Synchronization of complex networks has been extensively investigated by researchers due to its numerous practical applications. As one of the most celebrated periodic-oscillator models, Kuramoto model [1] and its variations have been widely used for explaining various synchronization phenomena, and they have attracted considerable attention from researchers in diverse fields ranging from biology [2, 3], mathematics [4], physics [5, 6] and engineering [7–10]. In the past decade, many progresses concerning on the synchronization of Kuramoto-oscillator networks have been made by researchers in the control community [10–20], where synchronization criteria with respect to constraints on coupling strengths and initial phases have been developed. For example, in [10], the relationship between the algebraic connectivity of a connected Kuramoto-oscillator network and critical coupling was revealed. In [11], Chopra and Spong showed that initial relative phases should be confined to π/2 and a critical coupling strength should be satisfied, which guaranteed the frequency synchronization of an all-to-all connected Kuramoto network.

In [14, 15, 17, 20], researchers have taken the pacemaker (i.e. the so-called leader) into consideration, where synchronization criteria were related to not only the constraint on coupling strengths and initial phases, but also the selection of direct controlled oscillators. Since the interactions between oscillators are usually in the sinusoidal form of phase differences, the theoretical results mentioned above were based on the requirements of initial phases, and only local stability analyses were provided. Based on the framework of cyber-physical systems [21, 22], distributed linear controllers have been adopted to synchronize Kuramoto-oscillator networks in [23, 24], where the derived stability conditions were independent of the initial phases such that the global synchronization was achieved. In [24], sufficient criteria were established for the Kuramoto-oscillator network with a pacemaker under distributed linear control.

The results aforementioned merely focused on the asymptotical synchronization, which indicated that synchronization was realized when t → ∞. Recently, in [18, 19, 25–27], more researchers have focused on the finite-time synchronization of Kuramoto-oscillator networks, which is also of significance in practical applications. For example, power girds need to get rid of local power failures as soon as possible in order to avoid the cascading failure. In [27], Wu and Li investigated the finite-time and fixed-time synchronization of Kuramoto-oscillator networks by employing a novel multiplex control. However, the finite-time synchronization of Kuramoto-oscillator network in present of a pacemaker has not been investigated so far.

Inspired by the above literatures, it is worth investigating the finite-time synchronization of Kuramoto-oscillator network with a pacemaker. In this paper, we aim to explore finite-time synchronization criteria of such network by adopting distributed schemes based on CPS. The main contributions of this paper are summarized as follows: Firstly, effective criteria are established to deal with finite-time phase agreement and frequency synchronization for Kuramoto-oscillator network with a pacemaker, and the upper bound of synchronization time is also provided; Secondly, synchronization can be achieved for arbitrary initial phases, which only influence the upper bound of synchronization time; Finally, the requirement on the connectivity of physical system is relaxed, even if it is an unconnected network.

The remainder of this paper is organized as follows. In Section 2, the framework of CPS is constructed, which consists of the physical Kuramoto-oscillator network system and the cyber (controlling) system. Furthermore, two definitions and some necessary mathematical preliminaries are encompassed in Section 2. Finite-time phase agreement in an identical Kuramoto-oscillator network and frequency synchronization in a non-identical Kuramoto-oscillator network cover the heart body of Section 3 and Section 4, respectively. Section 5 presents the numerical simulation results, and Section 6 concludes the whole paper.

In the framework of CPS, a Kuramoto-oscillator network consisting of N oscillators with control input u _i can be described as θ ̇ i = ω i + ∑ j = 1 N a i j ⁡ sin θ j − θ i + u i , i ∈ I , (1) where I = 1 , … , N , θ _i and ω _i are the phase and natural frequency of the ith oscillator, respectively. A = ( a i j ) N × N denotes the adjacency matrix of an undirected network, where a _ij = a _ji (i ≠ j) > 0 iff there is an edge between oscillator i and oscillator j; otherwise, a _ij = 0. Let L A = D A − A be the Laplacian matrix associated with the adjacency matrix A, where D ^A ∈ R ^N×N is a diagonal matrix with D i i A = ∑ j = 1 N a i j ( ∀ i ∈ I ) . The network associated with the adjacency matrix A is called physical network.

