Equations In Real Time Research Articles

Integration of Augmented Reality and Modeling Techniques in Dynamic Systems for Teaching Theorems and Demonstrations in University Mathematics Courses: A Differential Equations-Based Approach and Its Applications.

Teaching theorems and proofs in mathematics at the university level poses a considerable challenge due to their high abstraction and conceptual complexity. Augmented reality (AR) emerges as an innovative tool that transforms learning by facilitating the visualization and understanding of complex mathematical concepts, particularly in the realm of differential equations, which are fundamental for modeling dynamic systems. This technology allows students to interact with three-dimensional representations, enhancing their comprehension and analytical skills by visualizing the effects and solutions of equations in real time. Despite technological advancements in education, traditional teaching methods remain static and abstract, limiting the assimilation of concepts. The research employed a quantitative approach and a documentary methodology to analyze the effectiveness of integrating AR and modeling techniques, providing data that evidences their positive impact on academic performance. A thorough review of relevant literature was conducted, consolidating a theoretical framework that highlights the importance of these emerging technologies. The findings suggest that implementing AR in the classroom not only improves knowledge retention but also enriches the discussion on mathematics education in higher education.

A full-potential and multiscale computational scheme for interactions between ultrafast intense laser pulses and condensed medium

We have presented a multiscale computational scheme for the simulation of interactions between ultrafast intense laser pulses and condensed medium. This approach solves the coupled time-dependent Kohn-Sham (TDKS) and Maxwell equations in real time and provides a unified description of microscopic electron dynamics and macroscopic light propagation from first principles and applicable to wide ultrafast phenomena. Especially, as the TDKS equation was solved within the framework of all-electron full-potential and the augmented plane wave with local orbitals basis sets were adopted, this approach overcomes the drawbacks of real space grids and pseudopotential approximations and works well for some extreme conditions that the information near the nucleus is important. We demonstrated the approach by calculating the microscopic electron dynamics of crystalline silicon (Si) excited with an intense laser pulse and the macroscopic propagation of a laser pulse in a ∼3.5 μm free standing Si film with the feedback of microscopic electron dynamics coupled together. We also verified the availability of this approach under extreme condition with hundreds TPa (1 TPa = 1012 Pa) pressure in a metallic solid neon, and reasonable results were obtained. Our approach provides a new paradigm for multiscale simulations of ultrafast phenomena that induced by intense laser pulses in condensed medium.

Multi-Layer Fault-Tolerant Robust Filter for Integrated Navigation in Launch Inertial Coordinate System

As to an aerospace vehicle, the flight span is large and the flight environment is complex. More than that, the existing navigation algorithms cannot meet the needs to provide accurate navigation parameters for aerospace vehicles, which results in the decline of navigation accuracy. This paper proposes a multi-layer, fault-tolerant robust filtering algorithm of aerospace vehicle in the launch inertial coordinate system to address this problem. Firstly, the launch inertial coordinate system is used as the reference coordinate system for navigation calculation, and the state equation and measurement equation of the navigation system are established in this coordinate system to improve the modeling accuracy of the navigation system. On this basis, a multi-layer, fault-tolerant robust filtering algorithm is designed to estimate and compensate the unknown input in the state equation in real time and adjust the noise variance matrix in the measurement equation adaptively. Simulation results show that the errors about the integrated navigation system output parameters are reduced, through this algorithm, which improves the attitude, velocity and position estimation accuracy of the integrated navigation system. In addition, the algorithm enhances the fault tolerance and robustness of the filtering algorithm.

STM32F429 Discovery Board-Based Emulator for Lotka-Volterra Equations

Lotka-Volterra equations are commonly used in prey-predator population studies. Simulation programs are commonly used to produce solutions of Lotka-Volterra equations and to examine their initial value dependendence. In literature, chaotic waveform generators, ECG and EEG generators have been made and used for research and education. To the best of our knowledge, such an electrical circuit to produce the Lotka-Volterra waveforms does not exist. Such a circuit can be made using either analog or digital circuit components. However, such a device may be used for education in classroom and also to prove concepts by population researchers. In this study, implementation and experimental verification of the microcontroller-based circuit which solves LotkaVolterra equations in real time and produces its waveforms are presented. Euler method is used to solve the equation system in discrete time. Presented design has been implemented using an STM32F429 Discovery Board, two DACs and four opamps. The microcontroller sends the signals to the outputs of the circuit using digital-to-analog converters and opamps. The waveforms acquired experimentally from the implemented circuit outputs matches well with those obtained from numerical simulations.

