Nonlinear Function Approach Research Articles

A varying-parameter complementary neural network for multi-robot tracking and formation via model predictive control

In this paper, a varying-parameter complementary neural network (VPCNN) is designed and combined with model predictive control (MPC) to solve the multi-robot tracking and formation problems via a leader–follower strategy. First, multi-robot tracking and formation problems are transformed into quadratic programming (QP) problems employing an MPC approach. Second, a nonlinear complementary function approach, i.e., the Fischer–Burmeister function, is used to map the Karush–Kuhn–Tucker conditions of the QP problem with double-ended inequality constraints to a system of nonlinear equations. Finally, the VPCNN is designed to solve the multi-robot tracking and formation problem. The effectiveness of the proposed method is demonstrated by numerical simulations, and the advantage of VPCNN in terms of solution speed is indicated by comparisons with a primal–dual neural network.

Novel Nonlinear Complementarity Function Approach for Mechanical Analysis of Tensegrity Structures

Cable slacking may be frequently encountered in mechanical analysis of tensegrity structures. Traditional nonlinear finite element analysis in handling this issue may lead to numerical difficulty due to the dramatic changes of the stiffness matrix induced by cables switching between taut and slack states. To deal with the nonsmooth nature induced by cable slacking, we propose in this paper a novel nonlinear complementarity function-based framework for mechanical analysis of tensegrity structures. The core idea is that the constitutive relation of cables is reformulated as a set of equality equations. This is done by first introducing a parametric variable and expressing the constitutive relation with a complementarity condition, and then using the modified Fischer–Burmeister complementarity function. By considering the introduced parametric variables as additional degrees of freedom in the finite element model, it becomes possible to get rid of the nonsmooth nature, as well as to solve a continuous, smooth, differentiable problem by using the classical Newton–Raphson method. The proposed framework is first developed for static analysis and then extended to dynamic analysis, both under large displacement. The closed-form expressions of the Jacobian matrix are analytically derived. Multiple examples are presented to verify the robustness and effectiveness of the proposed framework.

Emergent behaviors of the generalized Lohe matrix model

We present a first-order aggregation model on space of complex matrices which can be derived from Lohe tensor model on space of tensors with same rank and size. We call such matrix-valued aggregation model as the generalized Lohe matrix model. For proposed matrix model with two cubic coupling terms, we study several structural properties such as conservation laws, solution splitting property. In particular, for case of only one coupling, we reformulate reduced Lohe matrix model into Lohe matrix model with a diagonal frustration, and provide several sufficient frameworks leading to complete and practical aggregations. For estimates of collective dynamics, we use a nonlinear functional approach using an ensemble diameter which measures degree of aggregation.

Nonlinear functional approach for mono-cluster flocking to the discrete Cucker–Smale model

Nonlinear functional approach for mono-cluster flocking to the discrete Cucker–Smale model

Under‐canopy turbulence and root water uptake of a Tibetan meadow ecosystem modeled by Noah‐MP

AbstractThe Noah‐MP land surface model adopts a multiparameterization framework to accommodate various alternative parameterizations for more than 10 physical processes. In this paper, the parameterizations implemented in Noah‐MP associated with under‐canopy turbulence and root water uptake are enhanced with: (i) an under‐canopy turbulence scheme currently adopted by the Community Land Model (CLM), (ii) two vertical root distribution functions, i.e., an exponential and an asymptotic formulation, and (iii) three soil water stress functions (βt) controlling root water uptake, e.g., a soil water potential (ψ)‐based function, a nonlinear soil moisture (θ)‐based power function and an empirical threshold approach considering preferential uptake from the moist part of the soil column. A comprehensive data set of in situ micrometeorological observations and profile soil moisture/temperature measurements collected from an alpine meadow site in the northeastern Tibetan Plateau is utilized to assess the impact of the augmentations on the Noah‐MP performance. The results indicate that (i) implementation of the CLM under‐canopy turbulence scheme greatly resolves the overestimation of sensible heat flux and underestimation of soil temperature across the profile, (ii) both exponential and asymptotic vertical root distribution functions better represent the Tibetan conditions enabling a better representation of the measured soil moisture dynamics, and (iii) the ψ‐based βt functions overestimate surface soil moisture, the default linear θ‐based βt function underestimates latent heat flux during the dry‐down, while both the nonlinear power function and empirical threshold approach simultaneously simulate well soil moisture, and latent and sensible heat fluxes. Additionally, the parameter uncertainty associated with soil water stress function and hydraulic parameterization is addressed.

Uniform Stability and the Propagation of Regularity for the Relativistic Boltzmann Equation

We present a priori uniform $L^q$-stability estimates of classical solutions to the relativistic Boltzmann equation. Our uniform stability analysis does not require any smallness in the amplitudes of solutions, but we need exponentially decaying far-field conditions on density functions in phase space. The uniform stability analysis follows directly from the integrable temporal decay of the collision frequencies along the particle trajectory. Our uniform stability results improve on the previous results [S. Y. Ha, E. Jeong, and R. M. Strain, Comm. Pure Appl. Anal., 12 (2013), pp. 1141--1161] from a uniform $L^1$-stability analysis of the relativistic Boltzmann equation, which relied crucially on the nonlinear functional approach and smallness in the amplitude of the distribution function.

