2023Activity reportProject-TeamHYCOMES

3.1 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 1. 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. A wider set of tools, both industrial and academic, now exists in this segment 2. In the Electronic Design Automation (EDA) sector, VHDL-AMS was developed as a standard  71 and also enables the use of 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 3. 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 the fact that 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, is indeed ambiguous. A main source of difficulty is the correct simulation of continuous-time dynamics, interacting with discrete-time dynamics: How the propagation of mode switchings 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  28, 20 and  21.

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.

3.2 Background on non-standard analysis

Non-Standard analysis plays a central role in our research on hybrid systems modeling  20, 28, 22, 21, 26, 3. The following text provides a brief summary of this theory and gives some hints on its usefulness in the context of hybrid systems modeling. This presentation is based on our paper 2, a chapter of Simon Bliudze's PhD thesis  35, and a recent presentation of non-standard analysis, not axiomatic in style, due to the mathematician Lindström  80.

Non-standard numbers allowed us to reconsider the semantics of hybrid systems and propose a radical alternative to the super-dense time semantics developed by Edward Lee and his team as part of the Ptolemy II project, where cascades of successive instants can occur in zero time by using ${ℝ}_{+}\times \mathbb{N}$ as a time index. In the non-standard semantics, the time index is defined as a set $𝕋=\{n\partial \mid n\in {}^{*}\mathbb{N}\}$, where $\partial$ is an infinitesimal and ${}^{*}\mathbb{N}$ is the set of non-standard integers. Remark that (1) $𝕋$ is dense in ${ℝ}_{+}$, making it “continuous”, and (2) every $t\in 𝕋$ has a predecessor in $𝕋$ and a successor in $𝕋$, making it “discrete”. Although it is not effective from a computability point of view, the non-standard semantics provides a framework that is familiar to the computer scientist and at the same time efficient as a symbolic abstraction. This makes it an excellent candidate for the development of provably correct compilation schemes and type systems for hybrid systems modeling languages.

Non-standard analysis was proposed by Abraham Robinson in the 1960s to allow the explicit manipulation of “infinitesimals” in analysis  92, 58, 54. Robinson's approach is axiomatic; he proposes adding three new axioms to the basic Zermelo-Fraenkel (ZFC) framework. While the need for non-standard analysis (in addition to the usual or standard analysis) has long agitated the mathematical community, it is not our purpose to debate such aspects. The important thing for us is that non-standard analysis allows the use of the non-standard discretization of continuous dynamics “as if” it was operational.

Not surprisingly, such an idea is not novel. Iwasaki et al.  73 first proposed using non-standard analysis to discuss the nature of time in hybrid systems. Bliudze and Krob  34, 35 have also used non-standard analysis as a mathematical support for defining a system theory for hybrid systems. They discuss in detail the notion of “system” and investigate computability issues. The formalization they propose closely follows that of Turing machines, with a memory tape and a control mechanism.

3.3 Structural Analysis of DAE Systems

The Modelica language is based on Differential Algebraic Equations (DAE). The general form of a DAE is given by:

where $F$ is a system of $n_e$ equations $\{f_1,\cdots ,f_{n_e}\}$ and $x$ is a finite list of $n_v$ independent real-valued, smooth enough, functions $\{x_1,\cdots ,x_{n_v}\}$ of the independent variable $t$. We use $x^{\text{'}}$ as a shorthand for the list of first-order time derivatives of $x_j$, $j=1,\cdots ,n_v$. High-order derivatives are recursively defined as usual, and $x^{\left(k\right)}$ denotes the list formed by the $k$-th derivatives of the functions $x_j$. Each $f_i$ depends on the scalar $t$ and some of the functions $x_j$ as well as a finite number of their derivatives.

Let ${\sigma }_{i,j}$ denote the highest differentiation order of variable $x_j$ effectively appearing in equation $f_i$, or $-\infty$ if $x_j$ does not appear in $f_i$. The leading variables of $F$ are the variables in the set

The state variables of $F$ are the variables in the set

A leading variable $x_j^{\left({\sigma }_j\right)}$ is said to be algebraic if ${\sigma }_j=0$ (in which case, neither $x_j$ nor any of its derivatives are state variables). In the sequel, $v$ and $u$ denote the leading and state variables of $F$, respectively.

DAE are a strict generalization of ordinary differential equations (ODE), in the sense that it may not be immediate to rewrite a DAE as an explicit ODE of the form $v=G\left(u\right)$. The reason is that this transformation relies on the Implicit Function Theorem, requiring that the Jacobian matrix $\frac{\partial F}{\partial v}$ to be full rank. This is, in general, not the case for a DAE. Simple examples, like the two-dimensional fixed-length pendulum in Cartesian coordinates  88, exhibit this behaviour.

For a square DAE of dimension $n$ (i.e., we now assume $n_e=n_v=n$) to be solved in the neighborhood of some $(v^{*},u^{*})$, one needs to find a set of non-negative integers $C=\{c_1,\cdots ,c_n\}$ such that system

can locally be made explicit, i.e., the Jacobian matrix of $F^{\left(C\right)}$ with respect to its leading variables, evaluated at $(v^{*},u^{*})$, is nonsingular. The smallest possible value of ${max}_i{c}_i$ for a set $C$ that satisfies this property is the differentiation index  46 of $F$, that is, the minimal number of time differentiations of all or part of the equations $f_i$ required to get an ODE.

In practice, the problem of automatically finding a minimal solution $C$ to this problem quickly becomes intractable. Moreover, the differentiation index may depend on the value of $(v^{*},u^{*})$. This is why, in lieu of numerical nonsingularity, one is interested in the structural nonsingularity of the Jacobian matrix, i.e., its almost certain nonsingularity when its nonzero entries vary over some neighborhood. In this framework, the structural analysis (SA) of a DAE returns, when successful, values of the $c_i$ that are independent from a given value of $(v^{*},u^{*})$.

A renowned method for the SA of DAE is the Pantelides method; however, Pryce's $\Sigma$-method is introduced also in what follows, as it is a crucial tool for our works.

3.4 Contract-Based Design, Interfaces Theories, and Requirements Engineering

System companies such as automotive and aeronautic companies are facing significant difficulties due to the exponentially raising complexity of their products coupled with increasingly tight demands on functionality, correctness, and time-to-market. The cost of being late to market or of imperfections in the products is staggering as witnessed by the recent recalls and delivery delays that many major car and airplane manufacturers had to bear in the recent years. The root causes of these design problems are complex and relate to a number of issues ranging from design processes and relationships with different departments of the same company and with suppliers, to incomplete requirement specification and testing.

