On the Hybrid Minimum Principle

TL;DR

This paper develops the Hybrid Minimum Principle (HMP) for deterministic hybrid systems with complex switching behaviors, providing explicit boundary conditions and demonstrating its application to electric vehicle control.

Contribution

It introduces the HMP for systems with autonomous and controlled switchings, including state dimension changes, with explicit boundary conditions and practical application examples.

Findings

Established the HMP for hybrid systems with dimension-changing jumps.

Derived explicit boundary conditions for Hamiltonians and adjoint processes.

Applied the HMP to optimize control in an electric vehicle powertrain.

Abstract

The Hybrid Minimum Principle (HMP) is established for the optimal control of deterministic hybrid systems with both autonomous and controlled switchings and jumps where state jumps at the switching instants are permitted to be accompanied by changes in the dimension of the state space. First order variational analysis is performed via the needle variation methodology and the necessary optimality conditions are established in the form of the HMP. A feature of special interest in this work is the explicit presentations of boundary conditions on the Hamiltonians and the adjoint processes before and after switchings and jumps. In addition to an analytic example, the HMP results are illustrated for the optimal control of an electric vehicle with transmission, where the modelling of the powertrain requires the consideration of both autonomous and controlled switchings accompanied by dimension changes.