Sufficient Conditions on Pole and Zero Locations for Rational Positive-Real Functions

Abstract

The problem of finding sufficient conditions on the pole and zero locations to insure that a rational function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(s)</tex> is positive-real has been an outstanding one in network theory. Several solutions to this problem are presented in this paper. In particular, assuming that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(s)</tex> has <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> poles and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> zeros, certain regions in the left-half <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> plane are constructed which have the following property: If these poles and zeros are placed in one of these regions in any arbitrary manner (with the restriction, of course, that complex elements appear in complex-conjugate pairs), the resulting W(s) will be positive-real. These results are then extended to the case where the number of poles and the number of zeros differ by one. In addition certain paths in these regions are derived which allow one to place any number of poles and zeros into any of these regions. That is, if the poles and zeros alternate in groups of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> elements on any such path, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W(s)</tex> will again be positive-real. The simple alternation of poles and zeros on the real negative axis and on a vertical line or circle in the closed left-half s plane, which is a known result, is a special case of these considerably more general conclusions.