We believe the most promising means to address the challenges in systems engineering is to employ formal design methodologies that seamlessly and coherently combine the various viewpoints of the design space (behavior, time, energy, reliability, ...), that provide the appropriate abstractions to manage the inherent complexity, and that can provide correct-by-construction implementations. The following issues must be addressed when developing new approaches to the design of complex systems:

• The overall design flows for heterogeneous systems and the associated use of models across traditional boundaries are not well developed and understood. Relationships between different teams inside a same company, or between different stake-holders in the supplier chain, are not supported by precise mathematical specifications of the components each party is expected to deliver.
• System requirements capture and analysis is in large part a heuristic process, where informal text and natural language-based techniques in use today are facing significant challenges  76. Formal requirements engineering is in its infancy: mathematical models, formal analysis techniques and links to system implementation must be developed.
• Dealing with variability, uncertainty, and life-cycle issues, such as extensibility of a product family, are not well-addressed using available systems engineering methodologies and tools.

The challenge is to address the entire process and not to consider only local solutions of methodology, tools, and models that ease part of the design.

Contract-based design has been proposed as a new approach to the system design problem that is rigorous and effective in dealing with the problems and challenges described before, and that, at the same time, does not require a radical change in the way industrial designers carry out their task as it cuts across design flows of different types. Indeed, contracts can be used almost everywhere and at nearly all stages of system design, from early requirements capture, to embedded computing infrastructure and detailed design involving circuits and other hardware. Intuitively, a contract captures two properties, respectively representing the assumptions on the environment and the guarantees of the system under these assumptions. Hence, a contract can be defined as a pair $C=(A,G)$ of assumptions and guarantees characterizing in a formal way 1) under which context the design is assumed to operate, and 2) what its obligations are. Assume/Guarantee reasoning has been known for a long time, and has been used mostly in software engineering  86. However, contract-based design is not limited to types and values in a piece of software. It can also be used to capture its performances (time, memory consumption, energy) and reliability. This amounts to enrich a component's interface with, on one hand, formal specifications of the behavior of the environment in which the component may be instantiated and, on the other hand, of the expected behavior of the component itself. To leverage contract-based reasoning as a technique of choice for system engineers, we aim to develop:

• mathematical foundations of contracts, that enable the design of formal verification frameworks;
• System engineering methodologies and tools, that focus on requirements modeling, contract specification and verification, at multiple abstraction levels.

A detailed bibliography on contract and interface theories for embedded system design can be found in 5. In a nutshell, contract and interface theories fall into two main categories:

• Assume/guarantee contracts.
  By explicitly relying on the notions of assumptions and guarantees, A/G-contracts are intuitive. This makes them appealing for the engineer. In A/G-contracts, assumptions and guarantees are just properties regarding the behavior of a component and of its environment. The typical case is when these properties are formal languages or sets of traces. This includes the class of safety properties  77, 50, 84, 18, 52. Contract theories were initially developed as specification formalisms able to refuse some inputs from the environment  59. A/G-contracts were advocated in  24 and are is still a very active research topic, with several contributions dealing with the timed  32 and probabilistic  42, 43 viewpoints in system design, and even hybrid systems design  87.
• Automata theoretic interfaces.
  Interfaces combine assumptions and guarantees in a single, automata theoretic specification. Most interface theories are based on Lynch's Input/Output Automata  83, 82. Interface Automata  13, 12, 14, 48 focus primarily on parallel composition and compatibility: two interfaces are compatible if there exists at least one environment where they can work together. The idea is that the resulting composition exposes as an interface the needed information to ensure that incompatible pairs of states cannot be reached. This can be achieved by using the possibility, for an Interface Automaton, to refuse some inputs from the environment in a given state. This amounts to the implicit assumption that the environment will never produce any of the refused inputs, when the interface is in this state. Modal Interfaces  90 inherit from both Interface Automata and the originally unrelated notion of Modal Transition System  79, 16, 36, 78. Modal Interfaces are strictly more expressive than Interface Automata by decoupling the I/O orientation of an event and its deontic modalities (mandatory, allowed or forbidden). Informally, a must transition is offered in every component that realizes the modal interface, while a may transition is optional. Research on interface theories is still very active. For instance, timed  15, 29, 31, 56, 55, 30, probabilistic  42, 57 and energy-aware  49 interface theories have been proposed recently.

Requirements Engineering is one of the major concerns in large systems industries today, particularly so in sectors where certification prevails  93. Most requirements engineering tools offer a poor structuring of the requirements and cannot be considered as formal modeling frameworks today. They are nothing less, but nothing more than an informal structured documentation enriched with hyperlinks.

We see Contract-Based Design and Interfaces Theories as innovative tools in support of Requirements Engineering. The Software Engineering community has extensively covered several aspects of Requirements Engineering, in particular:

• the development and use of large and rich ontologies; and
• the use of Model Driven Engineering technology for the structural aspects of requirements and resulting hyperlinks (to tests, documentation, PLM, architecture, and so on).

Behavioral models and properties, however, are not properly encompassed by the above approaches. This is the cause of a remaining gap between this phase of systems design and later phases where formal model based methods involving behavior have become prevalent. We believe that our work on contract-based design and interface theories is best suited to bridge this gap.

3.5 Efficient Symbolic Computation for Sparse Systems

This project consists in exploiting the parsimony of sparse systems to accelerate their symbolic manipulation (quantifiers elimination  53, differential-algebraic reductions  94 etc.). Let us cite two typical examples as a motivation: Boolean functions ($a\vee b\wedge \neg c$) and polynomial systems with inequalities ($x^2+y\le 1\wedge x+y=0$). We seek precisely to decompose these systems, automatically, in order to be able to manipulate them at an advantageous computational cost (in time and in memory) by attacking the pieces thus obtained rather than considering the system as a single monolithic block.

The current algorithms suffer from a theoretical complexity that is at best exponential (in the size of the input) limiting their use to instances of very modest size. The classic approach to overcome this problem is to develop/use numerical methods (with their limits and intrinsic problems) when possible of course. We aim to explore a different avenue.

In this project, we wish to exploit the structure of sparse systems to push the symbolic approach beyond its theoretical limits. The a priori limited application of our methods for dense systems is compensated by the fact that in practice, the problems are very often structured (in this regard, let us content ourselves with quoting the SAT solvers which successfully tackle industrial instances of a theoretically NP-complete problem).

The idea of exploiting the structure to speed up calculations that are a priori complex is not new. It has notably been developed and successfully used in signal processing via Factor Graphs  81, where one restricts oneself to local propagation of information, guided by an abstract graph which represents the structure of the system overall. Our approach is similar: we basically seek to use expensive algorithms sparingly on only subsystems involving only a small number of variables, thus hoping to reduce the theoretical worst case. One could then legitimately wonder why it is not enough to apply what has already been done on Factor Graphs? The difficulty (and the novelty for that matter) lies in the implementation of this idea for the problems that interest us. Let's start by emphasizing that the propagation of information has a significantly different impact depending on the operator (or quantifier) to be eliminated: a minimization or a summation do not look like a projection at all! This will obviously not prevent us from importing good ideas applicable to our problems and vice versa.

