## Section: Research Program

### Hybrid Systems Modeling

Systems industries today make extensive use of mathematical modeling tools to design computer controlled physical systems. This class of tools addresses the modeling of physical systems with models that are simpler than usual scientific computing problems by using only Ordinary Differential Equations (ODE) and Difference Equations but not Partial Differential Equations (PDE). This family of tools first emerged in the 1980's with SystemBuild by MatrixX (now distributed by National Instruments) followed soon by Simulink by Mathworks, with an impressive subsequent development.

In the early 90's control scientists from the University of Lund (Sweden) realized that the above approach did not support component based modeling of physical systems with reuse (http://www.lccc.lth.se/media/LCCC2012/WorkshopSeptember/slides/Astrom.pdf). For instance, it was not easy to draw an electrical or hydraulic circuit by assembling component models of the various devices. The development of the Omola language by Hilding Elmqvist was a first attempt to bridge this gap by supporting some form of Differential Algebraic Equations (DAE) in the models. Modelica quickly emerged from this first attempt and became in the 2000's a major international concerted effort with the Modelica Consortium (https://www.modelica.org/). A wider set of tools, both industrial and academic, now exists in this segment (SimScape by Mathworks, Amesim by LMS International, now Siemens PLM, and more.). In the EDA sector, VHDL-AMS was developed as a standard [12] and also allows for differential algebraic equations. Several domain-specific languages and tools for mechanical systems or electronic circuits also support some restricted classes of differential algebraic equations. Spice is the historic and most striking instance of these domain-specific languages/tools (http://bwrcs.eecs.berkeley.edu/Classes/IcBook/SPICE/MANUALS/spice3.html). The main difference is that equations are hidden and the fixed structure of the differential algebraic results from the physical domain covered by these languages.

Despite these tools are now widely used by a number of engineers, they raise a number of technical difficulties. The meaning of some programs, their mathematical semantics, can be tainted with uncertainty. A main source of difficulty lies in the failure to properly handle the discrete and the continuous parts of systems, and their interaction. How the propagation of mode changes and resets should be handled? How to avoid artifacts due to the use of a global ODE solver causing unwanted coupling between seemingly non interacting subsystems? Also, the mixed use of an equational style for the continuous dynamics with an imperative style for the mode changes and resets is a source of difficulty when handling parallel composition. It is therefore not uncommon that tools return complex warnings for programs with many different suggested hints for fixing them. Yet, these “pathological” programs can still be executed, if wanted so, giving surprising results — See for instance the Simulink examples in [19], [15] and [16].

Indeed this area suffers from the same difficulties that led to the development of the theory of synchronous languages as an effort to fix obscure compilation schemes for discrete time equation based languages in the 1980's. Our vision is that hybrid systems modeling tools deserve similar efforts in theory as synchronous languages did for the programming of embedded systems.