Network Synchronization With Nonlinear Dynamics and Switching Interactions

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 10, OCTOBER Network Synchronization With Nonlinear Dynamics an Switching Interactions Tao Yang, Ziyang Meng, Guoong Shi, Yiguang Hong, an Karl Henrik Johansson Abstract This technical note consiers the synchronization problem for networks of couple nonlinear ynamical systems uner switching communication topologies. Two types of nonlinear agent ynamics are consiere. The first one is non-expansive ynamics [stable ynamics with a convex Lyapunov function ϕ )] an the secon one is ynamics that satisfies a global Lipschitz conition. For the non-expansive case, we show that various forms of joint connectivity for communication graphs are sufficient for networks to achieve global asymptotic ϕ-synchronization. We also show that ϕ-synchronization leas to state synchronization provie that certain aitional conitions are satisfie. For the globally Lipschitz case, unlike the non-expansive case, joint connectivity alone is not sufficient for achieving synchronization. A sufficient conition for reaching global exponential synchronization is establishe in terms of the relationship between the global Lipschitz constant an the network parameters. Inex Terms Multi-agent systems, nonlinear agents, switching interactions, synchronization. I. INTRODUCTION We consier the synchronization problem for a network of couple nonlinear agents with agent set V = {1, 2,...,N}. Their interactions communications in the network) are escribe by a time-varying irecte graph G σt) =V, E σt) ), with σ :[0, ) Pas a piecewise constant signal, where P is a finite set of all possible graphs over V. The state of agent i V at time t is enote as x i t) R n an evolves as ẋ i = ft, x i ) j N i σt a ij t)x j x i ), i V 1) where ft, x i ):[0, ) R n R n is piecewise continuous in t an continuous in x i representing the uncouple inherent agent ynamics, N i σt = {j V:j, i) E σt) } is the set of agent i s neighbors Manuscript receive August 23, 2015; revise October 23, 2015; accepte November 2, Date of publication November 4, 2015; ate of current version September 23, This work was supporte in part by the Knut an Alice Wallenberg Founation an the Sweish Research Council. Recommene by Associate Eitor S. Zampieri. T. Yang is with the Pacific Northwest National Laboratory, Richlan, WA USA Tao.Yang@pnnl.gov). Z. Meng is with the Department of Precision Instrument an the State Key Laboratory of Precision Measurement Technology an Instruments, Tsinghua University, Beijing , China ziyangmeng@mail. tsinghua.eu.cn). G. Shi is with the College of Engineering an Computer Science, The Australian National University, Canberra, ACT 0200, Australia guoong.shi@anu.eu.au). Y. Hong is with the Key Laboratory of Systems an Control, Institute of Systems Science, Chinese Acaemy of Science, Beijing , China yghong@iss.ac.cn). K. H. Johansson is with the ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm, Sween kallej@kth.se). Digital Object Ientifier /TAC at time t, ana ij t) is a piecewise continuous function marking the weight of ege j, i) at time t. Note that N i σt may be an empty set at certain time intervals. Systems of the form 1) have attracte consierable attention. Most works focus on the case where the communication graph G σt) is fixe, e.g., [1] [7]. It is shown that for the case where ft, x i ) satisfies a Lipschitz conition, synchronization is achieve for a connecte graph provie that the coupling strength is sufficiently large. However, for the case where the communication graph is time-varying, the