On exponential synchronization of kuramoto Oscillators

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 2, FEBRUARY 2009
made contractively invariant, is proposed. The result is given explicitly in terms of the system matrices and P . Simultaneously, a necessary and sufficient conditions for 3 equalling 3 is presented. Furthermore, the maximal contractively invariant ellipsoid in minimal energy
control with guaranteed convergence rate problem is studied. Two examples are used to illustrate the effectiveness of the results. Along this
way, further research will attempt to reduce the conservatism without
destroying the method’s simplicity and to extend this method to multiple inputs case.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers and
the Associate Editor for their helpful comments and suggestions,
and Dr. Z. Lin for his helpful suggestions on the presentation of this
technical note.
353
On Exponential Synchronization of Kuramoto Oscillators
Nikhil Chopra and Mark W. Spong, Fellow, IEEE
Abstract—In this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators, the Kuramoto model. We consider the special case of finite oscillators
with distinct, bounded natural frequencies. Our first result derives a lower
bound on the coupling gain which is necessary for the onset of synchronization. This bound improves the one derived by Jadbabaie et al. [8]. We then
calculate a lower bound on the coupling gain that is sufficient to guarantee
oscillator synchronization and derive further sufficient conditions to ensure
exponential synchronization of the angular frequencies of all oscillators to
the mean natural frequency of the group. We also characterize the coupling
gain that is sufficient for the oscillator phase differences to approach any
desired compact set in finite time.
Index Terms—Cooperative control, exponential stabilization, Kuramoto
oscillators, multi-agent systems, synchronization.
REFERENCES
I. INTRODUCTION
[1] D. S. Bernstein and A. N. Michel, “A chronological bibliography
on saturating actuators,” Int. J. Robust Nonlin. Control, vol. 5, pp.
375–380, 1995.
[2] T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysis
and Design. Boston, MA: Birkhauser, 2001.
[3] G. J. Sussmann, E. D. Sontag, and Y. Yang, “A general result on the
stabilization of linear systems using bounded controls,” IEEE Trans.
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[4] Z. Lin, A. Saberi, and A. A. Stoorvogel, “Semiglobal stabilization of
linear discrete—Time systems subject to input saturation via linear
feedback—An ARE-based approach,” IEEE Trans. Automat. Control,
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[5] C. Burgat and S. Tarbouriech, Nonlinear Systems. London, U.K.:
Chapman & Hall, 1996, vol. 2, ch. 4, pp. 113–197, Appendices C, D,
E, pp. 217-239.
[6] R. Suarez, J. Alvarez-Ramirez, and J. Alvarez, “Linear systems with
single saturated input: Stability analysis,” in Proc. 30th IEEE-CDC,
1991, pp. 223–228.
[7] G. F. Wredenhagen and P. R. Belanger, “Piecewise linear LQ control
for systems with inputs constraints,” Automatica, vol. 30, no. 3, pp.
403–416, 1994.
[8] E. B. Castelan and J. C. Hennet, “On invariant polyhedra of continuoustime linear systems,” IEEE Trans. Automat. Control, vol. 38, no. 11, pp.
1680–1685, Nov. 1993.
[9] C. Pittet, S. Tarbouriech, and G. Burgat, “Stability regions for linear
systems with saturating controls via circle and Popov criteria,” in Proc.
36th Conf. Decision Control, San Diego, CA, 1997, pp. 4518–4523.
[10] J. M. Gomes da Silva, Jr. and S. Tarbouriech, “Anti-windup design
with guaranteed regions of stability: An LMI-basead approach,” IEEE
Trans. Automat. Control, vol. 50, no. 1, pp. 106–111, Jan. 2005.
[11] S. Tarbouriech, C. Prieur, and J. M. Gomes da Silva, Jr., “Stability analysis and stabilization of systems presenting nested saturations,” IEEE
Trans. Automat. Control, vol. 51, no. 8, pp. 1364–1371, Aug. 2006.
[12] E. B. Castelan, S. Tarbouriech, J. M. Gomes da Silva, Jr., and I.
Queinnec, “ L -stabilization of continuous-time linear systems with
saturating actuators,” Int. J. Robust Nonlin. Control, vol. 16, no. 18,
pp. 935–944, 2006.
