Maximally Resilient Controllers under Temporal Logic Specifications

Adnane Saoud; Negar Monir; Pushpak Jagtap; Ratnangshu Das; Sadegh Soudjani; Youssef Ait Si

REVIEW 2 major objections 5 minor 26 references

Maximally Resilient Controllers under Temporal Logic Specifications

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper defines resilience as the largest disturbance a controlled system can absorb while still satisfying a temporal logic specification, and shows it can be computed exactly for linear systems and with probabilistic guarantees for non

desk verdict Good theory, overstated exactness: the Farkas reformulation is real, but the reported resilience values are only local optima unless globally certified. read the letter →

arxiv 2509.01777 v2 pith:MK7HVUH7 submitted 2025-09-01 eess.SY cs.SYmath.OC

Youssef Ait Si , Ratnangshu Das , Negar Monir , Sadegh Soudjani , Pushpak Jagtap , Adnane Saoud This is my paper · ORCID

classification eess.SYcs.SYmath.OC MSC 93C0593C1090C26

keywords resiliencemetrictemporallogicspecificationscontrollersynthesisscenariooptimizationFarkaslemmarobustlinearsystemsnonlinear

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper defines the resilience of a controlled dynamical system as the largest disturbance level, measured in infinity norm, for which there exists a feedback controller such that all closed-loop trajectories satisfy a given finite-horizon temporal logic specification. The central claim is that for linear systems with linear controllers and polytopic specifications, this resilience value is exactly the optimum of a deterministic polynomial optimization problem, obtained by eliminating the universal quantifier over disturbance sequences with an affine Farkas lemma. For nonlinear systems, nonlinear controllers, and general specifications, the paper replaces the robust program by a scenario optimization and proves a probabilistic guarantee that the violation probability of the obtained controller is bounded by a computable quantity. If correct, the framework turns the question of how much disturbance a system can take into a concrete numerical value that can guide controller design.

What carries the argument

The load-bearing object is the resilience metric g_psi(x), defined as a supremum over disturbance levels epsilon and controller parameters alpha. For linear systems, the clean conversion is done by the affine form of Farkas' lemma: it replaces the infinite set of constraints 'E(alpha1) Y <= (1/epsilon) F(...) for all Y with ||Y||_inf <= 1' by the existence of a nonnegative matrix P with P A_b = E(alpha1) and epsilon P B_b <= F(...). For nonlinear systems, the machinery is the scenario approach: sample i.i.d. disturbance sequences, solve a nonconvex program (10) that enforces the constraints on the samples only, and bound the generalization error via the support-constraints count s*_M and the

What would settle it

Take a scalar linear system x(k+1)=x(k)+u(k)+d(k) with a simple polytopic specification over a short horizon, compute the true resilience by brute-force search over epsilon and controller parameters, then solve the polynomial program (4) with a local solver; if the solver returns a strictly smaller value, the claim that the optimization computes the exact resilience is falsified for that instance.

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Extended reading notes

Core claim

The core discovery is Theorem 1: for a discrete-time linear system x(k+1)=Ax(k)+Bu(k)+d(k) with linear feedback u=alpha1 x+alpha2 and a specification written as products of polytopes, the resilience metric g_psi(x) equals the maximum of epsilon over alpha1, alpha2 and a nonnegative matrix P satisfying P>=0, P A_b = E(alpha1), and epsilon P B_b <= F(x,alpha1,alpha2), where A_b, B_b, E, F are explicitly constructed from the system matrices and the specification. This reduces a robust 'for all disturbance sequences' condition to a single polynomial program. For the nonlinear case, Theorem 2 states that the solution theta*_M of the scenario program (10) satisfies P^M( V(theta*_M) < b(s*_M) ) > 1

Load-bearing premise

The exactness claim for linear systems assumes the nonconvex polynomial optimization in Theorem 1 is solved to global optimality; the paper does not provide such a certificate and uses a local solver in the case studies, so the reported resilience values are only guaranteed to be lower bounds.