More related to symbolic computation, to our knowledge, at least two recent attempts exist: chordal networks  51 which propose a representation of the ideals of the ring of polynomials (therefore algebraic sets), and triangular block shapes  97, initiated independently and under development in our team and which tackle Boolean functions, or, if you will, the algebraic sets over the field of Booleans. The similarity between the two approaches is striking and suggests that there is a common way of doing things that could be exploited beyond these two examples. It is this unification that interests us in the first place in this project.

We identify three research problems to explore: T1. Unify several optimization problems on graphs as a single problem parameterized by a cost function, we coin such a problem WAP, for weighted adjacency propagation. T2. Adapt (and possibly improve) the algorithm of  96 to WAP and consequently to all instances of the single problem detailed in T1. T3. Propose a unified and modular method consisting of: (1) an elimination algorithm, (2) a data structure and (3) an efficient algorithm to solve the problem (with an adequate cost function).

The work on chordal networks and our work on Boolean functions immediately become special cases. For example, for Boolean functions, one could use Binary Decision Diagrams (BDDs)  41 to represent each piece of the initial system. In fact, the final representation will no longer be a single monolithic BDD as is currently the case, but rather a graph of BDDs. In the same way, an algebraic set will be represented by a graph where each node is a Gröbner basis (or any other data structure used to represent systems of equations).

The structure of the system becomes thus apparent and is exploited to optimize the used representation, opening the way to a better understanding and therefore to a more efficient and better targeted manipulation. Let's remember a simple fact here: symbolic manipulation often solves the problem exactly (without approximation or compromise). Therefore, pushing the limits of applicability of these techniques to scale them can only be appreciated and will undoubtedly have a significant impact on all the areas where they apply and the list is as long as it is varied. (compilation, certification, validation, synthesis, etc.).

4.1 Modelica

Mathematical modeling tools are a considerable business, with major actors such as MathWorks, with Matlab/Simulink, or Wolfram, with Mathematica. However, none of these prominent tools are suitable for the engineering of large systems. The Modelica language has been designed with this objective in mind, making the best of the advantages of DAEs to support a component-based approach. Several industries in the energy sector have adopted Modelica as their main systems engineering language.

Although multimode features have been introduced in version 3.3 of the language  62, proper tool support of multimode models is still lagging behind. The reason is not a lack of interest from tool vendors and academia, but rather that multimode DAE systems poses several fundamental difficulties, such as a proper definition of a concept of solutions for multimode DAEs, how to handle mode switchings that trigger a change of system structure, or how impulsive variables should be handled. Our work on multimode DAEs focuses on these crucial issues  27.

Thanks to our IsamDAE software  45, 44, a larger class of Modelica models are expected to be compiled and simulated correctly. This should enable industrial users to have cleaner and simpler multimode Modelica models, with dynamically changing structure of cyberphysical systems. On the longer term, our ambition is to provide efficient code-generation techniques for the Modelica language, supporting, in full generality, multimode DAE systems, with dynamically changing differentiation index, structure and dimension.

The Hycomes team also focuses on scalability problems related to the compilation and simulation of large Modelica models. Digital twins developed by industrial Modelica users in the energy sector tend to be extremely large models, with up to ${10}^6$ equations. State-of-the-art Modelica compilers can not handle such models and users are forced to partition their model into smaller parts and use complex co-simulation techniques to produce executable digital twins. This puts a heavy burden on digital twin developers, since both the partitioning and the implementation of cosimulation methods are manual, finely tailored to the model, and require a high degree of expertise.

The Hycomes team is working on a new generation of algorithms for the compilation of the Modelica language, that can scale up to large models. The key contributations are modular index-reduction 9 and block-triangular equation sorting algorithms, that can be applied to incomplete (rectangular) DAE systems.

4.2 Dynamical Systems Verification

In addition to well-defined operational semantics for hybrid systems, one often needs to provide formal guarantees about the behavior of some critical components of the system, or at least its main underlying logic. To do so, we are actively developing new techniques to automatically verify whether a hybrid system complies with its specifications, and/or to infer automatically the envelope within which the system behaves safely. The approaches we developed have been already successfully used to formally verify the intricate logic of the ACAS X, a mid-air collision avoidance system that advises the pilot to go upward or downward to avoid a nearby airplane which requires mixing the continuous motion of the aircraft with the discrete decisions to resolve the potential conflict  74. This challenging example is nothing but an instance of the kind of systems we are targeting: autonomous smart systems that are designed to perform sophisticated tasks with an internal tricky logic. What is even more interesting perhaps is that such techniques can be often "reverted" to actually synthesize missing components so that some property holds, effectively helping the design of such complex systems.

7.1 New software

8.1 A Modular Structural Analysis of DAE Systems

Participants: Albert Benveniste, Benoît Caillaud, Mathias Malandain, Joan Thibault.

System modeling tools are key to the engineering of safe and efficient Cyber-Physical Systems (CPS). Although ODE-based languages and tools, such as Simulink  85, are widely used in industry, there are two main reasons why DAE-based modeling is best suited to the modeling of such systems: it enables a modeling based on first principles of the physics; it is physics-agnostic, and consequently accomodates arbitrary combinations of physics (mechanics, electrokinetics, hydraulics, thermodynamics, chemical reactions, etc.).

The pioneering work by Hilding Elmqvist  61 led to the emergence of the Modelica community in the 1990s, and the DAE-based modeling language of the same name  17 has become a de facto standard, with its object-oriented nature enabling a component-based modeling style. Its combined use with the port-Hamiltonian paradigm  91 results in a methodology that is instrumental to the scalable modeling of large systems, additionally ensuring that the model architecture preserves the system architecture, in stark contrast to ODE-based modeling  23, 25.

Consequently, DAE-based modeling requires that Modelica tools properly scale up to very large models. However, although Modelica enables the modeling of extremely large systems, its implementations  95, 66 are often not capable of compiling and simulating such large models. Scaling has been and still is a subject for sustained effort by the Modelica community  47, and although HPC issues belong to the landscape  37, a more specific issue is of uttermost importance for the Modelica language.