АНАЛИЗ УСТОЙЧИВОСТИ ЖЕСТКИХ СИСТЕМ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

A method for analyzing stability in the sense of Lyapunov for systems of ordinary differentialequations is proposed. The method is based on stability criteria in the form of necessary and sufficientconditions obtained on the basis of vector-matrix transformations of difference numericalintegration schemes. The varieties of criteria in multiplicative, additive and matrix form are presented.The design of the criteria implies the possibility of their programmatic realization. To increasethe reliability of the stability analysis, the approximations of the solution included in theconstruction of the criteria are based on piecewise interpolation approximation by Lagrange polynomialsconverted to a form with numerical coefficients. A programming and numerical experimentis carried out to analyze the stability of the Belousov-Jabotinsky periodic reaction model,which belongs to the class of rigid systems, under given initial conditions. The analysis is carriedout on the basis of the presented criteria and the results of the program clearly determine the natureof the stability in real time. Based on the results of the experiment, it can be argued that replacingthe difference approximations of the solution with piecewise interpolation approximationsincreases the reliability of the stability analysis, reduces the study time, and makes it possible todetermine the asymptotic properties of the solution. In general, the proposed approach is an alternativeto the methods of the qualitative theory of differential equations and makes it possible toreliably determine the stability of rigid systems of ordinary differential equations in real time.

Echo state network-based online optimal control for discrete-time nonlinear systems

This paper investigates the online optimal control problem of discrete-time nonlinear systems using echo state network (ESN)-based adaptive dynamic programming (ADP) method. An online iterative learning algorithm is proposed to solve the partial differential Hamilton–Jacobi–Bellman (HJB) equation in real time. A novel neural networks (NN) critic-actor architecture is presented using two ESNs to implement the ADP method. Then, two online learning laws of the output weights are designed for searching the optimal cost function and control policy. The stability of system and output weights is analysed using Lyapunov approach. Three simulations are given to show the feasibility and effectiveness of the designed algorithm.

Valley Depolarization Dynamics in Monolayer Transition-Metal Dichalcogenides: Role of the Satellite Valley.

The valley depolarization dynamics of free holes in monolayer transition-metal dichalcogenides are studied by solving the Boltzmann transport equation in real time fully ab inito. While monolayer MoSe2, WS2, WSe2, and MoTe2 possess long hole valley lifetimes due to the spin-valley locking effect, monolayer MoS2 unexpectedly shows ultrafast valley dynamics, with a hole valley lifetime two orders of magnitude shorter than those of the above four materials at room temperature. It is further revealed that the existence of the satellite Γ valley in MoS2 provides an additional hole relaxation path where the Γ valley acts as an intermediate in the hole relaxation between primary K' and K valleys, and moreover, the strong scattering between primary and satellite valleys ensures the ultrafast valley depolarization. By uncovering the pivotal role of the satellite valley, our results may have significant implications for finely controlling valley depolarization in the multivalley materials.

Design and Construction of A Two Axes Sun Tracker Prototype with A Real Time Controller and Hybrid Stepper Motors (Dept.M)

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In Memoriam: Mortimer L. Mendelsohn: (1 December, 1925-8 January, 2020).

In Memoriam: Mortimer L. Mendelsohn: (1 December, 1925-8 January, 2020).

