Perturbation Parameter Research Articles

Comprehensive Perturbation Approach to Nonlinear Viscous Gravity–Capillary Surface Waves at Arbitrary Wavelengths in Finite Depth

This study presents a comprehensive analysis of the second-order perturbation theory applied to the Navier–Stokes equations governing free surface flows. We focus on gravity–capillary surface waves in incompressible viscous fluids of finite depth over a flat bottom. The amplitude of these waves is regarded as the perturbation parameter. A systematic derivation of a nonlinear-surface-wave equation is presented that fully takes into account dispersion, while nonlinearity is included in the leading order. However, the presence of infinitely many over-damped modes has been neglected and only the two least-damped modes are considered. The new surface-wave equation is formulated in wave-number space rather than real space and nonlinear terms contain convolutions making the equation an integro-differential equation. Some preliminary numerical results are compared with computational-modelling data obtained via open source CFD software OpenFOAM.

Hyperacceleration of quantum thermalization dynamics by bypassing long-lived coherences: An analytical treatment.

We develop a perturbative technique for solving Markovian quantum dissipative dynamics, with the perturbation parameter being a small gap in the eigenspectrum. As an example, we apply the technique and straightforwardly obtain analytically the dynamics of a three-level system with quasidegenerate excited states, where quantum coherences persist for very long times, proportional to the inverse of the energy splitting squared. We then show how to bypass this long-lived coherent dynamics and accelerate the relaxation to thermal equilibration in a hyper-exponential manner, a Markovian quantum-assisted Mpemba-like effect. This hyperacceleration of the equilibration process manifests if the initial state is carefully prepared, such that its coherences precisely store the amount of population relaxing from the initial condition to the equilibrium state. Our analytical method for solving quantum dissipative dynamics readily provides equilibration timescales, and as such it reveals how coherent and incoherent effects interlace in the dynamics. It further advises on how to accelerate relaxation processes, which is desirable when long-lived quantum coherences stagnate dynamics.

A numerical method for a system of singularly perturbed Fredholm integro-differential reaction–diffusion equation

This article provides a dissection and analysis of a singularly perturbed reaction–diffusion second-order system of Fredholm integro-differential equation in qualitative and quantitative ways. To find a numerical solution to this system, the authors present an exponentially-fitted numerical approach that may be applied to a Shishkin mesh. They provide evidence that demonstrates that this approach converges evenly with respect to the perturbation parameter.

A Parameter-Robust Numerical Method Based on Defect-Correction Technique for Parabolic Singular Perturbation Problems with Discontinuous Convection Coefficient and Source

The paper presents a uniformly convergent finite difference method based on the defect correction technique to solve parabolic singular perturbation problems with discontinuous convection coefficient and source. The solution to the problem exhibits interior layer across the discontinuity and demonstrates turning point behavior. The simultaneous presence of perturbation parameters and discontinuity makes the problem stiff. A higher-order method is developed using an implicit difference scheme in time on a uniform mesh and a combination of the upwind difference method and the central difference scheme over an adaptive mesh in space. The method involves iteratively solving increasingly accurate discrete problems by computing and using the defect to correct the approximate solution. Parameter uniform error estimates show uniform convergence of first-order in time and second-order in space. Numerical experiments confirm the accuracy of the proposed scheme and support the theoretical analysis.

Robustness of Optimal Circadian Rhythm Entrainment under Model Perturbation.

The optimal circadian rhythm entrainment problem has been studied based on mathematical models, e.g. the Kronauer model. In this paper, we study the robustness of light-based circadian rhythm entrainment under model parameter perturbations as well as propose a feedback control law to improve the robustness of the entrainment strategy. Our study finds the model parameter whose perturbations affect the entrainment the most. We also find that feedback control reduces the sensivitity of the entrainment process to model parameter perturbations.Clinical relevance- Circadian misalignment has negative impacts on health, such as higher risks of cardiovascular disease and cancer. We present a numerical study of how well optimized circadian rhythm entrainment plans that are derived from a generic mathematical model work on personalized cases.

