Piecewise Linear Approximation and PID Control Optimization for Nonlinear Systems

Robert Vrabel

REVIEW 2 major objections 2 minor 17 references

Piecewise Linear Approximation and PID Control Optimization for Nonlinear Systems

T0 review · 2 major / 2 minor · reviewed 2026-05-23 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read PID parameters tuned on a piecewise linear approximation of a nonlinear system produce stable closed-loop behavior on the true nonlinear plant.

desk verdict This paper combines two standard techniques—piecewise linear approximation and PSO-tuned PID—without new theory or detailed validation of the transfer claim. read the letter →

arxiv 2501.00820 v2 submitted 2025-01-01 math.OC

Robert Vrabel This is my paper

classification math.OC

keywords piecewiselinearapproximationPIDcontrolnonlinearsystemsparticleswarmoptimizationITAEcriterionISOfrequencydomainanalysis

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 establishes that nonlinear system dynamics can be partitioned into linear segments over a compact domain, creating locally linear models that support standard transfer-function and frequency-domain tools. PID controller gains are then optimized on these models using particle swarm optimization with ITAE and ISO performance indices. Simulations show the resulting controllers deliver stable and accurate regulation when applied directly to the original nonlinear equations. The method trades off the number of segments against computational cost while preserving acceptable closed-loop performance. A reader would care because the approach supplies a concrete route for applying classical linear control design to nonlinear plants without requiring entirely new nonlinear synthesis methods.

What carries the argument

The piecewise linear approximation obtained by partitioning the nonlinear function into subregions over a compact domain, which converts the plant into a form amenable to transfer-function analysis and PID optimization.

What would settle it

A simulation in which the PID controller tuned on the piecewise linear model produces instability, excessive overshoot, or large steady-state error when applied to the original nonlinear equations.

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Extended reading notes

Core claim

Partitioning the nonlinear function into subregions yields an analytically tractable piecewise linear representation whose behavior converges to the original system as the number of segments grows. On these models, particle swarm optimization tunes PID parameters according to ITAE and ISO criteria. Numerical tests confirm that the optimized gains ensure stable and accurate control when transplanted to the true nonlinear dynamics.

Load-bearing premise

The piecewise linear model must be faithful enough that parameters optimized on it continue to stabilize and regulate the true nonlinear system without retuning or instability.

Editorial extensions

If this is right

As the number of linear segments increases, the approximated dynamics approach those of the original nonlinear system.
Frequency-domain and transfer-based analysis tools become applicable to the approximated models.
Particle swarm optimization using ITAE and ISO criteria produces controller parameters that transfer successfully to the nonlinear plant.
The method achieves a practical balance between computational effort and approximation accuracy.

Reading between the lines

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

The same partitioning idea could be used to design linear observers or state estimators that are later applied to the nonlinear plant.
Adaptive switching among different segment counts could be investigated for systems whose operating range changes over time.
The approach might be tested on hardware where the piecewise model runs in a digital controller while the plant remains continuous and nonlinear.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The one thing to know about this paper is that it takes two established methods—piecewise linear approximation of nonlinear functions and Particle Swarm Optimization for tuning PID controllers—and applies them in sequence to design controllers for nonlinear systems. There is no new mathematical framework or algorithm here. The paper does a decent job of explaining the process. It partitions the domain of the nonlinear function into subregions, approximates each with a linear model, and then optimizes the PID gains on this collection of linear models using PSO with ITAE and ISO criteria. It notes that more segments bring the approximation closer to the original system, which allows frequency-domain tools to be used. This is a reasonable practical approach for engineers who need to handle nonlinear plants with linear control techniques. Where it falls short is in the supporting evidence. The abstract claims that numerical simulations show the parameters transfer successfully to the nonlinear system, but it supplies no quantitative results, no error bounds on the approximation, no comparison with direct tuning on the nonlinear system or other methods, and no specifics on the test cases. Without this, it's impossible to tell how well the method actually works or how sensitive it is to the number of segments chosen. The key assumption is that the piecewise linear model is accurate enough that the optimized controller will be stable and perform well on the real system. The paper appears to rest on simulation results to support this, but the lack of detail leaves that open. This kind of work is for control practitioners looking for straightforward implementation ideas rather than for researchers seeking theoretical advances. A reader might find the workflow useful as an example, but the paper does not engage deeply with the literature on approximation errors or stability guarantees for such switched linear systems. I would not bring this to a reading group. I would not cite it. It does not seem ready for peer review because the central claim about successful transfer lacks the necessary supporting data and analysis to be evaluated properly.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a piecewise-linear approximation framework for nonlinear systems that partitions the nonlinearity into subregions to obtain locally linear models. PID controller parameters are then optimized on this approximation via Particle Swarm Optimization using ITAE and ISO performance indices; the central claim is that the resulting parameters transfer to the original nonlinear plant and yield stable, accurate closed-loop behavior, as purportedly verified by numerical simulations.

