arXiv subjects

Eric Goubault

Publications and source records attributed to Eric Goubault.

At least 37 records · Page 2Linked to original sources

arXiv2021

On-The-Fly Control of Unknown Systems: From Side Information to Performance Guarantees through Reachability

We develop data-driven algorithms for reachability analysis and control of systems with a priori unknown nonlinear dynamics. The resulting algorithms not only are suitable for settings with real-time requirements but also provide provable performance guarantees. To this end, they merge noisy data from only a single finite-horizon trajectory and, if available, various forms of side information. Such side information may include knowledge of the regularity of the dynamics, algebraic constraints on the states, monotonicity, or decoupling in the dynamics between the states. Specifically, we develop two algorithms, $\texttt{DaTaReach}$ and $\texttt{DaTaControl}$, to over-approximate the reachable set and design control signals for the system on the fly. $\texttt{DaTaReach}$ constructs a differential inclusion that contains the unknown dynamics. Then, in a discrete-time setting, it over-approximates the reachable set through interval Taylor-based methods applied to systems with dynamics described as differential inclusions. We provide a bound on the time step size that ensures the correctness and termination of $\texttt{DaTaReach}$. $\texttt{DaTaControl}$ enables convex-optimization-based control using the computed over-approximation and the receding-horizon control framework. Besides, $\texttt{DaTaControl}$ achieves near-optimal control and is suitable for real-time control of such systems. We establish a bound on its suboptimality and the number of primitive operations it requires to compute control values. Then, we theoretically show that $\texttt{DaTaControl}$ achieves tighter suboptimality bounds with an increasing amount of data and richer side information. Finally, experiments on a unicycle, quadrotor, and aircraft systems demonstrate the efficacy of both algorithms over existing approaches.

eess.SY↗

arXiv2021

Solving N-player dynamic routing games with congestion: a mean field approach

The recent emergence of navigational tools has changed traffic patterns and has now enabled new types of congestion-aware routing control like dynamic road pricing. Using the fundamental diagram of traffic flows - applied in macroscopic and mesoscopic traffic modeling - the article introduces a new N-player dynamic routing game with explicit congestion dynamics. The model is well-posed and can reproduce heterogeneous departure times and congestion spill back phenomena. However, as Nash equilibrium computations are PPAD-complete, solving the game becomes intractable for large but realistic numbers of vehicles N. Therefore, the corresponding mean field game is also introduced. Experiments were performed on several classical benchmark networks of the traffic community: the Pigou, Braess, and Sioux Falls networks with heterogeneous origin, destination and departure time tuples. The Pigou and the Braess examples reveal that the mean field approximation is generally very accurate and computationally efficient as soon as the number of vehicles exceeds a few dozen. On the Sioux Falls network (76 links, 100 time steps), this approach enables learning traffic dynamics with more than 14,000 vehicles.

math.DS↗

arXiv2021

Static analysis of ReLU neural networks with tropical polyhedra

This paper studies the problem of range analysis for feedforward neural networks, which is a basic primitive for applications such as robustness of neural networks, compliance to specifications and reachability analysis of neural-network feedback systems. Our approach focuses on ReLU (rectified linear unit) feedforward neural nets that present specific difficulties: approaches that exploit derivatives do not apply in general, the number of patterns of neuron activations can be quite large even for small networks, and convex approximations are generally too coarse. In this paper, we employ set-based methods and abstract interpretation that have been very successful in coping with similar difficulties in classical program verification. We present an approach that abstracts ReLU feedforward neural networks using tropical polyhedra. We show that tropical polyhedra can efficiently abstract ReLU activation function, while being able to control the loss of precision due to linear computations. We show how the connection between ReLU networks and tropical rational functions can provide approaches for range analysis of ReLU neural networks.

cs.LG↗

arXiv2021

Neural Network Based Model Predictive Control for an Autonomous Vehicle

We study learning based controllers as a replacement for model predictive controllers (MPC) for the control of autonomous vehicles. We concentrate for the experiments on the simple yet representative bicycle model. We compare training by supervised learning and by reinforcement learning. We also discuss the neural net architectures so as to obtain small nets with the best performances. This work aims at producing controllers that can both be embedded on real-time platforms and amenable to verification by formal methods techniques.

cs.RO↗

arXiv2021

Reinforcement Learning with Formal Performance Metrics for Quadcopter Attitude Control under Non-nominal Contexts

We explore the reinforcement learning approach to designing controllers by extensively discussing the case of a quadcopter attitude controller. We provide all details allowing to reproduce our approach, starting with a model of the dynamics of a crazyflie 2.0 under various nominal and non-nominal conditions, including partial motor failures and wind gusts. We develop a robust form of a signal temporal logic to quantitatively evaluate the vehicle's behavior and measure the performance of controllers. The paper thoroughly describes the choices in training algorithms, neural net architecture, hyperparameters, observation space in view of the different performance metrics we have introduced. We discuss the robustness of the obtained controllers, both to partial loss of power for one rotor and to wind gusts and finish by drawing conclusions on practical controller design by reinforcement learning.

