  Recently, various algorithms for data-driven simulation and control have been
proposed based on the Willems' fundamental lemma. However, when collected data
are noisy, these methods lead to ill-conditioned data-driven model structures.
In this work, we present a maximum likelihood framework to obtain an optimal
data-driven model, the signal matrix model, in the presence of output noise.
Data compression and noise level estimation schemes are also proposed to apply
the algorithm efficiently to large datasets and unknown noise level scenarios.
Two approaches in system identification and receding horizon control are
developed based on the derived optimal estimator. The first one identifies a
finite impulse response model. This approach improves the least-squares
estimator with less restrictive assumptions. The second one applies the signal
matrix model as the predictor in predictive control. The control performance is
shown to be better than existing data-driven predictive control algorithms,
especially under high noise levels. Both approaches demonstrate that the
derived estimator provides a promising framework to apply data-driven
algorithms to noisy data.
