arXiv subjects

Khalil Ghorbal

Publications and source records attributed to Khalil Ghorbal.

7 recordsLinked to original sources

arXiv2026

Structural Methods for handling mode changes in multimode DAE systems

Hybrid systems are an important concept in Cyber-Physical Systems modeling, for which multiphysics modeling from first principles and the reuse of models from libraries are key. To achieve this, DAEs must be used to specify the dynamics in each discrete state (or mode in our context). This led to the development of DAE-based equational languages supporting multiple modes, of which Modelica is a popular standard. Mode switching can be time- or state-based. Impulsive behaviors can occur at mode changes. While mode changes are well understood in particular physics (e.g., contact mechanics), this is not the case in physics-agnostic paradigms such as Modelica. This situation causes difficulties for the compilation of programs, often requiring users to manually smooth out mode changes. In this paper, we propose a novel approach for the hot restart at mode changes in such paradigms. We propose a mathematical meaning for hot restarts (such a mathematical meaning does not exist in general), as well as a combined structural and impulse analysis for mode changes, generating the hot restart even in the presence of impulses. Our algorithm detects at compile time if the mode change is insufficiently specified, in which case it returns diagnostics information to the user.

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arXiv2025

On Covering Euclidean Space with Q-arrangements of Cones

This paper is concerned with a covering problem of Euclidean space by a particular arrangement of cones that are not necessarily full and are allowed to overlap. The problem provides an equivalent geometric reformulation of the solvability of the linear complementarity problem defining the class of Q-matrices. Assuming feasibility, we rely on standard tools from convex geometry to study maximal connected uncovered regions, we term \emph{holes}. We then use our approach to fully characterize the problem for dimension $3$, regardless of degeneracy. We further provide, for $n \leq 3$, an algebraic characterization for the class of Q-matrices. That is, we show that, $M$ is a Q-matrix if and only if its entries belong to an explicit semi-algebraic set (in dimension $9$) where all the involved polynomials are subdeterminants of $M$. We showcase the usefulness of such a characterization by generating $3$-by-$3$ Q-matrices with specific interesting properties on the involved cones.

math.OC↗

arXiv2021

Characterizing Positively Invariant Sets: Inductive and Topological Methods

We present two characterizations of positive invariance of sets under the flow of systems of ordinary differential equations. The first characterization uses inward sets which intuitively collect those points from which the flow evolves within the set for a short period of time, whereas the second characterization uses the notion of exit sets, which intuitively collect those points from which the flow immediately leaves the set. Our proofs emphasize the use of the real induction principle as a generic and unifying proof technique that captures the essence of the formal reasoning justifying our results and provides cleaner alternative proofs of known results. The two characterizations presented in this article, while essentially equivalent, lead to two rather different decision procedures (termed respectively LZZ and ESE) for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations. The procedure LZZ improves upon the original work by Liu, Zhan and Zhao (EMSOFT 2011). The procedure ESE, introduced in this article, works by splitting the problem, in a principled way, into simpler sub-problems that are easier to check, and is shown to exhibit substantially better performance compared to LZZ on problems featuring semi-algebraic sets described by formulas with non-trivial Boolean structure.

cs.CG↗

arXiv2020

Ordered Functional Decision Diagrams: A Functional Semantics For Binary Decision Diagrams

We introduce a novel framework, termed $λ$DD, that revisits Binary Decision Diagrams from a purely functional point of view. The framework allows to classify the already existing variants, including the most recent ones like Chain-DD and ESRBDD, as implementations of a special class of ordered models. We enumerate, in a principled way, all the models of this class and isolate its most expressive model. This new model, termed $λ$DD-O-NUCX, is suitable for both dense and sparse Boolean functions, and is moreover invariant by negation. The canonicity of $λ$DD-O-NUCX is formally verified using the Coq proof assistant. We furthermore give bounds on the size of the different diagrams: the potential gain achieved by more expressive models can be at most linear in the number of variables n.

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arXiv2017

Formal Verification of Station Keeping Maneuvers for a Planar Autonomous Hybrid System

We formally verify a hybrid control law designed to perform a station keeping maneuver for a planar vehicle. Such maneuver requires that the vehicle reaches a neighborhood of its station in finite time and remains in it while waiting for further instructions. We model the dynamics as well as the control law as a hybrid program and formally verify both the reachability and safety properties involved. We highlight in particular the automated generation of invariant regions which turns out to be crucial in performing such verification. We use the theorem prover Keymaera X to discharge some of the generated proof obligations.

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arXiv2016

Formal Verification of Obstacle Avoidance and Navigation of Ground Robots

The safety of mobile robots in dynamic environments is predicated on making sure that they do not collide with obstacles. In support of such safety arguments, we analyze and formally verify a series of increasingly powerful safety properties of controllers for avoiding both stationary and moving obstacles: (i) static safety, which ensures that no collisions can happen with stationary obstacles, (ii) passive safety, which ensures that no collisions can happen with stationary or moving obstacles while the robot moves, (iii) the stronger passive friendly safety in which the robot further maintains sufficient maneuvering distance for obstacles to avoid collision as well, and (iv) passive orientation safety, which allows for imperfect sensor coverage of the robot, i. e., the robot is aware that not everything in its environment will be visible. We complement these provably correct safety properties with liveness properties: we prove that provably safe motion is flexible enough to let the robot still navigate waypoints and pass intersections. We use hybrid system models and theorem proving techniques that describe and formally verify the robot's discrete control decisions along with its continuous, physical motion. Moreover, we formally prove that safety can still be guaranteed despite sensor uncertainty and actuator perturbation, and when control choices for more aggressive maneuvers are introduced. Our verification results are generic in the sense that they are not limited to the particular choices of one specific control algorithm but identify conditions that make them simultaneously apply to a broad class of control algorithms.

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arXiv2010

A Logical Product Approach to Zonotope Intersection

We define and study a new abstract domain which is a fine-grained combination of zonotopes with polyhedric domains such as the interval, octagon, linear templates or polyhedron domain. While abstract transfer functions are still rather inexpensive and accurate even for interpreting non-linear computations, we are able to also interpret tests (i.e. intersections) efficiently. This fixes a known drawback of zonotopic methods, as used for reachability analysis for hybrid sys- tems as well as for invariant generation in abstract interpretation: intersection of zonotopes are not always zonotopes, and there is not even a best zonotopic over-approximation of the intersection. We describe some examples and an im- plementation of our method in the APRON library, and discuss some further in- teresting combinations of zonotopes with non-linear or non-convex domains such as quadratic templates and maxplus polyhedra.

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