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arxiv: 2604.21294 · v3 · submitted 2026-04-23 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Analytical PI Tuning for Second-Order Plants with Monotonic Response and Minimum Settling Time

Senol Gulgonul
Authors on Pith no claims yet

Pith reviewed 2026-05-15 07:25 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords PI controller tuningsecond-order plantsmonotonic step responseminimum settling timepole placementpole-zero cancellationmaximum sensitivityanalytical tuning
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The pith

Two closed-form PI tunings cover all second-order plants with real poles and deliver the minimum settling time for monotonic response.

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

The paper derives two analytical PI controller formulas for second-order plants with distinct real poles. One formula cancels the slower plant pole and reduces the closed loop to a critically damped second-order system. The second formula, used when the pole ratio is less than two, places all three closed-loop poles at the same real location without cancellation. The two formulas meet exactly at the pole ratio of two and together give a single continuous tuning rule for every possible plant pole ratio. The same derivation shows that the maximum sensitivity of the closed-loop transfer function a^n/(s+a)^n depends only on the integer n and is therefore constant for any choice of a.

Core claim

For any second-order plant with real poles the PI controller parameters can be written in closed form so that the step response is strictly monotonic and reaches its final value in the shortest possible time. When the ratio of the two plant poles is greater than or equal to two, the slower pole is cancelled by the controller zero and the remaining closed-loop poles are placed at equal real locations. When the ratio is less than two, all three closed-loop poles are placed at the same real value without cancellation, producing a transfer function with a triple real pole and one zero; this choice yields a strictly shorter settling time than the cancellation method inside its validity region. At

What carries the argument

Piecewise PI parameter selection that switches between pole-zero cancellation to a critically damped second-order loop and triple real-pole placement at a common location, with the switch occurring exactly at plant pole ratio two.

If this is right

  • Every second-order plant with real poles possesses an explicit PI tuning that produces a monotonic step response of provably minimum settling time.
  • The tuning law is continuous across the entire range of pole ratios because the two formulas coincide when the ratio equals two.
  • Closed-loop systems whose transfer function takes the form a^n/(s+a)^n have a maximum sensitivity Ms that is fixed for each integer n and independent of the speed parameter a.
  • Numerical checks on multiple plant pole placements confirm both the monotonicity and the exact settling-time values predicted by the formulas.

Where Pith is reading between the lines

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

  • The universal Ms constants for each n could be used directly as robustness specifications when the same closed-loop pole pattern is applied to higher-order or cascaded loops.
  • Because the formulas are algebraic, they can be embedded as direct calculations inside industrial controllers without requiring online optimization or iteration.
  • The monotonicity guarantee may allow the same tuning structure to be applied safely to plants whose step responses must never reverse direction, such as certain temperature or level control loops.
  • Scaling the closed-loop bandwidth a leaves relative stability margins unchanged for this specific pole pattern, which may simplify gain-scheduling design.

Load-bearing premise

The plant is known exactly as a second-order linear system with two distinct real poles and contains no unmodeled dynamics, time delays, actuator limits, or nonlinearities.

What would settle it

Measure the settling time and overshoot of a physical or simulated second-order plant driven by the proposed PI tuning; if the response exhibits overshoot or if settling time exceeds the analytically predicted minimum for that plant, the claim is falsified.

Figures

Figures reproduced from arXiv: 2604.21294 by Senol Gulgonul.

Figure 1
Figure 1. Figure 1: Step responses for all six plants under the proposed PI tuning. All responses are [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Nyquist diagram of T(jω) for all six plants. All curves lie on the Mt = 1 unit circle, [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

This study presents two analytical closed-form PI controller tuning solutions for second-order plants with real poles, each achieving monotonic step response and minimum settling time. The first solution employs pole-zero cancellation, placing the controller zero at the slower plant pole and reducing the closed-loop dynamics to a critically damped second-order system. The second solution, applicable when the plant pole ratio is less than two, places all three closed-loop poles at a common location without cancelling any plant pole, yielding a closed-loop transfer function with a triple real pole and a zero. Despite retaining a closed-loop zero, this solution achieves strictly faster settling time than the pole-zero cancellation method in its region of applicability. The two solutions coincide at the boundary pole ratio of two and together form a continuous piecewise-analytical tuning covering the full range of plant pole ratios. This study further establishes that closed-loop transfer functions of the form a^n/(s + a)^n possess a maximum sensitivity Ms that is independent of the pole location a and depends solely on the order n, yielding universal robustness constants for each n. Numerical verification confirms the analytical results across multiple plant 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.

