Principal gains and principal phases in the analysis of linear multivariable feedback systems

Multivariable control using the singular value decomposition in steel rolling with quantitative robustness assessment

“…Robustness measures based on singular values are widely reported in the literature, see (Doyle, 1979 ;Cruz et a/., 1981;Postlethwaite et a/., 1981). The analysis shown here follows the general spirit of these approaches.…”

Advanced Attitude and Position MIMO Robust Control Strategies for Telescope-Type Spacecraft with Large Flexible Appendages

“…However, the extension of the notion of phase, as understood in scalar systems, is not so straightforward. Several attempts have been made to define a multivariable phase, such as (Freudenberg & Looze, 1988;Hung & MacFarlane, 1982;MacFarlane & Hung, 1981;Postlethwaite et al, 1981). On the other hand, as (Wall et al, 1980) showed, transmission zeros contribute with extra phase lag in some directions, but not in others.…”

“…The theorem is applied to systems with unstructured uncertainty. When the phases of perturbations, rather than their gains, can be bounded, the small-phase theorem (Postlethwaite et al, 1981) can be used. However, the main drawback of this approach is the highly conservative results it may provide.…”

Robust Gain-Scheduled Nonlinear Control Design for Stability and Performance

“…Several authors have proposed such concepts. In [PEM81], the "principal phases" of a matrix are defined as the arguments of the eigenvalues of the unitary part of its polar decomposition, and a "small phase theorem" is derived that holds under rather stringent conditions. Hung and MacFarlane, in [HM82], propose a "quasi-Nyquist decomposition" in which the phase information of a transfer matrix is obtained by minimizing a measure of misalignment between the input and output singular vectors.…”

“…Median phase and phase spread are related to the concept of principal phases introduced by Postlethwaite et al in [PEM81]. Namely, for any square complex matrix T, MP(r) -PS(r) < ^m in (r) < Vwx(r) < MP(r) + ps(r)…”