Regularization of non-overshooting quasi-continuous sliding mode control for chattering suppression at equilibrium

Denis Efimov; Michael Ruderman

REVIEW 3 major objections 5 minor 18 references

Regularization of non-overshooting quasi-continuous sliding mode control for chattering suppression at equilibrium

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A max-value smoothing of a quasi-continuous sliding-mode controller removes chattering at the equilibrium while preserving robust stability properties and giving explicit residual error bounds.

desk verdict Useful regularization of a non-overshooting SMC, with a solid iISS part and a central strong-ISS theorem that is asserted rather than proven. read the letter →

arxiv 2506.05283 v1 pith:FOJIQWMU submitted 2025-06-05 math.OC cs.SYeess.SY

Michael Ruderman , Denis Efimov This is my paper · ORCID

classification math.OCcs.SYeess.SY MSC 93D2593D3093B12

keywords slidingmodecontrolchatteringsuppressionregularizationinput-to-statestabilityintegralsecond-ordersystemsnon-overshootingLyapunovfunction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies a controller for second-order systems that drives the state to zero without overshoot, with the control discontinuity located only at the origin; this is the non-overshooting quasi-continuous sliding mode control from the paper's reference [1]. The authors propose replacing the singular factor $1/|x_1|$ in the control law by $1/\max\{\mu, |x_1|\}$, which makes the closed-loop dynamics locally Lipschitz and free of chattering at the equilibrium while leaving the control unchanged outside a small $\mu$-neighborhood. They prove that the regularized system is 0-GAS, is integral-input-to-state stable (iISS), and, under a strengthened gain condition and a sufficiently small disturbance bound $D$, is strongly iISS, meaning it is also input-to-state stable for all disturbances with $\|d\|_\infty \le D$. They derive explicit residual error bounds for constant and resonant harmonic disturbances, and numerical simulations match those bounds. If the claims hold, the regularization gives a practically discretizable controller that keeps the robustness of sliding mode control without the high-frequency switching.

What carries the argument

The load-bearing object is the max-regularized control law in (3), $u = -(1/\max\{\mu, |x_1|\})(\gamma x_1 + |x_2|x_2)$ with $0<\mu\ll 1$. It is locally Lipschitz, hence discretizable by an explicit Euler scheme; for $|x_1| \ge \mu$ it reproduces the original discontinuous dynamics, while for $|x_1| < \mu$ it acts as a linear oscillator with state-dependent damping $|x_2|/\mu$. The proof machinery is a sequence of Lyapunov functions: an energy $E$ and its logarithmic extension $W = \ln(1+E)$ establish 0-GAS and iISS via LaSalle's theorem and the zero-output dissipativity characterization; the original control's Lyapunov function $V$, modified to the smoothed potential $z(x_1)$, shows strict decay for $|x_1| \ge \mu$; and the composite $U = \tilde{V}^3 + \sigma(W)$, with a bounded $C^1$ function $\sigma \in \mathcal{K}$ that is linear near the origin and constant for $|x_1| \ge \mu$, is meant to make the origin ISS for small disturbances by compensating the inner-region bias with quartic damping terms.

What would settle it

Find one admissible pair $(\mu, D)$ with $\gamma > \max\{4, 2D + 4\sqrt{2} D^{1.5}\}$ for which a numerical simulation of (3) with a bounded disturbance $\|d\|_\infty \le D$ produces an unbounded trajectory, or for which no choice of $\sigma$ and weighting makes $U = \tilde{V}^3 + \sigma(W)$ an ISS-Lyapunov function; either observation would refute Theorem 3.

Watch

Extended reading notes

Core claim

The central claim is that the max-regularized controller (3), $u = -(1/\max\{\mu, |x_1|\})(\gamma x_1 + |x_2|x_2)$, suppresses chattering at the equilibrium while preserving the essential stability and robustness properties of the original non-overshooting quasi-continuous sliding mode controller from [1]. The system is globally asymptotically stable for zero disturbance, iISS for bounded measurable disturbances, and, according to Theorem 3, strongly iISS whenever $\gamma > \max\{4, 2D + 4\sqrt{2} D^{1.5}\}$ and $D$ is sufficiently small for the chosen $\mu$. The residual regulation error is estimated by (14) for a constant disturbance, $x_1(t) \to \frac{\mu}{\gamma}\bar{d}$, and by (20) for a resonant harmonic disturbance, $\max|x_1| = \frac{\tilde{d}\,\mu}{\sqrt{\gamma\tilde{d}}}$. The proof combines the original Lyapunov function adapted to the smoothed potential $z(x_1)$ for the outer region with a quartic-and-power Lyapunov function $W$ for the inner region, united through a composite function $U = \tilde{V}^3 + \sigma(W)$.

