutions. We find out tumor-free singular point and interior steady states in our mathematical model. We have studied the local stability analysis of the biological feasible steady states in both of the delayed and non-delayed system. By using transversality condition, we have analyzed Hopf bifurcation by using time delay τ as a bifurcation parameter. We have estimated the length of time delay parameter applying Laplace transformation for preserving the stability of period-1 limit cycle that provides the idea about the mode of action in controlling oscillations in the growth of tumor cells. We performed numerical simulations and explored their biological implications to validate our theoretical analysis. We have also drawn bifurcation diagram of delayed model with reference to the intrinsic growth rate α of tumor cell, deactivation rate d1 of tumor cell, activation rate c2 of effector cell and death rate d2 of effector cell. Theoretical and numerical analysis show that in presence of IL-2 the effector cells can cause the tumor cell population to regress.

Eigenvalue optimisation-based centralised and decentralised stabilisation of time-delay systems

Both centralised and decentralised stabilisation of multi-input multi-output linear time-invariant (LTI) systems with discrete time-delays are considered. No restriction has been made on the number and type of the time-delays. The main objective is to design LTI finite-dimensional static/dynamic stabilising centralised/decentralised output feedback controllers. For this, the decentralised pole assignment algorithm developed for decentralised stabilisation of finite-dimensional systems is extended to the time-delay case. In this approach, decentralised controllers are designed sequentially, where a centralised design approach is used in each step. As the centralised design approach, the eigenvalue optimisation-based stabilisation method is utilised. Regarding the nonconvexity of the optimisation problem, an initialisation procedure, which determines the minimum possible dimension of the stabilising controller and the initial controller parameters, is developed. As a particular case of dynamic output feedback, the decentralised proportional-integral control is also considered. The proposed controller design methods are verified through numerical and practical examples.