{"id":"977e1540-780d-4a10-8c93-43422f3ecf39","arxiv_id":"2505.18336","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A sampled-data control loop is exponentially stable for small sampling periods, and even a single optimization iteration per sample suffices when the idealized continuous-feedback model is contractive.","lead":"The paper shows that a control loop whose controller is updated in discrete samples can stay exponentially stable with just one solver step per sample, as long as the sampling period is small. This matters for real-time model predictive control, where finishing the full optimization between samples is often impossible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The single-iteration MPC claim inherits a global-contractivity assumption that the paper's own constrained example violates; the local corollary relies on unverified forward invariance.","rationale":"The reader and I converge on the same weakest point: Theorem 3.6's quantitative bound rests on a global contraction rate ζ for the reduced model. Re-reading the proof, equations (39)-(41) indeed use ζ to control the contraction of the ideal flow, and the constants C1, C2 define the admissible sampling threshold. The paper handles the non-global case with Corollary 3.7, but that corollary only yields local exponential stability if the solution stays in the contractive region, and the paper verifies this by simulation for a single (n,T) pair rather than by a constructive certificate. The γ=100 contour in Fig. 6 explicitly shows positive log-norm regions, so the global hypothesis fails in a motivating MPC setting. I do not see an internal inconsistency in the main theorem; the proof is detailed and the unconstrained LTI experiment is consistent with it. The concern is therefore about scope: the single-iteration MPC claim is rigorous when the RM is globally contracting, and only heuristic/local otherwise. This matches the reader's CONDITIONAL verdict; no change is needed, but the paper should state the limitation more prominently and either prove forward invariance or restrict the claim. The minor T(n) sign/formula gap for ξ<0 and the Proposition 3.3 sign typo are secondary and addressable without altering the conclusion.","tokens_in":28898,"tokens_out":11785,"duration_ms":107330,"concrete_test":"Re-run the second experiment with γ=100, n=1, and T = 0.9·T(1) computed from (16) using the local ζ measured at the origin, with x0 placed at the point in X0=[-10,10]×[-3,3] where Fig. 6 (right) shows the largest positive µ. If the trajectory leaves X=[-20,20]×[-6,6] or fails to converge, Corollary 3.7's forward-invariance hypothesis is violated and the single-iteration guarantee is not established for that MPC regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.6 assumes osLip_x(f(x,z*(x))) <= -ζ uniformly on the forward-invariant set X, and the proof's per-interval bounds (39)-(41) are all anchored on this global ζ. When that condition fails, only Corollary 3.7 applies, and it requires an a priori forward-invariance certificate for X0×Z0. The paper's own second MPC experiment (Section IV-B, Fig. 6 with γ=100) shows that µ_{2,P}(A+BΠ1 Jz*(x)) is positive over part of the state space, so global contractivity is not satisfied for the constrained/soft-constrained MPC setting. Thus the advertised 'single iteration is sufficient' conclusion is, as stated, a theorem about globally contractive reduced models rather than about MPC with constraints; extending it to the constrained case requires checking local contraction and forward invariance numerically for every (n,T), which the paper does only for one (n,T) pair and without a formal certificate. This is a scope limitation rather than a proof error, but it is load-bearing for the MPC application claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the sampled-data interconnection of a continuous-time plant ẋ=f(x,z) with a discrete-time map z_k = G^n(x(kT), z_{k-1}), where z is held constant between sampling instants. It introduces a reduced model ẋ=f(x,z*(x)) obtained as T→0^+ and n→∞, and proves three groups of results: (i) Theorem 3.6, which states that if the DT map is contractive and the reduced model is strongly infinitesimally contracting, then for every n there is a threshold T(n) such that the CT-DT interconnection is globally