A nonlinear function approach for the normalized complementary relationship evaporation model

AbstractA nonlinear function approach for the normalized complementary relationship evaporation model that is different from the methodology maintaining the symmetric complementary relationship with appropriate definitions of potential and wet‐environment evaporation is proposed and verified. This approach employs the definitions used in the advection‐aridity model, wherein the potential is estimated using the Penman equation. Normalized by Penman potential evaporation, the complementary relationship model is expressed as a function describing the relationship between the evaporation ratio (the ratio of the actual to the Penman potential evaporation) and the proportion of the radiation term in Penman potential evaporation. The new nonlinear function proposed in the current study is approximately equivalent to the advection‐aridity and the modified Granger models under conditions that are neither too wet nor too dry, but is more reasonable under arid and wet conditions. The new nonlinear function model performs well in estimating actual evaporation, as verified by the observed data from four sites under different land covers. Copyright © 2011 John Wiley & Sons, Ltd.

A New Intelligent Estimation Model of Material Fatigue Limit

A fuzzy support vector machine kernel regression method of fatigue limit calculating is presented aiming at difficulty of material fatigue limit calculating and disadvantage of the other forecast model. Complicated and strong nonlinear fatigue limit of material is simulated by parameter optimized design and conformation of fuzzy support vector combinative algorithm, fuzzy support vector optimized algorithm has strong ability of nonlinear mapping and functional approach, it avoids availably phenomenon of partial minimum and overfitting, and gains high precision by comparing numerical value of network output with numerical value of experiment.

Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity

This paper investigates identification and inference in a nonparametric structural model with instrumental variables and non-additive errors. We allow for non-additive errors because the unobserved heterogeneity in marginal returns that often motivates concerns about endogeneity of choices requires objective functions that are non-additive in observed and unobserved components. We formulate several independence and monotonicity conditions that are sufficient for identification of a number of objects of interest, including the average conditional response, the average structural function, as well as the full structural response function. For inference we propose a two-step series estimator. The first step consists of estimating the conditional distribution of the endogenous regressor given the instrument. In the second step the estimated conditional distribution function is used as a regressor in a nonlinear control function approach. We establish rates of convergence, asymptotic normality, and give a consistent asymptotic variance estimator.

UNIFORM Lp-STABILITY ESTIMATE FOR THE SPATIALLY INHOMOGENEOUS BOLTZMANN EQUATION NEAR VACUUM

We present a new uniform Lp-stability theory for the spatially inhomogeneous Boltzmann equation near vacuum via the nonlinear functional approach proposed by the first author. Our stability analysis is based on new nonlinear functionals which are equivalent to the pth power of Lp-distance. The L1-nonlinear functionals play the key role of "modulators" which make the accumulative functional be non-increasing in time t along classical solutions.

Relationship between harmonic balance method and non-linear output frequency response function approach

The current paper is concerned with the investigation of the relationship between the harmonic balance method (HBM) and the non-linear output frequency response function (NOFRF) approach in the analysis of non-linear systems. Both are applied to the Duffing's oscillator to demonstrate their relationships. The results reveal that, if the Volterra series representation of a non-linear system is convergent, the harmonic components calculated by the NOFRFs are a solution of the HBM. Moreover, the simulation studies show that, in the convergent cases, the NOFRF method can give more accurate results for the higher-harmonic components than the HBM. The relationship investigated in the current study between the two methods should help researchers and engineers to understand the HBM and the NOFRF methods.

Global existence and stability of mild solutions to the Boltzmann system for gas mixtures

We present the global existence and stability of mild solutions to the Boltzmann system with inverse power molecular interactions for a binary gas mixture, when initial data are sufficiently small and decay exponentially in phase space. For the existence and stability of mild solutions, we employ a modified Kaniel-Shinbrot’s scheme and a weighted nonlinear functional approach. Time-asymptotic convergence toward the free molecular motion is established using a weighted collision potential, and we show that the weighted L 1 L^1 -distance between two mild solutions is uniformly controlled by that of initial data.

A Newton’s method for perturbed second-order cone programs

We develop optimality conditions for the second-order cone program. Our optimality conditions are well-defined and smooth everywhere. We then reformulate the optimality conditions into several systems of equations. Starting from a solution to the original problem, the sequence generated by Newton's method converges Q-quadratically to a solution of the perturbed problem under some assumptions. We globalize the algorithm by (1) extending the gradient descent method for differentiable optimization to minimizing continuous functions that are almost everywhere differentiable; (2) finding a directional derivative of the equations. Numerical examples confirm that our algorithm is good for "warm starting" second-order cone programs--in some cases, the solution of a perturbed instance is hit in two iterations. In the progress of our algorithm development, we also generalize the nonlinear complementarity function approach for two variables to several variables.

Stability estimates of the Boltzmann equation with quantum effects

We study uniform stability estimates to the Boltzmann equation for quantum particles such as Bose-Einstein particles and Fermi-Dirac particles. When the small amount of particles expands toward the vacuum, we show that continuous mild solutions are L1-stable and also satisfy BV-type estimates using a nonlinear functional approach.

