Antoine Girard

SpaceEx: scalable verification of hybrid systems

We present a scalable reachability algorithm for hybrid systems with piecewise affine, non-deterministic dynamics. It combines polyhedra and support function representations of continuous sets to compute an over-approximation of the reachable states. The algorithm improves over previous work by using variable time steps to guarantee a given local error bound. In addition, we propose an improved approximation model, which drastically improves the accuracy of the algorithm. The algorithm is implemented as part of SpaceEx, a new verification platform for hybrid systems, available at spaceex.imag.fr. Experimental results of full fixed-point computations with hybrid systems with more than 100 variables illustrate the scalability of the approach.

Approximation Metrics for Discrete and Continuous Systems

Established system relationships for discrete systems, such as language inclusion, simulation, and bisimulation, require system observations to be identical. When interacting with the physical world, modeled by continuous or hybrid systems, exact relationships are restrictive and not robust. In this paper, we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion, approximate simulation, and approximate bisimulation relations. We define a hierarchy of approximation pseudo-metrics between two systems that quantify the quality of the approximation, and capture the established exact relationships as zero sections. Our approximation framework is compositional for a synchronous composition operator. Algorithms are developed for computing the proposed pseudo-metrics, both exactly and approximately. The exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations, or dually, the solution of a static game whose cost is the so-called branching distance between the systems. Approximations for the pseudo-metrics can be obtained by considering Lyapunov-like functions called simulation and bisimulation functions. We illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

Reachability of uncertain linear systems using zonotopes

We present a method for the computation of reachable sets of uncertain linear systems. The main innovation of the method consists in the use of zonotopes for reachable set representation. Zonotopes are special polytopes with several interesting properties : they can be encoded efficiently, they are closed under linear transformations and Minkowski sum. The resulting method has been used to treat several examples and has shown great performances for high dimensional systems. An extension of the method for the verification of piecewise linear hybrid systems is proposed.

Dynamic Triggering Mechanisms for Event-Triggered Control

In this technical note, 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.

Hybridization methods for the analysis of nonlinear systems

In this article, we describe some recent results on the hybridization methods for the analysis of nonlinear systems. The main idea of our hybridization approach is to apply the hybrid systems methodology as a systematic approximation method. More concretely, we partition the state space of a complex system into regions that only intersect on their boundaries, and then approximate its dynamics in each region by a simpler one. Then, the resulting hybrid system, which we call a hybridization, is used to yield approximate analysis results for the original system. We also prove important properties of the hybridization, and propose two effective hybridization construction methods, which allow approximating the original nonlinear system with a good convergence rate.

Reachability analysis of nonlinear systems using conservative approximation

In this paper we present an approach to approximate reachability computation for nonlinear continuous systems. Rather than studying a complex nonlinear system x = g(x), we study an approximating system x = f(x) which is easier to handle. The class of approximating systems we consider in this paper is piecewise linear, obtained by interpolating g over a mesh. In order to be conservative, we add a bounded input in the approximating system to account for the interpolation error. We thus develop a reachability method for systems with input, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets. This method is then extended to the approximate piecewise linear systems arising in our construction. The final result is a reachability algorithm for nonlinear continuous systems which allows to compute conservative approximations with as great degree of accuracy as desired, and more importantly, it has good convergence rate. If g is a C2 function, our method is of order 2. Furthermore, the method can be straightforwardly extended to hybrid systems.

Reachability Analysis of Hybrid Systems Using Support Functions

This paper deals with conservative reachability analysis of a class of hybrid systems with continuous dynamics described by linear differential inclusions, convex invariants and guards, and linear reset maps. We present an approach for computing over-approximations of the set of reachable states. It is based on the notion of support function and thus it allows us to consider invariants, guards and constraints on continuous inputs and initial states defined by arbitrary compact convex sets. We show how the properties of support functions make it possible to derive an effective algorithm for approximate reachability analysis of hybrid systems. We use our approach on some examples including the navigation benchmark for hybrid systems verification.

Brief paper: Hierarchical control system design using approximate simulation

In this paper, we present a new approach for hierarchical control based on the recent notions of approximate simulation and simulation functions, a quantitative version of the simulation relations. Given a complex system that needs to be controlled and a simpler abstraction, we show how the knowledge of a simulation function allows us to synthesize hierarchical control laws by first controlling the abstraction and then lifting the abstract control law to the complex system using an interface. For the class of linear control systems, we give an effective characterization of the simulation functions and of the associated interfaces. This characterization allows us to use algorithmic procedures for their computation. We show how to choose an abstraction for a linear control system such that our hierarchical control approach can be used. Finally, we show the effectiveness of our approach on an example.

Approximate simulation relations for hybrid systems

Abstract Approximate simulation relations have recently been introduced as a powerful tool for the approximation of discrete and continuous systems. In this paper, we extend this notion to hybrid systems. Using the so-called simulation functions, we develop a computationally effective characterization of approximate simulation relations which can be used for hybrid systems approximation. An example of application in the context of safety verification is shown.

Recent progress in continuous and hybrid reachability analysis

Set-based reachability analysis computes all possible states a system may attain, and in this sense provides knowledge about the system with a completeness, or coverage, that a finite number of simulation runs can not deliver. Due to its inherent complexity, the application of reachability analysis has been limited so far to simple systems, both in the continuous and the hybrid domain. In this paper we present recent advances that, in combination, significantly improve this applicability, and allow us to find better balance between computational cost and accuracy. The presentation covers, in a unified manner, a variety of methods handling increasingly complex types of continuous dynamics (constant derivative, linear, nonlinear). The improvements include new geometrical objects for representing sets, new approximation schemes, and more flexible combinations of graph-search algorithm and partition refinement. We report briefly some preliminary experiments that have enabled the analysis of systems previously beyond reach.