Time-delay Equation Research Articles

A non-fragile robust [formula omitted] controller for continuous-time delayed models

In this paper, considering the uncertainties of the controller and the model, a non-fragile H∞ control law is robustly developed for time-delay equations with additive uncertainty. Hence, exploiting the linear matrix inequality (LMI), the robust compensator design is translated to an LMI one. In a continuous-time delayed model, the stabilization around equilibrium points and the rejection of the exogenous disturbance are discussed via the non-fragile robust H∞ controller. Such an aim is realized by checking the feasibility of the LMI. Then, through solving an LMI, the controller’s parameters would be found and updated directly. To show the superiority of the tactic in process control, the reported strategy is employed in a two-stage chemical reactor. The figurative results along with the quantitative findings consistently disclose the advantages of the addressed controller over conventional techniques.

Non-Markovian disentanglement dynamics in double-giant-atom waveguide-QED systems

We study the disentanglement dynamics of two giant atoms coupled to a common one-dimensional waveguide. We focus on the non-Markovian retarded effect in the disentanglement of the two giant atoms by taking the photon transmission time into account. By solving the time-delayed equations of motion for the probability amplitudes, we obtain the evolution of the entanglement of the two giant atoms, which are initially in the maximally entangled states in the single-excitation space. It is found that the retardation-induced non-Markovianity leads to non-exponential decay and revivals of entanglement. Concretely, we consider separate-, braided-, and nested-coupling configurations, and find that the disentanglement dynamics in these configurations exhibits different features. We demonstrate that the steady-state entanglement depends on the time delay under certain conditions in these three coupling configurations. We also study the dependence of the disentanglement of the two giant atoms on both the detuning effect and the initial-state phase effect. In addition, we consider the disentanglement dynamics of the two giant atoms, which are initially in the state superposed by zero-excitation and two-excitation components. This work will pave the way for the generation of stationary entanglement between two giant atoms, which may have potential applications in the construction of large-scale quantum networks based on the giant-atom waveguide-QED systems.

Existence Uniqueness and Stability of Nonlocal Neutral Stochastic Differential Equations with Random Impulses and Poisson Jumps

This manuscript aims to investigate the existence, uniqueness, and stability of non-local random impulsiveneutral stochastic differential time delay equations (NRINSDEs) with Poisson jumps. First, we prove theexistence of mild solutions to this equation using the Banach fixed point theorem. Next, we prove thestability via continuous dependence initial value. Our study extends the work of Wang and Wu [15] wherethe time delay is addressed by the prescribed phase space B (defined in Section 3). An example is given toillustrate the theory.

Iterative learning control for time-delay nonlinear hyperbolic distributed parameter systems

In this paper, we study the application of iterative learning control to a class of nonlinear varying time-delay equations. The handled procedure is the distributed parameter system, hyperbolic type. The selected plan is based on PD-type iterative learning control with initial state learning. Initially, we present the equation and the control law utilised. Subsequently, we presented some dilemmas. Then, sufficient conditions for monotone convergence of the error under the proper assumption are found. Also, a detailed convergence analysis is given based on the once-offered lemmas and Gronwall inequality. Finally, we show the usefulness of the method utilising a numerical model. Abbreviations DPS: distributed parameter system; ILC: iterative learning control; PDE:partial differential equation; DPS: partial difference system; PD: proportional derivative

FPGA implementation of nonlinear equations with delay

The study of dynamic systems has been the subject of growing interest in recent decades. Its purpose is to reveal the phenomena or behaviors of the systems, and even those hidden, for given initial conditions and parameter values. Their implementations allow them to be exploited in various fields of application. The present work deals with the study and FPGA implementation of nonlinear time-delay equations. These are the Johnson and Moon equation and the Mackey-Glass equation. The stability analysis of these two models allowed to plot the stability chart in the parameter space of each system, thus delimiting the areas of stability and instability. Numerical simulation for different values of the control parameter revealed a damped, periodic and even chaotic oscillation for the two systems. An architecture based on the fourth-order Runge-Kutta method (RK4) taking into account the delay, in which the calculations are performed according to the 32-bit IEEE-754–1985 floating-point numbers standard, has been proposed for the hardware description of the systems. The corresponding VHDL code in each case was synthesized on the Xilinx Virtex-6 family XCV6VLX130T-3FF484 FPGA chip and the maximum operating frequencies obtained are of the order of 431.727 MHz and 432.280 MHz respectively for the Johnson and Moon and Mackey-Glass equation. The results obtained from the Modelsim simulation of the FPGA implementation of the models are identical to those of the numerical simulation. This contribution highlighted a new approach for the FPGA implementation of time-delay systems and can be exploited for other time-delay systems.

