On the Expansion of Nonlinear Models Using Bell-Shaped Basis Functions

2003 ◽  
Author(s):  
Kiriakos Kiriakidis
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Model ◽  
Upper Bound ◽  
Nonlinear Models ◽  
Approximation Error ◽  
Basis Functions ◽  
Model Set

We propose a method that approximates any nonlinear model, without regard to complexity, by minimizing its distance from a rich model set. The method produces, potentially through an automated procedure, the approximation of the nonlinear dynamics in the form of a finite expansion associated with certain basis functions and provides an upper bound on the approximation error.

Nonlinear Modeling by Interpolation Between Linear Dynamics and Its Application in Control

2007 ◽  
Vol 129 (6) ◽  
pp. 813-824 ◽  
Author(s):  
Kiriakos Kiriakidis
Keyword(s):  
Nonlinear Dynamics ◽  
Linear Growth ◽  
Linear Models ◽  
Nonlinear Modeling ◽  
Approximation Error ◽  
Basis Functions ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Growth Bound ◽  
Dynamics Models

This paper proposes a finite series expansion to approximate general nonlinear dynamics models to arbitrary accuracy. The method produces an approximation of nonlinear dynamics in the form of an aggregate of linear models, weighted by unimodal basis functions, and results in a linear growth bound on the approximation error. Furthermore, this paper demonstrates that the proposed approximation satisfies the modeling assumptions for analysis based on linear matrix inequalities and hence widens the applicability of these techniques to the area of nonlinear control.

Multiple regression and nonlinear models

2017 ◽  
Author(s):  
Andrew Gelman ◽  
Deborah Nolan
Keyword(s):  
Regression Analysis ◽  
Social Science ◽  
Multiple Regression ◽  
Nonlinear Model ◽  
Nonlinear Models ◽  
Good Strategy ◽  
Science Journal ◽  
Golf Putting ◽  
Social Science Journal ◽  
To Come

This chapter covers multiple regression and links statistical inference to general topics such as lurking variables that arose earlier. Many examples can be used to illustrate multiple regression, but we have found it useful to come to class prepared with a specific example, with computer output (since our students learn to run the regressions on the computer). We have found it is a good strategy to simply use a regression analysis from some published source (e.g., a social science journal) and go through the model and its interpretation with the class, asking students how the regression results would have to differ in order for the study’s conclusions to change. The chapter includes examples that revisit the simple linear model of height and income, involve the class in models of exam scores, and fit a nonlinear model (for more advanced classes) for golf putting.

Dynamics of a Supported Cylinder in Axial Flow

2011 ◽  
Vol 99-100 ◽  
pp. 1059-1062
Author(s):  
Ji Duo Jin ◽  
Ning Li ◽  
Zhao Hong Qin
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Analysis ◽  
Numerical Simulations ◽  
Nonlinear Model ◽  
Dynamical Behavior ◽  
Axial Flow ◽  
Viscous Force ◽  
Friction Drag ◽  
Flutter Instability ◽  
Nonlinear Force

The nonlinear dynamics are studied for a supported cylinder subjected to axial flow. A nonlinear model is presented for dynamics of the cylinder supported at both ends. The nonlinear terms considered here are the quadratic viscous force and the structural nonlinear force induced by the lateral motions of the cylinder. Using two-mode discretized equation, numerical simulations are carried out for the dynamical behavior of the cylinder to explain the flutter instability found in the experiment. The results of numerical analysis show that at certain value of flow velocity the system loses stability by divergence, and the new equilibrium (the buckled configuration) becomes unstable at higher flow leading to post-divergence flutter. The effect of the friction drag coefficients on the behavior of the system is investigated.

scholarly journals Oscillatory behavior of two nonlinear microbial models of soil carbon decomposition

2014 ◽  
Vol 11 (7) ◽  
pp. 1817-1831 ◽  
Author(s):  
Y. P. Wang ◽  
B. C. Chen ◽  
W. R. Wieder ◽  
M. Leite ◽  
B. E. Medlyn ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Microbial Biomass ◽  
Soil Carbon ◽  
Nonlinear Model ◽  
Linear Models ◽  
Nonlinear Models ◽  
Oscillatory Behavior ◽  
Carbon Pools ◽  
Carbon Decomposition ◽  
Pool Model

Abstract. A number of nonlinear models have recently been proposed for simulating soil carbon decomposition. Their predictions of soil carbon responses to fresh litter input and warming differ significantly from conventional linear models. Using both stability analysis and numerical simulations, we showed that two of those nonlinear models (a two-pool model and a three-pool model) exhibit damped oscillatory responses to small perturbations. Stability analysis showed the frequency of oscillation is proportional to √(ϵ−1−1) Ks/Vs in the two-pool model, and to √(ϵ−1−1) Kl/Vl in the three-pool model, where ϵ is microbial growth efficiency, Ks and Kl are the half saturation constants of soil and litter carbon, respectively, and /Vs and /Vl are the maximal rates of carbon decomposition per unit of microbial biomass for soil and litter carbon, respectively. For both models, the oscillation has a period of between 5 and 15 years depending on other parameter values, and has smaller amplitude at soil temperatures between 0 and 15 °C. In addition, the equilibrium pool sizes of litter or soil carbon are insensitive to carbon inputs in the nonlinear model, but are proportional to carbon input in the conventional linear model. Under warming, the microbial biomass and litter carbon pools simulated by the nonlinear models can increase or decrease, depending whether ϵ varies with temperature. In contrast, the conventional linear models always simulate a decrease in both microbial and litter carbon pools with warming. Based on the evidence available, we concluded that the oscillatory behavior and insensitivity of soil carbon to carbon input are notable features in these nonlinear models that are somewhat unrealistic. We recommend that a better model for capturing the soil carbon dynamics over decadal to centennial timescales would combine the sensitivity of the conventional models to carbon influx with the flexible response to warming of the nonlinear model.

