IAE Optimized PID Tuning with Phase Margin and Crossover Frequency Constraints

Senol Gulgonul

REVIEW 3 major objections 5 minor 14 references

IAE Optimized PID Tuning with Phase Margin and Crossover Frequency Constraints

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A PID tuning method that pins phase margin and crossover frequency exactly while minimizing response error.

desk verdict A transparent PID tuning method that exactly enforces PM and wc while minimizing IAE; the 'guarantee' overstates feasibility, but the core idea is sound and worth a revision. read the letter →

arxiv 2506.00923 v2 pith:QLCT2ESW submitted 2025-06-01 eess.SY cs.SY

Senol Gulgonul This is my paper

classification eess.SYcs.SY

keywords PIDtuningphasemargincrossoverfrequencyIAEminimizationconstrainednonlinearprogrammingsequentialquadraticfrequency-domainconstraintsstepresponse

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

PMwc-Tune is a PID controller tuning method that treats the design as one constrained optimization: minimize the integral absolute error of the closed-loop step response while forcing the open-loop magnitude to be 1 at the target crossover frequency and the open-loop phase to be $-180^\circ$ plus the target phase margin. The paper tests it on the benchmark family $G(s)=1/(s+1)^n$ for $n=1,2,3$ with targets PM $=60^\circ$ and $\omega_c=1$ rad/s. In every test case the method reaches the frequency-domain targets to numerical tolerance, whereas the proprietary pidtune tool overshoots the phase margin for the first- and second-order plants. For the third-order plant the constrained controller also has lower IAE and a shorter settling time than pidtune, which is the paper's evidence that exact frequency-domain constraints and good transient response do not have to trade off against each other.

What carries the argument

The load-bearing object is the constrained nonlinear program of Eqs. (4)-(5): minimize IAE$(K_p,K_i,K_d)$ subject to the nonlinear equality constraints $|L(j\omega_c)|-1=0$ and $\angle L(j\omega_c)-(-180^\circ+\mathrm{PM})=0$, with $K_p,K_i,K_d\ge0$. The frequency-domain constraints are analytically differentiable, while the IAE is evaluated by simulating the closed loop on a fixed 20-second, 10-millisecond grid; Sequential Quadratic Programming solves the hybrid problem using quasi-Newton Hessian approximations. The gains are initialized from the plant's DC gain, $K_p=K_i=K_d=1/\mathrm{dcgain}(G)$, which the paper uses to promote convergence.

What would settle it

Run PMwc-Tune on a plant and target pair where the feasibility assumption is doubtful, for example a fifth-order lag $G(s)=1/(s+1)^5$ with a high requested crossover, or $G(s)=1/((s-1)(s+2))$ with PM $=60^\circ$, and record whether the optimizer returns a controller that satisfies both equality constraints inside tolerance, has nonnegative gains, and yields a stable closed loop. If the solver returns a 'successful' controller that violates either constraint, or if an independent grid search of $(K_p,K_i,K_d)$ shows the feasible set is empty while the optimizer reports convergence, the paper's guarantee of specification attainment fails.

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Extended reading notes

Core claim

The paper claims that a PID controller satisfying prescribed robustness margins can be found by solving a small nonlinear program in the three gains $(K_p,K_i,K_d)$. The objective is IAE $=\int_0^{20}|e(t)|\,dt$, computed from a fixed-step simulation, and the two equality constraints are the classical open-loop conditions $|L(j\omega_c)|=1$ and $\angle L(j\omega_c)=-180^\circ+\mathrm{PM}$. On the benchmark systems $G(s)=1/(s+1)^n$ with PM $=60^\circ$ and $\omega_c=1$ rad/s, the SQP solver converges to controllers with achieved PM and $\omega_c$ equal to the requested values, including exact $60.00^\circ$ and $1.0000$ rad/s in all three rows of Table 1. The third-order result is the paper's central evidence: an IAE of 1.1469 versus 1.1999 for pidtune, and a settling time of 4.22 s versus 6.11 s, while the first- and second-order cases trade a slightly larger IAE for a precisely met phase margin instead of pidtune's $69.3^\circ$-$69.4^\circ$ overshoot.

Load-bearing premise

The method assumes that a workable PID controller with no negative gains exists for the chosen plant and the requested phase margin and crossover frequency, and the paper does not prove or test that assumption beyond the three benchmark plants.

Editorial extensions

If this is right

Designers can request a specific phase margin and crossover frequency and get controllers that meet both, instead of accepting a conservative margin overshoot of roughly $9.4^\circ$ that pidtune produced on the first- and second-order plants.
Because IAE is minimized under the exact constraints, the closed loop can be faster without losing the requested robustness: the third-order benchmark settles in 4.22 s versus 6.11 s.
The method is transparent and open, so the same equations can be re-solved with different objectives, solvers, or plant models.
For stable low-order plants the method appears to give a predictable, reproducible trade between time-domain error and frequency-domain specifications.

