Biblio

We consider the problem of translating a deterministic \textbackslashemph\simulation model\ (like Matlab-Simunk, Modelica or Ptolemy models) into a \textbackslashemphěrification model\ expressed by a network of hybrid automata. The goal is to verify safety using reachability analysis on the verification model. Simulation models typically use transitions with urgent semantics, which must be taken as soon as possible. Urgent transitions also make it possible to decompose systems that would otherwise need to be modeled with a monolithic hybrid automaton. In this paper, we include urgent transitions in our verification models and propose a suitable adaptation of our reachability algorithm. However, the simulation model, due to its imperfections, may be unsafe even though the corresponding hybrid automata are safe. Conversely, set-based reachability may not be able to show safety of an ideal formal model, since complex dynamics necessarily entail overapproximations. Taken as a whole, the formal modeling and verification process can both falsely claim safety and fail to show safety of the concrete system. We address this inconsistency by relaxing the model as follows. The standard semantics of hybrid automata is a mathematical idealization, where reactions are considered to be instantaneous and physical measurements infinitely precise. We propose semantics that relax these assumptions, where guard conditions are sampled in discrete time and admit measurement errors. The relaxed semantics can be translated to an equivalent relaxed model in standard semantics. The relaxed model is realistic in the sense that it can be implemented on hardware fast and precise enough, and in a way that safety is preserved. Finally, we show that overapproximative reachability analysis can show safety of relaxed models, which is not the case in general.

We consider a continuous analogue of (Babai et al. 1996)'s and (Cai et al. 2000)'s problem of solving multiplicative matrix equations. Given k + 1 square matrices A1, ..., Ak, C, all of the same dimension, whose entries are real algebraic, we examine the problem of deciding whether there exist non-negative reals t1, ..., tk such that We show that this problem is undecidable in general, but decidable under the assumption that the matrices A1, ..., Ak commute. Our results have applications to reachability problems for linear hybrid automata. Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the Minkowski-Weyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability result is shown by reduction from Hilbert's Tenth Problem.