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On Model Predictive Funnel Control with Equilibrium Endpoint Constraints

Jens Göbel, Dario Dennstädt, Lukas Lanza, Karl Worthmann, Thomas Berger, Tobias Damm

TL;DR

This work introduces Model Predictive Funnel Control (MPFC), a bi- level scheme that merges high-gain funnel control with MPC to enforce prescribed transient performance for nonlinear MIMO systems. By optimizing a parameterized funnel shape $(c,T)$ over a receding horizon and applying a funnel-based input law between updates, MPFC achieves inter-sampling robustness while maintaining a fixed, reduced set of decision variables. The authors prove initial and recursive feasibility, establish bounded closed-loop costs, and show asymptotic convergence to the origin under a positive definite $Q$, leveraging equilibrium endpoint constraints for stability. A numerical example demonstrates convergence and reveals how optimal funnel parameters shrink over iterations, indicating improved constraint satisfaction without increasing computational burden compared to classical MPC.

Abstract

We propose model predictive funnel control, a novel model predictive control (MPC) scheme building upon recent results in funnel control. The latter is a high-gain feedback methodology that achieves evolution of the measured output within predefined error margins. The proposed method dynamically optimizes a parameter-dependent error boundary in a receding-horizon manner, thereby combining prescribed error guarantees from funnel control with the predictive advantages of MPC. On the one hand, this approach promises faster optimization times due to a reduced number of decision variables, whose number does not depend on the horizon length. On the other hand, the continuous feedback law improves the robustness and also explicitly takes care of the inter-sampling behavior. We focus on proving stability by leveraging results from MPC stability theory with terminal equality constraints. Moreover, we rigorously show initial and recursive feasibility.

On Model Predictive Funnel Control with Equilibrium Endpoint Constraints

TL;DR

This work introduces Model Predictive Funnel Control (MPFC), a bi- level scheme that merges high-gain funnel control with MPC to enforce prescribed transient performance for nonlinear MIMO systems. By optimizing a parameterized funnel shape over a receding horizon and applying a funnel-based input law between updates, MPFC achieves inter-sampling robustness while maintaining a fixed, reduced set of decision variables. The authors prove initial and recursive feasibility, establish bounded closed-loop costs, and show asymptotic convergence to the origin under a positive definite , leveraging equilibrium endpoint constraints for stability. A numerical example demonstrates convergence and reveals how optimal funnel parameters shrink over iterations, indicating improved constraint satisfaction without increasing computational burden compared to classical MPC.

Abstract

We propose model predictive funnel control, a novel model predictive control (MPC) scheme building upon recent results in funnel control. The latter is a high-gain feedback methodology that achieves evolution of the measured output within predefined error margins. The proposed method dynamically optimizes a parameter-dependent error boundary in a receding-horizon manner, thereby combining prescribed error guarantees from funnel control with the predictive advantages of MPC. On the one hand, this approach promises faster optimization times due to a reduced number of decision variables, whose number does not depend on the horizon length. On the other hand, the continuous feedback law improves the robustness and also explicitly takes care of the inter-sampling behavior. We focus on proving stability by leveraging results from MPC stability theory with terminal equality constraints. Moreover, we rigorously show initial and recursive feasibility.

Paper Structure

This paper contains 8 sections, 5 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System class and control objective
  3. Control Methodology
  4. Exact Funnel Control
  5. Model Predictive Funnel Control (MPFC)
  6. Main Results
  7. Numerical Example
  8. Conclusion and Outlook

Key Result

Lemma 3

[lemma]Thm:ExactFunnelControl For $\varphi$ given by eq:FunnelBoundaryExact, consider a system eq:system contained in $\mathcal{S}$ with $\|\hat{y}_0\| < \varphi(0) = cT$. Then, the application of the feedback law to system eq:system yields an absolutely continuous maximal closed-loop solution $y\colon [0,T)\to\mathbb{R}^m$ satisfying $\| y(t) \| < \varphi(t)$ for all $t \in [0,T)$, $\lim_{t \to

Figures (2)

  • Figure 1: Structure of the proposed bi-level controller, i.e., feedback control for unknown real system with optimal funnel function parameters $(c,T)$. The model is initialized with system output data $y(t_i)$. Then, the MPFC \ref{['alg:fmpc']} provides parameters $(c_i^*,T_i^*)$ to the funnel controller \ref{['eq:ExactFunnelControl']}. The latter is applied continuously to the system.
  • Figure 2: Simulation of system \ref{['eq:ExampleSys']} under the control generated by \ref{['alg:fmpc']}.

Theorems & Definitions (11)

  • Definition 1: High-gain property berger2021funnel
  • Definition 2: System class $\mathcal{S}$
  • Lemma 3
  • Theorem 5
  • proof
  • Lemma 6
  • proof
  • Corollary 7
  • proof
  • Theorem 8
  • ...and 1 more