Hardware Emulation of Energization of a Long Transmission Line by High-Frequency Power Electronic Converter

Hardware emulators, simulating a test environment in real time, are an essential tool in testing power system equipment. This article presents a hardware emulator of a programmable transmission line capable of simulating high-frequency transients. Emulation of, energization of long transmission line, requires the observer to solve transmission line or telegraphers equations in real time. This article identifies a continuous time model that captures the wave nature and suitable for real-time implementation. This article derives the discrete-time model to be solved in the observer of the hardware emulator. A step-by-step procedure is given to determine two key emulator parameters: observer sampling frequency, and switching frequency of the power amplifier considering different hardware, event, and software related constraints. For high bandwidth requirement, a high switching frequency, 100-kHz, 12-kVA silicon carbide (SiC) based voltage source converter is designed to operate under a current controller with a gain crossover frequency of 5 kHz. A system on chip that combines an Anvanced RISC Machine (ARM) processor along with an field programmable gate array (FPGA) is used to implement the observer. Experimental results verify accuracy of the designed emulator in the event of energization of a long transmission line. Though proposed transmission line emulator can only be used to study the energization of a long line with one end terminated with shunt reactor, the solution can be extended for emulation of a long line that links two sections of power grid.

Computationally Efficient Harmonic-Based Reactive Exploration

Although Harmonic Potential Fields constitute a powerful tool for tackling the autonomous robot exploration problem, yet their applicability is limited by the heavy computational load involved in solving the Laplace equation in real time. In this letter, we propose a computationally efficient exploration scheme employing a Fast Multipole accelerated Boundary Element Method, which enjoys both linear complexity w.r.t. the boundary's size as well as linear memory requirements. Furthermore, we devise an adaptive control law for the specified boundary conditions that allows us to tune the robot's behavior without affecting the inherent safety and convergence properties of the underlying potential field. Finally, we validate the performance of the proposed exploration scheme through extensive realistic simulations.

Time-Dependent Magnons from First Principles

We propose an efficient and non-perturbative scheme to compute magnetic excitations for extended systems employing the framework of time-dependent density functional theory. Within our approach, we drive the system out of equilibrium using an ultrashort magnetic kick perpendicular to the ground-state magnetization of the material. The dynamical properties of the system are obtained by propagating the time-dependent Kohn–Sham equations in real time, and the analysis of the time-dependent magnetization reveals the transverse magnetic excitation spectrum of the magnet. We illustrate the performance of the method by computing the magnetization dynamics, obtained from a real-time propagation, for iron, cobalt, and nickel and compare them to known results obtained using the linear-response formulation of time-dependent density functional theory. Moreover, we point out that our time-dependent approach is not limited to the linear-response regime, and we present the first results for nonlinear magnetic excitations from first principles in iron.

마이크로컨트롤러를 이용한 BLDC 모터 구동 Hardware-in-the-Loop Simulator 개발

In this paper, we deal with the development of a HIL (Hardware-in-the-Loop) simulator for BLDC motor driving using a microcontroller. We propose a modified BLDC model for the simulation of a BLDC motor using a microcontroller. The proposed HIL simulator is developed using various peripherals of STM32F407 (ARM cortex-M4 based microcontroller). The proposed HIL simulator consists of three parts. First, the part simulating the inverter analyzes the input PWM signals to calculate the corresponding input voltage. This part detects wrong PWM signals or shoot-throughs of the half-bridge. Second, the BLDC motor part is implemented by solving the modified model equations in real time. Finally, the third part generates the output of the BLDC motor. The HIL simulator generates the Back Electromotive Force (BEMF) waveform and signals of a rotary encoder and a hall sensor. Sensor signals are generated using a timer interrupt. The BEMF waveform is implemented using DAC. Moreover, a generation method using PWM and RC filter is presented. Through experiments, we illustrate that the developed HIL simulator accurately simulates the BLDC motor. Furthermore, we confirm that the BLDC motor drive algorithm designed with the developed HIL simulator can be successfully applied to real BLDC motor driving systems.

