On 'A Tutorial Comparison of Three Multivariable Stability Margins'

Abstract

A recent paper by A. Wilson and C.D. Johnson [1] compares the three multivariable stability margins developed by R.W. Bass, J.C. Doyle and M.K. Safonov, commonly represented by p, /spl mu/ and K/sub m/, respectively. The general discussion is regarded by three present author as correct, fair, and illuminating. However, the tutorial example, from which some 'conclusions' were drawn, consisted of a comparison of the allowable parameter regions for perturbed or uncertain parameters derived by the three approaches; unfortunately, this comparison utilized the structured versions of /spl mu/ and K/sub m/ and varied the natural frequency and damping ratio of an optimally tuned second-order oscillator to demonstrate that the unstructured (or 'norm-bounded') version of p gives results about 40 db worse than obtainable by either of the other two margins. This paper sets the record straight by pointing out that by use of a structured version of p one obtains parameter regions roughly a third the size of those obtainable with /spl mu/ and about a fourth the size of those obtainable with K/sub m/ (which are the exact linear stability boundaries). Also there is a reason why the /spl mu/-domain is smaller than the K/sub m/-domain (because it allows unmodeled linear dynamics and transport lag), and there is an acceptable reason why the p-domain is smaller than the /spl mu/-domain (because it allows arbitrarily time-varying coefficients and arbitrary [Lipschitzian] nonlinearities as well as true time-delays, and it guarantees invariance of the overshoot factor in the perturbed system). Each type of analysis discloses both overlapping and non-overlapping information, and so a thorough analysis benefits from the information provided by all three tools.