An Approximate Model of Load Frequency Control Systems with Time Delay

In this paper we present an approximate model for load frequency control system with time delay. The load frequency control is one of the conventional power system control problems. In order to secure the stability of the grid the frequency must remain within its limited range which is achieved through the load frequency control. The load frequency control signals experience time delay that could destabilize the power systems. The presence of the time delay complicates the analysis of the load frequency control system. In this paper we present a stability method based on the Direct Frequency Response approximation for the time delay. This approximation transforms the transcendental time delay equation into linear equation. This results in a simple stability criterion for the loadfrequency control system with time delay. A one-area load frequency control system is chosen as a case study. The effectiveness of the proposed approximation has been tested through simulation and comparison with the published research work. By tracking the eigenvalues or using Routh's criterion the maximum delay margin can be estimated. The proposed stability criterion has been compared with the most recent methods and showed it is merit. The range of the PI controller parameters for a given time delay can be determined which is very important in practice.

Representation of solutions of nonhomogeneous conformable fractional delay differential equations

• Linear and nonlinear time-delay equation with conformable fractional derivative is proposed. • Conformable delay perturbation of matrix exponential function is defined for the first time. • Solution of the equation is determined with the help of Conformable delay perturbation of matrix exponential function. • Its existence, uniqueness and stability are proved via Banach Fixed Point Theorem and Ulam-Hyers Stability. This paper is about the conformable fractional delay equations. We offer a conformable delay perturbation of matrix exponential function to give the representation of solutions for linear nonhomogeneous conformable fractional delay differential equations . Lastly, the existence and uniqueness of solutions and Ulam-Hyers stability of the equations are proved.

Novel Control-Oriented Models for Liquid Transport in Falling Film Evaporator Tubes

The paper presents novel control-oriented transport models for evaporating liquid films in the tubes of a falling film evaporator. In this context, our goal consists in qualitatively mapping the experimentally observed input-output behavior. Two transport models are proposed, where the difference between them is that one allows overtaking of liquid particles and the other does not. The transport models are equipped with two new models of evaporation which are different from the commonly assumed uniform evaporation. The models are initially developed from the conservation laws in the form of partial differential equations. Using the method of characteristics we then obtain the input-output relations for the proposed models in the form of time-delay equations. The time-delay representation is advantageous for simulation and for the future control design. In a simulation study, we observe the principal properties of the models and find that they correspond well with the experimentally observed input-output behavior.

Dissolved Oxygen-And Temperature-Dependent Simulation of the Population Dynamics of Moon Jellyfish (Aurelia coerulea) Polyps

As the extent of hypoxia in coastal waters increases, the survivorship of jellyfish polyps relative to that of competing sessile organisms often increases, enabling them to reproduce more prolifically, leading to a medusa bloom in the following year. Quantifying the population of polyps can be used to predict when these blooms will occur. We used a time-delayed logistic equation to quantify the response to variable dissolved oxygen (DO) concentrations and temperatures in a population of moon jellyfish (Aurelia coerulea) polyps on substrates that carried competing sessile organisms. The availability of substrate depends on the DO threshold for each competitor, and substrates only become available to the polyps during hypoxic periods. We used the median sublethal concentration (SLC50) thresholds of hypoxia for different groups of benthic organisms to calculate the DO-dependent survivorship of A. coerulea polyps competing on the substrate. Since the median lethal time (LT50) for cnidarians is close to 240 h, we chose a 10-day delay in the time-delayed logistic equation. The carrying capacity is determined every 10 days depending on DO concentrations and temperature. The polyps reproduce by budding at a temperature-dependent rate after settling on the substrate during the hypoxic period, and thus, the annual polyp reproduction rate is determined by multiplying the temperature-dependent budding rate by the DO-dependent survivorship. The duration of hypoxia is a key factor determining the polyp population, which can increase more as the duration of hypoxia increases. Modeling simulations were compared to observed data. In this model, the DO and temperature distribution data make it possible to quantify variations in the population of the A. coerulea polyps, which can be used to predict the abundance and appearance of medusa the following year.

A qualitative study and numerical simulations for a time-delayed dispersive equation

This paper deals with the stability analysis of a nonlinear time-delayed dispersive equation of order four. First, we prove the well-posedness of the system and give some regularity results. Then, we show that the zero solution of the system exponentially converges to zero when the time tends to infinity provided that the time-delay is small and the damping term satisfies reasonable conditions. Lastly, an intensive numerical study is put forward and numerical illustrations of the stability result are provided.

