Dissipativity analysis of SISO systems using Nyquist-Plot-Compatible (NPC) supply rates

In this paper we deal with a special class of quadratic supply rates, dissipativity with respect to which can be directly read off from a system´s Nyquist plot. These supply rates are called Nyquist-plot-compatible (NPC) supply rates [8]. The characterizing property of these supply rates is that to each of them a specific region in the complex plane can be associated: dissipativity w.r.t. these supply rates is equivalent to systems´ Nyquist plots being contained in the region. The classical results of small gain and passivity theorems are special cases of dissipativity w.r.t. NPC supply rates. We show in this paper, that apart from the aforementioned two special cases, there are many more such NPC supply rates. In particular, we construct supply rates for regions in the complex plane given by the right-half (or, left-half) of a vertical line, interiors (or, exteriors) of circles of various radii with centers on the real-axis. We then show that a system´s Nyquist plot being contained in the union of two regions is equivalent to dissipativity w.r.t. a frequency weighted combination of the corresponding two NPC supply rates. We finally give an algorithm for finding out these weighting polynomial functions.