Significance. If the simulation evidence were supplied with quantitative metrics, baselines, and explicit error bounds, the approach would offer a concrete route for applying frequency-domain and transfer-function tools to nonlinear plants while controlling the trade-off between approximation fidelity and computational cost. The explicit use of PSO on the piecewise-linear surrogate and the progressive convergence claim with increasing segment count are potentially useful contributions to practical nonlinear PID design.

major comments (2)

[Abstract / simulation results] Abstract and simulation-results section: the claim that 'controller parameters obtained from the piecewise linear model ensure stable and accurate control when applied to the original nonlinear system' is asserted without any reported quantitative metrics (overshoot, settling time, ITAE values on the true plant), baseline comparisons (e.g., direct PSO on the nonlinear system or Ziegler-Nichols), error bounds on the approximation, or even a description of the test systems, rendering the central transfer claim unevaluable.
[Abstract / simulation results] The weakest assumption—that the piecewise-linear model remains sufficiently faithful that PID parameters optimized on it require no retuning or produce instability on the true nonlinear plant—is load-bearing for the entire contribution yet is supported only by the qualitative statement in the abstract; no explicit validation (e.g., comparison of closed-loop poles or frequency responses between the two models) is referenced.

minor comments (2)

Clarify the precise partitioning procedure (e.g., uniform vs. adaptive breakpoints) and the exact definition of the ISO index in the optimization objective.
Add a table or figure that reports the number of segments, PSO hyperparameters, and resulting ITAE/ISO values for each example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We agree that the simulation results require additional quantitative support and explicit validation to make the transfer claim fully evaluable, and we will revise the manuscript to address these points.

read point-by-point responses

Referee: [Abstract / simulation results] Abstract and simulation-results section: the claim that 'controller parameters obtained from the piecewise linear model ensure stable and accurate control when applied to the original nonlinear system' is asserted without any reported quantitative metrics (overshoot, settling time, ITAE values on the true plant), baseline comparisons (e.g., direct PSO on the nonlinear system or Ziegler-Nichols), error bounds on the approximation, or even a description of the test systems, rendering the central transfer claim unevaluable.

Authors: We acknowledge that the current manuscript presents the simulation results primarily through qualitative statements and does not include the requested quantitative metrics, baseline comparisons, approximation error bounds, or explicit test-system descriptions. In the revised version we will expand the simulation section to report overshoot, settling time, and ITAE values evaluated on the original nonlinear plant, add comparisons against direct PSO optimization on the nonlinear system and Ziegler-Nichols tuning, provide approximation-error bounds as a function of segment count, and give a clear description of the benchmark systems used. revision: yes
Referee: [Abstract / simulation results] The weakest assumption—that the piecewise-linear model remains sufficiently faithful that PID parameters optimized on it require no retuning or produce instability on the true nonlinear plant—is load-bearing for the entire contribution yet is supported only by the qualitative statement in the abstract; no explicit validation (e.g., comparison of closed-loop poles or frequency responses between the two models) is referenced.

Authors: We agree that explicit validation of model fidelity is needed to support the transfer claim. The revised manuscript will include side-by-side comparisons of closed-loop step responses, settling behavior, and (where analytically feasible) pole locations or frequency-response characteristics between the piecewise-linear surrogate and the original nonlinear plant, thereby demonstrating that the optimized PID parameters remain stabilizing without retuning. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper approximates nonlinear dynamics via piecewise linear partitioning, optimizes PID gains via an external PSO algorithm using ITAE/ISO criteria on the approximation, and validates transfer to the original plant via numerical simulation. No derivation step reduces by construction to its inputs, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on self-citation chains. The argument remains self-contained against external benchmarks.

Assumptions & free parameters 2 free parameters · 1 assumptions · 0 invented entities

The abstract invokes standard approximation and control assumptions without stating new entities or fitted constants beyond the implicit choice of segment count and PSO hyperparameters.

free parameters (2)

number of linear segments
Determines approximation accuracy versus computational cost; value not reported in abstract.
PSO hyperparameters
Swarm size, inertia, and cognitive/social coefficients required for the optimizer but not specified.

assumptions (1)

domain assumption The nonlinear function is continuous on a compact domain so that it can be partitioned into subregions with well-defined linear approximations.
Required for the piecewise construction described in the abstract.

