Funnel Control in Output-Feedback Systems

• Funnel control is a continuous-time output-feedback method that uses a performance funnel to confine tracking errors within prescribed transient and asymptotic bounds.
• It employs an adaptive high-gain mechanism, making it applicable to a range of systems including those with arbitrary relative degree, infinite-dimensional dynamics, and nonlinearity.
• Recent developments focus on derivative-free designs using filter cascades, removing the need for explicit output derivative measurements in nonlinear MIMO systems.

Searching arXiv for recent and foundational papers on funnel control. Funnel control is a continuous-time output-feedback methodology for prescribed-performance tracking in which the tracking error is constrained to evolve within a preassigned time-varying “performance funnel.” Its central objective is not merely stabilization, but the enforcement of explicit transient and asymptotic error bounds through a simple adaptive high-gain mechanism that depends on the measured output and a designer-chosen funnel boundary. Across the literature, the method is developed for linear and nonlinear systems, systems of arbitrary relative degree, functional differential equations, differential-algebraic systems, and infinite-dimensional plants including PDEs, while typically requiring only structural information such as relative degree, stable internal dynamics, and sign or positivity properties of the high-frequency gain rather than precise plant parameters (Berger et al., 2023). More recent work extends the framework to derivative-free designs via input filters for unknown nonlinear MIMO systems of arbitrary relative degree (Schaa et al., 19 May 2026).

1. Concept and performance specification

In the standard tracking setup, the measured output is $y(t)\in\mathbb R^m$, the reference is $y_{\rm ref}(t)$, and the tracking error is

$e(t)=y(t)-y_{\rm ref}(t).$

A performance funnel is specified by a scalar boundary function $\varphi$, typically required to be positive and sufficiently regular. A representative formulation defines

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$

so that the admissible error magnitude is $1/\varphi(t)$ (Berger et al., 2023, Berger et al., 2023). Enforcing

$\varphi(t)\,\|e(t)\|<1$

for all $t\ge 0$ prescribes a time-varying error tube: narrow funnels impose aggressive transient accuracy, while bounded funnels encode practical tracking and unbounded funnels can yield asymptotic tracking under additional assumptions (Berger et al., 2020).

Common choices include exponential funnels such as

$\varphi(t)=(a\,e^{-b\,t}+d)^{-1},\quad a,b,d>0,$

which shrink the admissible error from an initially wide region to a strictly positive asymptotic tolerance (Berger et al., 2019). In broader formulations, $\varphi$ belongs to a class $y_{\rm ref}(t)$0 or similar function space guaranteeing positivity, bounded derivative behavior, and a positive limit inferior (Berger et al., 2023, Berger et al., 2020).

The defining feature of funnel control is that the performance specification is encoded directly in the feedback law. As the error approaches the funnel boundary, the gain increases and produces a barrier effect that prevents boundary violation (Berger et al., 2019). This mechanism is the basis for global-in-time feasibility results: global existence of closed-loop solutions, boundedness of all relevant signals, and strict interiority of the error trajectory with respect to the funnel (Berger et al., 2023).

A recurring distinction in the literature is between prescribed performance and asymptotic convergence. Classical funnel control ensures that the error remains within the funnel, but in general does not force $y_{\rm ref}(t)$1 when the funnel width tends to a nonzero limit (Berger et al., 2023). This has motivated several extensions, including internal-model augmentations and model-predictive funnel formulations.

2. Classical controller constructions

For relative degree one, the canonical funnel controller has the form

$y_{\rm ref}(t)$2

or equivalent high-gain variants using a funnel-transformed error (Berger et al., 2023, Berger et al., 2019). As $y_{\rm ref}(t)$3, the denominator tends to zero and the gain diverges, thereby repelling the trajectory from the funnel boundary.

For higher relative degree $y_{\rm ref}(t)$4, the classical design introduces recursively defined auxiliary errors. A representative recursion is

$y_{\rm ref}(t)$5

with gains

$y_{\rm ref}(t)$6

The control input is then taken as

$y_{\rm ref}(t)$7

or an equivalent variant depending on the structural assumptions and sign knowledge of the high-frequency gain (Berger et al., 2019, Berger et al., 2019, Berger et al., 2023). In nonlinear formulations with unknown control direction, Nussbaum-type or surjective gain functions are used instead of a fixed sign (Berger et al., 2020, Berger et al., 2023).

This recursive structure is closely linked to relative degree. The controller acts on the highest-order auxiliary error while lower stages shape the admissible behavior of successively differentiated tracking errors. The resulting closed-loop analysis typically establishes forward invariance of each funnel, boundedness of all gains and inputs, and strict separation from the boundary by some margin $y_{\rm ref}(t)$8 (Berger et al., 2019).

