Antoine Girard

Anthony Couthures, Athira Varma Jayakumar, Vineeth Satheeskumar Varma et al.

Conventional community detection requires centralized network data, making it unsuitable for distributed or privacy-preserving systems. In this paper, we demonstrate that macroscopic graph partitioning can emerge purely from strictly local, privacy preserving interactions driven by social learning. By reframing clustering as a symmetry-breaking process within nonlinear opinion dynamics, we show that exchanging saturated state dependent signal (like public actions) forces a network to naturally fracture along its sparsest cuts. We mathematically establish the spectral conditions under which dense core communities lock into stable, polarized states, robustly resisting external influence. To apply this mechanism, we propose three decentralized algorithms, leading up to the Score-based Edge Reliability (SER) framework. By evaluating network ties across multiple independent discussion topics, SER statistically bypasses the errors of traditional greedy bisections and naturally isolates structurally ambiguous frontier nodes. Validations on the ABCD benchmark and the real-world Ngogo chimpanzee network confirm that our fully decentralized approach matches the accuracy of globally optimized heuristics (e.g., Louvain, Leiden) up to a theoretical limit of detectable graphs.

Antoine Girard

In this paper, we present a new class of event triggering mechanisms for event-triggered control systems. This class is characterized by the introduction of an internal dynamic variable, which motivates the proposed name of dynamic event triggering mechanism. The stability of the resulting closed loop system is proved and the influence of design parameters on the decay rate of the Lyapunov function is discussed. For linear systems, we establish a lower bound on the inter-execution time as a function of the parameters. The influence of these parameters on a quadratic integral performance index is also studied. Some simulation results are provided for illustration of the theoretical claims.

Antoine Girard

In this paper, we consider the problem of controller design using approximately bisimilar abstractions with an emphasis on safety and reachability specifications. We propose abstraction-based approaches to solve both classes of problems. We start by synthesizing a controller for an approximately bisimilar abstraction. Then, using a concretization procedure, we obtain a controller for our initial system that is proved "correct by design". We provide guarantees of performance by giving estimates of the distance of the synthesized controller to the maximal (i.e the most permissive) safety controller or to the time-optimal reachability controller. Finally, we use the presented techniques combined with discrete approximately bisimilar abstractions of switched systems developed recently, for switching controller synthesis.

Mohamed Amin Ben Sassi, Antoine Girard

This paper deals with the computation of polytopic invariant sets for polynomial dynamical systems. An invariant set of a dynamical system is a subset of the state space such that if the state of the system belongs to the set at a given instant, it will remain in the set forever in the future. Polytopic invariants for polynomial systems can be verified by solving a set of optimization problems involving multivariate polynomials on bounded polytopes. Using the blossoming principle together with properties of multi-affine functions on rectangles and Lagrangian duality, we show that certified lower bounds of the optimal values of such optimization problems can be computed effectively using linear programs. This allows us to propose a method based on linear programming for verifying polytopic invariant sets of polynomial dynamical systems. Additionally, using sensitivity analysis of linear programs, one can iteratively compute a polytopic invariant set. Finally, we show using a set of examples borrowed from biological applications, that our approach is effective in practice.

Antoine Girard, Samuel Martin

In this paper, we propose an approach to controller synthesis for a class of constrained nonlinear systems. It is based on the use of a hybridization, that is a hybrid abstraction of the nonlinear dynamics. This abstraction is defined on a triangulation of the state-space where on each simplex of the triangulation, the nonlinear dynamics is conservatively approximated by an affine system subject to disturbances. Except for the disturbances, this hybridization can be seen as a piecewise affine hybrid system on simplices for which appealing control synthesis techniques have been developed in the past decade. We extend these techniques to handle systems subject to disturbances by synthesizing and coordinating local robust affine controllers defined on the simplices of the triangulation. We show that the resulting hybrid controller can be used to control successfully the original constrained nonlinear system. Our approach, though conservative, can be fully automated and is computationally tractable. To show its effectiveness in practical applications, we apply our method to control a pendulum mounted on a cart.

Antoine Girard

In this paper, we consider the problem of synthesizing low-complexity controllers for incrementally stable switched systems. For that purpose, we establish a new approximation result for the computation of symbolic models that are approximately bisimilar to a given switched system. The main advantage over existing results is that it allows us to design naturally quantized switching controllers for safety or reachability specifications; these can be pre-computed offline and therefore the online execution time is reduced. Then, we present a technique to reduce the memory needed to store the control law by borrowing ideas from algebraic decision diagrams for compact function representation and by exploiting the non-determinism of the synthesized controllers. We show the merits of our approach by applying it to a simple model of temperature regulation in a building.

Mohamed Amin Ben Sassi, Antoine Girard

In this paper, we consider a control synthesis problem for a class of polynomial dynamical systems subject to bounded disturbances and with input constraints. More precisely, we aim at synthesizing at the same time a controller and an invariant set for the controlled system under all admissible disturbances. We propose a computational method to solve this problem. Given a candidate polyhedral invariant, we show that controller synthesis can be formulated as an optimization problem involving polynomial cost functions over bounded polytopes for which effective linear programming relaxations can be obtained. Then, we propose an iterative approach to compute the controller and the polyhedral invariant at once. Each iteration of the approach mainly consists in solving two linear programs (one for the controller and one for the invariant) and is thus computationally tractable. Finally, we show with several examples the usefulness of our method in applications.

