A numerical approach to the co-design of PID controllers and low-pass filters for time-delay systems

Diego Torres-Garc\'ia; Wim Michiels

arxiv: 2604.16124 · v1 · submitted 2026-04-17 · 📡 eess.SY · cs.SY· math.OC

A numerical approach to the co-design of PID controllers and low-pass filters for time-delay systems

Diego Torres-Garc\'ia , Wim Michiels This is my paper

Pith reviewed 2026-05-10 08:04 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords PID controllerstime-delay systemslow-pass filtersspectral abscissanumerical optimizationcontroller designstabilityco-design

0 comments

The pith

Simultaneously tuning PID gains and derivative filter constant by minimizing the spectral abscissa produces improved controllers for time-delay systems.

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

The paper establishes a numerical framework for designing PID controllers that incorporates the low-pass filter on the derivative term from the start. Instead of optimizing an ideal unfiltered PID and adding the filter afterward, the gains and filter time constant are adjusted together to reduce the spectral abscissa of the actual closed-loop system that includes delays. This matters because real-world PID implementations always use such filters to curb noise, and the filter changes stability properties in ways that post-design filtering can degrade. Treating the filter as part of the design balances low-frequency tracking performance with high-frequency robustness and noise rejection for both SISO and MIMO cases.

Core claim

The central claim is that the PID gains and the filter constant should be optimized simultaneously by directly minimizing the spectral abscissa of the filtered closed-loop system, rather than optimizing the unfiltered system and treating the filter as a post-processing step. This co-design reconciles robustness at high frequencies with performance at low frequencies and helps mitigate measurement noise amplification. The method applies to general linear time-delay systems in both SISO and MIMO settings.

What carries the argument

Direct numerical minimization of the spectral abscissa for the closed-loop system whose characteristic equation includes the low-pass filter dynamics on the derivative term.

If this is right

The filtered closed-loop system can achieve a smaller spectral abscissa than would result from separate PID design followed by filter addition.
High-frequency fragility from approximate derivative action is reduced because the filter parameter is chosen with stability in mind.
The same numerical procedure works for systems containing both input delays and state delays.
The resulting controllers simultaneously address low-frequency performance and high-frequency noise rejection.

Where Pith is reading between the lines

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

Standard PID tuning software could embed this co-design step so that filter parameters are never chosen after the fact.
The approach might be combined with other objectives such as integral squared error to produce controllers that satisfy multiple specifications at once.
Because the optimization is numerical, it could be adapted to plants whose delay values are only known approximately or vary slowly.

Load-bearing premise

That the spectral abscissa of the filtered system alone is a reliable guide to practical closed-loop behavior without needing separate checks on noise sensitivity or other implementation details.

What would settle it

A time-domain simulation or experiment on a delayed plant where the jointly optimized controller shows larger overshoot, slower settling, or greater noise amplification than a PID tuned without the filter and then filtered afterward.

Figures

Figures reproduced from arXiv: 2604.16124 by Diego Torres-Garc\'ia, Wim Michiels.

**Figure 1.** Figure 1: Stable region on the control parameters plane of system (4) Example 3.1 (Small feedback delay). We start by considering the following single-input singleoutput linear time-invariant system: (4) Σ :=    x˙(t) = 0 1 −3 −4 x(t) + 0 1 u(t) y(t) = [1 −1]x(t) , with u(t) = −kpy(t) − kdy˙(t) a classical PD controller. The transfer function associated to the system reads: (5) Hyu(s) := C(sI − A) −1B … view at source ↗

**Figure 2.** Figure 2: Presence of a delay on the input/output channel of the system. 69.22 69.24 69.26 69.28 69.30 <(λ) −10000 −7500 −5000 −2500 0 2500 5000 7500 10000 =(λ) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Unstable solutions of (7) when kp = 1, kd = 2 and r = 0.01. Example 3.2 (Delay-difference approximations). Next, we consider the following system: (8) Σ :=    x˙(t) =   −1 1 1 1 0 0 0 1 0   x(t) +   −5 0 0   u(t) y(t) = [1 0 0]x(t) , which has a corresponding closed-loop characteristic function ∆ : C → C given by: (9) ∆(s) = s 3 + s 2 − s − 1 − 5s 2 (kp + kds). Let us consider kp = 3 and k… view at source ↗

