{"id":"7eae0846-3577-404c-b427-e2d9e43193f3","arxiv_id":"2505.01569","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A physics-informed Gaussian process model is combined with a modified matching equation to design a passivity-based tracking controller with probabilistic stability guarantees.","lead":"This paper designs a tracking controller for physical systems with unknown dynamics, using a Gaussian process model that preserves port-Hamiltonian energy structure. The controller comes with probabilistic stability and passivity guarantees and is demonstrated on a simulated electrostatic microactuator.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The stability claim rests entirely on dissipation inequality (8), which is never verified in the simulation and which the paper itself acknowledges is hard to satisfy; the numerical example therefore supports only boundedness, not the stable-equilibrium claim.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing point: the dissipation condition (8) is a strong state-dependent inequality that is acknowledged as difficult, not derived from the model or design choices, and never verified in the simulation. My independent reading of Theorem 1, Lemma 1, and the Section IV example confirms that the controller can only be said to render a bounded-error trajectory in the simulation, while the stronger stable-equilibrium statement remains conditional on an unverified inequality. The paper's own text admits the difficulty of (8), which strengthens rather than weakens this concern. I also note a secondary technical ambiguity in the confidence set: the stated bound uses βi var(˙xi|x,D) where the cited GP guarantee uses βi times the posterior standard deviation; this should be corrected or explicitly defined. Given that the reader's CONDITIONAL verdict already asks for clarification and stronger validation, my stress-test does not move the verdict: the identified concern is real but addressable by a computable verification of (8) or (15).","tokens_in":10100,"tokens_out":7209,"duration_ms":84652,"concrete_test":"Using the trained GP-PHS model from Section IV, compute the posterior standard deviation σi(x) and the confidence constants βi (e.g. from [23]) on a grid covering X = [−2,2]^3 and along the closed-loop trajectory. For the Hd and Rd chosen in (22)–(24), evaluate L(x) = [∇¯xHd]⊤η(x) − [∇¯xHd]⊤Rd(¯x)∇¯xHd with ηi(x) = ±βi σi(x). If max L(x) > 0 anywhere in X, Theorem 1 does not apply to this design; report the smallest ε such that (15) holds outside ∥x − xd∥ ≥ ε and the corresponding bounded-error radius. If max L(x) ≤ 0 everywhere, the stable-equilibrium claim is verified for the example.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 is conditional on inequality (8), i.e. [∇¯xHd]⊤η(x) ≤ [∇¯xHd]⊤Rd(¯x)∇¯xHd for every x ∈ X and every admissible model error η. This inequality is not derived from the GP-PHS construction, no design rule for Hd and Rd is given that would make it hold, and it is not checked in Section IV. The paper even concedes that (8) 'might be challenging to satisfy' because η must vanish as x → xd. In the simulation, the tracking claim is justified by Lemma 1 rather than Theorem 1, but Lemma 1's relaxed condition (15) is also never evaluated, so the demonstration does not establish the central stable-equilibrium guarantee. The conditional theorem itself may be logically sound, but the claim that the proposed controller ensures stable tracking is unsupported for the presented example unless either a constructive way to satisfy (8) is provided or the region where (15) holds and the corresponding ε are computed. A secondary issue compounds this: the confidence set is stated with βi var(˙xi|x,D) rather than the standard βi σi(x) used by the cited GP bound [23]; if the bound is not corrected, the η-set is mis-sized and condition (8) is being checked against the wrong uncertainty model.