{"id":"4040134b-b32b-4783-a4f3-080da77ec743","arxiv_id":"1908.06782","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"New explicit feedback controllers achieve prescribed-time stabilization of chains of integrators with input-to-state stability guarantees under measurement noise and unmatched disturbances.","lead":"This paper designs feedback laws that steer a chain of integrators to zero by a user-chosen deadline, even when the measurements are noisy and the model has unmatched disturbances. It recasts a classical missile-guidance method in a new time-varying homogeneity framework and then builds explicit sliding-mode-inspired controllers with guaranteed error bounds.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 34's matched-disturbance fixed-time convergence is false: for n=1, constant d2=c e1 gives a nonzero equilibrium, so the claim must be withdrawn; the ISS part is not contradicted.","rationale":"The reader's weakest assumption (gap between (63) and ISS via [17, Theorem 2]) is legitimate: Proposition 37 gives only a limsup bound and no ISS-Lyapunov function is constructed. However, Theorem 28 supplies unperturbed fixed-time stability and the appendix inequalities imply boundedness, so that gap may be repairable. The false matched-disturbance convergence is not repairable as stated and is a concrete counterexample to a sentence in the main theorem. It does not undermine the ISS-type statement, so the appropriate action is to keep the CONDITIONAL verdict: require the authors to delete or qualify the fixed-time convergence assertion and to complete the ISS-characterization argument. This is why I mark agreement as partial rather than full.","tokens_in":25716,"tokens_out":15917,"duration_ms":174216,"concrete_test":"Set n=1, b=1, d1=0, d2(t)=c e1, c≠0 in (62). Compute the scalar feedback from Definition 23: u=-ℓ1 |x|^{1+κ} sign(x), κ=±κ0. Solve -ℓ1 |x|^{1+κ} sign(x)+c=0; for every c≠0 there is a nonzero root x* = sign(c)(|c|/ℓ1)^{1/(1+κ)}, so the origin is not an equilibrium of the disturbed closed loop and fixed-time convergence to 0 is impossible. This analytic check settles the concern without simulation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 34's second assertion fails already for the scalar case. In (62) take n=1, b=1, d1=0, d2(t)=c e1 with c≠0. For the scalar chain, the feedback in Definition 23 (with β0=r2=1+κ) is u=-ℓ1 |x|^{1+κ} sign(x), and κ(x) is either -κ0 or κ0 away from the V0 transition band; both branches have 1+κ>0. The closed-loop scalar equation is ẋ = -ℓ1 |x|^{1+κ(x)} sign(x)+c. The function g(x)=ℓ1 |x|^{1+κ} sign(x) is continuous, strictly increasing, and tends to ±∞ as x→±∞, so there is a unique equilibrium x* = sign(c)(|c|/ℓ1)^{1/(1+κ)} ≠0. No trajectory can converge to 0, contradicting 'convergence occurs in fixed time'. The ISS assertion itself is not touched by this counterexample. Separately, the proof of Proposition 37 supplies only the limsup estimate (63); Theorem 28 gives unperturbed fixed-time stability, and the appendix inequalities would yield boundedness, but the paper does not verify the exact hypothesis of [17, Theorem 2] (e.g., an ISS-Lyapunov function or the AG+GS equivalence), so that derivation should be completed before the ISS claim is considered fully established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper treats prescribed-time stabilization of chains of integrators, both unperturbed and perturbed. The first part recasts proportional navigation feedback in the language of time-varying weighted homogeneity, recovering and simplifying previous results from Song et al. [16] and providing explicit statements about the effects of measurement noise. The second part constructs fixed-time stabilizing feedbacks from sliding-mode-type homogeneous controllers with a state-dependent homogeneity degree, gives explicit parameter choices in Section 4.2, and then claims in Section 4.3 that the closed-loop system (62) is ISS with respect to measurement noise and unmatched disturbances, with an additional fixed-time convergence guarantee for matched disturbances. A prescribed-time version is obtained by a homogeneity time rescaling.","tokens_in":25973,"tokens_out":11495,"duration_ms":117409,"significance":"If