Load-bearing premise

Strong iISS rests on the claim that a bounded smooth function $\sigma$, linear near the origin and constant outside the $\mu$-window, and a weighting of $\tilde{V}$ against $W$ can always be chosen so that the composite derivative is negative definite for sufficiently small disturbances; the paper states this construction exists but does not provide it.

Editorial extensions

If this is right

The regularized controller can be implemented with a standard explicit Euler discretization, since the right-hand side of (3) is locally Lipschitz.
For $|x_1| \ge \mu$ the control coincides with the original non-overshooting controller, so the original stability and convergence properties carry over outside the $\mu$-neighborhood.
The system is iISS, so trajectories stay bounded for disturbances with bounded integral; strong iISS extends this to bounded $L_\infty$ disturbances when $D$ is small enough.
Residual error obeys explicit formulas: a constant disturbance leads to $x_1 \to \frac{\mu}{\gamma}\bar{d}$, and a resonant harmonic disturbance leads to $\max|x_1| = \frac{\tilde{d}\,\mu}{\sqrt{\gamma\tilde{d}}}$.
The required gain $\gamma > \max\{4, 2D + 4\sqrt{2} D^{1.5}\}$ is more restrictive than the original condition in [1], the price paid for smoothing near the origin.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper leaves open is to make the small-$D$ condition in Theorem 3 quantitative by explicitly constructing $\sigma$ and the weighting; without that, practitioners cannot know how small $D$ must be for a given $\mu$.
The same max-regularization idea could be applied to higher-order quasi-continuous sliding-mode controllers, which also have discontinuities at the origin; the paper does not address that case.
The alternative regularization (22) is introduced and compared numerically but its stability analysis is deferred; if it follows the same Lyapunov pattern, it likely enjoys similar iISS and strong-iISS guarantees.
The resonant bound (20) suggests a tuning trade-off: decreasing $\mu$ reduces the residual error for a given disturbance but makes the control closer to the discontinuous original, so practical design would set $\mu$ just below the actuator's switching threshold.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper regularizes the discontinuous controller from the authors' own preprint [1] by replacing 1/|x1| with 1/max(μ,|x1|) (or 1/(|x1|+μ)) and analyzes the regularized system. What's genuinely new is the iISS/0-GAS analysis for the max-regularized system and the explicit residual-error formulas (14) and (20), neither of which appears in [1]. The simulations credibly show chattering suppression while preserving practical robustness.

The iISS and 0-GAS derivations are explicit and sound: the energy function E and log-energy W give the required dissipation and zero-output smooth dissipativity, and Theorem 2 then yields iISS. The local ISS analysis near the origin via W in (10) is also carried out with real inequalities, though it is local and confined to |x1|<μ. That part holds up.

The soft spot is Theorem 3, the central claim of strong iISS for perturbations bounded by D. The proof does not construct the ISS-Lyapunov function U=V~^3+σ(W). It asserts that some bounded C^1 σ∈K, linear near the origin and constant for |x1|≥μ, and some "proper weighting" of V~ and W will make U an ISS-Lyapunov function, provided D is sufficiently small. No σ is given, no weighting is shown, and no explicit small-D condition is derived. This is not a minor gap: Theorem 3 is the only result that gives ISS on a D-ball, and without a construction the claim is unproved. The iISS claim is unaffected, and the practical ISS claim for |x1|≥μ is also fine, but strong iISS as stated is not established. The residual bound (20) is likewise derived from the self-consistency relation max(x2)=sqrt(μ d~) rather than a rigorous estimate, so it should be treated as a heuristic upper bound, not a proven one.

That said, the core engineering claim—chattering is suppressed while preserving practical stability and convergence—is supported by the simulations and by the explicit parts of the analysis. The flaws are addressable: the authors could add a constructed σ and a quantitative D threshold, or relax Theorem 3 to a weaker statement. The reliance on [1] for the outside-μ Lyapunov derivative is acceptable because [1] is the source of the controller and is cited; it would be cleaner to include the estimate, but this is a minor issue.