[13] T. Hu and Z. Lin, “Exact characterization of invariant ellipsoids for
single input linear systems subject to actuator saturation,” IEEE Trans.
Automat. Control, vol. 47, no. 1, pp. 164–169, Jan. 2002.
[14] M. C. Turner and I. Postlethwaite, “Guaranteed stability regions of
linear systems with actuator saturation using the low-and-high gain
technique,” Int. J. Control, vol. 74, no. 14, pp. 1425–1434, 2001.
[15] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall,
1980.
[16] B. D. O. Anderson and J. B. Moore, Linear Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1971.
[17] A. R. Teel, “A nonlinear small gain theorem for the analysis of control
systems with saturation,” IEEE Trans. Automat. Control, vol. 41, no. 9,
pp. 1256–1270, Sep. 1996.
Collective synchronization has long been observed in biological,
chemical, physical and social systems. This phenomenon is observed
when the individual frequencies of coupled oscillators converge to a
common frequency despite differences in the natural frequencies of the
individual oscillators. Examples in biology and physics include groups
of synchronously flashing fireflies [1], crickets that chirp in unison
[20], and Josephson junctions [23]. Collective synchronization was first
studied by Wiener [22], who conjectured its involvement in the generation of alpha rhythms in the brain. It was then taken up by Winfree [24]
who used it to study circadian rhythms in living organisms. Winfree’s
model was significantly extended by Kuramoto [10], [11] who developed results for what is now popularly known as the Kuramoto model.
An elegant summary of Kuramoto’s work, and later attempts to answer
the questions that were raised by his formulations, can be found in [17]
and [18].
The Kuramoto model consists of a population of N oscillators whose
dynamics are governed by the following equations:
_i = !i + K
sin(k 0 i );
i = 1; . . . ; N
N k=1
N
(1)
where i is the phase of the ith oscillator, !i is its natural frequency,
and K > 0 is the coupling gain. The problem is then to characterize
the coupling gain K so that the oscillators synchronize.
Definition 1.1: The oscillators are said to synchronize if
lim
t!1
_i (t) 0 _j (t)
= 0
8i; j
;
= 1 ...
; N:
Manuscript received April 19, 2006; revised May 05, 2007 and January 06,
2008. Current version published February 11, 2009. This work was supported
in part by the Office of Naval Research under Grants N00014-02-1-0011 and
N00014-05-1-0186, and by the National Science Foundation under Grants ECS0122412, HS-0233314, and CCR-0209202. Recommended by Associate Editor
F. Bullo.
N. Chopra is with the Department of Mechanical Engineering and the Institute
for Systems Research, University of Maryland, College Park, MD 20742 USA
(e-mail: nchopra@umd.edu).
M. W. Spong is with the Erik Jonsson School of Engineering and Computer
Science, University of Texas at Dallas, Richardson, TX 75080 USA (e-mail:
mspong@utdallas.edu).
Digital Object Identifier 10.1109/TAC.2008.2007884
0018-9286/$25.00 © 2009 IEEE
354
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 2, FEBRUARY 2009
Recently, control theoretic methods have been used in [2], [3], [7], [8],
[14]–[16], [19], [21] to address the synchronization phenomenon. Stability analysis was carried out for a unidirectional ring of oscillators
with Kuramoto-type dynamics in [15]. In [14], the authors demonstrated that only phase locking solutions, where the phases differences
correspond to (0(=2); (=2)), can be locally asymptotically stable,
and a condition for guaranteeing local asymptotic stability was derived.
Control and graph theoretic methods were used in [8] to analyze Kuramoto oscillators for an arbitrary bidirectional graph topology. The
authors derived a coupling gain KL necessary for the existence of the
phase-locked state in the traditional Kuramoto model (all-to-all connectivity). It was also shown that there exists a large enough coupling
gain K so that the vector of phase differences locally (0(=2); (=2))
converges to an unique constant. Recently, local exponential stability
of the phase-locked state in the Kuramoto model was studied in [12].
There are some important questions still associated with the
Kuramoto model and they form the motivation for this technical note.
