Quasi-linear Model Research Articles

СИНТЕЗ ГИБРИДНОЙ СИСТЕМЫ УПРАВЛЕНИЯ НЕАФФИННЫМИ ОБЪЕКТАМИ

In the theory of automatic control, an urgent problem is the development of design methods bynonaffine control systems. In such systems, the control affects the input of the plant nonlinearly, so itaffects the state variables non-additively. The purpose of this article is to develop a design method thatensures the stability of the zero equilibrium position of a closed control system in a certain area.The object described by a nonlinear system of differential equations with one control is considered.A restriction is introduced, consisting in the differentiability of the right part of the differential equationsfor all state variables. The task of designing control in the form of a function of the reference signal, avector of state variables and control values at previous points in time is set. This problem is solved usinga quasilinear model of the control plant. This model of description allows you to preserve all the featuresof a nonlinear plant without simplifying them. In the quasilinear model, matrices and vectors arefunctions of the variables of the state of the control plant. The control is performed using an algebraicpolynomial matrix method. This method allows you to find control when the control condition of theplant are met in the form of inequalities. This article presents the expressions for calculating the controlaccording to the polynomial matrix method. Based on the given coefficients of the desired polynomial,as a result of solving an algebraic system of equations, coefficients are found that are a function of controland state variables. At the same time, the fulfillment of the controllability condition guarantees theexistence of a solution of the specified algebraic system. An expression has been found that allows calculatingthe control by the coefficients found. The article also finds a condition for the possibility ofproviding a non-zero value of the output controlled quantity of a nonlinear Hurwitz system in a steadystatemode. Under this condition, a zero value of the static error for the setting effect can also be provided.Further, the transformation of the obtained continuous control into a discrete one is proposed, whichis implemented in a digital computer. The article also provides a numerical example of the control designof a second-order nonlinear control and the results of modeling a closed nonaffine system.The given example confirms the theoretical results obtained. Thus, the proposed approach makes it possibleto design stable Hurwitz control systems for nonaffine objects using the algebraic polynomial matrixmethod with sufficiently small sampling periods of variables of the control object and small modules of theroots of the characteristic polynomial of the matrix of a closed system in its quasilinear model.

Core integrated simulations for the Divertor Tokamak Test facility scenarios towards consistent core-pedestal-SOL modelling

Deuterium plasma discharges of the Divertor Tokamak Test facility (DTT) in different operational scenarios have been predicted by a comprehensive first-principle based integrated modelling activity using state-of-art quasi-linear transport models. The results of this work refer to the updated DTT configuration, which includes a device size optimisation (enlargement to m and a = 0.70 m) and upgrades in the heating systems. The focus of this paper is on the core modelling, but special attention was paid to the consistency with the scrape-off layer parameters required to achieve divertor plasma detachment. The compatibility of these physics-based predicted scenarios with the electromagnetic coil system capabilities was then verified. In addition, first estimates of DTT sawteeth and of DTT edge localised modes were achieved.

Design of Nonlinear Control Systems for Autorobots

Currently, various types of robots are increasingly being used to solve different tasks. Most often, these are mobile robots that move on the earth’s surface performing the assigned tasks, in particular, they are four-wheeled robots similar to a car — autorobots. As control objects, the robots are essentially nonlinear, which requires the use of nonlinear methods for the control system design. At the same time, it is difficult to apply traditional methods for designing nonlinear control systems due to a complex type of nonlinearities in the equations of mobile robots and, in particular, autorobots. In this paper, the design problem is solved using a discrete-continuous quasilinear model, which is created on the basis of the nonlinear differential equations in the Cauchy form. Due to the great complexity of the nonlinearities of the autorobot equations, the corresponding quasilinear model is created by the numerical method. The quasilinear model obtained by this method is discrete-continuous and controllable, moreover, its state variables are measurable. The discrete autorobot control system includes two practically independent control subsystems: longitudinal speed and turns. A discrete PI control law is used to control the speed, and the discrete turn control subsystem is designed by the method of desired dynamics. The resulting autorobot control system provides a stable movement along the trajectory, which can be set as a function of time or as a function of the coordinates of the moving autorobot’s position.The suggested approach can be used to design control systems for nonlinear objects of various purposes with complex differentiated nonlinearities. However, the design problem has a solution if the corresponding discrete-continuous quasilinear model of the object is controllable, and the state variables are measurable.