**Figure 4.** Figure 4: Evolution of ¯s 7→ f(¯s; kd) with kd = 2. introducing the scaling variable s¯ = sr, and multiplying by r we can obtain the following expression: s¯ 3 + rs¯ 2 − r 2 s¯− 1 − 5¯s 2 (kpr + kd(1 − e −s¯ )) = 0. Note that by taking r → 0 +, we should expect the approximation of the derivative to improve, however, observe that with r → 0 +, the characteristic function converges on compact sets to: (11) ¯s 2 (¯s −… view at source ↗

**Figure 5.** Figure 5: Stability region in the (r, τ )-plane for τ a delay in the input channel, and r the filter constant from the closed-loop between system (12) and a PD controller A similar issue as the one presented with the inclusion of the approximation of the derivative action arises here, and stability can be destroyed if the filter acts as a destabilizing perturbation as discussed in [15]. Let us assume that is not the… view at source ↗

**Figure 6.** Figure 6: Stability crossing curves in the (T, τ0)-plane. −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 <(s) −0.4 −0.2 0.0 0.2 0.4 =(s) Right-most roots with optimized filter Right-most roots with Appeltans et al. controller [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: Comparison of the rightmost closed-loop roots obtained with fixed filtering and with joint controller–filter optimization. It is important to note that modifying kp and kd alters the stability region shown in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗

**Figure 8.** Figure 8: Stability region in the (T, τ0)-plane for the controller obtained using the classical design approach. given by (27) ∆(s) = s 4 + s 3 + 3s 2 + 2s + e −τ0s (1 − 0.5s 2 ) [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗

**Figure 9.** Figure 9: Stability region in the (T, τ0)-plane for the controller obtained using Algorithm 1. The shaded region indicates combinations of filter constant T and input delay τ0 for which the closed-loop system remains exponentially stable. In the delay-free case, these values place the dominant closed-loop roots at s = −0.1011 ± 1.6262 i, which correspond to a slower convergence rate than that obtained with the class… view at source ↗

**Figure 10.** Figure 10: Evolution of the rightmost characteristic roots as the input delay τ0 varies for the classical and filtered included PID designs. The filtered design exhibits significantly improved robustness with respect to delay variations. −0.4 −0.2 0.0 0.2 0.4 0.6 <(s) −4 −2 0 2 4 =(s) Optimized controller −0.4 −0.2 0.0 0.2 0.4 0.6 <(s) −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5 10.0 =(s) Controller without filter design 0… view at source ↗

**Figure 11.** Figure 11: Evolution of the rightmost characteristic roots as the filter constant T varies. The figure illustrates the sensitivity of the closed-loop spectrum to the filter dynamics. naturally motivates the use of the proposed methodology to tune the controller gains and filter constant such that robustness is increased, rather than just minimizing the spectral abscissa. Example 6.2. Delays are ubiquitous in contro… view at source ↗

**Figure 12.** Figure 12: Stability region in the (T, τ0)-plane for the initial set of parameters. where x(t) ∈ R 2 , u(t) ∈ R, and τ0 ≥ 0 denotes a constant input delay. In the delay-free case (τ0 = 0), the system cannot be stabilized using proportional feedback alone, nor by means of a proportional–integral (PI) controller. For this reason, we consider a proportional–derivative (PD) control law of the form u(t) = kpy(t) + kdz(… view at source ↗

**Figure 13.** Figure 13: Stability region corresponding to the locally optimal parameters obtained from Algorithm 1. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 T 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 τ0 Stable Region [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗

**Figure 14.** Figure 14: Stability region obtained after a second optimization with a restricted interval for T. T = 0.0216. The corresponding stability region, shown in [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗

**Figure 15.** Figure 15: Stability region obtained after a third iteration of Algorithm 1 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗

read the original abstract

This paper addresses the numerical optimization of proportional-integral-derivative (PID) controllers for linear time-invariant systems with delays, where the derivative action is implemented using a low-pass filter. While performance assessment is often based on the spectral abscissa of the ideal PID-controlled system, the inclusion of a derivative filter fundamentally alters the closed-loop spectral properties and cannot be treated as a post-processing step. In particular, the spectral abscissa of the filtered closed-loop system may differ significantly from that of its unfiltered counterpart, potentially affecting both stability and performance. We propose a systematic numerical design framework in which the PID gains and the filter constant are optimized simultaneously by directly minimizing the spectral abscissa of the filtered closed-loop system. Treating the filter as an integral part of the control design allows us to reconcile robustness at high frequencies, in the sense of mitigating fragility issues due to approximate identities, with performance at low frequencies, in addition to counter measurement noise amplification. At the end of the presentation, numerical examples illustrate the proposed approach and highlight the benefits of controller-filter co-design. The results apply to general linear systems with input and/or state delays and are valid for both single-input single-output (SISO) and multi-input multi-output (MIMO) configurations.

