Eigenvalues Of System Research Articles

A Consistent Comparison of Upper Subcritical Limit Methods

This study conducts a consistent and comprehensive comparison of several methods for estimating upper subcritical limits (USLs) for nuclear criticality safety analysis. To eliminate inconsistency caused by the discrepancy between the estimated covariance data and the true degree of uncertainty present in nuclear data, the experimental system eigenvalues k exp were assumed to equal the calculated eigenvalue for the systems using nominal ENDF/B-VII.1 cross sections, and the estimated calculated eigenvalue k calc was assumed to equal the calculated eigenvalue for the systems using nuclear data from one perturbed cross-section library. USLs are estimated for a variety of validation application cases using the Whisper, TSURFER, and USLSTATS tools and are compared to a reference 95/95 limit. The TSURFER approach produced the strongest agreement with the reference USLs; the Whisper produced USLs that were reliably conservative compared to the reference USL; and the USLSTATS approach experienced difficulty consistently producing accurate USL estimates. Last, this study observed a noteworthy discrepancy when using Cholesky decomposition to randomly sample neutron cross-section covariance data that are small in magnitude.

On the convergence of the fixed point method for solving neutron transport alpha eigenvalue problems

It was shown that the Fixed Point Method (also known as the Rayleigh Quotient Method) is several times faster than the Critical Search Method for solving neutron transport alpha eigenvalue problems. It was also shown that the Fixed Point Method is able to determine the alpha eigenvalues of sub-critical systems that are beyond the reach of the Critical Search Method. Despite these significant advances, the Fixed Point Method remains an unproven algorithm. This report provides a proof.

Solutions to the Schrödinger equation using deep neural networks for integrated photonics

Abstract This paper introduces a novel method for solving the Schrödinger equation through the use of deep neural networks (DNNs), presenting a significant departure from traditional techniques. Conventional approaches to solving the Schrödinger equation, such as analytical methods and numerical algorithms, often face challenges when dealing with complex quantum systems due to their inherent limitations. These traditional methods can become cumbersome or even infeasible as the complexity of the systems increases. In contrast, our approach harnesses the capabilities of deep neural networks to approximate both the wavefunction and the energy eigenvalues of quantum systems. By leveraging the flexible and powerful nature of DNNs, we provide a new pathway to solving the Schrödinger equation that can potentially overcome the constraints of classical methods. To validate the effectiveness of our approach, we apply it to the particle in a box problem – a fundamental quantum mechanics model with well-established analytical solutions. This benchmark problem serves as a useful test case, allowing us to demonstrate that DNNs can not only replicate the known results accurately but also offer insights into how these networks can handle more intricate quantum systems. Our results reveal that DNNs are capable of accurately reproducing the analytical solutions for the particle in a box, illustrating their potential as a versatile tool for quantum mechanics.

Angle stability improvement using optimised proportional integral derivative with filter controller

Load inconsistency has disrupted the power system, causing rotor angle fluctuation that leads to angle instability in the system. This research suggests an innovative proportional integral derivative with filter (PIDF)-based thyristor-controlled series compensator (TCSC) controller that utilise an evolutionary programming sine cosine algorithm (EPSCA) for hybrid optimisation to increase the angle stability of the power system. The challenge of the PIDF-TCSC design is transformed into an optimal control problem with respect to performance indices, such as the maximum imaginary part of system eigenvalues, damping ratio and damping factor, where another multi-objective function is utilised to determine the best stabiliser settings. Eigenvalue analysis is used to conduct the stability study in a linearised paradigm of the single-machine infinite-bus (SMIB) network. The resilience of the PIDF controller was tested using a SMIB power network under various operating circumstances. Simulation results are used to evaluate the system's effectiveness with the proposed optimised PIDF-TCSC controller to that of the system using the proportional integral derivative, proportional integral, and base case PIDF-TCSC approaches. The research findings demonstrate the efficacy of EPSCA in implementing PIDF-TCSC motif and its excellent resilient performance for improving power system stability as related to other strategies in various situations.

