{"id":"705a14cd-7984-4b0f-9230-b174b00fcfcf","arxiv_id":"2412.03901","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A data-driven sum-of-squares procedure certifies and constructs incremental ISS controllers for unknown polynomial systems from two trajectories.","lead":"This paper presents a data-driven method for designing Lyapunov certificates and controllers that guarantee incremental input-to-state stability for unknown polynomial systems, using only two input-state trajectories. If the certificate conditions are met, the controller provably makes all trajectories converge to each other under external disturbances.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Conditions (12a)-(12d) force the closed-loop state matrix Aℵ(x)+B K(x) to be constant, a hidden exact-feedback-linearization requirement not implied by the stated rank condition; the method can be infeasible for incrementally stabilizable systems.","rationale":"The reader's verdict is CONDITIONAL, and I agree with that overall disposition, but the decisive issue is sharper than the reader's stated weakest assumption. The reader points to missing feasibility/rank conditions and to derivative noise; both are real. The more load-bearing problem is that conditions (12a)-(12d) are not merely technical constraints on data richness: they enforce exact feedback linearization of the closed loop. This follows algebraically from (10), (12a), and (12c), and it means the method only applies to systems whose drift can be exactly cancelled by the input channel. The paper does not state, prove, or data-verify this condition. The stated rank condition on J0,T and J-tilde0,T is insufficient, and the SOS program can be infeasible for systems that are otherwise incrementally stabilizable, as the ẋ1 = x1^2 + x2, ẋ2 = u example shows. This does not invalidate the conditional theorem, but it substantially narrows the practical claim in the abstract and requires a revision that either adds an explicit feedback-linearizability/feasibility assumption or weakens the claim to 'whenever the SOS program is feasible.' The derivative-noise issue from Remark 3.1 is secondary and could also be addressed, but it is not the primary reason the practical guarantee fails.","tokens_in":11866,"tokens_out":13852,"duration_ms":125596,"concrete_test":"Run Algorithm 1's SOS feasibility problem (12) on synthetic exact-derivative data for ẋ1 = x1^2 + x2, ẋ2 = u (B = [0;1]^T, F = [x1^2; x2]) with a persistently exciting input so J0,T has full row rank. Because (12c) together with (12a) requires Aℵ(x) + B K(x) = ΣP and the (1,1) entry of Aℵ(x) is x1, no polynomial K can make that entry constant; the solver should report infeasibility. Then verify the same system is δ-ISS-stabilizable by a standard backstepping controller, which would show that the proposed data-driven certificate fails for a valid CTIA-NSP satisfying the stated rank assumption.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"From (10), X1,T = A J0,T + B U0,T. If (12a) holds, then J0,T Y(x) = ℵ(x)P^{-1}, so X1,T Y(x) = Aℵ(x)P^{-1} + B U0,T Y(x). Condition (12c) requires this to equal the constant matrix Σ for all x. With K(x) = U0,T Y(x)P, this is exactly Aℵ(x) + B K(x) = ΣP, a constant matrix. Thus any system satisfying (12) is feedback-linearized: the drift A F(x) = Aℵ(x)x is cancelled by the input channel B K(x)x, leaving closed-loop dynamics ẋ = ΣP x + Bû. Full row rank of J0,T (Remark 3.3) does not guarantee that Aℵ(x) lies in the column space of B, nor that a single polynomial Y(x) can satisfy (12a)-(12d) simultaneously. For a system such as ẋ1 = x1^2 + x2, ẋ2 = u with B = [0;1]^T, the first row of Aℵ(x) is [x1, 1], which no K(x) entering only the second channel can make constant; the SOS problem in Algorithm 1 is infeasible even though the system is incrementally stabilizable by backstepping. The abstract's claim that δ-ISS is 'ensured' therefore overstates the scope: (12) is a strong existence assumption, not a consequence of the stated data-rank condition, and the paper gives no verifiable condition under which it holds.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a direct data-driven method for synthesizing incremental input-to-state stability (delta-ISS) Lyapunov functions and controllers for continuous-time input-affine nonlinear systems with polynomial dynamics and unknown matrices A, B, and monomial dictionary F. The method collects two input-state trajectories, forms data matrices J0,T, J~0,T, X1,T, and X~1,T, and then enforces conditions (12a)-(12e) through a sum-of-squares program. Under these conditions, Theorem 3.5 proves that V(x,x~) = (x-x~)^T P (x-x~) is a delta-ISS Lyapunov function and that u = U0,T Y(x)P x + u-hat is the corresponding delta-ISS controller. A rotating rigid spacecraft example illustrates the approach. The algebraic derivation is internally valid, but the feasibility conditions are restrictive and are not implied by the stated data-rank assumption.","tokens_in":12251,"tokens_out":7550,"duration_ms":69479,"significance":"If the result