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arxiv: 2606.06642 · v1 · pith:CCNTQRMGnew · submitted 2026-06-04 · 📡 eess.SY · cs.SY

MPC for nonlinear systems: a comparative review of discretization methods

Guido Sanchez , Marina Murillo , Lucas Genzelis , Nestor Deniz , Leonardo Giovanini This is my paper

Pith reviewed 2026-06-27 23:56 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords model predictive controlnonlinear systemsdiscretization methodsdirect multiple shootingdirect collocationsuccessive linearizationsnumerical methods
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The pith

Three discretization methods for nonlinear MPC are compared on characteristics and simulation performance.

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

The paper reviews direct multiple shooting, direct collocation, and successive linearizations as ways to discretize continuous nonlinear dynamics for use inside model predictive control. It describes the main features of each approach and then runs them on two test cases to measure how they perform. A reader would care because the choice of discretization affects how accurately and quickly an MPC controller can be solved in practice. The simulations supply concrete data on the relative strengths of the methods for typical nonlinear problems.

Core claim

The paper establishes that an overview of the characteristics of direct multiple shooting, direct collocation, and successive linearizations can be given, and that their performance differences can be evaluated by simulating their application to two test cases involving nonlinear model predictive control.

What carries the argument

The three discretization methods—direct multiple shooting, direct collocation, and successive linearizations—for converting continuous-time nonlinear equations into forms usable by MPC solvers.

If this is right

  • Engineers can select among the three methods according to the accuracy and speed observed in the test cases.
  • Direct multiple shooting tends to preserve more of the original nonlinear structure during optimization.
  • Successive linearizations reduce computational load at the cost of approximation error.
  • Direct collocation embeds the dynamics constraints at chosen points, affecting both feasibility and solver effort.

Where Pith is reading between the lines

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

  • The comparison could be extended by measuring real-time feasibility on embedded hardware rather than simulation alone.
  • Hybrid schemes that switch between the three methods depending on operating region might combine their advantages.
  • The results suggest that benchmark suites with more varied nonlinearity and constraint types would strengthen future reviews.

Load-bearing premise

The two chosen test cases are representative enough of typical nonlinear MPC problems that performance differences will generalize.

What would settle it

Repeating the simulations on a third nonlinear system with qualitatively different dynamics and observing a reversal in the performance ranking of the three methods would undermine the general conclusions.

Figures

Figures reproduced from arXiv: 2606.06642 by Guido Sanchez, Leonardo Giovanini, Lucas Genzelis, Marina Murillo, Nestor Deniz.

Figure 1
Figure 1. Figure 1: State trajectory x1 for the Van der Pol oscillator. seconds for multiple shooting, 0.00290 seconds for direct collocation and 0.00195 seconds for the successive linearizations method. 4.2 Second example: Continuous stirred-tank reactor In the second example, we will consider a continuous stirred-tank reactor (CSTR) taken from the work of Rawlings and Mayne [13]. Here, an irreversible, first￾order reaction,… view at source ↗
Figure 2
Figure 2. Figure 2: State trajectory x2 for the Van der Pol oscillator. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Control trajectory u1 for the Van der Pol oscillator. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Execution time of each of the methods for the Van der Pol oscillator. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: State trajectory x1 for the CSTR. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: State trajectory x2 for the CSTR. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: State trajectory x3 for the CSTR. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Control trajectory u1 for the CSTR. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Control trajectory u2 for the CSTR. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Execution time of each of the methods for the CSTR. [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
read the original abstract

This work provides a comparative review of three different numerical methods generally used to discretize continuous-time non-linear equations appearing in model predictive control problems: direct multiple shooting, direct collocation and successive linearizations. An overview of the characteristics of each method is given and the performance of each method is evaluated through the simulation of two test cases.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The paper provides a comparative review of three discretization methods for continuous-time nonlinear systems in model predictive control—direct multiple shooting, direct collocation, and successive linearizations—by first outlining the characteristics of each method and then evaluating their performance through numerical simulations on two test cases.

Significance. If the simulation comparisons are conducted fairly and the test cases prove representative, the work could offer practical guidance to MPC practitioners on method selection. The direct simulation-based evaluation is a standard and potentially useful approach in the field, though the narrow scope limits broader impact.

major comments (1)
  1. [Abstract and simulation/results section] The central performance comparison rests on simulations of only two test cases (as stated in the abstract and detailed in the simulation/results section). No argument or additional evidence is provided that these cases span key variations in nonlinear MPC problems such as state dimension, degree of nonlinearity, stiffness, or constraint structure; therefore any observed rankings rest on an untested sampling assumption and do not support generalizable conclusions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and simulation/results section] The central performance comparison rests on simulations of only two test cases (as stated in the abstract and detailed in the simulation/results section). No argument or additional evidence is provided that these cases span key variations in nonlinear MPC problems such as state dimension, degree of nonlinearity, stiffness, or constraint structure; therefore any observed rankings rest on an untested sampling assumption and do not support generalizable conclusions.

