A variational and symplectic framework for model-free control: preliminary results

TL;DR

The paper addresses the robustness limitations of model-free control by introducing a variational, symplectic formulation that online-tunes the key parameter $\alpha$ (via $\gamma = 1/\alpha$) through a discrete Euler–Lagrange framework. By coupling the MF control law with a variational integrator, the approach preserves structural properties and enhances disturbance rejection, as demonstrated on a buck converter for both voltage and power control. Results show improved smoothness and stability under perturbations, with the dynamics of $\gamma$ playing a central role in adapting to changes. The work offers a principled path toward auto-tuning in model-free control, with implications for multi-rate and hybrid systems and potential extensions to broader control laws.

Abstract

The model-free control approach is an advanced control law that requires few information about the process to control. Since its introduction in 2008, numerous applications have been successfully considered, highlighting attractive robustness properties towards tracking efficiency and disturbance rejection. In this work, a variational approach of the model-free control is proposed in order to extend its robustness capabilities. An adaptive formulation of the controller is proposed using the calculus of variations within a symplectic framework, that aims to consider the control law as an optimization problem toward the auto-tuning of its main key parameter. The proposed formulation provides a coupling between the model-free control law and a variational integrator to improve the robustness of the tracking towards process changes and emphasize closed-loop stabilization. Some illustrative examples are discussed to highlight the rightness of the proposed approach.

Figures (13)

• Figure 1: Buck converter.
• Figure 2: Standard model-free control - controlled voltage $v$ and its reference $v^*$ (top) and associated duty-cycle $u$ (bottom) versus time (s).
• Figure 3: Variational-based model-free control - controlled voltage $v$ and its reference $v^*$ (top) and associated duty-cycle $u$ (bottom) versus time (s).
• Figure 4: Evolution of the parameter $\gamma$ versus time (s).
• Figure 5: Standard model-free control - controlled voltage $v$ and its reference $v^*$ (top) and associated duty-cycle $u$ (bottom) versus time (s) for a destabilizing perturbation.