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

Senol Gulgonul

arxiv: 2604.21294 · v5 · pith:JNVWINJDnew · submitted 2026-04-23 · 📡 eess.SY · cs.SY

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

Senol Gulgonul This is my paper

Pith reviewed 2026-05-21 09:36 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords PI controller tuningsecond-order plantsmonotonic step responseminimum settling timepole placementgain marginrobustness marginsanalytical design

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The pith

Two analytical PI tuning formulas achieve monotonic step responses with minimum settling time for second-order plants with real poles.

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

The paper develops two explicit formulas for setting the gains of a PI controller on a second-order plant that has two different real poles. One formula cancels the slower pole with the controller zero and reduces the closed loop to a critically damped second-order system. The other formula, useful when the poles differ by a factor less than two, collocates all three closed-loop poles at one location without cancellation and produces a faster monotonic response despite retaining a zero. The two formulas meet smoothly at a pole ratio of two and together give a complete analytical tuning rule for every such plant. The work also shows that the family of closed-loop systems a^n/(s+a)^n has maximum sensitivity, phase margin, and gain margin that stay constant regardless of the pole location a and depend only on the order n.

Core claim

This paper shows that for second-order plants with distinct real poles, there are two closed-form PI tuning solutions that guarantee a monotonic closed-loop step response and minimum settling time. The first uses pole-zero cancellation to obtain a critically damped second-order system. The second collocates the three closed-loop poles at one location without cancellation when the pole ratio is less than two, yielding a triple-pole transfer function with one zero that settles faster than the cancellation approach. These solutions form a continuous tuning rule across all pole ratios. It further establishes that the closed-loop systems a^n/(s+a)^n have maximum sensitivity, phase margin, and a n

What carries the argument

The two pole-placement strategies for the closed-loop characteristic equation (cancellation to a critically damped pair versus triple-pole collocation) together with the scale-invariant robustness properties of the monic all-pole family a^n/(s+a)^n.

If this is right

The resulting controller gains are given directly by algebraic expressions in the two plant pole locations.
When the plant poles differ by a factor less than two the non-cancellation method produces a strictly shorter settling time while keeping the response monotonic.
The robustness margins for the chosen closed-loop dynamics remain identical for any pole location and can be precomputed from order alone.
The piecewise definition supplies a complete analytical tuning procedure that covers the full range of second-order plants with real poles.

Where Pith is reading between the lines

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

The formulas could serve as initial analytical seeds for tuning plants that include small unmodeled dynamics or transport delays.
The constant robustness margins suggest that similar repeated-pole patterns may produce predictable margins in other controller structures.
Testing the tunings on plants whose poles are slightly complex would reveal how sensitive the monotonicity guarantee is to modeling error.

Load-bearing premise

The plant must be exactly second-order with two distinct real poles and the design goal must be strictly monotonic step response with minimum settling time under PI control.

What would settle it

A different PI tuning on any second-order real-pole plant that yields a strictly monotonic response with shorter settling time than either proposed method, or a measured gain margin that differs from the formula 1 + sec^n(π/n) for the corresponding closed-loop order n.

Figures

Figures reproduced from arXiv: 2604.21294 by Senol Gulgonul.

**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 together with phase margin and gain margin that are independent of the pole location a and depend solely on the order n, yielding universal robustness constants for each n. A closed-form expression GM(n) = 1 + sec^n(pi/n) is established for the gain margin of the family. 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.

Desk Editor's Note private letter to a colleague

This paper supplies two explicit closed-form PI tunings for monotonic responses on second-order real-pole plants plus a clean gain-margin formula that holds for the whole repeated-pole family.