I would send this to peer review. It deserves referee time: the problem is relevant, the iISS part is solid, and the unproved theorem is a specific, fixable gap rather than a sign of incoherence. A good referee could push for the missing construction. I'd probably not cite it immediately due to the incomplete Theorem 3, but if the authors fill that gap it becomes a useful reference.

Referee Report

3 major / 5 minor

Summary. The paper proposes two regularizations of the discontinuous quasi-continuous sliding-mode controller introduced in [1]: the max-regularized system (3) and the additive regularization (22). For system (3) the authors prove 0-GAS and iISS using the energy functions E and W=ln(1+E), establish a local ISS property near the origin with a constructed Lyapunov function W, derive residual error bounds (14) and (20) by linearization, and state a strong-iISS result (Theorem 3) for disturbances bounded by a sufficiently small D. Numerical simulations illustrate chattering suppression and agreement with the residual error estimates.

Significance. The iISS and 0-GAS analysis is clean and largely self-contained, with explicit Lyapunov functions, and the local ISS construction near the origin is concrete. The proposed regularization is simple and the numerical comparison with the discontinuous baseline is informative. The paper would be a useful practical contribution if the strong-iISS claim could be made rigorous. However, the proof of Theorem 3 currently rests on an unspecified Lyapunov-function construction, and the residual error estimate is heuristic; these are load-bearing gaps for the advertised robustness guarantees.

major comments (3)

[Section III-C, proof of Theorem 3] The main theorem is not proven. The candidate U(x)=V~^3(x)+sigma(W(x)) is introduced, but sigma is not constructed, the 'proper weighting' of V~ and W is not specified, and the condition 'sufficiently small D' is not quantified in terms of mu, gamma, and epsilon. The proof asserts that the O(||x||^4) bias inside |x1|<mu is compensated by negative terms in Wdot 'provided that the disturbances are sufficiently small', but no inequality for Udot on all of R^2 is given and the boundary |x1|=mu is not analyzed. It is also ambiguous whether W in U is the local Lyapunov function (10) or the log-energy W=ln(1+E); the required derivative bounds differ for the two choices. Definition 3 is therefore not verified, so the strong-iISS statement is unsupported.
[Section III-A, Eqs. (5)-(6)] The negativity of the Lyapunov derivative outside the mu-region is imported from the self-cited preprint [1], which is not published and whose proof is not reproduced. Since Theorem 3 relies on this negativity to conclude that V~ is strictly decreasing for |x1|>=mu, the estimate (5)-(6) must either be proved in the paper or stated as an explicit assumption with its exact domain and parameter range. As written, the main result inherits an unverified external condition.
[Section III-B2, Eqs. (16)-(20)] The residual error bound (20) is derived by setting sigma*=|max(bar x2)|/mu and then solving the self-consistency relation max(bar x2)=sqrt(d~ mu); this infers the damping coefficient from the solution's own maximum velocity rather than proving an estimate for the nonlinear system (3). In addition, (17) is the particular solution of the linearized oscillator (15) for a resonant harmonic disturbance, and no argument is given that this case dominates all bounded disturbances |d(t)|<=d~. The numerical agreement in Fig. 1 is suggestive, but (20) should be presented as a heuristic bound or derived from the Lyapunov analysis. The same caveat applies to (14), which is obtained from the linearized model (13).

minor comments (5)

[Section I and Section III] The notation 'L^1_infty' is confusing; the paper defines L^m_infty in the notation section, so the superscript 1 should be removed or explained as the dimension of the disturbance.
[Eqs. (5)-(6)] The displayed inequality appears to have a missing opening parenthesis before the term -epsilon(gamma - 1/2 - |d| - ...); please reformat to make the bracketing unambiguous.
[Fig. 1] The lower-row time axis label reads 't (t)' and should be 't (s)'.
[Section III-C] The phrase 'for |x1| < mu the constant negative term is obviously dominating for |x1| <= 2 mu sqrt(gamma)' is unclear because k(x1) is quadratic in x1 there; please state the intended bound explicitly.
[Section III-D] The alternative regularization (22) is introduced with a Lyapunov candidate, but the promised 'similar analysis' is omitted; if this scheme is meant to be more than an example, the authors should at least state whether it also enjoys 0-GAS and iISS.

