Funnel Schedule in Dynamic Systems

• Funnel schedule is a time-varying prescription that defines admissible errors, outputs, and motions across diverse applications including control systems and skyrmion transport.
• It dynamically allocates performance and safety constraints by adjusting invariant sets, drive protocols, or error envelopes in real time.
• The approach integrates methodologies such as DLMI-based synthesis, MPC handover, and staged diffusion to ensure continuous feasibility and robustness.

Funnel schedule denotes a time-parameterized prescription for how a funnel evolves and is enforced. In control, the funnel is a time-varying admissible set for outputs or errors, such as $\rho_i^L(t) < x_i(t) < \rho_i^U(t)$ or $\varphi(t)\|e(t)\|<1$; in funnel synthesis it is a family of controlled-invariant sets $F(t)$ around a nominal trajectory; and in skyrmion funnel geometries it is a protocol of AC drives that uses asymmetric channels and Magnus dynamics to route motion (Mehdifar et al., 2022, Berger et al., 2023, Kim et al., 2024, Souza et al., 2021). Across these uses, the common structure is temporal organization: the schedule specifies which states, errors, motions, or operations are admissible at each instant, and how those admissible sets or protocols are updated.

1. Conceptual scope and formal definitions

In prescribed-performance control, a funnel is a time-varying envelope in output or error space. For scalar outputs, one formulation is

$\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$

while a standard tracking formulation uses

$$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$

Equivalently, the admissible error magnitude is $\|e(t)\| < 1/\varphi(t)$, so the temporal choice of $\varphi$ is the schedule of allowable error over time (Mehdifar et al., 2022, Berger et al., 2023).

In finite-horizon funnel synthesis, the funnel is a time-varying controlled invariant set around a nominal trajectory $(\bar x(\cdot),\bar u(\cdot))$. With deviation $\eta(t)=x(t)-\bar x(t)$ and quadratic Lyapunov function

$V(t,\eta) = \eta(t)^\top Q(t)^{-1}\eta(t),$

the state funnel is the ellipsoid

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

and the full funnel is

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

Here the schedule is carried by the time dependence of $\varphi(t)\|e(t)\|<1$2 and $\varphi(t)\|e(t)\|<1$3, which determines the size, orientation, and feedback law associated with the moving invariant set (Kim et al., 2024).

A distinct but explicit use occurs in skyrmion transport through asymmetric channels. There, a “funnel schedule” is a time-dependent protocol specifying which AC drive components are active, their amplitudes, their relative phase, and the duration of each configuration, in order to realize transport, storage, or logic operations (Souza et al., 2021).

2. Prescribed-performance scheduling in constrained control

A central control-theoretic formulation is the constraint consistent funnel (CCF) for hard and soft output constraints. Hard constraints

$\varphi(t)\|e(t)\|<1$4

must never be violated, whereas soft constraints

$\varphi(t)\|e(t)\|<1$5

encode performance and may be relaxed when they conflict with safety. The CCF boundaries are defined by

$\varphi(t)\|e(t)\|<1$6

where the modification signals $\varphi(t)\|e(t)\|<1$7 are adapted online. When hard and soft constraints are compatible, the schedule coincides with their intersection; when they are incompatible, the planner inflates the funnel toward the hard constraints and then recovers the soft constraints exponentially when compatibility returns. The paper also gives smooth variants by replacing $\varphi(t)\|e(t)\|<1$8, $\varphi(t)\|e(t)\|<1$9, and $F(t)$0 with $F(t)$1 and soft approximations, yielding differentiable funnel schedules (Mehdifar et al., 2022).

Prescribed Performance Control (PPC) enforces this schedule by normalizing the output relative to the asymmetric funnel,

$F(t)$2

mapping $F(t)$3 to $F(t)$4 via

$F(t)$5

and designing the desired velocity as

$F(t)$6

A second PPC layer constrains the velocity tracking error inside a symmetric funnel, again via logarithmic barrier coordinates. For uncertain nonlinear Euler–Lagrange systems, the resulting controller is low-complexity, robust, and model-free with respect to the unknown matrices $F(t)$7, while guaranteeing that the outputs remain inside the planned CCFs (Mehdifar et al., 2022).

Input-constrained funnel control generalizes this idea by making the funnel boundaries themselves dynamic states. For a system of strict relative degree $F(t)$8, the schedule is generated by ODEs

$F(t)$9

and

$\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$0

where $\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$1 measures input saturation. When saturation is inactive, $\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$2 follows an exponentially decaying prescribed asymptotic shape; when saturation is active, the additional term widens the funnel. Once saturation ceases, the schedule relaxes exponentially back toward the desired shape. This makes funnel widening part of the control law rather than an exogenous redesign step (Berger, 2022).

