Simple yet Effective Anti-windup Techniques for Amplitude and Rate Saturation: An Autonomous Underwater Vehicle Case Study

arXiv:2601.01302 · eess.SY, cs.SY · Submitted 2026-01-03 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "Simple yet Effective Anti-windup Techniques for Amplitude and Rate Saturation".

Dev: The gist: Simple modifications to two popular controllers, PID and LQI, are proposed to enable them to handle both amplitude and rate saturation in autonomous underwater vehicle control problems,

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: We're looking at the paper "Simple yet Effective Anti-windup Techniques for Amplitude and Rate Saturation: An Autonomous Underwater Vehicle Case Study" again. It's important to understand that they aren't trying to solve every single anti-windup problem out there, but rather focusing on a specific area where classical methods often fall short.

Dev: They revisit the idea of simple anti-windup compensators, which are usually just static correction terms or limits on the integral operator. But this paper is proposing something different—modifications to PID and LQI controllers to handle both amplitude and rate saturation simultaneously.

Taro: So the core idea seems to be taking these established control structures and injecting a mechanism that addresses both types of saturation, which is what modern dynamic anti-windup controllers are supposed to do but often end up being too complicated for practical use.

Rosa: They're suggesting these specific modifications are simple enough that they can give results as good as Model Predictive Control in a simulation testing the REMUS AUV yaw control problem. That comparison against MPC is what makes the results so compelling for me.

Dev: The numbers on the metrics like tracking performance, which they measure with ISE, IACE, and IACER show that these modified controllers perform better than the unconstrained versions under saturation conditions. For instance, looking at their table, even the Gain Margin for PD AW is six point five and Delay Margin is zero point one two when compared to MPC's ten point five and one point four five respectively when constrained >

Taro: That comparison against MPC is key because MPC is inherently more complex, so if these simple modifications can match its performance, it really opens the door for engineers who want something easier to implement in real-time systems.

Rosa: It means we don't necessarily need the most intricate AI-driven anti-windup methods if a straightforward modification to a standard controller like PID or LQI does the job reliably.

Dev: But we have to remember what they say about their limitations; they point out that the complexity of deriving these modern dynamic compensators often imposes restrictive conditions and can be hard to apply in practical engineering problems.

Taro: So, while the paper offers a simple path forward for some common controllers, it flags that the derivation process for those more advanced methods isn't always straightforward or easy to implement in a real-world setting.

Rosa: It’s a trade-off: you get simpler tuning knobs and better performance on specific problems, but you have to accept that these are targeted solutions rather than a universal fix for every single control system challenge.

Dev: So, the main implication is providing an accessible toolset—these AW modifications for PID and LQI—for engineers who need reliable performance under saturation without having to wrestle with the complexity of full MAW theory right away.

The paper's summary: Rosa: Let's look at what the paper actually summarizes. They start by setting up the problem clearly, explaining how actuator amplitude and rate saturation create a windup issue that can lead to instability if you ignore it.

Dev: They review the existing literature on anti-windup solutions, categorizing them into classical methods like ad-hoc fixes and integrator resetting approaches like backcalculation. But they note that modern dynamic anti-windup controllers are often too complex for practical engineering application because of their derivation process.

Taro: Their summary seems to be a justification for why they are looking at simple modifications to PID and LQI controllers, suggesting that the potential of simple yet effective CAW schemes that handle both saturation types has been overlooked in the literature.

Rosa: So they argue that there's a gap between having powerful theoretical solutions and having easy-to-use solutions, and this paper aims to fill that gap by proposing specific changes to two very common controllers.

Dev: The summary then moves into introducing their two proposed AW modifications: the modified PD controller with an AW feedback gain, Kaw, and the modified LQI controller which uses an integrator resetting concept through back-calculating the full actuator input–output signal.

Taro: So they're not just suggesting a tweak; they are defining two concrete mathematical structures—one for PD and one for LQI—that incorporate this anti-windup logic directly into the control law.

Rosa: And the paper summarizes that when tested on the REMUS AUV yaw control problem, these modified techniques actually exhibit comparable performance to a constrained Model Predictive Control controller.

