Gain-Scheduled Control Synthesis Using Dynamic $D$-Scales

For systems that are affected by a priori unknown but on-line measurable dynamic components (such as parameters, nonlinearities or delays), we propose a novel design algorithm for controllers that are on-line scheduled by these so-called structured uncertainties in order to achieve closed-loop stability and a certain desired level of performance. Both the plant and the controller are assumed to admit a description in terms of a linear time-invariant system in feedback with the uncertainties as is standard in robust control. In contrast to the existing results in the literature, dynamic (i.e., frequency-dependent)
$D$
-scales are used to guarantee robust stability (and performance) of the closed-loop system in the form of frequency-dependent inequalities. Based on these well-known analysis results, it is shown in this paper how to completely reduce the synthesis of gain-scheduled controllers with dynamic
$D$
-scalings to a system of linear matrix inequalities. We sketch various potential applications of our main result and illustrate the advantages of frequency dependence in the
$D$
-scales by a simple numerical example.