Contraction Certification from Streaming Data: Wasserstein Robustness and Compositional Stability for Interconnected Nonlinear System

Streaming contraction certificates, which determine in real time whether observed data is sufficient to certify a safe control action, face two structural challenges: the disturbance distribution shifts during operation, and the system consists of coupled subsystems whose joint model is unavailable.

This paper addresses both.

First, we develop a Wasserstein-robust certificate beta_cert(t,epsilon) = beta_hat(t) - rho(t)(1+2epsilon(t)), where epsilon(t) is estimated online from the empirical excess kurtosis of recent residuals, so the certificate degrades gracefully under distributional shift rather than failing catastrophically.

Second, we prove that local certificates beta_A and beta_B, estimated independently from each subsystem's data, compose into a network-level guarantee via beta_net = (beta_A+beta_B)/2 - sqrt[(beta_A-beta_B)^2/4 + gamma^2] > 0 whenever gamma < sqrt(beta_Abeta_B), with no joint model required.

On a five-node G5 benchmark under three noise regimes - Gaussian, heavy-tailed Laplace, and spike events - the Wasserstein certificate remains valid in 73% of spike-regime timesteps versus 33% for the Gaussian baseline (2.2x improvement), while the Gaussian certificate never authorizes deployment during the spike window.

The compositional framework correctly identifies all three coupling regimes from local data alone, with gamma_warn = sqrt(beta_A*beta_B) is approximately 0.98, precisely predicting network-level contraction loss.