{"id":"2d11b5af-9a0b-4aba-8ee7-f3c40e8c0899","arxiv_id":"2506.05283","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Regularizing the singular control law with max(μ,|x1|) or |x1|+μ removes chattering at the origin while preserving iISS, and gives explicit residual error bounds.","lead":"This paper smooths a recently proposed sliding mode controller so it stops rapidly switching near the target, a problem called chattering, while keeping the system stable and robust to disturbances. It gives stability proofs and explicit residual error formulas for two smoothing schemes, with simulations showing the chattering disappears.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3 is unproved: the ISS-Lyapunov function U = V~^3 + σ(W) is asserted with no explicit σ, no weighting, and no quantified small-D threshold.","rationale":"The reader identified the missing construction of σ and the missing small-D condition as the weakest assumption, and my stress-test confirms that this is the load-bearing gap. The iISS and 0-GAS arguments for (3) are explicit and appear sound, and the simulations credibly show chattering suppression. However, Theorem 3 is the only result establishing ISS with respect to small bounded disturbances, and its proof ends with an existential assertion rather than a verifiable construction. One cannot certify strong iISS from the paper as written. This does not warrant rejection: the claim may be true, and the missing construction is a concrete, addressable step. The appropriate disposition remains CONDITIONAL, with the condition being an explicit Lyapunov construction and a quantitative D threshold. My recommendation is therefore to keep the reader's verdict unchanged.","tokens_in":9643,"tokens_out":13643,"duration_ms":112439,"concrete_test":"Provide a fully explicit construction: choose a smooth bounded σ with σ(s)=s near s=0 and σ'(s)=0 for s ≥ S, where S = inf{|W(x)| : |x1| ≥ μ}; set U = V~^3 + c σ(W) with c > 0; compute Udot along (3) and derive a closed-form threshold D*(μ, γ, c, S) such that Udot ≤ -α(||x||) + η(D) for all D ≤ D*. If no such (c, S, D*) exists for any γ > max{4, 2D + 4√2 D^1.5}, μ > 0, and D > 0 satisfying the theorem's hypotheses, then Theorem 3 is false; if it succeeds, the explicit D* removes the unquantified 'sufficiently small D'.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is strong iISS of (3) for ||d||∞ ≤ D. In the proof of Theorem 3, after deriving that V~ is only a practical ISS-Lyapunov function (a nonzero bias persists inside |x1| < μ), the authors assert U = V~^3 + σ(W) with σ ∈ K bounded, C^1, linear near the origin, and constant for |x1| ≥ μ, and claim that 'by a proper weighting' the O(||x||^4) bias is compensated by negative terms in Wdot 'provided that the disturbances are sufficiently small.' No σ is constructed, no weighting is specified, and no D threshold is computed. The proof also leaves W ambiguous: it may be the local Lyapunov function from (10) or the log-energy from Section III, and the compensation argument is never demonstrated for either choice. Since Definition 3 is not checked for the system on all of R^2, Theorem 3 — the only result giving ISS on a D-ball — is unsupported. The residual error bound (20) is also derived from the self-consistency relation max(x2) = sqrt(d~ μ) rather than from a rigorous equilibrium estimate, but that is secondary to the unproved strong-iISS claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9939,"tokens_out":5911,"duration_ms":64668,"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":[{"comment":"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":"Section III-C, proof of Theorem 3"},{"comment":"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":"Section III-A, Eqs. (5)-(6)"},{"comment":"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).","section":"Section III-B2, Eqs. (16)-(20)"}],"minor_comments":[{"comment":"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.","section":"Section I and Section III"},{"comment":"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.","section":"Eqs. (5)-(6)"},{"comment":"The lower-row time axis label reads 't (t)' and should be 't (s)'.","section":"Fig. 1"},{"comment":"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":"Section III-C"},{"comment":"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.","section":"Section III-D"}],"recommendation":"major_revision","confidential_remarks":"The paper leans heavily on the unpublished self-cited preprint [1] for the key derivative estimate (5)-(6), and the proof of Theorem 3 is incomplete as written. The central strong-iISS claim is defensible but needs a real construction of sigma, a specification of the weighting, and a quantitative small-D condition. I would advise requesting those before publication; the iISS and 0-GAS portions are sound and the numerical results are plausible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThat 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.