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arxiv: 2506.17655 · v3 · pith:I5OKBNV5new · submitted 2025-06-21 · 📡 eess.SY · cs.SY

PID Tuning via Desired Step Response Curve Fitting

Senol Gulgonul This is my paper

Pith reviewed 2026-05-22 00:17 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords PID tuningstep response fittingcurve fittingnonlinear optimizationtransient response shapingcontrol systemsreference tracking
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The pith

PID controllers are tuned by minimizing the error between their closed-loop step response and a user-chosen target curve.

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

The paper proposes a tuning procedure that treats PID design as a curve-fitting task: controller gains are adjusted until the simulated plant output under step input matches a pre-specified reference trajectory as closely as possible. The fit is measured by root-mean-square error, so any combination of settling time, overshoot, and steady-state behavior can be imposed simply by drawing or selecting the target curve. Because the method works directly with time-domain simulation of the closed loop, it avoids separate frequency-domain calculations or root-locus constructions. A reader would care if the resulting gains prove at least as reliable as classical rules while letting the designer state performance goals in the most intuitive language.

Core claim

Optimal PID parameters are obtained by solving a constrained nonlinear program whose objective is the L2 distance between the closed-loop step response produced by the plant-plus-PID system and an externally supplied desired step-response curve; first-order-plus-dead-time or second-order curves with explicit settling-time and overshoot values are the usual choices for the target.

What carries the argument

The PID-SRCF optimizer, which repeatedly simulates the closed-loop step response and adjusts the three PID coefficients to drive the root-mean-square mismatch to a minimum.

If this is right

  • The same fitting procedure can be used in place of Ziegler-Nichols, lambda, pole-placement or dominant-pole rules for any plant model that can be simulated.
  • Transient specifications become direct inputs: the designer draws or parametrizes the target curve rather than deriving gain formulas.
  • An open-source MATLAB implementation already exists, so the method can be tested on any linear or mildly nonlinear plant.
  • Comparative tests in the paper indicate that the fitted responses meet or exceed the tracking accuracy of standard analytical tuners.

Where Pith is reading between the lines

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

  • The approach could be extended to tune feed-forward terms or to incorporate actuator saturation directly inside the simulation used for fitting.
  • If the plant model contains significant uncertainty, the same optimizer could be wrapped in a robust or stochastic formulation that minimizes worst-case mismatch over a set of models.
  • Real-time retuning becomes conceivable by replacing the offline optimizer with a fast gradient step that uses recent step-test data.

Load-bearing premise

The nonlinear optimizer reaches a global or sufficiently good minimum that actually reproduces the chosen transient specifications when the plant model is known accurately enough to simulate the closed loop.

What would settle it

Apply the procedure to a plant whose step response under the returned PID gains deviates substantially from the supplied target curve in an independent high-fidelity simulation or on hardware.

Figures

Figures reproduced from arXiv: 2506.17655 by Senol Gulgonul.

Figure 1
Figure 1. Figure 1: Comparison of PID-SRCF and Ziegler Nichol’s PID Tuning 4.2. Lambda Tuning Case Lambda tuning example is a FOTD process transfer function having coefficients 𝐾𝑝 = 1, T=1 and L=1. 𝑃(𝑠) = 𝐾 1 + 𝑠𝑇 𝑒 −𝑠𝐿 (16) In Lambda tuning closed loop system is also expected to be a FOTD transfer function. Closed loop time constant is expected in a range for proper tuning as 𝑇 < 𝑇𝑐𝑙 < 3𝑇. We selected the midpoint as 𝑇𝑐𝑙 = 2… view at source ↗
read the original abstract

This paper presents a PID tuning method based on step response curve fitting (PID-SRCF) that utilizes L2-norm minimization for precise reference tracking and explicit transient response shaping. The algorithm optimizes controller parameters by minimizing the root-mean-square error between desired and actual step responses. The proposed approach determines optimal PID parameters by matching any closed-loop response to a desired system step response. Practically a first-order plus time delay model or a second-order system with defined settling time and overshoot requirements are preferred. The method has open-source implementation using constrained nonlinear optimization in MATLAB. Comparative evaluations demonstrate that PID-SRCF can replace known analytical methods like Ziegler Nichols, Lambda Tuning, Pole Placement, Dominant Pole and MATLAB proprietary PID tuning applications.

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

2 major / 2 minor

Summary. The manuscript proposes PID-SRCF, a tuning procedure that optimizes PID gains by minimizing the L2-norm (RMSE) between a user-specified desired closed-loop step response and the simulated response of a known plant under PID control. The desired response is typically taken from a first-order-plus-time-delay model or a second-order system with prescribed settling time and overshoot; parameters are found via constrained nonlinear optimization (MATLAB fmincon). The paper claims that this direct curve-fitting approach yields precise transient shaping and can replace or outperform classical analytical methods such as Ziegler-Nichols, Lambda tuning, pole placement, dominant-pole design, and MATLAB's built-in PID tuner.

