Structural, Dynamical and Symbolic Observability: From Dynamical Systems to Networks

TL;DR

This paper reviews various aspects of observability in dynamical systems and networks, clarifies concepts, and discusses methods to rank networks based on observability, supported by simulations.

Contribution

It provides a comprehensive review of observability concepts in nonlinear dynamics and networks, and proposes ways to rank networks by observability.

Findings

Different observability measures are compared and clarified.

Networks can be ranked based on their observability properties.

Simulations illustrate the application of these concepts.

Abstract

The concept of observability of linear systems initiated with Kalman in the mid 1950s. Roughly a decade later, the observability of nonlinear systems appeared. By such definitions a system is either observable or not. Continuous measures of observability for linear systems were proposed in the 1970s and two decades ago were adapted to deal with nonlinear dynamical systems. Related topics developed either independently or as a consequence of these. Observability has been recognized as an important feature to study complex networks, but as for dynamical systems in the beginning the focus has been on determining conditions for a network to be observable. In this relatively new field previous and new results on observability merge either producing new terminology or using terms, with well established meaning in other fields, to refer to new concepts. Motivated by the fact that twenty years have passed since some of these concepts were introduced in the field of nonlinear dynamics, in this paper (i)~various aspects of observability will be reviewed, and (ii)~it will be discussed in which ways networks could be ranked in terms of observability. The aim is to make a clear distinction between concepts and to understand what does each one contribute to the analysis and monitoring of networks. Some of the main ideas are illustrated with simulations.