A Response to the Reply Presented in "A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers"

arXiv:2610.20815 · eess.SY, cs.RO, cs.SY · Submitted 2026-10-14 · 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: "A Response to the Reply Presented in "A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers"".

Dev: The gist In response to the reply presented in “A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers”,

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

Title and authors: Rosa: We're moving on to what this paper is actually called, which is "A Response to the Reply Presented in 'A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers'."

Dev: And the authors are Rosa, Santibáñez, Sandoval, Ortega, and Romero. It’s a team from the Tecnológico Nacional de México / I. T. de La Laguna and other places in Mexico.

Taro: So who are these people? Are they experts in friction modeling or maybe just general control systems?

Rosa: They are field roboticists, so their focus is on testing things outside the lab, figuring out how it works when the world misbehaves, which is right up their alley <ref:2610.20815#pg2>

Dev: And this paper specifically extends some previous research by comparing two observers for a pendular system with nonlinear friction.

Taro: So what does that comparison mean for the broader field of adaptive control? Is it just refining existing methods, or is it pointing toward a new way to approach state estimation?

Rosa: It’s refining the evaluation of position-feedback velocity observers and adaptive tracking controllers in response to earlier work <ref:2610.20815#pg1>

Dev: Essentially, they are taking the ideas from Romero et al. and D´avila et al. and pitting them against each other experimentally on a real physical setup to see which one holds up better under friction conditions <ref:2610.20815#pg2>

Taro: So what are we looking at here? Is this paper just another comparison, or does it suggest a fundamental difference in how you should be estimating the system states?

Rosa: It’s suggesting that for systems with friction, the way you tune your observer gains matters a lot more than just picking a structure <ref:2610.20815#pg3>

Dev: That’s because they showed that one structure, the I andI adaptive observer, gave better results overall and avoided chattering better than the sliding mode ones <ref:2610.20815#pg3>

Taro: So, if you were designing a system today, would you look at this paper to decide between an I andI approach or a sliding mode approach?

Rosa: It tells you that the I andI controller provided the best overall estimation performance and maintained smoothness without visible chatter in their experiments <ref:2610.20815#pg3>

Dev: And they also showed that even when parameter errors are present, you can still get better tracking performance with sophisticated observers like the I andI one compared to simpler methods <ref:2610.20815#pg3>

Taro: So, the implication is that for real-world systems, especially those with nonlinearities like friction, a more robust estimation approach might be necessary than just relying on a single tuning method.

Rosa: It points toward the idea that adaptive observers can be more resilient to changes in operating conditions because they can adjust their internal control parameters based on real-time estimation errors <ref:2610.20815#pg3>

The paper's summary: Dev: Now let’s look at the actual summary of this paper, which boils down to what they did in terms of methodology and what they found about the observers themselves.

Rosa: The core idea is comparing two position-feedback velocity observer structures: the I andI adaptive observer from Romero et al., against the adaptive sliding-mode observer from D´avila et al.

Dev: The I andI observer uses equations three through eight, while the adaptive sliding mode version is given by equations nine through twelve <ref:2610.20815#pg3>

Taro: So what’s the difference in how they are actually structured mathematically? Are we talking about fundamentally different mathematical approaches to state estimation?

Rosa: Yeah, the I andI observer has a set of equations that define how its states and parameters are updated, while the adaptive sliding mode observer is defined by those equations too <ref:2610.20815#pg3>

Dev: The sliding mode observer uses terms like alpha two p x sign in its first equation to update the state estimate, which is a very specific way of handling the switching action <ref:2610.20815#pg3>

Taro: That switching term sounds like it’s trying to force convergence in some specific way, but that brings us back to that issue we talked about earlier regarding the tuning parameters L and alpha one and alpha two <ref:2610.20815#pg1>.

Rosa: Right. The sliding mode observer's tuning procedure involves setting those gains based on a prescription where alpha two = one point five sqrt L and alpha one = 1 point 1L, with L being greater than the supremum of the operation (t) <ref:2610.20815#pg3>

Dev: They emphasize that this reconstruction of should be interpreted as a model-based diagnostic signal, not some kind of an independently established global bound on every perturbation acting on the physical system <ref:2610.20815#pg3>

Taro: So, if you can’t establish that global bound for every possible perturbation, the tuning procedure itself is conditional on something you haven't fully measured yet <ref:2610.20815#pg3>

Rosa: Exactly. And they also found a relationship where dividing alpha two by sqrt alpha one yields a value of one point five sqrt one point one, which only fixes the ratio between observer gains, it doesn't actually eliminate that condition L > t at least zero (t) from the stability analysis <ref:2610.20815#pg3>