Abdalla Swikir, Antoine Girard, Majid Zamani

In this work, we introduce a compositional framework for the construction of finite abstractions (a.k.a. symbolic models) of interconnected discrete-time control systems. The compositional scheme is based on the joint dissipativity-type properties of discrete-time control subsystems and their finite abstractions. In the first part of the paper, we use a notion of so-called storage function as a relation between each subsystem and its finite abstraction to construct compositionally a notion of so-called simulation function as a relation between interconnected finite abstractions and that of control systems. The derived simulation function is used to quantify the error between the output behavior of the overall interconnected concrete system and that of its finite abstraction. In the second part of the paper, we propose a technique to construct finite abstractions together with their corresponding storage functions for a class of discrete-time control systems under some incremental passivity property. We show that if a discrete-time control system is so-called incrementally passivable, then one can construct its finite abstraction by a suitable quantization of the input and state sets together with the corresponding storage function. Finally, the proposed results are illustrated by constructing a finite abstraction of a network of linear discrete-time control systems and its corresponding simulation function in a compositional way. The compositional conditions in this example do not impose any restriction on the gains or the number of the subsystems which, in particular, elucidates the effectiveness of dissipativity-type compositional reasoning for networks of systems.

Antoine Aspeel, Antoine Girard, Thiago Alves Lima

We study the control of nonlinear constrained systems via over-approximations. Our key observation is that the over-approximation error, rather than being an unknown disturbance, can be exploited as input-dependent preview information. This leads to the notion of informed policies, which depend on both the state and the error. We formulate the concretization problem -- recovering a valid input for the true system from a preview-based policy -- as a fixed-point equation. Existence of solutions follows from the Brouwer fixed-point theorem, while efficient computation is enabled through closed-form, linear, or convex programs for input-affine systems, and through an iterative method based on the Banach fixed-point theorem for nonlinear systems.

Julien Calbert, Antoine Girard, Raphaël M. Jungers

Abstraction-based control design is a promising approach for ensuring safety-critical control of complex cyber-physical systems. A key aspect of this methodology is the relation between the original and abstract systems, which ensures that the abstract controller can be transformed into a valid controller for the original system through a concretization procedure. In this paper, we provide a comprehensive and systematic framework that characterizes various simulation relations, through their associated concretization procedures. We introduce the concept of interfaced system, which universally enables a feedback refinement relation with the abstract system. This interfaced system encapsulates the specific characteristics of each simulation relation within an interface, enabling a plug-and-play control architecture. Our results demonstrate that the existence of a particular simulation relation between the concrete and abstract systems is equivalent to the implementability of a specific control architecture, which depends on the considered simulation relation. This allows us to introduce new types of relations, and to establish the advantages and drawbacks of different relations, which we exhibit through detailed examples.

Zirui Niu, Antoine Girard, Giordano Scarciotti

Controlling complex dynamical systems to satisfy sophisticated specifications remains a significant challenge in modern engineering. A promising approach to this problem is the approximate simulation-based hierarchical control (ASHC) technique. In this method, a simplified representation of the complex system, called the abstract system, is first designed and controlled. An interface function is then designed to translate the control law into the input of the complex system, thereby achieving approximate control synthesis. However, most existing results in ASHC are only for linear systems. This paper proposes a constructive method for solving the ASHC problem for nonlinear systems. To this end, we propose invariance equation-based methods to achieve the two classical requirements of the ASHC technique, namely the bounded output discrepancy and the $m$-relation. We then study the solvability conditions of the problem and summarise the overall design procedures. We illustrate the results with a practical example, providing step-by-step solutions to the ASHC problem of a DC-to-DC Ćuk converter.

Pierre-Jean Meyer, Antoine Girard, Emmanuel Witrant

In this paper, we develop a compositional approach to abstraction and safety synthesis for a general class of discrete time nonlinear systems. Our approach makes it possible to define a symbolic abstraction by composing a set of symbolic subsystems that are overlapping in the sense that they can share some common state variables. We develop compositional safety synthesis techniques using such overlapping symbolic subsystems. Comparisons, in terms of conservativeness and of computational complexity, between abstractions and controllers obtained from different system decompositions are provided. Numerical experiments show that the proposed approach for symbolic control synthesis enables a significant complexity reduction with respect to the centralized approach, while reducing the conservatism with respect to compositional approaches using non-overlapping subsystems.

Jihene Ben Rejeb, Irinel-Constantin Morărescu, Antoine Girard et al.

Motivated by a real problem in steel production, we introduce and analyze a general class of singularly perturbed linear hybrid systems with both switches and impulses, in which the slow or fast nature of the variables can be mode-dependent. This means that, at switching instants, some of the slow variables can become fast and vice-versa. Firstly, we show that using a mode-dependent variable reordering we can rewrite this class of systems in a form in which the variables preserve their nature over time. Secondly, we establish, through singular perturbation techniques, an upper bound on the minimum dwell-time ensuring the overall system's stability. Remarkably, this bound is the sum of two terms. The first term corresponds to an upper bound on the minimum dwell-time ensuring the stability of the reduced order linear hybrid system describing the slow dynamics. The order of magnitude of the second term is determined by that of the parameter defining the ratio between the two time-scales of the singularly perturbed system. We show that the proposed framework can also take into account the change of dimension of the state vector at switching instants. Numerical illustrations complete our study.