OPTIMAL LOAD FREQUENCY CONTROL OF INTERLINKED NONLINEAR POWER SYSTEM INTEGRATED WITH SMES-TCSC AND HVDC

In this article Multi-Area Multi-Source Interconnected Power System (MAMS-IPS) incorporated with an Automatic Voltage Regulator (AVR) in both the areas is considered for Load Frequency Control (LFC) study. Nonlinearity such as Governor Rate Constraints (GRC), Governor Dead Band (GDB), and Boiler are introduced with the thermal reheat generating unit. The effect of energy storage device (SMES) and fact device (TCSC) on the dynamic responses of IPS is also tested. PD-PID controller is suggested along with PID/PI-PD controller for minimizing the Automatic Control Error (ACE). The parameters of the proposed controller are optimized through a nature-inspired algorithm (PSO). PD-PID is superior then other suggested PID/PI-PD controllers. Objective functions ITAE, IAE, ISE, and ITSE are taken into consideration for finding the optimized values of the suggested controller as well as of TCSC and SMES devices. A step load of 10% is applied in both IPS areas. The rigidness of the suggested controller is verified by varying the loading condition of the IPS. A 10% increment in step load form is introduced for each area. The effect of renewable energy sources such as solar and a wind unit is also taken into consideration in this article. Varying step input is applied to this renewable unit to absorb its effect on the output responses of MAMS-IPS. The superiority of the PD-PID controller is verified by comparing its result with the recently published optimal controllers results described in the literature. SMES helps in improving the dynamic responses of the MAMS-IPS. Based on performance indices and Settling-time (ST) results of proposed controllers are compared with each other. For commenting on the stability of the introduced Power system (PS) eigenvalue analysis is also conducted. MATLAB version 2018 @ simulation software is used for simulation and for creating .m files

Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability

Abstract Spacecraft models that mimic the Planck satellite’s behaviour have produced information on cosmic microwave background radiation, assisting physicists in their understanding of the composition and expansion of the universe. For achieving the intended formation, a framework for a discrete fractional difference spacecraft model is constructed by the use of a discrete nabla operator of variable order containing the Mittag–Leffler kernel. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as exterior disruptions, parameterized variations, time-varying feedback delays, and actuator defects. The implementation of the Banach fixed-point approach provides sufficient requirements for the presence of the solution as well as a distinctive feature for such mechanisms Furthermore, the consistent stability is examined. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of spacecraft systems to provide justification for structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the spacecraft system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional orders in the offered system. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the spacecraft chaotic model is an intriguing and crucial subject for research.

An efficient eigenvalue bounding method: CFL condition revisited

Direct and large-eddy simulations of turbulence are often solved using explicit temporal schemes. However, this imposes very small time-steps because the eigenvalues of the (linearized) dynamical system, re-scaled by the time-step, must lie inside the stability region. In practice, fast and accurate estimations of the spectral radii of both the discrete convective and diffusive terms are therefore needed. This is virtually always done using the so-called CFL condition. On the other hand, the large heterogeneity and complexity of modern supercomputing systems are nowadays hindering the efficient cross-platform portability of CFD codes. In this regard, our leitmotiv reads: relying on a minimal set of (algebraic) kernels is crucial for code portability and maintenance! In this context, this work focuses on the computation of eigenbounds for the above-mentioned convective and diffusive matrices which are needed to determine the time-step à la CFL. To do so, a new inexpensive method, that does not require to re-construct these time-dependent matrices, is proposed and tested. It just relies on a sparse-matrix vector product where only vectors change on time. Hence, both implementation in existing codes and cross-platform portability are straightforward. The effectiveness and robustness of the method are demonstrated for different test cases on both structured Cartesian and unstructured meshes. Finally, the method is combined with a self-adaptive temporal scheme, leading to significantly larger time-steps compared with other more conventional CFL-based approaches.

