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 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.

Guaranteed consistency between measurements and parameter systems with correlated additive and multiplicative uncertainties

“…Generally, the ellipsoidal set is privileged in literature for its simplicity. More recently, constrained zonotopes have been introduced [1][2][3] which provide a new representation of sets allowing to combine the flexibility of convex polytopes with the efficiency of zonotopes. For that reason they have been extensively used in several fields in automatic control that include in particular state estimation, reachability analysis, identification, fault detection and isolation, diagnosis [3][4][5][6][7][8] .…”

“…It is now necessary to structure this model in the form defined in (1). Following remark (2), after eliminating the states according to the measurements, we obtain :…”

“…This criterion is clearly a quadratic function of the magnitude of the uncertainties 𝛼 which include those affecting the model outputs and its parameters (1). In order to improve the accuracy of the model, it is therefore necessary to estimate the value of 𝛼 which makes the criterion 𝐽 minimal.…”