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System theory without transfer functions and state-space? Yes, it’s possible!

The paper demonstrates the claim in the title using missing data estimation as a generic example. The missing data estimation problem includes simulation, Kalman smoothing, and linear quadratic control as special cases. The solution method proposed uses an idea from subspace identification: under a persistency of excitation condition, the image of a Hankel matrix constructed from the data is equal to the behavior of the data-generating system. This fact allows construction of trajectories of the system directly from observed raw data. The construction of trajectories is the key for solving analysis, signal processing, and control problems without parametric model identification. The resulting methods require solution of systems of linear equations, however, the data is assumed exact and obtained from a linear time-invariant system.