Lp-Stability (1 ≤p≤ ∞) of multivariable non-linear time-varying feedback systems that are open-loop unstable†

Publisher Website
Google Scholar
Abstract
This paper considers a class of multivariable, non-linear time-varying feedback systems with an unstable convolution subsystem as feedforward and a time-varying non-linear gain as feedback. The impulse response of the convolution subsystem is the sum of (i) a finite number of increasing exponentials multiplied by non-negative powers of the time t, (ii) a term that is absolutely integrable and (iii) an infinite series of delayed impulses. The main result of the paper is theorem 1. It essentially states that (i) if the unstable convolution subsystem can be stabilized by a constant feedback gain F and (ii) if the incremental gain of the difference between the nonlinear gain function and F is sufficiently small, then the non-linear system is Lp instable for any p∊[l, ∞]; furthermore, the solutions of the non-linear system depend continuously on the inputs in any Lp -norm. The fixed point theorem is crucial in deriving the above theorem.