A Simplified Approach to Bode's Theorem for Continuous-Time and Discrete-Time Systems

Bing-Fei Wu, Edmond A. Jonckheere

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47 Scopus citations

Abstract

This note presents a simplified approach to Bode’s theorem for both continuous-time and discrete-time systems, along with some generalization. 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 in this note. The salient feature of this approach is that at no stage do we invoke either Cauchy’s theorem or the Poisson integral; the simplified proof relies only on elementary analysis. This approach carries over to the discrete-time case in a straightforward manner.

Original languageEnglish
Pages (from-to)1797-1802
Number of pages6
JournalIEEE Transactions on Automatic Control
Volume37
Issue number11
DOIs
StatePublished - 1 Jan 1992

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