Editorial extensions

If this is right

For linear systems with linear controllers and polytopic specifications, resilience and the maximizing controller can be obtained from one polynomial optimization program rather than a search over disturbance sequences.
Nonlinear control designs inherit a probabilistic certificate: with a chosen confidence, the probability that a new i.i.d. disturbance violates the specification is bounded by b(s*_M).
The same formulation covers finite-horizon safety, exact-time reachability, and convex linear temporal logic over finite traces within one framework.
Adding input constraints is straightforward in the scenario approach, at the cost of exactness.
The resilience value gives an operational meaning to safety margin: the largest infinity-norm disturbance level the closed loop tolerates while meeting its specification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The exactness of the linear result hinges on global optimality of a nonconvex polynomial program; if a local solver is used, the reported resilience is a certified lower bound but not necessarily the maximum.
The scenario bound could likely be sharpened by using adaptive sampling or wait-and-judge schemes, since the paper's bound depends only on the number of support constraints.
The resilience metric could serve as an objective in contract-based or assume-guarantee synthesis, since it quantifies how much disturbance a controller can promise to tolerate.
A natural testable extension is to compare the scenario-optimized controller's realized violation frequency on a large out-of-sample set against the bound b(s*_M).

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I skimmed arXiv:2509.01777. The one thing to know up front: the linear result is a genuine theoretical contribution, but 'exact' in the title and abstract is not earned by the numerical pipeline. The Farkas reformulation in Theorem 1 is a real step — it turns the robust constraint satisfaction problem into a polynomial optimization over the controller parameters and a multiplier matrix. That gives you a concrete object to maximize. The scenario extension for nonlinear systems is a sensible, competent application of Garatti–Campi nonconvex scenario theory.

What the paper does well: the proof of Theorem 1 in the appendix is algebraically solid. The definition of controlled resilience (Definition 1) is a natural and useful generalization of the uncontrolled notion in [16]. The case studies illustrate the intended use, especially the ACC example where input constraints matter. The LTL_f connection is clean.

The soft spots are about the gap between what is proved and what is computed. Theorem 1 equates g_psi(x) with the global maximum of (4). Problem (4) is nonconvex — alpha1 enters nonlinearly through A+B alpha1 and multiplies P — and the authors solve it with Ipopt, a local interior-point solver. No global optimality certificates, no dual bounds, no multistart. So the reported g_psi(x(0)) = 0.0686 is at best a lower bound on the true resilience. Calling it 'exact' overstates what is actually certified. The same concern applies to the scenario results: nonconvex scenario theory (Theorem 2) needs a deterministic, measurable selection rule for the local optimum. The authors assert such a rule exists ('using a deterministic solver and consistent initialization') but they don't specify the initialization or the selection rule, and they never explain how s*_M support constraints were computed for these nonconvex problems. These are addressable issues, but they are not cosmetic.

There are also a few smaller blemishes: the Figure 3 caption says eps <= 0.03 while Table II gives 0.0367 for M=100; Figure 4 caption says 0.045 while the text reports 0.078 for the polynomial controller. A careful reader will trip on these.

Overall, the theoretical framework holds up. The 'exact solutions' claim should be reworded to 'exact reformulation' with the numerical values presented as local optima unless certified globally. The paper deserves peer review; I'd send it back for revisions, asking for the supporting detail on the scenario solver and cleanup of the inconsistent numbers.

Referee Report

2 major / 5 minor

Summary. The paper defines a resilience metric for controlled discrete-time systems: the largest disturbance magnitude epsilon such that some parameterized feedback controller keeps all closed-loop trajectories within a finite-horizon temporal-logic specification psi. For linear dynamics with linear controllers and polytopic specifications, Theorem 1 reformulates the robust problem as a deterministic polynomial optimization via an affine Farkas lemma, and the paper calls this 'exact'. For nonlinear systems, nonlinear controllers, and general specifications including input constraints, the authors formulate a robust optimization and solve a scenario program (10), with a probabilistic satisfaction guarantee given in Theorem 2 using the non-convex scenario-optimization results of [21]. Two case studies, a mobile robot and an adaptive cruise control example, illustrate the approach.