In the first steps of the compilation of a Modelica model, its hierarchical structure is flattened, thanks to a recursive syntactic inlining of the objects composing it. See  17, Section 5.6 for a complete definition of this flattening process. The result is an unstructured DAE that can be exponentially larger than the source model. The structural analyses that are required for the generation of simulation code (namely, the index reduction of the DAE system, followed by a block-triangular form transformation of the reduced-index system) are then performed on this monolithic DAE model. As the compilation process does not fully take advantage of the hierarchical nature of the models it has to handle, the modeling capabilities offered by the Modelica language are undermined by performance issues on the structural analysis itself  70, 69. Additionally, model flattening poses a challenge when attempting to extend DAE-based modeling to higher-order modeling or dynamically changing systems  38, 40, 39.

In 9, a new modular structural analysis algorithm is proposed that takes full advantage of the object tree structure of a DAE model. The bedrock of this method is a novel concept of structural analysis-aware interface for components. The essence of a component interface is to capture the necessary information about a Modelica class that needs to be exposed, in order to perform the structural analysis of a component comprizing instances of the former class, while hiding away useless information regarding the equations and all protected features it may contain.

In order to compute a component interface, one has to be able to perform the structural analysis of the possibly non-square DAE system that this component encapsulates, and to use the interfaces of the components it aggregates in this analysis. We base our algorithm on Pryce's $\Sigma$-method for index reduction  89, which essentially consists in the successive solving of two dual linear integer programs. The striking difference with Pryce's algorithm is that these problems are solved by parts, in a scalable manner.

Putting all of this together, it is then possible to perform a modular structural analysis, in which structural analysis is performed at the class level, and the results can then be instantiated for each component of the system model, knowing its context. Hence, structural information at the system level is derived from composing the result of component-level analysis. Modular structural analysis yields huge gains in terms of memory usage and computational costs, as the analysis of a single large-scale DAE is replaced with that of multiple smaller subsystems. Moreover, the analysis is performed at the class level, meaning that a single structural analysis is needed for all system components that are instances of the same class.

8.2 Fault Diagnosability Analysis of Multi-Mode Systems

Participants: Benoît Caillaud, Mathias Malandain.

A new collaboration between the Hycomes and the University of Linköping (Sweden) has started this year on the topic of system diagnosis, based on multimode DAE systems.

Fault detection and diagnosis are important for the health monitoring of physical systems. Model-based approaches for single-mode, smooth, systems is a well-established field, supported by a large body of literature covering various approaches like structural methods  33, parity space techniques, and observer-based methods  72.

While single-mode systems are often described using differential algebraic equations (DAEs), the modeling of non-smooth physical systems yields switched DAEs, also known as multimode DAEs (mDAEs), which combine continuous behaviors, defined as solutions of a set of DAE systems, with discrete mode changes  98, 27. Direct application of traditional fault diagnosis methods to all possible configurations of multi-mode systems quickly becomes intractable, as the number of modes tends to be exponential in the size of the system. The method proposed by  75 works around this issue by coupling a mode estimation algorithm with a single-mode diagnosis methodology, akin to just-in-time compilation in computer science. This approach unfortunately puts the burden on solving mode estimation problems, which often turn out to be intractable for the same reason.

Structural fault detectability and isolability is a graph-based method to evaluate diagnosability properties on DAEs  65. It is based on the Dulmage-Mendelsohn decomposition (DM), a building block of the structural analysis of equation systems. In 11, we show how its extension to multimode systems, introduced in 4, can be applied in the context of structural fault detectability and isolability of mmDAEs  68. Building upon our previous research studies, the methods presented in this paper represent advancements in diagnostic methodologies for multi-mode systems, providing novel ways to study the diagnosability of multi-mode systems without enumerating their modes.

The case study used throughout this article is a model of a reconfigurable battery system, in which switching strategies enable to produce an AC output without relying on a central inverter  19. This model is parametrized by the number of battery cells, so that both the inherent complexity associated with the diagnostics of such systems and the scalability of our approaches can be addressed.

8.3 Mixed Nondeterministic-Probabilistic Automata

Participants: Albert Benveniste.

Graphical models in probability and statistics are a core concept in the area of probabilistic reasoning and probabilistic programming-graphical models include Bayesian networks and factor graphs. For modeling and formal verification of probabilistic systems, probabilistic automata were introduced. A coherent suite of models consisting of Mixed Systems, Mixed Bayesian Networks, and Mixed Automata is proposed in 8. This framework extends factor graphs, Bayesian networks, and probabilistic automata with the handling of nondeterminism. Each of these models comes with a parallel composition, and we establish clear relations between these three models. Also, we provide a detailed comparison between Mixed Automata and Probabilistic Automata.

8.4 On Continuous Solutions for Linear Complementarity Systems

Participants: Khalil Ghorbal.

Hybrid systems are dynamical systems alternating between continuous-time dynamics, called modes, and nonsmooth transitions between modes. Linear complementarity systems (LCS) form a special class of hybrid systems with an exponential number of modes and a linear differential algebraic equation in each mode. LCS are for instance used to describe mechanical and electrical systems featuring perfect contacts or ideal switches. For example, the ideal (Zener) diode is a 1-dimensional LCS with two modes: a passing mode in one direction and a blocking mode in the other direction. While seemingly simple, little is known about the existence, and eventually uniqueness, of continuous solutions (in the state space). The only known sufficient condition is too strong as it requires the existence and uniqueness of solutions for the underlying linear complementarity problem (LCP) which, for a fixed matrix $M$ and a given vector $q$, asks whether there exists a pair of vectors $(w,z)$ satisfying $w-Mz=q$, $w,z\ge 0$, and $w.z=0$. $M$ is said to be a Q-matrix when a solution exists for all $q$. It's worth noting that characterizing Q-matrices is an open problem since the sixties even for low dimensions. Motivated by generalizing the known sufficient conditions for the existence of continuous solutions for LCS, we were naturally led to better understand Q-matrices. In this work, we focused on the regions where no solution for a given LCP exists. We showed that such holes occur only in specific locations. We then exploited this property to fully characterize Q-matrices for $n\le 3$.

Characterizing Q-matrices for any finite dimension $n$ is still an open problem despite a large palette of attempts ranging from linear algebra to convex analysis all the way to the homology of simplicial sets. The novelty of our approach 10 relies on using geometric and topological intuitions to locate the regions for which the LCP doesn't have a solution. This property allowed us to reduce the spatial case to finite planar problems that we were able to enumerate and solve. Our characterization is a program enumerating a long list of (symbolic) constraints on the entries of the matrix $M$. The matrix is a Q-matrix if and only if all the constraints are satisfied. Such approach is for instance useful to generate examples (or counter-examples) to either solve existing conjectures or to improve our current understanding of the problem. For instance, we were able to find an example of a non-regular Q-matrix in dimension 3 (the smallest dimension for which such an example was known was $n=5$). This is a joint work with Christelle Kozaily.

