# Finite-Time Synchronization of Singular Hybrid Coupled Networks

Authors: Cong Zheng; Jinde Cao
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper investigates finite-time synchronization of the singular hybrid coupled networks. The singular systems studied in this paper are assumed to be regular and impulse-free. Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled networks under a state feedback controller by using finite-time stability theory. A numerical example is finally exploited to show the effectiveness of the obtained results.


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## Body

## 1. Introduction

In recent years, singular systems, also known as descriptor systems, generalized state-space systems, differential-algebraic systems, or semistate systems, are attracting more and more attentions from many fields of scientific research because they can better describe a larger class of dynamic systems than the regular ones. Many results of regular systems have been extended to the area about singular systems such as [1–21]. For example, stability (robust stability or quadratic stability) and stabilization for singular systems have been studied via LMI approach in [2–8]; robust control (or H2, H∞ control and robust dissipative filtering) for singular systems has been discussed in [9–16]; synchronization (or state estimation) for singular complex networks has been considered in [17–21].Synchronization is an interesting and important characteristic in the coupled networks. There are a lot of results in regular coupled networks. Recently, some authors study synchronization of the singular systems such as [17–21] and the references therein. In [17], Xiong et al. introduced the singular hybrid coupled systems to describe complex network with a special class of constrains. They gave a sufficient condition for global synchronization of singular hybrid coupled system with time-varying nonlinear perturbation based on Lyapunov stability theory. Synchronization issues are studied for singular systems with delays by using Linear Matrix Inequality (LMI) approach [18]. Koo et al. considered synchronization of singular complex dynamical network with time-varying delays [19]. Li et al. in [20] investigated synchronization and state estimation for singular complex dynamical networks with time-varying delays. Li et al. in [21] investigated robust H∞ control of synchronization for uncertain singular complex delayed networks with stochastic switched coupling.Finite-time synchronization or finite-time control is interesting topic for its practical application. There are some results on finite-time stability [22–26], finite-time synchronization [27–33], finite-time consensus or agreement [34–37], and finite-time observers [38]. However, these results are obtained for regular systems. Up to now, to the best of our knowledge, few authors studied finite-time synchronization of singular hybrid coupled systems whose structures are more complex than those in [27–33]. Considering the important role of synchronization of complex networks, the finite-time synchronization of singular hybrid coupled networks is worth studying.Motivated by the previous discussions, in this paper, we investigate finite-time synchronization of singular hybrid complex systems. Some sufficient conditions for it are obtained by the state feedback controller based on the finite-time stability theory. Finally, a numerical example is exploited to illustrate the effectiveness of the obtained result.The rest of this paper is organized as follows. In Section2, a singular hybrid coupled system is given, and some preliminaries are briefly outlined. In Section 3, some sufficient criteria are derived for the finite-time synchronization of the proposed singular system by the feedback controller. In Section 4, an example is provided to show the effectiveness of the obtained results. Some conclusions are finally drawn in Section 5.


