Parameter-dependent Lyapunov functions and the Popov criterion in robust analysis and synthesis

Many practical applications of robust feedback control involve constant real parameter uncertainty, whereas small gain or norm-bounding techniques guarantee robust stability against complex, frequency-dependent uncertainty, thus entailing undue conservatism. Since conventional Lyapunov bounding techniques guarantee stability with respect to time-varying perturbations, they possess a similar drawback. In this paper we develop a framework for parameter-dependent Lyapunov functions, a less conservative refinement of “fixed” Lyapunov functions. An immediate application of this framework is a reinterpretation of the classical Popov criterion as a parameter-dependent Lyapunov function. This result is then used for robust controller synthesis with full-order and reduced-order controllers