Referee Report

0 major / 2 minor

Summary. The paper presents two analytical closed-form PI controller tuning solutions for second-order plants with real poles, each achieving monotonic step response and minimum settling time. The first uses pole-zero cancellation to place the controller zero at the slower plant pole, reducing the closed-loop dynamics to a critically damped second-order system. The second, for plant pole ratios less than two, places all three closed-loop poles at a common location without cancellation, yielding a triple real pole closed-loop transfer function with a zero that settles strictly faster. The two solutions coincide continuously at pole ratio two. The paper also establishes that closed-loop transfer functions of the form a^n/(s+a)^n have maximum sensitivity Ms independent of a and dependent only on n, providing universal robustness constants.

Significance. If the derivations hold, the work supplies explicit, non-iterative tuning formulas that guarantee monotonicity and minimal settling time for the stated plant class, together with parameter-free robustness measures via the Ms result. This is valuable for analytical control design where optimization-based tuning is impractical, and the numerical verification across configurations strengthens the algebraic claims.

minor comments (2)
  1. [§2] §2 (plant and controller definitions): the pole ratio r = p2/p1 should be introduced with an explicit inequality direction (e.g., r > 1) to avoid ambiguity when stating the applicability regions of the two tunings.
  2. [Numerical verification] The numerical verification section would benefit from a table summarizing settling-time ratios between the two methods for several r values near the boundary r=2, to make the continuity and improvement claims immediately quantifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The referee summary accurately reflects the paper's contributions on the two analytical PI tuning methods and the universal Ms result for repeated-pole closed-loop systems.

read point-by-point responses

  1. Referee: The paper presents two analytical closed-form PI controller tuning solutions for second-order plants with real poles, each achieving monotonic step response and minimum settling time. The first uses pole-zero cancellation to place the controller zero at the slower plant pole, reducing the closed-loop dynamics to a critically damped second-order system. The second, for plant pole ratios less than two, places all three closed-loop poles at a common location without cancellation, yielding a triple real pole closed-loop transfer function with a zero that settles strictly faster. The two solutions coincide continuously at pole ratio two. The paper also establishes that closed-loop transfer functions of the form a^n/(s+a)^n have maximum sensitivity Ms independent of a and dependent only on n, providing universal robustness constants.

    Authors: We appreciate the referee's accurate and concise summary of the main results. No specific technical concerns, requests for additional proofs, or suggestions for changes were raised. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivations are direct algebraic pole placements

full rationale

The paper solves the PI tuning problem by direct algebraic manipulation of the closed-loop characteristic equation for a known second-order plant G(s) = 1/((s+p1)(s+p2)). The pole-zero cancellation rule places the controller zero at the slower pole and sets the remaining closed-loop poles equal for critical damping; the triple-pole rule equates the three closed-loop poles without cancellation when the plant pole ratio <2. Both rules are obtained by solving the resulting polynomial coefficient equations for the PI gains, with continuity enforced at ratio=2. The Ms independence result follows immediately from the normalized form T(s) = a^n/(s+a)^n by substituting s = a·jω and observing that the magnitude depends only on n. No parameters are fitted to data, no predictions are made from subsets of the same data, and no load-bearing steps rely on self-citations or prior ansatzes. All steps remain self-contained within the stated modeling assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters are introduced; the tunings are expressed directly in terms of the two known plant poles. Relies on standard linear control assumptions.

axioms (2)
  • domain assumption The plant is exactly second-order with two distinct real poles that are known precisely.
    Required for the pole-placement derivations to apply without modification.
  • domain assumption Monotonic step response is achieved by placing all closed-loop poles on the real axis.
    Standard assumption in linear control for eliminating overshoot.

pith-pipeline@v0.9.0 · 5493 in / 1283 out tokens · 82926 ms · 2026-05-15T07:25:07.954176+00:00 · methodology

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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    Tuning PI controllers to achieve satisfactory transient response has been an active research a rea for decades

    Introduction The proportional-integral (PI) controller remains the most widely used feedback controller in industrial process control due to its simplicity and effectiveness. Tuning PI controllers to achieve satisfactory transient response has been an active research a rea for decades. The first systematic tuning method was proposed by Ziegler and Nichols...