## 2. Model Formulation and Some Preliminaries

Consider a singular hybrid coupled system as follows:(1)Ex˙i(t)=Axi(t)+f(xi(t),t)+c∑j=1N‍bij(t)Γxj(t),i=1,2,…,N,
where xi(t)=(xi1(t),xi2(t),…,xin(t))T∈ℝn represents the state vector of the ith node, A,E∈ℝn×n are constant matrices, and E may be singular. Without loss of generality, we will assume that 0<rank(E)=r<n. f(xi(t),t) is a vector-value function. The constant c>0 denotes the coupling strength, and Γ=diag(γ1,γ2,…,γn)∈ℝn×n is inner-coupling matrix between nodes. B=(bij)N×N describes the linear coupling configuration of the network, which satisfies
(2)bij=bji,fori≠j,bii=-∑j=1,j≠iNbij,i=1,2,…,N.Remark 1.
Ifrank(E)=n, then system (1) is a general nonsingular coupled network. We will also give a sufficient condition of the finite-time synchronization for this circumstance. See Corollary 10.Definition 2.
The singular system (1) is said to be synchronized in the finite time, if for a suitable designed feedback controller, there exists a constant t*>0 (which depends on the initial vector value x(0)=(x1T(0),x2T(0),…,xNT(0))T), such that limt→t*∥xi(t)-xj(t)∥=0 and ∥xi(t)-xj(t)∥≡0 for t>t*, i,j=1,2,…,N.Assumption 3.
Assume that the singular system (1) is connected in the sense that there are no isolated clusters; that is, the matrix B is an irreducible matrix.With Assumption3, we obtain that zero is an eigenvalue of B with multiplicity 1, and all the other eigenvalues of B are strictly negative, which are denoted by 0=λ1>λ2≥⋯≥λN. At the same time, since B is a symmetric matrix, there exists a unitary matrix W=(W1,W2,…,WN)∈ℝn×n such that B=WΛWT with WWT=I and Λ=diag(λ1,λ2,…,λN).Lets(t) be a function to which all xi(t) are expected to synchronize in the finite time. That is, the synchronization state is s(t). Suppose that s(t) satisfies the equation Es˙(t)=As(t)+f(s(t),t). Let ei(t)=xi(t)-s(t),i=1,2,…,N. We can obtain the following singular error system:
(3)Ee˙i(t)=Aei(t)+f(xi(t),t)-f(s(t),t)+c∑j=1N‍bij(t)Γej(t),i=1,2,…,N.Lete(t)=(e1(t),e2(t),…,eN(t)), y(t)=e(t)W; then system (3) can be written as
(4)Ee˙(t)=Ae(t)+F(e(t),t)+cΓe(t)BT,Ey˙(t)=Ay(t)+F(e(t),t)W+cΓy(t)Λ,
where F(e(t),t) = (f(x1(t),t)-f(s(t),t),f(x2(t),t)-f(s(t),t),…,f(xN(t),t)-f(s(t),t)), y(t)=(y1(t),y2(t),…,yN(t)),and yi(t)=e(t)Wi∈ℝn, i=1,2,…,N. Then, system (4) can be written as
(5)Ey˙i(t)=Ayi(t)+F(e(t),t)Wi+cλiΓyi(t)=(A+cλiΓ)yi(t)+F(e(t),t)Wi.Therefore, the finite-time synchronization problem of system (1) is equivalent to the finite-time stabilization of system (5) at the origin under the suitable controllers ui, i=1,2,…,N.Assumption 4.
Assume that there exist nonnegative constantsLi such that
(6)∥f(xi(t),t)-f(s(t),t)∥≤Li∥xi(t)-s(t)∥,i=1,2,…,N.Assumption 5.
There exist matricesPi such that
(7)ETPi=PiTE≥0,i=1,2,…,N,(8)ATP1+P1TA<0,(A+cλiΓ)TPi+PiT(A+cλiΓ)≤-ηiI,i=2,…,N,
where ηi>2L(N-1)∥Pi∥, L=∑i=1N‍Li.Lemma 6 (see [26]).
Suppose that the functionV(t):[t0,∞)→[0,∞) is differentiable (the derivative of V(t) at t0 is in fact its right derivative) and V˙(t)≤-K(V(t))α, ∀t≥0, V(t0)≥0, where K>0, 0<α<1 are two constants. Then, for any given t0, V(t) satisfies the following inequality:
(9)V1-α(t)≤V1-α(t0)-K(1-α)(t-t0),t0≤t≤t*,V(t)≡0,∀t>t*,
with t* given by t*=t0+V1-α(t0)/K(1-α).Lemma 7 (Jensen’s Inequality).
Ifa1,a2,…,an are positive numbers and 0<r<p, then
(10)(∑i=1n‍aip)(1/p)≤(∑i=1n‍air)(1/r).