The general nonlinear framework in "Funnel control of nonlinear systems" formulates the plant as an $y_{\rm ref}(t)$9-th order functional differential equation

$e(t)=y(t)-y_{\rm ref}(t).$0

with a causal, locally Lipschitz, BIBO operator $e(t)=y(t)-y_{\rm ref}(t).$1 describing internal dynamics and a high-gain property for $e(t)=y(t)-y_{\rm ref}(t).$2 that generalizes sign-definite high-frequency gain (Berger et al., 2020). This formulation subsumes systems with delays, hysteresis, dead-zone effects, and infinite-dimensional internal dynamics. The survey emphasizes that the precise plant data are generally unknown; feasibility rests instead on structural assumptions such as relative degree, stable internal dynamics, and gain definiteness (Berger et al., 2023).

A common misconception is that funnel control is merely a tuning heuristic for high-gain tracking. The theory instead treats the funnel boundary as a formal performance specification and proves that the error remains strictly inside the prescribed set under explicit assumptions on the system class (Berger et al., 2023).

3. Structural assumptions and system classes

The literature organizes funnel control around structural rather than parametric assumptions. For finite-dimensional linear systems, the standard hypotheses are known relative degree, asymptotically stable zero dynamics, and sign-definite or positive-definite high-frequency gain $e(t)=y(t)-y_{\rm ref}(t).$3 (Berger et al., 2023). In nonlinear settings, the corresponding conditions are typically expressed in a Byrnes–Isidori-type input–output normal form with bounded-input bounded-output stable internal dynamics.

A representative arbitrary-relative-degree nonlinear MIMO model is

$e(t)=y(t)-y_{\rm ref}(t).$4

where $e(t)=y(t)-y_{\rm ref}(t).$5, the symmetric part of $e(t)=y(t)-y_{\rm ref}(t).$6 is positive definite, $e(t)=y(t)-y_{\rm ref}(t).$7 is a static nonlinearity, and $e(t)=y(t)-y_{\rm ref}(t).$8 is causal, locally Lipschitz on finite intervals, and BIBO (Schaa et al., 19 May 2026). Here “arbitrary relative degree $e(t)=y(t)-y_{\rm ref}(t).$9” means that the input appears only in the $\varphi$0-th derivative of the output.

These operator-theoretic formulations are central to extensions beyond ODEs. In "Funnel control in the presence of infinite-dimensional internal dynamics," the internal dynamics are permitted to be infinite-dimensional provided the operator $\varphi$1 remains causal, locally Lipschitz on bounded sets, and BIBO (Berger et al., 2019). The same paper treats systems whose internal dynamics can be modeled by a transport equation, showing prescribed performance despite the transport PDE not being exponentially stable.

For passive infinite-dimensional systems, funnel control has also been formulated in the system-node framework. In that setting, the plant is given by

$\varphi$2

with impedance passivity expressed through an energy balance and further assumptions such as surjectivity of $\varphi$3 and coercivity of $\varphi$4 (Govindaraj et al., 1 Oct 2025). The closed-loop analysis is then carried out via nonlinear evolution equations and nonlinear semigroup theory.

This structural emphasis explains the recurrent characterization of funnel control as “model-free” in the literature. The term does not mean that no assumptions are used; rather, the controller typically avoids parameter identification and relies only on qualitative properties such as relative degree, gain definiteness, and internal BIBO or passivity properties (Berger et al., 2023, Schaa et al., 19 May 2026).

4. Derivative-free and filtered funnel control

A major limitation of classical higher-relative-degree funnel control is the dependence on derivatives of the output or derivatives of intermediate errors. This issue has motivated pre-compensator, observer-like, and filter-based variants. The paper "Funnel control with input filter for nonlinear systems with arbitrary relative degree" develops a derivative-free extension for unknown nonlinear MIMO systems with arbitrary relative degree (Schaa et al., 19 May 2026).

The method introduces $\varphi$5 filter states $\varphi$6 governed by a chain of first-order systems driven by the control input: $\varphi$7 These filter states emulate unavailable output derivatives. The controller then defines auxiliary variables recursively,

$\varphi$8

$\varphi$9

and uses the final-stage law

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$0

Each denominator defines an inner funnel for an auxiliary variable, preventing singularity and enforcing boundedness (Schaa et al., 19 May 2026).

The main result states that for admissible initial conditions, the closed loop admits a global solution, there exists $$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$1 such that

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$2

and all filter states and the control input remain bounded (Schaa et al., 19 May 2026). The proof combines a reformulation as an ODE, boundedness of linear combinations of true and filtered derivatives, a Lyapunov-type funnel argument, and continuation to global existence.

