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

Luis Cervantes-Pérez; Víctor Santibáñez; Jesús Sandoval; Romeo Ortega; José Guadalupe Romero

eess.SYOct 14, 2026

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

Authors: Luis Cervantes-Pérez, Víctor Santibáñez, Jesús Sandoval, Romeo Ortega, José Guadalupe Romero

Organizations: Tecnológico Nacional de México / I. T. de La Laguna, Torreón, Coahuila, México · Tecnológico Nacional de México / I. T. de La Paz, La Paz, Baja California Sur, México · Departamento de Ingeniería Eléctrica y Electrónica / Instituto Tecnológico Autónomo de México, CDMX, Ciudad de México, México

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.

Figures & tables

Description	Notation	Value	Units
Length of Link	$l$	0.313	$\mathrm{m}$
Distance to the center of mass	$l_{b}$	0.0641	$\mathrm{m}$
Mass of Link	$m$	22.4466	$\mathrm{K_{g}}$
Inertia relative to center of mass	$J$	0.7013	$\rm Kg\ m^{2}$
Viscous friction coefficient	$\theta_{1}$	$5.317$	$\mathrm{Nm\ s/rad}$
Coulomb friction coefficient	$\theta_{2}$	$11.6403$	$\mathrm{Nm}$

Table 1: Physical parameters of the pendular device.

Figure 3: Behavior of the reconstructed perturbation $\dot{\delta}(t)$ for the four SM configurations using different values of $L$ .

Configuration	$L$	$\alpha_{1}$	$\alpha_{2}$	$\frac{\bar{\theta}_{1}}{J}$	$\frac{\bar{\theta}_{2}}{J}$
SM-H-SL	$25$	$27.50$	$7.50$	$0.01$	$0.01$
SM-H-BL	$30$	$33.00$	$8.22$	$0.01$	$0.01$
SM-NH-SL	$20$	$22.00$	$6.71$	$7$	$15$
SM-NH-BL	$80$	$88.00$	$13.42$	$7$	$15$

Table 2: Tuning parameters for the SM configurations.

Figure 4: Time behavior of the estimated velocity $\hat{x}_{2}(t)$ against the actual velocity

Figure 5: Velocity- and friction-parameter-estimation errors for all observer configurations in the open-loop test.

Figure 6: Velocity- and friction-parameter-estimation errors for the I $\&$ I implementation using different values of the damping ratio $\zeta$ .

Figure 7: Trajectory-tracking errors for the I $\&$ I implementation using different values of the damping ratio $\zeta$ .

Figure 8: Velocity- and friction-parameter-estimation errors for the SM implementation using different values of the damping ratio $\zeta$ .

Figure 9: Trajectory-tracking errors for the SM implementation using different values of the damping ratio $\zeta$ .

Figure 10: Velocity- and friction-parameter-estimation errors for both controllers following a change in the reference trajectory, with $L=32.4$ used for the SM observer.

Figure 11: Trajectory-tracking errors for both controllers following a change in the reference trajectory, with $L=32.4$ used for the SM observer.

Figure 12: Control-input behavior following a change in the reference trajectory for $L=32.4$ .

Figure 13: Behavior of the reconstructed perturbation $\dot{\delta}(t)$ following a change in the reference trajectory for $L=32.4$ .

Figure 14: Behavior of the reconstructed perturbation $\dot{\delta}(t)$ relative to its corresponding prescribed bound $L$ following a change in the reference trajectory, with the SM observer evaluated using different values of $L$ .

Figure 15: Control signals for the I $\&$ I and SM controller configurations following a change in the reference trajectory, with the SM controller evaluated using different values of $L$ .

Figure 16: Velocity- and friction-parameter-estimation errors for the I $\&$ I and SM implementations following a change in the reference trajectory, with the SM observer evaluated using different values of $L$ .

Figure 17: Trajectory-tracking error signals for the I $\&$ I and SM controller configurations following a change in the reference trajectory, with the SM controller evaluated using different values of $L$ .

Sliding Mode Control

Sliding mode control uses switching feedback to drive systems toward designed surfaces and maintain robust behavior despite disturbances and uncertainty.

See all 14 papers

Learned Lyapunov Shielding for Adaptive Control
May 7, 2026Giansalvo Cirrincione, Adriano FagioliniLyapunov-Based Control Safe RL
Laboratoire LTI, Université de Picardie Jules Verne, Amiens, France. · MIRPALab, Department of Engineering, University of Palermo, 90128 Palermo, Italy.
Temporal Attention for Adaptive Control of Euler-Lagrange Systems with Unobservable Memory
May 7, 2026Giansalvo Cirrincione, Adriano FagioliniRL Control Robot Trajectory Tracking
Laboratoire LTI, Université de Picardie Jules Verne, Amiens, France · MIRPALab, Department of Engineering, University of Palermo, 90128 Palermo, Italy
Passive Variable Impedance For Shared Control
Apr 22, 2026Maximilian Mühlbauer, Nepomuk Werner, Ribin Balachandran +4Impedance Control Feedback Control
School of Computation, Information and Technology, Sensor Based Robotic Systems and Intelligent Assistance Systems, Technical University Munich, Friedrich-Ludwig-Bauer-Str. 3, Garching, Germany. · German Aerospace Center (DLR), Robotics and Mechatronics Center (RMC), Münchener Str. 20, 82234 Weßling, Germany.

May 7, 2026J/K move · Enter open · S save

Learned Lyapunov Shielding for Adaptive Control

Giansalvo Cirrincione, Adriano Fagiolini

Laboratoire LTI, Université de Picardie Jules Verne, Amiens, France. · MIRPALab, Department of Engineering, University of Palermo, 90128 Palermo, Italy.

Lyapunov-Based Control Safe RL Adaptive Control Control Lyapunov Functions

Open paper AskPDF arXiv

We augment the Slotine--Li adaptive controller for Euler--Lagrange systems with three learned components: a structured-quadratic Lyapunov function

V_ψ

whose positive-definiteness follows from a Cholesky parameterization, a residual Soft Actor--Critic policy that adds bounded torque corrections to the analytic baseline, and a physics-informed neural network that estimates unmodeled dynamics. A closed-form safety filter, derived from the single affine constraint

\dot V_ψ+ αV_ψ\le 0

, projects every policy output onto the safe set without requiring an online QP solver. We prove: global feasibility of the filter under a drift-decay condition on the control-degeneracy set; exponential stability under exact shielding, with a robust extension whose margin depends on the PINN approximation error; almost-sure convergence of the three-timescale policy--certificate--multiplier updates to a KKT point; and a PAC generalization bound for the certificate over compacts. On a 2-DOF manipulator with nonlinear friction and variable payload, the learned certificate accounts for most of the empirical gain: tracking error drops by 41% on nominal friction and 24% on aggressive friction at the centroid of the training distribution. A 7-DOF scalability study on a Franka Emika Panda confirms clean convergence of the full pipeline at industrial scale, identifies the conditions under which gains over exact model-based baselines should and should not be expected, and documents a warm-start pathology of the learned certificate that has practical implications for deployment.

Abstract

Sliding Mode Control

Explore similar work

Learned Lyapunov Shielding for Adaptive Control

Temporal Attention for Adaptive Control of Euler-Lagrange Systems with Unobservable Memory

Passive Variable Impedance For Shared Control