An hybrid system approach to nonlinear optimal control problems

TL;DR

This paper presents a hybrid systems approach to nonlinear optimal control, replacing complex dynamics with piecewise affine models to facilitate analytical solutions and control synthesis.

Contribution

It introduces a novel hybrid automaton framework for approximating nonlinear control problems and derives a hybrid maximum principle for optimal control synthesis.

Findings

Developed a hybrid approximation of the nonlinear controllable domain

Proposed an algorithm for computing piecewise convex controllable regions

Derived a hybrid maximum principle for optimal control synthesis

Abstract

We consider a nonlinear ordinary differential equation and want to control its behavior so that it reaches a target by minimizing a cost function. Our approach is to use hybrid systems to solve this problem: the complex dynamic is replaced by piecewise affine approximations which allow an analytical resolution. The sequence of affine models then forms a sequence of states of a hybrid automaton. Given a sequence of states, we introduce an hybrid approximation of the nonlinear controllable domain and propose a new algorithm computing a controllable, piecewise convex approximation. The same way the nonlinear optimal control problem is replaced by an hybrid piecewise affine one. Stating a hybrid maximum principle suitable to our hybrid model, we deduce the global structure of the hybrid optimal control steering the system to the target.