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In this paper, we consider the problem of quantifying controllability andobservability of a nonlinear discrete time dynamical system. We introduce theKoopman operator as a canonical representation of the system and apply alifting technique to compute gramians in the space of full-state observables.We illustrate the properties of these gramians and identify severalrelationships with canonical results on local controllability andobservability. Once defined, we show that these gramians can be balancedthrough a change of coordinates on the observables space, which in turn allowsfor direct application of balanced truncation. Throughout the paper, wehighlight the aspects of our approach with an example nonlinear system.

A Koopman Operator Approach for Computing and Balancing Gramians for Discrete Time Nonlinear Systems