On Model Predictive Funnel Control with Equilibrium Endpoint Constraints

Dario Dennst\"adt; Jens G\"obel; Karl Worthmann; Lukas Lanza; Thomas Berger; Tobias Damm

REVIEW 2 major objections 3 minor 21 references

On Model Predictive Funnel Control with Equilibrium Endpoint Constraints

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Model predictive funnel control tunes its error tube online and proves that the closed-loop output converges to zero.

desk verdict A genuinely new funnel-MPC hybrid with solid feasibility/cost results and a fixable regularity gap in the stability theorem. read the letter →

arxiv 2505.20090 v1 pith:KWU5KOGH submitted 2025-05-26 math.OC

Jens Göbel , Dario Dennstädt , Lukas Lanza , Karl Worthmann , Thomas Berger , Tobias Damm This is my paper · ORCID

classification math.OC MSC 93C1093D1593C40

keywords modelpredictivecontrolfunnelprescribedtransientbehaviornonlinearoutputfeedbackadaptivestabilityrecursivefeasibilityequilibriumendpointconstraints

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces model predictive funnel control (MPFC), a two-layer controller in which a lower-level high-gain funnel law keeps the output inside a shrinking error tube, and an upper-level model predictive optimizer reshapes the tube at every sampling step by choosing just two parameters, the funnel slope $c$ and the end time $T$. The authors prove that the optimization is always feasible—initially and at every subsequent step—and that with a positive-definite state cost the closed-loop output asymptotically converges to zero. The central trick is a funnel boundary that reaches zero in finite time: the exact finite-time tracking property of the lower-level law supplies the equilibrium endpoint constraint that MPC stability theory needs, without adding a hard constraint to the optimization. A reader should care because the decision-variable count stays independent of the prediction horizon, inter-sampling behavior is explicitly controlled, and the scheme links the model-free robustness of funnel control to the predictive quality of MPC.

What carries the argument

The load-bearing object is the finite-time funnel pair $(c,T)$ defining $\varphi(t;c,T)=c(T-t)$, together with the feedback law $u(t)=(N\circ\alpha_c)(\|y(t)\|^2/\varphi(t)^2)\, y(t)/\varphi(t)$, where $\alpha_c:[0,1)\to[2c,\infty)$ is a bijection governing how quickly gain rises and $N:\mathbb{R}_{\ge0}\to\mathbb{R}$ is a surjection (the paper suggests $\alpha_c(s)=2c/(1-s)$ and $N(s)=s\cos s$). Lemma 3, taken from [8], guarantees that this law drives the output of every system in class $\mathcal{S}$ to zero exactly at time $T$ with bounded input, starting from any initial output inside the funnel. That exact finite-time property plays the role of the terminal equality constraint in MPC stability theory [17]: the horizon can be extended at zero cost, which produces the inequality in Lemma 6 that bounds each sample-interval cost by the drop in the horizon cost. Corollary 7 turns the bounded closed-loop cost into boundedness of the sequence of active parameters $c_i$. Theorem 8 then argues by contradiction: if the output norm kept returning to a fixed $\varepsilon$, the funnel law would have to generate unbounded control at the times when the output crosses $\varepsilon/3$ and $2\varepsilon/3$, but bounded $c_i$ and the distance-to-boundary structure of (4) give a uniform bound on the control, a contradiction.

What would settle it

A direct check: apply the feedback law (4) with $\alpha_c(s)=2c/(1-s)$ and $N(s)=s$ to the example system (18) from inside the funnel, and measure $\|y(T)\|$ and $\|u\|_\infty$; a nonzero $\|y(T)\|$ or an unbounded input would falsify Lemma 3 and therefore Theorems 5 and 8. Running Algorithm 4 on the same example with $Q=I_2$ should give $\|y_{\mathrm{cl}}(t)\to 0$; failure of that convergence would falsify Theorem 8.

