HEOL Feedback Control

• HEOL feedback setting is a unified control-theoretic framework that fuses flatness-based open-loop design with model-free controllers to achieve robust feedback in nonlinear systems.
• It employs algebraic decomposition and operational calculus to determine flat outputs and compute online estimations, reducing computational burden compared to RL and ANN methods.
• The approach has proven effective across industrial processes, autonomous vehicles, and multi-agent systems by ensuring robust tracking, disturbance rejection, and adaptive control.

The HEOL feedback setting denotes a unified control-theoretic methodology, recently formalized and extended, which fuses differential flatness-based control with intelligent model-free controllers. The resulting approach provides an algebraic, computational, and implementation framework enabling robust feedback loop closure in nonlinear, flat, or approximately flat dynamic systems. HEOL has been theoretically grounded via modern advances in module theory, differential algebra, and operational calculus, and has demonstrated efficient and robust performance in a variety of engineering domains—ranging from industrial process control to autonomous vehicles and multi-agent synchronization.

1. Algebraic Structure and Flatness Properties

The theoretical foundation of HEOL is built upon the intrinsic algebraic characterization of flatness. In flatness-based control, systems are modeled as finitely generated modules over rings of differential operators (e.g., the ring $k[d/dt]$ if $k$ is an ordinary differential field). The system module $\Lambda$ is decomposed into free ($\mathcal{F}$) and torsion ($\mathcal{T}$) components, $\Lambda = \mathcal{F} \oplus \mathcal{T}$, providing intrinsic definitions of controllability and observability without explicit state-space representations (Join et al., 21 Aug 2024). Flat outputs $\{y_1,\ldots,y_m\}$ are chosen so that every system variable is expressible as a function of these outputs and their derivatives, with no integration required. For both linear and nonlinear cases, flatness allows comprehensive open-loop trajectory generation.

The tangent (variational) linear system of a flat system is defined via the generalization of Kähler differentials to differential fields, giving rise to a module $\Omega_{L/K}$, whose rank is the differential transcendence degree—equal to the number of independent flat outputs.

2. Feedback Loop Closure via Homeostat and Intelligent Controllers

HEOL advances a solution to feedback loop closure that merges algebraically derived open-loop references with model-free, real-time adaptation. After establishing the flat outputs and nominal feedforward control, a tangent linearization yields the homeostat equation, e.g.:

$$\frac{d^\nu}{dt^\nu}(\Delta y) = F + \alpha \Delta u$$

where $\Delta y = y - y^*$, $F$ aggregates unmodeled dynamics and disturbances, and $\alpha$ is an algebraically computed coefficient dependent on the system's local structure. Intelligent controllers—typically proportional (iP) or proportional-derivative (iPD) regulators—use online estimates $F_{\text{est}}$ to realize feedback:

$$\Delta u = -\frac{F_{\text{est}} + K_P \Delta y}{\alpha}$$

This paradigm generalizes model-free control frameworks by accommodating multivariable coupling, optimizing the selection of differentiation order, and systematic calculation of $\alpha$. The feedback design remains robust to process uncertainty, parameter drift, and unforeseen disturbances.

3. Operational Calculus and Data-Driven Estimation

Central to the HEOL feedback setting is the operational calculus-based estimator for unknown dynamical contributions $F$. The Laplace transform is used to annhilate initial conditions and extract $F$:

$$F_{\text{est}} = -\frac{6}{T^3} \int_0^T [(T - 2\sigma)\Delta \hat{y}(\sigma) + \sigma(T - \sigma)\alpha(\sigma)\Delta u(\sigma)] d\sigma$$

where $T$ is the estimator window and $\Delta \hat{y}$, $\Delta u$ are the measured (possibly filtered) output and input deviations, respectively (Join et al., 21 Aug 2024, Delaleau et al., 14 Jan 2025). This closed-form, algebraic estimator enables real-time adaptation without system identification, providing both disturbance rejection and compensation for unmodeled dynamics.

4. Implementation and Comparative Computational Burden

HEOL is characterized by low computational complexity in both its algebraic calculations and online estimation. For instance, the estimation and controller expressions are comprised of short finite-memory integrals and elementary feedback laws, contrasting sharply with computationally intensive approaches such as reinforcement learning (RL), dynamic programming (DP), or neural-network-based identification (Join et al., 1 Feb 2025, Join et al., 3 Sep 2025). While HEOL requires some expertise in the process model—primarily for identification of flat outputs and formation of the tangent model—the incremental burden over purely model-free approaches (ultra-local models) is modest and yields only slight performance advantages in terms of transient accuracy.

A comparison of MFPC and HEOL reveals:

Setting	Model Knowledge	Computational Burden	Robustness	Performance Gain
HEOL	Moderate	Low	High	Slight
MFPC	Minimal	Very Low	High	Good
RL / ANN	High	Very High	Variable	Variable

5. Application Domains

HEOL has demonstrated efficacy in diverse domains:

• Unmanned Surface Vehicles (USVs): Flatness-based output parametrization with homeostat-based adaptive regulation, validated on both nominal hovercraft and canonical USV models, including the presence of wind-like disturbances (Degorre et al., 30 Apr 2025).
• Obstacle Avoidance in Autonomous Vehicles: Dubins’ car trajectory planning with on-the-fly reference synchronization, showing low computational burden and high robustness to random perturbations; superior performance to RL-based methods in terms of required trials and adaptation (Join et al., 3 Sep 2025).
• Chemical Reactor and Two-Tank Systems: Incremental model-based HEOL control yields only a slight advantage over model-free predictive control, indicating that a comprehensive process model may not be essential for satisfactory control performance (Join et al., 1 Feb 2025).
• Synchronization of Coupled Oscillators: Kuramoto networks with multiplicative control, leveraging differential flatness for reference generation and homeostat-based feedback for synchronization and adaptive frequency tracking even under mismatches and noise (Delaleau et al., 14 Jan 2025).
• Industrial Robotics, Crane Control, Climate Systems: Cited examples illustrate versatility and adaptability across nonlinear and multivariable systems (Join et al., 21 Aug 2024).

6. Robustness and Theoretical Implications

Simulation results show that HEOL achieves robust tracking even with parameter uncertainty, input disturbances, and model mismatches. The homeostat-based estimator effectively compensates such effects without explicit identification or parameter adaptation. Theoretical guarantees stem from the algebraic structure, differential module theory, and operational estimator design, establishing stability and convergence properties under realistic measurement and estimation error bounds.

The approach bypasses the “curse of dimensionality” characteristic of RL and DP, offering rapidly convergent, algebraically tractable controllers. The general framework is extensible to multi-agent, multivariable, and nonlinear systems so long as flatness or approximate flatness can be established.

7. Perspectives and Place within Cybernetics and AI

HEOL embodies the cybernetic principles championed by Wiener, positioning feedback, homeostasis, and adaptation as the central elements of control within artificial intelligence. The estimation and adaptation mechanisms in HEOL parallel real-time learning of discrepancies and model uncertainties, emphasizing the foundational role of advanced control-theoretic tools as integral components of modern AI (Delaleau et al., 14 Jan 2025). The HEOL framework situates control theory as a core discipline within intelligent systems, supporting robust decision-making, adaptation, and interaction within dynamic, uncertain environments.

---

In conclusion, the HEOL feedback setting marks a rigorous synthesis of flatness-based control and model-free intelligent controllers, equipped with robust algebraic estimation and feedback adaptation mechanisms. Its attractive computational and theoretical properties, proven robustness, and demonstrated versatility across a spectrum of nonlinear engineering problems reinforce its standing as a core methodology within contemporary control and AI disciplines.