Riemann Sphere Interpretation of the Passivity, Small Gain, and Conic Sector Theorems

One way to visualize the frequency response of a single-input single-output transfer function is to use the Nyquist plot. This plot gives insight into system properties, such as gain and phase properties, that are not immediately evident when looking at the transfer function itself. The Nyquist plot can also be used to assess closed-loop stability via the Nyquist stability criterion. There are well-known Nyquist interpretations of passive, finite gain, and conic system properties, as well as Nyquist interpretations of the passivity, small gain, and conic sector theorems. This paper provides Riemann sphere interpretations of passive, finite gain, and conic system properties, as well as Riemann sphere interpretations of the passivity, small gain, and conic sector theorems. These Riemann sphere interpretations provide control systems engineers with yet another tool to glean insight into plant and controller properties, closed-loop stability, performance, and robustness.