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Technical Briefs

Uncertainty Propagation in Abstracted Systems via the Liouville Equation

[+] Author and Article Information
Patricia Mellodge

Department of Electrical and Computer Engineering, University of Hartford, 200 Bloomfield Avenue, West Hartford, CT 06117mellodge@hartford.edu

Pushkin Kachroo

Department of Electrical and Computer Engineering, University of Nevada, Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154pushkin@unlv.edu

J. Dyn. Sys., Meas., Control 132(4), 044503 (Jun 17, 2010) (4 pages) doi:10.1115/1.4001705 History: Received October 02, 2007; Revised April 17, 2010; Published June 17, 2010; Online June 17, 2010

This technical brief shows that given a system and its abstraction, the evolution of uncertain initial conditions in the original system is, in some sense, matched by the evolution of the uncertainty in the abstracted system. In other words, it is shown that the concept of Φ-related vector fields extends to the case of stochastic initial conditions where the probability density function (pdf) for the initial conditions is known. In the deterministic case, the Φ mapping commutes with the system dynamics. In this paper, we show that in the case of stochastic initial conditions, the induced mapping Φpdf commutes with the evolution of the pdf according to the Liouville equation.

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Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

Grahic Jump Location
Figure 1

The relationship between Φ-related systems with known initial conditions

Grahic Jump Location
Figure 2

The relationship between Φ-related systems with uncertain initial conditions

Grahic Jump Location
Figure 3

The time evolution of the pdf ρY(y,t) with initial Gaussian distribution

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