Generalized Polynomial Chaos Theory Research Articles

Deterministic and Probabilistic Approaches to Model and Update Dynamic Systems

Updating simulation models, based on flight test or other data, or creating a model using only experimental data is a necessary process that can be costly and laborious. Simulation updates are necessary for high-fidelity simulations used in analysis, control development, handling qualities prediction, fleet training, etc. This paper presents methods to update aircraft models based on deterministic and probabilistic approaches. The deterministic approach is based on a collection of signal processing, statistical analysis, and maximum likelihood estimators embodied in the Algorithms to Update Simulation Parameters with Experimental Data tool. Mathematically sound and statistically rigorous techniques are used to calculate updates of current model parameters and suggest new terms capturing unmodeled dynamics, thereby improving correlation with observed vehicle response. The probabilistic approach is based on generalized polynomial chaos theory included in the Algorithms for Uncertainty Representation and Analysis (AURA) tool. The AURA tool allows the user to update probabilistic models based on experimental data, formulated as a Bayesian inference problem. This paper will present the methodology for both deterministic and probabilistic approaches and provide examples of their application to different aircraft designs, including tilt-rotor and rotary-wing unmanned air system vehicles. The results for both deterministic and probabilistic modeling approaches demonstrate excellent results. The probabilistic approach is novel in its expressive power to propagate and update probabilistic models and provides users with an intuitive and powerful modeling approach.

Regression-Based Stochastic Study of Electromagnetic Fields Due to Lightning Strikes

Electromagnetic (EM) fields triggered by lightning strikes are likely to display non-negligible variability in practice, with an apparent effect on various quantities of interest, such as induced voltages on transmission lines. This paper investigates the stochastic properties of lightning-generated fields, when the latter are affected by random ground parameters and/or nondeterministic lightning base current. In essence, the EM components are represented herein by truncated series of orthogonal polynomials, according to the generalized polynomial-chaos theory, and their expansion coefficients are computed via a regression approach. The necessary samplings of the random spaces are constructed with either tensor-based or sparse grids, and their performance is validated and compared. It is shown that reliable calculations of the expectation can be conducted with low-order polynomial approximations, while at least third-order expansions are required for the standard deviation. In addition, we conclude that nontrivial variability is induced in lightning-produced pulses, when variations of the considered stochastic parameters are taken into account.

A Flexible Polynomial Expansion Method for Response Analysis with Random Parameters

The generalized Polynomial Chaos Expansion Method (gPCEM), which is a random uncertainty analysis method by employing the orthogonal polynomial bases from the Askey scheme to represent the random space, has been widely used in engineering applications due to its good performance in both computational efficiency and accuracy. But in gPCEM, a nonlinear transformation of random variables should always be used to adapt the generalized Polynomial Chaos theory for the analysis of random problems with complicated probability distributions, which may introduce nonlinearity in the procedure of random uncertainty propagation as well as leading to approximation errors on the probability distribution function (PDF) of random variables. This paper aims to develop a flexible polynomial expansion method for response analysis of the finite element system with bounded random variables following arbitrary probability distributions. Based on the large family of Jacobi polynomials, an Improved Jacobi Chaos Expansion Method (IJCEM) is proposed. In IJCEM, the response of random system is approximated by the Jacobi expansion with the Jacobi polynomial basis whose weight function is the closest to the probability density distribution (PDF) of the random variable. Subsequently, the moments of the response can be efficiently calculated though the Jacobi expansion. As the IJCEM avoids the necessity that the PDF should be represented in terms of the weight function of polynomial basis by using the variant transformation, neither the nonlinearity nor the errors on random models will be introduced in IJCEM. Numerical examples on two random problems show that compared with gPCEM, the IJCEM can achieve better efficiency and accuracy for random problems with complex probability distributions.

An Efficient Technique Based on Polynomial Chaos to Model the Uncertainty in the Resonance Frequency of Textile Antennas Due to Bending

The generalized polynomial chaos theory is combined with a dedicated cavity model for curved textile antennas to statistically quantify variations in the antenna's resonance frequency under randomly varying bending conditions. The nonintrusive stochastic method solves the dispersion relation for the resonance frequencies of a set of radius of curvature realizations corresponding to the Gauss quadrature points belonging to the orthogonal polynomials having the probability density function of the random variable as a weighting function. The formalism is applied to different distributions for the radius of curvature, either using a priori known or on-the-fly constructed sets of orthogonal polynomials. Numerical and experimental validation shows that the new approach is at least as accurate as Monte Carlo simulations while being at least 100 times faster. This makes the method especially suited as a design tool to account for performance variability when textile antennas are deployed on persons with varying body morphology.