PDE-based observation and predictor-based control for linear systems with distributed infinite input and output delays

This paper considers output feedback stabilization of linear systems with both distributed infinite input and output delays. We first propose a PDE-based observer to deal with distributed infinite output delays. The main idea is to transform infinite-delayed output into a transport PDE with a semi-infinite domain. It is proved that the resulted error systems are exponentially stable and thus the observer is effective. Furthermore, relying on the PDE-based observer, a predictor-based output feedback stabilization controller is provided. This work removes a restrictive condition on eigenvalues of open-loop system in the previous work, which is a distinctive advantage of this work. A simulation example is given to illustrate the effectiveness of our proposed observer and controller.

Localized and delocalized topological modes of heat.

The topological physics has sparked intensive investigations into topological lattices in photonic, acoustic, and mechanical systems, powering counterintuitive effects otherwise inaccessible with usual settings. Following the success of these endeavors in classical wave dynamics, there has been a growing interest in establishing their topological counterparts in diffusion. Here, we propose an additional real-space dimension in diffusion, and the system eigenvalues are transformed from "imaginary" to "real." By judiciously tailoring the effective Hamiltonian with coupling networks, localized and delocalized topological modes are realized in heat transfer. Simulations and experiments in active thermal lattices validate the effectiveness of the proposed theoretical strategy. This approach can be applied to establish various topological lattices in diffusion systems, offering insights into engineering topologically protected edge states in dynamic diffusive scenarios.

Power system stability enhancement through optimal PSS design

Low-frequency oscillations (LFO) are generated in interconnected electric networks because of the weak tie lines between the parts of the networks. Such oscillations have been a significant challenge for engineers that may lead to system instability if appropriate measures are not taken. This article proposes two new metaheuristic algorithms, the jellyfish search algorithm (JSA) and the tunicate swarm algorithm (TSA), to design the power system stabilizers (PSS) for single machine infinite bus (SMIB) and multimachine power system (MPS) networks. The proposed method uses the JSA and TSA to dampen out the LFOs by modifying the vital parameters of lead-lag type conventional PSS (CPSS). A damping ratio-based objective function improves both models' system damping. These methods are tested on an SMIB system and two distinct MPS networks exposed to the three-phase fault under various loading conditions. The effectiveness of the suggested approach has been studied by comparing the system eigenvalues and minimum damping ratio (MDR) produced from JSA and TSA-tuned PSS and the CPSS. In addition, time domain simulation results are compared, demonstrating that the new approach outperforms the standard techniques, such as particle swarm optimization (PSO) and backtracking search algorithm (BSA). In the SMIB system, MDR produced by JSA and TSA-based PSS is 2.55 times higher than that of CPSS. Similarly, in comparison to PSO and BSA-based PSS, JSA, and TSA-tuned PSS offer >1.4 times greater MDR in the Two Area Four Machine network and up to 8.4 times larger MDR for the IEEE-39 Bus network. Furthermore, the efficacy of the suggested JSA and TSA approaches' prediction capacity is validated by satisfying the values of statistical performance metrics.

Power Regulation of a Three-Phase L-Filtered Grid-Connected Inverter Considering Uncertain Grid Impedance Using Robust Control

Uncertain grid impedance is often common in power distribution networks; therefore, it is crucial to design an efficient controller in this situation. An issue that frequently occurs is the problem of unpredictable grid impedance, which can cause voltage fluctuations, power quality problems, and potential damage to equipment. This work provides a systematic control strategy to tackle these issues by supplying well-regulated power from a DC source to an AC power grid. A linear matrix inequality (LMI)-based robust optimal control is proposed in this paper to provide stability to the inverter system without offset error at the output side. The convergence time to steady state is minimized by solving the LMI problem to maximize the eigen value of the closed-loop system with the inclusion of the uncertainty of the filter parameter and grid impedance. Furthermore, the uncertainties in this study include the potential variation of values for the filters and the grid's impedance. These uncertainties occur because the grid impedance can fluctuate fast in the event of a fault or termination of a transmission line, while the filter's impedance can also be affected by changes in operating temperature. The simulation study of this proposed control includes a comparison between wide and narrow uncertainty ranges, as well as a performance comparison under uncertain parameters. Furthermore, this approach exhibits a lower power ripple in comparison to existing PI control method.