synchronization problem becomes much more challenging an existing literature mainly focuses on a few special cases when ft, x i ) is linear, e.g., the single-integrator case [8] [10], the ouble-integrator case [11], an the neutrally stable case [12], [13]. Other stuies assume some particular structures for the communication graph [14] [16]. In particular, in [14], the authors focus on the case where the ajacency matrices associate with all communication graphs are simultaneously triangularizable. The authors of [15] consier switching communication graphs that are weakly connecte an balance at all times. A more general case where the switching communication graph frequently has a irecte spanning tree has been consiere in [16]. These special structures on the switching communication graph are rather restrictive compare to joint connectivity where the communication can be lost at any time. This technical note aims to investigate whether joint connectivity for switching communication graphs can yiel synchronization for the nonlinear ynamics 1). We istinguish two cases epening on whether the nonlinear agent ynamics ft, x i ) is expansive or not. For the non-expansive case, we focus on the case where the nonlinear agent ynamics is stable with a convex Lyapunov function ϕ ). We show that various forms of joint connectivity for communication graphs are sufficient for networks to achieve global asymptotic ϕ- synchronization, that is, the function ϕ of the agent state converges to a common value. We also show that ϕ-synchronization implies state synchronization provie that aitional conitions are satisfie. For the expansive case, we focus on when the nonlinear agent ynamics is globally Lipschitz, an establish a sufficient conition for networks to achieve global exponential synchronization in terms of a relationship between the Lipschitz constant an the network parameters. The remainer of this technical note is organize as follows. Section II presents the problem efinition an main results. Section III provies all technical proofs. Finally, concluing remarks are offere in Section IV. A. Problem Set-Up II. PROBLEM DEFINITION AND MAIN RESULTS In this technical note, we consier the network of couple nonlinear agents 1). The communications are moele by a time-varying irecte graph G σt) =V, E σt) ). The joint graph of G σt) in the time interval [t 1,t 2 ) with t 1 <t 2 is enote as G[t 1,t 2 = t [t1,t 2 )Gt) =V, t [t1,t 2 )E σt) ). For the communication graph, we introuce the following efinition IEEE. Personal use is permitte, but republication/reistribution requires IEEE permission. See for more information.

2 3104 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 10, OCTOBER 2016 Definition 1: i) G σt) is uniformly jointly strongly connecte if there exists a constant T>0 such that G[t, t T is strongly connecte for any t 0. ii) Assume that G σt) is unirecte for all t 0. G σt) is infinitely jointly connecte if G[t, is connecte for any t 0. We assume that there are constants 0 <a a such that a a ij t) a for all t 0. We also make a stanar well time assumption [17] on the switching signal σt): there is a lower boun τ D > 0 between two consecutive switching time instants of σt). We enote x =[x T 1,xT 2,...,xT N ]T R nn an assume that the initial time is t = t 0 0, an the initial state xt 0 )=x T 1 t 0 ),...,x T N t 0 T R nn. In this technical note, we are intereste in the following synchronization problems. Definition 2: The multi-agent system 1) achieves global asymptotic ϕ-synchronization, where ϕ : R n R is a continuously ifferentiable function, if for any initial state xt 0 ), there