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arxiv: 2604.10680 · v1 · submitted 2026-04-12 · 📡 eess.SY · cs.SY

Resilient and Effort-Optimal Controller Synthesis under Temporal Logic Specifications

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

Pith reviewed 2026-05-10 15:53 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords controller synthesisresilienceeffort metrictemporal logic specificationsrobust optimizationscenario optimizationdynamical systemsinput constraints
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The pith

Controllers can be synthesized to maximize the maximum disturbance a system can withstand while satisfying temporal logic specifications under input constraints.

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

This paper develops methods to synthesize controllers for dynamical systems that meet finite temporal logic specifications while maximizing the system's ability to withstand disturbances. It defines resilience as the maximum disturbance level the closed-loop system can handle without violating the spec, subject to input limits. A new effort metric is introduced to measure the smallest input bound required to satisfy the spec despite perturbations. The synthesis is cast as a robust optimization problem, solved exactly for linear systems with polytopic specs and approximately with probabilistic guarantees for nonlinear systems using scenario optimization. Trade-offs between higher resilience and lower control effort are also quantified.

Core claim

The paper establishes that a robust optimization program can compute the maximum resilience or minimal effort for controller synthesis under temporal logic specs. For linear systems and linear controllers with time-varying polytopic specifications, exact solutions are derived for both closed-loop and open-loop cases. For nonlinear systems and general specs, scenario optimization yields controllers with probabilistic satisfaction guarantees.

What carries the argument

A robust optimization program that maximizes resilience (or minimizes effort) while ensuring the system satisfies the temporal logic specification, using exact methods for linear cases and scenario sampling for nonlinear ones.

Load-bearing premise

The sampled scenarios in the scenario optimization accurately represent the uncertainty set so that the probabilistic guarantee holds for the synthesized controller.

What would settle it

A simulation or experiment where the synthesized controller fails to satisfy the temporal logic specification when subjected to a disturbance smaller than the computed resilience bound.

Figures

Figures reproduced from arXiv: 2604.10680 by Adnane Saoud, Negar Monir, Pushpak Jagtap, Ratnangshu Das, Sadegh Soudjani, Youssef Ait Si.

Figure 1
Figure 1. Figure 1: Illustration of the Pareto curve in blue which corresponds to the optimal trade-off for resilience and effort metric for [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the Pareto curve in blue which corresponds to the optimal trade-off for resilience and effort metric for [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sample trajectories for adaptive cruise control from Section VI-B with the optimal linear controller and with disturbances [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sample trajectories for adaptive cruise control from Section VI-B with the optimal polynomial controller and with [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the collision-avoidance scenario between the ego and the intruder cars. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: State trajectories showing rotor angle δ (rad), rotor speed ω (rad/s), and field voltage efd over the horizon N = 30 for the nominal and the control trajectories of the considered model. The shaded regions indicate constraint bounds that must be satisfied to respect the specification ψ in (31) by the controlled trajectories with the same color [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The normalized and synthesized open-loop controller values corresponding to inputs [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
read the original abstract

In this paper, we consider the notions of effort and resilience of a dynamical control system defined by the maximum disturbance the system can withstand while satisfying given finite temporal logic specifications. Given a dynamical system and a specification, the objective is to synthesize the controller such that the system satisfies the specification while maximizing its resilience, taking into account input constraints. In addition, we introduce a new metric, called the effort metric, which characterizes the minimal input bound necessary to satisfy a given specification for a perturbed system. The problem for both metrics is formulated as a robust optimization program where the objective is to compute the maximum resilience for the system with input constraints or the minimal effort while simultaneously synthesizing the corresponding controller parameters. Moreover, we study the trade-off between resilience and effort, where we seek to maximize resilience and minimize the control effort. For linear systems and linear controllers, exact solutions are provided for the class of time-varying polytopic specifications for the closed-loop and open-loop systems. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines resilience (maximum disturbance a closed-loop system can withstand while satisfying a finite-horizon temporal-logic specification) and effort (minimum input bound needed for a perturbed system to meet the same specification). It formulates both as robust optimization programs that jointly synthesize a controller and optimize the metric, subject to input constraints. Exact, closed-form solutions are claimed for linear systems under time-varying polytopic specifications in both open- and closed-loop settings. For nonlinear dynamics, nonlinear controllers, and general specifications, scenario optimization is invoked to obtain a probabilistic guarantee of satisfaction together with computational tractability. Trade-offs between resilience and effort are also considered, and the claims are illustrated on several case studies.