It has been observed via numerical simulations that for a large enough
coupling gain K , the oscillator frequencies exponentially synchronize
to the mean natural frequency of the group. Local exponential stability
of the Kuramoto model has recently been studied in [12]. However,
till date there is no global analysis that shows that the oscillator
frequencies in the original Kuramoto model (where the oscillators
have different natural frequencies) synchronize exponentially. In
this technical note we extend our preliminary results in [2] to solve
the exponential synchronization problem by studying the nonlinear
Kuramoto model (1).
In this technical note, we study the case of a finite number N of
Kuramoto oscillators with all-to-all connectivity. We assume that the
natural frequencies !i of the oscillators are arbitrarily chosen from the
set of reals. Our contribution in this technical note can be summarized
as follows:
• In Section II we derive a lower bound on the coupling gain K
that is necessary for the existence of the phase-locked state in the
Kuramoto model. This bound improves the result of Jadbabaie
et al. [8].
• In Section III we derive sufficient conditions for exponential synchronization of the oscillator frequencies to the mean natural frequency of the group, and then in Section IV a lower bound on the
coupling gain is developed to achieve the sufficient conditions.
We believe this is the first result that demonstrates almost global
exponential synchronization in the Kuramoto model for distinct
natural frequencies of the individual oscillators. Furthermore, we
characterize the coupling gain that guarantees convergence of the
oscillator phase differences to any desired set in (0; ), provided certain conditions on the initial phase differences are satisfied [9].
As we are interested in the evolution of the phase differences, we
write the phase difference dynamics using (1) as
0
=
K
! !+
N
0
0
2 sin(
0
)+
(sin(
0
) + sin(
0
!j0!i =
K
N
2 sin(j
0 i )+
N
(sin(k
0 i )+sin(j 0 k ))
:(3)
Assuming without loss of generality that !j > !i , to calculate a lower
bound on the coupling gain K satisfying (3), we need to maximize the
expression
N
Eij
= 2 sin(j
0 i ) +
(sin(k
0 i ) + sin(j 0 k )) :
(4)
The first order necessary conditions for maximizing Eij are given by
@Eij
@i
=
@Eij
@j
= 2 cos(j
@Eij
@k
= cos(k
N
0 2 cos(j 0 i ) 0
N
0 i ) +
cos(k
cos(j
0 i ) = 0
(5)
0 k ) = 0
(6)
0 i ) 0 cos(j 0 k ) = 0
(7)
where k = 1; . . . ; N , k 6= i; j . Equation (7) implies that either j =
i +2p or k = p +(i + j =2), p = 0; 1; 2; 3; . . .. The first solution
implies that Eij = 0. As we are looking for a maximum, we investigate
the other solution. Replacing k by the even solution (p = 0; 2; . . .) in
(5) yields
2 cos(j
0 i ) +
N
cos
j 0 i
=0
2
) 2 cos(j 0 i ) + (N 0 2) cos j 02 i = 0
) 4 cos2 j 0 i 0 2 + (N 0 2) cos j 0 i
2
2
= 0:
Solving the above quadratic equation we get
cos
j 0 i
2
=
0(N 0 2) 6
(N
8:
0 2)2 + 32
As cos(x) 18x 2 R, a well defined solution for all N is given by
(j
0 i )opt1 = 2 cos01 0(N 0 2) +
(N
0 2)2 + 32
8
: (8)
Similarly, it can be verified that the odd solutions (p = 1; 3; . . .) lead
to the solution
II. NECESSARY CONDITION FOR SYNCHRONIZATION
_ _
Proof: The oscillators can only synchronize if the expression in the right hand side of (2) has at least one fixed point
8i; j = 1; . . . ; N i 6= j . The fixed point equation can be written as
))
:
(2)
If the oscillators are to synchronize, i.e., _i (t) 0 _j (t) ! 0 as t ! 1,
the right hand side of (2) must go to zero. Our first result calculates a
lower bound on the coupling gain K necessary for the right-hand side
of (2) to equal zero. In other words, we calculate a necessary condition
for the onset of synchronization in (1).
Theorem 2.1: Consider the system of oscillators described by (1).
There exists a coupling gain K = Kc > 0 below which the oscillators
cannot synchronize.
