  The advent of easy access to large amount of data has sparked interest in
directly developing the relationships between input and output of dynamic
systems. A challenge is that in addition to the applied input and the measured
output, the dynamics can also depend on hidden states that are not directly
measured. The main contribution of this work is to identify the information
needed (in particular, the past history of the output) to remove the hidden
state dependence in Koopman-type inverse operators for linear systems.
Additionally, it is shown that the time history of the output should be
augmented with the instantaneous time derivatives of the output to achieve
precision of the inverse operator. This insight into the required output
(history and instantaneous derivative) information, to remove the hidden-state
dependence and improve the precision of data-enabled inverse operators, is
illustrated with an example system.