Significance. If the convexity and guarantee arguments hold, the work supplies two new, interpretable metrics that directly link temporal-logic satisfaction to disturbance rejection and control cost, together with synthesis procedures that are exact for an important linear-polytopic subclass and probabilistically sound for broader nonlinear cases. The explicit treatment of the resilience-effort Pareto front is a useful addition to the robust-control and STL-synthesis literature.

major comments (2)
  1. [Abstract / nonlinear-systems section] Abstract and the section presenting the nonlinear/scenario-optimization results: the probabilistic guarantee is stated to follow from standard scenario-optimization theory, yet the feasible set is defined by the robustness semantics of a temporal-logic formula (or equivalent predicate constraints) together with nonlinear dynamics. Standard sample-complexity bounds (Campi et al.) require convexity of the constraint set; no explicit convexity argument or alternative (e.g., VC-dimension) bound is supplied for the general-specification case. This is load-bearing for the central claim that the method “offers a probabilistic guarantee” for nonlinear systems.
  2. [Linear-systems exact-solution section] Linear-systems section (exact solutions for polytopic specifications): the derivation of the closed-form resilience and effort values appears to rely on the polytopic set remaining invariant under the linear closed-loop map. It is not shown whether the same closed-form expressions continue to hold when the specification is a general STL formula whose robustness margin is non-polytopic.
minor comments (2)
  1. [Introduction / metric definitions] Notation for the effort metric is introduced without an explicit comparison to existing input-energy or peak-input metrics; a short paragraph relating the new definition to prior work would improve clarity.
  2. [Numerical examples] The case-study figures would benefit from explicit labeling of the uncertainty samples used in the scenario approach and from reporting the empirical violation rate on a held-out test set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive review of our manuscript. The comments have identified important points for clarification and strengthening of our claims. We address each major comment below and will incorporate the necessary revisions in the next version of the paper.

read point-by-point responses

  1. Referee: [Abstract / nonlinear-systems section] Abstract and the section presenting the nonlinear/scenario-optimization results: the probabilistic guarantee is stated to follow from standard scenario-optimization theory, yet the feasible set is defined by the robustness semantics of a temporal-logic formula (or equivalent predicate constraints) together with nonlinear dynamics. Standard sample-complexity bounds (Campi et al.) require convexity of the constraint set; no explicit convexity argument or alternative (e.g., VC-dimension) bound is supplied for the general-specification case. This is load-bearing for the central claim that the method “offers a probabilistic guarantee” for nonlinear systems.

    Authors: We appreciate the referee highlighting this key requirement for the probabilistic guarantees. The manuscript invokes scenario optimization for the nonlinear and general-specification cases to achieve computational tractability and probabilistic guarantees, but does not supply an explicit convexity argument for the feasible set defined by general STL robustness semantics under nonlinear dynamics. We agree that the standard Campi et al. bounds rely on convexity, which does not hold universally for arbitrary STL formulas. We will revise the abstract and nonlinear-systems section to explicitly state that the probabilistic guarantees apply when the resulting optimization problem is convex (as is the case for our linear-polytopic results and certain convex nonlinear fragments), and for general non-convex cases we will clarify that the approach yields practical controllers with empirical validation on sampled scenarios while noting the limitation on theoretical sample-complexity bounds. We will also add a brief discussion of potential use of VC-dimension or non-convex scenario optimization extensions where relevant. This addresses the load-bearing concern directly. revision: yes

  2. Referee: [Linear-systems exact-solution section] Linear-systems section (exact solutions for polytopic specifications): the derivation of the closed-form resilience and effort values appears to rely on the polytopic set remaining invariant under the linear closed-loop map. It is not shown whether the same closed-form expressions continue to hold when the specification is a general STL formula whose robustness margin is non-polytopic.

    Authors: We thank the referee for this precise observation. The closed-form expressions for resilience and effort are derived specifically under the assumption of time-varying polytopic specifications, which ensures invariance of the polytopic structure under the linear closed-loop map and enables exact solutions. The manuscript limits these exact results to the polytopic class and does not claim or derive the same closed-forms for general STL formulas (where the robustness margin need not be polytopic); general STL cases are instead handled by the scenario optimization method. To eliminate any ambiguity, we will insert a clarifying sentence in the linear-systems section that explicitly delimits the scope of the closed-form results to polytopic specifications and cross-references the scenario approach for broader STL formulas. This is a straightforward clarification that strengthens the presentation without altering the technical claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are standard robust optimization without self-referential reductions

full rationale

The paper defines resilience and effort metrics directly from the system dynamics, input constraints, and temporal logic specifications, then formulates synthesis as a robust optimization program. Exact solutions for linear systems follow from the closed-form polytopic constraints without fitting or renaming. For nonlinear cases, scenario optimization is invoked as an external tool to obtain probabilistic guarantees on the sampled uncertainty set, with no evidence that the guarantee itself reduces to a self-citation chain, fitted parameter, or ansatz smuggled from prior author work. No step equates a claimed prediction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on standard assumptions from robust optimization and scenario theory plus the newly introduced definitions of resilience and effort; no free parameters are explicitly fitted in the abstract description.

axioms (1)
  • standard math Standard assumptions of robust optimization and scenario optimization theory hold for the probabilistic guarantees
    Invoked for the nonlinear case to obtain probabilistic feasibility
invented entities (2)
  • Resilience metric no independent evidence
    purpose: Quantify the maximum disturbance the closed-loop system can withstand while satisfying the temporal logic specification
    Newly defined in the paper as the core objective
  • Effort metric no independent evidence
    purpose: Characterize the minimal input bound required to satisfy the specification for a perturbed system
    Newly introduced metric

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discussion (0)

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