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

10.1 Major publications

• 1 articleA.Albert Benveniste, T.Timothy Bourke, B.Benoît Caillaud, J.-L.Jean-Louis Colaço, C.Cédric Pasteur and M.Marc Pouzet. Building a Hybrid Systems Modeler on Synchronous Languages Principles.Proceedings of the IEEE1069September 2018, 1568--1592HALDOI
• 2 articleA.Albert Benveniste, T.Timothy Bourke, B.Benoît Caillaud and M.Marc Pouzet. Non-standard semantics of hybrid systems modelers.Journal of Computer and System Sciences783This work was supported by the SYNCHRONICS large scale initiative of INRIA2012, 877-910HALDOIback to text
• 3 articleA.Albert Benveniste, B.Benoît Caillaud and M.Mathias Malandain. The mathematical foundations of physical systems modeling languages.Annual Reviews in Control502020, 72-118HALDOIback to text
• 4 articleA.Albert Benveniste, B.Benoît Caillaud, M.Mathias Malandain and J.Joan Thibault. Algorithms for the Structural Analysis of Multimode Modelica Models.Electronics1117September 2022, 1-63HALDOIback to text
• 5 articleA.Albert Benveniste, B.Benoît Caillaud, D.Dejan Nickovic, R.Roberto Passerone, J.-B.Jean-Baptiste Raclet, P.Philipp Reinkemeier, A.Albert Sangiovanni-Vincentelli, W.Werner Damm, T.Thomas Henzinger and K. G.Kim G. Larsen. Contracts for System Design.Foundations and Trends in Electronic Design Automation122-32018, 124-400HALDOIback to text
• 6 articleJ.-B.Jean-Baptiste Jeannin, K.Khalil Ghorbal, Y.Yanni Kouskoulas, A.Aurora Schmidt, R.Ryan Gardner, S.Stefan Mitsch and A.André Platzer. A Formally Verified Hybrid System for Safe Advisories in the Next-Generation Airborne Collision Avoidance System.International Journal on Software Tools for Technology Transfer196November 2017, 717-741HALDOI
• 7 articleA.Andrew Sogokon, K.Khalil Ghorbal and T. T.Taylor T Johnson. Operational Models for Piecewise-Smooth Systems.ACM Transactions on Embedded Computing Systems (TECS)165sOctober 2017, 185:1--185:19HALDOI