exponentially stable for all T<T(n), with an explicit formula for T(n); (ii) Theorem 3.4, a small-gain condition implying discrete-time contractivity and exponential stability for all T and n; and (iii) an application to suboptimal MPC, arguing that a single iteration of the solver per sampling period suffices for stability when T is small. The proofs are based on contraction theory, Lipschitz/one-sided Lipschitz constants, and Schur-stability analysis of a 2×2 nonnegative matrix.","tokens_in":29089,"tokens_out":8887,"duration_ms":83390,"significance":"If the main results hold, the paper gives a useful quantitative bridge between continuous-time contraction of a reduced model and robust stability of a sampled-data implementation with finite solver iterations. The explicit T(n) expression, its monotonicity in n, and the comparison between small-gain and reduced-model conditions are valuable design insights. The proof of Theorem 3.6 is detailed: the Schur-condition algebra is carried through carefully, and the identities C1C2=C12C21 and C1[Lip_z(G)]^n = Lip_z(f)C21 used in the proof check out. The MPC application addresses a timely question, and the numerical experiments illustrate the qualitative behavior. However, as detailed below, the explicit threshold formula has a domain issue, the sign convention in Proposition 3.3 makes a displayed decay estimate incorrect as written, and the MPC claim for constrained/soft-constrained cases is broader than what the stated theorems actually support.","major_comments":[{"comment":"The statement defines c := T^{-1} ln b. Since b∈(0,1), this c is negative, and the displayed bound ∥y(t)∥ ≤ r e^{-ct} ∥y(0)∥ is then an increasing exponential, not a decay. The proof repeats the same sign by writing c := ln b^{1/T} < 0 and then using e^{-c(kT+τ)}. The intended rate is certainly c := -T^{-1} ln b > 0, which is exactly what Remark 6 uses when it writes a := -T^{-1} ln ρ(A(n,T)). This is a load-bearing sign error in a stated implication, though it is locally fixable.","section":"III-C, Proposition 3.3 and Eq. (11)"},{"comment":"The formula for T(n) is not always defined or positive. When ξ<0, the equation h(T,ξ) = R with R := (1-[Lip_z(G)]^n)/(C2(n)+C1/ζ) has a positive solution only if ξ R + 1 > 0; if R ≥ 1/(-ξ), then the argument of the logarithm in (16) is non-positive and the formula is meaningless. In the latter case the stability condition h(T,ξ)<R actually holds for every T>0, because h(T,ξ) is bounded above by 1/(-ξ). The theorem should either state T(n) as the infimum of T satisfying the condition, allowing T(n)=+∞ in this case, or add a hypothesis ensuring ξ R + 1 > 0. As written, the claims after (16) that T(n)>0 for all n and that T(n) is bounded by (17) are not valid on the full stated domain.","section":"III-D, Theorem 3.6, Eq. (16)"},{"comment":"The constrained MPC claim goes beyond what the stated results establish. Theorem 3.6 assumes global exponential contraction of the reduced model: osLip_x(f(x,z*(x))) ≤ -ζ on the forward-invariant set X. The paper's own Fig. 6 (right panel, γ=100) shows that this condition fails for large constraint penalties, and the text explicitly states that contractivity is not global. Corollary 3.7 does not actually relax the contraction assumption to a local one: it still assumes osLip(f(x,z*(x))) ≤ -ζ and only localizes the initial conditions together with forward invariance of X0×Z0 in X×Z. In the γ=10 experiment, the paper verifies forward invariance only numerically for one (n,T) pair and does not verify the global (or even X0-local) contraction hypothesis that Corollary 3.7 requires. Consequently, the advertised conclusion that “single-iteration suboptimal MPC is stabilizing for sufficiently small T” is rigorously established only for globally contractive reduced models, e.g., the unconstrained LQ example in Section IV-B1, not for the constrained/soft-constrained setting presented as validation in Section IV-B2. The authors should either state a genuinely local version of Theorem 3.6 with osLip condition on X0 and forward invariance of X0, or restrict the MPC claim accordingly.","section":"IV-B2, Corollary 3.7 and Figs. 6-7"}],"minor_comments":[{"comment":"In the displayed definition of osLip_w(G), the denominator uses “∥v1-v2∥²” but v1 and v2 are not defined; it should be ∥w1-w2∥².","section":"II-A, definition of osLip_w(G)"},{"comment":"The statement uses the norm ∥·∥_U for the input difference, but no norm on an input space U has been introduced; this should be ∥·∥_Z (or a norm on the input space should be defined).","section":"V-A, Lemma 5.4"},{"comment":"The constant C1 depends on Lip_x(f), but Lip_x(f) is not defined in Assumptions 1-3; the authors should explicitly define it as the minimal uniform-in-z Lipschitz constant of x↦f(x,z).","section":"III-D, Theorem 3.6"},{"comment":"The proof of Proposition 3.8 is omitted with only a sketch; given that the paper otherwise provides complete proofs, a brief derivation or a precise reference to the LTI Schur-stability argument would improve completeness.","section":"III-E, Proposition 3.8"},{"comment":"There are several typos, including “Lipchitz” for “Lipschitz” and “assets” for “asserts”; these should be corrected in a revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the first author's contraction monograph [4] for standard lemmas and for the weighted-norm construction. This is not inappropriate, but the novelty relative to [4, Ch. 3] and to earlier contraction-based singular-perturbation work could be more sharply delineated. The main theoretical machinery appears sound aside from the sign and domain issues listed above; the central concern is the mismatch between the global assumptions of the theorems and the constrained-MPC claims in Section IV."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper deserves a serious referee. The genuinely new result is the single-iteration MPC stability guarantee: for a globally contractive reduced model, one solver iteration per sample is enough when T is small. That is a real advance over time-distributed MPC work that required n > 1. The explicit T(n) formula and the CT-DT small-gain theorem are useful extensions, and the proof of Theorem 3.6 is detailed; I checked the matrix Schur condition algebra and the identities C1C2 = C12C21 and C1 L^n = Lip_z(f) C21 do hold. The numerical experiments support the theory for the unconstrained case.\n\nThe soft spots are real but mostly minor. Formula (16) is not defined when xi < 0 and the log argument is non-positive; in that regime the relevant inequality actually holds for all T, so the theorem survives but the stated formula overclaims its domain. Proposition 3.3 has a sign typo: c should be -(1/T) ln b, not (1/T) ln b. Several auxiliary proofs are omitted (Prop 3.8, Lemma 5.4), though they look standard. The reliance on Bullo's contraction monograph, with one of the authors being the monograph's author, is noticeable but not circular; the cited lemmas are standard and independent.\n\nThe more substantive concern is the stress-test point, and I think it lands but with a narrower scope than the reader's note suggests. Theorem 3.6 assumes global contractivity of the reduced model on a forward-invariant set. The paper's own soft-constrained MPC example (Fig. 6, gamma = 100) shows that this fails when constraint penalties are large, so the paper correctly falls back to Corollary 3.7, which needs an a priori forward-invariance certificate. The paper verifies forward invariance numerically for one (n,T) pair, which is honest but not a general guarantee. The single-iteration claim is rigorously established for the unconstrained/globally contractive setting; for constrained MPC it is a local result with numerical support. The text mostly acknowledges this in Section VI, though the abstract's practical-sounding wording is broader than what is proved.\n\nWho is this for? Researchers in sampled-data control, contraction theory, and real-time MPC. I would cite it for the n=1 result and the explicit threshold. My recommendation: accept for peer review, with requests to fix the formula domain, correct the sign typo, and state the global-contractivity limitation in the abstract.","headline":"Worth refereeing: the n=1 MPC stability result is real, the core proof is mostly solid, and the known gaps are fixable presentation issues plus a global-contractivity scope limit.","tokens_in":29644,"tokens_out":3796,"would_cite":true,"duration_ms":34942,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C57","93D23","93D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"One optimization iteration per sample can