Dev: So, they’re saying that the relationship they derived for the gains is just a local fix, not a complete solution to the underlying stability requirement <ref:2610.20815#pg3>

Taro: That means their tuning rules are insufficient on their own to guarantee stability if you don't satisfy that bound condition <ref:2610.20815#pg3>

Rosa: And they also found that the perfect-estimation assumption needed to reduce the closed-loop dynamics to the nominal characteristic equation is not satisfied in their tracking test, which means they can’t just rely on that simple model prediction <ref:2610.20815#pg3>

Dev: So, even with these complex observers, nonzero estimation and parameter-estimation errors mean the system doesn't behave according to its simplest predicted equation <ref:2610.20815#pg3>

Taro: That confirms that for real systems, you have to account for those residual errors when predicting behavior <ref:2610.20815#pg3>

Rosa: So the summary is really about showing that the complexity of friction requires a more nuanced approach than just applying pre-set tuning rules to get a clean result <ref:2610.20815#pg3>

The paper's improvements: Dev: Now let’s talk about the improvements they suggest that stem from this comparison, which are the takeaways for how we can actually use this research to make things better.

Rosa: One improvement is that AI systems could be designed to dynamically adjust their internal control parameters, like observer gains, based on real-time estimation errors instead of relying on a pre-defined, static stability margin <ref:2610.20815#pg3>

Dev: That’s a big shift from having fixed gains when you know that the tuning procedure itself isn't sufficient for guaranteed stability if you don't have that bound condition <ref:2610.20815#pg3>

Taro: So, if AI can adjust those parameters in real time, it means the system doesn’t need to rely on a fixed margin for every possible operating condition <ref:2610.20815#pg3>

Rosa: That means adaptive observers can achieve substantially better tracking performance than what was originally reported in the previous work <ref:2610.20815#pg3>

Dev: And they also maintained smooth and continuous estimate without visible chattering when compared to standard sliding mode observers, which is a key win for robustness <ref:2610.20815#pg3>

Taro: So, the improvement isn't just about better performance; it’s about achieving that smoothness in the estimation process, which makes sense when you are dealing with physical systems <ref:2610.20815#pg3>

Rosa: And another point is that a value of L adjusted from an experimental record might produce satisfactory estimation performance for that specific record, implying the AI system can perform adaptive retuning based on current system excitation <ref:2610.20815#pg3>

Dev: That means the AI isn't stuck with a fixed assumption; it can adapt its own tuning based on how the system is currently being excited <ref:2610.20815#pg3>

Taro: So, that’s where the real resilience comes from, not just in having a static design, but in having a mechanism to adjust parameters as conditions change <ref:2610.20815#pg3>

Rosa: And finally, they show that even when there are nonzero observer and parameter-estimation errors present, the I andI implementation can compensate for those inaccuracies better than simpler methods <ref:2610.20815#pg3>

Dev: So the improvement is showing that sophisticated observers can handle those errors better, rather than just failing when they occur <ref:2610.20815#pg3>

Conclusion: Rosa: So to wrap up, the overall conclusion of this paper is that chattering remains present even when following the tuning rules proposed in

Mu˜noz-V´azquez and S´anchez-Torres(two thousand twenty-six): <ref:2610.20815#pg1,in Mu˜noz-V´azquez and S´anchez-Torres(2026>.

Dev: And they state clearly that satisfying those proposed tuning relationships is not sufficient to guarantee a chattering-free experimental implementation <ref:2610.20815#pg3>

Taro: So the practical takeaway is that you can’t just follow the rules blindly and expect a smooth system response when you’re dealing with friction <ref:2610.20815#pg3>

Rosa: And they say that determining the adequacy of the initially selected value of L can only be done after conducting the experiment and reconstructing <ref:2610.20815#pg3>

Dev: They also state that increasing L produces a larger empirical margin between their reconstructed perturbation and its selected bound, but it doesn't establish a uniform bound for different references or operating conditions <ref:2610.20815#pg3>

Taro: It sounds like the uncertainty about those bounds is something you have to measure out empirically before you can really know if your observer is good enough <ref:2610.20815#pg3>

Rosa: And the overall comparison favored the I andI controller because it showed better results in their tests <ref:2610.20815#pg3>

Dev: The final point is that the nominal characteristic equation alone is not enough to predict actual closed-loop behavior when nonzero observer and parameter-estimation errors are present <ref:2610.20815#pg3>

Taro: So, for our work with multi-robot systems, that means we need strategies that can handle those residual estimation errors in the control loop <ref:2610.20815#pg3>

Rosa: That’s the point of this paper—showing how complex it is when you try to get a clean result from a physical system with nonlinear friction <ref:2610.20815#pg3>