Watch

Extended reading notes

Core claim

The paper's claim, stated on its own terms, is that receding-horizon optimization of the funnel parameters $(c,T)$ over a fixed horizon $H$ stabilizes every system in the class $\mathcal{S}$. Theorem 8 makes this precise: under the assumptions of Theorem 5 and with $Q$ positive definite, $\lim_{t\to\infty} y_{\mathrm{cl}}(t) = 0$, so Algorithm 4 asymptotically stabilizes system (1). Theorem 5 establishes initial and recursive feasibility for every initial output with $\|y(0)\|<\psi(0)$ and shows that the closed-loop output remains inside the funnel boundary on each sampling interval. Because the funnel boundary $\varphi(t;c,T)=c(T-t)$ vanishes at $T$, Lemma 3 gives exact convergence to the equilibrium at the horizon end for free, which converts the classical MPC terminal-equality argument into a cost-decrease argument. The same proof yields bounded infinite-horizon closed-loop cost and bounded optimized parameters, and these bounds feed the contradiction argument that rules out any persistent deviation from zero.

Load-bearing premise

The whole argument rests on the borrowed finite-time exact-tracking guarantee—that for every system in class $\mathcal{S}$ the funnel law (4) drives the output to zero exactly at time $T$ with bounded input when the initial output lies inside the funnel—and if that guarantee failed for any admissible system, both the feasibility proof and the stability proof would collapse.

Editorial extensions

If this is right

Algorithm 4 is initially and recursively feasible for every system in class $\mathcal{S}$ with initial output inside the outer funnel, so the optimization can never get stuck at a sampling instant.
With $Q$ positive definite, the closed-loop output converges to zero, so the scheme asymptotically stabilizes the system while staying inside the user-prescribed error boundary at all times.
The infinite-horizon closed-loop cost is bounded by the first horizon cost, and the optimized funnel parameters $c_i$ remain bounded, so the controller's effort stays under control.
The optimization has exactly two decision variables, $c$ and $T$, regardless of the horizon length $H$, so the per-step problem size does not grow when predictions are made further ahead.
Between sampling instants the funnel feedback law continues to act, so constraint satisfaction and inter-sampling behavior are explicitly handled rather than left open-loop.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One consequence the paper leaves implicit is that the two-variable parameterization makes the online optimization independent of the horizon length in a way that could scale to long horizons; a benchmark against classical MPC on computation time would test how much of that promise is realized in practice.
Because the terminal equilibrium is enforced by the feedback law rather than by an explicit endpoint constraint, the same architecture might extend to time-varying reference tracking if the finite-time exact-tracking result is replaced by an analogous moving-target version.
The stability proof uses only two structural ingredients—positive definiteness of $Q$ and a uniform control bound from bounded $c_i$ and distance-to-boundary—so the same argument should apply to any parameterized feedback family sharing those properties.
A testable extension is to relax the outer funnel $\psi$ or the strict upper-bound requirement $\hat V_H > V_H$, for instance by penalizing the funnel parameters in the cost; the feasibility construction in Theorem 5 only needs $\psi$ smooth and positive to build a candidate funnel.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a genuinely new scheme, not a repackaging of funnel MPC. The authors optimize two funnel parameters (c,T) per receding-horizon step and let the inner funnel controller generate the input, which cuts the decision variables to two regardless of horizon length. That is a real departure from [18-20], which keep a static funnel and optimize control values.

The paper does solid work on the feasibility side. Theorem 5 gives explicit feasible candidates and the recursive argument via shifting (c,T) works. Lemma 6 telescopes correctly and gives the bounded closed-loop cost. The stability proof in Theorem 8 follows the standard terminal-equality route, and the contradiction structure is clever.

The soft spots are real but repairable. The main one is in Theorem 8, Steps 3 and 4: the control bounds use Nhat(alpha_c(4/9)) and Nhat(alpha_c(64/81)) assuming these are finite uniformly over the active parameter c. The stated assumptions only give pointwise continuous bijections alpha_c for each c; there is no joint continuity or uniform local boundedness in c. The stress-test example alpha_c(s) = 2c + (s/(1-s))/|c-1| (with a limit at c=1) satisfies every stated hypothesis but makes alpha_c(4/9) blow up as c->1. So the theorem as written is not proved. That said, the concrete family alpha_c(s)=2c/(1-s) satisfies the needed bound, so this is a gap in the statement, not a counterexample to the method. The fix is straightforward: add joint continuity on compact c-intervals, or directly assume sup over the relevant c-range of alpha_c(4/9) is finite.