Significance. The extension of temporal-logic resilience from uncontrolled systems [16] to controlled synthesis is a natural and valuable step. The Farkas-based reformulation in Theorem 1 is elegant and, if the global-optimality issue is handled, provides an exact characterization of resilience for a useful class of linear systems. The scenario-optimization route for nonlinear systems is practically relevant and gives a quantifiable, though approximate, guarantee. The mathematical derivation in the appendix appears sound at the symbolic level. The main caveat is that the numerical claims of 'exact' resilience values are not supported by the local NLP solver used, and the probabilistic theorem relies on a regularity assumption that is invoked but not verified in detail.

major comments (2)

[Section V.A, Theorem 1/Eq. (4)] Theorem 1 equates g_psi(x) with the value of the polynomial optimization (4), and the abstract and case study claim 'exact' solutions, reporting g_psi(x(0))=0.0686. Problem (4) is nonconvex: alpha1 and alpha2 appear nonlinearly, and the equality P A_b = E(alpha1) is bilinear in P and polynomial in alpha. The numerical results are obtained with Pyomo/Ipopt, a local interior-point solver, and no global optimality certificates, dual bounds, or multistart/branch-and-bound verification are provided. A local solution is only a feasible point, so the computed value is a lower bound on the true global optimum and hence on the actual resilience. Consequently the 'sharp transition' claimed in Figures 1-2 does not demonstrate tightness of the reported threshold. Please either solve (4) to certified global optimality or explicitly reframe the numerical values as lower bounds while keeping Theorem 1
[Appendix A-2, Theorem 2] The proof of Theorem 2 states that Property 1 of [21] is satisfied because a deterministic solver with consistent initialization is used. However, Property 1 is not stated in the manuscript, so the reader cannot verify this claim. For the nonconvex problem (10) solved by Ipopt, local solutions may be non-unique and dependent on initialization, and it is not obvious that repeatability is equivalent to the regularity condition required by [21]. Since Theorem 2 is the central guarantee for the nonlinear scenario approach, please state Property 1 explicitly and either prove it for problem (10) under the reported solver setup or state it as an explicit assumption. Without this, the probabilistic guarantee is not fully established.

minor comments (5)

[Section V, Figures 3-4] There are numerical inconsistencies between the text and the figure captions. The text reports g_psi(x(0))=0.036 and 0.078 for the M=100 linear and polynomial ACC cases, while Figure 3 caption says 0.03 and Figure 4 caption says 0.045. Table II lists epsilon*_M=0.0367 for M=100. Please align these values.
[Section IV-B, Eq. (12)] The displayed definition of b(k) and t(k) is garbled; the summation limits and the equation for t(k) are not readable as printed. Please rewrite Equation (12) with standard notation so the dependence on beta and M is unambiguous.
[Appendix B, Theorem 3] The statement says 'Suppose the set {x | Ex <= F} is not empty' before giving the Farkas equivalence. For the application in Section III, the nonempty set should be {Y | A_b Y <= B_b}; the conclusion Ex<=F for all such Y is what is being characterized. Please clarify which nonemptiness condition is intended, since the current wording is confusing.
[Section V.B] The sentence 'At four time steps, we observe that only few points fall outside the set B2, which confirms that... it is probabilistically guaranteed' conflates a finite-sample observation with the probabilistic bound. The guarantee in Theorem 2 is about the measure of violating disturbances, not about the displayed sample; please rephrase.
[Throughout] There are small typographical issues, e.g., 'takes 4s to converg' in Section V.B. The table formatting for the scenario results (Table II) is also hard to read; consider a clearer layout for alpha*1 and alpha*2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivations are self-contained transformations using external theorems (Farkas lemma, scenario optimization) and do not reduce the claims to their inputs.

full rationale

The paper's central result, Theorem 1, reformulates the robust optimization defining resilience into a polynomial optimization problem using the affine Farkas lemma (Theorem 3, citing Schrijver). The equivalence is derived explicitly from the closed-loop linear dynamics (equation 14) and the infinity-norm disturbance set, not from an assumed answer. The optimization variables (epsilon, alpha1, alpha2, P) are decision variables of the reformulation, not fitted parameters. The result genuinely generalizes the authors' prior non-controlled resilience definition [16], which is transparently cited as the origin of the metric rather than as a proof authority. Theorem 2 similarly invokes external scenario-optimization guarantees from Garatti and Campi, and the paper explicitly checks the required property for its nonconvex problem. No 'prediction' is renamed from a fitted value, no uniqueness claim is imported from self-citation, and no ansatz is smuggled via citation. The reported concern about global optimality of the nonconvex polynomial program (Ipopt being a local solver) is a correctness/rigor limitation about the numerical implementation, not a circularity in the mathematical derivation. Therefore the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard convex duality (Farkas) and on the scenario-optimization theory of Garatti and Campi. No data-fitting parameters are introduced in the theory; the reported numbers in the case studies depend on user-chosen M and unreported solver initialization.