exists a constant xt 0, such that lim t ϕx i t = xt 0 for any i V an any t 0 0. Definition 3: i) The multi-agent system 1) achieves global asymptotic synchronization if lim t x i t) x j t = 0 for any i, j V, any t 0 0 an any xt 0 ) R nn. Multi-agent system 1) achieves global exponential synchronization if there exist γ 1 an λ>0 such that x i t) x j t) 2 γe λt t 0) x i t 0 ) x j t 0 ) 2, t t 0 2) for any t 0 0 an any xt 0 ) R nn. Remark 1: ϕ-synchronization is a type of output synchronization where the output of agent i Vischosen to be ϕx i ). It is relate to butifferent fromχ-synchronization [18], [19] since ϕ is afunction of an iniviual agent state while χ is a function of all agent states. B. Non-Expansive Dynamics In this section, we focus on when the nonlinear agent ynamics is non-expansive as inicate by the following assumption. Assumption 1: ϕ : R n R is a continuously ifferentiable positive efinite convex function satisfying 1). lim η ϕη) = ; 2). ϕη),ft, η) 0 for any η R n an any t 0. The following lemma shows how Assumption 1 enforces nonexpansive ynamics. Lemma 1: Let Assumption 1 hol. Along the multi-agent ynamics 1), i V ϕx i t is non-increasing for all t 0. We now state main results for the non-expansive case. Theorem 1: Let Assumption 1 hol. The multi-agent system 1) achieves global asymptotic ϕ-synchronization if G σt) is uniformly jointly strongly connecte. Theorem 2: Let Assumption 1 hol. Assume that G σt) is unirecte for all t t 0. The multi-agent system 1) achieves global asymptotic ϕ-synchronization if G σt) is infinitely jointly connecte. Remark 2: For the linear time-varying case ft, x) =At)x, if there exists a matrix P = P T > 0 such that PAt)A T t)p 0, t 0 3) then ϕx) =x T Pxfor x R n satisfies Assumption 1. For the linear time-invariant case ft, x) =Ax, the conition 3) is equivalent to that the matrix A is neutrally stable [13]. C. ϕ-synchronization vs. State Synchronization The following result establishes conitions uner which ϕ- synchronization implies state synchronization. Theorem 3: Let G σt) Gwith G being a fixe, strongly connecte igraph uner which the multi-agent system 1) achieves global asymptotic ϕ-synchronization for some positive efinite function ϕ : R n R. Let Assumption 1 hol. Moreover, assume that 1) ft, η) is boune for any t 0 an any η R n. 2) c 1 η 2 ϕη) c 2 η 2 for some 0 <c 1 c 2 ; 3) ϕ ) is strongly convex. Then the multi-agent system 1) achieves global asymptotic synchronization. D. Lipschitz Dynamics We consier also the case when the nonlinear agent ynamics is possibly expansive. We focus on when the ynamics satisfies the following global Lipschitz conition. Assumption 2: There exists a constant L>0 such that ft, η) ft, ζ) L η ζ, η, ζ R n, t 0. 4) Our main result for this case is given below. Theorem 4: Let Assumption 2 hol. Assume that G σt) is uniformly jointly strongly connecte. The multi-agent system 1) achieves global exponential synchronization if L<ρ /2, where ρ is a constant epening on the network parameters. Remark 3: Assumption 2 an its variants have been consiere in the literature for fixe communication graphs, e.g., [1] [7]. Compare with the existing literature, we here stuy a more challenging case, where the communication graphs are time-varying. Unlike the fixe case where the global Lipschitz conition is sufficient to guarantee synchronization, Theorem 4 establishe a sufficient synchronization conition relate to the Lipschitz constant an the network parameters. III. PROOFS OF MAIN RESULTS In this section, we provie proofs of the main results. A. Proof of Lemma 1 Recall that the upper Dini erivative of