On the Study of Nyquist Contour Handling Sampled-Data Control System Real Poles and Zeros

The Nyquist plot is a crucial tool in the analysis and design of linear time-invariant (LTI) and linear shift-invariant (LSI) control systems such as in the relative stability analysis, gain margin, phase margin and robust stability analysis. In the Nyquist plot plane, certain positions must be determined, such as the real axis crossings. These positions are sometimes hidden or ambiguous because of the large span in the magnitude of the Nyquist plot over the entire frequency range. As a result, the Nyquist sketch is introduced as a guide to manually draw the Nyquist plot regarding the qualitative graphical representations such as high frequency asymptote, low frequency asymptote (DC gain point), real and imaginary axis crossings. For LTI control systems or continuous-time control systems, the Nyquist sketch is demonstrated in several studies. However, for LSI control systems or discrete-time control systems, few references mention the Nyquist sketch and only in a vague manner. This study delineates extensively the sketch of the Nyquist plot or the Nyquist sketch for discrete-time control systems extending to the sampled-data control systems when the loop pulse transfer function possesses some real poles and/or zeros outside the unit circle in the z-plane.