IEEE Circuits and Systems Magazine - Q3 2020 - 61

	

ui = G
N

N

/ z ij (i j - i i) + U i, (31)

j=1

.

.

where Uo i = k 1 R Nj = 1 p ij sig ( i j - i i) a1, or within a fixed time
.
.
T2l (estimated in [68]), where Uo i = k 1 R Nj = 1 p ij sig ( i j - i i) a1
.
.
N
+ k 2 R j = 1 q ij sig ( i j - i i) a2, provided that one of the following two connectivity criteria holds: (i) (local connectivity) Gz ij - Ka ij $ 0; (ii) (global co n ne c t i v it y)
m min (GL Z - KL A) $ 0. The control gains G, k 1, k 2 2 0; and
parameters 0 1 a 1 1 1, a 2 2 1 . Adjacency matrices
Z = [z ij], P = [p ij], and Q = [q ij] ! R N # N correspond to
the proportional control layer G Proportional, the finitetime control layer G Finite - time, and the fixed-time control layer G Fixed - time, respectively.
Not limited to the finite-time and fixed-time phasefrequency synchronization of generalized Kuramoto-oscillator networks achieved in [68], the requirements on
the initial phase diameter and the coupling strength are
relaxed effectively too, where the initial phases can be
dispersed over a circle, and the coupling strength can
be arbitrary weak. For saving the computation and communication resources, an event-triggered mechanism
has been adopted to achieve fixed-time synchronization
of a multilayer Kuramoto-oscillator network [69].
5.5. Cluster Synchronization
The latest efforts of [174]-[176] concentrate on the cluster phase-frequency synchronization of a generalized
Kuramoto-oscillator network
	 io i (t) = ~ i +

N

/ a ij

sin (i j (t) - i i (t)), i = 1, 2, f, N, (32)

j=1

where A = [a ij] ! R N # N is the weighted adjacency matrix
of graph G. The weighted digraph or undirected graph
G = (V, E) has been discussed in [174], [176] and [175],
respectively, in which the definitions of cluster of oscillators and cluster phase (frequency) synchronization
are as follows:
Cluster of oscillators: The set of oscillators C 3 V
is a cluster if there exists an angle 0 # c # r [174] (or
r/2 [175]) such that, whenever |i i (0) - i j (0)| # c, then
|i i (t) - i j (t)| # c for all i, j ! C and at all times t $ 0.
Cluster phase (frequency) synchronization [176]:
For the network of oscillators G = (V, E ), the partition P = {P1, f, Pm}, V = , im= 1 Pi and Pi + Pj = 4, is phase
(frequency) synchronizable if, for some initial phases
i i (0), f, i N (0) , it holds i i (t) = i j (t) (io i (t) = io j (t)) for all
times t ! R $ 0 and i, j ! Pk with k ! {1, f, m}.
Cluster synchronization conditions are derived based
on (integrating) edges weight and oscillators' natural
frequency in weighted digraph [174] and undirected
graph [175], respectively. The explicit analytical conditions are derived for the (local) stability of the cluster
THIRD QUARTER 2020 		

synchronization manifold in sparse and weighted networks of heterogeneous Kuramoto oscillators [7]. The
existing efforts of [7], [174], [175] show that how cluster
synchronization depends on a graded combination of
weak inter-cluster and strong intra-cluster connections,
similarity of the natural frequencies of the oscillators
within each group, and heterogeneity of the natural frequencies of coupled oscillators belonging to different
clusters. Furthermore, the sufficient and necessary conditions have been derived based on the network weights
and oscillators' natural frequencies for ensuring cluster
phase synchronization [176].
5.6. Stochastic Synchronization
In reality, Kuramoto-oscillator networks are often disturbed by various uncertain factors, e.g., the noisy sinusoidal coupling process [177]. Ha et al. [177] use the
stochastic mean-field limit and restrict the phase diameter within r/2, namely the local stochastic stability. We
further develop the global stochastic synchronization
of Kuramoto-oscillator networks via adopting a distributed control scheme [70], which constructs the duplex
network model as illustrated in Fig. 14.
The dynamics model is ruled by [70]:
di i (t) = =~ i + K / a ij sin (i j (t ) - i i (t)) + u i (t)G dt
N

	

j= 1


(33)

+ v i (t) dw i (t), i = 1, f, N,

where aij is the element of the adjacency matrix A ! R N # N
that describes the structure of the Kuramoto-oscillator
layer G k (see Fig. 14), K 2 0 is the coupling strength.
v i (t) is a continuous function modeling the diffusion
of noise through the network. wi is an M-dimensional
Brownian motion defined on a complete probability
space. The distributed control layer G c (see Fig. 14) is
designed as

Gc

Gc

Gk

Gk

(a)

(b)

Figure 14. The duplex networks consist of the Kuramotooscillator layer G k and the distributed control layer G c, in
which noise emerges in (a) G k and (b) G c, respectively [70].

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