Functional sets with typed symbols: Framework and mixed Polynotopes for hybrid nonlinear reachability and filtering

Verification and synthesis of Cyber-Physical Systems (CPS) are challenging and still raise numerous issues so far. In this paper, an original framework with mixed sets defined as function images of symbol type domains is first proposed. Syntax and semantics are explicitly distinguished. Then, both continuous (interval) and discrete (signed, boolean) symbol types are used to model dependencies through linear and polynomial functions, so leading to mixed zonotopic and polynotopic sets. Polynotopes extend sparse polynomial zonotopes with typed symbols. Polynotopes can both propagate a mixed encoding of intervals and describe the behavior of logic gates. A functional completeness result is given, as well as an inclusion method for elementary nonlinear and switching functions. A Polynotopic Kalman Filter (PKF) is then proposed as a hybrid nonlinear extension of Zonotopic Kalman Filters (ZKF). Bridges with a stochastic uncertainty paradigm are outlined. Finally, several discrete, continuous and hybrid numerical examples including comparisons illustrate the effectiveness of the theoretical results.Comment: 21 pages, 8 figure

Guaranteed Verification of Dynamic Systems

Diese Arbeit beschreibt einen neuen Spezifikations- und Verifikationsansatz für dynamische Systeme. Der neue Ansatz ermöglicht dabei Ergebnisse, die per Definition frei von Fehlern 2. Art sind. Dies bedeutet, dass das Ergebnis der Verifikation keine versteckten Fehler enthalten kann. Somit können zuverlässige Ergebnisse für die Analyse von sicherheitskritischen Systemen generiert werden. Dazu wird ein neues Verständnis von mengenbasierter Konsistenz dynamischer Systeme mit einer gegebenen Spezifikation eingeführt. Dieses basiert auf der Verwendung von Kaucher Intervall Arithmetik zur Einschließung von Messdaten. Konsistenz wird anhand der vereinigten Lösungsmenge der Kaucher Arithmetik definiert. Dies führt zu mathematisch garantierten Ergebnissen. Die resultierende Methode kann das spezifizierte Verhalten eines dynamischen System auch im Falle von Rauschen und Sensorungenauigkeiten anhand von Messdaten verifizieren. Die mathematische Beweisbarkeit der Konsistenz wird für eine große Klasse von Systemen gezeigt. Diese beinhalten zeitinvariante, intervallartige und hybride Systeme, wobei letztere auch zur Beschreibung von Nichtlinearitäten verwendet werden können. Darüber hinaus werden zahlreiche Erweiterungen dargestellt. Diese führen bis hin zu einem neuartigen iterativen Identifikations- und Segmentierungsverfahren für hybride Systeme. Dieses ermöglicht die Verfikation hybrider Systeme auch ohne Wissen über Schaltzeitpunkte. Die entwickelten Verfahren können darüber hinaus zur Diagnose von dynamischen Systemen verwendet werden, falls eine ausreichend schnelle Berechnung der Ergebnisse möglich ist. Die Verfahren werden erfolgreich auf eine beispielhafte Variation verschiedener Tanksysteme angewendet. Die neuen Theorien, Methoden und Algortihmen dieser Arbeit bilden die Grundlage für eine zuverlässige Analyse von hochautomatisierten sicherheitskritischen Systemen

Under-Approximate Reachability Analysis for a Class of Linear Uncertain Systems

Under-approximations of reachable sets and tubes have been receiving growing research attention due to their important roles in control synthesis and verification. Available under-approximation methods applicable to continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general, and/or suffer from high computational costs. In this note, we attempt to overcome these drawbacks for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes, utilizing approximations of the matrix exponential and its integral. In particular, we consider the class of continuous-time LTI systems with an identity input matrix and uncertain initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes, when implemented using zonotopes, with first-order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, we implement our approach in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered

Surveillance préventive des systèmes hybrides à incertitudes bornées

This thesis is dedicated to the development of generic algorithms for the set-membership observation of the continuous state and the discrete mode of hybrid dynamical systems in order to achieve fault detection. This thesis is organized into two parts. In the first part, we have proposed a fast and effective method for the set-membership guard crossing. It consists in carrying out bisection in the time direction only and then makes several contractors working simultaneously to reduce the domain of state vectors located on the guard during the study time slot. Then, we proposed a method for merging trajectories based on zonotopic enclosures. These methods, used together, allowed us to characterize in a guaranteed way the set of all hybrid state trajectories generated by an uncertain hybrid dynamical system on a finite time horizon. The second part focuses on set-membership methods for the parameters or the hybrid state (mode and continuous state) of a hybrid dynamical system in a bounded error framework. We started first by describing fault detection methods for hybrid systems using the parametric approach and the hybrid observer approach. Then, we have described two methods for performing fault detection tasks. We have proposed a method for computing in a guaranteed way all the parameters consistent with the hybrid dynamical model, the actual data and the prior error bound, by using our nonlinear hybrid reachability method and an algorithm for partition which we denote SIVIA-H. Then, for hybrid state estimation, we have proposed a method based on a predictor-corrector, which is also built on top of our non-linear method for hybrid reachability.Cette thèse est dédiée au développement d’algorithmes génériques pour l’observation ensembliste de l’état continu et du mode discret des systèmes dynamiques hybrides dans le but de réaliser la détection de défauts. Cette thèse est organisée en deux grandes parties. Dans la première partie, nous avons proposé une méthode rapide et efficace pour le passage ensembliste des gardes. Elle consiste à procéder à la bissection dans la seule direction du temps et ensuite faire collaborer plusieurs contracteurs simultanément pour réduire le domaine des vecteurs d’état localisés sur la garde, durant la tranche de temps étudiée. Ensuite, nous avons proposé une méthode pour la fusion des trajectoires basée sur l'utilisation des zonotopes. Ces méthodes, utilisées conjointement, nous ont permis de caractériser de manière garantie l'ensemble des trajectoires d'état hybride engendrées par un système dynamique hybride incertain sur un horizon de temps fini. La deuxième partie de la thèse aborde les méthodes ensemblistes pour l'estimation de paramètres et pour l'estimation d'état hybride (mode et état continu) dans un contexte à erreurs bornées. Nous avons commencé en premier lieu par décrire les méthodes de détection de défauts dans les systèmes hybrides en utilisant une approche paramétrique et une approche observateur hybride. Ensuite, nous avons décrit deux méthodes permettant d’effectuer les tâches de détection de défauts. Nous avons proposé une méthode basée sur notre méthode d'atteignabilité hybride non linéaire et un algorithme de partitionnement que nous avons nommé SIVIA-H pour calculer de manière garantie l'ensemble des paramètres compatibles avec le modèle hybride, les mesures et avec les bornes d’erreurs. Ensuite, pour l'estimation d'état hybride, nous avons proposé une méthode basée sur un prédicteurcorrecteur construit au dessus de notre méthode d'atteignabilité hybride non linéaire

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications