On Approximation of System Behavior From Large Noisy Data Using Statistical Properties of Measurement Noise

This article develops a method to determine an approximate behavior of a given linear time-invariant dynamical system from noise-corrupted data, which can be used for both data-driven simulation and predictive control using the behavioral systems theory. The system input and output are assumed to be measured subject to additive zero-mean white noise with known covariance. From the measured big data set, an approximated representation of the true behavior of the system is constructed using the statistical properties of measurement noise. The proposed construction method has no structural constraint on the representation. When the size of the measured dataset is large, the proposed approximate representation converges in probability to one that represents the true behavior of the system. This allows data-driven simulation and control to be performed using simple convex quadratic programming algorithms. Furthermore, a Kalman filter-like algorithm is developed for better prediction of future output. A numerical example is presented to illustrate the proposed method and its efficacy under high measurement noise levels.