keep MPC stable if the sampling period is small enough.","keywords":["sampled-data systems","contraction theory","reduced model","model predictive control","suboptimal MPC","exponential stability","small-gain condition","zero-order hold"],"falsifier":"Run a numerical continuation on the paper's first MPC example (double integrator with $A=[[0,1],[0,0]]$, $B=[0,1]^\\top$, $\\Delta=0.2$, horizon $H=5$, $R=1$, $Q=I$, $P$ from the DARE) with $n=1$: compute $T(1)$ explicitly from (16) and simulate the closed loop at $T=0.9\\,T(1)$ and $T=1.1\\,T(1)$ across a grid of initial conditions. Stability of all trajectories at $0.9\\,T(1)$ and instability at $1.1\\,T(1)$ supports the formula; conversely, any initial condition producing divergence at $T< T(1)$ under the stated assumptions would refute Theorem 3.6. A sharper falsification would search over the parameter space for any pair $(f,G)$ satisfying Assumptions 1-3 and contractivity of the reduced model for which the sampled loop is unstable for a sequence $T_k\\to0$.","tokens_in":2196,"feed_emoji":"⏱","tokens_out":2753,"duration_ms":45660,"temperature":0.7,"pith_summary":"This paper studies what happens when a continuous-time plant is controlled by a discrete-time algorithm that updates only every $T$ seconds, running just $n$ iterations of an optimization solver per update. Its central claim is that if the ideal, continuously applied control law produces a contractive closed-loop system (the reduced model), then for every finite $n$, including $n=1$, there exists a threshold sampling period $T(n)$ such that the real sampled-data loop is globally exponentially stable for all $T<T(n)$. The threshold is given explicitly in terms of the Lipschitz and contraction rates of the plant, the solver map, and the number of iterations. This matters for real-time control because it says a single imperfect solver step per sample period, not a converged solution, is enough to guarantee stability provided the computer can update fast enough. A separate small-gain theorem gives stability for any $n$ and any $T$, but under the stricter requirement that the plant itself be contractive.","feed_headline":"One solver iteration per sample can keep MPC stable","feed_subtitle":"A contraction-based threshold T(n) guarantees exponential stability whenever the sampling period is small enough.","key_machinery":"The load-bearing construct is the reduced model: the fictitious continuous-time system $\\dot{x}=f(x,z^*(x))$ in which $z^*(x)=\\lim_{n\\to\\infty}G^n(x,z)$ is the fixed point of the solver map, representing the ideal limit of infinitely fast sampling and infinitely many iterations. The argument then tracks the sampled-data dynamics through a $2\\times2$ nonnegative matrix $A(n,T)$ (defined in Remark 8) whose entries are products of the contraction rate $\\zeta$, the Lipschitz constants of $f$ and $G$, and the integral $h(T,\\xi)=\\int_0^T e^{\\xi(T-s)}ds$; stability is reduced to showing this matrix is Schur stable, which is exactly what the threshold inequality $h(T,\\xi)<(1-[\\mathrm{Lip}_z(G)]^n)/(C_2(n)+C_1/\\zeta)$ enforces. Solving this inequality for $T$ yields the explicit threshold $T(n)$ in (16), and the composite-norm machinery (weighted $\\ell^2$ norms built from Perron eigenvectors) converts Schur stability of $A(n,T)$ into a discrete-time contraction bound, which Proposition 3.3 then lifts to a global exponential stability bound for all $t\\ge0$.","core_discovery":"The paper's main theoretical result, Theorem 3.6, establishes that the sampled-data interconnection of a continuous-time system $\\dot{x}=f(x,z)$ and a discrete-time map $z_k=G^n(x(kT),z_{k-1})$ is globally exponentially stable whenever the discrete map $z\\mapsto G(x,z)$ is a contraction and the reduced model $\\dot{x}=f(x,z^*(x))$ -- the limiting system obtained as $T\\to0^+$ and $n\\to+\\infty$ -- is strongly infinitesimally contracting with rate $\\zeta>0$. For each iteration count $n\\ge1$, the paper produces a positive threshold $T(n)$, depending only on $n$ and on Lipschitz constants of $f$ and $G$, such that stability holds for all sampling periods $T<T(n)$; the threshold is explicitly computed in equation (16) and is shown to be increasing in $n$ and bounded above by a constant independent of $n$. Applied to model predictive control with a gradient-descent