Second, Algorithm 4 relies on an oracle that returns a strict upper bound Vhat_H > V_H and finds feasible points under it. No concrete optimizer is given. For a theory paper this is acceptable if flagged, but they should be clearer that the guarantees are conditional on that oracle.

Third, the claimed computational advantage is not benchmarked; the numerical example is one trajectory, and the paper itself defers performance studies. That is fine for a brief paper, but it keeps significance from being higher than reasonable.

Minor edge cases: in Theorem 5 Step 1, if psi_dot is identically zero, the candidate T divides by zero; use a max or separate case. There are also a few notation typos in Theorem 8 (t_crit vs hat t_crit).

Overall: the central idea is sound and the gap is patchable. This paper is for readers working on funnel control or MPC for constrained nonlinear systems. It deserves a serious referee and, after a revision that fixes the alpha_c regularity assumption and discusses the oracle, I would support publication.

Referee Report

2 major / 3 minor

Summary. The paper proposes model predictive funnel control (MPFC), a two-layer control scheme in which an outer MPC layer updates, at each sampling instant, the parameters (c,T) of a time-varying funnel boundary, while an inner finite-time funnel controller (from [8]) drives the output to zero within that boundary. The main theoretical results are Theorem 5 (initial and recursive feasibility, plus funnel constraint satisfaction), Lemma 6 (boundedness of the closed-loop infinite-horizon cost), Corollary 7 (boundedness of the optimized c-values), and Theorem 8 (asymptotic stability under positive definite Q). The proofs use the finite-time funnel control lemma from [8] and assume an optimization oracle that returns a strict upper bound on the optimal value at every step.

Significance. If the gap identified below in Theorem 8 is repaired, the paper makes a useful contribution: it combines model-free funnel control with model-based predictive optimization while keeping the number of optimization variables independent of the horizon, and it provides rigorous feasibility, cost-bound, and stability arguments. The proof of Theorem 8 is an intricate contradiction argument based on crossing times, and Lemma 6's telescoping of the cost is clean and correct. The main computational premise, the strict upper-bound oracle, is stated explicitly in Algorithm 4 rather than hidden, which is a strength. The numerical example illustrates the proposed scheme, though it is not a systematic comparison.

major comments (2)

[Theorem 8, Steps 3–4] The contradiction in Theorem 8 requires a bound on N-hat(alpha_c(4/9)) and N-hat(alpha_c(64/81)) that is uniform over the active parameters c = c_i^*. The stated hypotheses only require each alpha_c to be a continuous bijection [0,1) -> [2c,infty) pointwise in c, and Corollary 7 only establishes that {c_i^*} is bounded above, with no lower bound. An allowed family is alpha_c(s) = 2c + (s/(1-s))/|c-1| for c != 1 and alpha_1(s) = 2 + s/(1-s); each alpha_c is a continuous bijection with the required range, but alpha_c(4/9) tends to infinity as c -> 1, so the bound in Step 3 can blow up along a sequence of active parameters. The proof therefore does not establish the claimed contradiction as stated. I recommend adding an explicit uniformity assumption on the family (alpha_c), for example joint continuity of (c,s) |-> alpha_c(s) on compact c-intervals, or directly sup_{c in [c_min,c_max]} alpha_c(4/9) < infinity. This uniformity holds for the concrete choice alpha_c(s) = 2c/(1-s) used in the numerical section.
[Theorem 5, Step 1] The candidate construction in Step 1 divides by ||dot_psi|_{[t-hat, t-hat+H]}||_infty, which is zero whenever psi is constant on that interval (for example, psi == const > 0). In that case the expression for T-hat is undefined, so the proof does not cover all admissible outer funnel functions. The feasibility claim itself remains true, but the proof needs a separate case for ||dot_psi||_infty = 0, or an alternative construction such as T-hat = H and c-hat = (psi(t-hat) + ||y-hat||)/(2H).

minor comments (3)