High-order moment interval stability/stabilization and observability of stochastic impulsive T–S fuzzy systems

This article is concerned with the problems of high-order moment interval stability/stabilization and observability of stochastic impulsive T–S fuzzy systems (SITSFSs). Firstly, the definition of high-order moment interval stability is proposed of SITSFSs in this paper. In consideration of the connection between pole placement and performance of systems, sufficient conditions for the distribution of the eigenvalues of systems within a defined interval are provided, serving as the criterion for the interval stability in high-order moment of SITSFSs. Different from the general concept of stability, besides determining the stability in high-order moment of the system, high-order moment interval stability can also describe the dynamic performance in high-order moment of systems. Then, the proposed approach in combination with fuzzy control theory provides a reliable and effective method for designing an interval state feedback controller that can ensure the stability of systems and regulate convergence rate of systems. Furthermore, moment observability of SITSFSs has also been addressed in this article. Finally, the numerical simulation verifies the validity and suitability of the proposed method.

Scalar fields in Bonnor-Melvin-Lambda universe with potential: a study of dynamics of spin-zero particles-antiparticles

In this study, our primary focus is on exploring the relativistic quantum dynamics of spin-zero scalar particles in a magnetic space-time background. Our investigation revolves around solving the Klein–Gordon (KG) equation within the framework of an electrovacuum space-time, while incorporating an external scalar potential. Specifically, we consider a cylindrical symmetric Bonnor-Melvin magnetic universe featuring a cosmological constant, where the magnetic field aligns parallel to the symmetry axis. Our approach involves deriving the radial equation of the wave equation, initially considering a linear confining potential and subsequently incorporating a Cornell-type scalar potential. We successfully obtain an approximate analytical solution for the eigenvalues of the quantum system under examination. Worth noting is our observation that the energy spectrum and the corresponding radial wave function experience notable modifications due to the presence of various factors including the cosmological constant, the topological parameter characterizing the space-time geometry, and the potential parameters.

Enhancing the trustworthiness of chaos and synchronization of chaotic satellite model: a practice of discrete fractional-order approaches

Accurate development of satellite maneuvers necessitates a broad orbital dynamical system and efficient nonlinear control techniques. For achieving the intended formation, a framework of a discrete fractional difference satellite model is constructed by the use of commensurate and non-commensurate orders for the control and synchronization of fractional-order chaotic satellite system. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as Lyapunov exponent research, phase images and bifurcation schematics. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of satellite systems in order to provide justification for the structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the satellite system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional-orders in the offered system. Additionally, the sample entropy evaluation is employed in the research to determine complexities and endorse the existence of chaos. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the satellite chaotic model is an intriguing and crucial subject for research.

Boosting reservoir computer performance with multiple delays.

Time delays play a significant role in dynamical systems, as they affect their transient behavior and the dimensionality of their attractors. The number, values, and spacing of these time delays influences the eigenvalues of a nonlinear delay-differential system at its fixed point. Here we explore a multidelay system as the core computational element of a reservoir computer making predictions on its input in the usual regime close to fixed point instability. Variations in the number and separation of time delays are first examined to determine the effect of such parameters of the delay distribution on the effectiveness of time-delay reservoirs for nonlinear time series prediction. We demonstrate computationally that an optoelectronic device with multiple different delays can improve the mapping of scalar input into higher-dimensional dynamics, and thus its memory and prediction capabilities for input time series generated by low- and high-dimensional dynamical systems. In particular, this enhances the suitability of such reservoir computers for predicting input data with temporal correlations. Additionally, we highlight the pronounced harmful resonance condition for reservoir computing when using an electro-optic oscillator model with multiple delays. We illustrate that the resonance point may shift depending on the task at hand, such as cross prediction or multistep ahead prediction, in both single delay and multiple delay cases.