read the letter

The main thing to know is that the authors give two analytical PI tuning rules for second-order plants with distinct real poles that both produce monotonic step responses. The first cancels the slower pole and leaves a critically damped second-order closed loop. The second puts a triple real pole (plus a zero) when the pole ratio is below two. The rules meet at ratio two and together cover the full range. They also show that any closed loop of the form a^n/(s+a)^n has gain margin 1 + sec^n(π/n) that does not depend on a, and they give the corresponding Ms and phase margin as well. Numerical checks are said to back the formulas up across several plants. That is the concrete contribution. The derivations sit on standard pole-placement algebra and frequency-domain calculations, so they are internally consistent and the robustness result follows directly once you write the loop transfer function. For someone who wants an analytical starting point instead of running an optimizer or Ziegler-Nichols, these expressions are usable. The main limitations are the narrow scope and the modeling assumptions. Everything is restricted to exactly second-order plants with real poles, perfect model knowledge, and strictly monotonic responses. The minimum-settling-time claim is therefore the minimum among the monotonic PI options they consider, not a global optimum. No comparison appears against other controller structures or against responses that allow a small overshoot. The numerical verification is mentioned but the paper does not report how many plants were tested or what error tolerances were used, so the strength of that check is hard to judge from the abstract alone. This work is aimed at control engineers who tune PI loops on simple process plants and prefer closed-form rules they can implement without software. A reader in industrial or process control would get direct value from the formulas. It is worth sending out for peer review because the math is straightforward, the claims are verifiable, and the robustness result is neat enough to be checked by others.

Referee Report

0 major / 3 minor

Summary. The paper presents two analytical closed-form PI tuning rules for second-order plants with distinct real poles that guarantee monotonic closed-loop step responses and minimum settling time within the PI class. The first cancels the slower pole to obtain a critically damped second-order closed loop; the second places a triple real pole (with LHP zero) when the pole ratio is below two. The rules coincide at ratio two and cover the full range. It further shows that the family T(s) = a^n/(s+a)^n has maximum sensitivity, phase margin, and gain margin independent of a, with the explicit formula GM(n) = 1 + sec^n(π/n) derived from the phase-crossover condition ω/a = tan(π/n). Numerical verification is stated to confirm both the tuning formulas and the robustness expressions.

Significance. If the derivations hold, the work supplies practical, non-iterative tuning formulas that enforce monotonicity and settling-time optimality for a common plant class, together with parameter-independent robustness benchmarks for the all-real-pole closed-loop family. These results are directly usable in industrial control design and provide a clean benchmark for comparing robustness of other controllers.

minor comments (3)

[Abstract and §1] The abstract and introduction should state the plant model explicitly (e.g., G(s) = 1/((s+p1)(s+p2)), p1 < p2) and the precise definition of settling time used for the optimality claim.
[Numerical verification] The numerical verification section should report the exact range of pole ratios tested, the quantitative error metric between analytical and simulated settling times, and any cases where monotonicity was marginally violated.
[§3.2] Clarify whether the triple-pole placement remains strictly monotonic for all pole ratios below two or only for ratios sufficiently less than two; a short proof sketch or counter-example check would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the assessment of its significance, and the recommendation for minor revision. The referee's description accurately reflects the analytical tuning rules, the piecewise coverage of pole ratios, and the universal robustness results for the all-real-pole closed-loop family. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations are standard analytical pole placement and frequency-domain algebra

full rationale

The paper derives the two PI tuning rules via explicit pole-placement constructions: one cancels the slower plant pole to force a critically damped closed-loop pair, the other places a triple real pole (plus LHP zero) when the pole ratio <2. These are algebraic solutions to the desired closed-loop characteristic equation and monotonicity constraint, not reductions to fitted inputs or self-definitions. The GM(n) = 1 + sec^n(π/n) expression follows directly from solving for the phase-crossover frequency of L(s) = a^n / [(s+a)^n - a^n], yielding ω_pc/a = tan(π/n) and the magnitude relation by trigonometry; this is an independent frequency-domain calculation for the given family, not imported via self-citation or ansatz. No load-bearing self-citations, uniqueness theorems, or renamings of known results are present. The entire chain is self-contained against external benchmarks of pole placement and Bode analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text. The derivations presumably rest on standard linear control assumptions such as perfect plant model and linear time-invariant dynamics.

pith-pipeline@v0.9.0 · 5757 in / 1207 out tokens · 33963 ms · 2026-05-21T09:36:38.665115+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The optimal PI parameters are K=T1/(4KpT2), Ti=T1 ... closed-loop transfer function reduces to T(s)=1/(1+2T2 s)^2
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

closed-loop transfer functions of the form a^n/(s+a)^n ... GM(n)=1+sec^n(π/n)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

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...

work page
[2]

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 ...

work page
[3]

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...

work page
[4]

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...

