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A simplified approach to Bode's theorem for continuous-time and discrete-time systems

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2 Author(s)
Wu, Bing-Fei ; Dept. of Electr. Eng.-Syst., Univ. of Southern California, Los Angeles, CA, USA ; Jonckheere, E.A.

A simplified approach to W.H. Bode's (1945) theorem for both continuous-time and discrete-time systems, along with some generalization, are presented. For continuous-time systems, the constraints of open-loop stability and roll-off at s=∝ are removed. A counterexample shows that when the excess poles/zeros vanishes, the Bode integral drops from infinite to finite value when the open-loop gain crosses a critical value. A revised result is also developed. The salient feature of this approach is that at no stage are either Cauchy's theorem or the Poisson integral invoked; the simplified proof relies only on elementary analysis. This approach carries over to the discrete-time cases in a straightforward manner

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Automatic Control, IEEE Transactions on  (Volume:37 ,  Issue: 11 )