Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties.
Bibliographical noteFunding Information:
Manuscript received March 05, 2009; revised June 17, 2009. First published September 22, 2009; current version published October 07, 2009. This work was supported in part under a NASA Langley NRA NNH077ZEA001N entitled “Analytical Validation Tools for Safety Critical Systems” and by the Air Force Office of Scientific Research, USAF, under Grant FA9550-05-1-0266. Recommended by Associate Editor H. Ito.
- Parameter-dependent Lyapunov function (PDLF)