work page
[5]

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 =...

work page
[6]

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...

work page
[7]

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. ...

work page
[8]

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...

work page
[9]

Optimum settings for automatic controllers

Ziegler JG, Nichols NB. Optimum settings for automatic controllers. Trans ASME. 1942;64:759-768

work page 1942
[10]

Refinements of the Ziegler-Nichols tuning formula

Hang CC, Astrom KJ, Ho WK. Refinements of the Ziegler-Nichols tuning formula. IEE Proc D Control Theory Appl. 1991;138(2):111-118

work page 1991
[11]

Tuning PI controllers for integrator/dead time processes

Tyreus BD, Luyben WL. Tuning PI controllers for integrator/dead time processes. Ind Eng Chem. 1992;31:2625-2628

work page 1992
[12]

Modern Control Engineering

Ogata K. Modern Control Engineering. 5th ed. Prentice Hall; 2010

work page 2010
[13]

Advanced PID Control

Astrom KJ, Hagglund T. Advanced PID Control. ISA; 2006

work page 2006
[14]

Overshoot and settling time assignment with PID for first -order and second-order systems

Nguyen NH, Nguyen PD. Overshoot and settling time assignment with PID for first -order and second-order systems. IET Control Theory Appl. 2018;12(17):2407-2416

work page 2018
[15]

Generalized optimal and explicit PI/PID tuning formulas for underdamped second-order systems

Albatran S, Smadi IA, Bataineh HA. Generalized optimal and explicit PI/PID tuning formulas for underdamped second-order systems. Int J Control Autom Syst. 2020;18(4):1023-1032

work page 2020
[16]

Design of active inductor and stability test for passive RLC low-pass filter

Tran MT, Kuwana A, Kobayashi H. Design of active inductor and stability test for passive RLC low-pass filter. Comput Sci Inf Technol. 2020:203-224

work page 2020
[17]

Transient response counts when choosing phase margin

Basso C. Transient response counts when choosing phase margin. Electronic Design. November 2008

work page 2008

[1] [1]

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...

work page

[2] [2]

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 ...

work page

[3] [3]

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...

work page

[4] [4]

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...

work page

[5] [5]

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 =...

work page

[6] [6]

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...

work page

[7] [7]

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. ...

work page

[8] [8]

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...

work page

[9] [9]

Optimum settings for automatic controllers

Ziegler JG, Nichols NB. Optimum settings for automatic controllers. Trans ASME. 1942;64:759-768

work page 1942

[10] [10]

Refinements of the Ziegler-Nichols tuning formula

Hang CC, Astrom KJ, Ho WK. Refinements of the Ziegler-Nichols tuning formula. IEE Proc D Control Theory Appl. 1991;138(2):111-118

work page 1991

[11] [11]

Tuning PI controllers for integrator/dead time processes

Tyreus BD, Luyben WL. Tuning PI controllers for integrator/dead time processes. Ind Eng Chem. 1992;31:2625-2628

work page 1992

[12] [12]

Modern Control Engineering

Ogata K. Modern Control Engineering. 5th ed. Prentice Hall; 2010

work page 2010

[13] [13]

Advanced PID Control

Astrom KJ, Hagglund T. Advanced PID Control. ISA; 2006

work page 2006

[14] [14]

Overshoot and settling time assignment with PID for first -order and second-order systems

Nguyen NH, Nguyen PD. Overshoot and settling time assignment with PID for first -order and second-order systems. IET Control Theory Appl. 2018;12(17):2407-2416

work page 2018

[15] [15]

Generalized optimal and explicit PI/PID tuning formulas for underdamped second-order systems

Albatran S, Smadi IA, Bataineh HA. Generalized optimal and explicit PI/PID tuning formulas for underdamped second-order systems. Int J Control Autom Syst. 2020;18(4):1023-1032

work page 2020

[16] [16]

Design of active inductor and stability test for passive RLC low-pass filter

Tran MT, Kuwana A, Kobayashi H. Design of active inductor and stability test for passive RLC low-pass filter. Comput Sci Inf Technol. 2020:203-224

work page 2020

[17] [17]

Transient response counts when choosing phase margin

Basso C. Transient response counts when choosing phase margin. Electronic Design. November 2008

work page 2008