Optimal Experimental Design for Probabilistic Model Discrimination Using Polynomial Chaos

Building dynamic models is important in many applications including model-based design, optimization, and control. When multiple hypothesized models have predictions that are consistent with the measurements, experimental design is used to discriminate between the models. This task is particularly challenging for nonlinear systems subject to uncertainties. An optimal experimental design method for model discrimination for polynomial uncertain systems is presented that can be used to discriminate models based on dissimilarity of the probability densities of the model outputs. Generalized polynomial chaos theory in conjunction with Galerkin projection is used to derive an extended set of ordinary differential equations. Simulation of the extended system enables prediction of the propagation of probabilistic uncertainties associated with the model parameters and initial conditions, and to obtain the output probability densities. The simulation of the hypothetical models is embedded in a nonlinear optimization problem to determine an optimal input sequence that maximizes model dissimilarity. The experimental design method is demonstrated using a numerical example.

Linear Receding Horizon Control with Probabilistic System Parameters

In this paper we address the problem of designing receding horizon control algorithms for linear discrete-time systems with parametric uncertainty. We do not consider presence of stochastic forcing or process noise in the system. It is assumed that the parametric uncertainty is probabilistic in nature with known probability density functions. We use generalized polynomial chaos theory to design the proposed stochastic receding horizon control algorithms. In this framework, the stochastic problem is converted to a deterministic problem in higher dimensional space. The performance of the proposed receding horizon control algorithms is assessed using a linear model with two states.

Optimal Trajectory Generation With Probabilistic System Uncertainty Using Polynomial Chaos

In this paper, we develop a framework for solving optimal trajectory generation problems with probabilistic uncertainty in system parameters. The framework is based on the generalized polynomial chaos theory. We consider both linear and nonlinear dynamics in this paper and demonstrate transformation of stochastic dynamics to equivalent deterministic dynamics in higher dimensional state space. Minimum expectation and variance cost function are shown to be equivalent to standard quadratic cost functions of the expanded state vector. Results are shown on a stochastic Van der Pol oscillator.

Linear quadratic regulation of systems with stochastic parameter uncertainties

In this paper, we develop a theoretical framework for linear quadratic regulator design for linear systems with probabilistic uncertainty in the parameters. The framework is built on the generalized polynomial chaos theory. In this framework, the stochastic dynamics is transformed into deterministic dynamics in higher dimensional state space, and the controller is designed in the expanded state space. The proposed design framework results in a family of controllers, parameterized by the associated random variables. The theoretical results are applied to a controller design problem based on stochastic linear, longitudinal F16 model. The performance of the stochastic design shows excellent consistency, in a statistical sense, with the results obtained from Monte-Carlo based designs.

Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects

This study explores the use of generalized polynomial chaos theory for modeling complex nonlinear multibody dynamic systems in the presence of parametric and external uncertainty. The polynomial chaos framework has been chosen because it offers an efficient computational approach for the large, nonlinear multibody models of engineering systems of interest, where the number of uncertain parameters is relatively small, while the magnitude of uncertainties can be very large (e.g., vehicle-soil interaction). The proposed methodology allows the quantification of uncertainty distributions in both time and frequency domains, and enables the simulations of multibody systems to produce results with “error bars”. The first part of this study presents the theoretical and computational aspects of the polynomial chaos methodology. Both unconstrained and constrained formulations of multibody dynamics are considered. Direct stochastic collocation is proposed as less expensive alternative to the traditional Galerkin approach. It is established that stochastic collocation is equivalent to a stochastic response surface approach. We show that multi-dimensional basis functions are constructed as tensor products of one-dimensional basis functions and discuss the treatment of polynomial and trigonometric nonlinearities. Parametric uncertainties are modeled by finite-support probability densities. Stochastic forcings are discretized using truncated Karhunen-Loeve expansions. The companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part II: Numerical Applications” illustrates the use of the proposed methodology on a selected set of test problems. The overall conclusion is that despite its limitations, polynomial chaos is a powerful approach for the simulation of multibody systems with uncertainties.

Modeling multibody systems with uncertainties. Part II: Numerical applications

This study applies generalized polynomial chaos theory to model complex nonlinear multibody dynamic systems operating in the presence of parametric and external uncertainty. Theoretical and computational aspects of this methodology are discussed in the companion paper “Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects”.