Optimal Control for Nonlinear Hybrid Systems via Convex Relaxations

TL;DR

This paper introduces a convex relaxation method for solving the optimal control problem in nonlinear hybrid systems, enabling globally optimal solutions despite the combinatorial complexity of switching behaviors.

Contribution

It develops a convex relaxation framework using semidefinite programming that guarantees convergence to the global optimum for hybrid system control problems.

Findings

Method successfully computes globally optimal controllers.

Converges from below to the true solution through a sequence of SDPs.

Validated on examples with known solutions and compared favorably to commercial solvers.

Abstract

This paper considers the optimal control for hybrid systems whose trajectories transition between distinct subsystems when state-dependent constraints are satisfied. Though this class of systems is useful while modeling a variety of physical systems undergoing contact, the construction of a numerical method for their optimal control has proven challenging due to the combinatorial nature of the state-dependent switching and the potential discontinuities that arise during switches. This paper constructs a convex relaxation-based approach to solve this optimal control problem. Our approach begins by formulating the problem in the space of relaxed controls, which gives rise to a linear program whose solution is proven to compute the globally optimal controller. This conceptual program is solved by constructing a sequence of semidefinite programs whose solutions are proven to converge from below to the true solution of the original optimal control problem. Finally, a method to synthesize the optimal controller is developed. Using an array of examples, the performance of the proposed method is validated on problems with known solutions and also compared to a commercial solver.