Subharmonic Functions and Performance Bounds on Linear Time-Invariant Feedback Systems

In this paper we study multiple-input multiple-output (MEMO) linear time-invariant (LTI) control systems. We show that some well known constraints on the performance of single-input single-output (SISO) linear control systems, e.g. those expressed by the Paley-Wiener theorem, Bode's integral theorem, and more recently, Zames' inequality can be given a unified treatment using some elementary properties of subharmonic functions. Most importantly, results derived in this framework of subharmonic functions apply immediately to the MIMO case. Indeed the proofs of the MIMO generalizations axe often simpler than the original proofs of the SISO versions.