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where m i (i =1, 2, f, l ) denotes the distinct eigenvalues
of the matrix A, and n (m i) = N - rank (m i I N - A) is the
geometric multiplicity of m i . For undirected networks
with arbitrary edge weights, N D is determined by the
maximum algebraic multiplicity of the eigenvalues m i of
A, i.e.,
N D = max {t (m i)},
i

(16)

where t (m i) is the algebraic multiplicity of m i .
Note that condition (15) is rigorous and generally
valid without any limitation on the matrix A. Also, condition (16) is valid for general undirected networks with
diagonalizable matrix A.
3.4.1 Exact Controllability
Structural controllability provides a basic framework
for determining the controllability of directed networks
characterized by their structured matrices (Sec. 2.2).
This framework is ideal for many complex networked
systems for which only the underlying wiring diagram
is known (i.e., zero or nonzero values, indicating the
absence or presence of physical connections) but not
the edge characteristics (e.g., their weights). Yet, the
assumption of having independent free parameters is
quite strong, usually not satisfied by many real-world
systems, since system parameters are often related to
each other. For example, in undirected networks the matrix A in system (5) is symmetric, and in unweighted
networks all the edge weights are the same, both reflecting certain mutual dependence among parameters. In
these cases, the structural control theory could yield
misleading results on the minimum number of driver
nodes. Based on the PBH rank criterion in control theory, Yuan et al. [33] developed a so-called exact controllability paradigm as an alternative to the conventional
structural controllability framework, which offers a useful tool to study the controllability of complex dynamical networks with arbitrary structures and edge-weight
distributions, including directed, undirected, weighted
and unweighted networks, even with self-loops. The notion of exact controllability of complex networks is essentially the state controllability of complex systems,
but for the purpose of readability and uniformity, sometimes more convenient to use.
The exact controllability theory was applied to fractal
networks [34], bipartite networks [35] and multiplex networks [36]. Nie et al. [37] further investigated the effects
of degree correlation on the exact controllability of multiplex networks. They found that the minimal number of
driver nodes decreases with the correlation for networks
with lower density of interconnections, but conversely
for networks with higher density of interconnections.
sEcOnd QuartEr 2019

3.4.2 Node Self-Dynamics
It is obvious that, other than the network topology and
edge dynamics, the node system (nodal dynamics) is
also a crucial factor affecting the controllability. Cowan
et al. [38] pointed out that the main results in [24] depend strongly on a critical assumption about the model:
each node has an infinite time constant (i.e., each node is
treated as a pure integrator). However, several real networks considered therein, including food webs, power
grids, electronic circuits, regulatory networks, and neuronal networks, typically manifest more general dynamics at each node, for example with finite time constants.
In [38], the structural controllability of directed networks with linear nodal dynamics is analyzed, assuming that each node has a self-loop. They found that only
one single time-varying input is needed to guarantee
the network controllability. Zhao et al. [39] investigated
the synergistic effect of network topology and the d thorder individual dynamics on the exact controllability.
Interestingly, they found a global symmetry accounting
for the invariance of the controllability with respect to
the exchange of the densities between any two different
types of node dynamics, irrespective of the network topology. Recently, some more general and higher-dimensional node systems were investigated [40]-[42].
Consider a general directed and weighted network
consisting of identical multi-input multi-output (MIMO)
LTI node-systems [40],
N

xo i (t) =Wx i (t) + / a ij Hy j (t) + d i Bu i (t),
j =1

(17)

y i (t) = Cx i (t), i = 1, 2, f, N,
in which x i (t) ! R n is the state vector, u i (t) ! R p the input
vector and y i (t) ! R m the output vector of node i; H ! R n # m
is the inner-coupling matrix and A = (a ij) ! R N # N the
outer-coupling matrix, i.e., the adjacency matrix of the
network; C ! R m # n is the output matrix; B ! R n # p is the
input matrix; d i =1 if node i is under control, but d i = 0
otherwise. This networked system (17) can be rewritten
in a compact form as
xo (t) = (I N 7 W + A 7 HC ) x (t) + (D 7 B) u(t),

(18)

with D = diag {d 1, d 2, f, d N }.
The following result was obtained by Wang et al. [40],
based on the classic PBH rank condition.
Theorem 9 [40]
The networked system (18) with identical MIMO node
systems is state controllable if and only if the following
system of two matrix equations has a unique zero solution X = 0:
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