## 3. Main Results

In this section, we consider the finite-time synchronization of the singular coupled network (1) under the appropriate controllers. In order to control the states of all nodes to the synchronization state s(t) in finite time, we apply some simple controllers ui(t)∈ℝn,i=1,2,…,N, to system (1). Then, the controlled system can be written as
(11)Ex˙i(t)=Axi(t)+f(xi(t),t)+c∑j=1N‍bijΓxj(t)+ui,i=1,2,…,N.
Then, we have
(12)Ee˙i(t)=Aei(t)+f(xi(t),t)-f(s(t),t)+c∑j=1N‍bijΓej(t)+ui,(13)Ey˙i(t)=(A+cλiΓ)yi(t)+F(e(t),t)Wi+vi,
where vi=uWi, u=(u1,u2,…,uN).With Assumption5, it follows from the proof of Theorem  1 in [2] and Lemma  2.2 in [3] that the pair (E,A+cλiΓ) is regular and impulse-free; that is, there exist nonsingular matrices Mi,Qi∈ℝn×n satisfying that
(14)MiEQi=diag{Ir,0},Mi(A+cλiΓ)Qi=diag{Ai,In-r},
where Ai∈ℝr×r, i=1,2,…,N. So, system (13) is equivalent to
(15)y˙i1(t)=Aiyi1(t)+Mi1F(e(t),t)Wi+Mi1vi,(16)0=yi2(t)+Mi2F(e(t),t)Wi+Mi2vi,
where Qi-1yi(t)=(yi1(t)yi2(t)), yi1(t)∈ℝr, and yi2(t)∈ℝn-r. And Mi=(Mi1Mi2), Mi1∈ℝr×n, Mi2∈ℝ(n-r)×n, Qi=(Qi1Qi2), Qi1∈ℝn×r, and Qi2∈ℝn×(n-r).In order to achieve our aim, we design the following controllers:(17)vi=-kMi-1sign(MiEe(t)Wi)|MiEe(t)Wi|β,
where
(18)MiEe(t)Wi=MiEyi(t)=MiEQiQi-1yi(t)=(Ir000)(yi1yi2)=(yi10),|MiEyi(t)|β=(|yi11(t)|β,…,|yir1(t)|β,0,…,0︸n-r)T,sign(MiEyi(t))=diag((yir1(t)0,…,0︸n-r)sign(yi11(t)),sign(yi21(t)),…,sign(yir1(t)),0,…,0︸n-r).k>0 is a tunable constant, and the real number β satisfies 0<β<1. So, we obtain u=(v1,v2,…,vN)W-1=v(ξ1,ξ2,…,ξN). That is, ui=vξi.Remark 8.
From (17), the controllers ui are dependent not only on the coupled matrix B, but also on the singular matrix E. And from the shape of controllers, we only use the states yi1 of slow subsystems (15) in controllers vi, but we do not consider the states yi2 of fast subsystems (16). It is very special. It is interesting for our future research to design more general controller which makes the singular hybrid coupled networks synchronize in finite time.Theorem 9.
Suppose that Assumptions3, 4, and 5 hold. Under the controllers (17), the singular system (1) is synchronized in a finite time t*=t0+(V(1-β)/2(t0)/akb-(1+β)/2(1-β)), where V(t0)=∑i=1N‍yiT(t0)ETPiyi(t0)=∑i=1N‍(e(t0)Wi)TETPi(e(t0)Wi), e(t0) is the initial condition of e(t), and a and b are defined as (25).Proof.
Consider the following Lyapunov function:(19)V(t)=∑i=1N‍yiT(t)ETPiyi(t).
The derivative of V(t) along the trajectory of system (13) is
(20)V˙(t)=∑i=1N‍[+((A+cλiΓ)yi(t)+F(e(t),t)Wi+vi)TyiT(t)PiT((A+cλiΓ)yi(t)+F(e(t),t)Wi+vi)+((A+cλiΓ)yi(t)+F(e(t),t)Wi+vi)TPiyi(t)yiT(t)PiT((A+cλiΓ)yi(t)+F(e(t),t)Wi+vi)]=∑i=1N‍[2kyiT(t)PiTMi-1sign(MiEe(t)Wi)|MiEe(t)Wi|βyiT(t)(PiT(A+cλiΓ)+(A+cλiΓ)TPi)yi(t)+2yiT(t)PiTF(e(t),t)Wi-2kyiT(t)PiTMi-1sign(MiEe(t)Wi)|MiEe(t)Wi|β].
