Objective functions for falsification of signal temporal logic properties in cyber-physical systems

Cyber-physical systems, such as automation and automotive systems, are highly complex systems that are frequently also safety-critical. Thus, it is important that these systems behave as intended, as incorrectness can have serious consequences. Due to continuous dynamics that yield infinite state spaces, the systems cannot be exhaustively tested to guarantee correct behavior, but systematic testing can be used to systematically search for behavior that is inconsistent with the requirements. However, manual testing is a tedious and error-prone task, and thus it is desirable to automate as much of the testing process as possible to increase efficiency and eliminate manual errors. This paper considers falsification of requirements expressed as temporal logic formulae to find errors in the Model-in-the-Loop stage of model-based development. The falsification is an optimization procedure where the objective function is determined by the definition of a quantitative semantics for the temporal logic formalism, and the optimization is performed over an input parametrization. It is shown that in certain cases where the discreteness of the system under test manifests itself in the objective function, the robustness values for the temporal logic specifications need to be modified. This paper presents two alternative objective functions suitable for these cases, and illustrates their use with both a small example and a use case from Volvo Car Corporation