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Averaging with Exogenous Inputs and Electrical Networks

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Introduction to Averaging Dynamics over Networks

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 472))

Abstract

The dynamical models analyzed so far, with the exception of the noisy consensus models treated in Chap. 4, are autonomous systems with no input signals: Information enters the system only through the initial condition. Instead, there are a variety of different situations where it is natural to consider consensus models driven by exogenous input signals, including opinion dynamics in the presence of stubborn agents that do not modify their opinion, rendezvous problems with leader robots, and estimation algorithms based on pairwise measurements. A very useful tool to analyze these models is thinking of the graph as an electrical circuit with the exogenous signals interpreted as input currents or as nodes kept at a fixed voltage. In this chapter, we will first review the basic theory of electrical networks and their classical connection with reversible stochastic matrices: Sect. 5.1 concentrates on Green matrices and harmonic functions, while Sect. 5.2 is devoted to effective resistances. Afterward, we apply these tools to averaging dynamics with stubborn agents in Sect. 5.3 and to the problem of estimation from relative measurements in Sect. 5.4.

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Turin, Italy

    Fabio Fagnani

  2. GIPSA-lab, CNRS, Grenoble, France

    Paolo Frasca

Authors
  1. Fabio Fagnani
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  2. Paolo Frasca
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Correspondence to Paolo Frasca .

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Fagnani, F., Frasca, P. (2018). Averaging with Exogenous Inputs and Electrical Networks. In: Introduction to Averaging Dynamics over Networks. Lecture Notes in Control and Information Sciences, vol 472. Springer, Cham. https://doi.org/10.1007/978-3-319-68022-4_5

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  • DOI: https://doi.org/10.1007/978-3-319-68022-4_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68021-7

  • Online ISBN: 978-3-319-68022-4

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