Stability analysis Nyquist criterion

IV.74 Examine the stability of the closed-loop systems whose open-loop transfer functions are given in Problem IV.58. Employ the Nyquist criterion for your analysis. 

If approximation by the transfer function in the form of Eq. (13.17) is possible, then poles of the transfer function can be analyzed because the stable transfer function cannot have positive poles. When such an analysis is not possible, one can use the Nyquist stability criterion. Only validated data can be used for analysis. 

The Nyquist stability criterion is based on two concepts from complex variable theory, contour mapping and the Principle of the Argument, We briefly review these concepts in Appendix K. More detailed descriptions are available elsewhere (Brown and Churchill, 2008 Franklin et al., 2002). We now present one of the most important results of frequency domain analysis ... 

At this point, it is appropriate to summarize the relative advantages and disadvantages of the Bode and Nyquist plots. The Bode plot provides more information than the Nyquist plot, because the frequency is shown explicitly. In addition, it facihtates analysis over a wide range of frequencies due to its logarithmic frequency scale. Another advantage of the Bode plot is that it allows the open-loop frequency response characteristic to be graphically constructed from the characteristics for the individual transfer functions, G, Gy, Gp, and G, as shown in Chapter 14. The chief advantage of the Nyquist plot is that the Nyquist stability criterion is more widely applicable than the Bode stability criterion. 

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