Robust Discrete-Time Pontryagin Maximum Principle on Matrix Lie Groups

TL;DR

This paper develops a discrete-time Pontryagin maximum principle for robust optimal control on matrix Lie groups, accounting for disturbances, with applications to rigid body rotation.

Contribution

It introduces a novel PMP framework for min-max control problems on matrix Lie groups, including necessary conditions and a special Euclidean case.

Findings

Derived first-order necessary conditions for robust control on Lie groups.

Established a saddle point condition on the Hamiltonian.

Applied the theory to robust single-axis rigid body rotation.

Abstract

This article considers a discrete-time robust optimal control problem on matrix Lie groups. The underlying system is assumed to be perturbed by exogenous unmeasured bounded disturbances, and the control problem is posed as a min-max optimal control wherein the disturbance is the adversary and tries to maximise a cost that the control tries to minimise. Assuming the existence of a saddle point in the problem, we present a version of the Pontryagin maximum principle (PMP) that encapsulates first-order necessary conditions that the optimal control and disturbance trajectories must satisfy. This PMP features a saddle point condition on the Hamiltonian and a set of backward difference equations for the adjoint dynamics. We also present a special case of our result on Euclidean spaces. We conclude with applying the PMP to robust version of single axis rotation of a rigid body.