Phase transitions in distributed control systems with multiplicative noise

Allegra, N., Bamieh, B., Mitra, P., Sire, C. (January 2018) Phase transitions in distributed control systems with multiplicative noise. Journal of Statistical Mechanics-Theory and Experiment. ISSN 1742-5468

URL: http://iopscience.iop.org/article/10.1088/1742-546...
DOI: 10.1088/1742-5468/aa9bb3

Abstract

Contemporary technological challenges often involve many degrees of freedom in a distributed or networked setting. Three aspects are notable: the variables are usually associated with the nodes of a graph with limited communication resources, hindering centralized control; the communication is subject to noise; and the number of variables can be very large. These three aspects make tools and techniques from statistical physics particularly suitable for the performance analysis of such networked systems in the limit of many variables (analogous to the thermodynamic limit in statistical physics). Perhaps not surprisingly, phase-transition like phenomena appear in these systems, where a sharp change in performance can be observed with a smooth parameter variation, with the change becoming discontinuous or singular in the limit of infinite system size. In this paper, we analyze the so called network consensus problem, prototypical of the above considerations, that has previously been analyzed mostly in the context of additive noise. We show that qualitatively new phase-transition like phenomena appear for this problem in the presence of multiplicative noise. Depending on dimensions, and on the presence or absence of a conservation law, the system performance shows a discontinuous change at a threshold value of the multiplicative noise strength. In the absence of the conservation law, and for graph spectral dimension less than two, the multiplicative noise threshold (the stability margin of the control problem) is zero. This is reminiscent of the absence of robust controllers for certain classes of centralized control problems. Although our study involves a 'toy' model, we believe that the qualitative features are generic, with implications for the robust stability of distributed control systems, as well as the effect of roundoff errors and communication noise on distributed algorithms.

Item Type: Paper
Subjects: organs, tissues, organelles, cell types and functions > tissues types and functions > neural networks
CSHL Authors:
Communities: CSHL labs > Mitra lab
Depositing User: Matt Covey
Date: January 2018
Date Deposited: 20 Feb 2018 16:43
Last Modified: 20 Feb 2018 16:43
URI: http://repository.cshl.edu/id/eprint/36084

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