Reading between the lines

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

A natural next test is feasibility: for extreme targets, such as very high phase margin or crossover frequency relative to the plant's phase lag, no nonnegative PID may satisfy the equalities; adding a feasibility certificate or constraint relaxation would turn the method into a general design tool.
Because SQP is a local method and the IAE surface depends on simulation, different initializations could lead to different locally optimal gains that all meet the frequency constraints; restarting from several starting points would test how much the IAE comparison depends on initialization.
The paper's evidence is limited to three stable low-order plants with the same target pair, so the claim that exact PM/wc enforcement generally improves transient performance needs testing on plants with time delays, right-half-plane zeros, or higher order.
The same constraint structure could be extended to PI, PIDF, or setpoint-weighted structures by changing the controller transfer function inside the equalities, since the core formulation does not rely on the PID form being fixed.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take. The paper does one thing: it sets up a constrained optimization that ties the PID gains to exact phase-margin and crossover-frequency equalities while minimizing IAE, and it solves that problem with SQP. That's not a new idea in itself -- refs 7-9 do constrained optimization for PI/PID -- but the specific combination with equality constraints on both PM and wc, plus non-negative gains, is a clean, transparent formulation. The three benchmark examples are consistent with the math: the achieved PM and wc match the targets, and the IAE comparison against pidtune is an honest external checkpoint. The method is easy to understand and implement, which counts for something.

The soft spots are real but manageable. The word "guarantee" in the abstract and conclusion is not justified. At the target frequency, a PID with Kp,Ki,Kd >= 0 can only shift the phase of the loop by something in [-90°, 90°]. So for any plant where the required controller phase at wc falls outside that range, the constraints cannot be satisfied. The paper doesn't state this necessary condition, let alone test for feasibility. The three benchmarks are all feasible, so the validation never hits the boundary. That overclaim should be fixed, either by replacing "guarantee" with "for feasible targets" or by adding a feasibility check to the algorithm.

There are also a few numerical and presentation inconsistencies: the abstract says 4.6% IAE reduction, the results say 4.4% for the same case; settling time improvement appears as 44% in the Figure 1 discussion and 30.9% in the conclusion. And the paper calls the implementation open-source, but I don't see a link or repository. That needs to be addressed.

The pidtune comparison is not fully apples-to-apples, since pidtune is not trying to minimize IAE and it overshoots PM for the lower-order plants. The paper acknowledges this, but it should be more careful about claiming "improved transient response" when it actually wins on IAE for only one of the three cases.

Bottom line: the core method is sound for feasible problems, and the paper is a useful practical note for anyone who needs exact PM and wc constraints. I'd send it to review, but with the feasibility condition, the inconsistencies, and the code availability all flagged as required revisions.

Referee Report

3 major / 5 minor

Summary. The paper proposes PMwc-Tune, a PID tuning method that formulates controller design as a constrained nonlinear optimization problem: minimize the Integral Absolute Error (IAE) of the closed-loop step response subject to equality constraints on the phase margin (PM) and crossover frequency (ωc) at the open-loop transfer function. The optimization is solved with MATLAB's fmincon using the SQP algorithm, with nonnegative gain constraints on Kp, Ki, Kd. The method is validated on G(s)=1/(s+1)^n for n=1,2,3 with target PM=60° and ωc=1 rad/s, and compared against MATLAB's pidtune. The reported results show that PMwc-Tune achieves the frequency-domain targets exactly in all three cases, while pidtune overshoots the PM for n=1 and n=2. The IAE is lower for pidtune on the first- and second-order plants but lower for PMwc-Tune on the third-order plant. The paper claims that PMwc-Tune 'guarantees specification attainment within numerical tolerances' while improving transient performance compared to pidtune.

