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Safe continual learning in model predictive control with prescribed bounds on the tracking error

Lukas Lanza, Dario Dennstädt, Thomas Berger, Karl Worthmann

TL;DR

The paper tackles safe continual learning for prescribed-performance tracking in nonlinear systems by integrating three components: model-based funnel MPC, model-free funnel control, and online learning to update the surrogate model. It introduces a structured model class ${\mathcal M}_{\bar{u}, \bar{\gamma}}$ and explicit learning-scheme conditions to ensure that updates preserve robust funnel MPC feasibility. A central result proves initial and recursive feasibility, boundedness of all signals, and tracking error confinement within the prescribed funnel ${\mathcal F}_{\psi}$ under the three-component controller. Numerical simulations demonstrate the method on a relative degree one exothermic reactor and a higher relative degree mass-on-car system, showing improved model accuracy and sustained funnel adherence. The work lays groundwork for rigorously integrating data-driven model updates with safe MPC, with future directions including exploring concrete learning architectures and extending guarantees to broader system classes.

Abstract

We develop a three-component Model Predictive Control (MPC) algorithm to achieve output-reference tracking with prescribed performance for continuous-time nonlinear systems. One component is so-called funnel MPC, which achieves reference tracking with prescribed performance for the model output for suitable models. Recently, this MPC algorithm has been combined with a model-free reactive feedback controller (second component) to account for model-plant mismatches, bounded disturbances, and uncertainties. By construction, this two-component controller defines a robust funnel MPC algorithm. It achieves output-reference tracking within prescribed bounds on the tracking error for a class of unknown nonlinear systems. In this paper, we extend the robust funnel MPC by a machine learning component to adapt the underlying model to the system data and, thus, improve the contribution of MPC. We derive sufficient structural conditions to define a class of models for funnel MPC, and provide a characterization of suitable learning schemes. Since robust funnel MPC is inherently robust and the evolution of the tracking error in the prescribed performance funnel is guaranteed, the additional learning component is able to perform the learning task online -- even without an initial model or offline training.

Safe continual learning in model predictive control with prescribed bounds on the tracking error

TL;DR

The paper tackles safe continual learning for prescribed-performance tracking in nonlinear systems by integrating three components: model-based funnel MPC, model-free funnel control, and online learning to update the surrogate model. It introduces a structured model class and explicit learning-scheme conditions to ensure that updates preserve robust funnel MPC feasibility. A central result proves initial and recursive feasibility, boundedness of all signals, and tracking error confinement within the prescribed funnel under the three-component controller. Numerical simulations demonstrate the method on a relative degree one exothermic reactor and a higher relative degree mass-on-car system, showing improved model accuracy and sustained funnel adherence. The work lays groundwork for rigorously integrating data-driven model updates with safe MPC, with future directions including exploring concrete learning architectures and extending guarantees to broader system classes.

Abstract

We develop a three-component Model Predictive Control (MPC) algorithm to achieve output-reference tracking with prescribed performance for continuous-time nonlinear systems. One component is so-called funnel MPC, which achieves reference tracking with prescribed performance for the model output for suitable models. Recently, this MPC algorithm has been combined with a model-free reactive feedback controller (second component) to account for model-plant mismatches, bounded disturbances, and uncertainties. By construction, this two-component controller defines a robust funnel MPC algorithm. It achieves output-reference tracking within prescribed bounds on the tracking error for a class of unknown nonlinear systems. In this paper, we extend the robust funnel MPC by a machine learning component to adapt the underlying model to the system data and, thus, improve the contribution of MPC. We derive sufficient structural conditions to define a class of models for funnel MPC, and provide a characterization of suitable learning schemes. Since robust funnel MPC is inherently robust and the evolution of the tracking error in the prescribed performance funnel is guaranteed, the additional learning component is able to perform the learning task online -- even without an initial model or offline training.
Paper Structure (10 sections, 2 theorems, 31 equations, 6 figures)

This paper contains 10 sections, 2 theorems, 31 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. System class and model class
  4. Learning-based robust funnel MPC
  5. Control algorithm and main result
  6. On learning schemes
  7. Numerical simulation
  8. Relative degree one
  9. Second numerical example: higher relative degree.
  10. Conclusion

Key Result

Theorem 4.2

Consider a system eq:Sys with ${(P,\Gamma,Q,d)} \in \mathcal{N}$ and initial values $y^0\in\mathds{R}^m$ and $\zeta^0\in\mathds{R}^\kappa$. For given $\bar{u} \geq 0$ and $\bar{\gamma}>0$ such that $\mathcal{M}_{\bar{u}, \bar{\gamma}}\neq\emptyset$, choose an initial model eq:Mod with $(p_0,\gamma_0

Figures (6)

  • Figure 1: Evolution of the error $e$ in a funnel $\mathcal{F}_{\psi}$ with boundary $\psi$. The figure is based on BergLe18a, edited for present purpose.
  • Figure 2: Design of the proposed three-component controller. The gray box (containing the red (funnel MPC) and the blue (funnel control) structures) represents the two-component controller robust funnel MPC proposed in BergDenn23. The green box represents the learning component, which receives the four signals: system output $y$, model output $y_{\rm M}$, funnel MPC control signal $u_{\rm FMPC}$, and funnel control signal $u_{\rm FC}$. Structural requirements on the learning scheme are introduced in \ref{['Def:Model', 'Def:Learning']}.
  • Figure 3: Application of learning based robust funnel MPC to the exothermic chemical reactor system.
  • Figure 4: Application of learning based robust funnel MPC to the exothermic chemical reactor system in the presence of periodic disturbances.
  • Figure 5: Mass-on-car system. The figure is based on the respective figures in BergIlch21, and SeifBlaj13.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Definition 4.1
  • Theorem 4.2
  • proof
  • Proposition 4.3
  • proof