10.2 Publications of the year

10.3 Cited publications

• 12 inproceedingsL.Luca de Alfaro. Game Models for Open Systems.Verification: Theory and Practice2772Lecture Notes in Computer ScienceSpringer2003, 269-289DOIback to text
• 13 inproceedingsL.Luca de Alfaro and T. A.Thomas A. Henzinger. Interface automata.Proc. of the 9th ACM SIGSOFT International Symposium on Foundations of Software Engineering (FSE'01)ACM Press2001, 109--120DOIback to text
• 14 inproceedingsL.Luca de Alfaro and T. A.Thomas A. Henzinger. Interface-based design.In Engineering Theories of Software Intensive Systems, proceedings of the Marktoberdorf Summer SchoolKluwer2004DOIback to text
• 15 inproceedingsL.Luca de Alfaro, T. A.Thomas A. Henzinger and M.Mariëlle Stoelinga. Timed Interfaces.Proc. of the 2nd International Workshop on Embedded Software (EMSOFT'02)2491Lecture Notes in Computer ScienceSpringer2002, 108--122DOIback to text
• 16 articleA.Adam Antonik, M.Michael Huth, K. G.Kim G. Larsen, U.Ulrik Nyman and A.Andrzej Wasowski. 20 Years of Modal and Mixed Specifications.Bulletin of European Association of Theoretical Computer Science1942008, URL: https://dblp.org/rec/journals/eatcs/AntonikHLNW08.bibback to text
• 17 onlineM.Modelica Association. Modelica, A Unified Object-Oriented Language for Systems Modeling. Language Specification, Version 3.6.Accessed: 2023-06-122023, URL: https://modelica.org/documents/MLS.pdfback to textback to text
• 18 bookC.Christel Baier and J.-P.Joost-Pieter Katoen. Principles of Model Checking.MIT Press, Cambridge2008, URL: https://mitpress.mit.edu/9780262026499/principles-of-model-checking/back to text
• 19 inproceedingsA.Arvind Balachandran, T.Tomas Jonsson and L.Lars Eriksson. Design and Analysis of Battery-Integrated Modular Multilevel Converters for Automotive Powertrain Applications.2021 23rd European Conference on Power Electronics and Applications (EPE'21 ECCE Europe)IEEE2021, P--1back to text
• 20 articleA.Albert Benveniste, T.Timothy Bourke, B.Benoît Caillaud, J.-L.Jean-Louis Colaço, C.Cédric Pasteur and M.Marc Pouzet. Building a Hybrid Systems Modeler on Synchronous Languages Principles.Proceedings of the IEEE1069September 2018, 1568--1592HALDOIback to textback to text
• 21 miscA.Albert Benveniste, T.Timothy Bourke, B.Benoît Caillaud, B.Bruno Pagano and M.Marc Pouzet. A Type-Based Analysis of Causality Loops In Hybrid Systems Modelers.Deliverable D3.1_1 v 1.0 of the Sys2soft collaborative project ''Physics Aware Software''December 2013HALback to textback to text
• 22 miscA.Albert Benveniste, T.Timothy Bourke, B.Benoît Caillaud and M.Marc Pouzet. Semantics of multi-mode DAE systems.Deliverable D.4.1.1 of the ITEA2 Modrio collaborative projectAugust 2013HALback to text
• 23 incollectionA.Albert Benveniste, B.Benoît Caillaud, H.Hilding Elmqvist, K.Khalil Ghorbal, M.Martin Otter and M.Marc Pouzet. Multi-Mode DAE Models - Challenges, Theory and Implementation.Computing and Software Science: State of the Art and Perspectives10000Lecture Notes in Computer ScienceSpringerOctober 2019, 283-310HALDOIback to text
• 24 inproceedingsA.Albert Benveniste, B.Benoît Caillaud, A.Alberto Ferrari, L.Leonardo Mangeruca, R.Roberto Passerone and C.Christos Sofronis. Multiple Viewpoint Contract-Based Specification and Design.Proceedings of the Software Technology Concertation on Formal Methods for Components and Objects (FMCO'07)5382Revised Lectures, Lecture Notes in Computer ScienceAmsterdam, The NetherlandsSpringerOctober 2008DOIback to text
• 25 incollectionA.Albert Benveniste, B.Benôit Caillaud and M.Mathias Malandain. From Hybrid Automata to~DAE-Based Modeling.Principles of Systems Design13660Lecture Notes in Computer ScienceSpringer Nature SwitzerlandDecember 2022, 3-20HALDOIback to text
• 26 techreportA.Albert Benveniste, B.Benoît Caillaud and M.Mathias Malandain. Structural Analysis of Multimode DAE Systems: summary of results.RR-9387Inria Rennes -- Bretagne AtlantiqueJanuary 2021, 27HALback to text
• 27 articleA.Albert Benveniste, B.Benoît Caillaud and M.Mathias Malandain. The mathematical foundations of physical systems modeling languages.Annual Reviews in Control502020, 72-118HALDOIback to textback to text
• 28 inproceedingsA.Albert Benveniste, B.Benoît Caillaud, B.Bruno Pagano and M.Marc Pouzet. A type-based analysis of causality loops in hybrid modelers.HSCC '14: International Conference on Hybrid Systems: Computation and ControlProceedings of the 17th international conference on Hybrid systems: computation and control (HSCC '14)Berlin, GermanyACM PressApril 2014, 13HALDOIback to textback to text
• 29 inproceedingsN.Nathalie Bertrand, A.Axel Legay, S.Sophie Pinchinat and J.-B.Jean-Baptiste Raclet. A Compositional Approach on Modal Specifications for Timed Systems..11th International Conference on Formal Engineering Methods (ICFEM'09)5885LNCSRio de Janeiro, BrazilSpringerDecember 2009, 679-697HALback to text
• 30 articleN.Nathalie Bertrand, A.Axel Legay, S.Sophie Pinchinat and J.-B.Jean-Baptiste Raclet. Modal event-clock specifications for timed component-based design.Science of Computer Programming772012, 1212-1234HALDOIback to text
• 31 inproceedingsN.Nathalie Bertrand, S.Sophie Pinchinat and J.-B.Jean-Baptiste Raclet. Refinement and Consistency of Timed Modal Specifications..3rd International Conference on Language and Automata Theory and Applications (LATA'09)5457LNCSTarragona, SpainSpringerApril 2009, 152-163HALDOIback to text
• 32 inproceedingsP.Purandar Bhaduri and I.Ingo Stierand. A proposal for real-time interfaces in SPEEDS.Design, Automation and Test in Europe (DATE'10)IEEE2010, 441-446DOIback to text
• 33 inbookM.Mogens Blanke, M.Michel Kinnaert, J.Jan Lunze and M.Marcel Staroswiecki. Diagnosis and Fault-Tolerant Control.Springer Berlin, Heidelberg09 2006, 109-188DOIback to text
• 34 articleS.Simon Bliudze and D.Daniel Krob. Modelling of Complex Systems: Systems as Dataflow Machines.Fundamenta Informaticae9122009, 251-274HALDOIback to text
• 35 phdthesisS.S. Bliudze. Un cadre formel pour l'étude des systèmes industriels complexes: un exemple basé sur l'infrastructure de l'UMTS.Ecole Polytechnique2006back to textback to text
• 36 articleG.Gérard Boudol and K. G.Kim G. Larsen. Graphical versus logical specifications.Theoretical Computer Science10611992, 3-20URL: https://www.sciencedirect.com/science/article/pii/030439759290276LDOIback to text
• 37 inproceedingsW.Willi Braun, F.Francesco Casella and B.Bernhard Bachmann. Solving large-scale Modelica models: new approaches and experimental results using OpenModelica.Proceedings of the 12th International Modelica Conference2017back to text
• 38 inproceedingsD.David Broman and P.Peter Fritzson. Higher-Order Acausal Models.Proceedings of the 2nd International Workshop on Equation-Based Object-Oriented Languages and Tools, EOOLT 2008, Paphos, Cyprus, July 8, 200829Linköping Electronic Conference ProceedingsLinköping University Electronic Press2008, 59--69DOIback to text