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Summary: The paper proposes a data-driven passivity-based tracking controller for port-Hamiltonian systems with partially unknown dynamics. An unknown Hamiltonian is learned with a Gaussian process port-Hamiltonian system (GP-PHS) model, and a modified matching equation extends IDA-PBC to reference tracking. Theorem 1 states that if the desired Hamiltonian and damping satisfy the dissipation domination inequality (8), the tracking error equilibrium of the closed loop is stable with probability at least 1-p. Lemma 1 relaxes (8) to hold only outside an epsilon-ball, yielding probabilistic boundedness, and Corollary 2 gives semi-passivity under an external input. The approach is demonstrated on an electrostatic microactuator.","tokens_in":10395,"tokens_out":8159,"duration_ms":78496,"significance":"Significance: The paper addresses a genuine gap—extending IDA-PBC to tracking under unstructured uncertainty—and the use of GP-PHS is a principled way to carry uncertainty into the control design. The Lyapunov argument is transparent, and the relaxed result in Lemma 1 is a sensible fallback when the strong inequality (8) cannot be certified. The simulation illustrates the closed-loop behavior. However, the central stable-equilibrium claim is only conditional on (8), which is neither constructively designed nor verified; the demonstrated result is boundedness. With a verification procedure and a correction of the confidence-set scaling, the paper would make a solid contribution.","major_comments":[{"comment":"The stability guarantee in Theorem 1 depends entirely on the dissipation domination condition (8), but the paper provides no design rule for choosing H_d and R_d that makes (8) hold, and Section IV never verifies it. The simulation is justified with Lemma 1, yet the relaxed condition (15) is also not evaluated: the paper does not report the epsilon for which (15) holds on the set X=[-2,2]^3, nor the resulting bound on norm(x-x_d). As a consequence, the numerical example supports boundedness but not the stable-equilibrium claim of Theorem 1. Please add a verification procedure for (8) or, for the relaxed claim, compute the region where (15) holds and the corresponding epsilon.","section":"Section IV, Theorem 1 and Lemma 1"},{"comment":"The confidence set is stated as |eta_i(x)| <= beta_i var(xdot_i | x,D), but the standard GP error bound cited from [23] is of the form beta_i sigma_i(x), where sigma_i^2 = var(xdot_i | x,D). With the variance in place of the standard deviation, the admissible eta-set is mis-sized and the 'with probability at least 1-p' statement in Theorem 1 and Lemma 1 is not justified by the cited bound. Please replace var(xdot_i | x,D) by the posterior standard deviation, or provide a separate derivation for the variance-scaled bound.","section":"Section III-A, Theorem 1 and Lemma 1"},{"comment":"In Eq. (10), the model is decomposed as xdot = mu(xdot | x,D) + Ghat(x)u + eta(x), but mu(xdot | x,D) is the posterior mean of the GP-PHS model (6), which already contains the input-dependent term Ghat(x)u. If mu includes Ghat u, the control input is double-counted in (11), and the matching equation (7) must be re-derived. Please define mu precisely (e.g., as the posterior mean of the Hamiltonian drift with Ghat u separated out) and rewrite (7) and (10)-(12) consistently.","section":"Section III-B, proof of Theorem 1"},{"comment":"Property 1 requires only that H_d has its minimum at x=x_d, but the proof of Theorem 1 uses H_d as a Lyapunov function to conclude stability. Lyapunov stability at the equilibrium bar-x=0 also requires H_d to be positive definite with respect to bar-x (at least locally) and, for attractivity, to have an isolated minimum; these conditions are not stated or verified. Corollary 1 adds an isolated minimum only for asymptotic stability. Please state explicit positive-definiteness and regularity assumptions on H_d in Property 1.","section":"Section II-B, Property 1 and Theorem 1"}],"minor_comments":[{"comment":"The phrase 'than the conditions in Lemma 1 are satisfied' should read 'then the conditions in Lemma 1 are satisfied.'","section":"Section III-B, Corollary 2"},{"comment":"The desired trajectory components x_d,2 and x_d,3 are defined only implicitly as solutions of (23); the paper should give the explicit expressions or the numerical solution procedure so that the simulation is reproducible.","section":"Section IV"},{"comment":"The relationship between the plant damping b and the desired-dynamics parameter bhat used in (22) should be stated; the text says the only change is 1/r_d, but the matrix also contains bhat.","section":"Section IV, Eq. (22)"},{"comment":"The meaning of beta_i is left ambiguous; in [23] the confidence parameter depends on the information gain and the chosen failure probability. Please define beta_i explicitly and state how it is computed for the PHS