SALMON: Scalable Ab-initio Light–Matter simulator for Optics and Nanoscience

SALMON (Scalable Ab-initio Light–Mattersimulator for Optics and Nanoscience, http://salmon-tddft.jp) is a software package for the simulation of electron dynamics and optical properties of molecules, nanostructures, and crystalline solids based on first-principles time-dependent density functional theory. The core part of the software is the real-time, real-space calculation of the electron dynamics induced in molecules and solids by an external electric field solving the time-dependent Kohn–Sham equation. Using a weak instantaneous perturbing field, linear response properties such as polarizabilities and photoabsorptions in isolated systems and dielectric functions in periodic systems are determined. Using an optical laser pulse, the ultrafast electronic response that may be highly nonlinear in the field strength is investigated in time domain. The propagation of the laser pulse in bulk solids and thin films can also be included in the simulation via coupling the electron dynamics in many microscopic unit cells using Maxwell’s equations describing the time evolution of the electromagnetic fields. The code is efficiently parallelized so that it may describe the electron dynamics in large systems including up to a few thousand atoms. The present paper provides an overview of the capabilities of the software package showing several sample calculations. Program summaryProgram Title: SALMON: Scalable Ab-initio Light–Matter simulator for Optics and NanoscienceProgram Files doi:http://dx.doi.org/10.17632/8pm5znxtsb.1Licensing provisions: Apache-2.0Programming language: Fortran 2003Nature of problem: Electron dynamics in molecules, nanostructures, and crystalline solids induced by an external electric field is calculated based on first-principles time-dependent density functional theory. Using a weak impulsive field, linear optical properties such as polarizabilities, photoabsorptions, and dielectric functions are extracted. Using an optical laser pulse, the ultrafast electronic response that may be highly nonlinear with respect to the exciting field strength is described as well. The propagation of the laser pulse in bulk solids and thin films is considered by coupling the electron dynamics in many microscopic unit cells using Maxwell’s equations describing the time evolution of the electromagnetic field.Solution method: Electron dynamics is calculated by solving the time-dependent Kohn–Sham equation in real time and real space. For this, the electronic orbitals are discretized on a uniform Cartesian grid in three dimensions. Norm-conserving pseudopotentials are used to account for the interactions between the valence electrons and the ionic cores. Grid spacings in real space and time, typically 0.02 nm and 1 as respectively, determine the spatial and temporal resolutions of the simulation results. In most calculations, the ground state is first calculated by solving the static Kohn–Sham equation, in order to prepare the initial conditions. The orbitals are evolved in time with an explicit integration algorithm such as a truncated Taylor expansion of the evolution operator, together with a predictor–corrector step when necessary. For the propagation of the laser pulse in a bulk solid, Maxwell’s equations are solved using a finite-difference scheme. By this, the electric field of the laser pulse and the electron dynamics in many microscopic unit cells of the crystalline solid are coupled in a multiscale framework.

PMUs-Based Voltage Control of Isolated Power Systems With High Penetration of Wind Power

Isolated power systems are constructed to improve the wind power accommodation increasingly. This paper proposes a fast voltage control scheme based on phasor measurement units (PMUs) for isolated power systems with high penetration of wind power. The scheme is able to reduce the fast voltage variations and improve post fault voltage dynamics. Taking an actual isolated power system with high penetration of wind power for aluminum production as an example, a reduced system model has been derived by using PMU measurements. Then, the time delay of PMU measurements is compensated by applying Pade approximation. A fast voltage control model involving the reduced system and time delay compensation is established. The solution to the voltage control problem is formulated as a linear quadratic regulator. The voltage control scheme is mainly obtained by the procedures involving establishing the reduced system with PMU measurements, time delay compensation, and solution of Riccati equation in real time. The actual isolated power system is used to validate the effectiveness of the proposed scheme.

Quantum Monte Carlo solution of the dynamical mean field equations in real time

We present real-time inchworm quantum Monte Carlo results for single-site dynamical mean field theory on an infinite coordination number Bethe lattice. Our numerically exact results are obtained on the L-shaped Keldysh contour and, being evaluated in real-time, avoid the analytic continuation issues typically encountered in Monte Carlo calculations. Our results show that inchworm Monte Carlo methods have now reached a state where they can be used as dynamical mean field impurity solvers and the dynamical sign problem can be overcome. As non-equilibrium problems can be simulated at the same cost, we envisage the main use of these methods as dynamical mean field solvers for time-dependent problems far from equilibrium.