[Theorem 8, Step 4] There are two notation typos in the proof: the derivative in the mean-value theorem step should be applied to ||y_cl(t-hat_crit)||, not to ||y_cl(t_crit)||, and the expression 82/92 should be read as (8/9)^2 = 64/81.
[Section 4, closed-loop definitions] The sums defining y_cl, u_cl, and phi_cl start at i=1, which omits the first sampling interval [t_0,t_1) = [0,h). They should start at i=0, or the interval [0,h) should be included separately, to match the definition of y_i on [t_i,t_{i+1}] for i in N_0.
[Algorithm 4] The algorithm assumes that at every step an oracle returns a strict upper bound V-hat_H(t_i,y_i) > V_H(t_i,y_i). This is a genuine computational premise; the paper could usefully comment on how such a bound might be obtained in practice, for example by evaluating a feasible suboptimal candidate, and on what happens if the oracle only provides an approximate bound.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MPFC feasibility and stability proofs reduce to stated assumptions plus the independent finite-time funnel theorem from [8]; the uniformity gap in Theorem 8 is a correctness concern, not circularity.

full rationale

Algorithm 4's derivation chain is self-contained relative to its stated assumptions. The feasibility proof in Theorem 5 explicitly constructs feasible funnel parameters from the outer funnel ψ and the current output norm, and recursive feasibility is established either by shifting the previous funnel (c_i^*, T_i^*-h) or by using the already-reached equilibrium. Cost boundedness in Lemma 6 follows algebraically from the algorithm's descent condition (8) and the zero-extension of the horizon after the equilibrium is reached. Theorem 8 is a contradiction argument using funnel geometry and bounded input properties of the feedback law; it does not define its conclusion into its assumptions. The only external input is Lemma 3, quoted from the co-authored prior paper [8]. Although this is a load-bearing self-citation, the cited theorem is an independently published result about funnel control whose assumptions (system class S, ||y0||<cT) do not include the present paper's stability conclusion, and the present paper introduces no fitting relation or definitional identity that turns finite-time exact tracking into the MPFC convergence claim. The proof of Theorem 8 does contain a uniformity gap: the bounds such as N-hat(alpha_c(4/9)) and N-hat(alpha_c(64/81)) are used without a stated hypothesis that alpha_c(4/9) remains bounded over the active parameter set, and the stated pointwise hypotheses on alpha_c do not imply that. This is a genuine correctness risk in the proof as written, but it is not circularity, since the missing uniform bound is not among the theorem's inputs and the concrete family alpha_c(s)=2c/(1-s) does satisfy it. No circular step is exhibited.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data; the numbers in the paper (Q, R, h, H, N, alpha_c, psi) are declared user choices, and the theorems quantify over all admissible choices. The arguments rest on the system class S, on the quoted finite-time funnel controller of [8], and on an idealized optimization oracle; all are listed above. No new physical entities are introduced.

free parameters (4)

Stage-cost weights Q, R = Q=I2, R=0.2 I2 in Section 5.
User-chosen design weights. The theory only requires Q positive definite and R positive semi-definite, so the central claims do not depend on the exact values.
Sampling step h and horizon H = h=0.25, H=5 in Section 5.
User-chosen. The theory holds for any h>0 and H=nh with n>=2, so the feasibility and stability results are not tuned to these values.
Funnel shaping functions N and alpha_c = N(s)=s, alpha_c(s)=2c/(1-s) in Section 5.
User-chosen design functions. Theorem 8 implicitly needs the family alpha_c to be locally bounded in c for uniform control bounds; the example family satisfies this.
Outer funnel psi = psi=infinity in Section 5.
User-chosen constraint function. The existence construction in Theorem 5 Step 1 uses the derivative bound ||psi_dot||_infinity when psi is finite.

assumptions (5)

