Generalizing Nyquist criteria via conformal contours for internal stability analysis

Abstract

By contriving the regularized return difference relationship in linear time-invariant (LTI) feedback systems, we attempt to generalize and validate the Nyquist approach for such internal stability as Lyapunov stability/instability, asymptotic stability, exponential stability and district stability (or -stability), respectively, even when there exist decoupling zeros, by means of what we call the regularized Nyquist loci that are plotted with respect to a Nyquist contour and its conformal one(s). More precisely, miscellaneous open-loop/closed-loop pole cancellations in the return difference relationship that may complicatedly tangle our stability interpretation but usually neglected in most existing Nyquist criteria are scrutinized. And then, Nyquist-like criteria for internal stability are claimed with the regularized Nyquist loci. These criteria get rid of pole cancellations testing and can be implemented completely independent of open-loop pole distribution knowledge; moreover, the Nyquist criteria for asymptotic/exponential stability are necessary and sufficient, while those for -stability are sufficient. Internal stability of a cart system with an inverted pendulum is examined to illustrate the results.