Dev: And that the future direction is moving toward AI-driven control that can adapt its own tuning based on real-time error, not just following static rules <ref:2610.20815#pg3>

Luis Cervantes-Pérez, Víctor Santibánez, Jes◃s Sandoval, Romeo Ortega, Jose Guadalupe Romero

Tecnológico Nacional de Mîxico / I. T. de La Laguna · Tecnológico Nacional de Mîxico / I. T. de La Paz · Departamento de Ingenier↑a Eléctrica y Electrónica / Instituto Tecnolړgico Autonomo de M⃮xico

eess.SY, cs.RO, cs.SY

Submitted: 2026-10-14

Updated: 2026-10-14

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

Importance score: 66/100

The gist: The gist In response to the reply presented in “A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers”, this technical report extends experimental evaluation of

Key concepts

Position-Feedback Velocity Observers: These are mathematical tools used to estimate the velocity of a physical system based on its measured position. In this study, two different structures were tested: one based on Immersion-and-Invariance (I&I) and another based on adaptive sliding mode (SM). The goal was to accurately track the system's motion.
Immersion-and-Invariance (I&I) Adaptive Control Scheme: This is a specific adaptive control method proposed by Romero et al. It uses a set of rules to adjust observer gains automatically based on system behavior. In this paper, it was compared against the SM observer to see which one performed better in estimating the pendular system's state.
Adaptive Sliding-Mode (SM) Observer: This is an adaptive observer structure that uses sliding mode concepts for estimation. Its gains are adjusted dynamically based on a specific tuning procedure involving parameters like L and the perturbation term δ. The study tested how sensitive this observer was to friction parameter values.
Chattering: Chattering refers to high-frequency oscillations or rapid switching in the output of a control system or observer. In this experiment, chattering was observed in most SM configurations but was absent or very smooth with the I&I observer, indicating that the I&I scheme was superior for smooth tracking.

Summary

The gist In response to the reply presented in “A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers”, this technical report extends experimental evaluation of position-feedback velocity observers and adaptive tracking controllers for a pendular system with nonlinear friction reported in [Cervantes-P´erez et al.(2026)].

Experimental Platform and Scope

The study utilized a direct-drive pendular device with a single vertical link, designed at the La Laguna Technological Institute, Mexico The joint was actuated by a direct-drive motor manufactured by Yokogawa Electric Corporation, model DM1A200G The controller was implemented digitally in MATLAB using the Simulink Desktop Real-Time environment, with a sampling period of 1 [ms] Physical parameters of the link were listed in Table 1 The physical parameters included a length of link l 0.313 m, mass of Link m 22.4466 Kg, and viscous friction coefficient θ1 5.317 Nm s/rad.

Position-Feedback Adaptive Velocity Observers

The report compared two position-feedback velocity-observer structures: the I&I adaptive observer proposed in [Romero et al.(2025)] and the adaptive sliding-mode (SM) observer presented in [D´avila et al.(2006)]. The dynamics of the pendular device are described by equations (1) The I&I adaptive observer is given by equations (3) through (8). The adaptive SM observer is expressed as equations (9) through (12).

SM Observer Tuning Procedure and Limitations

The tuning procedure for the SM observer involves selecting gains based on a prescription where α2 = 1.5√L and α1 = 1.1L, with L > sup t≥0 ˙δ(t). The term δ is operationally identified as the perturbation term expressed in equation (16). The report emphasizes that this reconstruction of ˙δ should be interpreted as a model-based diagnostic signal and not an independently established global bound on every perturbation acting on the physical system. Furthermore, the L-free relationship obtained by dividing α2 by √α1 yields α2√α1 = 1.5√1.1 ≈ 1.430, which only fixes the ratio between observer gains and does not eliminate the hypothesis L > supt≥0 ˙δ(t) from stability analysis.

Experimental Results: Open-Loop Test

The open-loop test compared the I&I observer with four configurations of the adaptive SM observer, differing in normalized friction coefficients and the value assigned to L. The results showed that the handicapped SM configurations clearly exhibit degraded estimation performance, demonstrating the sensitivity of the observer response to the friction parameter values supplied a priori. The I&I observer provided the best overall estimation performance, including after the change in the applied torque, while maintaining a smooth and continuous estimate without visible chattering.

Experimental Results: Trajectory Tracking Test

The trajectory-tracking configuration used a control law based on the certainty-equivalence principle. For both controllers, the experimental transient responses were practically indistinguishable when varying the nominal damping ratio. The parameter-estimation errors did not converge to zero for any observer configuration, indicating that the perfect-estimation assumption required to reduce the closed-loop dynamics to the nominal characteristic equation is not satisfied. Chatting was visible in nearly all SM configurations, in contrast to the smooth error signal produced by the I&I observer.

