Reachability Analysis for High-Index Linear Differential Algebraic Equations

Abstract

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.

Decentralized Real-Time Safety Verification for Distributed Cyber-Physical Systems

“…Over the last two decades, although many methods have been developed to conduct reachability analysis and safety verification of CPS, such as the approaches proposed in [1,3,4,6,12,13,16,18,21,27,29], applying these techniques to real-time distributed CPS remains a big challenge. This is due to the fact that, 1) all existing techniques have intensive computation costs and are usually too slow to be used in a real-time manner and, 2) these techniques target the safety verification of a single CPS, and therefore they naturally cannot be applied efficiently to a distributed CPS where clock mismatches and communication between agents (i.e., individual systems) are essential concerns.…”

Linear Differential-Algebraic Equations (Benchmark Proposal)

“…This conversion and printing into the SpaceEx [15] input format is implemented in a tool called Daev 1 , that makes use of Python and its standard packages Numpy, Scipy, and Mathplotlib. A detailed description of the software tool Daev, discussion of decoupling techniques, and simulation of the benchmarks described in this paper can be found in [29]. Currently, the only DAEs that can be analyzed by the tools available in the research community are of index-1.…”

“…x (t) and v c y (t), of the fluid in the central cell of the spatial domain Ω and defined the unsafe set of the system to be U nsaf e −v c x (t) − v c y (t) ≤ 0.04. We refer readers to the following paper [29] for a more detailed description of the decoupling and benchmark construction process for the Semi-discretized Stokes Equation.…”

“…Although we have chosen to store all of our verification problems in a format used by the SpaceEx verification tool [15], only index-1 DAEs can be analyzed by the tools currently available in the research community. In order to construct the SpaceEx model file needed for verification for the RLC circuit benchmark, the DAE system given by (2.5) must be decoupled using the Marz decoupling method into an ODE subsystem coupled with algebraic constraints [23,29]. Once the DAE has been decoupled, we must ensure that the initial set of states and initial conditions are consistent and satisfy certain conditions.…”

“…Once the DAE has been decoupled, we must ensure that the initial set of states and initial conditions are consistent and satisfy certain conditions. We refer the reader to the following paper for an in-depth discussion of these requirements [23,29]. The final step in generating the SpaceEx model file is the construction of an automaton Figure 3.1: SpaceEx output for the RLC circuit benchmark reachable set over a ten second simulation with a single mode whose continuous evolution is specified by the ODE subsystem and governed by a set of invariants corresponding to the algebraic constraints.…”