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 H1 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 state-space formulae. This work was supported in part by the National Science Foundation, under contract ECS-9309387 1 Time-varying entropy G K z y w u oe oe oe - Figure 1: Closed-loop system. 1 Introduction Since the pioneering work of Zames [25], there has been much interest in finding stabilizing controllers which ensure that the H1 norm of a closed-loop transfer function is belo...