By using Assumption 4, one can get the following inequality:
(21)∥F(e(t),t)Wi∥=∥∑k=1N‍[f(xk(t),t)-f(s(t),t)]Wik∥≤∑k=1N‍Lk∥ek(t)∥=∑k=1N‍Lk∥y(t)ξk∥≤∑k=1N‍Lk∥y(t)∥=L∥y(t)∥≤L∑k=1N‍∥yk(t)∥,
where Wi=(Wi1,Wi2,…,Win)T and (ξ1,ξ2,…,ξN)=W-1=WT.
DefineMi-TPiQi=(Pi1Pi2Pi3Pi4), where Pi1∈ℝr×r, Pi2∈ℝr×(n-r), Pi3∈ℝ(n-r)×r, and Pi4∈ℝ(n-r)×(n-r). Using (7) and (8) (see [2]), one can obtain that Pi1=(Pi1)T>0 and Pi2=0; then,(22)V(t)=∑i=1N‍yiT(t)ETPiyi(t)=∑i=1N‍(yi1(t))TPi1yi1(t),(23)-2kyiT(t)PiTMi-1sign(MiEe(t)Wi)|MiEe(t)Wi|β=-2kyiT(t)PiTMi-1(sign(yi11(t))|yi11(t)|β,…,sign(yir1(t))|yir1(t)|β,0,…,0)=-2kyiT(t)Qi-T(Pi10Pi3Pi4)TMiMi-1×diag(sign(yi1(t)),0)(|yi1(t)|β0)=-2kyi1T(t)Pi1Tsign(yi1(t))|yi1(t)|β≤-2kλmin(Pi1T)|yi1(t)|1+β.
Substituting (8), (21), and (23) into (20) and letting η1=0, while ηi≥2L(N-1)∥Pi∥, i=2,…,N, one has
(24)V˙(t)≤∑i=1N‍[-ηi∥yi(t)∥2+2L∥Pi∥∑j=1N‍∥yj(t)∥∥yi(t)∥-2kλmin(Pi1T)|yi1(t)|1+β∑j=1N‍∥yj(t)∥]≤∑i=1N‍-2kλmin(Pi1T)|yi1(t)|1+β.
Let(25)a≜mini{λmin(Pi1T)},b≜maxi{λmax(Pi1T)}.
From (22), we get
(26)a∑i=1N∥yi1(t)∥2≤V(t)≤b∑i=1N∥yi1(t)∥2.
By the use of (24)–(26) and Lemma 7, we can obtain that
(27)V˙(t)≤-2ak∑i=1N|yi1(t)|1+β≤-2ak(∑i=1N∥yi(t)∥2)(1+β)/2≤-2ak(1bV(t))(1+β)/2=-2akb-(1+β)/2(V(t))(1+β)/2.
From Lemma6, we have that the solutions yi1(t) of system (15) are globally asymptotically stable with respect to yi1(t)=0 in the finite time t*; that is,
(28)limt→t*∥yi1(t)∥=0,∥yi1(t)∥=0fort≥t*,
where t*=t0+(V(1-β)/2(t0)/akb-(1+β)/2(1-β)).
In the following, we show thatyi2(t) are globally asymptotically stable with respect to yi2(t)=0 in the finite time t*. From (16), one has
(29)∥yi2(t)∥≤∥Mi2∥∥F(e(t),t)Wi∥+∥Mi2∥∥vi∥.
Similar to the proof of Lemma  2.2 in [3] and the proof of Theorem  1 in [17], let Mi2(Mi2)T=In-r, which implies that ∥Mi2∥=1. One has (30)∥yi2(t)∥≤∥F(e(t),t)Wi∥+∥uiWi∥≤L∑j=1N‍∥yj(t)∥+∥|MiEyi(t)|β-kMi-1sign(MiEyi(t))×|MiEyi(t)|β∥≤L∑j=1N‍∥Qj∥(∥yj1(t)∥+∥yj2(t)∥)+k∥Mi-1∥×|yi1(t)|β.
Then,
(31)∑i=1N‍∥yi2(t)∥≤∑i=1N[L∑j=1N‍∥Qj∥(∥yj1(t)∥+∥yj2(t)∥)+k∥Mi-1∥|yi1(t)|β∑j=1N‍];
that is,
(32)∑i=1N‍(1-NL∥Qi∥)∥yi2(t)∥≤∑i=1N(|yi1(t)|βNL∥Qi∥∥yj1(t)∥+k∥Mi-1∥|yi1(t)|β).
With Assumption4, there must exist nonsingular matrices Mi,Qi∈ℝn×n satisfying the equalities MiEQi=diag{Ir,0}, Mi(A+cλiΓ)Qi=diag{Ai,In-r}, where Ai∈ℝr×r, i=1,2,…,N. Moreover, nonsingular matrices Qi can be suitably chosen to satisfy 1-NL∥Qi∥>0, for ∀i∈{1,2,…,N}. Therefore, one can obtain limt→t*∥yi2(t)∥=0, and ∥yi2(t)∥=0 for t≥t* from (28) and (32), i=1,2,…,N. Consequently, limt→t*∥ei(t)∥=0, and ∥ei(t)∥=0 for t≥t*, i=1,2,…,N. The proof is completed.Ifrank(E)=n, system (1) is a general nonsingular coupled network. By using the controllers ui similar to vi in (17), we can derive the finite-time synchronization of system (1). For simplicity, let E=In. Then, we have the following.Corollary 10.
