IEEE Circuits and Systems Magazine - Q2 2019 - 9

is encountered more and more large-scale complex
networks [3]-[7], where nodes are higher-dimensional
dynamical systems and edges represent the interactions
among them. Typical examples include the Internet,
WWW, wireless communication networks, transportation networks, power grids, sensor networks, brain
neural networks, metabolic networks, gene regulatory
networks, social networks, and many others. Developments in engineering, physics and biology have recently
extended the classic concepts and notions on systems
control to networks control. It has been noted that perturbations on one node in a network can influence and
alter the states of many other nodes through their local
interactions. This interconnectedness can be exploited
to effectively control a complex network by manipulating the states of only a small fraction of nodes, in which
the underlying network structure plays a crucial role.
Therefore, it is of theoretical and practical importance
to explore the controllability of complex networked systems from a network-theoretic perspective. This can
help better understand, predict and optimize the collective behaviors of various networked dynamical systems
in practical applications.
In the past decade, research on network controllability has attracted increasing attention and, in effect,
become an exciting and rapidly developing research
direction. The goal of this article is to survey on the
current flourishing advances in the studies of the controllability of networked linear dynamical systems.
Fundamental concepts and selected theoretical results
on both state controllability and structural controllability for different types of complex networked systems
are reviewed and discussed. Several specific applications of network controllability are described. Finally,
a near-future research outlook will be highlighted.
2. Notions of Network Controllability
In the present literature, there are several notions of
network controllability, which strongly depend on the
types of the networked control systems and the forms of
admissible control inputs, but they can be classified into
two essential types of state controllability and structural controllability in general.
2.1 State Controllability
The basic concept of (complete) state controllability was
introduced by Kalman in the 1960s [1], for a linear timeinvariant (LTI) dynamical system of the form

xo (t) = A 0 x (t) + B 0 u (t)

(1)

where x (t) =[x 1 (t), x 2 (t), f, x n (t)] T ! R n is the internal
state vector of the system at time t, u (t) =[u 1 (t), u 2 (t), f,
u m (t)] T ! R m is the input vector at time t, A 0 ! R n # n is
the system matrix and B 0 ! R n # m is the input matrix.
Definition 1
The LTI system (1) is said to be (completely) state controllable if, for any initial state x (t 0) ! R n and any final
state x (t f ) ! R n, there exist a finite time t 1 and an input
u (t) ! R m, t ! [t 0, t 1], such that x (t 1; x (t 0), u) = x (t f ).
This definition implies that any initial state x (t 0) can
be steered to any final state x (t f ) in finite time. Here, the
finite time t 1 is not fixed, the trajectory of the dynamical
system (1) between t 0 and t 1 is not specified, and there
is no constraint on the input vector u (t).
The classic algebraic controllability criteria are given
as follows [2].
Theorem 1 (State Controllability Theorem)
The LTI system (1) is completely state controllable (state
controllable, or simply, controllable) if and only if one of
the following conditions is satisfied:
i) the controllability matrix
Q =[B 0, A 0 B 0, f, A n0 -1 B 0]

(2)

has full row rank; that is,
rank (Q) = n.

(3)

ii) rank [sI n - A 0, B 0] = n, 6s ! C.
iii) the relationship y T A 0 = my T implies y T B 0 ! 0, where
y is the nonzero left eigenvector of A 0 associated
with the eigenvalue m.
iv) the Gramian matrix
Wc =

#t t
0

1

T

e A0 t B 0 B T0 e A0 t dt

(4)

is nonsingular.
Conditions (i), (ii), (iii) and (iv) in Theorem 1 are referred to as the Kalman rank criterion, Popov-BelevitchHautus (PBH) rank criterion, PBH eigenvector test, and
Gramian matrix criterion, respectively. They are equivalent for the LTI system (1).
Today, strongly stimulated by the rapid and promising development of network science and engineering,

Linying Xiang and Fei Chen are with the School of Control Engineering, Northeastern University at Qinhuangdao, Qinhuangdao 066004, P. R. China,
(e-mails: xianglinying@neuq.edu.cn; xianglyhk@gmail.com). Fei Chen is also with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, P. R. China. Wei Ren is with the Department of Electrical and Computer Engineering, University of
California, Riverside, CA 92521, USA, Guanrong Chen is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong
Kong SAR, P. R. China.
sEcOnd QuartEr 2019

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