Observability of Network Systems: A Critical Review of Recent Results
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Observability of Network Systems: A Critical Review of Recent Results Arthur N. Montanari1
· Luis A. Aguirre2
Received: 13 December 2019 / Revised: 28 July 2020 / Accepted: 4 August 2020 © Brazilian Society for Automatics–SBA 2020
Abstract Observability is a property of a dynamical system that defines whether or not it is possible to reconstruct the trajectory temporal evolution of the internal states of a system from a given set of outputs (measurements). In the context of network systems, two important goals are: (i) to determine if a given set of sensor nodes is sufficient to render the network observable; and (ii) what is the best set of sensor nodes among different available combinations that provide a more accurate state estimation of the network state. Alongside Kalman’s classical definition of observability, a graph-theoretical approach to determine the observability of a network system has gathered a lot of attention in the literature despite several following works showing that, under certain circumstances, this kind of approach might underestimate, for practical purposes, the required number of sensor nodes. In this work, we review with a critical mindset the literature of observability of dynamical systems, counterpoising the pros and cons of different approaches in the context of network systems. Some future research directions for this field are discussed and application examples in power grids and multi-agent systems are shown to illustrate our main conclusions. Keywords Observability · Dynamical systems · Network systems · Sensor placement
1 Introduction The mathematical modeling of dynamical systems is a fundamental framework in engineering that provides a means to analyze aspects of a system, such as its stability, controllability or observability, and thereafter design control laws for practical applications (Chen 1999; Khalil 2002). However, being designed for the most part with systems of low-dimensional order in mind, classic control theory methods are not efficient, or even feasible, for large-scale systems, such as interconnected (networked) dynamical systems. This practical limitation has led control theory notions to be adapted, optimized, or even redefined, in the literature for high-dimensional applications (Chen 2014). A specific, but recurrent, type of high-dimensional system can be defined as networks. A network is a set of nodes inter-
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Arthur N. Montanari [email protected] Luis A. Aguirre [email protected]
1
Graduate Program in Electrical Engineering of the Universidade Federal de Minas Gerais (UFMG), Av. Antônio Carlos 6627, 31270-901 Belo Horizonte, Minas Gerais, Brazil
2
Depto. de Engenharia Eletrônica, UFMG, Belo Horizonte, Brazil
connected by edges, in which information flows among its elements through pairwise interactions. It can be mathematically modeled by graph structures, which allow a wide range of useful metrics and algorithms of graph theory (Newman 2010; Chen et al. 2013; Bullo 2016). For instance, graph theory can be used to assess the robustness to
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