ANR-25-CE48-4916: Generalized Filippov solutions for discontinuous DAEs: Control and Simulations (GFdDAE)
Published:
Differential-algebraic equations (DAEs) arise naturally when modeling dynamical systems from first principles. In many cases, physical laws are expressed as combinations of differential and algebraic equations. This modeling approach is common in constrained mechanics, chemical and biological processes, power systems, and especially analog circuit design—where idealized components (e.g., resistors, capacitors, inductors) and Kirchhoff’s laws define the system dynamics. When these systems experience abrupt changes—such as switching in electric circuits, mechanical contacts, or discontinuous control inputs—discontinuous DAEs emerge. However, there is currently no comprehensive theoretical foundation for studying such systems. Challenges include:
Their hybrid behaviors, which differ significantly from ODE counterparts, The inconsistent initialization problem caused by switching and algebraic constraints, The occurrence of Dirac impulses due to state jumps. Without a rigorous solution concept, tasks such as simulation, stability analysis, and control design lack solid justification.
Discontinuous DAEs are relevant across many research areas, including systems and control, hybrid systems, and computer-aided simulation. A notable example is switched DAEs. While time-dependent switching has been extensively studied [2-5], progress on state-dependent switching, a subclass of discontinuous DAEs, remains limited.
The Hycomes team at Inria Rennes has contributed to related research through the concept of multi-mode DAEs, in the context of the Modelica language [6-7]. Despite these advancements, challenges persist, including:
Computing consistent initial values and jumps, Managing sliding and chattering behaviors, Addressing scalability for large-scale, high-dimensional systems. These issues emphasize the need for refined mathematical foundations and advanced control methods compatible with Modelica-based simulation platforms.
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