kernel.","section":"Section III-A"},{"comment":"The caption 'Top 1 and 2: Closed-loop system with the proposed tracking control law' is grammatically unclear; consider separating the two panels or rewording the caption.","section":"Figure 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of the journal, and the conditional nature of Theorem 1 is not by itself a reason for rejection. My main concern is that the simulation does not certify the stability condition, so the headline claim is not demonstrated; acceptance should be contingent on adding the verification or explicitly reframing the contribution as boundedness. The variance-versus-standard-deviation issue in the confidence bound should be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real, incremental extension of the authors' earlier GP-PHS work from set-point regulation to trajectory tracking, with a coherent Lyapunov argument that is conditional on a strong dissipation inequality the simulation never verifies. I would send it to review, but the paper needs a corrected uncertainty bound and an honest evaluation before it can be published.\n\nWhat is actually new: the paper takes the modified IDA-PBC matching equation from Wang and Goldsmith and applies it to a learned GP-PHS model, producing a tracking controller with probabilistic stability, boundedness and semi-passivity statements. The split between Theorem 1 (stable equilibrium under a strict dissipation condition) and Lemma 1/Corollary 2 (bounded error and semi-passivity under a relaxed condition) is a useful and honest way to handle model error. The proof structure is standard and, once you read μ as the posterior mean of the Hamiltonian drift rather than the full derivative, the algebra from (7) to (12) is sound.\n\nThe soft spots are real but mostly addressable. The notation around μ is never fixed; equation (10) and the control law make the intended meaning clear, but a reader should not have to guess. More importantly, condition (8) is strong, state-dependent, and there is no design rule for H_d and R_d that guarantees it. The paper itself concedes this. The simulation then never verifies (8) or the relaxed condition (15), and the evaluation text explicitly relies on Lemma 1 rather than Theorem 1. So the numerical example supports boundedness, not the stable-equilibrium claim. That gap should be closed or the claims scaled down. A secondary technical issue: the stated GP confidence bound uses β_i var(˙x_i|x,D) while the cited bound [23] is of the form β_i^{1/2} σ_i(x). As written, the η-set is mis-sized, so condition (8) is checked against the wrong uncertainty model. The simulation also ships no code, data, or GP hyperparameters, which limits reproducibility.\n\nNet: this is a plausible starting point for data-driven passivity-based tracking control, and the relaxed boundedness/semi-passivity portion is worth keeping. With a corrected bound, a check (or design procedure) for (8)/(15), and a reproducible simulation, I would accept it. As it stands, it deserves a serious referee but is not ready as-is.","headline":"A credible but incremental tracking extension of GP-PHS; the stability theorem is conditional on a strong dissipation inequality that the simulation never verifies.","tokens_in":10857,"tokens_out":2990,"would_cite":false,"duration_ms":31031,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C10","93D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a Gaussian-process model of a partially unknown port-Hamiltonian system, combined with a modified matching equation and a damping condition, renders the tracking-error equilibrium stable with probability at least…","keywords":["passivity-based control","port-Hamiltonian systems","Gaussian process regression","tracking control","IDA-PBC","probabilistic stability","data-driven control","semi-passivity"],"falsifier":"For the microactuator simulation, evaluate inequality (8) at finitely many states along the closed-loop trajectory, using the trained GP-PHS posterior variances with $\\eta_i=\\beta_i\\operatorname{var}(\\dot{x}_i|x,\\mathcal{D})$ and the chosen $R_d$. If any sampled state violates the inequality, the conditions of Theorem 1 are not met at that point; a violation on a set of positive measure would refute the claimed probabilistic stability guarantee for that particular design.","tokens_in":9880,"feed_emoji":"🎛️","tokens_out":7728,"duration_ms":69345,"temperature":0.7,"pith_summary":"This