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    The plant is controlled by a PI controller 𝐶(𝑠) = 𝐾 (1 + 1 𝑠𝑇𝑖 ) (2) in a unity feedback configuration, where K > 0 is the controller gain and T i > 0 is the integral time

    Problem Statement Consider a stable second-order plant with transfer function 𝑃(𝑠) = 𝐾𝑝 (1 + 𝑠𝑇1)(1 + 𝑠𝑇2) (1) where Kp > 0 is the plant gain, and T1 ≥ T2 > 0 are the plant time constants, with slow pole at s = −1/T1 and fast pole at s = −1/T2. The plant is controlled by a PI controller 𝐶(𝑠) = 𝐾 (1 + 1 𝑠𝑇𝑖 ) (2) in a unity feedback configuration, where K ...

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    In the most general case, these roots are either three real poles or one real pole and a complex conjugate pair

    Analytical Solution The closed-loop characteristic polynomial (4) has three roots. In the most general case, these roots are either three real poles or one real pole and a complex conjugate pair. Both configurations are consistent with the Routh-Hurwitz stability conditions for positive K and Ti. From Vieta's formulas applied to (4), the three closed-loop...

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    Matching (4) to this form requires three degrees of freedom, but the PI controller provides only two

    (9) where 𝛼, 𝜁, and 𝜔0 are design parameters. Matching (4) to this form requires three degrees of freedom, but the PI controller provides only two. To reconcile this mismatch, the system order must be reduced. The PI controller introduces a zero at 𝑠 = −1/𝑇𝑖. Choosing 𝑇𝑖 = 𝑇1places this zero at the slow plant pole, yielding exact pole–zero cancellation. S...

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    Universal Robustness Properties After cancellation of the slow pole, the loop transfer function becomes 𝐿(𝑠)= 𝐶(𝑠)𝑃(𝑠)= KKp 𝑠𝑇1(1 + 𝑠𝑇2) (15) Substituting the optimal PI parameters from (12), 𝐾 = 𝑇1 4𝐾𝑝𝑇2 , 𝑇𝑖 = 𝑇1, yields 𝐿(𝑠)= 1 4𝑇2𝑠(1 + 𝑠𝑇2) (16) 4.1 Complementary Sensitivity The complementary sensitivity is 𝑇𝑐(𝑠)= 𝐿(𝑠) 1 + 𝐿(𝑠)= 1 4𝑇2 2𝑠2 + 4𝑇2𝑠 + 1 =...

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    Substituting back, ∣ 𝑆(𝑗𝜔)∣2= 16 ⋅ 1 2 ⋅ 3 2 (1 + 4 1 2) 2 = 12 9 (23) Hence, 𝑀𝑠 = √12 9 = 2 √3 ≈ 1.155 (24) 4.3 Phase Margin The phase margin is determined at the gain crossover frequency, where ∣ 𝐿(𝑗𝜔)∣= 1. From (15), 𝐿(𝑗𝜔)= 1 4𝑇2 𝑗𝜔(1 + 𝑗𝜔𝑇2) (25) The magnitude is ∣ 𝐿(𝑗𝜔)∣= 1 4𝑇2 𝜔√1 + (𝜔𝑇2)2 (26) Setting ∣ 𝐿(𝑗𝜔)∣= 1gives the gain crossover condition 4...

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    For each plant the controller parameters are computed directly from the closed-form expressions without any numerical optimization

    Numerical Verification The proposed tuning formulas (12) are verified on six second-order plants with different time constant ratios T1/T2. For each plant the controller parameters are computed directly from the closed-form expressions without any numerical optimization. The closed-loop step response is simulated and the performance metrics are recorded. ...

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    The optimal controller parameters K=T1/(4KpT2) and Ti=T1 are determined solely by the plant time constants T 1, T2 and plant gain Kp

    Conclusion A closed -form analytical PI tuning method is presented that guarantees monotonic step response with minimum settling time for second -order plants . The optimal controller parameters K=T1/(4KpT2) and Ti=T1 are determined solely by the plant time constants T 1, T2 and plant gain Kp. A further result is established: the proposed tuning yields un...

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