Data-Driven Estimation of Vinnicombe metric

Quantifying model mismatch in a control-relevant manner is fundamental in robust control. A well-known metric for this purpose is the $\nu$-gap, or Vinnicombe metric, which measures the discrepancy between a nominal model and the real system from a closed-loop viewpoint. However, its computation typically requires explicit knowledge of the true system. In this letter, we propose an identification-free, data-driven method to estimate the $\nu$-gap between discrete-time SISO systems directly from input-output experiments. The method is complemented by a data-driven winding-number test, based on Welch-type averaging, to verify a required topological condition for the computation of the metric. Numerical simulations on heavy-duty gas-turbine models and a textbook example show that the proposed estimate closely matches MATLAB$^\copyright$ \texttt{gapmetric}, while correctly detecting cases in which the admissibility conditions fail.