Finite gain and phase margins as dissipativity conditions

Abstract

It is well-known that the two situations: gain margin being infinite and phase margin being infinite are individually special cases of dissipativities with respect to suitable supply rates; the small gain theorem and passivity result being the respective theorems. However, in practice, the two margins being finite also allows concluding closed loop stability using the Nyquist stability criteria. We show in this paper that the finite and positive gain and phase margin condition is equivalent to dissipativity with respect to a convex combination of the supply rates arising from infinite gain margin and infinite phase margin supply rates. We formulate this situation into a more general framework of Nyquist Plot Compatible (NPC) supply rates and transformations that keep such supply rates invariant.