paper tries to extend passivity-based tracking control to physical systems whose dynamics are known only partially. It models the unknown part of a port-Hamiltonian system with a Gaussian process, forming a GP-PHS model, and then designs a controller from a modified matching equation rather than from the exact model. The central claim is Theorem 1: if the desired interconnections, damping, and Hamiltonian satisfy the dissipation inequality (8), the tracking error equilibrium $\\bar{x}=0$ is stable with probability at least $1-p$. Relaxing (8) to hold only outside a ball gives bounded tracking error and semi-passivity. A simulation on an electrostatic microactuator shows the controller keeping the tracking error in a small neighborhood.","feed_headline":"Partial physics knowledge is enough for stable tracking control","feed_subtitle":"A Gaussian-process port-Hamiltonian model plus a damping condition keeps the tracking error near zero with probability at least 1-p.","key_machinery":"The central object is the Gaussian process port-Hamiltonian system (GP-PHS), a Gaussian process whose kernel is built from the system's interconnection and dissipation matrices so that its sample paths are port-Hamiltonian dynamics with an unknown Hamiltonian. It supplies the mean prediction and per-component variances that define the confidence set for the model error. The argument is carried by the modified matching equation for tracking, which makes the designed control assign the desired port-Hamiltonian error dynamics, and by the dissipation inequality (8), which lets the desired Hamiltonian $H_d$ serve as a Lyapunov function despite the perturbation $\\eta$.","core_discovery":"The paper's own terms: data-driven IDA-PBC can be taken from set-point regulation to reference tracking while keeping probabilistic stability and passivity guarantees. With a GP-PHS model, the exact matching equation is replaced by $\\hat{G}^{\\perp}\\mu(\\dot{x}|x,\\mathcal{D})=\\hat{G}^{\\perp}\\left([J_d-R_d]\\nabla_{\\bar{x}}H_d+\\dot{x}_d\\right)$, and the control law cancels the learned dynamics and assigns the desired port-Hamiltonian dynamics. The closed-loop error equation becomes $\\dot{\\bar{x}}=[J_d-R_d]\\nabla_{\\bar{x}}H_d+\\eta$, with the GP model error $\\eta$ bounded by $\\beta_i\\operatorname{var}(\\dot{x}_i|x,\\mathcal{D})$ with probability $1-p$. If $[\\nabla_{\\bar{x}}H_d]^\\top\\eta\\le [\\nabla_{\\bar{x}}H_d]^\\top R_d\\nabla_{\\bar{x}}H_d$, then $H_d$ is a Lyapunov function and the error equilibrium is stable; weaker forms of the same inequality give boundedness and semi-passivity.","pith_inferences":["The paper leaves open whether the modified matching equation (7) can be solved for $H_d$ and $R_d$ in general; the simulation chooses $H_d$ by hand as the posterior mean evaluated at $\\bar{x}$, so the general solvability of the PDE is untested.","A direct test of the load-bearing condition would be to compute $\\beta_i\\operatorname{var}(\\dot{x}_i|x,\\mathcal{D})$ along the simulated trajectory and check (8) pointwise; the paper does not report this check.","The confidence set for $\\eta$ is symmetric and componentwise state-independent; if the true model error is biased or heteroscedastic, inequalities (8) and (15) could fail even when the GP marginal variances look small.","The approach suggests a design loop the paper does not pursue: jointly optimize $H_d$ and $R_d$ to satisfy both (7) and (8), which would make the damping condition easier to meet and the guarantees constructive."],"forward_implications":["If Theorem 1 holds, a tracking controller can be designed without knowing the Hamiltonian, provided training data make the GP error small and the damping satisfies inequality (8).","Stability is probabilistic with an explicit probability $1-p$, so data quantity and quality enter directly into the safety claim.","Under the relaxed condition (15), the tracking error is ultimately bounded in a neighborhood of the desired trajectory, and the neighborhood size can be estimated by level-set methods.","Corollary 2 means the closed-loop system is semi-passive with respect to an external input, allowing interconnection with other passive systems."],"supporting_citations":[{"why":"Introduces the GP-PHS model, its PHS-structured kernel, and the training algorithm used to obtain the mean and variance.","marker":"[21]"},{"why":"Provides the modified