IEEE Circuits and Systems Magazine - Q3 2020 - 59

region and derived the sufficient conditions for ensuring bounded phase synchronization and frequency synchronization, which can be applied into power grids for
guaranteeing frequency stability.
5.3. Synchronization with a Pacemaker
Exponential synchronization of a general undirected
Kuramoto-oscillator network in the presence of a pacemaker is firstly investigated by Wang and Doyle III [62],
where the dynamics model is governed by

*

io i = ~ i +

N

/ a ij sin (i j - i i) + g i sin (i 0 - i i), i = 1, 2, f, N,

(N - rank (L)) of pinned oscillators is determined under
given conditions [64], [65].
Besides a pacemaker, other additional control schemes
have been designed for achieving phase-frequency synchronization of Kuramoto-oscillator networks [58], [66].
Both the efforts of [58], [66] have constructed a multilayer network consisting of the Kuramoto-oscillator
layer and the control layer, where the dynamics model is govern by
	

*

N

io i = ~ i +
io 0 = ~ 0 .

/ a ij sin (i j - i i) + u i (t), i = 1, 2, f, N,

j=1

(26)

j=1

io 0 = ~ 0,
(25)
i 0 and ~ 0 are the pacemaker's phase and natural frequency, respectively. g i sin (i 0 - i i) denotes the force
of the pacemaker with the strength g i $ 0. The single
pacemaker connects all other oscillators in the network
for steering phase-frequency synchronization, and two
definitions are given as follows:
Phase agreement of Kuramoto-oscillator networks
with a pacemaker is achieved if

lim
(i i - i 0) = 0, 6i = 1, 2, f, N.
t"3
Frequency synchronization of Kuramoto-oscillator
networks with a pacemaker is achieved if
lim
(io i - io 0) = 0, 6i = 1, 2, f, N.
t"3
For the identical natural frequency case, phase agreement can be ensured even when phases are not constrained in an open half-circle [62]. For the non-identical
natural frequency case, frequency synchronization can
be achieved in the sense that phase differences can be
reduced to an arbitrary small by making the pacemaker
strength strong enough [62].
After that, the effort of [62] is extended as a special
case by investigating the weighted and directed Kuramoto-oscillator networks with a pacemaker [63], in
which the associated digraph is assumed to be weakly
connected. The sufficient conditions related to the initial phases and topologies are derived for guaranteeing
the phase agreement and frequency synchronization.
Rao et al. [64] further discuss the stability of synchronous solutions in a directed Kuramoto-oscillator network with a pacemaker, and the sufficient conditions
concerning the initial phases and coupling strength
are explicitly provided for obtaining such synchronous
solutions, which is under the assumption that the augmented digraph has a directed spanning tree rooted
at the pacemaker. In addition, the minimum number
THIRD QUARTER 2020 		

Rao and Li [58] implement the following pacemakerbased distributed control protocol:
	

N

u i (t) = K / b ij (i j - i i) + Kg i (i 0 - i i), (27)
j=1

where K is the coupling gain between neighboring oscillators in the control layer, g i and i 0 have
the same meaning as those in Eq. (25). Analogous to
A = [a ij] ! R N # N , B = [b ij] ! R N # N denotes the weighted
adjacency matrix of an undirected network. The sufficient criteria are established for achieving global
bounded and frequency synchronization of Kuramotooscillator networks, where there is no constraint on the
initial phases [58]. Tong et al. [66] employ the distributed multi-harmonic control
u i (t ) =

n

N

p=1

j=1

/ e / b ijp sin (p (i j - i i)) + g ip sin (p (i 0 - i i)) o,

where bij, gi and i 0 have the same meaning as those in Eq.
(27). Several conditions have been presented to achieve
exponential synchronization and phase locking of Kuramoto-oscillator networks with a pacemaker, where the
lower bounds for the synchronization and phase locking rate are estimated [66]. An integral consensus-based
controller is designed to force the network of oscillators
to follow a desired constant reference [67], in which the
communication constraints are taken into account. Even
when the Kuramoto-oscillator network is disconnected,
the disconnection can be compensated by the exchange
of information through the consensus-based controller
[67]. Additionally, even when the leader is connected to
few nodes, it is possible to reach a desired reference by
using a virtual leader [67].
5.4. Finite-Time and Fixed-Time Synchronization
All of the aforementioned synchronization of Kuramotooscillator networks are assumed to be realized asymptotically or exponentially [45]-[67], [159]-[168], in which
the time of achieving the synchronization tends to infinity, bringing some unsatisfactory system -performance in
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