Robust stabilization with positive real uncertainty: beyond the small gain theorem

The properties of positive real transfer functions are used to guarantee robust stability in the presence of positive real (but otherwise unknown) plant uncertainty. These results are then used for controller synthesis to address the problem of robust stabilization in the presence of positive real uncertainty. One of the principal motivations for these results is to utilize phase information in guaranteeing robust stability. In this sense these results go beyond the usual limitations of the small gain theorem and quadratic Lyapunov functions, which may be conservative when phase information is available. The results of this study are based upon a Riccati equation formulation of the positive real lemma and thus resemble certain Riccati-based approaches to bounded real (H_∞) control