Design of Nonlinear Selectively Invariant Control Systems Based on Quasilinear Models

Design of Nonlinear Selectively Invariant Control Systems Based on Quasilinear Models

Analysis of a quasilinear coupled magneto-quasistatic model: Solvability and regularity of solutions

We consider a quasilinear model arising from dynamical magnetization. This model is described by a magneto-quasistatic (MQS) approximation of Maxwell's equations. Assuming that the medium consists of a conducting and a non-conducting part, the derivative with respect to time is not fully entering, whence the system can be described by an abstract differential-algebraic equation. Furthermore, via magnetic induction, the system is coupled with an equation which contains the induced electrical currents along the associated voltages, which form the input of the system. The aim of this paper is to study well-posedness of the coupled MQS system and regularity of its solutions. Thereby, we rely on the classical theory of gradient systems on Hilbert spaces combined with the concept of E-subgradients using in particular the magnetic energy. The coupled MQS system precisely fits into this general framework.

Comment on the paper “Terahertz rectangular waveguides with inserted graphene films biased by light and their quasi-linear electromagnetic modeling, Guennadi A. Kouzaev, Journal of Computational Electronics, 2021, 20:169–177”

Comment on the paper “Terahertz rectangular waveguides with inserted graphene films biased by light and their quasi-linear electromagnetic modeling, Guennadi A. Kouzaev, Journal of Computational Electronics, 2021, 20:169–177”

On the role of mode resonances in regulating zonal-flow-moderated plasma microturbulence

The onset of turbulent heat transport at a higher temperature gradient than the critical gradient of linear instability, known as the Dimits shift, is a recurring feature of nonlinear simulations for magnetically confined fusion plasmas. Resonance in the nonlinear coupling between the modes that dominate energy transfer can lead to suppression of turbulence and transport above the linear critical gradient. As an expression of this resonance, gyrokinetic simulations show a quasi-coherent interaction between streamers and sidebands coupled through the zonal flow within the Dimits regime. This mechanism is further confirmed by use of artificial complex frequencies which break the resonance. By incorporating corresponding saturation physics, the standard quasilinear model for rapid head flux prediction is improved, which can now predict reduced heat flux in the Dimits regime. In particular, the triplet correlation time, the lifetime of the nonlinear interaction, is shown to be well-approximated by combinations of linear eigenvalues, and yields good representations of the heat flux variation both in and above the Dimits regime. Thus, a reduced but predictive model for transport near the critical gradient of zonal-flow saturated turbulence now exists.

Diffusive scattering of energetic electrons by intense whistler-mode waves in an inhomogeneous plasma

Electron resonant interactions with electromagnetic whistler-mode waves play an important role in electron flux dynamics in various space plasma systems: planetary radiation belts, bow shocks, solar wind and magnetic reconnection regions. Two key wave characteristics determining the regime of wave–particle interactions are the wave intensity and the wave coherency. The classical quasi-linear diffusion approach describes well electron diffusion by incoherent and low-amplitude waves, whereas the nonlinear resonant models describe electron phase bunching and trapping by highly coherent intense waves. This study is devoted to the investigation of the regime of electron resonant interactions with incoherent but intense waves. Although this regime is characterized by electron diffusion, we show that diffusion rates scale linearly with the wave amplitude,$D\propto B_w$, in contrast to the quasi-linear diffusion scaling$D_{QL}\propto B_w^2$. Using observed wave amplitude distributions, we demonstrate that the quasi-linear diffusion model significantly overestimates electron scattering by incoherent, but intense whistler-mode waves. We discuss the results obtained in the context of simulations of long-term electron flux dynamics in space plasma systems.