holds, the paper provides a data-based certificate that bypasses system identification for a class of polynomial systems, and the proof is explicit and checkable. The main limitation is that conditions (12) effectively require exact feedback linearization of the drift, which is a strong structural assumption that the stated rank condition does not guarantee. The paper also assumes exact state derivatives and gives no verifiable characterization of the systems for which the SOS program is feasible. These issues limit the scope of the claim that delta-ISS is 'ensured' for unknown polynomial dynamics and need to be addressed.","major_comments":[{"comment":"Conditions (12a)-(12d) force the closed-loop state matrix to be constant. Indeed, from (10), (12a), and (12c), one obtains (A*Aleph(x) + B*K(x)) P^{-1} = Sigma, so the closed-loop dynamics reduce to x_dot = Sigma P x + B*u-hat, which is an exact feedback linearization. Full row-rank of J0,T (Remark 3.3) does not imply that such a cancellation is possible. For example, the incrementally stabilizable system x_dot_1 = x_1^2 + x_2, x_dot_2 = u with B = [0;1]^T cannot satisfy (12), because the first row of A*Aleph(x) + B*K(x) is [x_1, 1] for any K(x) entering only the second channel. The abstract and Problem 2.4 claims that the method 'ensures' delta-ISS should therefore be qualified to the class of systems satisfying this hidden feedback-linearization condition.","section":"Section III, Theorem 3.5, conditions (12a)-(12d)"},{"comment":"Conditions (12c) and (12d) require the sampled derivative matrices X1,T and X~1,T to be exact. Remark 3.1 acknowledges that numerical differentiation or filtering introduces approximation errors but explicitly defers the noise analysis. Because the proof of Theorem 3.5 relies on the exact equalities (10) and (12c)-(12d), the 'certified' guarantee does not apply when derivatives are approximated by the procedures suggested in Remark 3.1. The paper should either provide a robustness result for bounded derivative errors or clearly restrict the certification claim to exact derivative measurements.","section":"Section III, Remark 3.1 and Theorem 3.5"},{"comment":"The theorem assumes the existence of global polynomial matrices Y(x) and Y(x~) satisfying (12a)-(12d), but no data-dependent condition beyond full row-rank is given that would certify this existence. Consequently, the SOS program in Algorithm 1 may be infeasible even when the rank condition is satisfied. The authors should provide a verifiable feasibility characterization or state explicitly that the method applies only when a solution to (12) is found.","section":"Section III, Theorem 3.5 and Algorithm 1"}],"minor_comments":[{"comment":"The notation F(x) is used both for the true monomial vector and for an exaggerated dictionary; the overloading is acknowledged, but it would help to denote the dictionary explicitly (for example, F_d(x)) to avoid ambiguity in conditions (12).","section":"Page 3, after Definition 2.1"},{"comment":"The input signal is typed as \"sin 2(t)\" and should be \"sin^2(t)\" for consistency with standard notation and the surrounding text.","section":"Section IV, simulation setup"},{"comment":"Remark 3.4 states that a transformation matrix Aleph(x) can always be constructed, but this matrix is not unique; since conditions (12a)-(12b) depend on the chosen Aleph, the paper should discuss how the choice of Aleph affects feasibility.","section":"Remark 3.4"},{"comment":"The y-axis label \"Norm of the difference\" is incomplete; it should specify which norm and between which trajectories (for example, |x(t)-x~(t)| in logarithmic scale).","section":"Figure 2"}],"recommendation":"major_revision","confidential_remarks":"The core theorem is logically sound, and the data-driven representation is a promising idea. My main concern is that the stated feasibility conditions are much stronger than the paper's claims suggest, and no verifiable condition is given for the class of systems covered. If the authors narrow the claims and add a feasibility discussion or sufficient conditions, the revised manuscript may be publishable. Otherwise, the contribution may be too limited for a general-audience control journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper has a real new result — the first direct data-driven synthesis of delta-ISS Lyapunov functions and controllers from two trajectories — and the proof of Theorem 3.5 is algebraically correct. But the result is narrower than the abstract claims. Conditions (12c)-(12d) force the closed-loop state matrix Aℵ(x)+BK(x) to be constant, i.e., the method implicitly requires exact feedback linearization. The stress-test example (x1' = x1^2 + x2, x2' = u) is incrementally stabilizable by backstepping, yet no polynomial Y can satisfy (12). So the abstract's 'ensuring delta-ISS' overstates the scope; the paper offers no verifiable check for feasibility beyond full row rank.\n\nWhat is genuinely new: the two-trajectory data representation in Lemma 3.2 and the synthesis theorem are not in [19], [28], or [30]. The proof is sound under the explicit existence assumptions; the Cauchy-Schwarz/Young step is fine.