Mathematical Problems in Engineering
Volume 3 (1998), Issue 5, Pages 413-477
The -transform: A tool for analysis of control systems robustness with respect to disturbance model uncertainty
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor 48109, MI, USA
Received 19 February 1997
Copyright © 1998 D. E. Davison et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This work is motivated by the need forr tools for the analysis of disturbance model uncertainty in feedback control systems. Such tools are developed in this paper for the case where the disturbance is modeled as the output of a first-order filter which is driven by white noise and whose bandwidth, , and gain, , are uncertain. An analytical expression for the steady-state output variance as a function of
is derived: This function is referred to as a -transform, and is denoted by
, where is the closed-loop transfer function from disturbance to output. Properties of -transforms are investigated and the notions of disturbance gain margin and disturbance bandwidth margin, both measures of robustness with respect to disturbance model uncertainty, are introduced. Using these new tools, it is shown that there is a fundamental robustness performance limitation if the plant has nonminimum-phase zeros, but no such limitation in the minimum-phase case.