energy-balancing matching equation for tracking, which the paper adapts to the GP-PHS setting.","marker":"[10]"},{"why":"Establishes IDA-PBC matching and the Lyapunov argument via the desired Hamiltonian; Theorem 1 relies on its Proposition 1 for asymptotic stability.","marker":"[6]"},{"why":"Supplies the RKHS-norm-based probabilistic bound on GP prediction error used to build the confidence set for $\\eta$.","marker":"[23]"},{"why":"Defines semi-passivity, the property claimed in Corollary 2.","marker":"[25]"},{"why":"Supplies the electrostatic microactuator model used in the simulation.","marker":"[26]"},{"why":"Provides the level-set methods cited for computing the size of the tracking-error neighborhood in Lemma 1.","marker":"[24]"}],"fun_headline_variants":["GP-PHS: Probabilistic guarantees for tracking control","From set-point to tracking with passivity-based learning","Uncertainty-aware passivity for reference tracking","Data-driven tracking with guaranteed passivity","Learning passivity-based tracking under uncertainty"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is condition (8) (and its relaxed form (15)): the injected damping $R_d$ must dominate the worst-case Gaussian-process model error $\\eta$ for every state and every $\\eta$ in the confidence set defined by $\\beta_i\\operatorname{var}(\\dot{x}_i|x,\\mathcal{D})$. This state-dependent inequality is not derived from the GP model or from $H_d$, is acknowledged as hard to satisfy, and is never verified in the simulation.","fun_headline_variants_meta":{"raw":{"variants":["GP-PHS: Probabilistic guarantees for tracking control","From set-point to tracking with passivity-based learning","Uncertainty-aware passivity for reference tracking","Data-driven tracking with guaranteed passivity","Learning passivity-based tracking under uncertainty"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000603,"raw_usage":{"total_tokens":2799,"prompt_tokens":913,"completion_tokens":1886,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":1816}},"tokens_in":529,"tokens_out":1886,"duration_ms":13527,"temperature":1.0,"reasoning_tokens":1816,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:15:47.350433+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the microactuator simulation, evaluate inequality (8) at finitely many states along the closed-loop trajectory, using the trained GP-PHS posterior variances with $\\eta_i=\\beta_i\\operatorname{var}(\\dot{x}_i|x,\\mathcal{D})$ and the chosen $R_d$. If any sampled state violates the inequality, the conditions of Theorem 1 are not met at that point; a violation on a set of positive measure would refute the claimed probabilistic stability guarantee for that particular design.","supporting_citations":[{"cited_title":"Gaussian process port-Hamiltonian systems: Bayesian learning with physics prior,","cited_arxiv_id":null,"evidence_quote":"Introduces the GP-PHS model, its PHS-structured kernel, and the training algorithm used to obtain the mean and variance."},{"cited_title":"Modified energy-balancing-based control for the tracking problem,","cited_arxiv_id":null,"evidence_quote":"Provides the modified energy-balancing matching equation for tracking, which the paper adapts to the GP-PHS setting."},{"cited_title":"Inter- connection and damping assignment passivity-based control of port- controlled Hamiltonian systems,","cited_arxiv_id":null,"evidence_quote":"Establishes IDA-PBC matching and the Lyapunov argument via the desired Hamiltonian; Theorem 1 relies on its Proposition 1 for asymptotic stability."},{"cited_title":"Information-theoretic regret bounds for Gaussian process optimiza- tion in the bandit setting,","cited_arxiv_id":null,"evidence_quote":"Supplies the RKHS-norm-based probabilistic bound on GP prediction error used to build the confidence set for $\\eta$."},{"cited_title":"Passivity based design of synchronizing systems,","cited_arxiv_id":null,"evidence_quote":"Defines semi-passivity, the property claimed in Corollary 2."},{"cited_title":"Nonlinear dynamic output feedback stabilization of electrostatically actuated mems,","cited_arxiv_id":null,"evidence_quote":"Supplies the electrostatic microactuator model used in the simulation."},{"cited_title":"Stability of dynamical systems,","cited_arxiv_id":null,"evidence_quote":"Provides the level-set methods cited for computing the size of the tracking-error neighborhood in Lemma 1."}],"review_version":1}