Elsevier

Automatica

Volume 61, November 2015, Pages 73-79
Automatica

Brief paper
Graph-theoretic analysis of network input–output processes: Zero structure and its implications on remote feedback control

https://doi.org/10.1016/j.automatica.2015.07.034Get rights and content

Abstract

The control of dynamical processes in networks is considered, in the case where measurement and actuation capabilities are sparse and possibly remote. Specifically, we study control of a canonical network dynamics, when only one network component’s state can be measured and only one (in general different) component can be actuated. To do so, we characterize the finite- and infinite-zeros of the resulting SISO system in terms of the graph topology. Using these results, we establish graph-theoretic conditions under which there are zeros in the closed right-half plane. These conditions depend on the length, strength, and number of the paths from the component where the input is applied to the component where the measurements are made. Then, we present the implications of these conditions on the controller design task focusing in stabilizations/destabilization of network processes under static negative feedback.

Introduction

Network synchronization and diffusion models are used to capture such diverse processes as vehicle coordination, biochemical reactions, and sensor-fusion algorithms (Watts & Strogatz, 1998). The temporal dynamics of these models have been extensively studied from a graph-theoretic perspective. First, emergence of synchronization has been shown under broad connectivity conditions, i.e. the models have been shown to have a stable manifold wherein all the network components’ states are identical (Barahona and Pecora, 2002, Wu and Chua, 1995). Further results have been developed that tie performance characteristics (e.g., the settling rate) to features of the network’s graph. As synchronization models have found wider application in engineering contexts, their design and control have also been of significant interest. Many of these studies consider decentralized control of multiple autonomous but communicating agents (e.g., vehicles), which yields a closed-loop dynamics that is a network synchronization process. In complement, several recent studies have considered topology design to shape the performance of synchronization processes (Abad Torres and Roy, 2013, Abad Torres and Roy, 2014, Roy et al., 2009, Wan et al., 2008).

While the literature has focused on emergence and design of whole-network behaviors such as synchronization, there is a growing need to understand input–output dynamics and feedback regulation of established network processes, when measurement and actuation are available at only a few network components. This need for an input–output analysis partially stems from challenges in security and vulnerability analysis of infrastructures and other complex dynamical networks (Belykh et al., 2004, Koh and Vinnicombe, 2012, Pasqualetti et al., 2009, Pasqualetti et al., 2011, Roy et al., 2012, Sandberg et al., 2010, Vidyasagar and Yamamoto, 2012). In these applications, an adversary can typically only make limited measurements and actuations of the dynamics, but may be able to initiate a significant propagative impact across the network (Roy et al., 2012, Sandberg et al., 2010). By the same token, system operators may often have limited measurement and actuation capabilities in reacting to threats/disturbances, and their ability to mitigate the wide-area threats via local feedback is of interest. The input–output dynamics of network processes are also germane to management and resource allocation problems in large-scale networks, where limited control resources must be placed to shape global network dynamics. A key need in these various application domains is to understand the implications of the network’s graph topology on the input–output dynamics and specifically its zeros, to achieve simple insights into propagative impacts and enable control design.

Recently, several studies have begun to study the input–output dynamics of network synchronization and spread processes from a graph-theoretic perspective. Our work is closely aligned with the study of Briegel et al., which characterizes the zeros of a single-input–single-output (SISO) system defined on a consensus (synchronization) process (Briegel, Zelazo, Burger, & Allgower, 2011). The authors focus particularly on symmetric unweighted network topologies, and give bounds on finite-zero locations and conditions for the presence of right-half-plane zeros. Meanwhile, our previous work pursues a structural decomposition of an input–output dynamics imposed on a synchronization process (Abad Torres & Roy, 2014), and uses this decomposition to achieve simple graphical characterizations of the zeros. Motivated by vulnerability-analysis goals, control theorists have also studied robustification of synchronization processes via feedback (Vidyasagar & Yamamoto, 2012), characterized disturbance propagation (Koh & Vinnicombe, 2012), and highlighted linkages between graph connectivity and network robustness (via the presence of non-trivial zero dynamics) (Belykh et al., 2004, Pasqualetti et al., 2009, Pasqualetti et al., 2011).