a function ht) :a, b) R at t is efine as D ht) = lim sup s 0 ht s) ht/s).the following lemma from [10], [20] is useful for the proof. Lemma 2: Let V i t, x): R R n R i =1,...,N) be continuously ifferentiable an V t, x) = i=1,...,n V i t, x). IfIt) = {i {1, 2,...,N} : V t, xt = V i t, xt} is the set of inices where the imum is reache at t, then D V t, xt = i It) V i t, xt. Denote It) ={i V: j V ϕx j t = ϕx i t}. Wefirst note that the convexity property of ϕ ) implies that [21, pp. 69] ϕη),ζ η ϕζ) ϕη), η, ζ R n. 5) It then follows from Lemma 2, Assumption 1ii) an 5) that: D ϕx it = ϕx i ),ft, x i ) a ij t)x j x i ) i V i It) i It) j N i σt j N i σt a ij t)ϕx j ) ϕx i 0

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 10, OCTOBER where the last inequality follows from ϕx j ) ϕx i ). This proves the lemma. B. Proof of Theorem 1 It follows from Lemma 1 that for any initial state xt 0 ) R nn, there exists a constant = xt 0 0, such that lim t i V ϕx i )=. We shall show that is the require constant in Definition 2 of ϕ-synchronization. We first note that, by Lemma 1, for all i V, there exist constants 0 α i β i, such that lim inf ϕ x it = α i, lim sup ϕ x i t = β i. t t Also note that it follows from lim t i V ϕx i t = that for any ε>0, there exists T 1 ε) > 0 such that: ϕ x i t [0, ε], i V, t T 1 ε). 6) The proof of Theorem 1 is base on a contraiction argument an relies on the following lemma. Lemma 3: Let Assumption 1 hol. Assume that G σt) is uniformly jointly strongly connecte. If there exists an agent k 0 V such that 0 α k0 <, then there exists 0 < ρ <1 an t such that for all i V, ϕx i t N 1)T 0 ρm 0 1 ρ) ε),where T 0 T 2τ D 7) with T given in Definition 1i) an τ D is the well time. Proof: LetusfirstefineM 0 α k0 β k0 )/2) <.Then there exists an infinite time sequence t 0 < t 1 < < t k < with lim k t k = such that ϕx t k = M 0 for all k =1, 2,...Let then t k be greater than or equal to T 1 ε), an enote it by t k0. We now prove the lemma by estimating an upper boun of the scalar function ϕx i ) agent by agent. The proof is base on a generalization of the metho propose in the proof of [22, Lemma 4.3] but with substantial ifferences on the ynamics an the Lyapunov function. Moreover, the convexity of ϕ ) plays an important role. Step 1. Focus on agent k 0. By using Assumption 1ii), 5), an 6), we obtain that for all t t k0 t ϕx k 0 t = ϕx k0 ),ft, x k0 ) a k0 jt)x j x k0 ) j N k0 σt j N k0 σt a k0 jt)ϕx j ) ϕ x k0 a N 1) ε ϕ x k0. It then follows that for all t t k0 : ϕ x k0 t e λ 1t t k0 ) ϕ xk0 t k0 1 e λ 1t t k0 ) ) ε) 8) where λ 1 = a N 1). Step 2. Consier agent k 1 k 0 such that k 0,k 1 ) E σt) for t [ t k0, t k0 T 0 ). The existence of such an agent can be shown as follows. Since G σt) is uniformly jointly strongly connecte, it is not har to see that there exists an agent k 1 k 0 Van t 1 t k0 such that k 0,k 1 ) E σt) for t [t 1,t 1 τ D ) [ t k0, t k0 T 0 ). From 8), we obtain that for all t [ t k0, t k0 N 1)T 0 ] ϕ x k0 t κ 0 ρm 0 1 ρ) ε) 9) where ρ = e λ 1N 1)T 0 = e a N 1) 2 T 0. We can similarly boun ϕx k1 t by consiering two ifferent cases: i) ϕx k1 t >ϕx k0 t for all t [t 1,t 1 τ D ) an ii) there exists a time instant t 1 [t 1,t 1 τ D ) such that ϕx k1 t 1 ϕx k0 t 1 κ 0. For both cases, we obtain that for all t [t 1 τ D, t k0 N 1)T 0 ] ϕ x k1 t ϕ 1 M 0 1 ϕ 1 ) ε). where ϕ 1 =1 μ)ρ 2. It then follows from 9) an 0 <ϕ 1 <ρ<1 that for all t [t 1 τ D, t k0 N 1)T 0 ]: ϕ x j tϕ 1 M 0 1 ϕ 1 ) ε), j {k 0,k 1 }. 