Circularity Check

1 steps flagged · score 4.0 of 10

One secondary prediction reduces by construction: the residual-error bound (20) is the fixed point of re-identifying the damping sigma* with the solution's own maximum velocity; the central 0-GAS, iISS, and Theorem 3 decay proofs are self-contained.

fitted input called prediction [Section III-B.2 (Analysis of linearization), Eqs. (13), (18)-(20)]
"From (18) and sigma* = |max(x2bar)| mu^-1 one obtains max(x2bar) = dtilde mu / max(x2bar), that leads to max(x2bar) = sqrt(dtilde mu). (19) Since for a forced harmonic oscillator (15) in steady-state, the maximal value of a periodic x2(t) leads to the correspondingly maximal |x1|, one can obtain from (17) and (19) max|x1| = dtilde mu / sqrt(gamma dtilde). (20) The estimate (20) constitutes the upper bound of the control error x1(t) for a bounded disturbance |d(t)| <= dtilde < D."

sigma* enters (13) as the damping coefficient |x2(t*)|/mu of the locally linearized model at a transient instant t*; in deriving (19) the paper silently re-identifies it as |max(x2bar)|/mu, i.e., the damping is defined in terms of the solution's own maximum velocity, the very quantity the bound is supposed to predict. Substituting that identification into the steady-state amplitude x2bar = dtilde/sigma* (18) yields the self-consistency equation max(x2bar) = dtilde mu / max(x2bar), whose solution (19) and consequent bound (20) are forced algebraically by the identification ansatz rather than derived from the nonlinear dynamics (3).

full rationale

The paper's central stability claims are not circular. The 0-GAS argument (Section III) uses E with Edot = -|x2|x2^2/max{mu,|x1|}, LaSalle invariance, and the observation that no nontrivial trajectory stays in {x2=0}; the iISS proof uses W = ln(1+E) with Wdot <= |d|/sqrt(2), establishing zero-output smooth dissipativity and invoking the external Theorem 2. Both are self-contained. In Theorem 3, the derivative of Vtilde is computed explicitly in the paper (the Young-inequality chain leading to k(x1) and e(x1,D) is written out), so the self-cited estimate (5)-(6) from [1] is not load-bearing for the main result; it is a non-load-bearing self-citation. The genuinely circular element is confined to Section III-B.2: the residual error bound (19)-(20) is obtained by re-identifying sigma*, first defined in (13) as the linearized damping |x2(t*)|/mu, with the solution's own steady-state maximum |max(x2bar)|/mu; substituting this identification into (18) gives max(x2bar) = dtilde mu / max(x2bar) as an algebraic fixed point, so (20) reduces to the self-consistency ansatz rather than to an estimate from the nonlinear dynamics. This is a fitted-input-called-prediction structure, but it concerns only the secondary performance estimate (the control error bound), not the main stability theorems. Separately, and per the reviewing rule on missing support: the final step of Theorem 3, where U = Vtilde^3 + sigma(W) is asserted to be an ISS-Lyapunov function 'by a proper weighting' with sigma, the weighting, and the small-D threshold never constructed, is an existence assertion rather than a demonstrated reduction; since no equation is exhibited that equals its own input, I treat this as a proof gap (a correctness risk), not as circularity. Overall, the 0-GAS, iISS, and strong-iISS structural claims retain independent content, and the main derivation does not reduce to its inputs; the score of 4 reflects the one secondary prediction that does reduce by construction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four assumptions: standard control-theoretic tools; the matched bounded disturbance model; the self-cited Lyapunov estimate from [1]; and the unconstructed ISS-Lyapunov weighting in Theorem 3. No new physical entities are introduced beyond design parameters (γ, μ, ε, D) whose values appear in simulations.

free parameters (4)

γ (control gain) = γ=100 in simulations; condition γ>max{4,2D+4√2 D^1.5}
Feedback gain of the controller; a design parameter, not fitted to data, but the main theorem and residual bounds depend on it.
μ (regularization width) = μ=0.0001 in simulations; μ∈{0.01,0.05} in residual-error simulations
Defines the boundary layer around the origin where chattering is removed; the proof of strong iISS requires D sufficiently small relative to μ, but no quantitative relation is given.
ε (cross-term weight in Lyapunov function) = ε=1/3 in Theorem 3 (admissible range 0<ε<√(2γ))
Introduced in Lyapunov function (4)/(Ṽ) and required to satisfy 0<ε<√(2γ); chosen as 1/3 to satisfy theorem conditions.
D (disturbance upper bound) = D=1 and D=10 in simulations
Assumed known bound on matched disturbance; enters the gain condition and residual error bounds.