An earlier relative-degree-two variant uses a single filter

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$3

together with the virtual funnel input

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$4

and the filter error

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$5

The control law

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$6

then enforces funnel invariance without direct use of $$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$7 (Dennstädt et al., 19 Dec 2025). The 2026 arbitrary-relative-degree construction can be understood as a systematic generalization of this idea.

These derivative-free designs suggest a notable shift in funnel-control research: instead of requiring direct derivative measurements or elaborate pre-compensators, simple filter cascades can recover the essential barrier structure with a small number of tuning parameters. The claim in (Schaa et al., 19 May 2026) that the controller maintains a simple structure with only a small number of tuning parameters is therefore significant within the evolution of the field.

5. Theoretical guarantees and proof mechanisms

The central guarantee in funnel control is feasibility of the funnel objective. In the survey formulation, this means that the closed-loop solution is global, all variables remain bounded, and the error evolves strictly inside the prescribed funnel (Berger et al., 2023). While details vary across system classes, several proof motifs recur.

A first ingredient is local well-posedness. Depending on the model class, the closed loop is cast as an ODE, a retarded functional differential equation, a differential-algebraic system, or a nonlinear evolution equation on a Banach or Hilbert space (Berger et al., 2019, Puche et al., 2019, Govindaraj et al., 1 Oct 2025). Standard local existence then holds on the open set where all funnel denominators are nonzero.

A second ingredient is a barrier or Lyapunov-type transformation. For instance, in the infinite-dimensional internal-dynamics setting one introduces coordinates

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$8

so that the transformed variables blow up at the funnel boundary (Berger et al., 2019). Differential inequalities of the form

$$\mathcal F_\varphi=\{(t,e)\mid \varphi(t)\,\|e\|<1\},$$9

then preclude finite-time escape and establish boundedness. In other formulations, the barrier function $1/\varphi(t)$0 is used in composite Lyapunov functions (Berger et al., 2019).

A third ingredient is boundedness of internal dynamics. For finite-dimensional minimum-phase systems, this comes from stable zero dynamics; for operator-based models, it comes from the BIBO property of $1/\varphi(t)$1; for passive systems, it is derived from dissipation inequalities and passivity (Berger et al., 2020, Berger et al., 2019, Govindaraj et al., 1 Oct 2025).

The output-funnel guarantee is often sharpened to strict interiority: $1/\varphi(t)$2 for some $1/\varphi(t)$3, rather than mere nonviolation (Berger et al., 2019). The 2026 filter-based arbitrary-relative-degree result obtains the analogous bound with a single $1/\varphi(t)$4 for the tracking error (Schaa et al., 19 May 2026).

A subtle point concerns asymptotic tracking. Classical funnel results yield prescribed performance but not necessarily convergence to zero. The paper "Asymptotic tracking by funnel control with internal models" addresses this by placing a linear internal model in series with the funnel controller for minimum-phase linear systems of strict relative degree $1/\varphi(t)$5 (Berger et al., 2023). Under a solvability condition on the reference class and sufficiently large $1/\varphi(t)$6, the closed loop satisfies both funnel invariance and

$1/\varphi(t)$7

This addresses the documented issue that classical funnel control can leave the error “close to the funnel boundary,” inducing comparatively large feedback gains and noise sensitivity (Berger et al., 2023).

6. Variants, interfaces, and application domains

Funnel control has generated a substantial family of variants that preserve the core barrier-based prescribed-performance mechanism while targeting additional objectives.

Selected variants of funnel control

Variant	Core idea	Representative paper
Infinite-dimensional internal dynamics	BIBO-stable operator $1/\varphi(t)$8 may represent PDE or transport dynamics	(Berger et al., 2019)
Internal-model funnel control	Add a linear internal model to obtain asymptotic tracking	(Berger et al., 2023)
Input-constrained funnel control	Dynamically widen the funnel when actuator saturation is active	(Berger, 2022)
Filter-based derivative-free control	Use input-driven filter states to emulate unavailable output derivatives	(Schaa et al., 19 May 2026)
Model predictive funnel control	Optimize funnel parameters $1/\varphi(t)$9 in receding horizon	(Göbel et al., 26 May 2025)
CBF interpretation	View funnel law as a zeroing-CBF-based model-free safety controller	(Lanza et al., 23 May 2025)

Input saturation has been addressed by dynamic funnel-boundary adaptation. In "Input-constrained funnel control of nonlinear systems," the plant input is

$\varphi(t)\,\|e(t)\|<1$0

while dynamic funnel radii $\varphi(t)\,\|e(t)\|<1$1 are widened via an extra term involving

$\varphi(t)\,\|e(t)\|<1$2

whenever saturation is active (Berger, 2022). This preserves the hard input constraint and relaxes the output specification only as needed. The paper proves global existence and error-in-funnel properties relative to the dynamically widened funnel.

