Bridging Batch and Streaming Estimations to System Identification under Adversarial Attacks

System identification in modern engineering systems faces emerging challenges from unanticipated adversarial attacks beyond existing detection mechanisms. In this work, we obtain a provably accurate estimate of the Markov parameter matrix of order $k$ to identify partially observed linear systems, in which the probability of having an attack at each time is $O(1/k)$. We show that given the batch data accumulated up to time $T^*$, the $\ell_2$-norm estimator achieves an error decaying exponentially as $k$ grows. We then propose a stochastic projected subgradient descent algorithm on streaming data that produces an estimate at each time $t<T^*$, in which case the expected estimation error proves to be the larger of $O(k/\sqrt{t})$ and an exponentially decaying term in $k$. This stochastic approach illustrates how non-smooth estimators can leverage first-order methods despite lacking recursive formulas. Finally, we integrate batch and streaming estimations to recover the Hankel matrix using the appropriate estimates of the Markov parameter matrix, which enables the synthesis of a robust adaptive controller based on the estimated balanced truncated model under adversarial attacks.