Class Of Nonlinear Models Research Articles

Solitary waves in a granular chain of elastic spheres: Multiple solitary solutions and their stabilities.

A granular chain of elastic spheres via Hertzian contact incorporates a classical nonlinear force model describing dynamical elastic solitary wave propagation. In this paper, the multiple solitary waves and their dynamic behaviors and stability in such a system are considered. An approximate KdV equation with the standard form is derived under the long-wavelength approximation and small deformation. The closed-form analytical single- and multiple-soliton solutions are obtained. The construction of the multiple-soliton solutions is analyzed by using the functional analysis. It is found that the multiple-soliton solution can be excited by the single-soliton solutions. This result is confirmed by the numerical analysis. Based on the soliton solutions of the KdV equation, the analytic solutions of multiple dark solitary waves are obtained from the original dynamic equation of the granular chain in the long-wavelength approximation. The stability of the single and multiple dark solitary wave solutions are numerically analyzed by using both split-step Fourier transform method and Runge-Kutta method. The results show that the single dark solitary wave solution is stable, and the multiple dark solitary waves are unstable.

Morphing control of a new bionic morphing UAV with deep reinforcement learning

With rapid development of aviation technology, materials science and artificial intelligence, aircraft design is pursuing higher requirements both in civil and military fields. The new generation of aircraft should have the autonomous capable of performing a variety of tasks (such as take-off and landing, cruising, maneuvering, hover, attack, etc.) under a highly variable flight environment (height, Mach number, etc.) and meanwhile maintaining good performance. Morphing aircraft can use smart materials and actuators to autonomously deform the shape according to the changes in flight environment and mission, and always maintain an optimal aerodynamic shape, therefore get flourished developments. Based on the ability of birds to stretch wings when flying at low speed and to constrict wings at high speed, a new bionic morphing UAV has been designed and developed as the study model by our team. In order to make this new aircraft be able to complete rapid autonomous morphing and aerodynamic performance optimization under different missions and flight conditions, we developed deep neural networks and reinforcement learning techniques as a control strategy. Considering the continuity of the state and action spaces for model, the Deep Deterministic Policy Gradient (DDPG) algorithm based on the actor-critic, model-free algorithm was adopted and verified on the classic nonlinear Pendulum model and Cart Pole game. After the feasibility was verified, morphing aircraft model was controlled to complete prescribed deformation using DDPG algorithm. Furthermore, on the condition that the DDPG algorithm can control morphing well, through training and testing on model using simulation data from wind tunnel tests and actual flight, the autonomous morphing control for the shape optimization of the bionic morphing UAV model could be realized.

Nonlinear models and bifurcation trees in quantum mechanics: a review of recent results

In this talk we discuss some recent results I obtained for a class of nonlinear models in quantum mechanics. In particular we focus our attention to the nonlinear one-dimensional Schrodinger equation with a periodic potential and a Stark-type perturbation. In the limit of large periodic potential the Stark–Wannier ladders of the linear equation become a dense energy spectrum because a cascade of bifurcations of stationary solutions occurs; for a detailed treatment we refer to Sacchetti (Phys Rev E 95:062212, 2017, SIAM J Math Anal 50(6):5783–5810, 2018) where this model has been studied.

Modeling the individual height and volume of two integrated crop-livestock-forest systems of <i>Eucalyptus</i> spp. in the Brazilian Savannah

