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

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. dL(t),dU (t) Lower and upper boundaries, scalar. Σ(t) State covariance matrix. Σy(t) Output covariance, scalar. Pa(t) Probability of no collision, scalar. Phit(t) Collision probability, 1−Pa(t), scalar.