Probabilistic Behavioral Distance and Tuning - Reducing and aggregating complex systems

Abstract

Given a complex system with a given interface to the rest of the world, what does it mean for a the system to behave close to a simpler specification describing the behavior at the interface? We give several definitions for useful notions of distances between a complex system and a specification by combining a behavioral and probabilistic perspective. These distances can be used to tune a complex system to a specification. We show that our approach can successfully tune non-linear networked systems to behave like much smaller networks, allowing us to aggregate large sub-networks into one or two effective nodes. Finally, we discuss similarities and differences between our approach and H∞ model reduction.

An Open Source Software Stack for Tuning the Dynamical Behavior of Complex Power Systems

“…In this publication, we will only use ProBeTune to demonstrate the capabilities of our approach. Hence we refer the interested reader to the original publication [9], which contains the full mathematical background, and only give a short introduction here.…”

“…al. [9] introduced a method that is used to optimize the internal parameters of a system p such that its dynamics will behave close to a specification. For this purpose, a distance measure between the system and specification is introduced.…”

“…This so-called behavioral distance d is the expected distance between the system and the specification output. In [9] it is shown that d can be approximated efficiently by solving a joint optimization problem in p and several copies of the specification parametrized by q i . We refer the reader to [9] for details.…”

“…In [9] it is shown that d can be approximated efficiently by solving a joint optimization problem in p and several copies of the specification parametrized by q i . We refer the reader to [9] for details. The key idea here is that it is not necessary to tune the system towards a particular parametrization of the specification but rather to achieve a certain dynamic behavior.…”

“…The resulting differential equation can be solved with the state-of-the-art solver algorithms available in the DifferentialEquations.jl ecosystem. Finally, we demonstrate the machine learning and automatic differentiation capabilities of the SciML ecosystem, by tuning the parameters of the system such that its dynamics match those of a given specification, according to the recently proposed probabilistic tuning framework [9] 2 .…”