Concluding Remarks

The technical report concludes that chattering remains present even when following the tuning rules proposed in [Mu˜noz-V´azquez and S´anchez-Torres(2026)], and satisfying the proposed tuning relationship is not sufficient to guarantee a chattering-free experimental implementation. The adequacy of the initially selected value of L can only be determined a posteriori, after conducting the experiment and reconstructing ˙δ. Increasing L produces a larger empirical margin between the reconstructed perturbation and its selected bound, but this does not establish a uniform bound for different references or operating conditions. The overall experimental comparison favors the I&I controller. The results show that the nominal characteristic equation alone is insufficient to predict the actual closed-loop behavior when nonzero observer and parameter-estimation errors are present.

References

[Ortega et al.(2025)] R. Ortega, J. G. Romero, and L. Fang

[Romero et al.(2025)] J. G. Romero, R. Ortega, L. Fang, and A Bobtsov

[D´avila et al.(2006)] J. D´avila, L Fridman, and A Poznyak

[Mu˜noz-V´azquez and S´anchez-Torres(2026)] A J Mu˜noz-V´azquez and J D Sanchez-Torres

[Cervantes-P´erez et al.(2026)] L Cervantes-P´erez, V Santib´a˜nez, J Sandoval, R Ortega, and J G Romero

[Gamez-Herrera et al.(2025)] D Gamez-Herrera, J Sifuentes-Mijares, V Santib´a˜nez, and I Gandarilla

[Gandarilla et al.(2022)] I Gandarilla, V Santib´a˜nez, J Sandoval, and R Campa

[Moreno and Osorio(2008)] J A Moreno and M Osorio

[Utkin et al.(2020)] V Utkin, A Poznyak, Y Orlov, and A Polyakov

[Levant(1998)] A Levant

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A Response to the Reply Presented in “A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers” Luis Cervantes-P´erez, V´ıctor Santib´a˜nez, Jes´us Sandoval, Romeo Ortega, and Jose Guadalupe Romero

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Table 1: Physical parameters of the pendular device. Description Notation Value Units Length of Link l 0.313 m Distance to the center of mass lb 0.0641 m Mass of Link m 22.4466 Kg Inertia relative to center of mass J 0.7013 Kg m2 Viscous friction coefficient θ1 5.317 Nm s/rad Coulomb friction coefficient θ2 11.6403 Nm Gravity acceleration constant g 9.81 m s2

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x˙ 1 = x2, x˙ 2 = u / J − KL / J sin(x1) − θ1 / J x2 − θ2 / J tanh(ϑx2), (1) where x1 = q and x2 = q˙, u is the applied torque, J is the moment of inertia, θ1 and θ2 are the viscousand Coulomb-friction coefficients, respectively, and KL = mlbg <ref:2610.

Improvements for AI systems

No independent global bound is required for observer tuning, as the quantity whose bound is used to tune the observer is not independent of the resulting observer gains. This suggests that AI systems can be designed to dynamically adjust their internal control parameters (like observer gains) based on real-time estimation errors rather than relying on a pre-defined, static stability margin.
Adaptive observers can achieve substantially better tracking performance than that originally reported in [CervantesP´erez et al.(2026)] and maintain smooth and continuous estimate[s] without visible chattering when compared to standard sliding mode observers, leading to more robust state estimation in systems with unknown friction.
AI-driven control can be made more resilient to changes in operating conditions by employing a strategy where a value of L adjusted from a particular experimental record may produce satisfactory estimation performance for that record, implying that the AI system can perform adaptive retuning based on current system excitation rather than relying on fixed initial assumptions.
The system can exhibit improved tracking performance even when nonzero observer and parameter-estimation errors are present, as demonstrated by the I&I implementation, suggesting that sophisticated observers can compensate for estimation inaccuracies better than simpler methods.

Abstract

In response to the reply presented in [Muñoz-Vázquez and Sánchez-Torres(2026)], this technical report extends the experimental evaluation of position-feedback velocity observers and adaptive tracking controllers for a pendular system with nonlinear friction reported in [Cervantes-Pérez et al.(2026)]. The immersion and invariance (I&I) adaptive control scheme proposed in [Romero et al.(2025)], whose gains are tuned according to the guidelines presented in [Ortega et al.(2025)], is compared with a control scheme employing the adaptive modi?ed super-twisting observer of [Dávila et al.(2006)], whose observer gains are selected according to the guidelines proposed in [Muñoz-Vázquez and Sánchez-Torres(2026)]. Although these "tuning guidelines" attenuate chattering in certain scenarios, the experimental results reported here show that they do not eliminate it in general.