Modeling of generating ultra-high frequency oscillations in VCSEL integrated with cascaded transverse coupled cavities

We report on the design and modeling of vertical-cavity surface-emitting laser (VCSEL) integrated in the lateral direction with a cascade of multiple passive cavities. The scheme is proposed to increase the bandwidth of the VCSEL in the mm-waveband to generate ultra-high-frequency oscillations with low-frequency chirp. The model treats the transverse feedback induced by the cascaded transverse coupled cavities (TTCs) as time delay of the transverse slow light due to round trips in these cavities. The simulation is based on numerical integration of the modified time-delay rate equations of the intensity and phase of the electric field. The simulation results are used to optimize the TCCs parameters, including the coupling ratio and TCC length, for generating ultra-high-frequency signals with high intensity, low chirping and low distortion. The obtained results show that the proposed structure could achieve 300% enhancement of the modulation bandwidth of the C-VCSEL due to either extended carrier-photon resonance (CPR) frequencies of photon–photon resonance (PPR). Signals with frequencies as high as (40–49 GHz) with second-harmonic distortion lower than − 30 dB are presented.

A Class of Shock Wave Solution to Singularly Perturbed Nonlinear Time-Delay Evolution Equations

Nonlinear singularly perturbed problem for time-delay evolution equation with two parameters is studied. Using the variables of the multiple scales method, homogeneous equilibrium method, and approximation expansion method from the singularly perturbed theories, the structure of the solution to the time-delay problem with two small parameters is discussed. Under suitable conditions, first, the outer solution to the time-delay initial boundary value problem is given. Second, the multiple scales variables are introduced to obtain the shock wave solution and boundary layer corrective terms for the solution. Then, the stretched variable is applied to get the initial layer correction terms. Finally, using the singularly perturbed theory and the fixed point theorem from functional analysis, the uniform validity of asymptotic expansion solution to the problem is proved. In addition, the proposed method possesses the advantages of being very convenient to use.

Variable-order fractal-fractional time delay equations with power, exponential and Mittag-Leffler laws and their numerical solutions

In this paper, a numerical method based on the Lagrangian piece-wise interpolation is proposed to solve variable-order fractal-fractional time delay equations with power law, exponential decay and Mittag-Leffler memories. These operators permit to describe physical phenomena with variable memory and fractal variable dimension. Numerical methods were applied to simulate the variable-order time delay Mackey–Glass and synaptically coupled FitzHugh–Nagumo models. Our numerical simulations display several new attractors.

On norm-based estimations for domains of attraction in nonlinear time-delay systems

For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.

A Closed-Form Location Algorithm Without Auxiliary Variables for Moving Target in Noncoherent Multiple-Input and Multiple-Output Radar System

This paper focuses on the moving target localization and velocity measurement in incoherent centralized multiple-input and multiple-output (MIMO) radar systems with widely separated antennas in 3D space. In this paper, we assume that parameters such as time-of-arrival (TOA), frequency-of-arrival (FOA), azimuth angles and elevation angles have already been measured. With these measurements, a final closed-form solution of the target position and velocity can be obtained via the weighted least squares (WLS) method. When the target is located in the far field, due to poor system observability, a first-order Taylor expansion based on the WLS solution is necessary to obtain a more accurate and unbiased solution. Unlike the preceding papers which were based on the two-stage weighted least squares (2SWLS) method [1]-[3], in this paper, the angle information is introduced into the time delay equations and the Doppler frequency equations, so that the intermediate variables in the estimator can be eliminated [4], [5]. Meanwhile, the time delay equations and the Doppler equations are transformed into linear equations only related to the position and the speed of the target. This method, unlike 2SWLS-based methods [1]-[3], does not introduce auxiliary variables, so it does not require the decorrelation procedure. Simulation results show that the root mean-square error (RMSE) of position and velocity can reach Cramer-Rao lower bound (CRLB) when the noise is at a moderate level before the thresholding effect occurs.

SOLVING AN INVERSE PROBLEM FOR A GENERALIZED TIME-DELAYED BURGERS-FISHER EQUATION BY HAAR WAVELET METHOD

In this paper, a numerical method consists of combining Haar wavelet method and Tikhonov regularization method to determine unknown boundary condition and unknown nonlinear source term for the generalized time-delayed Burgers-Fisher equation using noisy data is presented. A stable numerical solution is determined for the problem. We also show that the rate of convergence of the method is as exponential $ \Bigl(O\left(\frac{1}{2^{J+1}}\right)\Bigr) $, where $ J $ is maximal level of resolution of wavelet. Some numerical results are reported to show the efficiency and robustness of the proposed approach for solving the inverse problems.