cs.AI↗

arXiv2021

On-The-Fly Control of Unknown Smooth Systems from Limited Data

We investigate the problem of data-driven, on-the-fly control of systems with unknown nonlinear dynamics where data from only a single finite-horizon trajectory and possibly side information on the dynamics are available. Such side information may include knowledge of the regularity of the dynamics, monotonicity of the states, or decoupling in the dynamics between the states. Specifically, we develop two algorithms, $\texttt{DaTaReach}$ and $\texttt{DaTaControl}$, to over-approximate the reachable set and design control signals for the system on the fly. $\texttt{DaTaReach}$ constructs a differential inclusion that contains the unknown vector field. Then, it computes an over-approximation of the reachable set based on interval Taylor-based methods applied to systems with dynamics described as differential inclusions. $\texttt{DaTaControl}$ enables convex-optimization-based, near-optimal control using the computed over-approximation and the receding-horizon control framework. We provide a bound on its suboptimality and show that more data and side information enable $\texttt{DaTaControl}$ to achieve tighter suboptimality bounds. Finally, we demonstrate the efficacy of $\texttt{DaTaControl}$ over existing approaches on the problems of controlling a unicycle and quadrotor systems.

eess.SY↗

arXiv2021

Tractable higher-order under-approximating AE extensions for non-linear systems

We consider the problem of under and over-approximating the image of general vector-valued functions over bounded sets, and apply the proposed solution to the estimation of reachable sets of uncertain non-linear discrete-time dynamical systems. Such a combination of under and over-approximations is very valuable for the verification of properties of embedded and cyber-physical controlled systems. Over-approximations prove properties correct, while under-approximations can be used for falsification. Coupled, they provide a measure of the conservatism of the analysis. This work introduces a general framework relying on computations of robust ranges of vector-valued functions. This framework allows us to extend for under-approximation many precision refinements that are classically used for over-approximations, such as affine approximations, Taylor models, quadrature formulae and preconditioning methods. We end by evaluating the efficiency and precision of our approach, focusing on the application to the analysis of discrete-time dynamical systems with inputs and disturbances, on different examples from the literature.

eess.SY↗

arXiv2020

Directed Homotopy in Non-Positively Curved Spaces

A semantics of concurrent programs can be given using precubical sets, in order to study (higher) commutations between the actions, thus encoding the "geometry" of the space of possible executions of the program. Here, we study the particular case of programs using only mutexes, which are the most widely used synchronization primitive. We show that in this case, the resulting programs have non-positive curvature, a notion that we introduce and study here for precubical sets, and can be thought of as an algebraic analogue of the well-known one for metric spaces. Using this it, as well as categorical rewriting techniques, we are then able to show that directed and non-directed homotopy coincide for directed paths in these precubical sets. Finally, we study the geometric realization of precubical sets in metric spaces, to show that our conditions on precubical sets actually coincide with those for metric spaces. Since the category of metric spaces is not cocomplete, we are lead to work with generalized metric spaces and study some of their properties.

cs.LO↗

arXiv2020

Knowledge and simplicial complexes

Simplicial complexes are a versatile and convenient paradigm on which to build all the tools and techniques of the logic of knowledge, on the assumption that initial epistemic models can be described in a distributed fashion. Thus, we can define: knowledge, belief, bisimulation, the group notions of mutual, distributed and common knowledge, and also dynamics in the shape of simplicial action models. We give a survey on how to interpret all such notions on simplicial complexes, building upon the foundations laid in prior work by Goubault and others.

cs.DC↗

arXiv2019

A dynamic epistemic logic analysis of the equality negation task

In this paper we study the solvability of the equality negation task in a simple wait-free model where processes communicate by reading and writing shared variables or exchanging messages. In this task, two processes start with a private input value in the set {0,1,2}, and after communicating, each one must decide a binary output value, so that the outputs of the processes are the same if and only if the input values of the processes are different. This task is already known to be unsolvable; our goal here is to prove this result using the dynamic epistemic logic (DEL) approach introduced by Goubault, Ledent and Rajsbaum in GandALF 2018. We show that in fact, there is no epistemic logic formula that explains why the task is unsolvable. We fix this issue by extending the language of our DEL framework, which allows us to construct such a formula, and discuss its utility.

cs.LO↗

arXiv2018

Directed topological complexity

It has been observed that the very important motion planning problem of robotics mathematically speaking boils down to the problem of finding a section to the path-space fibration, raising the notion of topological complexity, as introduced by M. Farber. The above notion fits the motion planning problem of robotics when there are no constraints on the actual control that can be applied to the physical apparatus. In many applications, however, a physical apparatus may have constrained controls, leading to constraints on its potential future dynamics. In this paper we adapt the notion of topological complexity to the case of directed topological spaces, which encompass such controlled systems, and also systems which appear in concurrency theory. We study its first properties, make calculations for some interesting classes of spaces, and show applications to a form of directed homotopy equivalence.