(j
0 i )opt2 = 2 cos01
(N
0 2) 0
(N
8
0 2)2 + 32
:
0 i )opt1 , (j 0 i )opt2 satisfy =2 and (j 0 i )opt2 3=2 respectively.
Therefore, using (4) we have Eij [(j 0 i )opt1 ] > Eij [(j 0 i )opt2 ],
and hence the candidate optimal value of j 0 i is
(j 0 i )opt
=
(j 0 i )opt1 . To verify the second order
The solutions (j
(j
0 i )opt1 necessary condition, note that the off-diagonal terms of the Hessian
of Eij consists of sin(1) terms and the diagonal terms are simply the
negative sum of the off-diagonal terms. As =2 (j 0 i )opt1 < ,
it follows from the even solution that all phase differences
((j 0 i )opt ; (k 0 i )opt ; (j 0 k )opt ) modulo 2 belong to
the interval [(=4); ]. Thus, sin (j 0 i )opt , sin (k 0 i )opt ,
sin (j 0 k )opt 0. Using Gershgorin’s Theorem [6], the Hessian
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 2, FEBRUARY 2009
for the function Eij is negative semidefinite and hence Eij is a
concave function over the domain [(=4); ]. Thus, the necessary
conditions for maximization are sufficient as well and the optimal
value of Eij is given as
E
0
N0 :
max = 2 sin( j
2(
N0
i )opt + 2(
0
( j
2) sin
i )opt
2
0!
E
!
K
(
c =
N
min )
max
(9)
max
where !max , !min are the maximum and minimum frequencies in the
set of natural frequencies.
The phrase critical coupling gain required for onset of synchronization does not imply that at Kc the oscillators synchronize. The critical
gain Kc is the gain below which the oscillators cannot synchronize.
It is interesting to compare condition (9) with that for the critical gain
obtained in [8]. The value for the critical coupling in [8] is given as
!
K
(
L =
max
0!
N0
N:
min )
2(
1)
!
0! N
N0
: (11)
( 0 )
0
2
, from (8) we have 0 For the case N
01 p =
= : Therefore, using (11), the critical coupling is given as K
! 0 ! . Substituting N in (10) we
get that K
! 0 ! . Consequently, K K when there
(
c =
2 sin( j
32 8)
=
min )
2) sin
2
( j
max
max
!1 K
N
c=
=
!1
lim
N
!
(
=
= 2
min
L =
min
c
are only two oscillators in the system.
When N ! 1, (8) yields the solution (j 0 i )opt
tuting the optimal solution in (11) and taking limit
lim
i )opt
2
c =
L =
max
i )opt + 2(
=
(
2
max
!
(
0
sin(
)
+2
N
0!
min )
2
0!
0
min )
max
1
2
sin
N
(
=
0
)
2
:
!1 K
N
L =
=
N
2
lim
N
max
!1
!1 K :
lim
1
0!
0
min )
1
=
!
(
max
N
0!
min )
2
c
Thus, the critical gain condition proposed in this technical note is equivalent to the one proposed in [8] for the infinite oscillator case. The comparison between the two critical gains can be summarized as
K K
K >K
 =
i
;N ! 1
N:
c =
L
N
c
L
for all other
N
0 0 ; i
= 2
It is interesting to note that for the two oscillator system, KL = Kc =
!max 0 !min is also sufficient for oscillator synchronization. We refer
the reader to [8] for further details.
In this section we develop sufficient conditions for synchronization
of Kuramoto oscillators and characterize the rate of convergence.
_ )
i
i )( j
;
= 1 ...
; N:
(12)
2
0K
S 0K
N =1 =1 0 0 N L (13)
0 , L 0 0 8i; j
where, L
=1 6=
; ; N i 6 j . If all the phase differences are contained in the set
j 0 j < = , L > , i ; ; N . Using Gershgorin’s Theorem
[6], all eigenvalues of the matrix L are located in the union of the N
N
N
_ =
cos( i
j
_ )
j
i)
ij =
_T
=
_
N
k
k
cos( k
i
cos( i
j)
=
=
1 ...
i
_
j )( i
i
ii =
2
j
0
ii
= 1 ...
discs
[
z2C z0
N
i=1
N
0 :
cos( k
N
0
cos( k
i)
i)
:
Thus all eigenvalues are located in the right half plane, and hence the
matrix L is positive semidefinite. The row sums of the matrix L are
zero, which implies that the matrix L has a zero eigenvalue with 1N
as the corresponding N dimensional eigenvector of ones. The matrix
L is also popularly known as the weighted Laplacian [4] in the graph
theory literature.