Dev: The summary highlights that all non-constrained versions of these controllers become unstable in this condition, which emphasizes why their proposed modifications are necessary for stability under saturation.

Taro: It boils down to the idea that these two specific modifications are the key elements that allow these simple structures to handle both amplitude and rate saturation effectively enough for practical use.

The paper's improvements: Rosa: Now we get into the specifics of the improvements they propose. First, they detail how to modify a PD controller by augmenting its anti-windup capability with that AW feedback gain, Kaw.

Dev: The text gives us this specific control signal modification for the modified PD controller: uc(t) = Kpe(t)−Kdy˙(t)−Kaw∆uc (six). This is essentially adding that feedback term to adjust the output based on the saturation discrepancy ∆uc.

Taro: That structure, using Kaw to modify the control signal like that, is what allows the system to maintain tracking ability even when facing those demanding stepwise setpoint tests in an AUV environment.

Rosa: Then they show how they adapt a LQI controller by following an integrator resetting concept. It involves back-calculating and feeding back the full actuator input–output signal to the integrator, which leads to a modified law like uc(t) = KI e(t)−Kaw ∆uc(t)−Kxp xp(t), which is different from the PD AW control signal.

Dev: So they are proposing two distinct ways to apply this concept: one is a direct feedback modification for PD, and the other is a more integrated approach for LQI that involves feeding back both the error and the input history.

Taro: The paper suggests that these methods, each requiring only one additional gain—Kaw—are capable of handling both amplitude and rate saturation issues in these systems.

Rosa: So, to sum up the improvements, it's about giving PID and LQI controllers a specific mathematical structure to manage the saturation effects through feedback gains or input history correction.

Conclusion: Dev: So wrapping up this discussion on "Simple yet Effective Anti-windup Techniques for Amplitude and Rate Saturation: An Autonomous Underwater Vehicle Case Study," the main point is that these simple modifications to PID and LQI controllers can handle both amplitude and rate saturation with performance comparable to MPC.

Rosa: They are providing an accessible way for engineers seeking solutions that require only a few tuning knobs, which addresses the need for something simple yet effective without needing a huge amount of theoretical overhead.

Taro: I think what this means for autonomy is that we can get reliable control systems working in challenging environments like AUVs with standard controllers if we apply these targeted modifications instead of immediately jumping to the most complex dynamic compensators.

Dev: The paper shows that even when tested on the REMUS AUV yaw control problem, the modified PD AW and LQI AW controllers maintain stability where unconstrained versions fail.

Rosa: So ultimately, this work is about providing a practical set of tools—these modifications to PID and LQI—that can give you reliable performance under saturation without needing a massive theoretical framework.

Taro: I just think the focus on simplicity is what makes this interesting; it shows that you don't always need the most complex AI-driven approach to achieve good results in control engineering.

Dev: Yeah, these simple modifications to PID and LQI controllers for amplitude and rate saturation are a solid contribution, offering a direct path for practical implementation in systems like AUVs.

Rosa: That's what this paper does; it gives us concrete solutions that engineers can actually use right away, focusing on making existing controllers more robust.

School of Physics, Engineering and Computer Science, University of Hertfordshire

eess.SY, cs.SY

Submitted: 2026-01-03

Updated: 2026-10-09

Code: https://github.com/Psarhadi/Rsat-Classic-AW

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 77/100

The gist: The gist: Simple modifications to two popular controllers, PID and LQI, are proposed to enable them to handle both amplitude and rate saturation in autonomous underwater vehicle control problems,

Key concepts

Actuator Amplitude Saturation (A&RSat)
This occurs when the physical limits of an actuator are reached, meaning the requested control signal is too large or small for the motor to produce. Ignoring this saturation can cause instability or system failure in control systems.
Rate Saturation
This constraint limits how quickly an actuator can change its output, regardless of its total size limit. Failing to account for rate saturation often leads to oscillations and poor performance during rapid changes in the control input.
Classical Anti-Windup (CAW)
These are older methods that handle saturation by imposing simple limits on the integral term or resetting it when saturation occurs. They are considered simple but may not fully address both amplitude and rate limitations effectively.
Modern Dynamic Anti-Windup (MAW)
These advanced techniques use complex, dynamic compensators based on robust control theory to formally design anti-windup strategies. While theoretically powerful, their design process is often too restrictive and complicated for practical engineering application.