The aim of this study was to model the individual height and volume of eucalyptus wood in two integrated crop-livestock-forest systems (ICLF1 and ICLF2) in Campo Grande, a city in the state of Mato Grosso do Sul, Brazil. Classic nonlinear growth models were adjusted for height (Logistic, Gompertz, Richards, Weibull, Van Bertalanffy, Brody, Mitscherlich, and Chapman and Richards) and volume (Shumacher-hall nonlinear, Takata, Honner, Logistic, Gompertz, and Weibull) in two structural arrangements: ICLF1, with a spacing of 14 x 2 m and density of 357 trees ha-1, and ICLF2, with a spacing of 22 x 2 m and density of 227 trees ha-1. Diameter at Breast Height (DBH) measurements were performed in 100% of trees, with measurements of the total height of some individuals and a rigorous scaling procedure in diameter classes. According to the calculated value of Student's t-test, there was no significant evidence that DBH and the average height of the trees were different between ICLF1 and ICLF2. Based on the Akaike information criterion (AIC), the corrected Akaike information criterion (AICC) and the Bayesian information criterion (BIC), the Richards model was selected to estimate heights and the Takata model was selected to estimate the volume.

A Study of the Stress State of Underground Gas Pipelines in Mine Working Zones Using the Method of Internal Response Function

The stress state of pipelines in mine working zones, where the main loading factor is longitudinal soil displacements caused by the gradual filling of mine voids, has been calculated. Differential equations of pipe deformation have been formulated, for the solution of which the method of internal response function is proposed, which takes into account the mutual displacement of the pipe and soil in each iteration using a classical three-part nonlinear model of pipe–soil interaction. The type of interaction is determined on the basis of the concept of the basic displacement of the pipe, which is refined using displacements calculated in the previous iteration. The procedure of refining the basic solution is an important component of the method, where the basic solution in the next iteration is found as the sum of the previous solution and the difference between the calculated and previous solutions, multiplied by the convergence-control factor of the iterative procedure. To check the convergence accuracy and rate of the proposed iterative procedure, a comparative analysis with other existing procedures and results, obtained with the aid of commercial finite element programs, has been performed. To diagnose the stress state of pipelines during the working of longwall faces, a system for their continuous monitoring has been created. The correctness of the results of the performed numerical modeling of the stress state of a gas pipeline and the effectiveness of the measures that are developed to reduce the high stress level in its individual zones have been confirmed by relevant monitoring data, which are important both for refining the physical characteristics of pipe–soil interaction and for further taking into account the relaxation of these relations in time.

A Hammerstein–Wiener Model for Single-Electron Transistors

This paper proposes a new dynamic behavior model for single-electron transistors (SETs). A comprehensive review of modeling techniques and previous models was carried out aiming to remark the originality of the new proposed technique. Once established that SET is a nonlinear system, five classes of nonlinear models were simulated and compared to each other. Characteristic SET curves obtained using Simulation of Nanostructures were used as benchmark. The mean squared error (mse) was used as metric, and the simulation time was computed for each model. Regarding these two criteria—least mse and least simulation time—the Hammerstein–Wiener model outperformed the others models.

Model-Free Identification of Nonlinear Restoring Force with Modified Observation Equation

Nonlinearity exists widely in civil engineering structures; for example, the initiation and growth of damage under dynamic loadings is a typical nonlinear process. To date, for the purpose of structural evaluation and a better understanding nonlinear characteristics of complicated structures, a number of parametric and nonparametric methods have been developed for the identification of nonlinear restoring force (NRF). However, due to the highly individualistic nature of nonlinear systems, it would be inefficient to attempt to express the structural NRF in a general parametric form. For many nonparametric techniques, their nonparametric models or approximations may result in undesirable results or oscillations around unsmooth points. In this paper, on the basis of extended Kalman filter (EKF), a model-free NRF identification approach is proposed to circumvent the limitations mentioned above. The NRF to be identified was treated as ‘unknown fictitious input’, and thus, no prior assumptions or approximations for the NRF model were required. With the aid of a projection matrix, a modified version of observation equation was obtained. Based on the principle of EKF, the recursive solution of the proposed approach was analytically derived. The NRFs provided by the nonlinear components were identified by means of least squares estimation (LSE) at each time step. Numerical examples, including building structures equipped with magnetorheological (MR) damper and shape memory alloy (SMA) damper, demonstrated that the proposed approach is capable of satisfactorily identifying NRF without knowledge or intuitive assumptions of any nonlinear model class in advance.

