An entropy formula for time-varying discrete-time control systems

The results of this paper generalize the formula for the entropy of a transfer function to time-varying systems. This is done through the use of some results on spectral factorizations due to Arveson and properties of the W-transform which generalizes the usual Z-transform for time-varying systems. Using the formula defined, it is shown that for linear fractional transformations like those that arise in time-varying H-infinity control, there exists a unique, bounded contraction which maximizes the entropy. This generalizes known results in the time-invariant case. Possible extensions are discussed, along with stare-space formulae.