• 39 articleD.David Broman. Interactive Programmatic Modeling.ACM Trans. Embed. Comput. Syst.2042021, 33:1--33:26DOIback to text
• 40 phdthesisD.David Broman. Meta-Languages and Semantics for Equation-Based Modeling and Simulation.Linköping University, Sweden2010, URL: https://nbn-resolving.org/urn:nbn:se:liu:diva-58743back to text
• 41 articleR. E.R. E. Bryant. Graph-Based Algorithms for Boolean Function Manipulation.IEEE Trans. Comput.358August 1986, 677--691URL: http://dx.doi.org/10.1109/TC.1986.1676819DOIback to text
• 42 inproceedingsB.Benoît Caillaud, B.Benoît Delahaye, K. G.Kim G. Larsen, A.Axel Legay, M. L.Mikkel L. Pedersen and A.Andrzej Wasowski. Compositional design methodology with constraint Markov chains.QEST 2010Williamsburg, Virginia, United StatesSeptember 2010, URL: http://hal.inria.fr/inria-00591578/enDOIback to textback to text
• 43 articleB.Benoît Caillaud, B.Benoît Delahaye, K. G.Kim G. Larsen, A.Axel Legay, M. L.Mikkel L. Pedersen and A.Andrzej Wasowski. Constraint Markov Chains.Theoretical Computer Science41234May 2011, 4373-4404URL: http://hal.inria.fr/hal-00654003/enDOIback to text
• 44 miscB.Benoit Caillaud, M.Mathias Malandain and J.Joan Thibault. Demo: IsamDAE, an Implicit Structural Analysis Tool for Multimode DAE Systems.PosterApril 2020, 1HALback to text
• 45 inproceedingsB.Benoît Caillaud, M.Mathias Malandain and J.Joan Thibault. Implicit structural analysis of multimode DAE systems.HSCC 2020 - 23rd ACM International Conference on Hybrid Systems: Computation and ControlSydney New South Wales Australia, FranceACMApril 2020, 1-11HALDOIback to textback to textback to text
• 46 articleS. L.S. L. Campbell and C. W.C. W. Gear. The index of general nonlinear DAEs.Numerische Mathematik722dec 1995, 173--196URL: http://dx.doi.org/10.1007/s002110050165DOIback to text
• 47 inproceedingsF.Francesco Casella and A.Adrien Guironnet. ScalableTestGrids - An Open-Source and Flexible Benchmark Suite to Assess Modelica Tool Performance on Large-Scale Power System Test Cases.Proceedings of the 14th International Modelica ConferenceLinköping Electronic Conference Proceedings181Linköping, SwedenModelica Association and Linköping University Electronic PressSeptember 2021, 351--358DOIback to text
• 48 phdthesisA.Arindam Chakrabarti. A Framework for Compositional Design and Analysis of Systems.EECS Department, University of California, BerkeleyDec 2007, URL: http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-174.htmlback to text
• 49 inproceedingsA.Arindam Chakrabarti, L.Luca de Alfaro, T. A.Thomas A. Henzinger and M.Mariëlle Stoelinga. Resource Interfaces.Embedded Software, Third International Conference, EMSOFT 2003, Philadelphia, PA, USA, October 13-15, 2003, Proceedings2855Lecture Notes in Computer ScienceSpringer2003, 117-133DOIback to text
• 50 inproceedingsE. Y.Edward Y. Chang, Z.Zohar Manna and A.Amir Pnueli. Characterization of Temporal Property Classes.ICALP623Lecture Notes in Computer ScienceSpringer1992, 474-486DOIback to text
• 51 articleD.Diego Cifuentes and P. A.Pablo A. Parrilo. Chordal Networks of Polynomial Ideals.SIAM J. Appl. Algebra Geom.112017, 73--110URL: https://doi.org/10.1137/16M106995XDOIback to text
• 52 bookE.E. Clarke, O.O. Grumberg and D.D. Peled. Model Checking.MIT Press1999, URL: https://mitpress.mit.edu/9780262038836/model-checking/back to text
• 53 articleG. E.George E. Collins and H.H. Hong. Partial Cylindrical Algebraic Decomposition for Quantifier Elimination.J. Symb. Comput.1231991, 299--328DOIback to text
• 54 bookN. J.N. J. CutlandNonstandard analysis and its applications.Cambridge Univ. Press1988DOIback to text
• 55 inproceedingsA.Alexandre David, K. G.Kim G. Larsen, A.Axel Legay, U.Ulrik Nyman and A.Andrzej Wasowski. ECDAR: An Environment for Compositional Design and Analysis of Real Time Systems.Automated Technology for Verification and Analysis - 8th International Symposium, ATVA 2010, Singapore, September 21-24, 2010. Proceedings2010, 365-370DOIback to text
• 56 inproceedingsA.Alexandre David, K. G.Kim G. Larsen, A.Axel Legay, U.Ulrik Nyman and A.Andrzej Wasowski. Timed I/O automata: a complete specification theory for real-time systems.Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control, HSCC 2010, Stockholm, Sweden, April 12-15, 20102010, 91-100DOIback to text
• 57 inproceedingsB.Benoît Delahaye, J.-P.Joost-Pieter Katoen, K. G.Kim G. Larsen, A.Axel Legay, M. L.Mikkel L. Pedersen, F.Falak Sher and A.Andrzej Wasowski. Abstract Probabilistic Automata.Verification, Model Checking, and Abstract Interpretation - 12th International Conference, VMCAI 2011, Austin, TX, USA, January 23-25, 2011. Proceedings6538Lecture Notes in Computer Science2011, 324-339DOIback to text
• 58 bookF.F. Diener and G.G. Reeb. Analyse non standard.Hermann1989, URL: https://www.editions-hermann.fr/livre/analyse-non-standard-francine-dienerback to text
• 59 bookD. L.David L. Dill. Trace Theory for Automatic Hierarchical Verification of Speed-Independent Circuits.ACM Distinguished DissertationsMIT Press1989DOIback to text
• 60 articleJ.J. Edmonds and R. M.R. M. Karp. Theoretical improvements in algorithmic efficiency for network flow problems.Journal of the ACM1921972, 248--264URL: http://dx.doi.org/10.1145/321694.321699DOIback to text
• 61 phdthesisH.Hilding Elmqvist. A structured model language for large continuous systems.Universiteit i Lund1978back to text
• 62 inproceedingsH.H. Elmqvist, S. E.S. E. Mattsson and M.M. Otter. Modelica extensions for Multi-Mode DAE Systems.Proceedings of the 10th International Modelica Conference, March 10-12, 2014, Lund, SwedenLinköping University Electronic Pressmar 2014DOIback to text
• 63 inproceedingsH.Hilding Elmqvist, A.Andrea Neumayr and M.Martin Otter. Modia-dynamic modeling and simulation with julia.Juliacon'18University College London, UKAugust 2018, URL: https://elib.dlr.de/124133/back to text
• 64 inproceedingsH. J.H. J. Ferreau, S.S. Almér, H.H. Peyrl, J. L.J. L. Jerez and A.A. Domahidi. Survey of industrial applications of embedded model predictive control.2016 European Control Conference (ECC)2016, 601-601DOIback to text
• 65 articleE.Erik Frisk, A.Anibal Bregon, J.Jan Aslund, M.Mattias Krysander, B.Belarmino Pulido and G.Gautam Biswas. Diagnosability analysis considering causal interpretations for differential constraints.IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans4252012, 1216--1229back to text
• 66 articleP.Peter Fritzson, A.Adrian Pop, K.Karim Abdelhak, A.Adeel Ashgar, B.Bernhard Bachmann, W.Willi Braun, D.Daniel Bouskela, R.Robert Braun, L.Lena Buffoni, F.Francesco Casella, R.Rodrigo Castro, R.Rüdiger Franke, D.Dag Fritzson, M.Mahder Gebremedhin, A.Andreas Heuermann, B.Bernt Lie, A.Alachew Mengist, L.Lars Mikelsons, K.Kannan Moudgalya, L.Lennart Ochel, A.Arunkumar Palanisamy, V.Vitalij Ruge, W.Wladimir Schamai, M.Martin Sjölund, B.Bernhard Thiele, J.John Tinnerholm and P.Per Östlund. The OpenModelica Integrated Environment for Modeling, Simulation, and Model-Based Development.Modeling, Identification and Control4142020, 241--295DOIback to text
