A Direct Proof of the Equivalence of Side Conditions for Strictly Positive Real Matrix Transfer Functions

Augusto Ferrante,Bernard Brogliato,Alexander Lanzon

doi:10.1109/tac.2019.2918123

Abstract

This brief note proves in a direct way that two different side conditions, which have been used in the literature to characterize strictly positive real matrix transfer functions in the frequency domain, are equivalent.

Highlights

The frequency domain conditions characterizing the fact that a matrix transfer function F is strictly positive real involve a positivity constraint at infinite frequency
A different, but valid, condition at infinite frequency was proposed in the second edition of the book by Khalil published in 1996; such a condition, that reads as follows, was recently used in [4] to establish a counterpart result for negative imaginary systems: ∃ δ > 0, σ0 > 0 such that σ ω2 F + F (−jω) ≥ σ0 ∀|ω| ≥ δ
While this could be deduced from [3] and [6], our results provide a direct proof of such equivalency

Summary

INTRODUCTION

The frequency domain conditions characterizing the fact that a matrix transfer function F is strictly positive real involve a positivity constraint at infinite frequency. SPR matrix transfer functions can be characterized in the frequency domain by three conditions: the first two conditions are 1 and 2 in the proposition, the third is the side condition and it has been stated in two different manners: side condition (3a) in Proposition 2.1 can be found in [4] and [6], while side condition (3b) can be found in [3] These side conditions can be interpreted as apparently different conditions on how F (jω) + F (−jω) approaches zero for sufficiently large |ω| in directions where it looses rank.

Now let

Using the fact that