L2-stability of distributed feedback systems: Singular perturbation

Abstract

A continuous time, single input-single output, linear, time-invariant, distributed feedback system F sup epsilon, containing a small delay of length epsilon in the loop, is considered. Conditions are given under which L2-stability and L2-instability of this feedback system can be deduced from those of the reduced model obtained by neglecting the delay. The two system models associated with F sup epsilon are the low-frequency model F and the high frequency model F. The condition for neglecting the small delay is the L2-stability of the family of high-frequency models, where epsilon or = 0 is sufficiently small. A lemma and a theorem are given. The lemma gives sharp Nyquist-type conditions for the L2-stability and L2-instability of the family of high frequency models for sufficiently small epsilon or = 0, while the Theorem gives explicit conditions under which the small delay may or may not be neglected.