Assume that there is a pacemaker with dynamics θ ̇ 0 = ω 0 , where θ ₀ and ω ₀ are the phase and natural frequency of the pacemaker, respectively.

In this paper, we concern phase agreement and frequency synchronization with respect to the pacemaker in finite time.

Definition 1

Network Eq. 1 with control input u _i achieves (pacemaker-based) finite-time phase agreement, if there exists a settling time T > 0 depending on the initial states θ _i (0) (i ∈ {0}⋃I), such that lim t → T θ i − θ 0 = 0 , i ∈ I , (2) and θ _i − θ ₀ ≡ 0 for t ≥ T.

Definition 2

Network Eq. 1 with control input u _i achieves (pacemaker-based) finite-time frequency synchronization, if there exists a settling time T > 0 depending on the initial states θ ̇ i ( 0 ) ( i ∈ { 0 } ⋃ I ) , such that lim t → T θ i ̇ − θ 0 ̇ = 0 , i ∈ I , (3) and θ i ̇ − θ 0 ̇ ≡ 0 for t ≥ T.

In order to obtain the sufficient conditions, the following Lemmas are needed.

Lemma 1

[28] For an undirected graph G with N nodes, x T L x = 1 2 ∑ i = 1 N ∑ j = 1 N a i j ( x i − x j ) 2 holds, where x = ( x 1 , x 2 , … , x N ) T and L is the Laplacian matrix of G .

Lemma 2

[29] Consider the system of differential equation x ̇ t = f x t (4) where f : D → R n is continuous on an open neighborhood D ⊆ R n of the origin and f (0) = 0. A continuously differentiable function x : I → D is said to be a solution of Eq. 4 on the interval I ⊂ R if x satisfies Eq. 4 for all t ∈ I .

If there exists a continuous function V(x): D → R such that

(1) V(x) is positive definite;

(2) There exist real numbers c > 0, 0 < ρ < 1, and an open neighborhood D 0 ⊆ D of the origin such that V ̇ ( x ) ≤ − c V ρ ( x ) , x ∈ D 0 \ { 0 } .

Then, the origin is a finite-time stable equilibrium of Eq. 4 and the finite settling time T satisfies T ≤ V 1 − ρ x 0 c 1 − ρ .

If in addition D 0 = D = R N , the origin is globally finite-time stable equilibrium.

For the sake of convenience, let ξ _i = θ _i − θ ₀, then ξ i ̇ = θ i ̇ − θ 0 ̇ = θ i ̇ − ω 0 . For a real symmetric matrices L , let λ min ( L ) be the minimum eigenvalue of matrix L . Denote s i g ( x ) α = sign ( x ) x α , where the signum function sign(x) is defined as s i g n x = 1 , ∀ x > 0 , 0 , x = 0 , − 1 , ∀ x < 0 .

In this section, we first concentrate on the case of oscillators with identical natural frequency, i.e., ω _i = ω ₀, ∀i ∈ I. Thus, network Eq. 1 with control input u _i becomes θ ̇ i = ω + ∑ j = 1 N a i j ⁡ sin θ j − θ i + u i , i ∈ I . (5)

For achieving finite-time phase agreement, a distributed control strategy is constructed as u i = ∑ i = 1 N b i j θ j − θ i + f i s i g θ 0 − θ i α , (6) where B = ( b i j ) N × N denotes the adjacency matrix of an undirected network with elements b _ij defined similar to a _ij , f _i ≥ 0, and parameter 0 < α < 1. The network associated with the connections between the oscillators in the controller Eq. 6 is called cyber network. Let L B be the Laplacian matrix associated with the adjacency matrix B, where its elements are defined similar to L A .

