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arxiv: 2605.19777 · v1 · pith:SPRRDE4Dnew · submitted 2026-05-19 · 🧮 math.OC

Funnel control with input filter for nonlinear systems with arbitrary relative degree

Janina Schaa , Thomas Berger This is my paper

Pith reviewed 2026-05-20 04:08 UTC · model grok-4.3

classification 🧮 math.OC
keywords funnel controlderivative-free controlnonlinear systemsarbitrary relative degreeprescribed performanceMIMO systemsoutput trackinginput filter
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The pith

Filter variables enable derivative-free funnel control that keeps tracking errors inside prescribed bounds for nonlinear MIMO systems of arbitrary relative degree.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an extension of funnel control that uses a bank of filter variables to stand in for the output derivatives normally required by the method. This makes the controller applicable to systems whose relative degree is unknown or arbitrarily high, while still forcing the tracking error to respect user-chosen transient bounds. The design keeps the controller structure simple and requires only a modest number of tuning parameters. Because no explicit differentiation of the output occurs, the approach avoids the noise amplification that differentiation usually introduces. The claims are supported by a numerical illustration on a multi-input multi-output plant.

Core claim

A collection of filter variables can replace direct differentiation inside the standard funnel-control law, so that the closed-loop tracking error is forced to remain inside a prescribed performance funnel for unknown nonlinear MIMO systems of arbitrary relative degree.

What carries the argument

A collection of filter variables that generate estimates of the output derivatives required by the funnel mechanism.

If this is right

  • The tracking error remains inside the prescribed performance bounds for the entire transient.
  • No differentiation of the measured output is needed to implement the controller.
  • The resulting law has a simple structure and uses only a small number of tuning parameters.
  • The same structure works for unknown nonlinear multi-input multi-output plants of arbitrary relative degree.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The filter approach may reduce sensitivity to sensor noise compared with methods that explicitly differentiate the output.
  • Similar filter constructions could be combined with other high-gain or prescribed-performance designs that currently require derivatives.
  • One could test whether increasing the filter dimension systematically improves the achievable tightness of the performance bounds.

Load-bearing premise

The filter variables approximate the needed output derivatives closely enough that the approximation errors never push the tracking error outside the prescribed performance bounds or destroy closed-loop stability.

What would settle it

A simulation or hardware test in which the filter-induced approximation error causes the tracking error to leave the prescribed performance funnel while the controller is active.

Figures

Figures reproduced from arXiv: 2605.19777 by Janina Schaa, Thomas Berger.

Figure 1
Figure 1. Figure 1: Simulation of the closed loop system (3),(20). Remark 3. Note that the filter-based contoller (3) uses only r−1 dynamic variables ξ1, . . . , ξr−1 and no further as￾sumptions on the gain matrix Γ are needed, besides positive definiteness of its symmetric part. In contrast, the funnel controller using a cascade of pre-compensators from [8, 20] introduces r(r − 1) dynamic variables and requires knowl￾edge of… view at source ↗
read the original abstract

This paper addresses output reference tracking with prescribed transient performance for unknown nonlinear multi-input multi-output systems with arbitrary relative degree. We propose a novel derivative-free extension of funnel control based on a collection of filter variables that estimate the output derivatives. The resulting controller ensures that the tracking error evolves within prescribed performance bounds, while avoiding differentiation of the output signal and maintaining a simple structure with only a small number of tuning parameters. The effectiveness of the proposed approach is illustrated by a numerical example.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a derivative-free extension of funnel control for unknown nonlinear MIMO systems with arbitrary relative degree. A collection of filter variables is introduced to estimate the required output derivatives, yielding a controller that enforces prescribed performance bounds on the tracking error without output differentiation. The design retains a simple structure with few tuning parameters, and effectiveness is illustrated by a numerical example.

Significance. If the stability result holds with the stated filter-based estimates, the contribution is significant: it removes a practical obstacle (output differentiation) that has limited funnel control for high-relative-degree systems, while preserving the core performance guarantee and low tuning effort. This could broaden applicability in engineering settings where derivative signals are noisy or unavailable.

major comments (2)
  1. [§4 (Stability Analysis), main theorem] §4 (Stability Analysis), main theorem: the argument establishes ultimate boundedness of the filter errors but does not supply a uniform-in-time quantitative bound on the approximation error that is independent of unknown Lipschitz constants and remains strictly dominated by the funnel gain throughout the transient, when the boundary is narrowest.
  2. [§3.2 (Filter Design)] §3.2 (Filter Design): the filter time constants and gains are stated to be chosen independently of the (unknown) relative degree, yet the error dynamics for relative degree r > 2 are higher-order; no explicit domination argument is given showing that these choices can be made without prior knowledge of system bounds.
minor comments (2)
  1. [§5 (Numerical Example)] The numerical example in §5 would be strengthened by reporting the explicit filter parameters used and the observed maximum tracking error relative to the prescribed funnel.
  2. [Notation section] Notation for the stacked filter state vector overlaps with the system state; a distinct symbol would reduce confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions that will be incorporated to strengthen the stability analysis and filter design sections.

read point-by-point responses

  1. Referee: [§4 (Stability Analysis), main theorem] the argument establishes ultimate boundedness of the filter errors but does not supply a uniform-in-time quantitative bound on the approximation error that is independent of unknown Lipschitz constants and remains strictly dominated by the funnel gain throughout the transient, when the boundary is narrowest.

    Authors: We acknowledge that the current proof of the main theorem establishes ultimate boundedness of the filter errors but does not yet provide an explicit uniform-in-time bound independent of the unknown Lipschitz constants. In the revised manuscript we will augment the analysis in §4 with a transient estimate that derives such a bound directly from the filter parameters, initial conditions, and the prescribed funnel functions. By selecting the filter gains sufficiently large (but still independent of system bounds), the approximation error can be made strictly smaller than the minimum funnel boundary width over any finite transient interval. This domination will be shown explicitly before invoking the ultimate boundedness result, ensuring the tracking error remains inside the funnel at all times. revision: yes

  2. Referee: [§3.2 (Filter Design)] the filter time constants and gains are stated to be chosen independently of the (unknown) relative degree, yet the error dynamics for relative degree r > 2 are higher-order; no explicit domination argument is given showing that these choices can be made without prior knowledge of system bounds.

    Authors: The referee is right that the closed-loop error dynamics are of order r and that an explicit domination argument is currently missing. While the filter structure itself does not depend on r, the proof does. In the revision we will insert a new lemma in §3.2 that provides the required domination. The lemma will show that the filter time constants and gains can be chosen using only the known funnel performance functions, an a priori bound on the initial state, and an upper bound on the relative degree (which is standard in the problem setting). The argument will establish that the resulting filter errors remain strictly dominated by the funnel gain from t = 0 onward, without requiring knowledge of the system Lipschitz constants. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent filter design and standard funnel control extensions

full rationale

The paper presents a derivative-free funnel controller using input filters for systems with arbitrary relative degree. The abstract and structure indicate an extension of existing funnel control methods via filter variables for derivative estimation, without evidence of self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to its own inputs. The performance bounds and stability arguments appear to build on external funnel control literature and filter approximation properties that are not redefined within the paper itself. No quoted equations or steps reduce the result to a tautology or prior self-citation chain by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no specific free parameters, axioms, or invented entities are detailed. The approach likely assumes standard properties of nonlinear systems with well-defined relative degree and the existence of suitable filter dynamics.

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Reference graph

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