the ISS claim in Theorem 34 is established, it would be a meaningful step beyond prior work: [16] suggested poor behavior under measurement noise and [12] only obtained ISpS. The paper's transparent time-varying homogeneity viewpoint, its recovery of known results with shorter LMI-based arguments, and its explicit determination of the controller parameters are genuine strengths. The false matched-disturbance convergence assertion and the incomplete justification of the ISS implication mean that the main theorem as stated is not proven, but the underlying construction appears plausible and worth repairing.","major_comments":[{"comment":"The second assertion of Theorem 34 is false as stated. Take n=1, b=1, d1=0, and d2(t)=c e1 with c≠0. With Definition 23 and Definition 27, the closed-loop scalar equation is \\dot{x} = -\\ell_1 |x|^{1+\\kappa(x)} sign(x) + c, where 1+\\kappa(x) ∈ [1-\\kappa_0,1+\\kappa_0] ⊂ (0,∞). The function g(x)=\\ell_1 |x|^{1+\\kappa(x)} sign(x) is continuous, strictly increasing, satisfies g(0)=0 and g(x)→±∞ as x→±∞. Hence there is a unique nonzero equilibrium x* with g(x*)=c, and this equilibrium is attracting. No trajectory converges to 0, contradicting the claimed fixed-time convergence for matched disturbances. This assertion should be withdrawn or replaced by an ultimate-boundedness statement such as limsup |x(t)| ≤ g^{-1}(||d2||∞).","section":"Section 4.3, Theorem 34"},{"comment":"The inference from the limsup estimate (63) to the ISS property is not demonstrated. The sentence preceding Proposition 37 says that, by invoking [17, Theorem 2], it is enough to prove (63), but the paper neither states the exact characterization being used nor verifies its hypotheses. In particular, the proof does not establish global existence for (62) under arbitrary bounded inputs, does not prove the unforced system is 0-GAS in the precise sense required by the equivalence, and does not exhibit the ISS-Lyapunov function that the standard Lyapunov characterization would require. Since the ISS conclusion is the central robustness claim of the paper, this step must be completed with either a direct proof of the [17] implication adapted to this system or a separate ISS-Lyapunov construction.","section":"Section 4.3, Proposition 37 and proof of Theorem 34"}],"minor_comments":[{"comment":"The definition of 'class KL' is incorrect: a single-variable function F: R_+ → R_+ that is increasing with F(0)=0 and F(s)→∞ is a class K∞ function, not a class KL function. This affects the notation used for F, F1, F2, and F3 in and around inequalities (65)-(67).","section":"Section 4.3, before Proposition 37"},{"comment":"The admissible range of κ0 is inconsistent: Definition 27 states κ0 ∈ (0, 1/(2n)), Theorem 28 states κ0 ∈ (0, 1/n), and Proposition 33 contains the interval [−1/(2n),−1/(2n)], which is a typo. These ranges should be harmonized.","section":"Definition 27 and Theorem 28"},{"comment":"The notation ⌈x⌋^α or ⌊x⌉^α for signed fractional powers is used throughout without a definition; since the construction relies heavily on the sign convention, a short definition would substantially improve readability.","section":"Equations (32)-(33) and (42)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope. The main obstacles are the false matched-disturbance convergence claim and the incomplete verification of the ISS implication; neither appears to be a fundamental obstruction, so a major revision rather than rejection is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Know this: the paper's real contribution is the time-varying homogeneity reframing of proportional navigation feedback, plus an explicit state-dependent homogeneity design with workable parameter estimates. The ISS-type result advertised in Section 4.3 is not established as written, and one of its subclaims is false.\n\nWhat is new and good. Section 3.1 recovers [16]'s prescribed-time results with a cleaner LMI/linear argument and shows transparently why measurement noise defeats PNF. Section 3.2 and Section 4.1 adapt [8] and [12]'s continuous fixed-time feedback in a credible way; the parameter estimates in Section 4.2 are genuinely useful. The paper is honest about provenance: [3], [8], [9], [12], [16] are standard, two co-authored by Chitour, and the new consequences do not collapse into those inputs. No circularity problem.