By transforming θ _i into ξ _i , network Eq. 5 with distributed control strategy Eq. 6 becomes ξ ̇ i = ∑ j = 1 N a i j ⁡ sin ξ j − ξ i + ∑ j = 1 N b i j ξ j − ξ i − f i s i g ξ i α . (7)

Theorem 1

Network Eq. 1 with identical oscillator under distributed control strategy Eq. 6 achieves finite-time phase agreement with the settling time bounded by T 1 ≤ ξ 0 1 − α 1 − α f min , (8) if λ min cos ⁡ γ ⋅ L A + L B ≥ 0 , where f _min = min{f ₁, …, f _N }, ‖ ξ ( 0 ) ‖ 2 = ∑ i = 1 N [ ξ i ( 0 ) ] 2 and γ ∈ (π, 2π) satisfies tan γ = γ.

Proof 1

Consider the following Lyapunov functional candidate V 1 = 1 2 ξ T ξ = 1 2 ∑ i = 1 N ξ i 2 .

The derivation of V ₁ along trajectories Eq. 7 gives V ̇ 1 = ∑ i = 1 N ξ i ξ ̇ i = ∑ i = 1 N ∑ j = 1 N a i j ξ i ⁡ sin ξ j − ξ i + ∑ i = 1 N ∑ j = 1 N b i j ξ i ⁡ sin ξ j − ξ i − ∑ i = 1 N f i ξ i s i g ξ i α . (9)

According to Lemma 1 and the fact sin ( θ j − θ i ) ( θ j − θ i ) ∈ [ cos ⁡ γ , 1 ] , we can obtain ∑ i = 1 N ∑ j = 1 N a i j ξ i ⁡ sin ξ j − ξ i = 1 2 ∑ i = 1 N ∑ j = 1 N a i j ξ i − ξ j sin ξ j − ξ i = − 1 2 ∑ i = 1 N ∑ j = 1 N a i j sin ξ j − ξ i ξ j − ξ i ξ j − ξ i 2 ≤ − cos ⁡ γ 2 ⋅ ∑ i = 1 N ∑ j = 1 N a i j ξ j − ξ i 2 = − cos ⁡ γ ⋅ ξ T L A ξ . (10)

And, ∑ i = 1 N ∑ j = 1 N b i j ξ i ξ j − ξ i = − 1 2 ∑ i = 1 N ∑ j = 1 N b i j ξ j − ξ i 2 = − ξ T L B ξ , (11) ∑ i = 1 N f i ξ i s i g ξ i α = ∑ i = 1 N f i ξ i sign ξ i ξ i α ≤ ∑ i = 1 N f i ξ i 1 + α . (12)

Combining Eqs. 10–12, Eq. 9 yields V ̇ 1 ≤ − ξ T cos ⁡ γ ⋅ L A + L B ξ − ∑ i = 1 N f i ξ i 1 + α .

If λ min ( cos ⁡ γ ⋅ L A + L B ) ≥ 0 , we get V ̇ 1 ≤ − ∑ i = 1 N f i ξ i 1 + α = − ∑ i = 1 N f i ξ i 2 1 + α 2 ≤ − f min 2 1 + α 2 1 2 ∑ i = 1 N ξ i 2 1 + α 2 = − 2 1 + α 2 f min V 1 1 + α 2 .

By Lemma 2 and Definition 1 , network Eq. 1 with identical oscillator under distributed control strategy Eq. 6 achieves finite-time phase agreement with the settling time bounded by T 1 ≤ ∑ i = 1 N ξ i 0 2 1 − α 2 1 − α f min = ξ 0 1 − α 1 − α f min .

This completes the proof.

Remark 1

According to (8), we find that the upper bound of synchronization time is proportionate to initial state ‖ξ(0)‖, and is inversely proportional to f _min . According to Theorem 1 , it is sufficient to achieve finite-time phase agreement if λ min ( cos ⁡ γ ⋅ L A + L B ) ≥ 0 . Therefore, even if the physical network is not connected, phase agreement could be also achieved with the help of cyber network, which relaxes the requirement on the connectivity of the physical network.