_ = Ni=1 !i . Therefore,
It is to be noted that (see [5], [14]) N
i=1 i
N
N
_
define = (1=N ) i=1 i = (1=N ) i=1 !i which implies that
is an invariant quantity. Following [13], the vector _ can be written as
_ =
1N +
(14)
where 1N is the N dimensional vector of ones associated with the zero
= 0 (as Ni=1 _i =
eigenvalue of the matrix L, satisfies N
i=1 i
N ). The vector is orthogonal to 1N and was referred to as the group
disagreement vector in [13]. Substituting (14) in (13) yields
d
dt
) dtd
L
2
(
1
2
+
T
1N +
(
T
)
T
1N
1N +
(
1N +
T
) =
0K
N
L L :
d=dt
T
1N +
1N
1N +
+
T
L
T
(
1N +
T
1N
T
+
)
) =
(
1N +
0K
N
(
T
1N
)
L
1N
)
T
Using invariance of ((
) 1N
1N = 0), orthogonality of the disagreement vector ( T 1N = 0), and the fact that 1N is an eigenvector
T
associated with the zero eigenvalue of L (1N
L = 0) we have
d dt
(
T
)
0 NK L:
2
=
T
(15)
In order to get a bound on the convergence rate of , it is easy to
verify that the smallest non-zero eigenvalue of the matrix L (using the
fact that the smallest nonzero eigenvalue of the N 2 N Laplacian of a
complete graph is N, and minfcos(i 0 j ) : ji 0 j j (=2) 0 g=
cos((=2) 0 ) = sin()), is lower bounded by N sin(). Using this
fact in (15) yields
d 0 K L 0 K dt
N
) kk e0 sin( )( 0 )
) j 0 j j j e0 sin( )( 0 ) 8i ; ; N
(
III. SUFFICIENT CONDITIONS FOR SYNCHRONIZATION
_
cos( j
j =1
Consider the positive function, S = (1=2)_ T _ where _ =
_ ...
_ ]T . Differentiating S along trajectories of (1) and rear[
1
N
ranging terms we have that
1
!
(
K
N
0
. Substi-
Comparing the above estimate to the critical gain condition proposed
by [8]
lim
0
(10)
As Emax 2(N 0 1), it follows that synchronization is not possible
for all coupling gains K satisfying KL K < Kc .
The critical gain condition (9) can be written as
2 cos
Theorem 3.1: Consider the system of oscillators described by (1).
If there exists T 0 such that 8t T , ji 0 j j < =2 0 8i; j ,
where 0 < < =2, then the oscillator frequencies _i synchronize
exponentially to the mean frequency = (1=N ) N
! and satisfy
i=1 i
j_i 0 j T e K sin()(t T ) , T > 0 8i = 1; . . . ; N .
Proof: Differentiating the Kuramoto model (1) we have
1)
Consequently, using (3) the critical coupling gain required for the onset
of synchronization in (2) is given by Kc = (!j 0 !i )N=Emax , 8i; j
If the natural frequencies belong to a compact set, then the critical coupling gain required for onset of synchronization in (1) is given as
K
355
T
)
min (
2
K
T
_
i
=
i
T
)
t
K
T
2
sin( )
T
T
t
T
=1 . . .
(16)
356
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 2, FEBRUARY 2009
where the notation k 1 k denotes the Euclidean norm of . Thus the
exponential convergence rate for synchronization is no worse that
K sin().
It also follows from the above result that the order parameter [17]
exponentially converges to a constant. The underlying assumption in
the previous result is that 9T 0s:t:fji 0 j j < (=2) 0 8i; j 8t T . In the next section we use the coupling gain K to steer the phase
differences to this desired set in finite time.
