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Proportional-Retarded (PR) Protocol for a Large Scale Multi-agent Network with Noisy Measurements; Stability and Performance

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Book cover Delays and Interconnections: Methodology, Algorithms and Applications

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 10))

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Abstract

A distributed Proportional-Retarded (PR) protocol is studied for leaderless coordination of a large scale multi-agent network with undirected links and noisy measurements. The protocol, already developed for SISO systems, introduces a time delay to mimic a derivative action allowing improvement of agents’ transient responses but being insensitive to high-frequency signals. The main contribution of this chapter is the stability analysis of the closed-loop network system of arbitrarily large scale, in terms of PR protocol parameters. Next, this is connected to our recent results in [28] from which we summarize how PR protocol can be tuned for the network system to achieve certain performance as dictated by its rightmost eigenvalues. Over a case study, the results are demonstrated.

This work was supported in part by the US National Science Foundation Award 1536397.

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Notes

  1. 1.

    It is worthy of mention that the radius of convergence of the series is \(e^{-1}\). For practical computation, the reader is referred to [31] where additional asymptotic formulae can be found considering all the branches of the Lambert W function.

  2. 2.

    The scalar Lambert W function is available as embedded function in MATLAB, see the function lambertw.

  3. 3.

    Note that Proposition 6 uses \(\lambda _{\min }\) to guarantee the placement of \(\gamma ^*\) at a desired location \(\gamma _d\) through a stabilizing pair \((k_p,k_r)\). Since \(\gamma _d<0\) is a necessary and sufficient condition for the stability of the consensus network, it can be conjectured that the stabilizing pair \((k_p,k_r)\) must lie within the stability domain associated with \(\lambda _{\max }\). Therefore, one may consider Proposition 6 as a link between two important Laplacian eigenvalues, namely, \(\lambda _{\min }\) and \(\lambda _{\max }\).

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA, 02115, USA

    Adrián Ramírez & Rifat Sipahi

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  1. Adrián Ramírez
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  2. Rifat Sipahi
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Correspondence to Rifat Sipahi .

Editor information

Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Giorgio Valmorbida

  2. Laboratoire d’Analyse et d’Architecture de Systèmes-CNRS, Toulouse, France

    Alexandre Seuret

  3. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Islam Boussaada

  4. Mechanical and Industrial Engineering, Northeastern University, Boston, MA, USA

    Rifat Sipahi

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Ramírez, A., Sipahi, R. (2019). Proportional-Retarded (PR) Protocol for a Large Scale Multi-agent Network with Noisy Measurements; Stability and Performance. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_16

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