Design and implementation of n-scroll chaotic attractors from a general jerk circuit

This paper proposes a novel nonlinear modulating function approach for generating n-scroll chaotic attractors based on a general jerk circuit. The systematic nonlinear modulating function methodology developed here can arbitrarily design the swings, widths, slopes, breakpoints, equilibrium points, shapes, and even the general phase portraits of the n-scroll chaotic attractors by using the adjustable sawtooth wave, triangular wave, and transconductor wave functions. The dynamic mechanism and chaos generation condition of the general jerk circuit are further investigated by analyzing the system stability. A simple block circuit diagram, including integrator, sawtooth wave and triangular wave generators, buffer, switch linkages, and voltage-current conversion resistors, is designed for the hardware implementations of various 3-12-scroll chaotic attractors via switchings of the switch linkages. This is the first time to experimentally verify a 12-scroll chaotic attractor generated by an analog circuit. In particular, the recursive formulas of system parameters and real physical circuit parameters are rigorously derived for the hardware implementations of the n-scroll chaotic attractors. Moreover, the adjustability of the nonlinear modulating function and the rigorous recursive formulas together provide a theoretical principle for the hardware implementations of various chaotic attractors with a large number of scrolls.

L1 Stability for Systems of Hyperbolic Conservation Laws with a Resonant Moving Source

In this paper, we study the L1 stability for the system ut + f(u)x = g(x-ct, u) when one of the characteristic fields has resonance with the moving source. The nonlinear resonance occurs when the speed of the source can coincide with one of the characteristic speeds of the hyperbolic conservation laws. In this situation, a wave pattern can be either stable or unstable. By employing a nonlinear functional approach, we prove the L1 stability of a transonic shock wave under the stability conditions introduced in [W.-C. Lien, Comm. Pure Appl. Math., 52 (1999), pp. 1075--1098; T.-P. Liu, Comm. Math. Phys., 83 (1982), pp. 243--260].

A nonlinear functional approach to LFT model validation

Model validation provides a useful means of assessing the ability of a model to account for a specific experimental observation, and has application to modeling, identification and fault detection. In this paper, we consider a new approach to the model validation problem by deploying quadratic functionals, and more generally nonlinear functionals, to specify noise and dynamical perturbation sets. Specifically, we consider a general linear fractional transformation framework for the model structure, and use constraints involving nonlinear functional inequalities to specify model non-linearities and unknown perturbations, and characteristics of noise and disturbance signals. Sufficient conditions for invalidation of such models are provided in terms of semidefinite programming problems.

Dynamical theory of strongly coupled two-dimensional Coulomb fluids in the weakly degenerate quantum domain.

We study the problem of dynamical response and plasma mode dispersion in strongly coupled two-dimensional Coulomb fluids (2DCFs) in the weakly degenerate quantum domain. Adapting the nonlinear response function approach of Golden and Kalman [Phys. Rev. A 19, 2112 (1979)] to the 2DCF, we construct a self-consistent approximation scheme for the calculation of the density response functions and plasma mode dispersion at long wavelengths. The basic ingredients in the construction are (i). the first kinetic equation in the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, (ii). the velocity-average-approximation (VAA) hypothesis, (iii.) the quadratic fluctuation-dissipation theorem, and (iv). the dynamical superposition approximation (DSA) closure hypothesis. The reliability of the VAA-DSA theory can be assessed by observing that the principal coupling correction to the 2D temperature-dependent Lindhard function is identified as being precisely the part of the third-frequency-moment sum-rule coefficient proportional to the potential energy.

Electronic stopping power for Monte Carlo simulation of ion implantation into SiC

A new electronic stopping power model for Monte Carlo simulation of ion implantation into 6H–SiC is presented. This model is based on the nonlinear density functional approach of Echenique et al. for the energy loss of slow ions moving through an electron gas and the ab initio pseudopotential calculations of Park et al. for the map of valence electrons of 6H–SiC crystal. A modified linear response theory has been used to treat the core electrons stopping. This model does not need fitting parameters and allows to fit the experimental distributions of implanted dopants in SiC both the peak region and the channeling tail of the SIMS profiles.

Multicritical unbinding phenomena and nonlinear functional renormalization group.

Using a nonlinear functional renormalization-group approach, we study the problem of unbinding phenomena both for interfaces and membranes. Our numerical and analytical solutions of the differential recursion relation confirm the existence of the ``unusual bifurcation'' of the fixed-point potentials describing unbinding at the upper critical dimension d=${\mathit{d}}^{\mathrm{*}}$. However, in addition to one critical and one purely repulsive potential, we find an infinite family of multicritical potentials bifurcating from the drifting fixed point. The values for the scaling indices, which are singular near ${\mathit{d}}^{\mathrm{*}}$, are calculated here exactly to leading orders in \ensuremath{\epsilon}=${\mathit{d}}^{\mathrm{*}}$-d and are shown to behave as ${\mathrm{\ensuremath{\epsilon}}}^{2/3}$.