Proof: Let a positive definite Lyapunov function for the dynamic
system governed by (18) be given as V = (1=2K )(i 0 j )2 . The
derivative of V along trajectories of the system (18) is given as
V_ =
1
K
= (i
(i
0 j )(_i 0 _j )
0 j )
!i 0 !j
0 sin(i 0 j )
K
2
IV. CONTROLLING THE PHASE DIFFERENCES
In this section, a lower bound on the coupling gain K is developed
which is sufficient to trap the oscillator phase differences within any desired arbitrary compact subset of (0(=2); (=2)), and hence by Theorem 3.1 guarantee oscillator synchronization. We consider two possible cases
1) The initial phase of all oscillators lies within the set described by
D = fi ; j j ji 0 j j (=2) 0 g;8i; j where < =2 is an
arbitrary positive number.
2) Let p = [1 0 2 . . . 1 0 N . . . N 01 0 N ]T , be the (N (N 0
1)=2)2 1 vector of phase differences. The initial phase differences
satisfy the condition kp k 0 where is an arbitrary small
strictly positive number.
The first assumption allows for the case when all the initial phase differences lie within a compact set in (0(=2); (=2)). The second condition relaxes this constraint but imposes a norm constraint on the phase
differences. However, it is evident that the above cases do not span
(0; ) and we will address this in our future work. For both of the
above scenarios, we show that there exists a large enough gain to trap
the phase differences within any desired arbitrary compact subset of
(0(=2); (=2)).
Starting with the first case, we develop a lower bound on the coupling
gain K , denoted by Kinv , that makes the set D positively invariant for
all oscillators, i.e. i (0) 0 j (0) 2 D ) i (t) 0 j (t) 2 D8t > 0.
The phase difference dynamics, as described by (2), can be rewritten as
!i 0 !j
0 sin(i 0 j )
K
_i 0 _j = K
+
1
N
N
(sin(i 0 j )+sin(k 0 i )+sin(j 0 k ))
:
(17)
Consider the term under the summation in (17). This can be rewritten
using standard trigonometric manipulations as (1=N ) sin(i 0 j )Ck
where Ck = (1 0 (cos(k 0 ((i + j )=2)=(cos(i 0 j )=2))). It
is easy to see that 8(i 0 j ) 2 D , jCk j < 1. Using the transformed
summation, (17) can be rewritten as
_i 0 _j = K f
1
!i 0 !j
0 sin(i 0 j ) +
K
N
ji 0 j j
N
Ck
10
Ck
N
!i 0 !j
0 (i 0 j )
K
N 02
2 sin(i 0 j ) 1 0
N
ji 0 j j
where we have used the fact that jCk j < 1. Thus the derivative can be
written as
V_ ji 0 j j
2
!i 0 !j
0 (i 0 j ) sin(i 0 j ) :
K
N
It is to be noted that the function sin(i 0 j )(i 0 j ) is always
nonnegative in the considered domain. Therefore, if K > (N j!i 0
!j j=2 cos()), the derivative of the Lyapunov function is negative at
ji 0 j j = (=2) 0 , and thus the phase difference i 0 j cannot
leave the set D . Finally, if K = Kinv > (N j!max 0 !min j=2 cos()),
all phase differences i 0 j 8i = 1; . . . ; N are positively invariant
with respect to the compact set D .
As the phase differences are trapped within the set D by appropriately choosing the coupling gain, using Theorem 3.1 the oscillators are
guaranteed to synchronize.
Consider the second case when the initial phase differences satisfy
the condition kp (0)k 0 . Our next result shows that for a large
enough coupling gain K , the vector p is ultimately bounded.
Theorem 4.2: Consider the system dynamics as described by (17).
Let the initial vector of phase differences be contained in the set kp k 0 . Then there exists a coupling gain K (s) > 0, for some s > 0
such that the phase differences enters the set described by kp k s in
finite time.
Proof: A positive definite function of the phase differences is
given as Vp = (1=2)kp k2 . Differentiating Vp along solutions of (17)
yields
(_i
V_ p =
0 _j )(i 0 j )
i<j
((!i
=
0 !j )(i 0 j ) 0 K sin(i 0 j )(i 0 j )
i<j
+
N
(18)
K
N
N
(sin(i
+ sin(j
We are now in a position to state the first result of this section.
Theorem 4.1: Consider the system dynamics as described by (18).
