REVIEW 5 major objections 3 minor 22 references
An I + PI Controller Structure for Integrating Processes with Dead-Time: Application to Depth Control of an Autonomous Underwater Vehicle
T0 review · 5 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A feedforward I controller plus a feedback PI controller, tuned from damping and natural frequency, controls integrating processes with dead time and extends to AUV depth control.
desk verdict Standard two-DOF PI pole placement with an ad hoc dead-time knob; useful for practitioners but not new, and the reported simulations are not reproducible as written. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the I+PI configuration: $G_{ff}(s)=K_c/(T_i s)$ in the feedforward path and $G_{fb}(s)=K_c(1+1/(T_i s))$ in the feedback path, applied to the plant $G(s)=K_p e^{-ds}/s$. The design works by matching the delay-free closed-loop denominator to the standard second-order form $s^2+2\zeta\omega_n s+\omega_n^2$, producing the explicit gain formulas $K_c=2\zeta\omega_n/K_p$ and $T_i=2\zeta/\omega_n$. The dead time is then folded back in through the tuning rule $\omega_n=4k/(\zeta(T_s+d))$ with $T_s=d/K_p$, where $k$ is a free aggressiveness factor; this is the mechanism that turns user-level specifications (damping, settling speed, and delay) into two controller gains.
What would settle it
Run the proposed tuning on an IPDT process whose dead time is comparable to the designed settling time, for example $K_p=0.0506$ with $d=60$ and the same $\zeta=0.7071$, and compare the simulated overshoot and settling time with the specified values; if the response either no longer follows the $\zeta$/ $\omega_n$ behavior or becomes unstable as $k$ is raised, the neglected delay is the cause. The same check can be done analytically by computing the phase margin of $L(s)=K_p K_c(1+1/(T_i s))e^{-ds}/s$ at its gain crossover and observing whether it vanishes for large $d$.
Extended reading notes
Core claim
The central claim is that an integrating plus dead-time process can be controlled by a feedforward I controller placed in series with a feedback PI controller, both using the same two gains, so that setpoint tracking and disturbance rejection are handled by parallel paths. The paper derives the PI settings by setting $d=0$ in the closed-loop transfer function, comparing the denominator with $s^2 + 2\zeta\omega_n s + \omega_n^2$, and solving for $K_c$ and $T_i$; it then reintroduces the delay through the heuristic $\omega_n = 4k/(\zeta(T_s+d))$ with $T_s=d/K_p$. With these settings the closed-loop response is claimed to follow the chosen damping factor and natural frequency, and simulations show a benchmark IPDT process ($K_p=0.0506$, $d=6$) tracking a setpoint with 7.68% overshoot and settling in 60.1 s, responses to dead-time variations, and workable depth control for a nonlinear AUV model whose depth dynamics are approximated as an IPDT system with $K_p=0.7918$.
Load-bearing premise
The design assumes the dead time can be ignored when deriving the closed-loop dynamics and only re-enters through the natural-frequency formula, so the promised damping and settling behavior depends on the actual delay being small compared with the designed response time.
Editorial extensions
If this is right
- An operator who knows $K_p$ and $d$ for an IPDT plant can tune the controller from two user choices, $\zeta$ and $\omega_n$, with no derivative mode to cause jerky actuator action.
- Because the same gains appear in the feedforward and feedback paths and both paths share the closed-loop denominator, setpoint tracking and disturbance rejection are specified together rather than separately.
- On the benchmark plant $K_p=0.0506$, $d=6$, the claimed tuning reaches the setpoint with 7.68% overshoot and a 60.1 s settling time, keeping overshoot below the compared PID tunings while avoiding derivative action.
- The scheme extends to depth control: once depth dynamics are approximated as an IPDT process with $K_p=0.7918$, the gain formulas with $\omega_n=0.03$ and $\zeta=0.7$ produce a working depth response with acceptable stern-plane actuator profiles.
- Varying $\zeta$ and $k$ changes the transient response in a predictable way, so the same formulas can be used to dial in more or less aggressive control.
Reading between the lines
- The delay-neglected derivation and heuristic $\omega_n$ choice suggest the promised $\zeta$-and-$\omega_n$ response will only hold when $d$ is small relative to the designed closed-loop time constant; a phase-margin or robust-stability bound on $d/(T_s+d)$ would be a natural extension the paper leaves implicit.
- Because $k$ is a free aggressiveness knob, one could map $k$ to a target gain or phase margin, converting the heuristic into a robustness-based tuning rule.
