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

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.


I. 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.This constraintusually referred to as side condition -has been a source of confusion and controversy in the literature for more than a decade.As pointed out in [3], the side conditions used in [6], [7], [8] were incorrect as they had some inconsistencies.To fix the problem, [3] proposed a new condition at infinite frequency, i.e.
where ρ is the dimension of ker F (∞) + F (∞) .On the other hand, a different, but equally valid, condition at infinite frequency was proposed the second edition of the book by Khalil published in 1996 (see [5,Lemma 10.1]); such a condition, that reads as follows, was recently used in [4] to establish a counterpart result for negative imaginary systems (see [4,Remark 1] This note is devoted to the analysis of the two sideconditions (1) and (2).We will prove that while they are in general not equivalent at infinite frequency, they are indeed equivalent under the other conditions guaranteeing that F is strictly positive real.Hence both conditions at infinite frequency are equally valid.While this could be deduced from [3] and [5], our results provide a direct proof of such equivalency.Notation: Let the set of real (resp.complex) numbers be denoted by R (resp.C) and the corresponding sets of matrices bernard.brogliato@inria.fr of dimension m×n be denoted by R m×n (resp.C m×n ).Given M ∈ C m×m , M denotes the complex conjugate transpose of matrix M (i.e., if M = A + jB for real matrices A and B, then M = A − jB ).A matrix M is said to be Hermitian if M = M and M > 0 denotes that the matrix M is Hermitian and positive definite.The smallest singular value of M is denoted by σ(M ).We recall that the singular values of a positive semi-definite Hermitian matrix are its nonzero eigenvalues [2, p.649].

II. MAIN RESULT
The following definition, adapted from [4, Definitions 1 and 2], is the standard definition for strictly positive real systems.It essentially states that a transfer function matrix F (s) is strictly positive real if for some > 0, the transfer function matix F (s − ) is positive real and F (s) + F (−s) has full normal rank.See also [4,Lemma 2] for an equivalent re-characterization.
Definition 2.1: Let F : C −→ C m×m be a real transfer function.Then F (s) is said to be Strictly Positive Real (SPR) if there exists a real scalar > 0 such that F (s) is analytic in {s ∈ C : Re{s} > − }, F (s) + F (s) ≥ 0 for all s ∈ {s ∈ C : Re{s} > − } and F (s) + F (−s) has full normal rank.SPR matrix transfer functions can be characterized in the frequency domain by three conditions: the first two are conditions 1 and 2 in the next 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 [5], [4], while side condition (3b) can be found in [3].These side conditions can be interpreted as apparently different conditions on how approaches zero for sufficiently large |ω| in directions where it looses rank.
Then, the following two side conditions are equivalent: where for some state-space realization (A, B, C, D).Near s = ∞, the expansion holds (with with Then it is easy to see that If Q is positive definite, in view of the Hermitian symmetry of both H and K(ω), both conditions 3a) and 3b) are clearly satisfied and the result is obvious.
Assume than that Q is singular with rank m − ρ.Since Q is symmetric, there exists an orthogonal matrix U such that Hence, without changing the essence of the problem, we assume that Q has the form with Q 1 ∈ R (m−ρ)×(m−ρ) nonsingular, and hence Q 1 > 0.
Then Φ(ω) has m strictly positive eigenvalues for all ω ∈ R, ρ of which are going down to zero at least as fast as 1/ω 2 as ω → ∞.The remaining (m − ρ) eigenvalues tend to the strictly positive eigenvalues of Q 1 as ω → ∞.Now, order the eigenvalues of Φ(ω) in non-decreasing size.
The pathological case, corresponding to the situation in which some of the eigenvalues of the spectrum go to zero faster than 1 ω 2 , as ω tends to infinity, is more interesting: let (1 + ω 2 ) 3 = 0.