\n\nSoft spots. First, Theorem 34 states that for d1=0 and d2 parallel to en, convergence occurs in fixed time. That is false already for n=1. With d2=c e1, c≠0, the scalar closed loop is x'=-ℓ1|x|^{1+κ(x)} sign(x)+c. The right side is continuous, strictly increasing, and has a unique nonzero equilibrium, so no trajectory can go to zero. The matched-disturbance bonus claim must be withdrawn or substantially weakened.\n\nSecond, Proposition 37 gives the limsup estimate (63), then invokes [17, Theorem 2] to conclude ISS for system (62). The paper never constructs the required ISS-Lyapunov function nor verifies the underlying equivalence. A limsup bound is a gain/boundedness statement; it does not by itself imply ISS. This is a gap, not a contradiction, but it sits in the main robustness theorem, so the headline 'ISS under measurement noise and unmatched disturbances' is unproven until that step is filled. The proof of Proposition 37 also needs tightening in places.\n\nWhat holds up. The homogeneous-framework material and the explicit fixed-time design are solid enough. The ISS claim may survive a careful repair; I see no obvious counterexample to it, only to the exact-convergence add-on.\n\nWho this is for: control theorists working on prescribed-time and fixed-time stabilization. The paper deserves a serious referee, but it needs revision, not desk rejection. Send it out and tell the referee to focus on Theorem 34 and Proposition 37.","headline":"The time-varying homogeneity reframing is the genuinely useful part; the advertised ISS theorem is not yet proved and its matched-disturbance fixed-time claim is false already in the scalar case.","tokens_in":26558,"tokens_out":5099,"would_cite":false,"duration_ms":54122,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93D15","93D30","93D40","93C10","93C73"],"pacs":[],"model":"deepseek-v4-flash","headline":"A sliding-mode feedback whose homogeneity degree depends on the state stabilizes a chain of integrators in prescribed time and is input-to-state stable under measurement noise and unmatched disturbances.","keywords":["prescribed-time stabilization","chain of integrators","input-to-state stability","sliding mode control","time-varying homogeneity","fixed-time stability","measurement noise","unmatched disturbances"],"falsifier":"Take n=2, fix a horizon T, set the measurement noise to the constant d1=(0,epsilon) and the unmatched disturbance to d2=(epsilon,0), and compute the trajectory of (62) with the explicit parameters of Section 4.2. If the limiting size of x(t) does not go to zero as epsilon tends to zero, or if it grows faster than linearly in epsilon, then the ISS statement of Theorem 34 is false; equivalently, trying to build the ISS-Lyapunov function required by the cited ISS characterization for this two-dimensional case would settle whether the proof gap is fatal.","tokens_in":25451,"feed_emoji":"🕒","tokens_out":12511,"duration_ms":115581,"temperature":0.7,"pith_summary":"The paper tackles the problem of driving a chain of n integrators to zero within a user-chosen horizon T, even when the model is perturbed. In the first part, the authors recast the classical proportional navigation feedback as a time-varying homogeneity transformation, recovering earlier prescribed-time results with simpler proofs and making their robustness limits transparent. The central contribution is a sliding-mode feedback whose homogeneity degree depends on the current state; it stabilizes the pure chain in prescribed time without control gains that blow up as t approaches T. The paper's strongest assertion is that the same feedback makes the perturbed system (62) input-to-state stable with respect to any bounded measurement noise d1 and unmatched disturbance d2, and that exact convergence in fixed time persists when d1=0 and d2 acts along the input direction.","feed_headline":"Feedback tames integrator chains on schedule, even with noise","feed_subtitle":"Sliding-mode design stabilizes any chain in prescribed time and tolerates