On positive para-odd and complex discrete reactance functions

Abstract

Positive para-odd functions have been extensively used in the stability analysis of continuous-time systems of differential equations. Their discrete-time counterparts, called complex discrete reactance functions are much less famous. In the present work, some of the major results on stability of continuous-time systems are recalled. They are presented in terms of special partial fraction and continued fraction expansions of a positive para-odd function associated with the system. By means of suitable conformal mappings, these results are converted to discrete-time systems of difference equations. We establish a characterization of complex discrete reactance functions in terms of Schur polynomials, and we develop a continued fraction expansion for complex discrete reactance functions associated with systems which are Schur stable.

Keywords:

• Positive para-odd functions
• Complex discrete reactance functions
• Conformal mappings
• Hurwitz polynomials
• Schur polynomials
• Stability analysis

2010 AMS Subject Classification:

• Primary 37C75
• 93C15. Secondary 93D
• 30E