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A low-complexity funnel control approach for non-linear systems of higher-order

Dario Dennstädt

TL;DR

This paper presents a low-complexity funnel control method for unknown nonlinear MIMO systems of higher order. By employing constant-gain auxiliary error variables and a model-free, high-gain framework, it achieves output reference tracking within a prescribed performance funnel without relying on time-varying reciprocal penalties. The main theoretical result guarantees a global solution, bounded actuation, and funnel-adherent tracking under modest initial-condition requirements, with a proof structure rooted in Carathéodory solutions and the system's high-gain property. Simulations on a mass-on-car example demonstrate favorable tracking within the funnel and reduced numerical complexity compared to prior designs, albeit at the cost of reduced adaptability to changing funnel boundaries. The work suggests future directions to reintroduce adaptive funnel responsiveness while maintaining the simplified controller structure.

Abstract

We address the problem of output reference tracking for unknown non-linear multi-input, multi-output systems described by functional differential equations. This class of systems includes those with a strict relative degree, and bounded-input bounded-output (BIBO) stable internal dynamics. The objective is to ensure that the tracking error evolves within a prescribed performance funnel. To achieve this, we propose a novel model-free adaptive controller with lower complexity than existing funnel control methods, avoiding the use of non-linearities in intermediate error signals. We establish the feasibility and effectiveness of the proposed design through theoretical analysis and demonstrate its performance with simulations, comparing it to previous approaches.

A low-complexity funnel control approach for non-linear systems of higher-order

TL;DR

This paper presents a low-complexity funnel control method for unknown nonlinear MIMO systems of higher order. By employing constant-gain auxiliary error variables and a model-free, high-gain framework, it achieves output reference tracking within a prescribed performance funnel without relying on time-varying reciprocal penalties. The main theoretical result guarantees a global solution, bounded actuation, and funnel-adherent tracking under modest initial-condition requirements, with a proof structure rooted in Carathéodory solutions and the system's high-gain property. Simulations on a mass-on-car example demonstrate favorable tracking within the funnel and reduced numerical complexity compared to prior designs, albeit at the cost of reduced adaptability to changing funnel boundaries. The work suggests future directions to reintroduce adaptive funnel responsiveness while maintaining the simplified controller structure.

Abstract

We address the problem of output reference tracking for unknown non-linear multi-input, multi-output systems described by functional differential equations. This class of systems includes those with a strict relative degree, and bounded-input bounded-output (BIBO) stable internal dynamics. The objective is to ensure that the tracking error evolves within a prescribed performance funnel. To achieve this, we propose a novel model-free adaptive controller with lower complexity than existing funnel control methods, avoiding the use of non-linearities in intermediate error signals. We establish the feasibility and effectiveness of the proposed design through theoretical analysis and demonstrate its performance with simulations, comparing it to previous approaches.

Paper Structure

This paper contains 10 sections, 2 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Nomenclature:
  3. Problem formulation
  4. Control objective
  5. Controller design
  6. Funnel control -- main result
  7. Proof of the main result
  8. Simulations
  9. Conclusion
  10. Acknowledgements

Key Result

Theorem 4.1

Consider system eq:Sys with $(d,f,\mathbf{T})\in\mathcal{N}^{m,r}$ and initial data $y^0 \in \mathcal{C}^{r-1}([0,t_0],\mathds{R}^m)$. Let $\psi\in\mathcal{G}$ with associated constants $\alpha, \beta >0$ and choose parameter Furthermore, let $y_{\mathop{\mathrm{ref}}\limits}\in W^{r,\infty}(\mathds{R}_{\geq0},\mathds{R}^{m})$ such that Then, the application of the funnel controller eq:Definiti

Figures (3)

  • Figure 1: Error evolution in a funnel $\mathcal{F}_{\psi}$ with boundary $\psi(t)$. The figure is based on BergLe18a, edited for present purpose.
  • Figure 2: Mass-on-car system.The figure is based on SeifBlaj13
  • Figure 3: Simulation of system \ref{['eq:ExampleMassOnCarSystem']} under control law \ref{['eq:DefinitionFC']} and funnel control from BergLe18a and BergIlch21.

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.1
  • Lemma 5.1