assumptions (4)

domain assumption Disturbance d is Lebesgue measurable, essentially bounded, and matched; its bound D is known and equal to the worst-case amplitude.
Stated in the problem statement and Section II; all stability and residual error results are conditional on this assumption.
ad hoc to paper For |x1| ≥ μ, dynamics (3) coincide with (1), and the Lyapunov derivative estimate (5)-(6) from the self-cited preprint [1] is valid.
This identification is used in Section III.A to conclude stability outside the μ-region; the present paper does not re-derive (5)-(6), so the argument leans on [1].
ad hoc to paper There exists a bounded C^1 function σ∈K, linear near the origin and constant for |x1| ≥ μ, and a weighting of Ṽ and W such that U = Ṽ^3 + σ(W) is an ISS-Lyapunov function for sufficiently small D.
This existence is asserted in the proof of Theorem 3 (Section III.C); no explicit construction or quantitative small-D condition is provided, and the strong-iISS conclusion depends on it.
standard math Standard nonlinear control results: LaSalle's invariance principle, Young's inequality, final value theorem, and the ISS/iISS Lyapunov characterizations (Theorems 1 and 2).
These are invoked throughout the derivations and are accepted background results in nonlinear control theory.

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Cite this review

Pith. "Pith review of Regularization of non-overshooting quasi-continuous sliding mode control for chattering suppression at equilibrium." pith.science (2026). https://pith.science/paper/FOJIQWMU

@misc{pith2026250605283,
  author       = {Pith},
  title        = {Pith review of: Regularization of non-overshooting quasi-continuous sliding mode control for chattering suppression at equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FOJIQWMU}},
  note         = {Machine review of arXiv:2506.05283}
}

read the original abstract

Robust finite-time feedback controller introduced for the second-order systems in [1] can be seen as a non-overshooting quasi-continuous sliding mode control. The paper proposes a regularization scheme to suppress inherent chattering due to discontinuity of the control [1] in the origin, in favor of practical applications. A detailed analysis with ISS and iISS proofs are provided along with supporting numerical results.

Figures

Figures reproduced from arXiv: 2506.05283 by the authors.

**Figure 2.** Phase portrait of the regularized and not regularized closed-loop control systems: convergence of trajectories with x(0) = [0.0001, 1]⊤ in (a), zoom in the [−µ, µ] vicinity to origin in (b), and control signal in (c). The numerical simulations with the first-order Euler solver and the fixed step sampling of 0.00001 sec, the assigned control gain γ = 100 and regularization factor µ = 0.0001, and the initial condition… view at source ↗

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Non-overshooting continuous in convergence sliding mode control of second-order systems

M. Ruderman and D. Efimov, “Non-overshooting quasi-continuous sliding mode control for second-order systems,”arXiv preprint arXiv:2406.14000, 2024

work page Pith review arXiv 2024

[2] [2]

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H. Kwakernaak, “Robust control and H-infinity optimization – tutorial paper,”Automatica, vol. 29, no. 2, pp. 255–273, 1993

work page 1993

[3] [3]

High-gain robust control,

Y . Tsypkin and B. T. Polyak, “High-gain robust control,”European Journal of Control, vol. 5, no. 1, pp. 3–9, 1999

work page 1999

[4] [4]

˚Astr¨om and T

K. ˚Astr¨om and T. H ¨agglund,Advanced PID control. ISA - Instru- mentation, Systems and Automation Society, 2006

work page 2006

[5] [5]

Universal integral controllers for minimum-phase nonlinear systems,

H. Khalil, “Universal integral controllers for minimum-phase nonlinear systems,”IEEE Tran. on Aut. Con., vol. 45, no. 3, pp. 490–494, 2000

work page 2000

[6] [6]

Hippe,Windup in control: its effects and their prevention

P. Hippe,Windup in control: its effects and their prevention. Springer, 2006

work page 2006

[7] [7]

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A. Ilchmann and E. Ryan, “High-gain control without identification: a survey,”GAMM-Mitteilungen, vol. 31, no. 1, pp. 115–125, 2008

work page 2008

[8] [8]

Shtessel, C

Y . Shtessel, C. Edwards, L. Fridman, and A. Levant,Sliding mode control and observation. Springer, 2014

work page 2014