The Shapes of Nyquist Plots Connections with Classical Plane Curves

The Nyquist criterion is a valuable design tool with applications to control systems and circuits [1], [2]. In this article, we show that many Nyquist plots are classical plane curves. Surprisingly, this connection seems to have gone unnoticed. We determine the precise shapes of several Nyquist curves and relate them to the shapes of the classical plane curves. Some classical plane curves are related to exactly proper or improper loop transfer functions, which do not roll off at high frequencies and thus are not physical. Classical plane curves are used for robustness analysis in [3]. In addition, the area enclosed by the Nyquist curve is related to the Hilbert-Schmidt-Hankel norm of a linear system [4]. Therefore, knowledge of the precise shape of the Nyquist curve can provide additional useful information about the properties of a system. The organization of this article is as follows. We first give a brief history of plane curves and then describe various plane curves. We then state some results that relate Nyquist plots to plane curves and present various illustrative examples. We end with some concluding remarks.