3. Invariant funnel synthesis and MPC handover

In nonlinear trajectory-centric synthesis, funnel scheduling is the computation of $\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$3 and $\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$4 so that the ellipsoids $\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$5 are invariant under bounded disturbances. A differential linear matrix inequality (DLMI),

$\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$6

encodes the continuous-time invariance condition. A central contribution of the copositivity-based approach is that the DLMI is not enforced only at temporal nodes: finite LMI conditions are derived that imply the continuous-time DLMI on every subinterval, thereby preserving invariance over the full horizon rather than only at collocation points. The method parameterizes $\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$7, $\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$8, and the multipliers by first-order hold interpolation and proves that the resulting funnel is invariant over $\rho_i^L(t) < x_i(t) < \rho_i^U(t), \quad t \ge 0,$9 (Kim et al., 2024).

A later convex framework for incrementally quadratic nonlinear systems keeps the same invariance objective but reformulates the DLMI into a differential matrix equality (DME) plus a pointwise LMI. In one formulation,

$$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$0

with a residual LMI; in another,

$$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$1

and the full invariance burden is shifted to the pointwise matrix inequality. This makes the funnel schedule explicit as a dynamical system in $$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$2. The same framework handles Lipschitz, $$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$3-smooth, and sector-bounded nonlinearities through incremental quadratic constraints, and enforces continuous-time state and input constraints by either intermediate checking points or successive convexification with subgradients (Kim et al., 12 Nov 2025).

Model Predictive Control uses funnel schedules differently: funnel control is employed first to guarantee initial and recursive feasibility, and then as a practical bridge to a model-based controller. In the output-constrained nonlinear MPC formulation, the optimization imposes

$$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$4

over the prediction horizon, while a feasibility margin derived from the funnel controller ensures that the next MPC problem remains feasible. A learning-based variant applies the model-free funnel controller during an initial learning phase, identifies a model from the measured data, and then hands control over to Funnel-MPC. The reported result is that the resulting feedback controller outperforms the funnel controller both with respect to the required sampling rate for a zero-order-hold implementation and the required control action (Berger et al., 2019).

4. Output-feedback, internal models, stochastic dynamics, and PDE–delay systems

For high-relative-degree systems with output feedback only, the funnel pre-compensator supplies a schedule not on the plant output directly but on the approximation error between the plant output $$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$5 and an auxiliary signal $$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$6. A single pre-compensator stage uses

$$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$7

and a cascade of $$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$8 such stages yields an output $$\mathcal{F}_{\varphi} := \left\{(t,e)\in\mathbb{R}_{\ge 0} \times\mathbb{R}^m \;\big|\; \varphi(t)\,\|e\| < 1 \right\}.$$9 whose derivatives up to order $\|e(t)\| < 1/\varphi(t)$0 are explicitly known. The conjunction of the cascade with a minimum phase system of relative degree $\|e(t)\| < 1/\varphi(t)$1 yields a new system of the same relative degree that is minimum phase as well, so standard funnel controllers can be applied without measuring derivatives of the original output (Lanza, 2022).

Internal-model extensions change the purpose of the schedule. Classical funnel control imposes

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

with $\|e(t)\| < 1/\varphi(t)$3 bounded and bounded away from zero, so the funnel width $\|e(t)\| < 1/\varphi(t)$4 stays strictly positive. In that setting, asymptotic tracking does not come from collapsing the funnel. Instead, the internal model principle is combined with funnel control, and for a class $\|e(t)\| < 1/\varphi(t)$5 of minimum-phase linear systems with positive definite high-frequency gain, sufficiently large $\|e(t)\| < 1/\varphi(t)$6 yields

$\|e(t)\| < 1/\varphi(t)$7

while the prescribed funnel constraints remain in force. The paper notes that this arrangement is intrinsically less sensitive to measurement noise because the error need not hover near the funnel boundary (Berger et al., 2023).

For overdamped Langevin dynamics,

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

the schedule acts on the mean output $\|e(t)\| < 1/\varphi(t)$9. The funnel is

$\varphi$0

and the controller is

$\varphi$1

Under structural assumptions on $\varphi$2, the closed-loop SDE has a unique solution and there exists $\varphi$3 such that

$\varphi$4

The numerical example uses a constant funnel $\varphi$5, showing that a funnel schedule need not be shrinking (Berger et al., 2022).

For nonlinear drill strings, the schedule explicitly compensates propagation delay. The corrected error is

$\varphi$6

with reference adjustment

$\varphi$7

and boundary input

$\varphi$8

Theorem 4.2 guarantees

$\varphi$9

so the schedule is respected even though the effective reference is dynamically modified to compensate for large wave traveling times (Berger et al., 12 May 2026).