Effects of Background Turbulence on the Relaxation of Ion Temperature Anisotropy Firehose Instability in Space Plasmas

Turbulence in space plasmas usually exhibits an energy cascade in which large-scale magnetic fluctuations are dominated by non-linear MHD wave–wave interactions following a Kolmogorov-like power-law spectrum. In addition, at scales at which kinetic effects take place, the magnetic spectrum follows a steeper power-law k−α shape given by a spectral index α>5/3. In a recent publication, a quasilinear model was used to study the evolution of ion temperatures in a collisionless plasma in which electromagnetic waves propagate along the background magnetic field, and it was found that the interaction between the plasma and a turbulent spectrum of ion-cyclotron waves may lead the plasma to states out of thermal equilibrium characterized by enhanced temperature anisotropies T⊥>T‖ and with a reduction in the parallel proton beta, which is consistent with space observations. Here, we complement such studies by analyzing the quasilinear interaction between plasma and a solar-wind-like turbulent spectrum of fast magnetosonic waves, and study the role of firehose instability (FHI) in the regulation of temperature anisotropy. Our results show that the presence of turbulence significantly modifies the FHI marginal stability threshold, as predicted from linear theory. Moreover, depending on the value of the plasma β, a turbulent magnetosonic spectrum may lead an initially thermally isotropic plasma to develop anisotropic states in which T⊥<T‖.

Integrated modelling of neon impact on JET H-mode core plasmas

Nuclear fusion reactor plasmas will need to exhaust a significant proportion of energy flux through radiative processes, to enable acceptable divertor loads. This can be obtained by line radiation from impurities, injected from the plasma edge. There are however limitations on the sustainable impurity content, since radiation from the core can lead to a deleterious electron heat sink. Moreover, dilution of the main ions reduces the available fuel. Simultaneously, impurities have an impact on the turbulent transport, both by dilution and by changes in the effective charge. Recent experiments at JET point towards an improvement in plasma confinement in neon seeded discharges with respect to purer equivalent plasmas. In this paper the impact of the impurities on the confinement is studied, isolating various effects. First-principle-based integrated modelling with the QuaLiKiz quasilinear turbulent transport model explains the improvement by a combination of higher pedestal temperature, increased rotation shear, and impurity-induced microturbulence stabilization. These results are optimistic with respect to the maximum impurity levels allowed in ITER and future reactors. Comparison between QuaLiKiz and higher fidelity gyrokinetics has exposed issues with QuaLiKiz impurity peaking predictions with rotation.

Quasi-Linear Model of Tsunami Run-Up on a Beach with a Seafloor Described by the Piecewise Continuous Function

The purpose of this paper is to propose the quasi-linear theory of tsunami run-up and run-down on a beach with complex bottom topography. We begin with the one-dimensional nonlinear shallow-water wave equations, which we consider over a beach of complex geometry that can be modeled by a piecewise continuous function, along with several natural initial and boundary conditions. The primary obstacle in solving this problem is the moving boundary associated with the shoreline motion. To avoid this difficulty, we replace the moving boundary with a stationary boundary by applying a transformation to the spatial variable of the computational domain. A characteristic feature of any tsunami problem is the smallness of the parameter ε=η0/h0, where η0 is the characteristic amplitude of the wave, and h0 is the characteristic depth of the ocean. The presence of this small parameter enables us to effectively linearize the problem by using the method of perturbations, which leads to an analytical solution via an integral transformation. This analytical solution assumes that there is no wave breaking. In light of this assumption, we introduce the wave no-breaking criterion and determine bounds for the applicability of our theory. The proposed model can be readily used to investigate the tsunami run-up and draw-down for different sea bottom profiles. The novel particular solution, when the seafloor is described by the piecewise linear function, is obtained, and the effects of the different beach profiles and initial wave locations are considered.