This short paper is concerned with the input–output dynamics of a class of continuous-time linear network processes defined on a general (directed, weighted) graph, which is actuated at a single network component and measured at another component (see Section  2). For this SISO model, a full characterization of the infinite- and finite-zeros in terms of the network’s graph topology and the actuation/measurement locations is undertaken (Section  4), using a structural decomposition for the input–output dynamics (reviewed in Section  3). A significant result is that networks with weak short paths as well as alternative long strong paths between the input and output have non-minimum-phase zeros. Some implications on feedback control of the network dynamics are briefly discussed, particularly focusing on destabilization through remote feedback.

Section snippets

Problem formulation

We are concerned about the input–output behavior of a dynamical-network process that is actuated at a single network component, and measured at a single component (which in general may be remote from the stimulation location). Formally, a network with n components, labeled 1,2,,n, is considered. Each component is assumed to have a scalar state x̃i associated with it. These states evolve according to the differential equations: x̃̇=Ax̃+eiũ where x̃=[x̃1x̃n] is the full state of the network, e

Background

In Abad Torres and Roy (2014), we used the special coordinate basis (SCB) for linear systems (Sannuti & Saberi, 1987) to obtain some preliminary structural results on the zeros of the SISO network model ((1), (2)), which are foundational to the graph-theoretic results developed here. The SCB is a convenient tool for the graph-theoretic analysis of zeros, because it provides an explicit matrix-algebraic characterization of a system’s zero structure. Specifically, the SCB expresses a linear

Results

Graph-theoretic characterizations of the infinite and finite zeros are presented. Implications on control, including on stabilization and destabilization through static feedback, are discussed. Finally, an example is given. Graph-theoretic terminologies are introduced as needed.

Conclusions

We have characterized the infinite and finite zeros of a SISO network process in terms of its underlying (asymmetric) graph topology and input/output locations. A key outcome is the development of conditions for the presence of nonminimum-phase zeros, in terms of the lengths and strengths of paths between the input and output. Implications of the graph-theoretic analyses on control of the network dynamics are briefly discussed. A particular focus is on remote regulation/manipulation of the

Jackeline Abad Torres (M’15) received the B.S. degree from Escuela Politécnica Nacional, Quito, Ecuador, and the M.S. and Ph.D. degrees in Electrical Engineering from Washington State University, Pullman. She joined the Department of Automation and Industrial Control, Escuela Politécnica Nacional, Quito, in February 2015. She held a Fulbright grant in 2010. Her current interests include power systems network control, sensor/vehicle networking, and epidemic control.

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    Jackeline Abad Torres (M’15) received the B.S. degree from Escuela Politécnica Nacional, Quito, Ecuador, and the M.S. and Ph.D. degrees in Electrical Engineering from Washington State University, Pullman. She joined the Department of Automation and Industrial Control, Escuela Politécnica Nacional, Quito, in February 2015. She held a Fulbright grant in 2010. Her current interests include power systems network control, sensor/vehicle networking, and epidemic control.

    Sandip Roy (M’04) received the B.S. degree in Electrical Engineering from the University of Illinois at Urbana-Champaign and the M.S. and Ph.D. degrees in Electrical Engineering from the Massachusetts Institute of Technology, Cambridge. He joined the School of Electrical Engineering and Computer Science, Washington State University, Pullman, in September 2003 as an Assistant Professor. He has also held visiting summer appointments with the University of Wisconsin-Madison and the Ames Research Center, National Aeronautics and Space Administration, Moffett Field, CA. His current interests include controller and topology design for dynamical networks, with applications to airtraffic control, computational biology, and sensor networking.

    This work was partially supported by the National Science Foundation grants ECS-0901137, CNS-1035369, and CNS-1058124. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Antonis Papachristodoulou under the direction of Editor Christos G. Cassandras.

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