A data-driven approach to system invertibility and input reconstruction

We consider the problems of system invertibility and input reconstruction for linear time-invariant (LTI) systems using only measured data. The two problems are connected in the sense that input reconstruction is possible provided that the system is left invertible. To verify the latter property without model knowledge, we leverage behavioral systems theory and develop two data-driven algorithms: one based on input/state/output data and the other based only on input/output data. We then consider the problem of input reconstruction for both noise-free and noisy data settings. In the case of noisy data, a statistical approach is leveraged to formulate the problem as a maximum likelihood estimation (MLE) problem. The proposed approaches are finally illustrated with numerical examples that show: exact input reconstruction in the noise-free setting; and the better performance of the MLE-based approach compared to the standard least-norm solution.