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.

Desk Editor's Note private letter to a colleague

The paper gives a numerical co-design for PID plus filter in delay systems by minimizing the filtered spectral abscissa, but the non-convex nature of the problem and limited validation on the optimizer leave the reliability open to question.

read the letter

The paper's main point is that you should optimize the PID gains and the low-pass filter constant together by minimizing the spectral abscissa of the filtered closed-loop system, rather than designing the controller first and adding the filter later. This matters for time-delay systems because the filter changes the high-frequency poles and can shift the stability margin. What the work does well is to lay out a direct numerical framework for this co-design. It applies to general linear systems with input or state delays and handles both SISO and MIMO cases. The authors correctly note that ignoring the filter during design can lead to fragility or excessive noise amplification, and their approach tries to address that by treating the filter parameter as a design variable from the beginning. The numerical examples show concrete cases where the joint optimization improves the achieved spectral abscissa. The soft spot is the reliance on numerical minimization without strong evidence that the optimizer finds good solutions reliably. The spectral abscissa for delay systems is a non-convex, non-smooth function of the parameters. Local methods can land in suboptimal points, and the paper only presents illustrative examples. There is no discussion of multiple random initializations, success rates, or comparisons with global search techniques. This makes it hard to judge how systematic the framework really is in practice. This paper is for people working on PID tuning for processes with delays, such as in process control or networked systems. A reader who already knows spectral methods for time-delay systems will see the value in the co-design idea, while someone new to the area might find the examples helpful as a starting point. I would send this to peer review. The core proposal is sensible and fills a practical gap, even if the numerical robustness needs more attention in revision.

Referee Report

1 major / 1 minor

Summary. The paper proposes a numerical co-design framework for PID controllers and their low-pass derivative filters in linear time-delay systems (input and/or state delays, SISO or MIMO). Instead of optimizing the ideal PID spectral abscissa and then adding a filter, the method jointly tunes the three PID gains plus the filter time-constant by direct numerical minimization of the spectral abscissa of the filtered closed-loop quasi-polynomial. Numerical examples are used to illustrate that the co-design can yield better closed-loop poles than post-filtering an ideal PID design while also mitigating high-frequency fragility and noise amplification.

Significance. If the numerical procedure reliably locates controllers with improved spectral abscissa, the work would supply a practical, systematic alternative to the common two-step design practice for delay systems. Treating the filter parameter as an explicit decision variable directly addresses the mismatch between ideal and implemented controllers, which is a recognized source of fragility. The approach is applicable to both SISO and MIMO cases and does not rely on ad-hoc parameter tuning.

major comments (1)

[Numerical examples] Numerical examples section: the reported improvements rest on single optimization runs of a non-convex, non-smooth objective (spectral abscissa of a quasi-polynomial in five design parameters). No multi-start statistics, basin-of-attraction analysis, or comparison against a global solver are provided, so it is unclear whether the observed gains over separate design are reproducible or merely initialization-dependent. This directly affects the claim that the procedure constitutes a 'systematic numerical design framework.'

minor comments (1)

[Abstract] Abstract: the phrase 'at the end of the presentation' is imprecise; replace with 'in the numerical examples' or 'in Section X' for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The major comment raises a valid point about the presentation of numerical results, which we address below. We will revise the manuscript accordingly to strengthen the evidence for the proposed co-design framework.

read point-by-point responses

Referee: [Numerical examples] Numerical examples section: the reported improvements rest on single optimization runs of a non-convex, non-smooth objective (spectral abscissa of a quasi-polynomial in five design parameters). No multi-start statistics, basin-of-attraction analysis, or comparison against a global solver are provided, so it is unclear whether the observed gains over separate design are reproducible or merely initialization-dependent. This directly affects the claim that the procedure constitutes a 'systematic numerical design framework.'