scholarly journals An efficient nonlinear cardinal B-spline model for high tide forecasts at the Venice Lagoon

2006 ◽  
Vol 13 (5) ◽  
pp. 577-584 ◽  
Author(s):  
H. L. Wei ◽  
S. A. Billings
Keyword(s):  
Nonlinear Models ◽  
High Tide ◽  
Venice Lagoon ◽  
Basis Functions ◽  
Short Term ◽  
B Spline

Abstract. An efficient class of nonlinear models, constructed using cardinal B-spline (CBS) basis functions, are proposed for high tide forecasts at the Venice lagoon. Accurate short term predictions of high tides in the lagoon can easily be calculated using the proposed CBS models.

Nonlinear Modeling of a Rigid Rotor Supported on Rolling Element Bearings With Localized Defects

2010 ◽  
Author(s):  
Karthik Kappaganthu ◽  
C. Nataraj
Keyword(s):  
Nonlinear Model ◽  
Nonlinear Models ◽  
Nonlinear Modeling ◽  
Rolling Element Bearings ◽  
Rigid Rotor ◽  
Rolling Element ◽  
Localized Defects ◽  
Bearing Race ◽  
Overall Performance ◽  
Bearing System

In this paper a nonlinear model for defects in rolling element bearings is developed. Detailed nonlinear models are useful to detect, estimate and predict failure in rotating machines. Also, accurate modeling of the defect provides parameters that can be estimated to determine the health of the machine. In this paper the rotor-bearing system is modeled as a rigid rotor and the defects are modeled as pits in the bearing race. Unlike the previous models, the motion of the rolling element thorough the defect is not modeled as a predetermined function; instead, it is dynamically determined since it depends on the clearance and the position of the shaft. Using this nonlinear model, the motion of the shaft is simulated and the effect of the rolling element passing through the defect is studied. The effect of shaft parameters and the defect parameters on the precision of the shaft and the overall performance of the system is studied. Finally, suitable measures for health monitoring and defect tracking are suggested.

scholarly journals Estimate of Alabama argillacea (Hübner) (Lepidoptera: Noctuidae) development with nonlinear models

2003 ◽  
Vol 63 (4) ◽  
pp. 589-598 ◽  
Author(s):  
R. S. Medeiros ◽  
F. S. Ramalho ◽  
J. C. Zanuncio ◽  
J. E. Serrão
Keyword(s):  
Relative Humidity ◽  
Gossypium Hirsutum ◽  
Nonlinear Model ◽  
Nonlinear Models ◽  
Gossypium Hirsutum L ◽  
Developmental Rates ◽  
Alabama Argillacea ◽  
The Relationship ◽  
Lepidoptera Noctuidae

The objective of this work was to evaluate which nonlinear model [Davidson (1942, 1944), Stinner et al. (1974), Sharpe & DeMichele (1977), and Lactin et al. (1995)] best describes the relationship between developmental rates of the different instars and stages of Alabama argillacea (Hübner) (Lepidoptera: Noctuidae), and temperature. A. argillacea larvae were fed with cotton leaves (Gossypium hirsutum L., race latifolium Hutch., cultivar CNPA 7H) at constant temperatures of 20, 23, 25, 28, 30, 33, and 35ºC; relative humidity of 60 ± 10%; and photoperiod of 14:10 L:D. Low R² values obtained with Davidson (0.0001 to 0.1179) and Stinner et al. (0.0099 to 0.8296) models indicated a poor fit of their data for A. argillacea. However, high R² values of Sharpe & DeMichele (0.9677 to 0.9997) and Lactin et al. (0.9684 to 0.9997) models indicated a better fit for estimating A. argillacea development.

scholarly journals Radial Basis Functions Intended to Determine the Upper Bound of Absolute Dynamic Error at the Output of Voltage-Mode Accelerometers

Sensors ◽  
2019 ◽  
Vol 19 (19) ◽  
pp. 4154 ◽  
Author(s):  
Krzysztof Tomczyk ◽  
Marcin Piekarczyk ◽  
Grzegorz Sokal
Keyword(s):  
Mathematical Model ◽  
Radial Basis Functions ◽  
Upper Bound ◽  
Dynamic Error ◽  
Basis Functions ◽  
Voltage Mode ◽  
Radial Basis ◽  
Operational Frequency ◽  
Mc Method

In this paper, we propose using the radial basis functions (RBF) to determine the upper bound of absolute dynamic error (UAE) at the output of a voltage-mode accelerometer. Such functions can be obtained as a result of approximating the error values determined for the assumed-in-advance parameter variability associated with the mathematical model of an accelerometer. This approximation was carried out using the radial basis function neural network (RBF-NN) procedure for a given number of the radial neurons. The Monte Carlo (MC) method was also applied to determine the related error when considering the uncertainties associated with the parameters of an accelerometer mathematical model. The upper bound of absolute dynamic error can be a quality ratio for comparing the errors produced by different types of voltage-mode accelerometers that have the same operational frequency bandwidth. Determination of the RBFs was performed by applying the Python-related scientific packages, while the calculations related both to the UAE and the MC method were carried out using the MathCad program. Application of the RBFs represent a new approach for determining the UAE. These functions allow for the easy and quick determination of the value of such errors.

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