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Controllability of Complex Networks at Minimum Cost
Linköping University, Faculty of Science & Engineering. Linköping University, Department of Electrical Engineering, Automatic Control.ORCID iD: 0000-0002-4049-6018
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: urn:nbn:se:liu:diva-165258DOI: 10.3384/diss.diva-165258ISBN: 9789179298470 (print)OAI: oai:DiVA.org:liu-165258DiVA, id: diva2:1425446
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: 2020-04-30Bibliographically approved
List of papers
1. Controllability of complex networks with unilateral inputs
Open this publication in new window or tab >>Controllability of complex networks with unilateral inputs
2017 (English)In: Scientific Reports, ISSN 2045-2322, E-ISSN 2045-2322, Vol. 7, article id 1824Article in journal (Refereed) Published
Abstract [en]

In this paper, we study the problem of controlling complex networks with unilateral controls, i.e., controls which can assume only positive or negative values, not both. Given a complex network represented by the adjacency matrix A, an algorithm is developed that constructs an input matrix B such that the resulting system (A, B) is controllable with a near minimal number of unilateral control inputs. This is made possible by a reformulation of classical conditions for controllability that casts the minimal unilateral input selection problem into well known optimization problems. We identify network properties that make unilateral controllability relatively easy to achieve as compared to unrestricted controllability. The analysis of the network topology for instance allows us to establish theoretical bounds on the minimal number of controls required. For various categories of random networks as well as for a number of real-world networks these lower bounds are often achieved by our heuristics.

Place, publisher, year, edition, pages
NATURE PUBLISHING GROUP, 2017
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-138253 (URN)10.1038/s41598-017-01846-6 (DOI)000401262400017 ()28500342 (PubMedID)
Available from: 2017-06-13 Created: 2017-06-13 Last updated: 2020-04-21
2. Minimum energy control for complex networks
Open this publication in new window or tab >>Minimum energy control for complex networks
2018 (English)In: Scientific Reports, ISSN 2045-2322, E-ISSN 2045-2322, Vol. 8, article id 3188Article in journal (Refereed) Published
Abstract [en]

The aim of this paper is to shed light on the problem of controlling a complex network with minimal control energy. We show first that the control energy depends on the time constant of the modes of the network, and that the closer the eigenvalues are to the imaginary axis of the complex plane, the less energy is required for complete controllability. In the limit case of networks having all purely imaginary eigenvalues (e.g. networks of coupled harmonic oscillators), several constructive algorithms for minimum control energy driver node selection are developed. A general heuristic principle valid for any directed network is also proposed: the overall cost of controlling a network is reduced when the controls are concentrated on the nodes with highest ratio of weighted outdegree vs indegree.

Place, publisher, year, edition, pages
NATURE PUBLISHING GROUP, 2018
National Category
Other Electrical Engineering, Electronic Engineering, Information Engineering
Identifiers
urn:nbn:se:liu:diva-145766 (URN)10.1038/s41598-018-21398-7 (DOI)000425285200018 ()29453421 (PubMedID)
Note

Funding Agencies|Swedish Research Council [2015-04390]

Available from: 2018-03-22 Created: 2018-03-22 Last updated: 2020-04-21

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Lindmark, Gustav

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