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Cite this review

Pith. "Pith review of Piecewise Linear Approximation and PID Control Optimization for Nonlinear Systems." pith.science (2026). https://pith.science/paper/2501.00820

@misc{pith2026250100820,
  author       = {Pith},
  title        = {Pith review of: Piecewise Linear Approximation and PID Control Optimization for Nonlinear Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2501.00820}},
  note         = {Machine review of arXiv:2501.00820}
}

read the original abstract

This paper investigates the control of nonlinear systems using a piecewise linear approximation framework. The proposed approach combines a PID controller with locally linearized models obtained by partitioning the nonlinear function into subregions over a compact domain. This approximation yields an analytically tractable representation of the system dynamics, enabling the application of transfer-based and frequency-domain analysis tools that are not directly applicable to nonlinear systems. As the number of linear segments increases, the approximated system progressively approaches the behavior of the original nonlinear system, allowing for a meaningful frequency-domain interpretation of the dynamics. The PID controller parameters are optimized using the Particle Swarm Optimization method with performance criteria based on ITAE (Integral of Time-weighted Absolute Error) and ISO (Integral of Squared Overshoot). Numerical simulations confirm the effectiveness of the proposed method, demonstrating that controller parameters obtained from the piecewise linear model ensure stable and accurate control when applied to the original nonlinear system, while maintaining a balance between computational effort and approximation accuracy.

Figures

Figures reproduced from arXiv: 2501.00820 by the authors.

**Figure 2.** Convergence of piecewise linear approximation to the original nonlinear control [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** Dirac delta with σ = 0.01 − − − − [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: C(s) = 3.72 + 10.0 s + 0.0s. PID-Controlled Output (PSO), ITAE+ISO criterion (α = 2000): error 0.1327, 5 iterations, swarm size=30, (Kp,h, Ki,h, Kd,h) ∈ [0, 10]3 . These issues are typically related to how the Dirac delta and its derivative are handled in the Laplace …

**Figure 8.** Figure 8: C(s) = 4.65 + 10 s + 0.0s. PID-Controlled Output (PSO), ITAE+ISO criterion (α = 2000): error 0.1121, 10 iterations, swarm size=30, (Kp,h, Ki,h, Kd,h) ∈ [0, 10]3 . The overall piecewise linear approximation is denoted as Linh(y) ≜ {Linh=6,i(y)| i = 1, 2, . . . , 6}. 32…

**Figure 8.** Figure 8: Expanding D dynamically ensures that the simulation remains valid for a wide range of system responses. However, it also increases computational complexity. Choosing an appropriate initial domain D is important to minimize unnecessary expansions. Dividing D into 6 equ…

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Pith tools

Reviewed May 23, 2026 · model on record in the stance chip above.

[1] [1]

K. J. Astrom and T. Hagglund. PID Controllers: Theory, Design, and Tuning. ISA - The Instrumentation, Systems and Automation Society, Research Triangle Park, North Carolina, 1995

work page 1995

[2] [2]

K. J. Astrom and T. Hagglund. Advanced PID Control. ISA-The Instru- mentation, Systems, and Automation Society, Research Triangle Park, North Carolina, 2006

work page 2006

[3] [3]

S. Banach. Sur les operations dans les ensembles abstraits et leur ap- plication aux equations integrales. Mathematische Annalen, 98(1):1–88, 1922

work page 1922

[4] [4]

R. L. Burden and J. D. Faires. Numerical Analysis. Brooks/Cole, Cen- gage Learning, Boston, MA, 9th edition, 2010. Youngstown State Uni- versity

work page 2010

[5] [5]

C. A. Coello Coello and M. S. Lechuga. Mopso: A proposal for multi- ple objective particle swarm optimization. In Proceedings of the IEEE Congress on Evolutionary Computation , pages 1051–1056, 2002

work page 2002

[6] [6]

Deb and H

K. Deb and H. Jain. An evolutionary many-objective optimization algo- rithm using reference-point-based nondominated sorting approach, part i: Solving problems with box constraints. IEEE Transactions on Evo- lutionary Computation, 18(4):577–601, 2013. 34

work page 2013

[7] [7]

R. C. Eberhart and J. Kennedy. Particle swarm optimization. In Pro- ceedings of IEEE International Conference on Neural Networks , pages 1942–1948, 1995

work page 1942

[8] [8]

T. H. Gronwall. Note on the derivatives with respect to a parame- ter of the solutions of a differential equation. Annals of Mathematics , 20(1):292–296, 1919

work page 1919