The role of non-minimum phase zeros in stability of infinite dimensional systems

Abstract

In this article, the authors examine a class of transfer functions (the Nevanlinna class) and present a well known theorem about the location of the non-minimum phase zeros for the Nevanlinna class. Next, they relate the Nevanlinna class to the class of stabilizable transfer functions and find two necessary conditions for a transfer function to be stabilizable. One of the necessary conditions is a constraint on the non-minimum phase zeros of a plant and the other is a constraint on the unstable poles of the plant. The necessary condition is automatically satisfied by every finite dimensional plant, so our result is only interesting for infinite dimensional plants. Finally, the authors construct a transfer function that does not satisfy our condition and is not stabilizable. Upon examination of our example, they see a counterintuitive result: the transfer function does not have any unstable poles and the only reason it is unstable (and not stabilizable) is because of the non-minimum phase zeros.