Stabilization of Optimum Trajectory Costate Differential Equations

Abstract

Introduction I N trajectory optimizationand synthesis of initial-valueand twopoint boundary-valueguidance and control laws, most applications of the Euler–Lagrange Ž rst-variation necessary conditions of the calculusof variationshave led to asymptoticallyunstablecostate differentialequationsand, hence,unstablebehaviorof systemphysical states.This instabilityhas beenmost troublesomewhen solving for long-durationextremal arcs, particularlywhen signiŽ cant statedependent nonlinear forces act on the  ight vehicle. For example, the purposemight be to designminimum fuel consumptiontrajectories for cruising  ight of a high L=D aircraft; the optimumsolutions may involve sustainedphugoidal oscillations, but that behavior can be obscured by rapid costate divergence. ThisEngineeringNote introducesamethod for stabilizingcostate differential equations via analytical phase adjustment of the timevarying costates. The method employs a time-varying phaseadjustment parameter. For a given model of the vehicle dynamics, a necessary lower bound on the phase-adjustment parameter is derived, and a stronger sufŽ cient condition is suggested. For simplicity, the subject is here presented in the context of a Ž rst-order one-dimensionalsystem,but extensiontomultidimensionalsystems is straightforward. Consider Ž rst a one-dimensionalguided object (such as a sled or railcar) that moves horizontallyon a straight path and has the linear, time-invariant state equations: