## Section: New Results

### Structural analysis of multimode DAE systems

Participants : Albert Benveniste, Benoît Caillaud, Khalil Ghorbal, Mathias Malandain.

The Hycomes team has obtained two results related to the structural analysis of multimode DAE systems.

#### Impulsive behavior of multimode DAE systems

A major difficulty with multimode DAE systems are the commutations from one mode to another one when the number of equations may change and variables may exhibit impulsive behavior, meaning that not only the trajectory of the system may be discontinuous, but moreover, some variables may be Dirac measures at the instant of mode changes. In [7] , we compare two radically different approaches to the structural analysis problem of mode changes. The first one is a classical approach, for a restricted class of DAE systems, for which the existence and uniqueness of an impulsive state jump is proved. The second approach is based on nonstandard analysis and is proved to generalize the former approach, to a larger class of multimode DAE systems. The most interesting feature of the latter approach is that it defines the state-jump as the standardization of the solution of a system of system of difference equations, in the framework of nonstandard analysis.

#### An implicit structural analysis method for multimode DAE systems

Modeling languages and tools based on Differential Algebraic Equations (DAE) bring several specific issues that do not exist with modeling languages based on Ordinary Differential Equations. The main problem is the determination of the differentiation index and latent equations. Prior to generating simulation code and calling solvers, the compilation of a model requires a structural analysis step, which reduces the differentiation index to a level acceptable by numerical solvers.

The Modelica language, among others, allows hybrid models with multiple modes, mode-dependent dynamics and state-dependent mode switching. These Multimode DAE (mDAE) systems are much harder to deal with. The main difficulties are (i) the combinatorial explosion of the number of modes, and (ii) the correct handling of mode switchings.

The focus of the paper [31] is on the first issue, namely: How can one perform a structural analysis of an mDAE in all possible modes, without enumerating these modes? A structural analysis algorithm for mDAE systems is presented, based on an implicit representation of the varying structure of an mDAE. It generalizes J. Pryce's $\Sigma $-method [56] to the multimode case and uses Binary Decision Diagrams (BDD) to represent the mode-dependent structure of an mDAE. The algorithm determines, as a function of the mode, the set of latent equations, the leading variables and the state vector. This is then used to compute a mode-dependent block-triangular decomposition of the system, that can be used to generate simulation code with a mode-dependent scheduling of the blocks of equations.

This method has been implemented in the IsamDAE software. This has allowed the Hycomes team to evaluate the performance and scalability of the method on several examples. In particular, it has been possible to perform the structural analysis of systems with more than 750 equations and ${10}^{23}$ modes.