Light-weight hybrid model checking facilitating online prediction of temporal properties

We address the question of system-in-the-loop monitoring of dynamic systems with a moderately fast sampling rate. The objective is, given a partial time series of (possibly inexact) data obtained from online measurements of observable variables of the system, to safely predict the truth value of a Linear-time Temporal Logic (LTL) formula including future modalities, in realtime while the system is evolving. We therefore develop an online monitoring tool providing early warning based on safe extrapolation from system history, where the history is recorded as a time series of observations and, in order to achieve sound predictions, the future is safely approximated using differential equations and other constraints pertinent to the system dynamics. The systems considered are heterogeneous systems from the automotive domain, featuring a seamless integration of cognitive models for human behavior, traffic models, model-based designs of driver assistance systems, and models describing the dynamic behavior of the car. Tool support for online monitoring of hybrid systems with respect to requirements expressed using temporal logics has previously been studied in [NM07], where a property based tool (named ATM) for monitoring analogue systems is described. The properties are expressed as formulae of STL/PSL, an extension of MITL [AFH96] and STL [MN04]. A serious limitation of AST, however, is that formulae are just interpreted over a history of past events, forcing online monitoring to fall behind real-time. Our work can be seen as an extension of that in the sense that we interpret formulae over infinite extensions of the monitored traces, where the future events are safely approximated using laws describing the dynamics of the system. To arrive at useful safe overapproximations of the possible infinite extensions of the sampled finite trace, interval constraint propagation [BG06] is used to efficiently estimate ranges of observations in the bounded future based on ordinary differential equations (ODE) governing parts of the system dynamics, thereby exploiting a mean-value form of the ODEs akin to that used in the hybrid-system model-checker HSolver [RS05]. An offline preprocessing step is used to permit efficient online computation of estimates covering the unbounded future. Following the intuition that dynamical systems gradually turn their state to the good or bad, we let our monitor output a quantitative figure in the form of a numerical interval providing a safe estimate of how severe a property violation can be, instead of a qualitative yes/no answer. To this end, we resort to a robust semantics for LTL, where the meaning of a formula is the degree of accuracy within which it is satisfied. This notion of robustness originates from [Rat00], where it is applied to arithmetic first-order constraints, and has been lifted to the linear-time temporal logics Duration Calculus [FH05] and LTL [FLS08]. We adopt the latter and extend it to an interval-valued interpretation over interval-valued traces, thus being able to accommodate inexact measurements as well as not fully determined extrapolated values in the semantics.