IEEE Circuits and Systems Magazine - Q3 2020 - 58

nonlinear orbital stability of phase-locked states are
addressed with providing an explicit lower bound for
a coupling strength [160], which improves previous efforts of [45] and [46] on the formation of phase-locked
states because their attention is focused only on the
latter relaxation stage. Several latest efforts of [36],
[37], [50]-[52] extend the phase space from a circle to
the Euclidean space.
Besides the all-to-all interaction, Ha et al. [161] have
discussed the local coupling topology which is connected
and symmetric. Dong and Xue [162] have studied the allto-all Kuramoto-oscillator network and the generalized
Kuramoto-oscillator network with directed coupling topology, respectively. Based on the Łojasiewicz inequality
for gradient systems of analytic functions, for the all-to-all
Kuramoto model with identical oscillators, frequency synchronization can occur for all initial phase configurations
distributed over the whole circle. For the generalized
Kuramoto model with directed coupling topology, frequency synchronization can be guaranteed if the phases
of oscillators are distributed over the half circle, and the
coupling strength is sufficiently large [162]. Considering
that the network topologies are not static all the time, Lu
and Atay [163] have explored the stability of Kuramotooscillator networks with time-varying and intrinsic frequencies, where the coupling coefficients might be negative, and the sufficient conditions are provided. Moreover,
Schmidt et al. [164] have investigated phase-frequency
synchronization in Kuramoto-oscillators networks with
delayed couplings, where a delay-dependent lower bound
on the coupling gain is given. Jafarpour and Bullo [165]
have provided the first rigorous synchronization test
based on the 3-norm, which is less conservative than
any 2-norm, with compelling numerical comparisons on
the IEEE test cases. Recent investigations have took the
effect of frustration [166], [167] and inertia [168] of Kuramoto oscillators into consideration.
Generally speaking, it is difficult to realize the synchronization of Kuramoto-oscillator networks spontaneously due to the requirement on connectivity, coupling
strength, and phase differences. Mozafari et al. [59]
have adopted the distributed sine control protocol for
guaranteeing frequency synchronization with arbitrary
undirected and connected network topologies. Mao and
Zhang [60] have introduced a distributed proportionalintegral control to achieve frequency synchronization
and phase-difference tracking simultaneously, where
the individual frequencies can asymptotically synchronize to the average natural frequencies of the group,
and the phase differences can asymptotically track the
given references. Moreover, the obtained results in [60]
hold for an arbitrary weak coupling strength and dispersed initial phases. Recently, the network topology is
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taken as a control variable for the asymptotic frequency
synchronization [61], where a sufficient condition is derived for the Kuramoto model with step force generated by the topology switching, and there is no
constraint on the magnitude of the coupling strength
or the phase -differences.
5.2. Bounded Synchronization
Compared with the complete synchronization, the
bounded synchronization is relatively weak, which
means that the phase differences of a Kuramoto-oscillator network are restricted by a positive constant d when
time goes to infinity:
lim
|i i - i j| # d, 6i, j = 1, 2, f, N.
t"3
Bounded synchronization of Kuramoto-oscillator
networks without control has been applied into power
systems [53], where each generator is modeled as a
second-order Kuramoto model. By using the contraction and partial contraction theory, a sufficient condition is obtained for guaranteeing the bounded phase
synchronization exponentially [53], in which each
pairwise phase difference starts from (r/2, r] and
reaches [0, d], d 1 r/2 , eventually. By adopting the distributed impulsive control, Zhang et al. [54] have studied the bounded synchronization of Kuramoto-oscillator networks with phase lags, in which the boundary
d can be arbitrary small, and there is no restriction
on the initial phase difference, namely global exponentially stable. Besides the single first-order [54] and single second-order [53] Kuramoto models, Wu et al. [55]
have explored the bounded phase synchronization
of a multirate Kuramoto-oscillator network (9)-(10)
with deriving conditions on the edge weights, in which
there is no external control, and the network converges to a locally stable equilibrium point. Although
condition (11) for ensuring complete synchronization
of (9)-(10) is simple, its accuracy is only empirically
established. Later on, Jafarpour et al. [26] have made
a clearer mathematical explanation why condition (11)
works, namely it is a first order term in the series expansion to compute the equilibrium. Then, a multirate
Kuramoto-oscillator network with control is modeled
as follows [56], [57]:
Z
N
] M i ip i + D i io i = ~ i + / a ij sin (i j - i i) + u i, i = 1, 2, f, l,
]
j=1
[
N
]] D i io i = ~ i + / a ij sin (i j - i i) + u i, i = l + 1, l + 2, f, N,
j=1
\
where ~ i is the natural frequency, ui is the designed controller. Wu et al. [55]-[57] have estimated the attraction

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

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