Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle and Popov theorems and their application to robust stability

Abstract

Lyapunov function proofs of sufficient conditions for asymptotic stability are given for feedback interconnections of bounded real and positive real transfer functions. Two cases are considered: a proper bounded real (resp., positive real) transfer function with a bounded real (resp., positive real) time-varying memoryless nonlinearity; and two strictly proper bounded real (resp., positive real) transfer functions. A similar treatment is given for the circle and Popov theorems. Application of these results to robust stability with time-varying bounded real, positive real, and sector-bounded uncertainty is discussed. >