Significance. If the results hold, the method provides a transparent, reproducible alternative to proprietary PID tuning tools for enforcing frequency-domain specifications while optimizing time-domain performance. The formulation is clear, and the inclusion of code snippets supports reproducibility. However, the validation is thin: only three stable, low-order benchmark plants and a single target (PM=60°, ωc=1 rad/s) are tested. The comparison with pidtune is only unconfounded for the third-order case, where both methods achieve the same PM; the first- and second-order comparisons are confounded by different achieved phase margins. The central claims of a 'guarantee' and general IAE improvement are overstated relative to the evidence. The method's core idea is plausible, but the manuscript requires correction of technical errors and qualification of its claims.

major comments (3)

[Section 2, Eqs. (1)-(2)] The equations for the PID frequency response are written incorrectly. In Eq. (1), the controller term 'jωc Ki' should be 'Ki/(jωc)' (equivalently -j Ki/ωc), and the same error appears in Eq. (2). As printed, the expressions are dimensionally inconsistent and do not represent C(jωc) = Kp + j(Kd ωc - Ki/ωc). The implementation in Section 3 uses the correct expression, so the numerical results are unaffected, but the theoretical foundation must be corrected.
[Section 2 and Abstract] The claim that the method 'guarantees specification attainment within numerical tolerances' is not qualified by a feasibility condition. For a PID controller with nonnegative gains, C(jω) has phase in [-90°, 90°], so a necessary condition for the constraints to be solvable is that θ = -180° + PM - arg G(jωc) lies in that interval. The paper neither states this condition nor implements a feasibility check. For example, G(s)=1/(s+1)^5, ωc=1 rad/s, PM=60° gives θ=105°, so no feasible PID exists. The three benchmark plants are feasible, so the validation does not probe the boundary. Please add a feasibility analysis or qualify the guarantee.
[Table 1 and Section 4] The IAE comparison is not apples-to-apples for n=1 and n=2, because pidtune achieves phase margins of 69.31° and 69.44°, respectively, rather than the target 60°. Since IAE generally depends on the phase margin, the lower IAE of pidtune in those rows may be due to the larger robustness margin rather than to better time-domain optimization. The only unconfounded comparison is the n=3 case, where both methods meet PM=60°. Consequently, the abstract's claim of a '4.6% IAE reduction' (and the 4.4% in Table 1) should be presented as specific to the third-order benchmark, not as a general result.

minor comments (5)

[Abstract] The abstract states a '4.6% IAE reduction' compared to pidtune, but Table 1 shows a 4.4% reduction for the third-order plant (1.1469 vs 1.1999). Please reconcile these numbers.
[Section 4, Figure 1 description] The text says 'the first order plant G(s)=1/(s+1)^3'; this is contradictory. It should be the third-order plant.
[Section 4 and Section 5] The results section reports a '44% improvement in transient response speed' while the conclusion states a '30.9% improvement' for the same settling-time change (6.11 s to 4.22 s). The correct relative reduction is 30.9%, so the 44% figure is erroneous.
[General] The paper repeatedly calls the method 'open-source' but provides no repository link or access instructions, and the implementation relies on MATLAB's proprietary fmincon solver. Please clarify the availability of the code and the meaning of 'open-source' in this context.
[Section 3] The IAE simulation horizon and step size are said to be 'selected through numerical sensitivity analysis,' but no such analysis is presented. Please either show the sensitivity data or delete the claim.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: the PM/wc 'achievement' is the optimization's own equality constraints, a benign tautology; the IAE comparison with pidtune is an external benchmark.

other [Section 2, Eq. (4); Section 4, Table 1]
"minimize IAE(Kp, Ki, Kd) subject to |L(jωc)| − 1 = 0; ∠L(jωc) − (−180° + PM) = 0; Kp ≥ 0, Ki ≥ 0, Kd ≥ 0 ... Table 1: ... PMwc-Tune ... 60.00 ... 1.0000"

The reported phase margin and crossover frequency are not independent predictions: they are the equality constraints imposed on the optimizer. The solver must satisfy |L(jωc)|=1 and ∠L(jωc)=−180°+PM, so the Table 1 entries PM=60.00° and wc=1.0000 rad/s are the constraints themselves, reproduced by construction. This is a mild tautology, not a damaging circularity; the substantive result is the external IAE comparison against MATLAB pidtune, which does not reduce to these constraints.

full rationale

The paper derives its controllers by solving Eq. (4), which directly enforces the target PM and wc as nonlinear equality constraints. Table 1 then reports exact satisfaction of those same targets; this is expected by construction and is not an independent validation of the frequency-domain claim. However, this is not circular reasoning in the pejorative sense: the method's central contribution is the IAE comparison with pidtune (an external, independent benchmark), and no fitted parameters are renamed as predictions. No load-bearing self-citations appear; all references are to standard control literature or external benchmarks. The main weakness is a correctness/feasibility gap, not circularity: the paper gives no condition under which nonnegative Kp, Ki, Kd exist to meet both equality constraints for an arbitrary plant/target pair, so the unqualified 'guaranteeing specification attainment' is overbroad. That concern belongs to correctness risk, and it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The method introduces no new physical entities or fitted constants. It relies on standard control-theoretic definitions, the assumption that the chosen benchmark plants are representative, and the practical assumption that SQP finds a feasible solution for the constrained optimization problem.

assumptions (3)

standard math The magnitude and phase conditions (Eqs. 1 and 2) correctly define crossover frequency and phase margin for the open-loop transfer L(s).
This is textbook control theory, used in the constraint formulation.
domain assumption The benchmark plants G(s)=1/(s+1)^n for n=1,2,3 are representative of systems where the method's claims hold.
The validation is limited to these three stable plants; the method's generalization is assumed.
domain assumption SQP converges to a feasible solution satisfying the equality constraints for these problems.
The paper assumes fmincon with SQP will find a feasible point; no proof of global or even local convergence is given.