domain assumption The high-gain property (Definition 1) and f(0,0)=0 define the system class S.
Invoked in Lemma 3 (from [8]) and in every theorem; restricts the results to square, relative-degree-one systems with high-gain structure.
domain assumption The funnel feedback law (4) from [8, Thm 3.1] drives the output exactly to zero at time T with bounded input.
Quoted as Lemma 3; this is the load-bearing external result. The terminal equality property and the control bounds in Theorem 8 both depend on it.
standard math Caratheodory solutions to (1) exist and are unique for continuous f and measurable L^infinity inputs.
Implicitly assumed in Lemma 3 and in the concatenation of solutions across sampling intervals in Theorem 5 and Lemma 6.
ad hoc to paper An optimization oracle returns a strict upper bound V_hat_H(t_i, y_i) > V_H(t_i, y_i) at every step.
Algorithm 4 requires this strict bound for (8) to be solvable, but no construction of such an oracle is provided.
ad hoc to paper The family alpha_c is regular enough for the control bounds in Theorem 8 to be uniform over the active parameter values.
The proof bounds the input by N_hat(alpha_c(4/9)) and N_hat(alpha_c(64/81)) uniformly over bounded c; this holds for alpha_c(s)=2c/(1-s) and under joint continuity with c in a compact interval away from zero, but it is not stated.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On Model Predictive Funnel Control with Equilibrium Endpoint Constraints." pith.science (2026). https://pith.science/paper/KWU5KOGH

@misc{pith2026250520090,
  author       = {Pith},
  title        = {Pith review of: On Model Predictive Funnel Control with Equilibrium Endpoint Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KWU5KOGH}},
  note         = {Machine review of arXiv:2505.20090}
}

read the original 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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 20 canonical work pages

[8]

Exact output tracking in prescribed finite time via funnel control,

L. Lanza, “Exact output tracking in prescribed finite time via funnel control,” Automatica, vol. 170: 111873, 2024

work page 2024
[1]

Tracking with prescribed transient behaviour,

A. Ilchmann, E. P. Ryan, and C. J. Sangwin, “Tracking with prescribed transient behaviour,” ESAIM: Control Optim. Calc. Var., vol. 7, pp. 471–493, 2002

work page 2002
[2]

Funnel control of nonlinear systems,

T. Berger, A. Ilchmann, and E. P. Ryan, “Funnel control of nonlinear systems,” Math. Control Signals Syst., vol. 33, pp. 151–194, 2021

work page 2021
[3]

C. M. Hackl,Non-identifier Based Adaptive Control in Mechatronics. Springer International Publishing, 2017

work page 2017
[4]

Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance,

C. P. Bechlioulis and G. A. Rovithakis, “Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance,”IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2090–2099, 2008

work page 2008
[5]

A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems,

——, “A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems,”Automatica, vol. 50, no. 4, pp. 1217–1226, 2014

work page 2014
[6]

On the equivalence between prescribed performance control and control barrier functions,

R. Namerikawa, A. Wiltz, F. Mehdifar, T. Namerikawa, and D. V. Dimarogonas, “On the equivalence between prescribed performance control and control barrier functions,” inProc. American Control Conference (ACC), 2024, pp. 2458–2463

work page 2024
[7]

Asymptotic tracking via funnel control,

J. G. Lee and S. Trenn, “Asymptotic tracking via funnel control,” in2019 IEEE 58th Conference on Decision and Control (CDC), 2019, pp. 4228–4233

work page 2019

Show all 21 references

[9]

Model predictive control: Review of the three decades of development,

J. H. Lee, “Model predictive control: Review of the three decades of development,” Int. J. Control Autom. Syst., vol. 9, no. 3, 2011

2011
[10]

Model predictive control in practice,

T. A. Badgwell and S. J. Qin, “Model predictive control in practice,” inEncyclopedia of Systems and Control. Springer, 2021, pp. 1239–1252

2021
[11]

Embedded online optimization for model predictive control at megahertz rates,

J. L. Jerez, P. J. Goulart, S. Richter, G. A. Constantinides, E. C. Kerrigan, and M. Morari, “Embedded online optimization for model predictive control at megahertz rates,” IEEE Trans. Autom. Control, vol. 59, no. 12, pp. 3238–3251, 2014

2014
[12]

From linear to nonlinear MPC: bridging the gap via the real-time iteration,

S. Gros, M. Zanon, R. Quirynen, A. Bemporad, and M. Diehl, “From linear to nonlinear MPC: bridging the gap via the real-time iteration,”Int. J. Control, vol. 93, no. 1, pp. 62–80, 2020

2020
[13]

Robust model predictive control of constrained linear systems with bounded disturbances,

D. Q. Mayne, M. M. Seron, and S. Raković, “Robust model predictive control of constrained linear systems with bounded disturbances,”Automatica, vol. 41, no. 2, pp. 219–224, 2005. 13