solver, this yields the claim that a single solver iteration per sample ($n=1$) preserves closed-loop stability whenever $T$ is sufficiently small, which the authors note is the first rigorous guarantee of this kind and is in contrast to earlier time-distributed MPC analyses that required more iterations. Under the stronger conditions that both the CT map and the DT map are contractive and satisfy the small-gain inequality $-\\mathrm{osLip}_x(f)(1-\\mathrm{Lip}_z(G))>\\mathrm{Lip}_z(f)\\mathrm{Lip}_x(G)$, Theorem 3.4 gives exponential stability and discrete-time contractivity for every $n$ and every $T>0$, with the small-gain condition shown in Theorem 3.5 to imply contractivity of the reduced model.","pith_inferences":["The same threshold logic should carry over to other operator-splitting or proximal solvers whose iteration maps are contractions with known Lipschitz constants, such as ADMM loops, suggesting a general recipe: certify contractivity of the ideal feedback law, then bound the sampling period by $T(n)$ computed from the solver's contraction factor.","The monotonicity of $T(n)$ in $n$ suggests a natural online adaptation law: if the available compute time per sample grows, the controller can either increase $n$ or increase $T$ while preserving stability; the explicit trade-off curve $T(n)$ makes this a one-dimensional scheduling problem.","Because contractivity of the reduced model is typically only local for constrained MPC (as the paper's own $\\gamma=100$ experiment shows), the practically useful statement will be the local version, Corollary 3.7, whose forward-invariance condition currently must be checked numerically; a Lyapunov-function or barrier certificate that certifies the required set invariance would turn the result into","The paper's Example 3.1, showing that any fixed $T$ can be destabilizing for an unstable open-loop plant, implies that the threshold $T(n)$ necessarily depends on the plant's unstable dynamics; controllers that aggressively stabilize may require extremely small $T$ at $n=1$, so the practical value of the single-iteration guarantee hinges on whether the required $T(1)$ is achievable in hardware."],"forward_implications":["For any MPC problem whose ideal closed-loop map is contractive, stability of the online implementation holds for every fixed iteration count $n$, including $n=1$, once the sampling period is below the computable threshold $T(n)$.","Increasing the number of solver iterations per sample strictly enlarges the admissible sampling period, since $T(n)$ is increasing in $n$; the payoff of extra computation is a less demanding real-time update rate.","The small-gain condition of Theorem 3.4 guarantees stability with no constraint on $T$ or $n$ at all, at the price of requiring the open plant map $x\\mapsto f(x,z)$ to be contractive uniformly in $z$.","For linear time-invariant systems, Theorem 3.6 upgrades to discrete-time contractivity of the sampled loop, not merely exponential stability, whenever the reduced-model matrix $A+B(I-D)^{-1}C$ is Hurwitz and $D$ is Schur.","The explicit threshold formula gives a quantitative engineering guideline: the maximum allowable sensing-and-computation delay is expressible directly from plant and solver Lipschitz data, without solving LMIs or simulating the nonlinear loop."],"supporting_citations":[{"why":"Supplies the contraction-theory toolkit (weak pairings, log-norms, matrix Schur-stability lemmas, composite norms) that all main proofs are built on.","marker":"[4]"},{"why":"Provides the classical singular-perturbation reduced-model idea that motivates defining the reduced model as the $T\\to0^+, n\\to+\\infty$ limit.","marker":"[1]"},{"why":"Extends the singular-perturbation and order-reduction framework whose two-time-scale interpretation the paper adapts to the CT-DT setting.","marker":"[2]"},{"why":"Baseline time-distributed MPC work whose stability analysis requires more than one solver iteration per sample, against which the paper's $n=1$ claim is contrasted.","marker":"[19]"},{"why":"Establishes closed-loop stability conditions for linear MPC based on time-distributed optimization with $n>1$, the prior result the paper's single-iteration guarantee is compared with.","marker":"[20]"},{"why":"Provides the framework for stabilizing nonlinear sampled-data systems via