- The structure might carry over to other IPDT-like processes, such as liquid-level or motion-control plants, where a smooth derivative-free control action is desirable; an experimental validation beyond simulation would test that transfer.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a feedforward I plus feedback PI controller structure for integrating plus dead-time (IPDT) processes. The feedback PI gains are derived by matching the delay-free closed-loop transfer function to a standard second-order form, yielding Kc = 2ζωn/Kp and Ti = 2ζ/ωn; a heuristic rule (Eqs. 11–12) then selects ωn via an assumed settling-time relation Ts = d/Kp. The scheme is compared by simulation with three PID tuning methods on an IPDT example (Kp = 0.0506, d = 6) and applied to a nonlinear AUV depth-control simulation. The paper claims smooth control, disturbance rejection, and robustness to dead-time variations.
Significance. The delay-free pole-placement derivation is correct, and the two-degree-of-freedom I+PI structure is clean and easy to implement. However, the central tuning rule for the dead-time case is heuristic rather than derived, the delayed closed-loop is never analyzed for stability or performance guarantees, and the simulation evidence is incomplete because key tuning parameters and a robustness figure are missing. If the missing analysis and reproducible tuning data were supplied, the method could be a useful simple alternative to existing IPDT PID tuning rules; as it stands, the contribution is incremental and not fully substantiated.
major comments (5)
- [Section II, Eqs. (4)-(10)] The controller settings are derived from Eq. (5), which sets d = 0, while the actual closed loop contains e^{-ds}. The paper does not quantify when the dead time is negligible or provide a stability margin for the delayed loop. Please give the actual characteristic equation 1 + (Kp Kc / s)(1 + 1/(Ti s)) e^{-ds} = 0 and the admissible range of k or ωn for which stability and the specified ζ/ωn behavior are retained.
- [Section II, Eqs. (11)-(12)] The tuning rule is dimensionally inconsistent: with Kp in s^{-1} and d in seconds, Ts = d/Kp has units of s^2, so the ωn obtained from Eq. (11) is not in rad/s. Neither the derivation of Ts = d/Kp nor the values of k and ζ used for the simulation reported in Table I and Fig. 3 are stated, making the 7.68% overshoot and 60.10 s settling time non-reproducible from the published rules. Please specify k and ζ for the example and provide a dimensionally consistent, justified settling-time relationship.
- [Table I] The comparison in Table I is not controlled for design specifications: the proposed I+PI method has a rise time of 18.91 s versus 4.1–14.63 s for the three PID methods, so its lower overshoot (7.68%) may reflect a more conservative tuning rather than an advantage of the controller structure. Please compare all methods at matched settling-time or robustness targets, or explicitly discuss the trade-off between speed and overshoot.
- [Section II, robustness paragraph] The claim that the proposed scheme is robust to variations in the modeled dead time is supported only by the sentence 'Robustness... can be seen from the plots in Fig. ??.' This figure does not exist in the manuscript. Since robustness to dead-time mismatch is closely tied to the paper's stated focus on IPDT processes, please provide the missing figure or a quantitative robustness analysis.
- [Section III] For the AUV example, ωn = 0.03 and ζ = 0.7 are stated, but the dead-time d estimated from Fig. 8 is not reported, and no rule or rationale is given for choosing ωn = 0.03. This prevents verification of the controller settings from Eq. (10) and leaves the link between the AUV demonstration and the proposed tuning rule incomplete.
minor comments (3)
- [Section II, text near Fig. 3] There is a typo: 'The the plot showing setpoint tracking responses' should read 'The plot showing setpoint tracking responses'.
- [Section II, D-mode motivation] The statement that the D mode 'cannot be successfully used for regulatory control' is asserted without a reference or demonstration; please qualify the claim or support it with a citation.
- [Fig. 7] In Fig. 7, the caption says that k varies while ζ is fixed, but the actual k values used are not printed on the plot or in the caption, making the effect of k difficult to assess.
