Quantifying conformance using the Skorokhod metric

Abstract

The conformance testing problem for dynamical systems asks, given two dynamical models (e.g., as Simulink diagrams), whether their behaviors are “close” to each other. In the semi-formal approach to conformance testing, the two systems are simulated on a large set of tests, and a metric, defined on pairs of real-valued, real-timed trajectories, is used to determine a lower bound on the distance. We show how the Skorokhod metric on continuous dynamical systems can be used as the foundation for conformance testing of complex dynamical models. The Skorokhod metric allows for both state value mismatches and timing distortions, and is thus well suited for checking conformance between idealized models of dynamical systems and their implementations. We demonstrate the robustness of the metric by proving a transference theorem: trajectories close under the Skorokhod metric satisfy “close” logical properties in the timed linear time logic FLTL (Freeze LTL) containing a rich class of temporal and spatial constraint predicates involving time and value freeze variables. We provide efficient window-based streaming algorithms to compute the Skorokhod metric for both piecewise affine and piecewise constant traces, and use these as a basis for a conformance testing tool for Simulink. We experimentally demonstrate the effectiveness of our tool in finding discrepant behaviors on a set of control system benchmarks, including an industrial challenge problem.