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

Abstract

In many applications of feedback control, phase information is available concerning the plant uncertainty. For example, lightly damped flexible structures with colocated rate sensors and force actuators give rise to positive real transfer functions. Closed-loop stability is thus guaranteed by means of negative feedback with strictly positive real compensators. In this paper, 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 these sense results go beyond the usual limitations the small gain theorem and quadratic Lyapunov functions which may be conservative when phase information is available. The results of this paper are based upon a Riccati equation formulation of the positive real lemma and thus are in the spirit of recent Riccati-based approaches to bounded real (H ∞) control.