Circularity Check
No circularity: the I+PI gain derivation is a self-contained pole-matching exercise; the dead-time handling is heuristic and approximate, not a fitted input disguised as a prediction.
full rationale
The derivation from Eq. (4) to Eq. (10) is a standard coefficient-matching exercise. With d=0, the closed-loop denominator is s^2 + Kp Kc s + Kp Kc / Ti, and equating it with s^2 + 2 zeta omega_n s + omega_n^2 gives Kc = 2 zeta omega_n / Kp and Ti = 2 zeta / omega_n. These are algebraic identities from the stated plant and controller models, not fits to the simulation results. The later formulas, omega_n = 4/(zeta(Ts+d)) * k and Ts = d/Kp, are explicitly introduced as a heuristic way to choose omega_n ('may be indirectly specified', 'k is a constant multiplying factor'), and the simulations in Table I and Figs. 3-7 are demonstrations for chosen design parameters rather than inferences from output data. No load-bearing self-citation is present: references [3] and [10] are prior works by the author but are only listed among existing tuning methods, not used to justify the proposed controller settings. The paper's genuine weakness is that dead time d is neglected in Eq. (5) and reintroduced only heuristically in Eqs. (11)-(12), with no stability margin or admissible dead-time range for the delayed loop; that is a correctness and reproducibility concern, not circularity. The central derivation is therefore self-contained and not circular.
Assumptions & free parameters
free parameters (3)
- k =
1 (used in simulations)
- zeta (damping factor) =
0.7071 in IPDT comparison; 0.7 in AUV
- omega_n (AUV case) =
0.03 rad/s
assumptions (5)
- ad hoc to paper Dead time d is negligible for controller design; d=0 in Eq. (5).
- standard math The closed loop should match a standard second-order transfer function (Eq. 8).
- standard math The feedforward controller should cancel the zero at s=-1/Ti introduced by the PI feedback (Eq. 7 and 13).
- ad hoc to paper Settling time is given by Ts=d/Kp (Eq. 12).
- domain assumption Process model parameters Kp and d are known exactly.
Cite this review
Pith. "Pith review of An I + PI Controller Structure for Integrating Processes with Dead-Time: Application to Depth Control of an Autonomous Underwater Vehicle." pith.science (2026). https://pith.science/paper/5MSFU3V6
@misc{pith2026190809250,
author = {Pith},
title = {Pith review of: An I + PI Controller Structure for Integrating Processes with Dead-Time: Application to Depth Control of an Autonomous Underwater Vehicle},
year = {2026},
howpublished = {\url{https://pith.science/paper/5MSFU3V6}},
note = {Machine review of arXiv:1908.09250}
}
read the original abstract
The paper presents a feedforward plus feedback controller structure with I and PI controllers for control of an integrating process with dead time. Guidelines for controller gain selection based on time domain specifications of damping factor and natural frequency are provided along with simulations indicating the selectivity of process response. The utility of proposed controller structure is shown by simulating the depth control of a nonlinear autonomous underwater vehicle system by the proposed controller structure.
Figures
Figures from the paper (5 more)
Reference graph
Works this paper leans on
-
[1]
A pi tuning rule for integrating plus dead time processes with parametric uncertainty,
P. Mercader and A. Banos, “A pi tuning rule for integrating plus dead time processes with parametric uncertainty,” ISA transactions, vol. 67, pp. 246–255, 2017
work page 2017
-
[2]
Extending spectrum of filtered controllers for ipdt plant models,
M. Huba, “Extending spectrum of filtered controllers for ipdt plant models,” in 2018 Cybernetics & Informatics (K&I) , pp. 1–6, IEEE, 2018
work page 2018
-
[3]
Disturbance Observer based Control of Integrating Processes with Dead-Time using PD controller
S. D. Kadam, “Disturbance observer based control of integrat- ing processes with dead-time using pd controller,” arXiv preprint arXiv:1711.11250, 2017
work page Pith review arXiv 2017
-
[4]
Comparing controllers based on ipdt plant models,
M. Huba and K. Žáková, “Comparing controllers based on ipdt plant models,” IF AC-PapersOnLine, vol. 50, no. 1, pp. 15710–15715, 2017
work page 2017
-
[5]
Limits of a simplified controller design based on ipdt models,
M. Huba and I. Bélai, “Limits of a simplified controller design based on ipdt models,” ProcIMechE Part I: J. Systems and Control Engineering, vol. 232, no. 6, pp. 728–741, 2018
