Minimizing the memory of a system

Consider a stable linear time-invariant system G(x) with tunable parameters x ϵ R
ⁿ
, which maps inputs w ϵ L
²
(R
^m
) to outputs z ϵ L
²
(R
^p
). Our goal is to find a choice of the tunable parameters x
^*
which avoids undesirable responses of the system to past excitations known as system ringing. We address this problem by minimizing the Hankel norm ∥G(x)∥
_H
of G(x), which quantifies the influence of past inputs on future outputs.