Doppler-shifted laser self-mixing interferometry for enhanced detection sensitivity

ABSTRACTLaser self-mixing interferometry (SMI) is a well-known measurement technique having been applied to the fields of geometrical quantities detection, medical treatment and industry manufacturing. Its detection sensitivity can be improved by frequency shifting towards the laser's relaxation oscillation frequency (ROF), which has always been implemented by a pair of acousto-optic modulators (AOM). This manuscript presents a novel method based on Doppler effect for frequency shifting, which is based on a rotating disk and has the advantages of convenient adjustment and low cost. The theoretical analysis in the form of Doppler shift and time-delayed rate equations is followed, and simulative and experimental results are included to prove its validity. To our knowledge, this technique has not been used in practice till now, and the proposed structure, with these advantages, can find significant applications in designing high performance SMI sensors, and can extend SMI measurement to more kinds of non-cooperative surfaces.

Generalized model-independent characterization of strong gravitational lenses V: reconstructing the lensing distance ratio by supernovae for a general Friedmann universe

ABSTRACT We determine the cosmic expansion rate from supernovae of type Ia to set up a data-based distance measure that does not make assumptions about the constituents of the universe, i.e. about a specific parametrization of a Friedmann cosmological model. The scale, determined by the Hubble constant H0, is the only free cosmological parameter left in the gravitational lensing formalism. We investigate to which accuracy and precision the lensing distance ratio D is determined from the Pantheon sample. Inserting D and its uncertainty into the lensing equations for given H0, especially the time-delay equation between a pair of multiple images, allows to determine lens properties, especially differences in the lensing potential (Δϕ), without specifying a cosmological model. We expand the luminosity distances into an analytic orthonormal basis, determine the maximum-likelihood weights for the basis functions by a globally optimal χ2-parameter estimation, and derive confidence bounds by Monte Carlo simulations. For typical strong lensing configurations between z = 0.5 and 1.0, Δϕ can be determined with a relative imprecision of 1.7 per cent, assuming imprecisions of the time delay and the redshift of the lens on the order of 1 per cent. With only a small, tolerable loss in precision, the model-independent lens characterisation developed in this paper series can be generalised by dropping the specific Friedmann model to determine D in favour of a data-based distance ratio. Moreover, for any astrophysical application, the approach presented here, provides distance measures for z ≤ 2.3 that are valid in any homogeneous, isotropic universe with general relativity as theory of gravity.

Generalised model-independent characterisation of strong gravitational lenses – VI. The origin of the formalism intrinsic degeneracies and their influence on H0

We give a physical interpretation of the formalism intrinsic degeneracies of the gravitational lensing formalism that we derived on a mathematical basis in part IV of this series. We find that all degeneracies occur due to the partition of the mass density along the line of sight. Usually, it is partitioned into a background (cosmic) density and a foreground deflecting object. The latter can be further partitioned into a main deflecting object and perturbers. Weak deflecting objects along the line of sight are also added, either to the deflecting object or as a correction of the angular diameter distances, perturbing the cosmological background density. A priori, this is an arbitrary choice of reference frame and partition. They can be redefined without changing the lensing observables which are sensitive to the integrated deflecting mass density along the entire line of sight. Reformulating the time delay equation such that this interpretation of the degeneracies becomes easily visible, we note that the source can be eliminated from this formulation, which simplifies reconstructions of the deflecting mass distribution or the inference of the Hubble constant, $H_0$. Subsequently, we list necessary conditions to break the formalism intrinsic degeneracies and discuss ways to break them by model choices or including non-lensing observables like velocity dispersions along the line of sight with their advantages and disadvantages. We conclude with a systematic summary of all formalism intrinsic degeneracies and possibilities to break them.

Analysis, Design, and Implementation of Accurate ZVS Angle Control for EV Battery Charging in Wireless High-Power Transfer

To acquire high efficiency and reduce electromagnetic interference in wireless high-power charging for electric vehicles (EVs), it is of great importance to realize accurate zero voltage switching angle (ZVSA) control of the inverter. However, the traditional zero-crossing detection cannot be applied in high power because of considerable noise and poor accuracy in wide power range applications. In this paper, to suppress the noise in high power, a current hysteresis comparator is adopted, and an enhanced phase detection methodology is proposed to measure the phase of resonant current by using a reference signal produced by a processor. Meanwhile, to improve the accuracy of ZVSA, a uniform time delay compensation method (UTDCM) by considering the whole time delays synthetically is proposed, and a uniform time delay equation can be obtained to guarantee the high accuracy of phase detection, especially in the wide power range. Finally, a novel control strategy with the ZVSA loop based on UTDCM is proposed for battery charging in wireless high-power transfer system. A 500-W wireless power transfer prototype is built to verify the accuracy of ZVSA based on UTDCM and the system efficiency can achieve 94.17% with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> = 0.22.