Minimizing the memory of a system

Consider a stable linear time-invariant system G(x) with tunable parameters x ϵ Rⁿ, which maps inputs w ϵ L²(R^m) to outputs z ϵ L²(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.