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

S. Boyd and C. A. Desoer

IMA Journal of Mathematical Control and Information, Oxford University Press, 2:153-170, 1985. Conference paper appeared in Proceedings IEEE Conference on Decision and Control, pages 311-312, December 1984.

In this paper we study multi-input multi-output (MIMO) 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 junctions. Most importantly, results derived in this framework of subharmonic functions apply immediately to the MIMO case. Indeed the proofs of the MIMO generalizations are often simpler than the original proofs of the SISO versions.