Nyquist plots, finite gain/phase margins & dissipativity

Abstract

The relation between the small gain theorem and ‘infinite phase margin’ is classical; in this paper we formulate a novel supply rate, called the ‘not-out-of-phase’ supply rate, to first prove that ‘infinite gain margin’ (i.e. non-intersection of the Nyquist plot of a transfer function and the negative half of the real axis) is equivalent to dissipativity with respect to this supply rate. Capturing non-intersection of half-line makes the supply rate system-dependent: a novel feature unobserved in the supply rates considered in the literature so far.We then show that the traditional finite and positive gain/phase margin conditions for stability are equivalent to dissipativity with respect to a frequency weighted convex combination of the not-out-of-phase supply rate and the small-gain supply rate; both frequency weightings and combining two supply-rates/performance-indices have been investigated in the literature in different contexts, but only as sufficient conditions.