\n\nI 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.","headline":"Useful regularization of a non-overshooting SMC, with a solid iISS part and a central strong-ISS theorem that is asserted rather than proven.","tokens_in":10446,"tokens_out":2186,"would_cite":false,"duration_ms":22666,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93D25","93D30","93B12"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["sliding mode control","chattering suppression","regularization","input-to-state stability","integral input-to-state stability","second-order systems","non-overshooting control","Lyapunov function"],"falsifier":"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.","tokens_in":9356,"feed_emoji":"⚙️","tokens_out":9756,"duration_ms":105174,"temperature":0.7,"pith_summary":"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.","feed_headline":"Chattering at equilibrium tamed by max-value regularization","feed_subtitle":"A locally Lipschitz smoothing preserves convergence and robustness without high-frequency switching.","key_machinery":"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.","core_discovery":"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 $U = \\tilde{V}^3 + \\sigma(W)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original non-overshooting quasi-continuous sliding-mode controller, its Lyapunov function V in (4), and the gain condition that the regularization extends.","marker":"[1]"},{"why":"Motivates the nonlinear damping inverse to the output distance that underlies the control structure.","marker":"[15]"},{"why":"Provides the ISS and iISS Lyapunov characterizations used to convert the paper's Lyapunov functions into stability properties.","marker":"[16]"},{"why":"Defines the class of quasi-continuous high-order sliding-mode controllers whose origin chattering this paper removes.","marker":"[13]"},{"why":"Supplies the alternative additive regularization form (21) used in the numerical comparison.","marker":"[18]"}],"fun_headline_variants":["Max-value regularization eliminates sliding mode chattering","Smooth sliding mode control: no chattering at equilibrium","Non-overshooting SMC regularized for chattering-free equilibrium","Max-regularized sliding mode: Lipschitz smooth, chattering gone"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Max-value regularization eliminates sliding mode chattering","Smooth sliding mode control: no chattering at equilibrium","Non-overshooting SMC regularized for chattering-free equilibrium","Max-regularized sliding mode: Lipschitz smooth, chattering gone"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000626,"raw_usage":{"total_tokens":2852,"prompt_tokens":858,"completion_tokens":1994,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":474,"completion_tokens_details":{"reasoning_tokens":1924}},"tokens_in":474,"tokens_out":1994,"duration_ms":18637,"temperature":1.0,"reasoning_tokens":1924,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:22:11.345206+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Non-overshooting continuous in convergence sliding mode control of second-order systems","cited_arxiv_id":"2406.14000","evidence_quote":"Supplies the original non-overshooting quasi-continuous sliding-mode controller, its Lyapunov function V in (4), and the gain condition that the regularization extends."},{"cited_title":"Optimal nonlinear damping control of second-order systems,","cited_arxiv_id":null,"evidence_quote":"Motivates the nonlinear damping inverse to the output distance that underlies the control structure."},{"cited_title":"Input to state stability: Basic concepts and results,","cited_arxiv_id":null,"evidence_quote":"Provides the ISS and iISS Lyapunov characterizations used to convert the paper's Lyapunov functions into stability properties."},{"cited_title":"Quasi-continuous high-order sliding-mode controllers,","cited_arxiv_id":null,"evidence_quote":"Defines the class of quasi-continuous high-order sliding-mode controllers whose origin chattering this paper removes."},{"cited_title":"Convergent dynamics of optimal nonlinear damping control,","cited_arxiv_id":null,"evidence_quote":"Supplies the alternative additive regularization form (21) used in the numerical comparison."}],"review_version":1}