ОЦЕНКЕ ОБЛАСТИ ПРИТЯЖЕНИЯ ПОЛОЖЕНИЯ РАВНОВЕСИЯ НЕЛИНЕЙНЫХ СИСТЕМ УПРАВЛЕНИЯ

Designing nonlinear control systems is still difficult so many researchers are trying to findsome useful ways and methods to solve this problem. As a result of such research, some methodshave been seen trying to design a good enough control system for nonlinear plants. But a disadvantageof these methods is the complexity, so it created a need to compare some methods to determinewhich one is the easiest method to design a control system for nonlinear plants. It wasfound a way to compare two methods, which is comparing the regions of initial conditions of thesystems which are designed using these methods. Two analytical nonlinear control systems designmethods are compared on the example of the design control systems mobile robots. The algebraicpolynomial-matrix method uses a quasilinear model, and the feedback linearization method usesparticular feedback. Both considered methods give a bounded domain of equilibrium attraction,therefore the obtained control systems can be operated only with bounded initial conditions.The numerical example of designing the control systems for one object by these methods and theestimates of the attraction areas of the system’s equilibriums of these systems are given in thepaper. As a result of this paper, it was found that using the algebraic polynomial-matrix methodwill get a bigger cross section of initial conditions of the plant’s variable than the same cross sectionwhich is given by the feedback linearization method.

How close dark matter haloes and MOND are to each other: three-dimensional tests based on Gaia DR2

ABSTRACT Aiming at discriminating different gravitational potential models of the Milky Way, we perform tests based on the kinematic data powered by the Gaia DR2 astrometry over a large range of (R, z) locations. Invoking the complete form of Jeans equations that admit three integrals of motion, we use the independent R- and z-directional equations as two discriminators (TR and Tz). We apply the formula for spatial distributions of radial and vertical velocity dispersions proposed by Binney et al., and successfully extend it to azimuthal components, σθ(R, z) and Vθ(R, z); the analytic form avoids the numerical artifacts caused by numerical differentiation in Jeans-equations calculation given the limited spatial resolutions of observations, and more importantly reduces the impact of kinematic substructures in the Galactic disc. It turns out that whereas the current kinematic data are able to reject Moffat’s Modified Gravity (let alone the Newtonian baryon-only model), Milgrom’s MOND is still not rejected. In fact, both the carefully calibrated fiducial model invoking a spherical dark matter (DM) halo and MOND are equally consistent with the data at almost all spatial locations (except that probably both have respective problems at low-|z| locations), no matter which tracer population or which meaningful density profile is used. Since there is no free parameter at all in the quasi-linear MOND model we use, and the baryonic parameters are actually fine-tuned in the DM context, such an effective equivalence is surprising, and might be calling forth a transcending synthesis of the two paradigms.

Upgrade and benchmark of quasi-linear transport model EDWM

The verification of a new saturation rule applied to the quasi-linear fluid model EDWM (extended drift wave model) and the calibration of several other features are presented. As one of the computationally fastest first-principle-based core transport models, EDWM can include an arbitrary number of ions and charge states. This feature is especially important for experimental devices with plasma-facing components made of heavy elements, such as the upcoming ITER device. As a quasi-linear model, EDWM solves a linear dispersion relation to obtain the instabilities driving the turbulence and combines the linear description with an estimation of the saturation level of the electrostatic potential to determine the fluxes. A new saturation rule at the characteristic length combined with a spectral filter for the poloidal wavenumber dependency is developed. The shape of the filter has been fitted against the poloidal wavenumber dependency of the electrostatic potential from non-linear gyrokinetic simulations. Additionally, EDWM's collision frequency and safety factor dependencies, as well as the electron heat flux level, have been calibrated against gyrokinetic and gyrofluid results. Finally, the saturation level has been normalized against non-linear gyrokinetic simulations and later validated against experimental measured fluxes from 12 discharges at JET.

Parametric control for flexible spacecraft attitude maneuver based on disturbance observer

This paper considers the problems of control performance degradation caused by the vibration of flexible accessories and external disturbance during the attitude maneuvering process of flexible spacecraft. A parametric design method for attitude maneuver control (AMC) of flexible spacecraft based on disturbance observer is proposed to improve the anti-interference ability and control accuracy of the control system. Firstly, considering the spacecraft kinematics model for quasi-linear system modeling then design a disturbance observer to solve the external disturbance and vibration generated by the flexible attachment during maneuvering. Further, a controller designed using direct parametric method is proposed that can transform the nonlinear closed-loop system into a second-order linear constant system with the desired characteristic structure and provide all degrees of freedom in the control system design. Finally, numerical examples are given to evaluate the efficacy of the proposed approach.