Significance. If the optimizer reliably attains a global minimum that exactly reproduces the target transient metrics, the method supplies an intuitive, specification-driven alternative to pole-placement or rule-based tuning for plants whose models are known to sufficient accuracy. The open-source MATLAB implementation is a concrete strength that would facilitate reproducibility and further testing.

major comments (2)
  1. [Optimization procedure (Section 3)] The central claim that L2-norm minimization produces PID parameters realizing any prescribed settling time and overshoot rests on the assumption that fmincon reaches the global minimizer of the non-convex cost surface. No multi-start statistics, basin-of-attraction analysis, Hessian evaluation at reported solutions, or convergence diagnostics from varied initial conditions are supplied to substantiate that the published performance numbers are not artifacts of favorable starting points.
  2. [Comparative results (Section 4)] In the comparative evaluations (Tables 2–4), the desired-response parameters (settling time, overshoot) used to generate the target curve for PID-SRCF are not stated explicitly for each benchmark plant, nor is it shown how the competing methods (Ziegler-Nichols, Lambda, etc.) were tuned to the same transient specifications. Without this information the reported RMSE or IAE improvements cannot be interpreted as independent evidence of superiority rather than a consequence of the chosen fitting criterion.
minor comments (2)
  1. [Abstract] The abstract asserts that the method 'can replace known analytical methods'; this phrasing should be softened to 'provides a viable alternative' or accompanied by the qualifier 'under the conditions examined'.
  2. [Method] The exact mathematical expression for the desired step-response curve (e.g., the analytic form of the FOPTD target) should be written out once in the method section to remove ambiguity about how overshoot and settling time are encoded.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript 'PID Tuning via Desired Step Response Curve Fitting'. The points raised regarding optimization reliability and transparency in comparative evaluations are important for strengthening the paper. We address each major comment below and will revise the manuscript to incorporate additional analysis and details as outlined.

read point-by-point responses

  1. Referee: [Optimization procedure (Section 3)] The central claim that L2-norm minimization produces PID parameters realizing any prescribed settling time and overshoot rests on the assumption that fmincon reaches the global minimizer of the non-convex cost surface. No multi-start statistics, basin-of-attraction analysis, Hessian evaluation at reported solutions, or convergence diagnostics from varied initial conditions are supplied to substantiate that the published performance numbers are not artifacts of favorable starting points.

    Authors: We acknowledge that the underlying cost surface is non-convex and that fmincon is a local optimizer that does not guarantee global optimality. In the original work, initial conditions were chosen using standard PID heuristics derived from plant models, and solutions consistently produced step responses closely matching the targets across the tested plants. To substantiate robustness, the revised Section 3 will include a multi-start study with 50 randomly sampled initial conditions per benchmark (within physically reasonable bounds), reporting the fraction of runs attaining the published RMSE values, the variance in converged PID gains, and basic convergence diagnostics. This will provide quantitative evidence that the reported results are not dependent on specially chosen starting points. revision: yes

  2. Referee: [Comparative results (Section 4)] In the comparative evaluations (Tables 2–4), the desired-response parameters (settling time, overshoot) used to generate the target curve for PID-SRCF are not stated explicitly for each benchmark plant, nor is it shown how the competing methods (Ziegler-Nichols, Lambda, etc.) were tuned to the same transient specifications. Without this information the reported RMSE or IAE improvements cannot be interpreted as independent evidence of superiority rather than a consequence of the chosen fitting criterion.

    Authors: We agree that explicit documentation of the target transient specifications and the tuning procedures applied to baseline methods is required for unambiguous interpretation. In the revised manuscript we will add a dedicated table (or expanded caption for Tables 2–4) listing the exact desired settling time and overshoot used to construct the reference step response for each plant. We will also describe the concrete tuning rules employed for each competing method (e.g., Ziegler-Nichols ultimate-gain formulas, Lambda tuning with a specific lambda value, pole-placement pole locations) and note how these were selected to achieve the closest feasible match to the same transient metrics. These additions will allow readers to evaluate whether the observed improvements arise from the direct curve-fitting objective or from differences in specification adherence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method is direct optimization to user target

full rationale

The paper proposes an explicit numerical procedure: select a desired step-response shape (e.g., first-order-plus-delay or second-order with prescribed settling time/overshoot), then minimize the sampled L2 error between that target and the closed-loop response produced by the plant under candidate PID gains using constrained nonlinear optimization (MATLAB fmincon). This is a standard fitting definition of optimality rather than a derivation that reduces to its own inputs. Comparative tables against Ziegler-Nichols, Lambda tuning, pole placement, and MATLAB's pidtune supply external benchmarks. No self-citations, uniqueness theorems, or ansatzes imported from prior author work appear in the load-bearing steps. The approach is therefore self-contained as an engineering algorithm whose performance can be falsified by the reported numerical comparisons.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method depends on user-chosen desired-response parameters and the assumption that the plant model permits accurate closed-loop simulation; no new physical entities are introduced.

free parameters (1)
  • Desired-response parameters (settling time, overshoot, time constant)
    User-specified values that define the target curve and therefore directly determine the optimization objective.
axioms (1)
  • domain assumption The plant dynamics are known sufficiently well to simulate the closed-loop step response under candidate PID parameters.
    Invoked implicitly when the optimizer evaluates the error between desired and actual responses.

pith-pipeline@v0.9.0 · 5639 in / 1386 out tokens · 76776 ms · 2026-05-22T00:17:22.808924+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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