Authors: We agree that the spectral abscissa minimization is a non-convex, non-smooth optimization problem and that single-run examples leave open questions about reproducibility. The core contribution of the framework is the formulation of a single, joint optimization problem over the PID gains and filter time-constant, which is solved numerically; the examples serve to illustrate potential benefits rather than to statistically validate global optimality. To address the concern, the revised manuscript will include multi-start experiments: for each example we will report results from 50 independent random initializations, providing the best, mean, and standard deviation of the achieved spectral abscissa for both the co-design and the conventional two-step approach. This will demonstrate that the observed improvements are not isolated to particular initial guesses. A full basin-of-attraction study or comparison with a global solver is computationally prohibitive given the cost of evaluating the spectral abscissa of the quasi-polynomial; however, the multi-start local optimization provides a practical and reproducible assessment of the method's reliability in the context of the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical minimization of spectral abscissa

full rationale

The paper's central contribution is a numerical co-design method that simultaneously optimizes PID gains and filter constant by minimizing the spectral abscissa of the filtered closed-loop system for time-delay plants. This is a direct application of an established stability metric (spectral abscissa of a quasi-polynomial) via standard numerical optimization; no quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and the abstract and claim contain no load-bearing self-citation or imported uniqueness theorem that reduces the result to prior author work. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from control theory for delay systems and the effectiveness of spectral abscissa minimization.

axioms (2)

domain assumption Linear time-invariant systems with delays can be analyzed using spectral properties of the closed-loop operator.
This is a standard assumption in the theory of time-delay systems.
domain assumption Minimizing the spectral abscissa leads to improved stability and performance.
Common in spectral optimization approaches for control design.

pith-pipeline@v0.9.0 · 5530 in / 1298 out tokens · 41295 ms · 2026-05-10T08:04:37.799916+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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C.-F. M´ endez-Barrios, J.-D. Torres-Garc´ ıa, and S.-I. Niculescu. Delay-difference approximations of pd- controllers. improperly-posed systems in multiple delays case.International Journal of Robust and Non- linear Control, 34(10):6431–6454, 2024

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A. H. Mhmood and M. N. Mahyuddin. H∞sliding mode control: A recent review of applications and design methods.IEEE Access, 2025

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Michiels

W. Michiels. To filter or not to filter? impact on stability of delay-difference and neutral equations with multiple delays.IEEE Transactions on Automatic Control, 68(6):3687–3693, 2022

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Michiels and S.-I

W. Michiels and S.-I. Niculescu.Stability, control, and computation for time-delay systems: an eigenvalue- based approach. SIAM, 2014

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Michiels, S.-I

W. Michiels, S.-I. Niculescu, and I. Boussaada. A complete characterization of minima of the spectral abscissa and rightmost roots of second-order systems with input delay.IMA Journal of Mathematical Control and Information, 40(3):403–428, 2023

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Michiels, S.-I

W. Michiels, S.-I. Niculescu, I. Boussaada, and G. Mazanti. On the relations between stability optimization of linear time-delay systems and multiple rightmost characteristic roots.Mathematics of Control, Signals, and Systems, 37(1):143–166, 2025

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A. Ram´ ırez, S. Mondi´ e, R. Garrido, and R. Sipahi. Design of proportional-integral-retarded (pir) controllers for second-order lti systems.IEEE Transactions on Automatic Control, 61(6):1688–1693, 2015

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R. Sipahi, S.-I. Niculescu, C. T. Abdallah, W. Michiels, and K. Gu. Stability and stabilization of systems with time delay.IEEE Control Systems Magazine, 31(1):38–65, 2011

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[1] [1]

Appeltans and W

P. Appeltans and W. Michiels. Analysis and controller-design of time-delay systems using tds-control. a tutorial and manual.arXiv preprint arXiv:2305.00341, 2023

work page arXiv 2023

[2] [2]

Appeltans, S.-I

P. Appeltans, S.-I. Niculescu, and W. Michiels. Analysis and design of strongly stabilizing pid controllers for time-delay systems.SIAM Journal on Control and Optimization, 60(1):124–146, 2022. 20 7 CONCLUDING REMARKS

work page 2022

[3] [3]

K. J. ˚Astr¨ om and R. M. Murray.Feedback systems: an introduction for scientists and engineers. Princeton university press, 2021

work page 2021

[4] [4]

Breda, S

D. Breda, S. Maset, and R. Vermiglio.Stability of linear delay differential equations: A numerical approach with MATLAB. Springer, 2014

work page 2014

[5] [5]

J. Chen, D. Ma, Y. Xu, and J. Chen. Delay robustness of pid control of second-order systems: Pseu- doconcavity, exact delay margin, and performance tradeoff.IEEE Transactions on Automatic Control, 67(3):1194–1209, 2021

work page 2021

[6] [6]

L. R. da Silva, R. C. C. Flesch, and J. E. Normey-Rico. Controlling industrial dead-time systems: When to use a pid or an advanced controller.ISA transactions, 99:339–350, 2020

work page 2020

[7] [7]