    Authors: We agree that the manuscript presents performance comparisons based solely on two test cases without providing explicit arguments or evidence that these cases are representative across dimensions such as state dimension, degree of nonlinearity, stiffness, or constraint structure. The cases were chosen as standard benchmarks from the MPC literature to demonstrate the methods, but this does not constitute a comprehensive sampling. In a revised version we will modify the abstract, introduction, and conclusions to clearly frame the results as illustrative examples rather than generalizable rankings, and we will add an explicit discussion of the scope limitations with respect to the variations mentioned by the referee. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation or evaluation chain

full rationale

The paper is a comparative review of three standard discretization methods for nonlinear MPC. It provides overviews of method characteristics and evaluates relative performance via direct simulation on two explicit test cases. No derivations, parameter fits presented as predictions, self-citations used as load-bearing premises, or ansatzes smuggled via prior work are present. The evaluation rests on external simulation benchmarks rather than reducing to the paper's own inputs by construction, satisfying the self-contained criterion for a score of 0.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the standard definitions of the three discretization methods (taken from prior literature) and the assumption that the two chosen test cases are informative for general nonlinear MPC.

pith-pipeline@v0.9.1-grok · 5579 in / 979 out tokens · 19629 ms · 2026-06-27T23:56:12.870211+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 4 canonical work pages

  1. [1]

    A survey of numerical methods for optimal control

    Anil Rao. “A survey of numerical methods for optimal control”. In:Ad- vances in the Astronautical Sciences135.1 (2009), pp. 497–528.issn: 1569- 3953.doi:10.1515/jnum-2014-0003

    work page doi:10.1515/jnum-2014-0003 2009

  2. [2]

    Fast direct multiple shooting algorithms for optimal robot control

    Moritz Diehl et al. “Fast direct multiple shooting algorithms for optimal robot control”. In:Fast motions in biomechanics and robotics. Springer, 2006, pp. 65–93

    2006

  3. [3]

    Springer, 2011

    Christian Kirches.Fast numerical methods for mixed-integer nonlinear model-predictive control. Springer, 2011

    2011

  4. [4]

    A multiple shooting algorithm for direct solution of optimal control problems

    Hans Bock and Karl-Josef Plitt. “A multiple shooting algorithm for direct solution of optimal control problems”. In: Proceedings of the IFAC World Congress. 1984

    1984

  5. [5]

    Multiple shooting method for two-point boundary value problems

    David Morrison, James Riley, and John Zancanaro. “Multiple shooting method for two-point boundary value problems”. In:Communications of the ACM5.12 (1962), pp. 613–614

    1962

  6. [6]

    A collocation method for two-point boundary value problems

    John Ahlberg and T Ito. “A collocation method for two-point boundary value problems”. In:Mathematics of Computation29.131 (1975), pp. 761– 776

    1975

  7. [7]

    Survey of direct transcription for low-thrust space trajectory optimization with applications

    F Topputo and C Zhang. “Survey of direct transcription for low-thrust space trajectory optimization with applications”. In:Abstract and Applied Analysis. Vol. 2014. Hindawi Publishing Corporation. 2014. 18

    2014

  8. [8]

    Generalized nonlinear optimal predictive control using iterative state-space trajectories: Applications to autonomous flight of UAVs

    Marina Murillo et al. “Generalized nonlinear optimal predictive control using iterative state-space trajectories: Applications to autonomous flight of UAVs”. In:International Journal of Control, Automation and Systems 13.2 (2015), pp. 361–370.issn: 2005-4092.doi:10.1007/s12555- 013- 0416-y

    work page doi:10.1007/s12555- 2015

  9. [9]

    Iterated non-linear model predictive control based on tubes and contractive constraints

    M. Murillo, G. S´ anchez, and L. Giovanini. “Iterated non-linear model predictive control based on tubes and contractive constraints”. In:ISA Transactions62 (2016), pp. 120–128.issn: 0019-0578.doi:https : / / doi.org/10.1016/j.isatra.2016.01.008

    work page doi:10.1016/j.isatra.2016.01.008 2016

  10. [10]

    Pearson education, 2002

    Jan Maciejowski.Predictive control: with constraints. Pearson education, 2002

    2002

  11. [11]

    Efficient numerical methods for nonlinear MPC and moving horizon estimation

    Moritz Diehl, Hans Ferreau, and Niels Haverbeke. “Efficient numerical methods for nonlinear MPC and moving horizon estimation”. In:Non- linear Model Predictive Control: Towards New Challenging Applications. Ed. by Lalo Magni, Davide Martino Raimondo, and Frank Allg¨ ower. Vol. 384. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. Chap. Ef- ficient, p...

    work page doi:10.1007/978-3-642-01094- 2009

  12. [12]

    Prentice-Hall, Inc., 1989

    Raymond DeCarlo.Linear systems: A state variable approach with nu- merical implementation. Prentice-Hall, Inc., 1989

    1989

  13. [13]

    Nob Hill Publishing, LLC, 2009, p

    James Rawlings and David Mayne.Model predictive control: Theory and Design. Nob Hill Publishing, LLC, 2009, p. 669.isbn: 978-0-9759377-0-9

    2009

  14. [14]

    A General-Purpose Software Framework for Dynamic Optimization

    Joel Andersson. “A General-Purpose Software Framework for Dynamic Optimization”. PhD thesis. Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center, Kasteelpark Arenberg 10, 3001- Heverlee, Belgium: Arenberg Doctoral School, KU Leuven, Oct. 2013

    2013

  15. [15]

    Risbeck and J

    M. Risbeck and J. Rawlings.MPCTools: Nonlinear model predictive con- trol tools for CasADi (Python interface).https : / / bitbucket . org / rawlings-group/mpc-tools-casadi. 2015. 19

    2015