  In this work, we propose a data-driven approach for the construction of
finite abstractions (a.k.a., symbolic models) for discrete-time deterministic
control systems with unknown dynamics. We leverage notions of so-called
alternating bisimulation functions (ABF), as a relation between each unknown
system and its symbolic model, to quantify the mismatch between state behaviors
of two systems. Accordingly, one can employ our proposed results to perform
formal verification and synthesis over symbolic models and then carry the
results back over unknown original systems. In our data-driven setting, we
first cast the required conditions for constructing ABF as a robust
optimization program (ROP). Solving the provided ROP is not tractable due to
the existence of unknown models in the constraints of ROP. To tackle this
difficulty, we collect finite numbers of data from trajectories of unknown
systems and propose a scenario optimization program (SOP) corresponding to the
original ROP. By establishing a probabilistic relation between optimal values
of SOP and ROP, we formally construct ABF between unknown systems and their
symbolic models based on the number of data and a required confidence level. We
verify the effectiveness of our data-driven results over two physical case
studies with unknown models including (i) a DC motor and (ii) a nonlinear jet
engine compressor. We construct symbolic models from data as appropriate
substitutes of original systems and synthesize policies maintaining states of
unknown systems in a safe set within infinite time horizons with some
guaranteed confidence levels.