WhenE=In, under Assumptions 3 and 4, let the controllers ui(t) be as follows:
(33)ui(t)=-ksign(ei(t))|ei(t)|β,i=1,2,…,N;
system (1) is synchronized in a finite time.Remark 11.
Since the conditions in Assumption5 are not strict LMIs problems, they cannot be solved directly by the LMI Matlab Toolbox. According to Lemma  1 in [17], Lemma  1 in [9], and Remark  3 in [18], if matrix E has the decomposition as
(34)E=U(Ir000)(Ξr00In-r)VT,
where U=(U1,U2), V=(V1,V2), and Ξr=diag{ϱ1,…,ϱr} with ϱi>0 for i=1,2,…,r, then Assumption 5 can be transformed into a strict LMIs problem.Corollary 12.
Suppose that Assumptions3 and 4 hold, and matrix E has the decomposition as (34) in Remark 11. By the controllers (17), the singular hybrid coupled network (1) can be synchronized in the finite time in the sense of Definition 2, if there exist matrices Ti∈ℝr×r, Ti≥0, and Si∈ℝ(n-r)×r, i=1,2,…,N, such that
(35)AT(U1T1U1TE+U2S1)+(U1T1U1TE+U2S1)TA<0,(A+cλiΓ)T(U1TiU1TE+U2Si)+(U1TiU1TE+U2Si)T(A+cλiΓ)≤-ηiI,
where ηi>2L(N-1)∥Pi∥, Pi=(U1TiU1TE+U2Si),i=2,…,N, and L=∑i=1N‍Li.Suppose that we choose the average state of all node states as synchronized state; that is,s(t)=(1/N)∑k=1N‍xk(t). We have similar results. Before giving these results, we need some assumptions as follows:Assumption2′. Assume that there exist nonnegative constants Lij such that
(36)∥f(xi(t),t)-f(xj(t),t)∥≤Lij∥xi(t)-xj(t)∥,i,j=1,2,…,N.Assumption3′. There exist matrices Pi such that
(37)ETPi=PiTE≥0,i=1,2,…,N,ATP1+P1TA<0,(A+cλiΓ)TPi+PiT(A+cλiΓ)≤-ςiI,i=2,…,N,
where ςi>(4L/N)(N-1)∥Pi∥,L=∑i=1N‍Li,  andLi=∑k=1,k≠iN‍Lik.Theorem 13.
Suppose that Assumptions3, 2′, and 3′ hold. By the controllers (17), the singular hybrid coupled network (1) can be synchronized to the average state of all node states in the finite time in the sense of Definition 2.Corollary 14.
Suppose that Assumptions3 and 2′ hold, and matrix E has the decomposition as (34) in Remark 11. By the controllers (17), if there exist matrices Ti∈ℝr×r, Ti≥0, and Si∈ℝ(n-r)×r, i=1,2,…,N, such that
(38)AT(U1T1U1TE+U2S1)+(U1T1U1TE+U2S1)TA<0,(A+cλiΓ)T(U1TiU1TE+U2Si)+(U1TiU1TE+U2Si)T(A+cλiΓ)≤-ηiI,
where ηi>(4L/N)(N-1)∥Pi∥, Pi=(U1TiU1TE+U2Si), i=2,…,N, L=∑i=1N‍Li, and Li=∑k=1,k≠iN‍Lik, the singular hybrid coupled network (1) can be synchronized to the average state of all node states in the finite time.Remark 15.
In this paper, we study finite-time synchronization of the singular hybrid coupled networks when the singular systems studied in this paper are assumed to be regular and impulse-free. However, it may be more complicated when we do not assume in advance that the systems are regular and impulse free. Synchronization or finite-time synchronization of singular coupled systems is worth discussing without the assumption that the considered systems are regular and impulsive free.