Let all initial phase differences at t=0 be contained in the compact set
D = fi ; j j ji 0 j j (=2) 0 8i; j = 1; . . . ; N g where 0 < <
=2. Then there exists a coupling gain Kinv > 0 such that (i (t) 0
j (t)) 2 D8t > 0.
N
N
Ck sin(i 0 j )g
Ck ):g
1
!i 0 !j
0 (i 0 j )
K
2 sin(i 0 j )
N
!i 0 !j
1
= Kf
0 sin(i 0 j ) 2 (1 0
K
N
10
0 j ) + sin(k 0 i )
0 k )) (i 0 j )) :
The term under the double summation can be simplified as
N
(N
0 2)
sin(i
0 j )(i 0 j ) +
i<j
+ sin(j
0 k ))(i 0 j ):
(sin(k
0 i )
i<j
(19)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 2, FEBRUARY 2009
Consider any two oscillators u; v , with v < u, among the N oscillators. Corresponding to these two oscillators, the first term in the above
equation contains the expression (N 0 2) sin(v 0 u )(v 0 u ). Now
consider the second term in (19). As both the oscillators are interconnected to N 0 2 other oscillators, the second term contains the two
sums r=1;r6=u;v sin(u 0 v )(v 0 r ) and r=1;r6=u;v sin(v 0
u )(u 0 r ). Adding the two sums results in the expression (N 0
2) sin(u 0 v )(v 0 u ). From the above discussion it follows that:
N
(sin(i
i<j
0 j ) + sin(k 0 i )+ sin(j 0 k )) (i 0j ) = 0:
The derivative of Vp is then given as
V_ p =
i<j
((!i
0 !j )(i 0 j ) 0 K sin(i 0 j )(i 0 j )) :
Using the fact that at time t = 0, sinc(i 0 j ) = (sin(i 0 j )=i 0
j ) 8i; j where 0 < 1 (as kp k 0 and sinc is a
decreasing even function in (0; )), the derivative reduces to
V_ p i<j
(!i
0 !j )(i 0 j ) 0 K
(i
0 j )2
:
Let !d = [!1 0 !2 . . . !1 0 !N . . . !N 01 0 !N ]T denote the vector of
the difference of natural frequencies. Then using the Cauchy–Schwarz
inequality
V_ p k!d kkp k 0 K
kp k2 kp k (k!d k 0 K kp k) :
Choose K (s) = (k!d k= s) + 2m, m > 0. The derivative of the
Lyapunov function reduces to
k!d k 0 k!sd k + 2m kp k
0 2m kp k2 if kp k s
) V_ p 0 4m Vp :
V_ p kp k
From the above inequality it is evident that Vp , and hence p , decays
at an exponential rate. Therefore, the phase differences enter the set
kp k s in finite time.
It is to be noted that the vector p is always decreasing until it enters the desired set. To ensure synchronization, choose s < =2 and
the coupling gain K (s) appropriately, then 9T 0 s:t: ji 0 j j <
(=2) 8i; j 8t T . Consequently, oscillator synchronization follows
from Theorem 3.1.
V. CONCLUSION
In this technical note we studied the phenomenon of synchronization
in the traditional Kuramoto model with an arbitrary but finite number
of oscillators. A necessary condition in the form of a lower bound on
the coupling gain K = Kc was established for the existence of a
phase-locked state in the Kuramoto model. This bound is tighter than
the result provided by Jadbabaie et al. [8]. Sufficient conditions for exponential synchronization of the oscillator frequencies to the mean natural frequency of the group were also developed. To the best of the authors’ knowledge this is the first result that demonstrates almost global
exponential synchronization in the finite oscillator Kuramoto model
with different natural frequencies, thereby complementing the recent
local stability results in [12]. A lower bound on the coupling gain was
also developed to guarantee oscillator synchronization. Furthermore,
the coupling gain was characterized so as to trap the oscillator phase
differences within any desired compact set in (0; ), provided the initial oscillator phase differences satisfy certain conditions. Future work
involves studying synchronization for probabilistic distribution of the
natural frequencies and incorporating time delays in communication.
357
ACKNOWLEDGMENT
The authors are grateful to the anonymous reviewers and to
Dr. S. H. Strogatz and Dr. B. A. Francis for their detailed and helpful
comments.
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