Spatio-temporal constrained zonotopes for validation of optimal control problems | IEEE Conference Publication | IEEE Xplore

Spatio-temporal constrained zonotopes for validation of optimal control problems


Abstract:

A controlled system subject to dynamics with unknown but bounded parameters is considered. The control is defined as the solution of an optimal control problem, which ind...Show More

Abstract:

A controlled system subject to dynamics with unknown but bounded parameters is considered. The control is defined as the solution of an optimal control problem, which induces hybrid dynamics. A method to enclose all optimal trajectories of this system is proposed. Using interval and zonotope based validated simulation and Pontryagin’s Maximum Principle, a characterization of optimal trajectories, a conservative enclosure is constructed. The usual validated simulation framework is modified so that possible trajectories are enclosed with spatio-temporal zonotopes that simplify simulation through events. Then optimality conditions are propagated backward in time and added as constraints on the previously computed enclosure. The obtained constrained zonotopes form a thin enclosure of all optimal trajectories that is less susceptible to accumulation of error. This algorithm is applied on Goddard’s problem, an aerospace problem with a bang-bang control.
Date of Conference: 14-17 December 2021
Date Added to IEEE Xplore: 01 February 2022
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Conference Location: Austin, TX, USA

I. INTRODUCTION

Optimal control of aerospace systems is performed by modeling the considered system by dynamics depending on multiple parameters (for example the engine thrust parameter) and an initial state (initial velocity, mass,…). Pontryagin’s Maximum Principle (PMP) provides necessary optimality conditions for the resolution of optimal control problems by transforming an optimal control problem into a two point boundary value problem which can be solved accurately [1], [2]. However, some of the parameters and initial states may be subject to uncertainties, that is, their exact value is subject to perturbations and estimation errors. Since the solution of an optimal control problem can rarely be explicitly expressed in a closed loop form, the effect of those perturbations can be hard to predict. This is complicated further by the fact that optimal trajectories tend to have hybrid behaviors: they may be subject to discrete events that abruptly change dynamics and state. For instance, Goddard’s problem [1], which consists in launching a rocket to a given position while minimizing its fuel consumption, alternates between a full throttle mode and a free fall mode. The duration of these phases depends on the parameters.

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