Fault-tolerant control strategy of six-phase permanent magnet synchronous motor based on deadbeat current prediction.

The fault-tolerant control after phase loss is crucial in the studies of the six-phase permanent magnet synchronous motor (PMSM), and the one phase loss is the most frequent phase loss fault. To improve the system instability caused by nonlinear and time-varying perturbations of inductance parameters in a double-Y phase shifted 30° six-phase PMSM, an improved deadbeat predictive current fault-tolerant control (DPC-FTC) method is proposed in this study. The transformation matrix after single-phase open-phase is first reduced and reconstructed, and the reduced-dimensional voltage equation is derived. Based on this equation, the deadbeat current predictive control is then used to predict the expected voltage using the current feedback value and the reference value, so as to shorten the response time and improve the overall control effect. The voltage equation after parameter perturbation is rewritten, and the current discrete transfer function under constant expected voltage before and after parameter perturbation is calculated. Afterwards, to further improve the low stability of fault-tolerant control after phase loss, which is caused by the inductance parameter perturbation of the control system, the weight coefficient is introduced in order to enhance the deadbeat predictive current control so that it splits and optimizes the direct-quadrature axis current. The stability of the system is then analyzed. By changing the weight coefficient, the fault-tolerant control system has a wider stable working range. Finally, the simulation model and experimental platform are completed. The results show that the improved DPC-FTC method improves the permissible inductor parameter uptake range by a factor of 1/β, reduces the current static difference by 32.05% and 46.02% when the inductor parameter is mismatched by a factor of 2, reduces the current oscillation and effectively reduces the sensitivity of system stability to inductor parameter uptake.

An efficient numerical approach for solving singularly perturbed parabolic differential equations with large negative shift and integral boundary condition

In this study, we consider singularly perturbed large negative shift parabolic reaction–diffusion with integral boundary condition. The continuous solution's properties are discussed. On a non-uniform Shishkin mesh, the spatial derivative is discretized using the tension spline method, and the temporal derivative is discretized using the Crank–Nicolson method. In order to handle the integral boundary condition, Simpson's rule is used. After conducting an error analysis, it was determined that the method was uniformly convergent. To support the theoretical findings, numerical examples are taken into account and solved for different values of the perturbation parameter and mesh sizes. It is proved to be uniformly convergent to almost second order.

An improved time accurate numerical estimation for singularly perturbed semilinear parabolic differential equations with small space shifts and a large time lag

The objective of this work is to provide a robust numerical scheme for solving a singularly perturbed semilinear parabolic differential–difference equation having both time delay along with spatial delay, and shift parameters. The small delay and shift arguments in spatial direction are treated with Taylor’s approximation. The technique of quasilinearization is used to treat the semilinearity and the temporal direction is handled with the θ-method. To handle the layer behavior caused by the presence of perturbation parameter, the problem is solved using the upwind scheme on two-layer resolving meshes in the spatial direction providing a first-order accurate result for 0.5≤θ≤1. For θ=0.5, we have the second-order accurate Crank–Nicolson scheme in time. In order to have a higher-order accuracy, the Richardson extrapolation technique is applied in the spatial direction, which in turn provides a global second-order accurate result. Thomas algorithm is used to solve the tridiagonal system of equations formed in the fully discrete scheme. The numerical approach presented is shown to be parameter uniform and convergent for 0.5≤θ≤1. Some numerical tests are presented to show the efficacy of the proposed scheme.

