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Simulation-Based Reachability Analysis for High-Index Large Linear Differential Algebraic Equations
Reachability analysis is a fundamental problem for safety verification and
falsification of Cyber-Physical Systems (CPS) whose dynamics follow physical
laws usually represented as differential equations. In the last two decades,
numerous reachability analysis methods and tools have been proposed for a
common class of dynamics in CPS known as ordinary differential equations (ODE).
However, there is lack of methods dealing with differential algebraic equations
(DAE) which is a more general class of dynamics that is widely used to describe
a variety of problems from engineering and science such as multibody mechanics,
electrical cicuit design, incompressible fluids, molecular dynamics and
chemcial process control. Reachability analysis for DAE systems is more complex
than ODE systems, especially for high-index DAEs because they contain both a
differential part (i.e., ODE) and algebraic constraints (AC). In this paper, we
extend the recent scalable simulation-based reachability analysis in
combination with decoupling techniques for a class of high-index large linear
DAEs. In particular, a high-index linear DAE is first decoupled into one ODE
and one or several AC subsystems based on the well-known Marz decoupling method
ultilizing admissible projectors. Then, the discrete reachable set of the DAE,
represented as a list of star-sets, is computed using simulation. Unlike ODE
reachability analysis where the initial condition is freely defined by a user,
in DAE cases, the consistency of the inititial condition is an essential
requirement to guarantee a feasible solution. Therefore, a thorough check for
the consistency is invoked before computing the discrete reachable set. Our
approach sucessfully verifies (or falsifies) a wide range of practical,
high-index linear DAE systems in which the number of state variables varies
from several to thousands