Terminology

Summary

The gist: Simple modifications to two popular controllers, PID and LQI, are proposed to enable them to handle both amplitude and rate saturation in autonomous underwater vehicle control problems, achieving performance comparable to Model Predictive Control.

Problem Background

Actuator amplitude and rate saturation (A&RSat), along with their consequent windup problem, have long been recognized as challenges in control systems <ref:2601.01302#pg2>. Two types of constraints, amplitude and rate saturation, can occur in actuators. Neglecting the latter in control design has been shown to cause instability and, in some cases, catastrophic failures. The problem of rate saturation has received relatively little attention in the literature, particularly within classical anti-windup approaches.

Classical Anti-Windup Approaches

Classical methods for handling windup are usually ad hoc solutions that impose limits on the integral operator or utilize feedback to manage saturation <ref:2601.01302#pg6>. Two popular categories of CAW are presented:

  1. Ad-hoc methods: these involve imposing limits on the integral operator or freezing the integrator by setting it to zero once saturation occurs <ref:2601.01302#pg6>.

  2. Anti-reset windup or integrator resetting: this approach operates based on the difference between the computed control signal and the actuated control signal, and is described as backcalculation <ref:2601.01302#pg6>. The concept involves reducing the input to the integrator in proportion to the discrepancy between the two sides of the actuator, i.e., ∆uc = uc − uac <ref:2601.01302#pg6>.

Modern Dynamic Anti-Windup Controllers (MAW)

Modern Anti-Windup (MAW) techniques are comparatively new and aim to formalize anti-windup design using robust and nonlinear control theory <ref:2601.01302#pg4>. MAWs are often dynamic compensators rather than simple static correction terms, represented by equations such as x˙aw(t) = (Ap +BpFaw)xaw(t) +Bp ∆uc(t) <ref:2601.01302#pg4>. These controllers utilize robust and optimal control theories in their design, offering formal stability proofs and applicability to MIMO systems <ref:2601.01302#pg4>. However, due to the relative complexity of their design process, they are not widely adopted within the engineering community <ref:2601.01302#pg4>.

Proposed Anti-windup Solutions

The paper proposes two techniques to modify PID and LQI controllers to enhance their performance in dealing with rate saturation <ref:2601.01302#pg5>.

A) Modified PD Controller with AW (PD AW): This method augments the AW capability for handling A&RSat in the cascade PD controller, modifying the controller output through an AW feedback gain, Kaw <ref:2601.01302#pg5>. The control signal is given by uc(t) = Kpe(t)−Kdy˙(t)−Kaw∆uc (6).

B) Modified LQI Controller with AW (LQI AW): This approach follows the integrator resetting concept, back-calculating and feeding back the full actuator input–output signal to the integrator <ref:2601.01302#pg6>. The modified law is written as uc(t) = KI e(t)−Kaw ∆uc(t)−Kxp xp(t), which differs from the PD AW control signal in Eq. (6).

Simulation Results and Comparison

The results of the three controllers—PD AW, LQI AW, and a constrained MPC controller—for the REMUS AUV yaw control problem demonstrate that they exhibit comparable performance <ref:2601.01302#pg7>. All non-constrained versions of the controllers become unstable in this condition <ref:2601.01302#pg6>. Metrics such as tracking performance (ISE), control activity (IACE), and robustness (GM, DM) are used for comparison <ref:2601.01302#pg6>. The results highlight that the simple yet effective AW modifications applied to the PD and LQI controllers, each requiring only one additional gain, are capable of handling both amplitude and rate saturation <ref:2601.01302#pg6>.

Conclusion

The paper concludes that these simple modifications to PID and LQI controllers can handle both amplitude and rate saturation with performance comparable to MPC <ref:2601.01302#pg6>. The intention is to address the needs of the engineering community seeking simple yet effective solutions that can modify a controller with a few (here, one) tuning knobs <ref:2601.01302#pg6>. Further theoretical research into stability analysis and extension to MIMO systems is encouraged <ref:2601.01302#pg6>.