10) Step 3. Consier agent k 2 {k 0,k 1 } such that there exists an ege from the set {k 0,k 1 } to the agent k 2 in E σt) for t [t 2,t 2 τ D ) [ t k0 T 0, t k0 2T 0 ). The existence of such an agent k 2 an t 2 follows similarly from the argument in Step 2. Similarly, we can boun ϕx k2 t by consiering two ifferent cases an obtain that for all t [t 2 τ D, t k0 N 1)T 0 ] ϕ x k2 t ϕ 2 M 0 1 ϕ 2 ) ε) 11) where ϕ 2 = 1 μ)ρ 2 ) 2. By combining 10) an 11), an using 0 <ϕ 2 <ϕ 1 < 1, we obtain that for all t [t 2 τ D, t k0 N 1)T 0 ] ϕ x j t ϕ 2 M 0 1 ϕ 2 ) ε), j {k 0,k 1,k 2 }. Step 4. By repeating the above process on time intervals [ t k0 2T 0, t k0 3T 0 ),..., [ t k0 N 2)T 0, t k0 N 1)T 0 ),weeventually obtain that for all i V ϕ x i t k0 N 1)T 0 ϕn 1 M 0 1 ϕ N 1 ) ε). where ϕ N 1 = 1 μ)ρ 2 ) N 1. The result of the lemma then follows by choosing ρ = ϕ N 1 an t = t k0. We are now reay to prove Theorem 1 by contraiction. Suppose that there exists an agent k 0 V such that 0 α k0 <.Itthen follows from Lemma 3 that ϕx i t N 1)T 0 < for all i V, provie that ε< ρ M 0 )/1 ρ. This contraicts the fact that lim t i V ϕx i )=. Thus, there oes not exist an agent k 0 V such that 0 α k0 <. Hence, lim t ϕx i t = for all i V. C. Proof of Theorem 2 The proof relies on the following lemma. Lemma 4: Let Assumption 1 hol. Assume that G σt) is infinitely jointly connecte. If there exists an agent k 0 Vsuch that 0 α k0 <, then there exist 0 < ρ <1 an t such that ϕ x i t τ D ) ) ρm 0 1 ρ) ε), i V. The proof of Lemma 4 is similar to that of Lemma 3 an base on estimating an upper boun for the scalar quantity ϕx i ) agent by agent. However, the metho by which we obtaine the sequence of agents k 0,k 1,...,k N 1 in the proof of Theorem 1 oes not apply uner the conition that G σt) is infinitely jointly connecte. We can however apply the strategy in [22]. We omit the etails an refer to [23] for a complete proof. The remaining proof of Theorem 2 follows from a contraiction argument an Lemma 4 in the same way as the proof of Theorem 1.

4 3106 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 10, OCTOBER 2016 D. Proof of Theorem 3 If the multi-agent system 1) reaches asymptotic ϕ-synchronization, i.e., lim t ϕx i t = for all i V, then for any ɛ>0, there exists a T ɛ > 0 such that Step 3. We first note that the strong convexity of ϕ ) implies that [21, pp. 459] there exists an m>0 such that ϕη),ζ η ϕζ) ϕη) m 2 η ζ 2, η, ζ R n. 14) ɛ ϕ x i t ɛ, i V, t T ɛ. 12) If =0 the esire conclusion hols trivially, i.e., lim t x i t) =0for all i Vue to the positive efiniteness of ϕ ). Inthe remainer of the proof we assume > 0. We shall prove the result by contraiction. Suppose that state synchronization is not achieve. Then there exist two agents i 0,j 0 V such that lim sup t x i0 t) x j0 t) > 0. In other wors, there exist an infinite time sequence t 1 < <t k < with lim k t k =, an a constant >0 such that x i0 t k ) x j0 t k ) =2 N 1 /c 1 for all k 1, wherec 1 is given in conition ii) of Theorem 3. We ivie the following analysis into three steps. Step 1. In this step, we prove the following crucial claim. Claim.Foranyt k, there are two agents i,j Vwith i,j ) E such that x i t k ) x j t k ) /c 1. We establish this claim via a recursive analysis. If either i 0,j 0 ) E or j 0,i 0 ) E then the result follows trivially. Otherwise we pick up another agent k 0 satisfying that there is an ege between k 0 an {i 0,j 0 }.Thisk 0 always exists since G is strongly connecte. Then either x k0 t k ) x i0 t k ) 2 N 2 /c 1 or x k0 t k ) x j0 t k ) 2 N 2 /c 1 must hol. Thus, we have again either establishe the claim, or we can continue to select another agent ifferent from i 0, j 0,ank 0 an repeat the argument. Since we have a finite number of agents, the esire claim hols. Furthermore, since