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The Euler-Lagrange equation and optimal control: Preliminary results

Cédric Join, Emmanuel Delaleau, Michel Fliess

TL;DR

This work reframes optimal control for controllable LTI systems within a module-theoretic framework, where controllability corresponds to freeness and a flat output enables an Euler-Lagrange–based open-loop trajectory. It shows that when the Lagrangian is quadratic, the resulting Euler-Lagrange equations are linear and the cost can be expressed as a function of flat outputs and their derivatives, linking to classical LQR theory while enabling horizon optimization via a two-point boundary-value formulation. The closed-loop is achieved by augmenting the EL-based trajectory with a model-free, ultra-local controller (HEOL) using iP or iPD schemes, yielding strong robustness to disturbances and model mismatches. The approach is demonstrated on a simplified DC motor, and the framework is positioned to extend to nonlinear flat systems with non-quadratic Lagrangians and shooting methods for boundary-value problems.

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.

The Euler-Lagrange equation and optimal control: Preliminary results

TL;DR

This work reframes optimal control for controllable LTI systems within a module-theoretic framework, where controllability corresponds to freeness and a flat output enables an Euler-Lagrange–based open-loop trajectory. It shows that when the Lagrangian is quadratic, the resulting Euler-Lagrange equations are linear and the cost can be expressed as a function of flat outputs and their derivatives, linking to classical LQR theory while enabling horizon optimization via a two-point boundary-value formulation. The closed-loop is achieved by augmenting the EL-based trajectory with a model-free, ultra-local controller (HEOL) using iP or iPD schemes, yielding strong robustness to disturbances and model mismatches. The approach is demonstrated on a simplified DC motor, and the framework is positioned to extend to nonlinear flat systems with non-quadratic Lagrangians and shooting methods for boundary-value problems.

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.

Paper Structure

This paper contains 17 sections, 3 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A module-theoretic setting for LTI systems: A short review
  3. Basics on modules
  4. LTI systems
  5. Controllability
  6. Observability
  7. Transfer matrix
  8. Linear Euler-Lagrange equations
  9. Peculiar Lagrangians
  10. A connection with classic LQR
  11. Two-point boundary-value problem
  12. Closing the loop
  13. Homeostat
  14. iP and iPD controller
  15. Computer experiments
  16. ...and 2 more sections

Key Result

Proposition 1

If the Lagrangian $\mathcal{L}$ is a time-invariant positive-definite quadratic form in the variables $(z_{\lambda}^{(\mu_\lambda)} - \chi_{\lambda}^{\mu_\lambda})$, $\chi_{\lambda}^{\mu_\lambda} \in \mathbb{R}$, the Euler-Lagrange equation may be written where $H_1, \dots, H_m \in {\rm span}_{{\mathbb{R}}[\frac{d}{dt}]}({\bf z})$. The above system of linear differential equations is thus homogen

Figures (1)

  • Figure :

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3