Section: New Results

Hybrid Systems Modeling and Verification

Hybrid systems modeling languages that mix discrete and continuous time signals and systems are widely used to develop Cyber-Physical systems where control software interacts with physical devices. Compilers play a central role, statically checking source models, generating intermediate representations for testing and verification, and producing sequential code for simulation and execution on target platforms. In [5], Albert Benveniste, Timothy Bourke (PARKAS team Inria/ENS Paris), Benoît Caillaud, Jean-Louis Colaço, Cédric Pasteur (ANSYS/Esterel Technologies, Toulouse) and Marc Pouzet (PARKAS team Inria/ENS Paris) propose a comprehensive study of hybrid systems modeling languages (formal semantics, causality analysis, compiler design, ...). This paper advocates a novel approach to the design and implementation of these languages, built on synchronous language principles and their proven compilation techniques. The result is a hybrid systems modeling language in which synchronous programming constructs can be mixed with Ordinary Differential Equations (ODEs) and zero-crossing events, and a runtime that delegates their approximation to an off-the-shelf numerical solver. We propose an ideal semantics based on non standard analysis, which defines the execution of a hybrid model as an infinite sequence of infinitesimally small time steps. It is used to specify and prove correct three essential compilation steps: (1) a type system that guarantees that a continuous-time signal is never used where a discrete-time one is expected and conversely; (2) a type system that ensures the absence of combinatorial loops; (3) the generation of statically scheduled code for efficient execution. Our approach has been evaluated in two implementations: the academic language Zélus, which extends a language reminiscent of Lustre with ODEs and zero-crossing events, and the industrial prototype Scade Hybrid, a conservative extension of Scade 6.

The modeling and simulation of Cyber-Physical Systems (CPS) such as robots, vehicles, and power plants often require models with a time varying structure, due to failure situations or due to changes in physical conditions. These are called multi-mode models. In  [17], Albert Benveniste, Benoît Caillaud, Hilding Elmqvist (Mogram AB, Lund, Sweden), Khalil Ghorbal, Martin Otter (DLR-SR, Oberpfaffenhofen, Germany) and Marc Pouzet (PARKAS team, Inria/ENS Paris) are interested in multi-domain, component-oriented modeling as performed, for example, with the modeling language Modelica that leads naturally to Differential Algebraic Equations (DAEs). This paper is thus about multi-mode DAE systems. In particular, new methods are introduced to overcome one key problem that was only solved for specific subclasses of systems before: How to switch from one mode to another one when the number of equations may change and variables may exhibit impulsive behavior? An evaluation is performed both with the experimental modeling and simulation system Modia, a domain specific language extension of the programming language Julia, and with SunDAE, a novel structural analysis library for multi-mode DAE systems.

Vector Lyapunov functions are a multi-dimensional extension of the more familiar (scalar) Lyapunov functions, commonly used to prove stability properties in systems of non-linear ordinary differential equations (ODEs). In [7], Kahlil Ghorbal and Andrew Sogokon (CMU, Pittsburgh, USA) explore an analogous vector extension for so-called barrier certificates used in safety verification. As with vector Lyapunov functions, the approach hinges on constructing appropriate comparison systems, i.e., related differential equation systems from which properties of the original system may be inferred. The paper presents an accessible development of the approach, demonstrates that most previous notions of barrier certificate are special cases of comparison systems, and discusses the potential applications of vector barrier certificates in safety verification and invariant synthesis.