Now we further concentrate on the case of oscillators with non-identical natural frequencies, i.e., there exists some i ∈ I such that ω _i ≠ ω ₀. For achieving finite-time frequency synchronization, a distributed control strategy u _i is designed as u i = ∑ j = 1 N b i j θ j − θ i + U i , (13) where U i ̇ = f i s i g ( θ ̇ 0 − θ ̇ i ) α , f _i ≥ 0, parameter 0 < α < 1, and b _ij denotes the same as that in Eq. 6. Let L B be the Laplacian matrix associated with the adjacency matrix B, where its elements are defined similar to L A .

Theorem 2

Network Eq. 1 with non-identical oscillators under distributed control strategy Eq. 13 achieves frequency synchronization with the settling time bounded by T 2 ≤ ξ ̇ 0 1 − α 1 − α f min , (14) if λ min L B − L A ≥ 0 , where ‖ ξ ̇ ( 0 ) ‖ 2 = ∑ i = 1 N [ ξ ̇ i ( 0 ) ] 2 .

Proof 2

By taking the derivation of Eq. 1 , we obtain ξ i ̈ = θ i ̈ = ∑ j = 1 N a i j ⁡ cos ξ j − ξ i ξ ̇ j − ξ ̇ i + u ̇ i . (15)

Consider the following Lyapunov functional candidate V 2 = 1 2 ξ T ̇ ξ ̇ = 1 2 ∑ i = 1 N ξ i ̇ 2 .

The derivation of V ₂ along trajectories Eq. 15 gives V ̇ 2 = ∑ i = 1 N ξ ̇ i ξ ̈ i = ∑ i = 1 N ξ ̇ i ∑ j = 1 N a i j ⁡ cos ξ j − ξ i ξ ̇ j − ξ ̇ i + u ̇ i = ∑ i = 1 N ξ ̇ i ∑ j = 1 N a i j ⁡ cos ξ j − ξ i ξ ̇ j − ξ ̇ i + ∑ j = 1 N b i j ξ ̇ j − ξ ̇ i − f i s i g ξ ̇ i α = ∑ i = 1 N ∑ j = 1 N ξ ̇ i a i j ⁡ cos ξ j − ξ i ξ ̇ j − ξ ̇ i + ∑ i = 1 N ∑ j = 1 N b i j ξ ̇ i ξ ̇ j − ξ ̇ i − ∑ i = 1 N ξ ̇ i s i g ξ ̇ i α . (16)

According to Lemma 1 and the fact cos ( θ j − θ i ) ≤ 1 , we can obtain ∑ i = 1 N ∑ j = 1 N ξ ̇ i a i j ⁡ cos ξ j − ξ i ξ ̇ j − ξ ̇ i = − 1 2 ∑ i = 1 N ∑ j = 1 N a i j ξ ̇ j − ξ ̇ i 2 ⁡ cos ξ j − ξ i ≤ 1 2 ∑ i = 1 N ∑ j = 1 N a i j ξ ̇ j − ξ ̇ i 2 = ξ ̇ T L A ξ ̇ . (17)

And, ∑ i = 1 N ∑ j = 1 N b i j ξ ̇ i ξ ̇ j − ξ ̇ i = − 1 2 ∑ i = 1 N ∑ j = 1 N b i j ξ ̇ j − ξ ̇ i 2 = − ξ ̇ T L B ξ ̇ , (18) ∑ i = 1 N ξ ̇ i s i g ξ ̇ i α = ∑ i = 1 N f i ξ ̇ i sign ξ ̇ i ξ ̇ i α ≤ ∑ i = 1 N f i ξ ̇ i α + 1 . (19)

Combining Eqs. 17–19, Eq. 16 yields V ̇ 2 ≤ − ξ ̇ T L B − L A ξ ̇ − ∑ i = 1 N f i ξ ̇ i α + 1 .

If λ min ( L B − L A ) ≥ 0 , we get V ̇ 2 ≤ − ∑ i = 1 N f i ξ ̇ i α + 1 = − ∑ i = 1 N f i ξ ̇ i 2 1 + α 2 = − f min 2 1 + α 2 1 2 ∑ i = 1 N ξ ̇ i 2 1 + α 2 = − 2 1 + α 2 f min V 2 1 + α 2 .

