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Research Papers

A Single Differential Equation for First-Excursion Time in a Class of Linear Systems

[+] Author and Article Information
Matthew B. Greytak

 Control Systems Engineer Naval Surface Warfare Center, Carderock Division Bethesda, MD 20817 matthew.greytak@navy.mil

Franz S. Hover

Department of Mechanical Engineering,  Massachusetts Institute of Technology, Cambridge, MA 02139 hover@mit.edu

The fundamental theorem of calculus as used here is:

If N(a,A) and N(b,B) are normal distributions with means a and b and variances A and B, respectively, then the product of the two distributions is:

J. Dyn. Sys., Meas., Control 133(6), 061020 (Nov 23, 2011) (7 pages) doi:10.1115/1.4004602 History: Received October 25, 2010; Revised April 14, 2011; Published November 23, 2011; Online November 23, 2011

First-excursion times have been developed extensively in the literature for oscillators; one major application is structural dynamics of buildings. Using the fact that most closed-loop systems operate with a moderate to high damping ratio, we have derived a new procedure for calculating first-excursion times for a class of linear continuous, time-varying systems. In several examples, we show that the algorithm is both accurate and time-efficient. These are important attributes for real-time path planning in stochastic environments, and hence the work should be useful for autonomous robotic systems involving marine and air vehicles.

FIGURES IN THIS ARTICLE
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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

The pdf of the output of a dynamic system diffuses past a gate, an instantaneous constraint. Our complete solution is the limiting case as the distance between consecutive gates approaches zero.

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Figure 2

The system shown in Eq. 10 with two time-varying constraints, Eq. 11. The first plot shows 100 trajectories from a 50,000-point Monte Carlo simulation. The trajectories drawn in black violate one of the constraints, while the gray trajectories do not. The dashed curves show the mean and standard deviation of the Monte Carlo simulation. The second plot shows the particle absorption fraction Phit as given by the SDE estimate (dashed) and the Monte Carlo estimate (solid). The error between these is shown at the bottom.

Grahic Jump Location
Figure 3

A time-varying second-order system, Eq. 12, representing the lateral dynamics of a rocket. The first plot shows 100 trajectories from a 50,000-point Monte Carlo simulation. The second plot shows the particle absorption fraction Phit as given by the SDE estimate (dashed) and the Monte Carlo estimate (solid). The error between these is shown at the bottom.

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Figure 4

The domain of integration for P(γ < c) Eq. 16, bounded by dL  < y < dL and Δy < cy. The first term is represented by the shaded triangle and the second term is represented by the open rectangle.

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Figure 5

p(γ) after Δt = 0.2, 0.5, and 1.0 s with the system in example 1 and walls at d = ±0.2. The result is validated with the histogram from a 100,000-point Monte Carlo simulation (gray bars). The dashed line is the pdf p(y) at t0 .

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