Phase information has largely been neglected in robust control theory but is essential for maximizing achievable performance in controlling complex multivariable systems. Phase information, here, refers to the characterization of the phase of the modeling uncertainty in the frequency domain. Real parameter uncertainty modeling in the time domain provides phase information in the frequency domain. The objective is to develop effective methods for utilizing phase information in the analysis and design of robust controllers. A proposal is made to study numerous interrelated ideas that are relevant to phase properties of dynamical systems in general and feedback control systems in particular. Although quadratic Lyapunov functions provide the foundation for much robust control theory, connections with small-gain (H-infinity) theory illustrate limitations of this approach in the presence of phase information. Extensions of quadratic Lyapunov functions, such as parameter-dependent Lyapunov functions, and structured Lyapunov functions are some of the ideas being proposed to overcome these limitations. Also proposed, is to establish a theoretical foundation of robust stabilization with positive real uncertainty. Recent Riccati-based robustness results for positive real uncertainty obtained by the author form the starting point for these investigations. Finally, it is proposed to generalize classical frequency-domain stability criteria, such as the Popov and circle criteria, to state-space controller synthesis using a Riccati equation approach. This development will provide the necessary tools for controller synthesis for sector-bounded nonlinearities. This research will provide a new approach to robust stabilization--one that involves phase information that can be significantly less conservative than an H-infinity small-gain approach.
