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

Abstract

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 timeefficient. 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. Nomenclature x(t)State vector. y(t)System output, scalar.A(t), G(t), C System matrices. w(t)Unit Wiener noise process, vector.Lower and upper boundaries, scalar. Σ Σ Σ(t)State covariance matrix.Σ y (t) Output covariance, scalar.P a (t) Probability of no collision, scalar.P hit (t) Collision probability, 1 − P a (t), scalar.