Optimal Trajectory Generation With Probabilistic System Uncertainty Using Polynomial Chaos

2010 ◽  
Vol 133 (1) ◽  
Author(s):  
James Fisher ◽  
Raktim Bhattacharya
Keyword(s):  
Optimal Trajectory ◽  
Polynomial Chaos ◽  
Stochastic Dynamics ◽  
Trajectory Generation ◽  
Van Der Pol Oscillator ◽  
Probabilistic Uncertainty ◽  
Higher Dimensional ◽  
Generalized Polynomial ◽  
Dimensional State Space ◽  
Polynomial Chaos Theory

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.

Wiener–Haar Expansion for the Modeling and Prediction of the Dynamic Behavior of Self-Excited Nonlinear Uncertain Systems

2012 ◽  
Vol 134 (5) ◽  
Author(s):  
Lyes Nechak ◽  
Sébastien Berger ◽  
Evelyne Aubry
Keyword(s):  
Nonlinear Systems ◽  
Dynamic Behavior ◽  
Polynomial Chaos ◽  
Probabilistic Uncertainty ◽  
Processing Theory ◽  
Modeling And Prediction ◽  
Generalized Polynomial ◽  
Nonlinear Uncertain Systems ◽  
Weighting Coefficients ◽  
Polynomial Chaos Theory

This paper deals with the modeling and the prediction of the dynamic behavior of uncertain nonlinear systems. An efficient method is proposed to treat these problems. It is based on the Wiener–Haar chaos concept resulting from the polynomial chaos theory and it generalizes the use of the multiresolution analysis well known in the signal processing theory. The method provides a powerful tool to describe stochastic processes as series of orthonormal piecewise functions whose weighting coefficients are identified using the Mallat pyramidal algorithm. This paper shows that the Wiener–Haar model allows an efficient description and prediction of the dynamic behavior of nonlinear systems with probabilistic uncertainty in parameters. Its contribution, compared to the representation using the generalized polynomial chaos model, is illustrated by evaluating the two models via their application to the problems of the modeling and the prediction of the dynamic behavior of a self-excited uncertain nonlinear system.

Mathematics of Probabilistic Uncertainty Modeling

2014 ◽  
pp. 173-204 ◽  
Author(s):  
D. Datta
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Sample Size ◽  
Chaos Theory ◽  
Polynomial Chaos ◽  
Probabilistic Method ◽  
Uncertainty Modeling ◽  
Probabilistic Uncertainty ◽  
Log Normal ◽  
Polynomial Chaos Theory

This chapter presents the uncertainty modeling using probabilistic methods. Probabilistic method of uncertainty analysis is due to randomness of the parameters of a model. Randomness of parameters is characterized by specified probability distribution such as normal, log normal, exponential etc., and the corresponding samples are generated by various methods. Monte Carlo simulation is applied to explore the probabilistic uncertainty modeling. Monte Carlo simulation being a statistical process is based on the random number generation from the specified distribution of the uncertain random parameters. Sample size is generally very large in Monte Carlo simulation which is required to have small errors in the computation. Latin hypercube sampling and importance sampling are explored in brief. This chapter also presents Polynomial Chaos theory based probabilistic uncertainty modeling. Polynomial Chaos theory is an efficient Monte Carlo simulation in the sense that sample size here is very small and dictated by the number of the uncertain parameters and by choice of the order of the polynomial selected to represent the uncertain parameter.

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