Analysis of dynamic wave model for unsteady flow and sediment transport in alluvial rivers

The coupling interactions between flood propagation, sediment transport, and river morphology in alluvial rivers are mathematically described by the high-order dynamic wave model. The coupling capability of currently used dynamic wave models is systematically conducted. The results indicate that the propagation of a dynamic flood wave only depends on the Froude number, but is independent of the coupling of sediment transport and river mobility. Furthermore, based on the continuum hypothesis, the dynamic equations describing the motion of the active bed layer are obtained. A renewed dynamic wave model is established. Four families of asymptotic solutions to the eigenvalues of the renewed four-order hyperbolic system are obtained by means of the singular-perturbation technology. The results demonstrate that the interactions between flood propagation, sediment transport, and riverbed mobility are coupled. Propagation of the main dynamic flood wave and the dynamic sediment wave will be slower with the increasing deposition rate, but will be faster when the erosion intensity is enhanced. These mainly occur in the lower flow regime. In the process of deposition, the second dynamic flood wave and the dynamic bed wave will propagate both upward and downstream. Besides, the dynamic bed wave will propagate downstream and the second dynamic flood wave will only propagate upstream, regardless of the flow regime.

Radiative loss and ion-neutral collisional effects in astrophysical plasmas.

In this paper, we study the role of radiative cooling (RC) in a two-fluid model consisting of coupled neutrals and charged particles. We first analyse the linearized two-fluid equations where we include radiative losses in the energy equation for the charged particles. In a one-dimensional geometry for parallel propagation and in the limiting cases of weak and strong coupling, it can be shown analytically that the instability conditions for the thermal mode and the sound waves, the isobaric and isentropic criteria, respectively, remain unchanged with respect to one-fluid radiative plasmas. For the parameters considered in this paper, representative for the solar corona, the RC produces growth of the thermal mode and damping of the sound waves. In the weak coupling limit, the growth of the thermal instability and the damping of the sound waves is as derived in Field (Field 1965 Astrophys. J. 142, 531(doi:10.1086/148317)) using the charged fluid properties. When neutrals are included and are sufficiently coupled to the charges, the thermal mode growth rate and the wave damping both reduce by the same factor, which depends on the ionization fraction only. For a heating function that is constant in time, we find that the growth of the thermal mode and the damping of the sound waves are slightly larger. The numerical calculation of the eigenvalues of the general system of equations in a three-dimensional geometry confirm the analytic results. We then run two-dimensional fully nonlinear simulations that give consistent results: a higher ionization fraction or lower coupling will increase the growth rate. The magnetic field contribution is negligible in the linear phase. Ionization-recombination effects might play an important role because the RC produces a large range of temperatures in the system. In the numerical simulation, after the first condensation phase, when the minimum temperature is reached, the fraction of neutrals increases four orders of magnitude because of the recombination. This article is part of the theme issue 'Partially ionized plasma of the solar atmosphere: recent advances and future pathways'.

Optical Darboux Transformer.

The optical Darboux transformer for solitons is introduced as a photonic device that performs the Darboux transformation directly in the optical domain. This enables two major advances for optical signal processing based on the nonlinear Fourier transform: (i)the multiplexing of solitonic waveforms corresponding to different discrete eigenvalues of the Zakharov-Shabat system, and (ii)the selective filtering of an arbitrary number of individual solitons too. The optical Darboux transformer can be built using existing commercially available photonic technology components and constitutes a universal tool for signal processing, optical communications, optical rogue waves generation, and waveform shaping and control in the nonlinear Fourier domain.

On the explicit Hermitian solutions of the continuous‐time algebraic Riccati matrix equation for controllable systems

AbstractThis paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution.

The Impact of Frame Transformations on Power System EMT Simulation

This article investigates the impact of frame transformations on the accuracy of numerical discretization in power system transient and stability studies. As analysed, frame transformations influence the convergence of the numerical discretization. Specifically, for an explicit discretization method (e.g., forward Euler method), the stability of the original system is best preserved in the frame where the system eigenvalue is closer to the origin of the complex plane, e.g., in the stationary frame for inductors and capacitors, and in the synchronous frame for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq-</i> frame controllers of inverters. Simulation results are given to validate the theoretical analysis.