2005
[14]

Tube-based robust nonlinear model predictive control,

D. Q. Mayne, E. C. Kerrigan, E. Van Wyk, and P. Falugi, “Tube-based robust nonlinear model predictive control,”Int. J. Robust & Nonlinear Control, vol. 21, no. 11, pp. 1341–1353, 2011

2011
[15]

Input to state stability of min– max mpc controllers for nonlinear systems with bounded uncertainties,

D. Limón, T. Alamo, F. Salas, and E. F. Camacho, “Input to state stability of min– max mpc controllers for nonlinear systems with bounded uncertainties,”Automatica, vol. 42, no. 5, pp. 797–803, 2006

2006
[16]

Data-driven min-max mpc for linear systems: Robustness and adaptation,

Y. Xie, J. Berberich, and F. Allgöwer, “Data-driven min-max mpc for linear systems: Robustness and adaptation,”arXiv preprint arXiv:2404.19096, 2024

2024 arXiv
[17]

Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations,

S. S. Keerthi and E. G. Gilbert, “Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations,” Journal of optimization theory and applications, vol. 57, pp. 265– 293, 1988

1988
[18]

Learning-based funnel-mpc for output- constrained nonlinear systems,

T. Berger, C. Kästner, and K. Worthmann, “Learning-based funnel-mpc for output- constrained nonlinear systems,”IFAC-PapersOnLine, vol. 53, no. 2, pp. 5177–5182, 2020

2020
[19]

Funnel Model Predictive Control for Nonlinear Systems with Relative Degree One,

T. Berger, D. Dennstädt, A. Ilchmann, and K. Worthmann, “Funnel Model Predictive Control for Nonlinear Systems with Relative Degree One,”SIAM Journal on Control and Optimization, vol. 60, no. 6, pp. 3358–3383, 2022

2022
[20]

Funnel MPC for nonlinear systems with arbitrary relative degree,

T. Berger and D. Dennstädt, “Funnel MPC for nonlinear systems with arbitrary relative degree,”Automatica, vol. 167, 2024

2024
[21]

Constrained nonlinear output regulation using model predictive control,

J. Köhler, M. A. Müller, and F. Allgöwer, “Constrained nonlinear output regulation using model predictive control,”IEEE Trans. Autom. Control, vol. 67, no. 5, pp. 2419–2434, 2022. 14

2022

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [8]

Exact output tracking in prescribed finite time via funnel control,

L. Lanza, “Exact output tracking in prescribed finite time via funnel control,” Automatica, vol. 170: 111873, 2024

work page 2024

[2] [1]

Tracking with prescribed transient behaviour,

A. Ilchmann, E. P. Ryan, and C. J. Sangwin, “Tracking with prescribed transient behaviour,” ESAIM: Control Optim. Calc. Var., vol. 7, pp. 471–493, 2002

work page 2002

[3] [2]

Funnel control of nonlinear systems,

T. Berger, A. Ilchmann, and E. P. Ryan, “Funnel control of nonlinear systems,” Math. Control Signals Syst., vol. 33, pp. 151–194, 2021

work page 2021

[4] [3]

C. M. Hackl,Non-identifier Based Adaptive Control in Mechatronics. Springer International Publishing, 2017

work page 2017

[5] [4]

Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance,

C. P. Bechlioulis and G. A. Rovithakis, “Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance,”IEEE Trans. Autom. Control, vol. 53, no. 9, pp. 2090–2099, 2008

work page 2008

[6] [5]

A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems,

——, “A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems,”Automatica, vol. 50, no. 4, pp. 1217–1226, 2014

work page 2014

[7] [6]

On the equivalence between prescribed performance control and control barrier functions,

R. Namerikawa, A. Wiltz, F. Mehdifar, T. Namerikawa, and D. V. Dimarogonas, “On the equivalence between prescribed performance control and control barrier functions,” inProc. American Control Conference (ACC), 2024, pp. 2458–2463

work page 2024

[8] [7]

Asymptotic tracking via funnel control,

J. G. Lee and S. Trenn, “Asymptotic tracking via funnel control,” in2019 IEEE 58th Conference on Decision and Control (CDC), 2019, pp. 4228–4233

work page 2019