A Compositional Dissipativity Approach for Data-Driven Safety Verification of Large-Scale Dynamical Systems

Abstract

This work is concerned with a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">compositional data-driven approach</i> for formal safety verification of large-scale continuous-time dynamical systems with unknown models. The proposed framework enjoys the interconnection matrix and joint dissipativity-type properties of subsystems, described by the notion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic storage certificates</i> . In the first part of the paper, we cast the required conditions for constructing storage certificates as a robust optimization program (ROP). Since the proposed ROP is not tractable due to the unknown model appearing in one of its constraints, we propose a scenario optimization program (SOP) corresponding to the original ROP by collecting finite numbers of data from trajectories of each subsystem. By establishing a probabilistic relation between the optimal value of SOP and that of ROP, we construct a storage certificate for each unknown subsystem based on the number of data and a required level of confidence. We accordingly propose a compositional technique based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dissipativity reasoning</i> to construct <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stochastic barrier certificates</i> of interconnected systems based on storage certificates of individual subsystems. By leveraging the acquired barrier certificate, we quantify a lower bound on the probability that an interconnected system never reaches a certain unsafe region in finite time horizons with an a-priori guaranteed confidence. We also propose an auxiliary compositional approach without requiring any compositionality condition but at the cost of providing a potentially conservative safety guarantee. In the second part of the paper, we propose our approaches for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deterministic</i> continuous-time systems with unknown dynamics. We verify our results over an unknown room temperature network containing 100 rooms.