Index-2 hybrid DAE: a case study with well-posedness and numerical analysis

In this work, we study differential algebraic equations with constraints defined in a piecewise manner using a conditional statement. Such models classically appear in systems where constraints can evolve in a very small time frame compared to the observed time scale. The use of conditional statements or hybrid automata are a powerful way to describe such systems and are, in general, well suited to simulation with event driven numerical schemes. However, such methods are often subject to chattering at mode switch in presence of sliding modes, or can result in Zeno behaviours. In contrast, the representation of such systems using differential inclusions and method from non-smooth dynamics are often closer to the physical theory but may be harder to interpret. Associated time-stepping numerical methods have been extensively used in mechanical modelling with success and then extended to other fields such as electronics and system biology. In a similar manner to the previous application of non-smooth methods to the simulation of piecewise linear ODEs, non-smooth event-capturing numerical scheme are applied to piecewise linear DAEs. In particular, the study of a 2-D dynamical system of index-2 with a switching constraint using set-valued operators, is presented

Online Cycle Detection for Models with Mode-Dependent Input and Output Dependencies

In the fields of co-simulation and component-based modelling, designers import models as building blocks to create a composite model that provides more complex functionalities. Modelling tools perform instantaneous cycle detection (ICD) on the composite models having feedback loops to reject the models if the loops are mathematically unsound and to improve simulation performance. In this case, the analysis relies heavily on the availability of dependency information from the imported models. However, the cycle detection problem becomes harder when the model's input to output dependencies are mode-dependent, i.e. changes for certain events generated internally or externally as inputs. The number of possible modes created by composing such models increases significantly and unknown factors such as environmental inputs make the offline (statical) ICD a difficult task. In this paper, an online ICD method is introduced to address this issue for the models used in cyber-physical systems. The method utilises an oracle as a central source of information that can answer whether the individual models can make mode transition without creating instantaneous cycles. The oracle utilises three types of data-structures created offline that are adaptively chosen during online (runtime) depending on the frequency as well as the number of models that make mode transitions. During the analysis, the models used online are stalled from running, resulting in the discrepancy with the physical system. The objective is to detect an absence of the instantaneous cycle while minimising the stall time of the model simulation that is induced from the analysis. The benchmark results show that our method is an adequate alternative to the offline analysis methods and significantly reduces the analysis time.Comment: \c{opyright} 2021. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

Index-2 hybrid DAE: a case study with well-posedness and numerical analysis

International audienceIn this work, we study differential algebraic equations with constraints defined in a piecewise manner using a conditional statement. Such models classically appear in systems where constraints can evolve in a very small time frame compared to the observed time scale. The use of conditional statements or hybrid automata are a powerful way to describe such systems and are, in general, well suited to simulation with event driven numerical schemes. However, such methods are often subject to chattering at mode switch in presence of sliding modes, or can result in Zeno behaviours. In contrast, the representation of such systems using differential inclusions and method from non-smooth dynamics are often closer to the physical theory but may be harder to interpret. Associated time-stepping numerical methods have been extensively used in mechanical modelling with success and then extended to other fields such as electronics and system biology. In a similar manner to the previous application of non-smooth methods to the simulation of piecewise linear ODEs, non-smooth event-capturing numerical scheme are applied to piecewise linear DAEs. In particular, the study of a 2-D dynamical system of index-2 with a switching constraint using set-valued operators, is presented

Index-2 hybrid DAE: a case study with well-posedness and numerical analysis

In this work, we study differential algebraic equations with constraints defined in a piece-wise manner using a conditional statement. Such models classically appear in systems where constraints can evolve in a very small time frame compared to the observed time scale. The use of conditional statements or hybrid automata are a powerful way to describe such systems and are, in general, well suited to simulation with event driven numerical schemes. However, such methods are often subject to chattering at mode switch in presence of sliding modes, and can result in Zeno behaviours. In contrast, the representation of such systems using differential inclusions and method from non-smooth dynamics are often closer to the physical theory but may be harder to interpret. Associated time-stepping numerical methods have been extensively used in mechanical modelling with success and then extended to other fields such as electronics and system biology. In a similar manner to the previous application of non-smooth methods to the simulation of piece-wise linear ODEs, we want to apply non-smooth numerical scheme to piece-wise linear DAEs. In particular, the study of a 2-D dynamical system of index-2 with a switching constraint using set-valued operators, is presented

Efficient implementations of expressive modelling languages

This thesis is concerned with modelling languages aimed at assisting with modelling and simulation of systems described in terms of differential equations. These languages can be split into two classes: causal languages, where models are expressed using directed equations; and non-causal languages, where models are expressed using undirected equations. This thesis focuses on two related paradigms: FRP and FHM. FRP is an approach to programming causal time-aware applications that has successfully been used in causal modelling applications; while FHM is an approach to programming non-causal modelling applications. However, both are built on similar principles, namely, the treatment of models as first-class entities, allowing for models to be parametrised by other models or computed at runtime; and support for structurally dynamic models, whose behaviour can change during the simulation. This makes FRP and FHM particularly flexible and expressive approaches to modelling, especially compared to other mainstream languages. Because of their highly expressive and flexible nature, providing efficient implementations of these languages is a challenge. This thesis explores novel implementation techniques aimed at improving the performance of existing implementations of FRP and FHM, and other expressive modelling languages built on similar ideas. In the setting of FRP, this thesis proposes a novel embedded FRP library that uses the implementation approach of synchronous dataflow languages. This allows for significant performance improvement by better handling of the reactive network's topology, which represents a large portion of the runtime in current implementations, especially for applications that make heavy use of continuously varying values, such as modelling applications. In the setting of FHM, this thesis presents the modular compilation of a language based on FHM. Due to inherent difficulties with the simulation of systems of undirected equations, previous implementations of FHM and similarly expressive languages were either interpreted or generated code on the fly using just-in-time compilation, two techniques which have runtime overhead over ahead-of-time compilation. This thesis presents a new method for generating code for equation systems which allows for the separate compilation of FHM models. Compared with current approaches to FRP and FHM implementation, there is greater commonality between the implementation approaches described here, suggesting a possible way forward towards a future non-causal modelling language supporting FRP-like features, resulting in an even more expressive modelling language

Structural Analysis of Multi-Mode DAE Systems

International audienceDifferential Algebraic Equation (DAE) systems constitute the mathematical model supporting physical modeling languages such as Modelica, VHDL-AMS, or Simscape. Unlike 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 does 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 this paper 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