\n\nSoft spots, in order of severity: first, the hidden linearization condition is the main issue and should be stated and discussed, ideally with a sufficient feasibility condition. Second, derivatives are assumed noiseless; Remark 3.1 acknowledges this and defers it, which is honest but leaves the practical claim incomplete. Third, no code or data is shipped; the spacecraft example works because B is invertible, so the condition happens to hold. Fourth, global SOS feasibility is asserted but not certified.\n\nThis paper deserves a serious referee. The core conditional result appears new and correct, but the scope must be narrowed and the feasibility question addressed. If it comes to you for review, ask for a realistic feasibility condition and an honest statement of the linearization requirement.","headline":"A genuinely new two-trajectory delta-ISS synthesis result with a sound proof, but the existence condition secretly demands exact feedback linearization and the abstract overstates the scope.","tokens_in":12748,"tokens_out":2542,"would_cite":true,"duration_ms":68135,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93D25","93D30","93C10","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that for continuous-time input-affine nonlinear systems with polynomial dynamics but unknown matrices A, B, and monomial map F, a δ-ISS Lyapunov function and controller can be designed directly from two recorded…","keywords":["incremental input-to-state stability","delta-ISS","data-driven control","polynomial dynamics","sum-of-squares optimization","Lyapunov function","controller synthesis","unknown dynamics"],"falsifier":"Record real trajectories of a known polynomial input-affine system using finite-difference derivative estimates, solve the SOS program for $P$ and $K(x)$, then test the closed loop under the true dynamics with a small difference between $\\hat u$ and $\\hat{\\tilde u}$; if the state deviation fails to obey the δ-ISS bound, the exact-derivative equalities are load-bearing and the certificate is not noise-robust.","tokens_in":11666,"feed_emoji":"🎛️","tokens_out":10574,"duration_ms":93906,"temperature":0.7,"pith_summary":"This paper claims that incremental input-to-state stability (δ-ISS), a robustness property comparing two trajectories under different inputs, can be certified and enforced for a class of unknown nonlinear systems using only two recorded input-state trajectories. The systems treated are continuous-time input-affine systems with polynomial dynamics, where the coefficient matrices and the exact monomial dictionary are unknown. Under a data-rank condition, the paper constructs a quadratic Lyapunov function in the state difference and a state-feedback controller by solving a sum-of-squares feasibility program. If the conditions in Theorem 3.5 are satisfied, the designed controller renders the closed-loop system δ-ISS with respect to an external input and incrementally globally asymptotically stable when the external inputs coincide. This matters because δ-ISS controllers are normally built from a model; the paper's claim is that the model can be bypassed entirely.","feed_headline":"Two data trajectories yield certified incremental ISS control","feed_subtitle":"With no plant equations, two recorded runs and one optimization program yield a Lyapunov function and controller.","key_machinery":"The load-bearing object is the data-driven closed-loop representation of Lemma 3.2: from the recorded trajectories one has $X_{1,T} = A J_{0,T} + B U_{0,T}$ and $\\tilde X_{1,T} = A \\tilde J_{0,T} + B U_{0,T}$, and by choosing $G(x)$ with $J_{0,T} G(x) = \\aleph(x)$, the unknown drift $AF(x) + Bu$ becomes $X_{1,T} G(x) x + B\\hat u$. Here $\\aleph(x)$ is the monomial factor matrix satisfying $F(x) = \\aleph(x) x$, and full row-rank of the monomial data matrices $J_{0,T}$ and $\\tilde J_{0,T}$ is the persistence-of-excitation condition that makes the factorization possible. The certificate itself is the simultaneous set of equalities and inequality in (12), which is posed as a sum-of-squares feasibility program over polynomial matrices $Y(x)$, $Y(\\tilde x)$ and a constant matrix $\\Sigma$, with $\\Theta = P^{-1}$ required symmetric positive definite.","core_discovery":"The central claim is Theorem 3.5: for a continuous-time input-affine nonlinear system with polynomial dynamics whose matrices A, B and monomial map F are unknown, if recorded data satisfy conditions (12a)–(12e), then $V(x, \\tilde x) = (x - \\tilde x)^\\top P (x - \\tilde x)$ with $P \\succ 0$ is a δ-ISS Lyapunov function and $u = U_{0,T} Y(x) P x + \\hat u$ is a δ-ISS controller with respect to the external input $\\hat u$. Conditions (12a)–(12b) match the data to the factorization $F(x) = \\aleph(x) x$, conditions (12c)–(12d) force the sampled closed-loop drift on both trajectories to equal a common constant matrix $\\Sigma$, and condition (12e) is a dissipation inequality that yields $\\mathcal{L}V \\le -\\epsilon V + (\\|B\\|^2/\\vartheta)|\\hat u - \\hat{\\tilde u}|^2$. The theorem also states that when $\\hat u = \\hat{\\tilde u}$, the closed loop is incrementally globally asymptotically stable. The proof replaces the unknown open-loop matrices with the data-driven representation $X_{1,T} G(x) x + B\\hat u$ and then bounds the cross term with Young's inequality.","pith_inferences":["The noiseless-derivative assumption is the practical bottleneck: applying finite differences to real sampled data will generally violate the exact equalities in (12c)–(12d), and a noise-aware reformulation would need to relax these to set-membership or robustness constraints.","Full row-rank of $J_{0,T}$ and $\\tilde J_{0,T}$ guarantees existence of some data factor $G(x)$, but it does not by itself certify global polynomial solutions $Y(x)$, $Y(\\tilde x)$ to all of (12); feasibility of the SOS program is the real existence test.","The same data representation could be adapted to certify other trajectory-pair properties, such as contraction or incremental dissipativity, by keeping (12a)–(12d) and only replacing the dissipation inequality (12e).","Because the gain constant $\\rho = \\|B\\|^2/\\vartheta$ is never needed to construct the controller, the method never requires estimating $B$; if the gain value is later wanted, only an upper bound on the induced norm of $B$ is needed."],"forward_implications":["No system identification step is needed: the controller and Lyapunov function are read directly from two recorded trajectories and the corresponding derivative samples.","The certificate is checkable by semidefinite programming, so the practical bottleneck is the size of the monomial dictionary rather than the dimension of the state alone.","The designed controller guarantees a quantitative robustness bound: bounded differences between the external inputs $\\hat u$ and $\\hat{\\tilde u}$ produce state deviations bounded by a class-K function of the input difference.","When the external inputs coincide, all trajectories converge to each other, giving incremental global asymptotic stability for an unknown plant.","The method works with an exaggerated dictionary containing the active monomials plus irrelevant terms, so exact knowledge of the dynamics is never required."],"supporting_citations":[{"why":"Defines δ-ISS and supplies the Lyapunov sufficient condition in Theorem 2.3 that the paper's Theorem 3.5 verifies.","marker":"[1]"},{"why":"Supplies the data-driven closed-loop representation and rank-condition idea for polynomial systems that Lemma 3.2 and the certificate conditions build on.","marker":"[19]"},{"why":"Provides the sum-of-squares optimization toolbox used in Algorithm 1 to enforce the conditions in (12).","marker":"[35]"},{"why":"Provides the semidefinite programming solver used together with the SOS toolbox for feasibility.","marker":"[36]"},{"why":"Supplies the rotating rigid spacecraft model used to demonstrate the data-driven δ-ISS controller design.","marker":"[37]"}],"fun_headline_variants":["Two data runs certify incremental ISS control","Data-only certified delta-ISS controllers","No model? Two runs suffice for delta-ISS","Certified control from just two data sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the sampled state derivatives $X_{1,T}$ and $\\tilde X_{1,T}$ are exact, noiseless data, because conditions (12c)–(12d) are equalities and any numerical differentiation or filtering error makes the certified inequality apply to a data set different from the one measured.","fun_headline_variants_meta":{"raw":{"variants":["Two data runs certify incremental ISS control","Data-only certified delta-ISS controllers","No model? Two runs suffice for delta-ISS","Certified control from just two data sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000497,"raw_usage":{"total_tokens":2465,"prompt_tokens":1007,"completion_tokens":1458,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":623,"completion_tokens_details":{"reasoning_tokens":1403}},"tokens_in":623,"tokens_out":1458,"duration_ms":10082,"temperature":1.0,"reasoning_tokens":1403,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:58:51.309027+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Record real trajectories of a known polynomial input-affine system using finite-difference derivative estimates, solve the SOS program for $P$ and $K(x)$, then test the closed loop under the true dynamics with a small difference between $\\hat u$ and $\\hat{\\tilde u}$; if the state deviation fails to obey the δ-ISS bound, the exact-derivative equalities are load-bearing and the certificate is not noise-robust.","supporting_citations":[{"cited_title":"Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,","cited_arxiv_id":null,"evidence_quote":"Provides the semidefinite programming solver used together with the SOS toolbox for feasibility."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the rotating rigid spacecraft model used to demonstrate the data-driven δ-ISS controller design."}],"review_version":1}