## Section: New Results

### Semantics, Static or Runtime Analysis of Hybrid Systems

#### Structural Analysis of Multi-Mode DAEs

Differential Algebraic Equation (DAE) systems constitute the
mathematical model supporting physical modeling languages such as
Modelica or Simscape. Unlike Ordinary Differential Equations, or
ODEs, they exhibit subtle issues because of their implicit
*latent equations* and related *differentiation index*.
Multi-mode DAE (mDAE) systems are much harder to deal with, not only
because of their mode-dependent dynamics, but essentially because of
the events and resets occurring at mode transitions. Unfortunately,
the large literature devoted to the numerical analysis of DAEs do
not cover the multi-mode case. It typically says nothing about mode
changes. This lack of foundations cause numerous difficulties to
the existing modeling tools. Some models are well handled, others
are not, with no clear boundary between the two classes.
In [11], we develop a comprehensive
mathematical approach to the *structural analysis* of mDAE
systems which properly extends the usual analysis of DAE systems. We
define a constructive semantics based on nonstandard analysis and
show how to produce execution schemes in a systematic way. This work
has been accepted for presentation at the HSCC 2017
conference [18] in April 2017.

#### Operational Models for Piecewise-Smooth Systems

In [7], we study ways of constructing meaningful operational models of piecewise-smooth systems (PWS). The systems we consider are described by polynomial vector fields defined on non-overlapping semi-algebraic sets, which form a partition of the state space. Our approach is to give meaning to motion in systems of this type by automatically synthesizing operational models in the form of hybrid automata (HA). Despite appearances, it is in practice often difficult to arrive at satisfactory HA models of PWS. The different ways of building operational models that we explore in our approach can be thought of as defining different semantics for the underlying PWS. These differences have a number of interesting nuances related to phenomena such as chattering, non-determinism, so-called mythical modes and sliding behaviour.

#### Accelerated Simulation of Hybrid Systems: Method combining static analysis and runtime execution analysis

Ayman Aljarbouh has defended his PhD [4] on September 13th 2017. His PhD has been partially funded by an ARED grant of the Brittany Regional Council. His doctoral work took place in the context of the Modrio (completed in 2016) and Sys2Soft (completed in 2015) projects on hybrid systems modeling. Ayman Aljarbouh has been working on accelerated simulation techniques for hybrid systems. In particular, he has contributed, and implemented in a software prototype, a regularisation method transforming automatically at runtime a chattering behaviour into a semantics preserving smooth behaviour. He has also contributed a method for the approximation of Zeno behaviour. This method enables to jump past an accumulation of an infinite number of zero-crossing events, and to continue the simulation of a large class of Zeno hybrid systems, after accumulation points.

#### A Type-based Analysis of Causality Loops in Hybrid Systems Modelers

Explicit hybrid systems modelers like Simulink/Stateflow allow for programming both discrete- and continuous-time behaviors with complex interactions between them. A key issue in their compilation is the static detection of algebraic or causality loops. Such loops can cause simulations to deadlock and prevent the generation of statically scheduled code. In [5], we addresses this issue for a hybrid modeling language that combines synchronous data-flow equations with Ordinary Differential Equations (ODEs). We introduce the operator last(x) for the left-limit of a signal x. This operator is used to break causality loops and permits a uniform treatment of discrete and continuous state variables. The semantics relies on non-standard analysis, defining an execution as a sequence of infinitesimally small steps. A signal is deemed causally correct when it can be computed sequentially and only changes infinitesimally outside of announced discrete events like zero-crossings. The causality analysis takes the form of a type system that expresses dependences between signals. In well-typed programs, signals are provably continuous during integration provided that imported external functions are also continuous. The effectiveness of this system is illustrated with several examples written in Zélus, a Lustre-like synchronous language extended with hierarchical automata and ODEs.