IEEE Circuits and Systems Magazine - Q2 2019 - 21

Based on Theorem 11, several specific results (w 1 =
w 2 =f = w N = w) can be obtained, as follows:
i) A directed path is controllable if the beginning
node is selected to be the only leader.
ii) A directed cycle with a single leader is controllable.
iii) A complete digraph with a single leader is uncontrollable.
iv) A star digraph (with more than 2 nodes) is uncontrollable if the center node is the only leader.
3.6 Switched Systems Theory
In the real world, the interaction topologies of networked systems may be varying with time [45]. For
example, in p2p communication networks, contact
patterns among individuals such as telephoning and
emailing, are typically temporal [46]. Recently, attention was attracted to the controllability of temporal
networks [47]-[49]. Temporal networks are networks
in which nodes and edges may appear and disappear
at various time instants. Although temporal networks
are essentially time-varying systems, the irreversibility of time and the deterministic chronological order
in these networks distinguish them from general timevarying systems. In the framework of conventional
switching networks [50], [51], the edges can periodically switch to repeat some topological forms, but
this is not the case in temporally networks due to
time causality, rendering the existing methodologies
of switched systems theory inapplicable to temporal
networks in general.
A temporally switching network G is characterized
by a node set {1, 2, f, N }, a static (time-invariant) network topology set represented by G 1, G 2, f, G m in a
specified order, where G k (and G m) exists only on the
time interval [t k -1, t k), k =1, 2, f, m -1 (and [t m -1, t m]) .
The adjacency matrix A k =[a ij (k)] ! R N # N of G k, is defined by
a ij (k) '

inputs can be simply represented by the matrix pair
(A o (t), B ) .
Definition 6
The linear temporally switching system (22) is state
controllable on the time interval [t 0, t m] if, for any initial
state x 0 ! R N at t 0 $ 0, there exists an input u(t) ! R M ,
t ! [t 0, t m] such that x (t m) = 0.
The following result was obtained by Hou et al. [48].
Theorem 12 [48]
The linear temporally switching system (22) is state
controllable on the time interval [t 0, t m] if and only if its
controllability matrix
6e (t m -t m -1) Aom g e (t 2 -t 1) Ao2 Q 1, f, e (t m -t m -1) Aom Q m -1, Q m@

has full row rank, where Q i =[B, A oi B, f, A oiN -1 B] is
the controllability matrix of the static network on
the i th time interval [t i -1, t i), i =1, 2, f, m -1, and
[t m -1, t m].
For notational simplicity, a temporally switching netu , B ).
work with fixed external inputs is denoted by (A
u
Note that (A, B ) can be described by the matrix pairs
(A k, B ), k =1, 2, f, m, in a time sequence. A linear temu , B ) and a linear temporalporally switching network (A
ly switching system (A o (t), B ) have the same structure
if and only if they have the same number of system
matrices, A 1, A 2, f, A m and A o1, A o2, f, A om, and moreover they have the same (fixed) zero and (parametric)

1
.
xi = wi
+
3

2
(a)

! 0, edge ( j, i, [t k -1, t k)) ! Q,
= 0,
otherwise,

wi

1 0
x
-1 2 i

-wi
xi 1

(b)

xi 2

2wi

where a ij (k) denotes the directed edge from node j to
node i on the time interval [t k -1, t k).
Consider a linear temporally switching system of the
form [48]
xo (t) = A o (t) x (t) + Bu (t), x (t 0) = x 0

(22)

where x (t) ! R N is the state vector, u(t) ! R M is the input
vector, B ! R N # M is the constant input matrix, and the elements of the state matrix A o (t) : R " R N # N are piecewise
constant functions of t ! [t 0, t 1) ' [t 1, t 2) ' g' [t m -1, t m].
This temporally switching system with fixed external
sEcOnd QuartEr 2019

(c)
Figure 8. Integration of network topology and node selfdynamics. (a) network topology (b) dynamic unit (c) Integrated network.

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