Fading-memory feedback systems and robust stability

This paper considers fading memory for nonlinear time-varying systems and associated problems of robust stability. We define two notions of fading memory for stable dynamical systems: uniform and pointwise. We then provide conditions under which stable linear or nonlinear systems exhibit uniform or pointwise fading memory. In particular, we show that (1) all stable discrete-time linear time-varying (LTV) systems have uniform fading-memory, (2) all stable continuous-time LTV systems have pointwise fading-memory, and (3) stable finite-dimensional continuous-time LTV systems have uniform fading-memory. We then show that a version of the small gain theorem which employs the asymptotic gain of a fading-memory system is necessary for the stable invertibility of certain feedback operators. These results are presented for both continuous-time and discrete-time systems using general ℓ^p or L^p notions of input/output stability and generalize existing results for ℓ² stability. We further investigate fading memory in a closed-loop context. For linear plants, we parametrize all nonlinear controllers which lead to closed-loop pointwise fading memory.