Finite-Horizon $H_{\infty }$ Tracking Control for Unknown Nonlinear Systems With Saturating Actuators.

In this paper, a neural network (NN)-based online model-free integral reinforcement learning algorithm is developed to solve the finite-horizon optimal tracking control problem for completely unknown nonlinear continuous-time systems with disturbance and saturating actuators (constrained control input). An augmented system is constructed with the tracking error system and the command generator system. A time-varying Hamilton-Jacobi-Isaacs (HJI) equation is formulated for the augmented problem, which is extremely difficult or impossible to solve due to its time-dependent property and nonlinearity. Then, an actor-critic-disturbance NN structure-based scheme is proposed to learn the time-varying solution to the HJI equation in real time without using the knowledge of system dynamics. Since the solution to the HJI equation is time-dependent, the form of NNs representation with constant weights and time-dependent activation functions is considered. Furthermore, an extra error is incorporated in order to satisfy the terminal constraints in the weight update law. Convergence and stability proofs are given based on the Lyapunov theory for nonautonomous systems. Two simulation examples are provided to demonstrate the effectiveness of the designed algorithm.

A nonlinearly activated neural dynamics and its finite-time solution to time-varying nonlinear equation

This paper proposes a nonlinearly activated neural dynamics to solve time-varying nonlinear equations in real time. Different from most existing neural dynamics for solving time-varying nonlinear equations, the proposed neural dynamics can converge in finite time. In addition, the upper bound of convergence time is estimated analytically in theory. Simulations are performed to evaluate the performance of the proposed neural dynamics, which substantiate the effectiveness and superiority of the finite-time convergent neural dynamics for solving time-varying nonlinear equations in real time, as compared with the conventional gradient-based neural dynamics and the recently proposed Zhang neural dynamics.

Synthesis of Optimal Distributed Measurement Systems for Control of an Elastic Object

Effective control of an elastic space vehicle is possible only on the basis of exact information on its elastic oscillations. Exact estimation of these oscillations is possible via a distributed system of measuring instruments installed in different points of the elastic design. In recent years, new types of inertial measuring devices came into use like micro-electro-mechanical (MEMS) gyroscopes and accelerometers. Significant progress in the development of on-board computers allows the use of modern means of filtering algorithms, based on numerical integration of a large number of differential equations in real time. Miniaturization of sensors and the decrease of their power consumption increases admissible quantity of used sensitive elements and reduces restrictions for their placement. This article presents new results on optimization of placement of system of sensors on an elastic controlled object.

Electron dynamics in complex environments with real-time time dependent density functional theory in a QM-MM framework.

This article presents a time dependent density functional theory (TDDFT) implementation to propagate the Kohn-Sham equations in real time, including the effects of a molecular environment through a Quantum-Mechanics Molecular-Mechanics (QM-MM) hamiltonian. The code delivers an all-electron description employing Gaussian basis functions, and incorporates the Amber force-field in the QM-MM treatment. The most expensive parts of the computation, comprising the commutators between the hamiltonian and the density matrix-required to propagate the electron dynamics-, and the evaluation of the exchange-correlation energy, were migrated to the CUDA platform to run on graphics processing units, which remarkably accelerates the performance of the code. The method was validated by reproducing linear-response TDDFT results for the absorption spectra of several molecular species. Two different schemes were tested to propagate the quantum dynamics: (i) a leap-frog Verlet algorithm, and (ii) the Magnus expansion to first-order. These two approaches were confronted, to find that the Magnus scheme is more efficient by a factor of six in small molecules. Interestingly, the presence of iron was found to seriously limitate the length of the integration time step, due to the high frequencies associated with the core-electrons. This highlights the importance of pseudopotentials to alleviate the cost of the propagation of the inner states when heavy nuclei are present. Finally, the methodology was applied to investigate the shifts induced by the chemical environment on the most intense UV absorption bands of two model systems of general relevance: the formamide molecule in water solution, and the carboxy-heme group in Flavohemoglobin. In both cases, shifts of several nanometers are observed, consistently with the available experimental data.