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Cite this review

Pith. "Pith review of IAE Optimized PID Tuning with Phase Margin and Crossover Frequency Constraints." pith.science (2026). https://pith.science/paper/QLCT2ESW

@misc{pith2026250600923,
  author       = {Pith},
  title        = {Pith review of: IAE Optimized PID Tuning with Phase Margin and Crossover Frequency Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLCT2ESW}},
  note         = {Machine review of arXiv:2506.00923}
}

read the original abstract

This paper presents PMwc-Tune, a novel PID tuning method that uniquely combines frequency-domain robustness constraints with time-domain performance optimization through constrained nonlinear programming. The key contribution is a unified formulation that simultaneously enforces phase margin and crossover frequency requirements (via nonlinear equality constraints) while minimizing the Integral Absolute Error (IAE) of the closed-loop response. The algorithm employs Sequential Quadratic Programming (SQP) to solve this constrained optimization problem, guaranteeing specification attainment within numerical tolerances while optimizing transient performance. Numerical validation on benchmark systems demonstrates precise convergence to design targets (phase margin and crossover frequency errors <1%) with a 4.6% IAE reduction compared to MATLAB's pidtune. The open-source implementation provides both methodological transparency and practical design flexibility, enabling PID controllers that rigorously balance frequency-domain robustness and time-domain performance.

Figures

Figures reproduced from arXiv: 2506.00923 by the authors.

**Figure 1.** shows the closed-loop step response comparison of both methods for the first order plant 𝐺(𝑠) = 1/(𝑠 + 1) 3 . The PMwc-Tune controller achieves a settling time of 4.22 seconds compared to 6.11 seconds for pidtune, representing a 44% improvement in transient response speed. This comes at the cost of a marginally higher overshoot (9.07% vs 7.27%), which is consistent with the design trade-off between response speed an… view at source ↗

**Figure 2.** presents the Bode plot comparison of the open-loop systems for both methods. The frequency response analysis reveals that both PMwc-Tune and the reference controller achieve the design specifications of 60° phase margin at the target crossover frequency of 1.000 rad/s, as indicated by the markers on the plot. The magnitude curves show close agreement in the critical frequency range around the crossover point, with b… view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

The future of PID control

Åström, Karl Johan, and Tore Hägglund. "The future of PID control." Control engineering practice 9.11 (2001): 1163-1175

work page 2001

[2] [2]

Desbourough, R

L. Desbourough, R. Miller, Increasing customer value of industrial control performance monitoring––Honeywell’s experience, in: Sixth International Conference on Chemical Process Control, AIChE Symposium Series Number 326, vol. 98, 2002

work page 2002

[3] [3]

Optimum Settings for Automatic Controllers

Ziegler, J. G., and Nichols, N. B. (June 1, 1993). "Optimum Settings for Automatic Controllers." ASME. J. Dyn. Sys., Meas., Control. June 1993; 115(2B): 220–222

1993

[4] [4]

Theoretical consideration of retarded control

Cohen, GHp, and G. A. Coon. "Theoretical consideration of retarded control." Transactions of the American Society of Mechanical Engineers 75.5 (1953): 827-834

1953

[5] [5]

PID controllers 2nd edition

Astrom, K., and T. Hagglung. "PID controllers 2nd edition." ISA: The Instrumentation, Systems and Automation Society (1995)

work page 1995

[6] [6]

Design of PI Controller by Minimization of IAE

Andreas, Helena, and Karl Johan Åström. "Design of PI Controller by Minimization of IAE." (1997)

work page 1997

[7] [7]

Design of PID controllers based on constrained optimisation

Panagopoulos, Hélène, K. J. Astrom, and T. Hagglund. "Design of PID controllers based on constrained optimisation." IEE Proceedings-Control Theory and Applications 149.1 (2002): 32- 40

work page 2002

[8] [8]

Optimal tuning for PI controller, Automatica, 2000, V ol

Wang, Ya-Gang, Shao, H -H. Optimal tuning for PI controller, Automatica, 2000, V ol. 36, No. I, 147-152

work page 2000