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The \(V\)-transform: A tool for analysis of control systems robustness with respect to disturbance model uncertainty. (English) Zbl 0892.93024

The robustness of control system performance with respect to plant model uncertainty has been studied in the control community for decades. In contrast, robustness of performance with respect to disturbance model uncertainty has received far less attention. Typically, it is assumed that the disturbance belongs to a given class of signals, and a controller is designed to accommodate in a desirable manner disturbances generated by such a model. The question of what happens if the actual disturbance violates the assumption is, typically, ignored. Consequently, if the disturbance does indeed violate the assumption, unsatisfactory performance may result. An example of such a case is a recent incident on January 7, 1997, in which a jetliner cruising over the Atlantic Ocean encountered unexpected turbulence, resulting in six injured persons and minor aircraft damage. The authors’s interpretation of the root cause of this incident is that the encountered turbulence was of a different nature than the turbulence model used in the autopilot design and evaluation. The main goal of this paper is to introduce measures of performance robustness with respect to disturbance model uncertainty and to use these measures for control systems analysis. “Performance” here refers to the disturbance rejection ability of a controller. The particular issues addressed include the following: How does disturbance rejection performance vary with changes in the disturbance model? How can disturbance rejection robustness be quantified? Are there fundamental limits on disturbance rejection robustness? The tools for the analysis of disturbance model uncertainty in feedback control systems are developed 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, \(\omega_d\), and gain, \(K\), are uncertain. An analytical expression for the steady-state output variance as a function of \(\omega_d\), is derived: This function is referred to as a \(V\)-transform. Properties of \(V\)-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.

MSC:

93B35 Sensitivity (robustness)
93B51 Design techniques (robust design, computer-aided design, etc.)
93C99 Model systems in control theory
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