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Centrality Measures and the Role of Non-Normality for Network Control Energy Reduction
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-4049-6018
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0003-4142-6502
2021 (English)In: IEEE Control Systems Letters, E-ISSN 2475-1456, Vol. 5, no 3, p. 1013-1018Article in journal (Refereed) Published
Abstract [en]

Combinations of Gramian-based centrality measures are used for driver node selection in complex networks in order to simultaneously take into account conflicting control energy requirements, like minimizing the average energy needed to steer the state in any direction and the energy needed for the worst direction. The selection strategies that we propose are based on a characterization of the network non-normality. We show that the concept is also related to the idea of balanced realization.

Place, publisher, year, edition, pages
IEEE, 2021. Vol. 5, no 3, p. 1013-1018
Keywords [en]
Controllability; networks; centrality measures
National Category
Control Engineering
Identifiers
URN: urn:nbn:se:liu:diva-168078DOI: 10.1109/LCSYS.2020.3008325ISI: 000554885800001OAI: oai:DiVA.org:liu-168078DiVA, id: diva2:1458166
Available from: 2020-08-14 Created: 2020-08-14 Last updated: 2023-08-25
In thesis
1. Controllability of Complex Networks at Minimum Cost
Open this publication in new window or tab >>Controllability of Complex Networks at Minimum Cost
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The control-theoretic notion of controllability captures the ability to guide a system toward a desired state with a suitable choice of inputs. Controllability of complex networks such as traffic networks, gene regulatory networks, power grids etc. can for instance enable efficient operation or entirely new applicative possibilities. However, when control theory is applied to complex networks like these, several challenges arise. This thesis considers some of them, in particular we investigate how a given network can be rendered controllable at a minimum cost by placement of control inputs or by growing the network with additional edges between its nodes. As cost function we take either the number of control inputs that are needed or the energy that they must exert.

A control input is called unilateral if it can assume either positive or negative values, but not both. Motivated by the many applications where unilateral controls are common, we reformulate classical controllability results for this particular case into a more computationally-efficient form that enables a large scale analysis. Assuming that each control input targets only one node (called a driver node), we show that the unilateral controllability problem is to a high degree structural: from topological properties of the network we derive theoretical lower bounds for the minimal number of unilateral control inputs, bounds similar to those that have already been established for the minimal number of unconstrained control inputs (e.g. can assume both positive and negative values). With a constructive algorithm for unilateral control input placement we also show that the theoretical bounds can often be achieved.

A network may be controllable in theory but not in practice if for instance unreasonable amounts of control energy are required to steer it in some direction. For the case with unconstrained control inputs, we show that the control energy depends on the time constants of the modes of the network, the longer they are, the less energy is required for control. We also present different strategies for the problem of placing driver nodes such that the control energy requirements are reduced (assuming that theoretical controllability is not an issue). For the most general class of networks we consider, directed networks with arbitrary eigenvalues (and thereby arbitrary time constants), we suggest strategies based on a novel characterization of network non-normality as imbalance in the distribution of energy over the network. Our formulation allows to quantify network non-normality at a node level as combination of two different centrality metrics. The first measure quantifies the influence that each node has on the rest of the network, while the second measure instead describes the ability to control a node indirectly from the other nodes. Selecting the nodes that maximize the network non-normality as driver nodes significantly reduces the energy needed for control.

Growing a network, i.e. adding more edges to it, is a promising alternative to reduce the energy needed to control it. We approach this by deriving a sensitivity function that enables to quantify the impact of an edge modification with the H2 and H norms, which in turn can be used to design edge additions that improve commonly used control energy metrics.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. p. 38
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2074
National Category
Control Engineering
Identifiers
urn:nbn:se:liu:diva-165258 (URN)10.3384/diss.diva-165258 (DOI)9789179298470 (ISBN)
Public defence
2020-06-05, Ada Lovelace, B-Building, Entrance 25, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2020-04-30 Created: 2020-04-21 Last updated: 2020-08-14Bibliographically approved

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Lindmark, GustavAltafini, Claudio

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