measurement noise and unmatched disturbances.","key_machinery":"The mechanism is a two-step construction. First, prescribed-time stabilization is converted into fixed-time stabilization by the time-varying homogeneity change of coordinates: with $\\lambda(t)=1/\\int_t^T a(\\xi)d\\xi$ and $s(t)=\\int_0^t \\lambda(\\xi)d\\xi$, the state $y(s)=D^r_{\\lambda(t)}x(t)$ obeys $y'=(a(s)D_r+J_n)y+(bu+d)e_n$, where the diagonal term $a(s)D_r$ is the cost of the time change. Second, for the robust design the controller is the recursive sliding-mode law $u=\\omega^H_{\\kappa(x)}(x)$ whose homogeneity degree $\\kappa(x)$ switches continuously between $-\\kappa_0$ and $\\kappa_0$ depending on the level set of $V_0(x)$; the associated Lyapunov functions $V_\\kappa$ satisfy $\\dot V_\\kappa \\le -C V_\\kappa^{1+\\alpha(\\kappa)}$, so positive degree gives fixed-time convergence and negative degree finite-time convergence. The parameters $\\ell_j$, $m$, $\\kappa_0$, and the rescaling factor $\\mu$ are chosen explicitly so the settling time is bounded above by a computable expression. The ISS proof then works with the minimum $Z(x)=\\min(V_0(x), V_{\\kappa_0}^{1+\\alpha(\\kappa_0)}(x), V_{-\\kappa_0}^{1-\\alpha(\\kappa_0)}(x))$ and absorbs the disturbance terms into a fraction of the decay plus a class-KL function of $\\|d_1\\|_\\infty+\\|d_2\\|_\\infty$.","core_discovery":"The paper's main result is Theorem 34: for the perturbed chain $\\dot x = J_n x + b\\, \\omega^H_{\\kappa(x+d_1)}(x)e_n + d_2$, with $b$ bounded above and below away from zero, the closed loop is ISS for every bounded disturbance pair $(d_1,d_2)$ measuring the feedback noise and the unmatched perturbation. When $d_1=0$ and $d_2$ is parallel to $e_n$, the disturbed trajectory converges exactly to zero in fixed time. By rescaling the state through the dilation $D^r_\\mu$ and choosing $\\mu$ proportional to $T(m,\\kappa_0)/T$, the same statement holds for any prescribed horizon $T$. The proof establishes three differential inequalities on the regions where the homogeneity degree is positive, zero, and negative, and uses them to deduce an eventual bound on a minimum of Lyapunov functions; from that bound ISS is concluded via a characterization rather than by displaying an ISS-Lyapunov function.","pith_inferences":["The ISS statement is asymptotic, but the three-region argument also yields finite-horizon bounds, so one could extract a quantitative settling-time-versus-noise formula for practical use, something the paper leaves implicit.","The same glueing of a positive-homogeneity fixed-time region and a negative-homogeneity finite-time region, with continuous interpolation in between, is not tied to chains of integrators and should transfer to any control-affine system admitting explicit homogeneous Lyapunov pairs.","Because the feedback has explicit parameters, one could test the ISS bound numerically by computing the gain function in (63) for n=2 and comparing it with the linear scaling in the disturbance norm; the paper does not give such a numerical validation."],"forward_implications":["For any n and any prescribed T, the explicit controller of Section 4.1 makes the pure chain converge to zero in time at most T, with no control gain diverging as t approaches T.","The same controller keeps trajectories bounded with respect to measurement noise and unmatched disturbances; the ultimate bound is governed by the sum of the disturbance norms, giving an ISS guarantee rather than only practical stability.","If there is no measurement noise and the disturbance is matched, the closed-loop system converges exactly to zero in fixed time, not merely to a neighborhood.","The parameter choices (m, kappa_0, mu, and the gains ell_j) are explicit, so the settling-time upper bound can be computed before implementation, and smaller horizons can be achieved by a larger rescaling factor.","The time-varying homogeneity viewpoint recovers previous proportional navigation feedback results without taking time derivatives of lambda, giving simpler proofs and more freedom in selecting the convergence rate."],"supporting_citations":[{"why":"Defines prescribed-time input-to-state stability with time-varying gains and gives