A Fair Comparison of Sliding-Mode and Immersion-and-Invariance Observers

Abstract

Cervantes-P\'erez et al. (arXiv:2601.12545) claim, on experimental grounds, that high-gain injection is practically inadmissible, taking the SM observer of Davila, Fridman & Poznyak as a prototypical example. We re-simulate their plant, trajectory and gains with explicit Euler integration at \Delta t = 1 ms and 0.1 ms, and implement both observers on independent hardware. We got three findings: i) the controller gains k_p=1600, k_v=1100 place the tracking-error poles at the "pathological" locations -1.455 and -1098.5 (\zeta = 13.7); the super-twisting scheme lands on the resulting nominal response and is invariant to its own observer gain over a fifty-fold range; ii) the I&I margin is monotone in k_v and reverses near k_v \approx 170, where the loop pole crosses the induced observer pole at -9.0 rad/s; iii) refining the sampling step tenfold moves every I&I figure by a factor of three, while the SM figures move by 0.2%. On hardware, a signum-based super-twisting observer tuned at the Levant-Moreno perturbation bound outperforms I&I on ISE, ITAE and ISC simultaneously, without audible chattering, on a $60 motor with a 0.043^\circ encoder at 500 Hz. The gains of Cervantes-Perez et al. exceed the same prescription by two orders of magnitude; the chattering the authors report is the designed consequence of that excess.