• 67 inproceedingsA. V.A. V. Goldberg and R. E.R. E. Tarjan. A new approach to the maximum flow problem.Proceedings of the eighteenth annual ACM symposium on Theory of computing (STOC'86)1986, URL: http://dx.doi.org/10.1145/12130.12144DOIback to text
• 68 inproceedingsF.Fatemeh Hashemniya, E.Erik Frisk and M.Mattias Krysander. Hierarchical Diagnosis Algorithm for Component-Based Multi-Mode Systems.22nd IFAC World CongressIFAC2023back to text
• 69 phdthesisC.Christoph Höger. Compiling Modelica: about the separate translation of models from Modelica to OCaml and its impact on variable-structure modeling.TU Berlin2019DOIback to text
• 70 articleC.Christoph Höger. Faster Structural Analysis of Differential-Algebraic Equations by Graph Compression.IFAC-PapersOnLine4818th Vienna International Conference on Mathematical Modelling2015, 135--140URL: https://www.sciencedirect.com/science/article/pii/S2405896315001019DOIback to text
• 71 miscIEEE Standard VHDL Analog and Mixed-Signal Extensions, Std 1076.1-1999.1999, URL: http://dx.doi.org/10.1109/IEEESTD.1999.90578DOIback to text
• 72 bookR.R. Isermann. Fault-Diagnosis Systems: An Introduction from Fault Detection to Fault Tolerance.Springer Berlin Heidelberg2006back to text
• 73 inproceedingsY.Y. Iwasaki, A.A. Farquhar, V.V.A. Saraswat, D.D.G. Bobrow and V.V. Gupta. Modeling Time in Hybrid Systems: How Fast Is ``Instantaneous''?IJCAI1995, 1773--1781URL: https://www.ijcai.org/Proceedings/95-2/Papers/097.pdfback to text
• 74 inproceedingsJ.-B.Jean-Baptiste Jeannin, K.Khalil Ghorbal, Y.Yanni Kouskoulas, R.Ryan Gardner, A.Aurora Schmidt, E.Erik Zawadzki and A.André Platzer. Formal verification of ACAS X, an industrial airborne collision avoidance system.2015 International Conference on Embedded Software, EMSOFT 2015, Amsterdam, Netherlands, October 4-9, 2015Amsterdam, NetherlandsIEEE2015, 127--136HALDOIback to text
• 75 articleH.Hamed Khorasgani and G.Gautam Biswas. Structural fault detection and isolation in hybrid systems.IEEE Transactions on Automation Science and Engineering1542017, 1585--1599back to text
• 76 phdthesisA.Aurélien Lamercerie. Principe de transduction sémantique pour l'application de théories d'interfaces sur des documents de spécification.Université Rennes 1 ; Rennes 1April 2021HALback to text
• 77 articleL.Leslie Lamport. Proving the Correctness of Multiprocess Programs.IEEE Trans. Software Eng.321977, 125--143DOIback to text
• 78 inproceedingsK. G.Kim G. Larsen, U.Ulrik Nyman and A.Andrzej Wasowski. On Modal Refinement and Consistency.Proc. of the 18th International Conference on Concurrency Theory (CONCUR'07)Springer2007, 105--119DOIback to text
• 79 inproceedingsK. G.Kim G. Larsen and B.Bent Thomsen. A Modal Process Logic.Proceedings of the Third Annual Symposium on Logic in Computer Science (LICS'88)IEEE1988, 203-210DOIback to text
• 80 incollectionT.T. Lindstr\om. An Invitation to Nonstandard Analysis.Nonstandard Analysis and its ApplicationsCambridge Univ. Press1988, 1--105DOIback to text
• 81 articleH.-A.Hans-Andrea Loeliger. An introduction to factor graphs.IEEE Signal Processing Magazine2112004, 28-41DOIback to text
• 82 inproceedingsN. A.Nancy A. Lynch. Input/Output Automata: Basic, Timed, Hybrid, Probabilistic and Dynamic.CONCUR 2003 - Concurrency Theory, 14th International Conference, Marseille, France, September 3-5, 2003, Proceedings2761Lecture Notes in Computer ScienceSpringer2003, 187-188DOIback to text
• 83 articleN. A.Nancy A. Lynch and E. W.Eugene W. Stark. A Proof of the Kahn Principle for Input/Output Automata.Inf. Comput.8211989, 81-92DOIback to text
• 84 bookZ.Zohar Manna and A.Amir Pnueli. Temporal verification of reactive systems: Safety.Springer1995DOIback to text
• 85 onlineI.Inc. MathWorks. Simulink official webpage.Accessed: 2023-06-122023, URL: https://www.mathworks.com/products/simulink.htmlback to text
• 86 articleB.Bertrand Meyer. Applying ``Design by Contract''.Computer2510October 1992, 40--51URL: http://dx.doi.org/10.1109/2.161279DOIback to text
• 87 articleP.P. Nuzzo, A. L.Alberto L. Sangiovanni-Vincentelli, X.X. Sun and A.A. Puggelli. Methodology for the Design of Analog Integrated Interfaces Using Contracts.IEEE Sensors Journal1212Dec. 2012, 3329--3345DOIback to text
• 88 articleC.C. Pantelides. The consistent initialization of differential-algebraic systems.SIAM J. Sci. Stat. Comput.921988, 213--231DOIback to textback to text
• 89 articleJ. D.J. D. Pryce. A Simple Structural Analysis Method for DAEs.BIT Numerical Mathematics412March 2001, 364--394URL: http://dx.doi.org/10.1023/a:1021998624799DOIback to textback to textback to text
• 90 articleJ.-B.Jean-Baptiste Raclet, E.Eric Badouel, A.Albert Benveniste, B.Benoît Caillaud, A.Axel Legay and R.Roberto Passerone. A Modal Interface Theory for Component-based Design.Fundamenta Informaticae1081-22011, 119-149URL: http://hal.inria.fr/inria-00554283/enDOIback to text
• 91 articleR.Ramy Rashad, F.Federico Califano, A.Arjan van der Schaft and S.Stefano Stramigioli. Twenty years of distributed port-Hamiltonian systems: a literature review.IMA J. Math. Control. Inf.3742020, 1400--1422DOIback to text
• 92 bookA.A. Robinson. Non-Standard Analysis.Princeton Landmarks in Mathematics1996, URL: https://press.princeton.edu/books/paperback/9780691044903/non-standard-analysisback to text
• 93 articleE.Ernst Sikora, B.Bastian Tenbergen and K.Klaus Pohl. Industry needs and research directions in requirements engineering for embedded systems.Requirements Engineering172012, 57--78URL: http://link.springer.com/article/10.1007/s00766-011-0144-xDOIback to text
• 94 inproceedingsW. Y.William Y. Sit. The Ritt–Kolchin theory for differential polynomials.Differential Algebra and Related Topics2002, 1-70DOIback to text
• 95 onlineD.Dassault Systèmes. Dymola official webpage.Accessed: 2023-06-122023, URL: https://www.3ds.com/products-services/catia/products/dymola/back to text
• 96 articleH.Hisao Tamaki. Positive-instance driven dynamic programming for treewidth.J. Comb. Optim.3742019, 1283--1311DOIback to text
• 97 articleJ.Joan Thibault and K.Khalil Ghorbal. Leveraging Structural Analysis for Quantified Boolean Formulae.Summer School on Modelling and Verification of Parallel Processes, Grenoble, France6http://khalilghorbal.info/assets/pdf/papers/RBTF_movep.pdf2020back to text
• 98 inbookS.Stephan Trenn. Switched Differential Algebraic Equations.Dynamics and Control of Switched Electronic Systems: Advanced Perspectives for Modeling, Simulation and Control of Power ConvertersF.Francesco VascaL.Luigi IannelliLondonSpringer London2012, 189--216DOIback to text