Flatness-based control revisited: The HEOL setting
Cédric Join, Emmanuel Delaleau, Michel Fliess
TL;DR
The paper addresses the practical challenge of closing loops in flatness-based control and model-free control by embedding the tangent linear system via Kähler differentials to define a homeostat. It develops an algebraic framework based on differential algebra and differential geometry to intrinsically define tangent dynamics, derives data-driven estimators $F_{\rm est}$, and designs intelligent controllers (iP, iPD) that robustly track reference trajectories in both monovariable and multivariable settings. A computer experiment demonstrates the approach on a flat two-output system, showing robustness to disturbances and initial-condition mismatches. The HEOL framework is positioned to simplify the implementation of flat systems, potentially aid optimal control problems, and motivate further exploration and applications.
Abstract
We present the algebraic foundations of the HEOL setting, which combines flatness-based control and intelligent controllers, two advances in automatic control that have been proven in practice, including in industry. The result provides a solution to many pending questions on feedback loops concerning flatness-based control and model-free control (MFC). Elementary module theory, ordinary differential fields and the generalization of Kähler differentials to differential fields provide an intrinsic definition of the tangent linear system. The algebraic manipulations associated with the operational calculus lead to homeostat and intelligent controllers. They are illustrated via some computer simulations.