Geometric Links Among Classical Controls Tools

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Institute of Electrical and Electronics Engineers (IEEE)

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IEEE Transactions on Education

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DOI: 10.1109/13.275186


This paper develops a geometric perspective that ties together a number of graphically based techniques from classical control theory. In particular, in the frequency domain, a connection between the Nyquist diagram and the Bode plots is unfolded via a sequence of three-dimensional representations. A parallel development in the "gain-domain" begins with the Evans root locus plot and leads to a set of gain plots that portray eigenvalue behavior as an explicit function of forward gain. The gain plots extend the standard root locus plot by depicting explicitly the influence of gain (or any system parameter) on the closed-loop system eigenvalues. This is similar to the way the Bode plots embellish the information of the Nyquist diagram by exposing frequency explicitly. The gain plots enable direct determination of gain values for which the closed-loop system is stable or unstable. By exposing the correspondence of gain values to specific eigenvalues, the plots serve as a pole-placement tool for identifying closed-loop designs meeting performance specifications. Furthermore, the gain plots reveal by inspection information about the closed-loop root sensitivity. The authors have found the gain plots as well as the underlying geometric development in both the frequency and gain domains invaluable in undergraduate and graduate controls education.


IEEE Transactions on Education, Vol. 37, No. 1 (February 1994): 77-83. DOI.

Mark Nagurka was affiliated with Carnegie Mellon University at the time of publication.