Locally D-optimal designs for a wider class of non-linear models on the k-dimensional ball

In this paper we extend the results of Radloff and Schwabe ( arXiv:1806.00275 , 2018), which could be applied for example to Poisson regression, negative binomial regression and proportional hazard models with censoring, to a wider class of non-linear multiple regression models. This includes the binary response models with logit and probit link besides others. For this class of models we derive (locally) D-optimal designs when the design region is a k-dimensional ball. For the corresponding construction we make use of the concept of invariance and equivariance in the context of optimal designs as in our previous paper. In contrast to the former results the designs will not necessarily be exact designs in all cases. Instead approximate designs can appear. These results can be generalized to arbitrary ellipsoidal design regions.

Nonlinear Classical Elasticity Model for Materials with Fluid and Matrix Phases

Materials with fluid and matrix phases present different acoustic responses in each phase. While longitudinal waves propagate in both phases, shear waves do it only through the solid matrix. Longitudinal waves are mainly described by volumetric propagation and shear waves by deviatoric processes. In the case of nonlinear propagation cross effects occur between both components. This paper presents a new classical nonlinear model proposing a constitutive equation that separates volumetric and deviatoric effects. Four nonlinear constants of third order are defined. The formulation is compared to constitutive equations with Landau constants for weakly elasticity and both types of nonlinear constants related. Some reinterpretation of the Landau’s constants arises in terms of parallel or cross nonlinear effects between volumetric and deviatoric components.

Spline-Based Motion Planning for Autonomous Guided Vehicles in a Dynamic Environment

Autonomous vehicles require a collision-free motion trajectory at every time instant. This brief presents an optimization-based method to calculate such trajectories for autonomous vehicles operating in an uncertain environment with moving obstacles. The proposed approach applies to linear system models, as well as to a particular class of nonlinear models, including industrially relevant vehicles, such as autonomous guided vehicles with front wheel, differential wheel, and rear-wheel steering. The method computes smooth motion trajectories, satisfying the vehicle’s kinematics, by using a spline parameterization. Furthermore, it exploits spline properties to keep the resulting nonlinear optimization problem small scale and to guarantee constraint satisfaction, without the need for time gridding. The resulting problem is solved sufficiently fast for online motion planning, dealing with uncertainties and changes in the environment. This brief demonstrates the potential of the method with extensive numerical simulations. In addition, it presents an experimental validation in which a KUKA youBot , steered as a holonomic or differential drive vehicle, drives through an environment with moving obstacles. To facilitate the further development and the numerical and experimental validation of the presented method, it is embodied in a user-friendly open-source software toolbox.

Speed Tracking Control of Permanent Magnet Synchronous Motor by a Novel Two-step Internal Model Control Approach

The control problem of permanent magnet (PM) synchronous motor has attracted extensive attentions in both control society and industrial applications. Much recently, a local speed tracking and nonlinear disturbance rejection problem of PM synchronous motor was investigated by nonlinear internal model design. In this paper, we propose a novel two-step controller design strategy to achieve speed tracking and nonlinear disturbance rejection of PM synchronous motor with wide speed range, which combines the advantages of classical double-loop control and nonlinear internal model control. It is worth mentioning that the proposed two-step controller design strategy can not only guarantee exact speed tracking with wide speed range, but also reject small nonlinear external disturbances in the load torque, generated by a so-called nonlinear exosystem. Simulation results demonstrate the effectiveness of our design.

Finite element modeling of nonlinear reaction–diffusion–advection systems of equations

PurposeThis paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed.Design/methodology/approachA stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation.FindingsThe stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model.Originality/valueA stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.

Characterization ofc-,L- andϕk-optimal Designs for a Class of Non-Linear Multiple-Regression Models

SummaryOptimal designs for multiple-regression models are determined. We consider a general class of non-linear models including proportional hazards models with different censoring schemes, the Poisson and the negative binomial model. For these models we provide a complete characterization of c-optimal designs for all vectors c in the case of a single covariate. For multiple regression with an arbitrary number of covariates, c-optimal designs for certain vectors c are derived analytically. Using some general results on the structure of optimal designs for multiple regression, we determine L- and ϕk-optimal designs for models with an arbitrary number of covariates.