A model-predictive extension, "On Model Predictive Funnel Control with Equilibrium Endpoint Constraints," optimizes the finite-time funnel parameters $\varphi(t)\,\|e(t)\|<1$3 over a prediction horizon while using the exact funnel controller as the feedback law between sampling times (Göbel et al., 26 May 2025). The feasible set requires the inner funnel $\varphi(t)\,\|e(t)\|<1$4 to remain inside an outer funnel $\varphi(t)\,\|e(t)\|<1$5, and the main results establish initial feasibility, recursive feasibility, and convergence $\varphi(t)\,\|e(t)\|<1$6 under $\varphi(t)\,\|e(t)\|<1$7. The reduced online optimization dimension—two scalars independent of the horizon length—is emphasized in that work (Göbel et al., 26 May 2025).

Another important interface is with control barrier functions. "A model-free approach to control barrier functions using funnel control" defines

$\varphi(t)\,\|e(t)\|<1$8

so that $\varphi(t)\,\|e(t)\|<1$9 is equivalent to $t\ge 0$0, and shows that the funnel-type control family

$t\ge 0$1

belongs to the zeroing-CBF admissible set under structural assumptions for relative-degree-one systems (Lanza et al., 23 May 2025). This establishes a direct connection between funnel invariance and forward invariance of a safe set.

Beyond tracking, funnel concepts have been adapted to motion planning and reach-avoid problems. In kinodynamic planning, KDF combines a geometric planner with a low-level funnel controller that guarantees safe tracking of a planned path for uncertain fully actuated systems (Verginis et al., 2021). In reach-avoid-stay control, a time-varying funnel is combined with circumvent functions and adaptive boundary updates to steer trajectories to a target set while avoiding unsafe regions (Das et al., 2023). These developments suggest that “funnel” now functions both as a prescribed-performance tracking device and as a geometry-aware safety envelope.

Applications span finite-dimensional and infinite-dimensional domains. The survey enumerates servo drives, synchronous machines, cruise control, multi-agent synchronization, robotic manipulators, mass-on-car benchmarks, water-tank sloshing systems, heat and wave equations, and cardiac monodomain equations (Berger et al., 2023). Specific PDE applications include the moving water tank modeled by linearized Saint-Venant equations (Berger et al., 2019), boundary control systems on Hilbert spaces (Puche et al., 2019), monodomain equations with FitzHugh–Nagumo dynamics (Berger et al., 2019), and passive Euler–Bernoulli beam systems (Govindaraj et al., 1 Oct 2025). There is also a stochastic extension to Langevin dynamics, where the objective is to keep the mean state in a prescribed funnel around a reference (Berger et al., 2022).

7. Limitations, misconceptions, and research directions

A persistent misconception is that funnel control guarantees exact asymptotic tracking by default. The literature explicitly rejects this: classical funnel control enforces $t\ge 0$2 but may allow a nonzero residual error when the funnel boundary levels off (Berger et al., 2023). The internal-model extension was developed precisely because the error may “slide” near the boundary, causing large gains and poor noise behavior (Berger et al., 2023).

Another misconception is that the method is assumption-free because it is called model-free. In fact, every rigorous result depends on structural assumptions such as relative degree, stable or BIBO internal dynamics, positive-definite or sign-definite high-frequency gain, passivity, or solvability conditions for the reference class (Berger et al., 2023, Berger et al., 2019, Govindaraj et al., 1 Oct 2025). The “model-free” descriptor refers to the absence of parameter identification and the use of output-based feedback rather than to complete agnosticism about plant structure.

Current limitations and open directions are explicitly identified in the literature. The survey lists partial observability and sampled-data funnel control, unstable zero dynamics, learning within funnel-MPC, input and state constraints, and extensions to non-co-located boundary control in PDEs as ongoing research topics (Berger et al., 2023). The infinite-dimensional internal-dynamics paper highlights nonlinear PDE internal dynamics with unbounded observation operators, non-integer or time-varying relative degree, and robustness to perturbations of the infinite-dimensional part as open problems (Berger et al., 2019). The 2025 relative-degree-two input-filter paper indicates future work on constraints, sampled-data implementations, and time-varying high-frequency gain matrices (Dennstädt et al., 19 Dec 2025).

The 2026 arbitrary-relative-degree input-filter result points to a particularly active frontier: derivative-free funnel control for unknown nonlinear MIMO systems of high relative degree (Schaa et al., 19 May 2026). Its combination of filter-based derivative emulation, simple recursive barrier structure, and boundedness guarantees suggests a consolidation of two long-standing objectives in the area: preserving the low-complexity spirit of funnel control while removing the need for differentiated measurements. A plausible implication is that future work will increasingly connect funnel control with observer-lite, filter-based, and optimization-based architectures rather than treat it as an isolated high-gain design paradigm.