work page 2018
-
[6]
Pidmncontrol for ipdt plants. part 1: Disturbance response,
M. Huba, D. Vran ˇciˇc, and P. Bisták, “Pidmncontrol for ipdt plants. part 1: Disturbance response,” in 2018 26th Mediterranean Conference on Control and Automation (MED) , pp. 1–6, IEEE, 2018
work page 2018
-
[7]
Introduction to the discrete time pidmn control for the ipdt plant,
M. Huba and D. Vran ˇciˇc, “Introduction to the discrete time pidmn control for the ipdt plant,”IF AC-PapersOnLine, vol. 51, no. 6, pp. 119– 124, 2018
work page 2018
-
[8]
Performance measures, performance limits and optimal pi control for the ipdt plant,
M. Huba, “Performance measures, performance limits and optimal pi control for the ipdt plant,” Journal of Process Control , vol. 23, no. 4, pp. 500–515, 2013
work page 2013
Show all 22 references
-
[9]
Setpoint versus disturbance responses of the ipdt plant,
M. Huba, “Setpoint versus disturbance responses of the ipdt plant,” IF AC Proceedings V olumes, vol. 45, no. 3, pp. 412–417, 2012
2012
-
[10]
Control of integrating processes with dead-time using pid controllers with disturbance observer based smith predictor,
S. Kadam and L. Waghmare, “Control of integrating processes with dead-time using pid controllers with disturbance observer based smith predictor,” in Control Applications (CCA), 2013 IEEE International Conference on, pp. 1265–1269, Aug 2013
2013
-
[11]
Tuning PID controllers for integrating processes,
L. Wang and W. Cluett, “Tuning PID controllers for integrating processes,” Control Theory and Applications, IEE Proceedings - , vol. 144, pp. 385 –392, Sep 1997
1997
-
[12]
Identification and Tuning of Integrating Processes with Deadtime and Inverse Response,
W. L. Luyben, “Identification and Tuning of Integrating Processes with Deadtime and Inverse Response,” Industrial & Engineering Chemistry Research, vol. 42, no. 13, pp. 3030–3035, 2003
2003
-
[13]
Internal model control: 4.PID controller design,
D. E. Rivera, M. Morari, and S. Skögestad, “Internal model control: 4.PID controller design,” Industrial & Engineering Chemistry Process Design and Development , vol. 25, no. 1, pp. 252–265, 1986
1986
-
[14]
PI settings for integrating processes based on ultimate cycle information,
E. Poulin and A. Pomerleaú, “PI settings for integrating processes based on ultimate cycle information,” Control Systems Technology, IEEE Transactions on , vol. 7, pp. 509 –511, Jul 1999
1999
-
[15]
Robust tuning of dead-time compensators for processes with an integrator and long dead-time,
J. Normey-Rico and E. Camacho, “Robust tuning of dead-time compensators for processes with an integrator and long dead-time,” Automatic Control, IEEE Transactions on , vol. 44, pp. 1597 –1603, Aug 1999
1999
-
[16]
A simple method of tuning PID controllers for integrator/dead-time processes,
M. Chidambaram and R. P. Sree, “A simple method of tuning PID controllers for integrator/dead-time processes,” Computers &; Chemical Engineering , vol. 27, no. 2, pp. 211 – 215, 2003
2003
-
[17]
Control of integral processes with dead-time. 1. Disturbance observer-based 2 DOF control scheme,
Q.-C. Zhong and J. Normey-Rico, “Control of integral processes with dead-time. 1. Disturbance observer-based 2 DOF control scheme,” Control Theory and Applications, IEE Proceedings - , vol. 149, pp. 285 – 290, Jul 2002
2002
-
[18]
A Simple and Robust Method of Tuning PID Controllers for Integrator/Dead Time Processes,
R. P. Sree and M. Chidambaram, “A Simple and Robust Method of Tuning PID Controllers for Integrator/Dead Time Processes,” Chem- ical Engineering Japan, Journal of - , vol. 38, no. 2, pp. 113–119, 2005
2005
-
[19]
A pi-pd controller design for control of unstable and integrating processes,
I. Kaya, “A pi-pd controller design for control of unstable and integrating processes,” ISA Transactions , vol. 42, no. 1, pp. 111 – 121, 2003
2003
-
[20]
PID controller tuning for integrating processes,
A. Ali and S. Majhi, “PID controller tuning for integrating processes,” ISA Transactions, vol. 49, no. 1, pp. 70 – 78, 2010
2010
-
[21]
Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles,
A. Healey and D. Lienard, “Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles,” Oceanic Engineering, IEEE Journal of , vol. 18, no. 3, pp. 327–339, 1993
1993
-
[22]
Fossen, Guidance and control of ocean vehicles
T. Fossen, Guidance and control of ocean vehicles . Wiley, 1994
1994
Reviewed August 14, 2026 · model on record in the stance chip above.
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