Modeling of anisotropic remodeling of trabecular bone coupled to fracture

As a living tissue, bone is subjected to internal evolutions of its trabecular architecture under normal everyday mechanical loadings leading to damage. The repeating bone remodeling cycle aims at repairing the damaged zones in order to maintain bone structural integrity; this activity of sensing the peak stress at locations where damage or microcracks have occurred, removing old bone and apposing new bone is achieved thanks to a complicated machinery at the cellular level involving specialized cells (osteocytes, osteoclasts, and osteoblasts). This work aims at developing an integrated remodeling-to-fracture model to simulate the bone remodeling process, considering trabecular bone anisotropy. The effective anisotropic continuum mechanical properties of the trabecular bone are derived from an initially discrete planar hexagonal structure representative of femur bone microstructure, relying on the asymptotic homogenization technique. This leads to scaling laws of the effective elastic properties of bone versus effective density at an intermediate mesoscopic scale. An evolution law for the local bone apparent density is formulated in the framework of the thermodynamics of irreversible processes, whereby the driving force for density evolutions is identified as the local strain energy density weighted by the locally accumulated microdamage. We adopt a classical nonlinear damage model for high cycle fatigue under purely elastic strains, where the assumed homogeneous damage is related to the number of cycles bone experiences. Based on this model, we simulate bone remodeling for the chosen initial microstructure, showing the influence of the external mechanical stimuli on the evolution of the density of bone and the incidence of this evolution on trabecular bone effective mechanical properties.

Forecasting spikes in electricity return innovations

This paper evaluates the accuracy of several hundred one-day-ahead value at risk (VaR) forecasts for predicting Australian electricity returns. We propose a class of observation-driven time series models referred to as asymmetric exponential generalised autoregressive score (AEGAS) models. The mechanism to update the parameters over time is provided by the scaled score of the likelihood function in the AEGAS model. Based on this new approach, the results provide a unified and consistent framework for introducing time-varying parameters in a wide class of non-linear models. The Australian energy markets is known as one of the most volatile and, when compared to some well-known models in the recent literature as benchmarks the fitting and forecasting results demonstrate the superior performance and considerable flexibility of proposed model for electricity markets.

Time fractional super-diffusion model and its application in peak-preserving smoothing

The super-diffusion model is suggested for peak-preserving smoothing. In this model, the time derivative on the left of the classical diffusion model is replaced with the time fractional derivative. Because of the weight property of the fractional derivative, the super-diffusion model can further improve the smooth performance of the classical nonlinear diffusion model. An explicit difference scheme and an implicit difference scheme are given. Then some comparisons between the proposed model and the classical nonlinear diffusion model are done. The results indicate the proposed model outperforms the classic nonlinear diffusion model. In the end, the proposed method is used to smooth a nuclear magnetic resonance spectroscopy and a mass spectrometry.

Structural stability of nonlinear population dynamics.

In population dynamics, the concept of structural stability has been used to quantify the tolerance of a system to environmental perturbations. Yet, measuring the structural stability of nonlinear dynamical systems remains a challenging task. Focusing on the classic Lotka-Volterra dynamics, because of the linearity of the functional response, it has been possible to measure the conditions compatible with a structurally stable system. However, the functional response of biological communities is not always well approximated by deterministic linear functions. Thus, it is unclear the extent to which this linear approach can be generalized to other population dynamics models. Here, we show that the same approach used to investigate the classic Lotka-Volterra dynamics, which is called the structural approach, can be applied to a much larger class of nonlinear models. This class covers a large number of nonlinear functional responses that have been intensively investigated both theoretically and experimentally. We also investigate the applicability of the structural approach to stochastic dynamical systems and we provide a measure of structural stability for finite populations. Overall, we show that the structural approach can provide reliable and tractable information about the qualitative behavior of many nonlinear dynamical systems.