Engelborghs, T

K. Engelborghs, T. Luzyanina, and D. Roose. Numerical bifurcation analysis of delay differential equations using dde-biftool.ACM Transactions on Mathematical Software (TOMS), 28(1):1–21, 2002

work page 2002

[8] [8]

T. T. Georgiou and M. C. Smith. w-stability of feedback systems.Systems & Control Letters, 13(4):271–277, 1989

work page 1989

[9] [9]

M. A. Gomez, A. V. Egorov, S. Mondi´ e, and W. Michiels. Optimization of the h 2 norm for single-delay systems, with application to control design and model approximation.IEEE Transactions on Automatic Control, 64(2):804–811, 2018

work page 2018

[10] [10]

J. K. Hale and S. M. V. Lunel. Strong stabilization of neutral functional differential equations.IMA Journal of Mathematical Control and Information, 19(1 and 2):5–23, 2002

work page 2002

[11] [11]

J. K. Hale and S. V. Lunel. Stability and control of feedback systems with time delays.International Journal of Systems Science, 34(8-9):497–504, 2003

work page 2003

[12] [12]

Ma and J

D. Ma and J. Chen. Delay margin of low-order systems achievable by pid controllers.IEEE Transactions on Automatic Control, 64(5):1958–1973, 2018

work page 1958

[13] [13]

M´ endez-Barrios, J.-D

C.-F. M´ endez-Barrios, J.-D. Torres-Garc´ ıa, and S.-I. Niculescu. Delay-difference approximations of pd- controllers. improperly-posed systems in multiple delays case.International Journal of Robust and Non- linear Control, 34(10):6431–6454, 2024

work page 2024

[14] [14]

A. H. Mhmood and M. N. Mahyuddin. H∞sliding mode control: A recent review of applications and design methods.IEEE Access, 2025

work page 2025

[15] [15]

Michiels

W. Michiels. To filter or not to filter? impact on stability of delay-difference and neutral equations with multiple delays.IEEE Transactions on Automatic Control, 68(6):3687–3693, 2022

work page 2022

[16] [16]

Michiels and S.-I

W. Michiels and S.-I. Niculescu.Stability, control, and computation for time-delay systems: an eigenvalue- based approach. SIAM, 2014

work page 2014

[17] [17]

Michiels, S.-I

W. Michiels, S.-I. Niculescu, and I. Boussaada. A complete characterization of minima of the spectral abscissa and rightmost roots of second-order systems with input delay.IMA Journal of Mathematical Control and Information, 40(3):403–428, 2023

work page 2023

[18] [18]

Michiels, S.-I

W. Michiels, S.-I. Niculescu, I. Boussaada, and G. Mazanti. On the relations between stability optimization of linear time-delay systems and multiple rightmost characteristic roots.Mathematics of Control, Signals, and Systems, 37(1):143–166, 2025

work page 2025

[19] [19]

Niculescu.Delay Effects on Stability: A Robust Control Approach

S.-I. Niculescu.Delay Effects on Stability: A Robust Control Approach. Springer, 2001

work page 2001

[20] [20]

O’Dwyer.Handbook of PI and PID Controller Tuning Rules

A. O’Dwyer.Handbook of PI and PID Controller Tuning Rules. Imperial College Press (ICP), London, 3rd edition, 2009

work page 2009

[21] [21]

Peichl, A

A. Peichl, A. Saldanha, and D. Torres-Garcia. tdcpy, 2026

work page 2026

[22] [22]

Ramirez, D

A. Ramirez, D. Breda, and R. Sipahi. A scalable approach to compute delay margin of a class of neutral-type time delay systems.SIAM Journal on Control and Optimization, 59(2):805–824, 2021

work page 2021

[23] [23]

Ram´ ırez, S

A. Ram´ ırez, S. Mondi´ e, R. Garrido, and R. Sipahi. Design of proportional-integral-retarded (pir) controllers for second-order lti systems.IEEE Transactions on Automatic Control, 61(6):1688–1693, 2015

work page 2015

[24] [24]

Sipahi, S.-I

R. Sipahi, S.-I. Niculescu, C. T. Abdallah, W. Michiels, and K. Gu. Stability and stabilization of systems with time delay.IEEE Control Systems Magazine, 31(1):38–65, 2011

work page 2011

[25] [25]

Torres-Garc´ ıa, C.-F

D. Torres-Garc´ ıa, C.-F. M´ endez-Barrios, and S.-I. Niculescu. Stabilization of second-order non-minimum phase system with delay via pi controllers. spectral abscissa optimization.IEEE Access, 2024

work page 2024

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Virtanen, R

P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- terson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henrikse...

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