Combining local model calibration with the emergent constraint approach to reduce uncertainty in the tropical land carbon cycle feedback

Abstract. The role of the land carbon cycle in climate change remains highly uncertain. A key source of the projection spread is related to the assumed response of photosynthesis to warming, especially in the tropics. The optimum temperature for photosynthesis determines whether warming positively or negatively impacts photosynthesis, thereby amplifying or suppressing CO2 fertilisation of photosynthesis under CO2-induced global warming. Land carbon cycle models have been extensively calibrated against local eddy flux measurements, but this has not previously been clearly translated into a reduced uncertainty in terms of how the tropical land carbon sink will respond to warming. Using a previous parameter perturbation ensemble carried out with version 3 of the Hadley Centre coupled climate–carbon cycle model (HadCM3C), we identify an emergent relationship between the optimal temperature for photosynthesis, which is especially relevant in tropical forests, and the projected amount of atmospheric CO2 at the end of the century. We combine this with a constraint on the optimum temperature for photosynthesis, derived from eddy covariance measurements using the adjoint of the Joint UK Land Environment Simulator (JULES) land surface model. Taken together, the emergent relationship from the coupled model and the constraint on the optimum temperature for photosynthesis define an emergent constraint on future atmospheric CO2 in the HadCM3C coupled climate–carbon cycle under a common emissions scenario (A1B). The emergent constraint sharpens the probability density of simulated CO2 change (2100–1900) and moves its peak to a lower value of 497 ± 91 compared to 607 ± 128 ppmv (parts per million by volume) when using the equal-weight prior. Although this result is likely to be model and scenario dependent, it demonstrates the potential of combining the large-scale emergent constraint approach with a parameter estimation using detailed local measurements.

Aesthetics and neural network image representations

We analyze the spaces of images encoded by generative neural networks of the BigGAN architecture. We find that generic multiplicative perturbations of neural network parameters away from the photo-realistic point often lead to networks generating images which appear as “artistic renditions” of the corresponding objects. This demonstrates an emergence of aesthetic properties directly from the structure of the photo-realistic visual environment as encoded in its neural network parametrization. Moreover, modifying a deep semantic part of the neural network leads to the appearance of symbolic visual representations. None of the considered networks had any access to images of human-made art.

Modelling biogeochemical reactions triggered by graphene's addition in a fertilized calcareous sandy soil

Graphene production has dramatically increased in the last years and new ways to recycle this engineered material need to be investigated. To this purpose, a reactive model network was developed using PHREEQC-3 code to quantify the relevant biogeochemical reactions induced by graphene scraps' incorporation in a calcareous sandy soil. The numerical model was calibrated versus a complete dataset of column experiments in water saturated conditions using two different fertilizers, a synthetic NPK fertilizer and fertigation water produced in a wastewater treatment plant. Column experiments consisted of 50 cm columns filled with a mixture of graphene scraps (0.015 % dry weight) and soil in the first 10 cm, while the remaining 40 cm had only soil. The model performance was tested using classical statistical indices (R2, Modelling Efficiency, and Index of Agreement), resulting to be satisfactory. Besides, a simple sensitivity analysis via the perturbation of relevant parameters showed a low degree of uncertainty. The main outcome of this study was the quantification of the increased denitrification rate triggered by graphene incorporation into the soil. Moreover, graphene incorporation substantially increased soil CEC and DOC sorption capacity, demonstrating a good adsorption capacity for ammonium and organic compounds, thus decreasing nutrients leaching that represents a major concern related to agricultural practice. Indeed, Graphene incorporation increased by 40 % the CEC in the first 10 cm of the CSG_NPK column (2.50e−02 mol/L) respect to the CS_NPK column (1.75e−02 mol/L) and increased it by 150 % in the first 10 cm of the CSG_FW column (2.50e−02 mol/L) in comparison with the CS_FW column 1.00e−02 (mol/L). pH fluctuations were most likely due to the precipitation of Ca5(PO4)3OH, indeed the consumption of H+ ions could have triggered the pH lowering during the experiment. These results could be relevant for future graphene applications as a soil improver or as suitable material to enhance soil bioremediation in order to include graphene in a circular economy loop.

Cahn–Hilliard equation with regularization term

We will study in this article the nonlinear Cahn–Hilliard equation with proliferation and regularization terms with regular and logarithmic potentials. First, we consider the regular potential case, we show that the solutions blow up in finite time or exist globally in time. Furthermore, we prove that the model possess a global attractor. In addition, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. In the second part, we consider the logarithmic potential case and show the existence of a global solution.

Excellent predictive-performances of photonic reservoir computers for chaotic time-series using the fusion-prediction approach.

In this work, based on two parallel reservoir computers realized by the two polarization components of the optically pumped spin-VCSEL with double optical feedbacks, we propose the fusion-prediction scheme for the Mackey-Glass (MG) and Lorenz (LZ) chaotic time series. Here, the direct prediction and iterative prediction results are fused in a weighted average way. Compared with the direct-prediction errors, the fusion-prediction errors appear great decrease. Their values are far less than the values of the direct-prediction errors when the iteration step-size are no more than 15. By the optimization of the temporal interval and the sampling period, under the iteration step-size of 3, the fusion-prediction errors for the MG and LZ chaotic time-series can be reduced to 0.00178 and 0.004627, which become 8.1% of the corresponding direct-prediction error and 28.68% of one, respectively. Even though the iteration step-size reaches to 15, the fusion-prediction errors for the MG and LZ chaotic time-series can be reduced to 55.61% of the corresponding direct-prediction error and 77.28% of one, respectively. In addition, the fusion-prediction errors have strong robustness on the perturbations of the system parameters. Our studied results can potentially apply in the improvement of prediction accuracy for some complex nonlinear time series.

Overview of Models and Methods for Control of Stepper Motors

A review of models and algorithms for control of a stepper motor (SM) is presented. Due to high accuracy, improved power density, economy and reliability compared to other synchronous motors, stepper motors are widely used in various practical applications and scientific equipment. In aviation and space technology, step motors are actively used in actuating systems, such as drives for the movement of elements of large-sized structures, guidance, and stabilization systems, etc. The article describes some existing stepper motor control algorithms, which are both based on the knowledge of the parameters of the stepper motor model, and on the absence of this or that information. Of the many described algorithms, four were selected (PID controller, exact feedback linearization algorithm, adaptive control with partially unknown parameters and adaptive control with completely unknown parameters), which showed the best results of transient processes in tracking the angle of the rotor of the SM behind the reference value. A comparative numerical analysis among these four algorithms is also given, which showed that the best results of transients are demonstrated by adaptive controllers (in the sense of the smallest error in steady state), while the worst results are demonstrated by the PID controller. It is noted that the studied PID controller contains much fewer feedback loops compared to other algorithms, which simplifies the choice of adjustable parameters and reduces the dynamic order of the closed system, however, the design is based on knowing the exact parameters of the drive and is also sensitive to external disturbances. On the contrary, adaptive approaches successfully solve the problem of estimating parametric and functional perturbations, but their implementation is associated with significant difficulties.

Upper and lower bounds for the maximal Lyapunov exponent of singularly perturbed linear switching systems

In this paper we consider the problem of determining the stability properties, and in particular assessing the exponential stability, of a singularly perturbed linear switching system. One of the challenges of this problem arises from the intricate interplay between the small parameter of singular perturbation and the rate of switching as both tend to zero. Our approach consists in characterizing suitable auxiliary linear systems that provide lower and upper bounds for the asymptotics of the maximal Lyapunov exponent of the linear switching system as the parameter of the singular perturbation tends to zero.

Constrained Model Predictive Control With Integral Action for Twin Rotor MIMO Systems

Abstract This paper describes the design and successful implementation of a constrained model predictive controller with integral action for the control of a Twin Rotor MIMO System (TRMS). The integral action guarantees zero steady-state error in set-point tracking with robustness toward perturbations of the nominal system parameters. In addition to saturation constraints on the input variables, hard constraints are imposed on the controlled output variables, i.e., on pitch and yaw angular positions, to avoid collisions with obstacles. The model predictive controller was designed using a high-fidelity nonlinear model of the TRMS developed in previous work. As an intermediate step, exact linearized models of the TRMS are obtained and their closed-form expressions are reported. The controller was tested experimentally, also showing its effectiveness in ensuring actual collision avoidance by the TRMS when physical obstacles were present.