approximate discrete-time models, which the paper's treatment of $T$ as a stability-relevant parameter builds on.","marker":"[33]"}],"fun_headline_variants":["One solver iteration per sample: stable MPC","Single optimizer step per sample preserves MPC stability","Sampled-data MPC: single iteration suffices for stability","Contractivity threshold guarantees sampled-data stability","Small sampling period lets one MPC iteration keep stability"],"cache_read_input_tokens":31872,"weakest_assumption_plain":"The proof requires the reduced model to be globally uniformly contracting on the whole forward-invariant state set, meaning the ideal feedback law shrinks the distance between any two trajectories at a rate at least $\\zeta$ everywhere; if this fails even locally, the paper's main theorem only gives a local stability result whose region of validity must be verified numerically.","fun_headline_variants_meta":{"raw":{"variants":["One solver iteration per sample: stable MPC","Single optimizer step per sample preserves MPC stability","Sampled-data MPC: single iteration suffices for stability","Contractivity threshold guarantees sampled-data stability","Small sampling period lets one MPC iteration keep stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000954,"raw_usage":{"total_tokens":4168,"prompt_tokens":1143,"completion_tokens":3025,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":759,"completion_tokens_details":{"reasoning_tokens":2956}},"tokens_in":759,"tokens_out":3025,"duration_ms":17984,"temperature":1.0,"reasoning_tokens":2956,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:33:21.078537+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a numerical continuation on the paper's first MPC example (double integrator with $A=[[0,1],[0,0]]$, $B=[0,1]^\\top$, $\\Delta=0.2$, horizon $H=5$, $R=1$, $Q=I$, $P$ from the DARE) with $n=1$: compute $T(1)$ explicitly from (16) and simulate the closed loop at $T=0.9\\,T(1)$ and $T=1.1\\,T(1)$ across a grid of initial conditions. Stability of all trajectories at $0.9\\,T(1)$ and instability at $1.1\\,T(1)$ supports the formula; conversely, any initial condition producing divergence at $T< T(1)$ under the stated assumptions would refute Theorem 3.6. A sharper falsification would search over the parameter space for any pair $(f,G)$ satisfying Assumptions 1-3 and contractivity of the reduced model for which the sampled loop is unstable for a sequence $T_k\\to0$.","supporting_citations":[{"cited_title":"Bullo,Contraction Theory for Dynamical Systems, 1.2 ed","cited_arxiv_id":null,"evidence_quote":"Supplies the contraction-theory toolkit (weak pairings, log-norms, matrix Schur-stability lemmas, composite norms) that all main proofs are built on."},{"cited_title":"Singular perturbation method for reducing the model order in optimal control design,","cited_arxiv_id":null,"evidence_quote":"Provides the classical singular-perturbation reduced-model idea that motivates defining the reduced model as the $T\\to0^+, n\\to+\\infty$ limit."},{"cited_title":"Singular perturba- tions and order reduction in control theory—an overview,","cited_arxiv_id":null,"evidence_quote":"Extends the singular-perturbation and order-reduction framework whose two-time-scale interpretation the paper adapts to the CT-DT setting."},{"cited_title":"Time- distributed optimization for real-time model predictive control: Stability, robustness, and constraint satisfaction,","cited_arxiv_id":null,"evidence_quote":"Baseline time-distributed MPC work whose stability analysis requires more than one solver iteration per sample, against which the paper's $n=1$ claim is contrasted."},{"cited_title":"An analysis of closed-loop stability for linear model predictive control based on time-distributed optimization,","cited_arxiv_id":null,"evidence_quote":"Establishes closed-loop stability conditions for linear MPC based on time-distributed optimization with $n>1$, the prior result the paper's single-iteration guarantee is compared with."},{"cited_title":"A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models,","cited_arxiv_id":null,"evidence_quote":"Provides the framework for stabilizing nonlinear sampled-data systems via approximate discrete-time models, which the paper's treatment of $T$ as a stability-relevant parameter builds on."}],"review_version":1}