math.AT↗

arXiv2018

Time-reversal homotopical properties of concurrent systems

Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which respect this directedness have been introduced to classify directed spaces. In this work we study the properties of such invariants with respect to the reversal of the flow of time in directed spaces. Known invariants, natural homotopy and homology, have been shown to be unchanged under this time-reversal. We show that these can be equipped with additional algebraic structure witnessing this reversal. Specifically, when applied to a directed space and to its reversal, we show that these refined invariants yield dual objects. We further refine natural homotopy by introducing a notion of relative directed homotopy and showing the existence of a long exact sequence of natural homotopy systems.

math.CT↗

arXiv2017

On directed homotopy equivalences and a notion of directed topological complexity

This short note introduces a notion of directed homotopy equivalence and of "directed" topological complexity (which elaborates on the notion that can be found in e.g. Farber's book) which have a number of desirable joint properties. In particular, being dihomotopically equivalent implies having bisimilar natural homologies (defined in Dubut et al. 2015). Also, under mild conditions, directed topological complexity is an invariant of our directed homotopy equivalence and having a directed topological complexity equal to one is (under these conditions) equivalent to being dihomotopy equivalent to a point (i.e., to being "dicontractible", as in the undirected case). It still remains to compare this notion with the notion introduced in Dubut et al. 2016, which has lots of good properties as well. For now, it seems that for reasonable spaces, this new proposal of directed homotopy equivalence identifies more spaces than the one of Dubut et al. 2016.

math.AT↗

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.

eess.SY↗

arXiv2017

Models of fault-tolerant distributed computation via dynamic epistemic logic

The computability power of a distributed computing model is determined by the communication media available to the processes, the timing assumptions about processes and communication, and the nature of failures that processes can suffer. In a companion paper we showed how dynamic epistemic logic can be used to give a formal semantics to a given distributed computing model, to capture precisely the knowledge needed to solve a distributed task, such as consensus. Furthermore, by moving to a dual model of epistemic logic defined by simplicial complexes, topological invariants are exposed, which determine task solvability. In this paper we show how to extend the setting above to include in the knowledge of the processes, knowledge about the model of computation itself. The extension describes the knowledge processes gain about the current execution, in problems where processes have no input values at all.

cs.DC↗

arXiv2017

A simplicial complex model of dynamic epistemic logic for fault-tolerant distributed computing

The usual epistemic S5 model for multi-agent systems is a Kripke graph, whose edges are labeled with the agents that do not distinguish between two states. We propose to uncover the higher dimensional information implicit in the Kripke graph, by using as a model its dual, a chromatic simplicial complex. For each state of the Kripke model there is a facet in the complex, with one vertex per agent. If an edge (u,v) is labeled with a set of agents S, the facets corresponding to u and v intersect in a simplex consisting of one vertex for each agent of S. Then we use dynamic epistemic logic to study how the simplicial complex epistemic model changes after the agents communicate with each other. We show that there are topological invariants preserved from the initial epistemic complex to the epistemic complex after an action model is applied, that depend on how reliable the communication is. In turn these topological properties determine the knowledge that the agents may gain after the communication happens.

cs.DC↗

arXiv2013

Robustness analysis of finite precision implementations

A desirable property of control systems is to be robust to inputs, that is small perturbations of the inputs of a system will cause only small perturbations on its outputs. But it is not clear whether this property is maintained at the implementation level, when two close inputs can lead to very different execution paths. The problem becomes particularly crucial when considering finite precision implementations, where any elementary computation can be affected by a small error. In this context, almost every test is potentially unstable, that is, for a given input, the computed (finite precision) path may differ from the ideal (same computation in real numbers) path. Still, state-of-the-art error analyses do not consider this possibility and rely on the stable test hypothesis, that control flows are identical. If there is a discontinuity between the treatments in the two branches, that is the conditional block is not robust to uncertainties, the error bounds can be unsound. We propose here a new abstract-interpretation based error analysis of finite precision implementations, which is sound in presence of unstable tests. It automatically bounds the discontinuity error coming from the difference between the float and real values when there is a path divergence, and introduces a new error term labeled by the test that introduced this potential discontinuity. This gives a tractable error analysis, implemented in our static analyzer FLUCTUAT: we present results on representative extracts of control programs.

cs.SE↗

arXiv2013

Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs

The problem of computing the smallest fixed point of an order-preserving map arises in the study of zero-sum positive stochastic games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount rate may be negative. We characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential. We apply this characterization to design a policy iteration algorithm, which applies to the case of finite state and action spaces. The algorithm returns a locally minimal fixed point, which turns out to be globally minimal when the discount rate is nonnegative.

math.OC↗