the previous feedback construction that the paper revisits and extends.","marker":"[16]"},{"why":"Supplies the LMI result from which the linear prescribed-time feedback and its ISS estimates are derived.","marker":"[3]"},{"why":"Gives the recursive homogeneous feedback laws and Lyapunov functions used throughout the robust construction.","marker":"[9]"},{"why":"Introduces the fixed-time stabilization idea via state-dependent homogeneity degree on which the new controller and its settling-time bound are built.","marker":"[8]"},{"why":"Provides the perturbation trick that makes the homogeneity degree continuous and yields the earlier ISpS result that Theorem 34 strengthens to ISS.","marker":"[12]"},{"why":"Supplies the ISS characterization the proof invokes to pass from the limsup bound of Proposition 37 to the ISS conclusion.","marker":"[17]"},{"why":"Gives the ultimate-boundedness argument used inside the proof of Proposition 37.","marker":"[2]"}],"fun_headline_variants":["Prescribed-time stabilizer for all integrator chains, noise-proof","Arbitrary integrator chain stabilizes on schedule under noise","Sliding mode gives fixed-time stability with noise robustness","Chain of integrators tamed in prescribed time, even noisy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole ISS claim rests on the step where a bound on the eventual size of the state, through the minimum of the three Lyapunov functions, is taken as sufficient to prove input-to-state stability, even though the associated Lyapunov function is never constructed and the equivalence with the usual ISS definition is not shown.","fun_headline_variants_meta":{"raw":{"variants":["Prescribed-time stabilizer for all integrator chains, noise-proof","Arbitrary integrator chain stabilizes on schedule under noise","Sliding mode gives fixed-time stability with noise robustness","Chain of integrators tamed in prescribed time, even noisy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000522,"raw_usage":{"total_tokens":2501,"prompt_tokens":894,"completion_tokens":1607,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":1537}},"tokens_in":510,"tokens_out":1607,"duration_ms":15099,"temperature":1.0,"reasoning_tokens":1537,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:36:39.931523+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take n=2, fix a horizon T, set the measurement noise to the constant d1=(0,epsilon) and the unmatched disturbance to d2=(epsilon,0), and compute the trajectory of (62) with the explicit parameters of Section 4.2. If the limiting size of x(t) does not go to zero as epsilon tends to zero, or if it grows faster than linearly in epsilon, then the ISS statement of Theorem 34 is false; equivalently, trying to build the ISS-Lyapunov function required by the cited ISS characterization for this two-dimensional case would settle whether the proof gap is fatal.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines prescribed-time input-to-state stability with time-varying gains and gives the previous feedback construction that the paper revisits and extends."},{"cited_title":"Chitour and M","cited_arxiv_id":null,"evidence_quote":"Supplies the LMI result from which the linear prescribed-time feedback and its ISS estimates are derived."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the recursive homogeneous feedback laws and Lyapunov functions used throughout the robust construction."},{"cited_title":"Harmouche, S","cited_arxiv_id":null,"evidence_quote":"Introduces the fixed-time stabilization idea via state-dependent homogeneity degree on which the new controller and its settling-time bound are built."},{"cited_title":"Lopez-Ramirez, D","cited_arxiv_id":null,"evidence_quote":"Provides the perturbation trick that makes the homogeneity degree continuous and yields the earlier ISpS result that Theorem 34 strengthens to ISS."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the ISS characterization the proof invokes to pass from the limsup bound of Proposition 37 to the ISS conclusion."},{"cited_title":"Chitour, M","cited_arxiv_id":null,"evidence_quote":"Gives the ultimate-boundedness argument used inside the proof of Proposition 37."}],"review_version":1}