By Lemma 2 and Definition 2 , network Eq. 1 with non-identical oscillators under distributed control strategy Eq. 13 achieves finite-time frequency synchronization with the settling time bounded by T 2 ≤ ∑ i = 1 N ξ i ̇ 0 2 1 − α 2 1 − α f min = ξ ̇ 0 1 − α 1 − α f min .

This completes the proof.

Remark 2

According to Eq. 14 , we find that the upper bound of synchronization time is proportionate to initial state ‖ ξ ̇ ( 0 ) ‖ , and is inversely to the f _min . According to Theorem 2 , it is sufficient to achieve finite-time frequency synchronization if λ min ( L B − L A ) ≥ 0 . Therefore, even if the physical network is not connected, frequency synchronization could be also achieved with the help of cyber network, which relaxes the requirement on the connectivity of the physical network.

In this section, we assume networks associated with adjacency matrices A and B as shown in Figures 1A,B, respectively.

We first verify Theorem 1. Obviously, λ min ( cos ⁡ γ ⋅ L A + L B ) = 0 . For simplicity, set ω _i = 0 (i = 0, 1, 2, 3, 4, 5), α = 0.5, f _i = 2 (i = 1, 2, 3, 4, 5), and (θ ₀ (0), θ ₁ (0), θ ₂ (0), θ ₃ (0), θ ₄ (0), θ ₅ (0)) = (0.25,−0.1028,28.8866,−10.0289,5.3575,−17.3534) ^T . In Figure 2A, phase differences θ _i − θ ₀ converge to zero, which means finite-time phase agreement is achieved. Besides, it also shows phase agreement is achieved about 2.2s, which is less than the upper bound of settling time T ₁ = 5.9600s. Time evolutions of the controller Eq. 6 of each oscillator are showed in Figure 2B.

Secondly, we verify Theorem 2. Obviously, λ min ( L B − L A ) = 0 . Set α = 0.5, f _i = 2 (i = 1, 2, 3, 4, 5), θ ̇ 0 ( 0 ) = ω 0 = 2 , (ω ₁, ω ₂, ω ₃, ω ₄, ω ₅) = (−10,−4,0,4,10) ^T , and ( θ ̇ 1 ( 0 ) , θ ̇ 2 ( 0 ) , θ ̇ 3 ( 0 ) , θ ̇ 4 ( 0 ) , θ ̇ 5 ( 0 ) ) = ( 44.5324 , − 4.2299 , 17.8956 , − 39.9282 , − 18.2699 ) T . In Figure 3Afrequency differences θ i ̇ − θ 0 ̇ converge to zero, which means finite-time frequency synchronization is achieved. Besides, it also shows frequency synchronization is achieved about 0.825s, which is less than the upper bound of settling time bound T ₂ = 2.1147s. Time evolutions of the controller Eq. 13 of each oscillator are showed in Figure 3B.

Finally, we move to see the influence of parameter α on synchronization time. In the simulations, we set α = 0.1, 0.3, 0.5, 0.7, 0.9. In Figure 4, it is showed that the synchronization time decreases as α grows.

In this paper, the problems of finite-time phase agreement and frequency synchronization of Kuramoto-oscillator networks with a pacemaker have been investigated. Two distributed control strategies, based on the CPS, have been designed to drive the Kuramoto-oscillator networks. In the light of finite-time stability theory, the sufficient criteria have been derived for guaranteeing the phase agreement and frequency synchronization of identical and non-identical Kuramoto-oscillator networks with a pacemaker. At the same time, the upper bounds estimation of convergence time of Kuramoto-oscillator networks have been given accordingly. Numerical examples have validated the effectiveness of the derived theoretical results.

However, the convergence time estimations of this paper are heavily related to initial phases and/or frequencies of oscillators. Therefore, it is urgent to explore the fixed-time synchronization of Kuramoto model with a pacemaker in the future.