there is a finite number of agent pairs, without loss of generality, we assume that the given agent pair i,j oes not vary for ifferent t k otherwise we can always select an infinite subsequence of t k for the following iscussions). Step 2. In this step, we establish a lower boun of x i t) x j t) 2 for a small time interval after a particular t k satisfying t k > T ɛ. From 12) an lim x ϕx) = giveninassumption1i), we see that x i t) an ϕx i t) x j t are boune for all i, j V an for all t T ɛ. It then follows from conition i) of Theorem 3 an 1) that: t ϕ x i x j ) ϕ x i x j ),ft, x i ) f t, x j ) ϕ x i x j ), ϕ x i x j ), k 1 N i σt k 2 N j σt a i k 1 t)x k1 x i ) a j k 2 t)x k2 x j ) L for all t t k an some L > 0. Without loss of generality we assume that ɛ 1. ThenL will be inepenent of ɛ. By plugging in the fact that x i t k ) x j t k ) /c1 an using the conition ii) of Theorem 3, we obtain that x i t) x j t) 2 2c 2, t [ t k,t k 2L ]. 13) By using Assumption 1ii) an 14), we obtain for t T ɛ t ϕx j ta j i t) ϕ x i ) ϕx j ) m 2 a j i t) x i t) x j t) 2 k N j σt\{i } a j kt) ϕx k ) ϕ x j ). 15) By using 12), 13), 15), an conition ii) of Theorem 3, we obtain that for t [t k,t k /2L )] which yiels ϕ x j t k t ϕ x j t 2N 1)a ɛ a m 4c 2 2L It is then straightforwar to see that if we take ϕ x j t k [ ɛ 2N 1)a ɛ a ] m. 4c 2 2L 2L { a m 2 ɛ<min, 32c 2 L < a m 2 a m 32c 2 L 16N 1)a c 2 However, this contraicts the efinition of ϕ-synchronization since t k is arbitrarily chosen. This completes the proof an the esire conclusion hols. E. Proof of Theorem 4 }. The proof is base on the convergence analysis of the scalar quantity where V t, xt = V ij t, xt= 1 2 e 2Lt t 0) x i t) x j t) 2, V ij t, xt 16) {i, j} V V. 17) Unlike the contraiction argument use in the proof of Theorem 1, where the convergence rate is unclear, here we explicitly characterize the convergence rate. The proof relies on the following lemmas. Lemma 5: Let Assumption 4 hol. Along the multi-agent ynamics 1), V t, xt is non-increasing for all t 0.

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 10, OCTOBER Proof: Let V 1 V 2 be the set containing all the noe pairs that reach the imum at time t, i.e., V 1 t) V 2 t) ={{i, j} V V: V ij t) =V t)}. It is not har to obtain that D V = 1 2 {i,j} V 1 V 2 e 2Lt t 0) x i x j ) T ft, x i ) ft, x j e 2Lt t 0) x i x j ) T e 2Lt t 0) x i x j ) T k 1 N i σt k 2 N j σt Le 2Lt t0) x i x j 2 {i,j} V 1 V 2 e 2Lt t 0) k 1 N i σt {i,j} V 1 V 2 a ik1 t) x j x k1 2 x i x j 2) e 2Lt t 0) k 2 N j σt a ik1 t)x k1 x i ) a jk2 t)x k2 x j ) a jk2 t) x i x k2 2 x i x j 2) k 1 N i σt k 2 N j σt a ik1 t)v jk1 V ij ) a jk2 t)v ik2 V ij ) 0 18) where the first equality follows from Lemma 2 an 1), the first inequality follows from 4) an ±ab a 2 b 2 )/2) for all a, b R, an the secon inequality follows from 17). Lemma 6: Let Assumption 4 hol. Assume that G σt) is uniformly jointly strongly connecte. Then there exists 0 < β <1 such that V ij NT0,xNT 0 ) ) βv, {i, j} V V where N = N 1, T 0 given by 7) an V = V t 0,xt 0. Proof: The proof is base on the convergence analysis of V ij t, xt) for all agent pairs {i, j} V Vin several steps, which is similar to the proof of Lemma 3. Without loss of generality, we assume that t 0 =0. We also sometimes enote V t, xt an V ij t, xt as V an V ij, respectively, for notational simplification. Step 1. We begin by consiering any agent i 1 V.SinceG σt) is uniformly jointly strongly connecte, we know that i 1 is the root an that there exists a time t 1 ananagenti 2 V\{i 1 } such that i 1,i 2 ) Euring t [t 1,t 1 τ D ) [0,T 0 ]. We first note that it follows from 16) an