Automatica 115 (2020) 108900
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Automatica
journal homepage: www.elsevier.com/locate/automatica
Brief paper
Characterizing controllable subspace and herdability of signed
weighted networks via graph partition
✩
Baike She
a
, Zhen Kan
b,
∗
a
Department of Mechanical Engineering, The University of Iowa, Iowa City, IA, 52246, USA
b
Department of Automation, University of Science and Technology of China, Hefei, Anhui, 230052, China
a r t i c l e i n f o
Article history:
Received 30 April 2019
Received in revised form 19 January 2020
Accepted 8 February 2020
Available online 26 February 2020
a b s t r a c t
Herdability is a variant of controllability, and is an indicator of the ability to drive system states to a
specific subset of the state space. This paper characterizes the controllable subspace and herdability
of signed weighted networks. Specifically, a dynamic signed leader–follower network is considered,
in which a small subset of the network nodes (i.e., the leaders) is endowed with exogenous control
input and the remaining nodes are influenced by the leaders via the underlying network connectivity.
The considered network permits positive and negative edges to capture cooperative and competitive
interactions, resulting in a signed graph. Motivated by practical application, the system states are
required to be driven by the leaders to be element-wise above a positive threshold, i.e., a specific
subset rather than the entire state space as in classical controllability. Graph partitions are exploited
to characterize the controllable subspace of the system, from which sufficient conditions are derived to
render the system herdable. It is revealed that the quotient graph can be used to infer the herdability
of the original graph, wherein criteria of the herdability of quotient graphs are developed based on
positive systems. Examples are provided to illustrate the developed topological characterizations.
© 2020 Elsevier Ltd. All rights reserved.
1. Introduction
Networked multi-agent systems are increasingly applied in bi-
ological science (Muldoon et al., 2016), social science (Kan, Klotz,
Jr, & Dixon, 2015; Mirtabatabaei & Bullo, 2011), and engineer-
ing (Asimakopoulou, Dimeas, & Hatziargyriou, 2013; Klotz, Kan,
Shea, Pasiliao, & Dixon, 2015; Klotz, Obuz, Kan, & Dixon, 2018). A
subject of expanding interest, network controllability in these ap-
plications is indicative of the ability to arbitrarily control system
states. When a networked system is fully controllable, the system
states can be driven to any desired states. However, requiring a
system to be fully controllable is often restrictive and unnecessary
in many practical applications. For example, in applying adaptive
cruise control to a platoon of autonomous vehicles, the vehicles
are often required to maintain a desired positive speed. In a
political election, a candidate aims to drive the supportive rate
above a positive percentage in order to win. In these applications,
fully controllable systems become unnecessary, since driving the
vehicles’ speed or the candidate’s supportive rate to be negative
✩
The material in this paper was not presented at any conference. This paper
was recommended for publication in revised form by Associate Editor Michael
M. Zavlanos under the direction of Editor Christos G. Cassandras.
∗
Corresponding author.
E-mail addresses: baike-she@uiowa.edu (B. She), zkan@ustc.edu.cn
(Z. Kan).
does not make any physical sense. Instead, the relaxed controlla-
bility that drives the system states to a specific subset, rather than
the entire state space, as in classical controllability, is of more
practical significance. Such a relaxed controllability is referred to
as herdability (Ruf, Egerstedt and Shamma, 2018). To this end,
this work is practically motivated to characterize the herdability
of networked systems from graph topological perspectives.
Since both herdability and controllability concern the abil-
ity to drive system states, the literature on the controllability
of networked systems is first reviewed. Based on the type of
interactions, a network can be classified as either cooperative
or non-cooperative. Cooperative networks are often modeled as
unsigned graphs in which only positive edge weights are allowed
to represent cooperative interactions between network compo-
nents, while non-cooperative networks are often modeled as
signed graphs wherein both positive and negative edge weights
represent cooperative and competitive interactions, respectively.
The controllability of cooperative networks has long been a re-
search focus. For instance, the influence of network topological
structures on network controllability has been investigated using
a variety of tools, such as graph theoretic approaches (Haghighi
& Cheah, 2017; Liu, Slotine, & Barabási, 2011; Yazıcıoğlu, Abbas,
& Egerstedt, 2016), structural controllability (Liu, Lin, & Chen,
2013a, 2013b; Tang, Wang, Gao, Qiao, & Kurths, 2014), and con-
sensus based results (Aguilar & Gharesifard, 2015; Commault &
https://doi.org/10.1016/j.automatica.2020.108900
0005-1098/© 2020 Elsevier Ltd. All rights reserved.
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