Approximating the System Behavior with Input Uncertainty Using Big Data

This paper aims to construct a representation to approximate the behavior of linear time-invariant systems from a large data set whose input contains unmeasured uncertainty and output is subject to measurement noise. Using big data combined with the statistical properties of the input uncertainty and measurement noise, the covariance of the input uncertainty and the output can be approximated. This enables the construction of an approximate covariance of the trajectories in the data set, through which a representation for the approximation of system behavior is obtained. The behavior of this representation is shown to converge in probability to the true behavior. An illustrative example is provided to show that the proposed representation is able to predict the system trajectory to a satisfactory accuracy. The result of this paper provides a potential basis for the development of data-based trajectory estimation and predictive control algorithm when the system input uncertainty is unmeasured.