Lemma 5 that for all t [0, NT 0 ] V ij t, xt V t, xt V, {i, j} V V. 19) Taking the erivative of V ij along the trajectories of 1), we obtain that for all t [t 1,t 1 τ D ) V i1 i 2 = Le 2Lt t0) x i1 x i2 2 e 2Lt t0) x i1 x i2 ) T a i1 k 1 t)x k1 x i1 ) k 1 N i1 σt a i2 k 2 t)x k2 x i2 )f t, x i1 ) ft, x i2 k 2 N i2 σt a i1 kt)v i2 k 1 V i1 i 2 ) a i2 i 1 t)v i1 i 2 k 1 N i1 σt k 2 N i2 σt\{i 1 } a i2 k 2 t)v i1 k 2 V i1 i 2 ) N 1)a V V i1 i 2 ) a V i1 i 2 N 2)a V V i1 i 2 ) ) 2N 3)a = α V i1i2 V α where α =2N 3)a a. The first inequality follows from 4) an 17), while the secon inequality follows from 19). It thus follows that: V i1 i 2 t 1 τ D,xt 1 τ D ˆα 1 V 20) where ˆα 1 =1 a /α)1 e ατ D) 0, 1). Similarly we obtain that for all t [t 1 τ D, NT 0 ], Vi1 i 2 αv V i1 i 2 ),whereα =2N 1)a. It then follows from 20) that: V i1 i 2 t, xt α 1V, t [t 1 τ D, NT 0 ] 21) where α 1 =1 1 e ατ D )a /α)e αnt 0 0, 1). Step 2. Since G σt) is uniformly jointly strongly connecte, we know that that there exists a time instant t 2 an an arc from h V 1 {i 1,i 2 } to i 3 V\V 1 uring [t 2,t 2 τ D ) [T 0, 2T 0 ]. We then estimate an upper boun for V hi3 by consiering two ifferent cases: h = i 1 an h = i 2. We eventually obtain that for all t [t 2 τ D, NT 0 ] where V hi3 t, xt 1 β 2 ) V, h V 1. 22) β =1 e ατ D ) a α e αnt 0 0, 1). 23) It then follows from 21), 22), an 1 β 2 > 1 β = α 1 that for all t [2T 0, NT 0 ]: V i1 k t, xt 1 β 2 ) V, k V 2 \{i 1 } where V 2 {i 1,i 2,i 3 }. Step 3. By continuing the above process, we obtain that for all k V\{i 1 } V i1 k NT0,xNT 0 ) ) ) 1 β N V. 24) Step 4. Since G σt) is uniformly jointly strongly connecte, 24) hols for any i 1 V. By using the same analysis, we eventually obtain that for all i, j V, V ij NT 0,xNT 0 1 β N )V. Hence the result follows by choosing β =1 β N with β given by 23). We are now reay to prove Theorem 4. By using Lemma 6 an 16), we obtain that V t, xt β t/nt 0 V 1/ β)e ρ t V,where

6 3108 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 61, NO. 10, OCTOBER 2016 t/nt 0 enotes the largest integer that is not greater than t/nt 0 ) an ρ =1/NT 0 )ln1/ β). It then follows from 16) an 17) that: x i t) x j t) 2 1 β e ρ 2L)t x i 0) x j 0) 2. Hence, global exponential synchronization is achieve with γ = 1/ β) an λ = ρ 2L provie that ρ > 2L. This conclues the proof of the esire theorem. IV. CONCLUSION In this technical note, synchronization problems for networks with nonlinear agent ynamics an switching topologies have been investigate. Two types of nonlinear ynamics were consiere: nonexpansive an globally Lipschitz. For the non-expansive case, we foun that the convexity of the Lyapunov function plays a crucial rule in the analysis an showe that the uniformly joint strong connectivity is sufficient for achieving global asymptotic ϕ-synchronization. When communication graphs are unirecte, the infinitely joint connectivity is a sufficient synchronization conition. Moreover, we establishe conitions uner which ϕ-synchronization implies state synchronization. For the globally Lipschitz case, we foun that joint connectivity alone is not sufficient to achieve synchronization but establishe a sufficient synchronization conition. The propose conition reveals the relationship between the Lipschitz constant an the network parameters. The results of this technical note can be extene to leaerfollower networks, see [23] for etails. An interesting future irection is to stuy the synchronization problem for couple non-ientical nonlinear ynamics uner general switching topologies. REFERENCES [1] C. W. Wu an L. O. Chua, Application of graph theory to the synchronization in an array of couple nonlinear oscillators, IEEE Trans. Circuits Syst. I, Funam. Theory Appl., vol. 42, no. 8, pp , Aug [2] C.W.Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems. Singapore: Worl Scientific: Singapore, [3] V. Belykh, I. Belykh, an M. Hasler, Connection graph stability metho for synchronize couple chaotic systems, Physica D, vol. 195, no. 1/2, pp , Aug [4] P. DeLellis, M. D. Bernaro, an G. Russo, On QUAD, Lipschitz, contracting vector fiels for consensus an synchronization of networks, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 58, no. 3, pp , Mar [5] W. Yu, G. Chen, an M. Cao, Consensus in irecte networks of agents with nonlinear ynamics, IEEE Trans. Autom. Control, vol. 56, no. 6, pp , Jun [6] U. Münz, A. Papachristooulou, an F. Allgöwer, Consensus in multiagent systems with coupling elays an switching topology, IEEE Trans. Autom. Control, vol. 56, no. 12, pp , Dec [7] H. Liu, M. Cao, an C. W. Wu, Coupling strength allocation for synchronization in complex networks using spectral graph theory, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 61, no. 5, pp , May [8] A. Jababaie, J. Lin, an A. S. Morse, Coorination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Autom. Control, vol. 48, no. 6, pp , Jun [9] L. Moreau, Stability of multiagent systems with time-epenent communication links, IEEE Trans. Autom. Control, vol. 50, no. 2, pp , Feb [10] Z. Lin, B. Francis, an M. Maggiore, State agreement for continuoustime couple nonlinear systems, SIAM J. Control Optim., vol. 46, no. 1, pp , [11] W. Ren an R. W. Bear, Consensus seeking in multiagent systems uner ynamically changing interaction topologies, IEEE Trans. Autom. Control, vol. 50, no. 5, pp , May [12] L. Scarovi an R. Sepulchre, Synchronization in networks of ientical linear systems, Automatica, vol. 45, no. 11, pp , Nov [13] Y. Su an J. Huang, Stability of a class of linear switching systems with applications to two consensus problem, IEEE Trans. Autom. Control, vol. 57, no. 6, pp , Jun [14] J. Zhao, D. J. Hill, an T. Liu, Synchronization of complex ynamical networks with switching topology: A switche system point of view, Automatica, vol. 45, no. 11, pp , Nov [15] J. Qin, H. Gao, an W. Zheng, Exponential synchronization of complex networks of linear systems an nonlinear oscillators: A unifie analysis, IEEE Trans. Neural Netw. Learn. Syst., vol. 61, no. 2, pp , Mar [16] G. Wen, Z. Duan, G. Chen, an W. Yu, Consensus tracking of multi-agent systems with Lipschitz-type noe ynamics an switching topologies, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 61, no. 2, pp , Feb [17] D. Liberzon an A. S. Morse, Basic problem in stability an esign of switche systems, IEEE Control Syst. Mag., vol. 19, no. 5, pp , Oct [18] J. Cortés, Distribute algorithms for reaching consensus on general functions, Automatica, vol. 44, no. 3, pp , Mar [19] X. Wang an Y. Hong, Distribute finite-time χ-consensus algorithms for multi-agent systems with variable coupling topology, J. Syst. Sci. Complexity, vol. 23, no. 2, pp , [20] J. Danskin, The theory of -min, with applications, SIAM J. Appl. Math., vol. 14, no. 6, pp , [21] S. Boy an L. Vanenberghe, Convex Optimization. New York, NY, USA: Cambrige Univ. Press, [22] G. Shi, K. H. Johansson, an Y. Hong, Reaching an optimal consensus: Dynamical systems that compute intersections of convex sets, IEEE Trans. Autom. Control, vol. 58, no. 3, pp , Mar [23] T. Yang, Z. Meng, G. Shi, Y. Hong, an K. H. Johansson, Network synchronization with nonlinear ynamics an switching interactions, Systems an Control, pp. 1 27, [Online]. Available: abs/