free parameters (2)

Scenario count M = 10, 100, 500
User-chosen number of sampled disturbances in Section V-B; the reported resilience g_psi(x0) decreases with M (0.039 for M=10 to 0.0305 for M=500), so M is a tuning knob that affects the result.
Ipopt initial guess
Not reported; for nonconvex problems the local solution depends on the starting point, so the values in Section V may not be reproducible.

assumptions (6)

standard math Affine Farkas' lemma (Theorem 3 in Appendix B)
Used to prove Theorem 1, converting robust constraints over all disturbances into existence of a dual matrix P. Reference [20].
domain assumption Disturbance set is an infinity-norm ball centered at zero (D = Omega_eps(0))
Assumed throughout (Section II-B). The resilience metric and all theorems depend on this specific disturbance geometry.
domain assumption Dynamics and controller are continuous, allowing max instead of sup for closed specifications
Remark 1; needed for Definition 1 to have a maximum rather than a supremum.
domain assumption Scenarios are i.i.d. from (D,F,P) for any probability measure P
Section IV-B; the probabilistic guarantee in Theorem 2 is with respect to this sampling distribution.
ad hoc to paper Property 1 of [21] (deterministic solver with consistent initialization) holds for the solver
Appendix A.2; required for Theorem 2 but not verified for Ipopt in the case studies.
domain assumption Specification is a polytopic set at each time step for Theorem 1
Theorem 1 requires Gamma_k = {x | G_k x <= H_k}. This excludes union-type reachability specs.

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Cite this review

Pith. "Pith review of Maximally Resilient Controllers under Temporal Logic Specifications." pith.science (2026). https://pith.science/paper/MK7HVUH7

@misc{pith2026250901777,
  author       = {Pith},
  title        = {Pith review of: Maximally Resilient Controllers under Temporal Logic Specifications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MK7HVUH7}},
  note         = {Machine review of arXiv:2509.01777}
}

read the original abstract

In this paper, we consider the notion of resilience of a dynamical system, defined by the maximum disturbance a controlled dynamical system can withstand while satisfying given temporal logic specifications. Given a dynamical system and a specification, the objective is to synthesize the controller such that the closed-loop system satisfies this specification while maximizing its resilience. The problem is formulated as a robust optimization program where the objective is to compute the maximum resilience while simultaneously synthesizing the corresponding controller parameters. For linear systems and linear controllers, exact solutions are provided for the class of time-varying polytopic specifications. For the case of nonlinear systems, nonlinear controllers and more general specifications, we leverage tools from the scenario optimization approach, offering a probabilistic guarantee of the solution as well as computational feasibility. Different case studies are presented to illustrate the theoretical results.

Figures

Figures reproduced from arXiv: 2509.01777 by the authors.

**Figure 2.** Illustration of two controlled trajectories of the robot [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Illustration of Adaptive Cruise Control trajectories [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Illustration of adaptive cruise control trajectories with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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——, “Temporal logic resilience for dynamical systems,”arXiv preprint arXiv:2404.19223, 2024

work page Pith review arXiv 2024

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S. Garatti and M. C. Campi, “Non-convex scenario optimization,” Mathematical Programming, pp. 1–52, 2024

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H. Zhang, W. Xiang, W. Lin, and J. Wen, “Grid forming converters in renewable energy sources dominated power grid: Control strategy, sta- bility, application, and challenges,”Journal of modern power systems and clean energy, vol. 9, no. 6, pp. 1239–1256, 2021

work page 2021

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Robustness of temporal logic spec- ifications for continuous-time signals,

G. E. Fainekos and G. J. Pappas, “Robustness of temporal logic spec- ifications for continuous-time signals,”Theoretical Computer Science, vol. 410, no. 42, pp. 4262–4291, 2009

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M. Kloetzer and C. Belta, “A fully automated framework for control of linear systems from temporal logic specifications,”IEEE Transactions on Automatic Control, vol. 53, no. 1, pp. 287–297, 2008. Fig. 4: Illustration of adaptive cruise control trajectories with the optimal polynomial controller and with disturbances less than the maximum disturbance given...

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2015