Robust Fault Estimation Design for Discrete-Time Nonlinear Systems via A Modified Fuzzy Fault Estimation Observer

The paper provides relaxed designs of fault estimation observer for nonlinear dynamical plants in the Takagi–Sugeno form. Compared with previous theoretical achievements, a modified version of fuzzy fault estimation observer is implemented with the aid of the so-called maximum-priority-based switching law. Given each activated switching status, the appropriate group of designed matrices can be provided so as to explore certain key properties of the considered plants by means of introducing a set of matrix-valued variables. Owing to the reason that more abundant information of the considered plants can be updated in due course and effectively exploited for each time instant, the conservatism of the obtained result is less than previous theoretical achievements and thus the main defect of those existing methods can be overcome to some extent in practice. Finally, comparative simulation studies on the classical nonlinear truck-trailer model are given to certify the benefits of the theoretic achievement which is obtained in our study.

Comparison of least squares and exponential sine sweep methods for Parallel Hammerstein Models estimation

Abstract Linearity is a common assumption for many real-life systems, but in many cases the nonlinear behavior of systems cannot be ignored and must be modeled and estimated. Among the various existing classes of nonlinear models, Parallel Hammerstein Models (PHM) are interesting as they are at the same time easy to interpret as well as to estimate. One way to estimate PHM relies on the fact that the estimation problem is linear in the parameters and thus that classical least squares (LS) estimation algorithms can be used. In that area, this article introduces a regularized LS estimation algorithm inspired on some of the recently developed regularized impulse response estimation techniques. Another mean to estimate PHM consists in using parametric or non-parametric exponential sine sweeps (ESS) based methods. These methods (LS and ESS) are founded on radically different mathematical backgrounds but are expected to tackle the same issue. A methodology is proposed here to compare them with respect to (i) their accuracy, (ii) their computational cost, and (iii) their robustness to noise. Tests are performed on simulated systems for several values of methods respective parameters and of signal to noise ratio. Results show that, for a given set of data points, the ESS method is less demanding in computational resources than the LS method but that it is also less accurate. Furthermore, the LS method needs parameters to be set in advance whereas the ESS method is not subject to conditioning issues and can be fully non-parametric. In summary, for a given set of data points, ESS method can provide a first, automatic, and quick overview of a nonlinear system than can guide more computationally demanding and precise methods, such as the regularized LS one proposed here.

Particle filters for partially-observed Boolean dynamical systems

Partially-observed Boolean dynamical systems (POBDS) are a general class of nonlinear models with application in estimation and control of Boolean processes based on noisy and incomplete measurements. The optimal minimum mean square error (MMSE) algorithms for POBDS state estimation, namely, the Boolean Kalman filter (BKF) and Boolean Kalman smoother (BKS), are intractable in the case of large systems, due to computational and memory requirements. To address this, we introduce approximate MMSE filtering and smoothing algorithms based on the auxiliary particle filter (APF) method, which are called APF–BKF and APF–BKS, respectively. For joint state and parameter estimation, the APF–BKF is used jointly with maximum-likelihood (ML) methods for simultaneous state and parameter estimation in POBDS models. In the case the unknown parameters are discrete, the proposed ML adaptive filter consists of multiple APF–BKFs running in parallel, in a manner reminiscent of the Multiple Model Adaptive Estimation (MMAE) method in classical linear filtering theory. In the presence of continuous parameters, the proposed ML adaptive filter is based on an efficient particle-based expectation maximization (EM) algorithm for the POBDS model, which is based on a modified Forward Filter Backward Simulation (FFBSi) in combination with the APF–BKS. The performance of the proposed particle-based adaptive filters is assessed through numerical experiments using a POBDS model of the well-known cell cycle gene regulatory network observed through noisy RNA-Seq time series data.