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Research Papers

State Uncertainty Propagation in the Presence of Parametric Uncertainty and Additive White Noise

[+] Author and Article Information
Umamaheswara Konda1

Puneet Singla

Tarunraj Singh

 Department of Mechanical and Aerospace Engineering, University at Buffalo, Buffalo, NY 14260tsingh@buffalo.edu

Peter D. Scott

 Department of Computer Science and Engineering, University at Buffalo, Buffalo, NY 14260peter@buffalo.edu

1

Corresponding author.

J. Dyn. Sys., Meas., Control 133(5), 051009 (Aug 05, 2011) (10 pages) doi:10.1115/1.4004072 History: Received October 12, 2009; Revised March 14, 2011; Published August 05, 2011; Online August 05, 2011

The focus of this work is on the development of a framework permitting the unification of generalized polynomial chaos (gPC) with the linear moment propagation equations, to accurately characterize the state distribution for linear systems subject to initial condition uncertainty, Gaussian white noise excitation and parametric uncertainty which is not required to be Gaussian. For a fixed value of parameters, an ensemble of moment propagation equations characterize the distribution of the state vector resulting from Gaussian initial conditions and stochastic forcing, which is modeled as Gaussian white noise. These moment equations exploit the gPC approach to describe the propagation of a combination of uncertainties in model parameters, initial conditions and forcing terms. Sampling the uncertain parameters according to the gPC approach, and integrating via quadrature, the distribution for the state vector can be obtained. Similarly, for a fixed realization of the stochastic forcing process, the gPC approach provides an output distribution resulting from parametric uncertainty. This approach can be further combined with moment propagation equations to describe the propagation of the state distribution, which encapsulates uncertainties in model parameters, initial conditions and forcing terms. The proposed techniques are illustrated on two benchmark problems to demonstrate the techniques’ potential in characterizing the non-Gaussian distribution of the state vector.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

State and pdf transition

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Figure 2

Proposed ideas for uncertainty propagation through a stochastic linear dynamical system: (a) method 1: conditioning first on uncertain parameters and (b) method 2: conditioning first on Gaussian stochastic forcing

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Figure 3

Spring-mass system: (a) mass–spring–damper system and (b) evolution of nominal states

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Figure 4

Propagation of moments of the actual states: (a) propagation of mean, (b) propagation of variance, and (c) propagation of third central moment

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Figure 5

Histograms of the states at t = 10 s

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Figure 6

Hovering helicopter system: (a) hovering helicopter 31 and (b) evolution of nominal states

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Figure 7

Propagation of moments of the states: (a) propagation of mean, (b) propagation of variance, (c) propagation of third central moment, and (d) third central moment with 10,000 Monte Carlo runs

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Figure 8

Histograms of the states at t = 10 s

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