Controller synthesis with guaranteed closed-loop phase constraints

In this paper, we present an analysis and synthesis framework for guaranteeing that the phase of a single-input, single-output closed-loop transfer function is contained in the interval [-alpha, alpha] for a given alpha>0 at all frequencies. Specifically, we first derive a sufficient condition involving a frequency domain inequality for guaranteeing a given phase constraint. Next, we use the Kalman-Yakubovich-Popov theorem to derive an equivalent time domain condition. In the case where alpha=pi/2, we show that frequency and time domain sufficient conditions specialize to the positivity theorem. Furthermore, using linear matrix inequalities, we develop a controller synthesis framework for guaranteeing a phase constraint on the closed-loop transfer function. Finally, we extend this synthesis framework to address mixed gain and phase constraints on the closed-loop transfer function.