## 4. An Illustrative Example

In this section, a numerical example will be given to verify the theoretical results obtained earlier.Example 16.
Consider the following singular hybrid coupled network which is similar to one given in [18]:
(39)Ex˙i(t)=Axi(t)+f(xi(t),t)+c∑j=16‍bijΓxj(t)+ui,i=1,2,…,6,
where xi(t)=(xi1(t),xi2(t))T, f(xi(t),t)=((1/15)tanh(xi1(t)),tanh(xi2(t)))T, s(t)=(0,0)T, Li=1/15, L=2/5, c=1, and
(40)E=(8000),A=(-1011-10),Γ=(1001),B=(-5111111-4111011-4101111-4101101-4110101-3).
Since B is symmetric matrix and its six eigenvalues are λ1=0,λ2=-3,λ3=-4,λ4=-5,λ5=-6,andλ6=-6, there exists a unitary matrix
(41)W=(W1,…,W6)=(-16000-16-26-16160-12160-160-120-1616-1616012160-160120-1616-16-2600160)
such that B=WΛWT and Λ=diag(0,-3,-4,-5,-6,-6).
Choose(42)Mi=(1110-λi01),Qi=(18018(10-λi)1λi-10),Pi=(1001),ηi>4, i=1,2,…,6, and  β=1/2 satisfying Assumptions 3, 4, and 5. Under the controllers ui defined in Theorem 9, the singular hybrid system (39) can be synchronized in the finite time t*=7.1858 according to Theorem 9 if k=1. If the controller gain k=5, t*=1.4372. Corresponding simulation results are shown in Figures 1 and 2 with initial conditions e(0)=(1320.72.50.30.5-10.8-21.5-0.5).Figure 1
Error variableei1(t)(i=1,2,…,6) of system (39) with k=1.Figure 2
Error variableei2(t)(i=1,2,…,6) of system (39) with k=1.


## 5. Conclusions

In this paper, we discuss finite-time synchronization of the singular hybrid coupled networks with the assumption that the considered singular systems are regular and impulsive-free. Some sufficient conditions are derived to ensure finite-time synchronization of the singular hybrid coupled networks under a state feedback controller by finite-time stability theory. A numerical example is finally exploited to show the effectiveness of the obtained results. It will be an interesting topic for the future researches to extend new methods to study synchronization, robust control, pinning control, and finite-time synchronization of singular hybrid coupled networks without the assumption that the considered singular systems are regular and impulsive-free.



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