Additive-state-decomposition-based model predictive tracking control for PMSM servo system with multiple disturbances

This paper presents an additive-state-decomposition-based model predictive tracking control and disturbance rejection method for a permanent magnet synchronous motor (PMSM) servo system subject to unknown parameter perturbations, unmodeled dynamics, and time-varying load torque. The basic idea of this method is to equivalently decompose the original system into a primary system for handling the tracking control subproblem and a secondary system for dealing with the robust stabilization subproblem. A model predictive controller is designed for the primary system to achieve high-accuracy tracking of the reference speed. As for the secondary system, a novel high-order generalized extended state observer (HGESO) is constructed to estimate the multiple disturbances simultaneously, and a state feedback control law incorporating a disturbance compensator is developed to eliminate the adverse effect of the multiple disturbances on the system output. By combining the control inputs of the two subsystems together, the control objectives of the original system can be achieved. Both the stability criterion and design procedure of the closed-loop control system are developed. Finally, hardware-in-the-loop-based comparative experiments are conducted to demonstrate that the proposed method effectively suppresses the influence of the multiple disturbances on motor speed tracking accuracy and that the control system has both satisfactory dynamic performance and robustness.

Efficacy of velocity feedback control for suppression of the friction-induced oscillation due to mode-coupling instability in a two degrees-of-freedom mechanical system

The present article focuses on suppressing mode-coupling instability in frictional systems, which can cause unwanted vibrations such as squeal in braking systems. Mode-coupling instability occurs when two or more modes of a system approach each other. This article proposes the use of resonant velocity feedback control to eliminate friction-induced oscillations generated by mode-coupling instability. A two degree-of-freedom linear model is used to capture the instability due to friction-induced mode coupling. The efficacy of the control is demonstrated through linear stability analysis, and by optimizing control parameters using pole crossover method to minimize the transient time of the response. The robustness analysis demonstrates the ability of the control to effectively mitigate the instability even under parametric perturbations. The controlled system is simulated in MATLAB Simulink to validate analytical results. The effects of time delay, which is commonly present in feedback systems, are also investigated. Finally, the effectiveness of the control is studied in the presence of nonlinearity in the system. The nonlinear dynamics are analyzed by creating bifurcation diagrams with different bifurcation parameters.

Strong Purcell effect of magnetic quasi-BICs in the dielectric metasurface

The ultra-strong light field trapped in micro- and nanostructures, and derived from optical bound states in the continuums (BICs), provides a new platform for the facilitation of light–matter interactions. In this work, the spontaneous emission enhancement of quantum emitters using magnetic quasi-BICs metasurfaces is numerically studied. It is found that ultrahigh Purcell factor (PF) () can be easily achieved when the asymmetry parameter decreases. In theory, infinite PF may emerge due to the infinite Q-factor at the BICs. In addition, the PF exhibits a strong dependence on the polarization direction of dipole emitter and the local field distribution in the metasurface. Finally, the photoluminescence enhancement for different perturbation parameters is calculated to confirm strong Purcell effect in the quasi-BICs metasurface. These results show that the magnetic dipole BICs mode holds an enormous potential in manipulating spontaneous emission.

Physics-Assisted Learning Scheme for Quantitative Imaging via Near-Field Scanning Microwave Microscopy

Quantitative determination of the dielectric property is important to the analysis and evaluation of the investigated samples’ performance. This article proposes a physics-assisted learning scheme (PALS) for the quantitative imaging via the near-field scanning microwave microscopy (NFSMM) in a nondestructive way. To avoid using the machine learning in a black-box way, an efficient input generation is realized by utilizing the Rayleigh approximation method. Numerical and experimental examples demonstrate that the PALS recovers the dielectric parameters of the subsurface perturbation more accurately and shows a better generalization ability. The proposed learning scheme provides a reasonable way to combine the machine learning with underlying physics in the NFSMM, rather than utilizing machine learning in a black-box way. The same principle is also potential to be applied in other types of scanning microscopies, such as optical and thermal ones.