Minimum-Energy Control For Control-Affine Systems

TL;DR

This paper develops a method to compute minimum-energy controls for control-affine systems using a fixed-point equation and Gramian-like matrices, with convergence guarantees and practical examples.

Contribution

Introduces a fixed-point based approach for minimum-energy control in control-affine systems, including scalable algorithms and controllability conditions for specific models.

Findings

Convergent fixed-point iteration for control synthesis

Energy bounds established for control solutions

Applicability demonstrated on unicycle model

Abstract

In this letter, we derive minimum-energy controls for a broad class of control-affine systems using a Lagrange multiplier fixed-point equation and a generally non-symmetric Gramian-like matrix. In feasible coercivity classes, this fixed point is unique and can be computed by standard Picard iteration. These iterates converge with factorial decay, yielding an implementable, highly scalable synthesis with an intrinsic energy bound. As a demonstration of concept, we use uniform complete controllability results for linear time-varying systems to derive a bracket-generating condition ensuring complete controllability for time-dependent planar control-affine systems with scalar inputs. Special treatment for the unicycle kinematic model is also provided, and numerical examples illustrate the approach's effectiveness.