Analytic Optimal Control for a Class of Driftless x-Flat Systems

TL;DR

This paper develops an analytic optimal control method for driftless x-flat systems, providing explicit feedback laws and avoiding numerical boundary-value problem solutions.

Contribution

It derives a closed-form solution for optimal control of driftless x-flat systems using geometric properties and specific matrix relations.

Findings

Explicit feedback law for control inputs is obtained.

Avoids numerical solutions of boundary-value problems.

Demonstrated on a steerable axle model with accurate tracking.

Abstract

This paper studies optimal trajectory-tracking for driftless, x-flat nonlinear systems with three states and two inputs. The tracking problem is formulated in Bolza form with a quadratic cost of the tracking error and its derivative. Applying Pontryagin's maximum principle yields a mixed regular-singular optimal control problem. By exploiting geometric properties and a specific relation between the weighting matrices, a closed-form expression for the costate and an explicit feedback law for both inputs is derived. Thereby, the numerical solution of a two-point boundary-value problem is avoided. The singular input leads to a bang-singular-bang optimal control structure, while on the singular arc, the tracking error dynamics reduces to a linear dynamics of order two. The approach is illustrated for the kinematic model of a steerable axle, demonstrating accurate trajectory-tracking.