5. Protocol schedules in transport physics, network diffusion, and concurrent algorithms

In skyrmion transport through periodic arrays of asymmetric funnels, the schedule is a protocol of AC drives

$(\bar x(\cdot),\bar u(\cdot))$0

Three primitive classes are studied: unidirectional parallel drive ($(\bar x(\cdot),\bar u(\cdot))$1), unidirectional perpendicular drive ($(\bar x(\cdot),\bar u(\cdot))$2), and biharmonic drive ($(\bar x(\cdot),\bar u(\cdot))$3). Parallel drive produces easy-direction ratcheting with quantized average velocity in units of the funnel length $(\bar x(\cdot),\bar u(\cdot))$4; perpendicular drive produces a Magnus-induced hard-direction ratchet with $(\bar x(\cdot),\bar u(\cdot))$5 over a broad range; symmetric biharmonic drive can generate reentrant pinning; and asymmetric biharmonic drive can switch the motion between easy, hard, and pinned regimes. The guided-motion example makes the schedule operational by specifying amplitudes and durations to move a skyrmion between chosen funnels (Souza et al., 2021).

On networks, the phrase is used for staged diffusion strategies. In the discrete Bass model, the pairwise nonadoption inequality

$(\bar x(\cdot),\bar u(\cdot))$6

and the funnel inequality

$(\bar x(\cdot),\bar u(\cdot))$7

relate adoption in the full network to adoption in subnetworks. The notions of influential node and funnel node determine when these inequalities are strict or exact. The associated schedule is a phased rollout in which adoption is built first in subnetworks $(\bar x(\cdot),\bar u(\cdot))$8 and $(\bar x(\cdot),\bar u(\cdot))$9, then propagated through nodes $\eta(t)=x(t)-\bar x(t)$0 whose position in the graph makes the product structure informative (Fibich et al., 2023).

In concurrent algorithms, the term appears in a still different sense: a schedule for routing fetch-and-add operations through aggregators rather than a single hot spot. Aggregating Funnels use one principal variable Main and $\eta(t)=x(t)-\bar x(t)$1 aggregators. The entry rule $V(t,\eta) = \eta(t)^\top Q(t)^{-1}\eta(t),$0 maps threads into groups of size $\eta(t)=x(t)-\bar x(t)$2, so worst-case contention per location drops from $\eta(t)=x(t)-\bar x(t)$3 to $\eta(t)=x(t)-\bar x(t)$4. A delegate then performs one hardware fetch-and-add on Main for the whole batch, while the other operations recover their return values from batch metadata. Recursive application reduces worst-case contention to

$\eta(t)=x(t)-\bar x(t)$5

and the implementation is proved strongly linearizable (Roh et al., 2024).

6. Interpretation, design choices, and recurrent issues

The literature does not use “funnel schedule” for a single formal object. In one group of papers it is the temporal evolution of admissible error or output sets; in another it is the sequence of invariant ellipsoids around a nominal trajectory; in skyrmion transport it is a drive protocol; in network diffusion it is a staged seeding plan; and in concurrent computing it is a rule for routing operations through aggregators. This suggests a unifying interpretation: a funnel schedule is a rule that allocates admissible behavior over time while preserving a specified property such as safety, invariance, tracking performance, directed transport, or throughput.

A common misconception is that funnel schedules are necessarily monotone shrinkage laws. The cited work shows otherwise. Hard/soft output funnels are widened online when constraints conflict and then recovered exponentially (Mehdifar et al., 2022). Input-constrained funnels widen when saturation is active (Berger, 2022). The class $\eta(t)=x(t)-\bar x(t)$6 for drill-string and Langevin problems allows any bounded, Lipschitz, strictly positive $\eta(t)=x(t)-\bar x(t)$7, including constant or non-monotone choices (Berger et al., 12 May 2026, Berger et al., 2022). In skyrmion funnels, increasing the drive amplitude can even re-enter a pinned phase, so the relation between actuation level and motion is not monotone (Souza et al., 2021).

A second misconception is that prescribed-performance funnels automatically imply asymptotic zero error. In the internal-model formulation, $\eta(t)=x(t)-\bar x(t)$8 is bounded and $\eta(t)=x(t)-\bar x(t)$9, so the funnel width stays strictly positive; asymptotic tracking is obtained by augmenting the controller with an internal model rather than by forcing the funnel to collapse (Berger et al., 2023). The drill-string controller likewise achieves funnel feasibility for a delayed and corrected tracking error by dynamically adjusting the effective reference, not by driving the admissible bound to zero (Berger et al., 12 May 2026).

A third recurring issue concerns continuous-time validity. Node-based funnel synthesis may check inequalities only at temporal grid points, leaving open the possibility of between-node violation. Copositivity-based conditions for DLMI satisfaction on every subinterval and later DME-based continuous-time constrained synthesis address precisely this point by making invariance a genuinely continuous-time property rather than a nodewise surrogate (Kim et al., 2024, Kim et al., 12 Nov 2025).

These themes explain why funnel schedules recur across otherwise distant fields. Whether the object being scheduled is an error envelope, an invariant tube, a driven particle trajectory, a diffusion pathway, or a contention-management hierarchy, the technical problem is the same in structure: to prescribe a temporally ordered admissible region or protocol and to construct dynamics or algorithms that keep the evolution inside it.