The Euler-Lagrange equation and optimal control: Preliminary results

TL;DR

This paper explores the algebraic structure of LTI systems as modules, deriving optimal control strategies through Euler-Lagrange equations, and discusses extensions to nonlinear systems with model-free control approaches.

Contribution

It introduces a module-based algebraic framework for LTI systems and derives open-loop control strategies using Euler-Lagrange equations, with preliminary insights on nonlinear extensions.

Findings

Linear systems can be modeled as modules with controllability as freeness.

Optimal control can be formulated via Euler-Lagrange equations in this framework.

Model-free control provides robust closed-loop strategies against disturbances.

Abstract

Algebraically speaking, linear time-invariant (LTI) systems can be considered as modules. In this framework, controllability is translated as the freeness of the system module. Optimal control mainly relies on quadratic Lagrangians and the consideration of any basis of the system module leads to an open-loop control strategy via a linear Euler-Lagrange equation. In this approach, the endpoint is easily assignable and time horizon can be chosen to minimize the criterion. The loop is closed via an intelligent controller derived from model-free control, which exhibits excellent performances concerning model mismatches and disturbances. The extension to nonlinear systems is briefly discussed.