IEEE Circuits and Systems Magazine - Q3 2020 - 60

practice. For instance, in power grids, asymptotical (or
exponential) frequency synchronization is not optimal,
while electrical engineers prefer to restore the system
frequency to the nominal value as fast as possible. The
so-called finite-time control scheme, as an effective
and feasible way, has the optimality in the convergence
time [169], [170]. Dong and Xue [171] have explored the
finite-time synchronization of Kuramoto-type oscillators, and give an upper bound of the convergence time,
which possesses higher control accuracy with the robustness to the disturbance rejection and uncertainty.
Recently, Zhang et al. [172] have extended it into the
case of continuous phase couplings with the constraint
of the initial phases of all oscillators on a half-circle.
Jia and Mwanandiye [173] only discuss the finite-time
phase agreement of coupled Kuramoto oscillators for
the identical case, where the coupling is assumed to be
switched on and off frequently. Nevertheless, the past
works [171]-[173] excessively modify the conventional
Kuramoto model (1) and restrict the initial phase diameter less than r/2 or r.
With implementing multiplex control scheme consisting of the proportional control layer G Proportional ,
the finite-time control layer G Finite - time , and the fixedtime control layer G Fixed - time (see Fig. 13), a generalized

Kuramoto-oscillator network can achieve finite-time
synchronization (regardless of initial conditions)
[6 8], whose dy na mics is governed by the following equation
	

Figure 13. A multiplex network representation that the Kuramoto-oscillator layer G Kuramoto is controlled by the proportional control layer G Proportional, finite-time control layer G Finite - time,
and fixed-time control layer G Fixed - time, respectively [68].

60 	

ui = G
N

N

N

j=1

j=1

/ z ij (i j - i i) + c 1 / b ij sig (i j - i i)

ui = G
N

, (29)

a1

N

N

j=1
N

j=1

/ z ij (i j - i i) + c 1 / b ij sig (i j - i i)

+ c2

GFinite-Time

GKuramoto

j=1

or within a fixed time T2 (estimated in [68]) under the
triplex control protocol

	

GProportional

N

/ a ij sin (i j - i i) + u i, i = 1, 2, f, N, (28)

where A = [a ij] ! R N # N is the a djace nc y m at r i x o f
Kuramoto-oscillator layer G Kuramoto, and ui is a multiplex
control protocol of system dynamics and information transmission among these oscillators. The role of
the proportional control layer G Proportional is to drive the
Kuramoto-oscillator layer G Kuramoto for achieving asymptotical phase-frequency synchronization, which is
prerequisite to implement the finite-time control layer
G Finite - time for realizing finite-time phase-frequency synchronization with estimating the upper bound of synchronization time. Nevertheless, the time estimation
heavily dependents on the initial conditions, and this
drawback can be overcome by implementing the fixedtime control layer G Fixed - time .
In the identical Kuramoto-oscillator case, i.e., ~ i = ~ j,
i, j = 1, 2, f, N, 6K 2 0, 6i i (0), i j (0) ! T, , the undirected
Kuramoto-oscillator network (28) can achieve the phase
agreement, i i = ir, i = 1, 2, f, N, within a finite time T1 (estimated in [68]) under the duplex control protocol
	

GFixed-Time

K
N

io i = ~ i +

/ d ij sig (i j - i i)

,

a2

a1


(30)

j=1

provided that one of the following two connectivity criteria holds: (i) (local connectivity) cos zKa ij + Gz ij $ 0; (ii)
(global connectivity) m min (GL Z + cos zKL A) $ 0. The control gains G, c 1, c 2 2 0 and parameters 0 1 a 1 1 1, a 2 2 1.
Z = [z ij], B = [b ij], and D = [d ij] ! R N # N are adjacency
matrices corresponding to the proportional control layer G Proportional, the finite-time control layer G Finite - time, and
the fixed-time control layer G Fixed - time, respectively. LZ
and LA represent the Laplacian matrices of the graphs
G Proportional, G Kuramoto . m min (·) represents the smallest eigenvalue of matrix, z ! ! (r, 2r) satisfies tanz = z.
I n t h e c a s e o f o s c i l l a to r s w it h n o n - i d e nt ic a l
n at ural frequencies, i.e., ~ i ! ~ j (i ! j ), i, j = 1, 2, f, N,
6K 2 0, 6i i (0), i j (0) ! T, the undirected Kura moto
network (28) can achieve frequency synchronization,
io i = io j, i, j = 1, 2, f, N, within a finite time T1l (estimated
in [68]) under the control scheme

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q3 2020