TABLE I

Criteria PD-AW LQI-AW MPC

ISE = R Tf 0 e 2 (t)dt/Tf 21885.7 21933.1 22762.3 <ref:2601.01302#pg6>

IACE = R Tf 0 uac(t)dt/Tf 342.69 393.93 316.69 <ref:2601.01302#pg6>

IACER = R Tf 0 u˙ac(t)dt/Tf 445.35 366.18 280.38 <ref:2601.01302#pg6>

Gain Margin - GM(s) 6.5 >10.5 <ref:2601.01302#pg6>

Delay Margin - DM(s) 0.12 0.37 1.45 <ref:2601.01302#pg6>

References

[1] A. Tustin, “A method of analysing the effect of certain kinds of non-linearity in closed-cycle control systems,” Journal of the Institution of Electrical Engineers-Part IIA: Automatic Regulators and Servo Mechanisms, vol. 94, no. 1, pp. 152–160, 1947

[2] J. Lozier, “A steady state approach to the theory of saturable servo systems,” IRE Transactions on Automatic Control, vol. 1, no. 1, pp. 19–39, 1956

[3] D. S. Bernstein and A. N. Michel, “A chronological bibliography on saturating actuators,” International Journal of Robust and Nonlinear Control, vol. 5, no. 5, pp. 375–380, 1995

[4] H. A. Fertik and C. W. Ross, “Direct digital control algorithm with anti-windup feature,” ISA transactions, vol. 6, no. 4, p. 317, 1967

[5] K. J. Astrom and L. Rundqwist, “Integrator windup and how to avoid it,” in 1989 American Control Conference. IEEE, 1989, pp. 1693–1698 <ref:2601.01302#pg6>

[6] K. J. Åström and R. M. Murray, Feedback systems: an introduction for scientists and engineers, Princeton university press, 2021

[7] J. C. Doyle, R. S. Smith, and D. F. Enns, “Control of plants with input saturation nonlinearities,” in 1987 American control conference IEEE, 1987, pp. 1034–1039

[8] M. V. Kothare, P. J. Campo, M. Morari, and C. N. Nett, “A unified framework for the study of anti-windup designs,” Automatica, vol. 30, no. 12, pp.

Improvements for AI systems

  1. A modified PD controller (PD AW) can be implemented to handle both amplitude and rate saturation by using an AW feedback gain, Kaw, such that The PD AW control signal is given by: uc(t) = Kpe(t)−Kdy˙(t)−Kaw∆uc (6). This allows the system to maintain tracking ability under demanding conditions while accommodating actuator limits.

  2. An LQI controller (LQI AW) can be extended to handle A&RSat by back-calculating and feeding back the full actuator input–output signal, resulting in the law: uc(t) = KI e(t)−Kaw ∆uc(t)−Kxp xp(t), (12). This provides a simple yet effective structure for handling both amplitude and rate saturation in state-space systems.

  3. The proposed methods can be applied to control autonomous underwater vehicles (AUVs), such as the REMUS AUV yaw control problem, allowing the system to maintain tracking ability even under demanding stepwise setpoint tests where non-constrained versions become unstable.

Abstract

Actuator amplitude and rate saturation (A&RSat), together with their consequent windup problem, have long been recognised as challenges in control systems. Anti-windup (AW) solutions have been developed over the past decades, which can generally be categorised into two main groups: classical and modern anti-windup (CAW and MAW) approaches. Classical methods have provided simple and effective results, mainly addressing amplitude saturation. In contrast, modern approaches offer powerful and theoretically sound solutions capable of handling both amplitude and rate saturations. However, MAW's derivation process often imposes restrictive conditions and can be complex to apply in practical engineering problems. Nevertheless, the literature has paid limited attention (if not entirely ignored) to the potential of simple yet effective CAW schemes that can operate in the presence of both A&RSat elements. This paper revisits this issue and proposes modifications to two well-known controllers: PID and LQI. The obtained results, benchmarked on the REMUS AUV yaw control problem and compared with constrained MPC, indicate that these classical techniques can still provide simple yet effective solutions with comparable performance, at least for SISO systems. These findings may stimulate further research into solutions that achieve comparable performance with only one (or a limited number of) additional tuning parameters and straightforward implementation.

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