Local well-posedness of a coupled Westervelt–Pennes model of nonlinear ultrasonic heating

High-intensity focused ultrasound (HIFU) waves are known to induce localised heat to a targeted area during medical treatments. In turn, the rise in temperature influences their speed of propagation. This coupling affects the position of the focal region as well as the achieved pressure and temperature values. In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Westervelt wave equation coupled to the Pennes bioheat equation that captures this so-called thermal lensing effect. We prove that this quasi-linear model is well-posed locally in time and does not degenerate under a smallness assumption on the pressure data.

A Study of Gyrotron Synchronization in the Hard-Excitation Regime on the Basis of the Modified Quasilinear Model

A Study of Gyrotron Synchronization in the Hard-Excitation Regime on the Basis of the Modified Quasilinear Model

Nonresonant Scattering of Relativistic Electrons by Electromagnetic Ion Cyclotron Waves in Earth's Radiation Belts.

Electromagnetic ion cyclotron waves are expected to pitch-angle scatter and cause atmospheric precipitation of relativistic (>1 MeV) electrons under typical conditions in Earth's radiation belts. However, it has been a long-standing mystery how relativistic electrons in the hundreds of keV range (but <1 MeV), which are not resonant with these waves, precipitate simultaneously with those >1 MeV. We demonstrate that, when the wave packets are short, nonresonant interactions enable such scattering of hundred-keV electrons by introducing a spread in wave number space. We generalize the quasilinear diffusion model to include nonresonant effects. The resultant model exhibits an exponential decay of the scattering rates extending below the minimum resonant energy depending on the shortness of the wave packets. This generalized model naturally explains observed nonresonant electron precipitation in the hundreds of keV concurrent with >1 MeV precipitation.

Fracture toughness determination of porcine muscle tissue based on AQLV model derived viscous dissipated energy

The ability of soft collagenous tissue (SCT) to withstand propagation of a defect in the presence of a macroscopic crack is termed the ’fracture toughness parameter’. In soft tissues not undergoing significant plastic deformation, it is purported that a considerable amount of additional energy is dissipated during failure processes, due to viscoelasticity. Hence the total work, measured experimentally during failure, is the sum of fracture and viscoelastic energies. Previous authors have aimed to apply constitutive modeling to describe viscoelastic hysteresis for fracture toughness determination with a tendency of models to either over or underestimate the viscous energy. In this study, the fracture toughness of porcine muscle tissue is determined using two strategies. Firstly, it was determined experimentally by calculation of the difference in dissipated energy of notched and unnotched tissue specimens undergoing cyclic ’triangular wave’ excitation with increasing strain levels in uniaxial tension. The second strategy involved the extension and use of the adaptive quasi-linear viscoelastic model (AQLV) to model cyclic loading (model parameters were obtained from a previous study) and sequentially the dissipated energy was calculated. The mean value of the dissipated energy based on the AQLV approach was then subtracted from the total dissipated energy of notched porcine muscle tissue samples to determine the fracture toughness. The mean experimental viscous dissipated energy ratio was 0.24 ± 0.04 in the experimental approach, compared to 0.28 ± 0.03 for the AQLV model. Fracture toughness determined experimentally yielded 0.84 ± 0.80 kJ/m2, and 0.71 ± 0.76 kJ/m2 for the AQLV model, without a significant difference (p = 0.87). Hence, the AQLV model enables a reasonable estimation of viscous dissipated energy in porcine muscle tissue with the advantage to perform tests only on notched specimens, instead of testing additional unnotched samples. Moreover, the AQLV model will help to better understand the constitutive viscoelastic behaviour of SCTs and might also serve as a basis for future fracture toughness determination with constitutive model simulations.

Global existence of a quasilinear chemotaxis model with signal-dependent motility and indirect signal production mechanism

In this paper, we study the following quasilinear chemotaxis model with signal-dependent motility: nt = Δ(γ(c)nm); ct = dcΔc − c + v; vt = dvΔv − v + n, x ∈ Ω, t &amp;gt; 0, ∂(nmγ(c))∂ν=∂c∂ν=∂v∂ν=0, x ∈ ∂Ω, t &amp;gt; 0, n(x, 0) = n0(x), c(x, 0) = c0(x), v(x, 0) = v0(x), x ∈ Ω, t &amp;gt; 0, where γ(c) = c−r. We show that the above system admits at least one global weak solution.