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Robust Network Synchronization of Time-Delayed Coupled Systems
6th IFAC International Workshop on Periodic Control Systems 6th IFAC Periodic Control Systems June 29 -International July 1, 2016. Workshop Eindhoven,on The Netherlands 6th IFAC IFAC International Workshop on Periodic Control Systems Systems 6th Periodic Control June 29 -International July 1, 2016. Workshop Eindhoven,on The Netherlands Available online at www.sciencedirect.com June 29 July 1, 2016. Eindhoven, The Netherlands June 29 - July 1, 2016. Eindhoven, The Netherlands
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Robust Network Network Synchronization Synchronization of of Robust Robust Network Synchronization Time-Delayed Coupled Coupled Systems Systems of Time-Delayed Time-Delayed Coupled Systems
∗∗ Carlos Murguia ∗∗ Justin Ruths ∗∗ Henk Nijmeijer ∗∗ Carlos Murguia ∗ Justin Ruths ∗ ∗ Henk Nijmeijer ∗∗ ∗∗ ∗ Carlos Murguia Justin Ruths Henk Nijmeijer Carlos Murguia Justin Ruths Henk Nijmeijer ∗ Singapore University of Technology and Design, Engineering ∗ ∗ Singapore University of Technology and Design, Engineering ∗ Singapore University of Center Technology and Design, Design, Engineering Systems and Design (ESD), for Research in Cyber Security Singapore University of Technology and Engineering Systems and Design (ESD), Center for Research in Cyber Security Systems and Design (ESD), Center for Research in Cyber Security (iTrust), e-mails: murguia [email protected] and Systems (iTrust), and Design (ESD), Center for Research in Cyber Security murguia (iTrust), e-mails: e-mails: murguia [email protected] [email protected] and and [email protected] (iTrust), e-mails: murguia [email protected] and [email protected] [email protected] ∗∗ [email protected] ∗∗ Eindhoven University of Technology, Mechanical Engineering Eindhoven University of Mechanical ∗∗ ∗∗ Eindhoven University e-mail: of Technology, Technology, Mechanical Engineering Engineering Department, [email protected] Eindhoven University of Technology, Mechanical Engineering Department, e-mail: [email protected] Department, e-mail: [email protected] Department, e-mail: [email protected] Abstract: We address the problem of controlled synchronization in networks of time-delayed Abstract: We the of synchronization in of Abstract: We address address the problem problem of controlled controlled synchronization in networks networks of time-delayed time-delayed coupled nonlinear systems. In particular, we prove that, under some mild conditions, there Abstract: We address the problem of controlled synchronization in networks of time-delayed coupled nonlinear systems. In particular, we prove that, under some mild conditions, there coupled nonlinear systems. In particular, we prove that, under some mild conditions, there always exists a unimodal region in the parameter space (coupling strength γ versus time-delay coupled nonlinear systems. In particular, we prove that, under some mild conditions, there always parameter space (coupling strength versus always exists unimodal region intothe the parameter space (coupling strength γ γWe versus time-delay τ ), suchexists that a ifunimodal γ and τ region belongin this region, the systems synchronize. showtime-delay how this always exists aaif unimodal region in the parameter space (coupling strength γ versus time-delay ττunimodal ), such that γ and τ belong to this region, the systems synchronize. We show how this ), such that if γ and τ belong to this region, the systems synchronize. We show how this region scales with the network topology, which, in turn, provides useful insights of τunimodal ), such that if γscales and with τ belong to this region, thewhich, systems synchronize. We showinsights how this region the network topology, in turn, provides useful of unimodal region scales withtopology the network network topology,robustness which, in in against turn, provides provides usefulThe insights of how to design thescales network to maximize time-delays. results unimodal region with the topology, which, turn, useful insights of how to the network to maximize against time-delays. The how to design design by thecomputer network topology topology to of maximize robustness againstneural time-delays. The results results are illustrated simulations coupled robustness Hindmarsh-Rose chaotic oscillators. how to design the network topology to maximize robustness against time-delays. The results are illustrated by computer simulations of coupled Hindmarsh-Rose neural chaotic oscillators. are illustrated by computer simulations of coupled Hindmarsh-Rose neural chaotic oscillators. are illustrated by computer simulations of coupled Hindmarsh-Rose neural chaotic oscillators. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Robustness, Delays, Synchronization, Control of Networks, Nonlinear Systems. Keywords: Robustness, Delays, Delays, Synchronization, Control Control of Networks, Networks, Nonlinear Systems. Systems. Keywords: Keywords: Robustness, Robustness, Delays, Synchronization, Synchronization, Control of of Networks, Nonlinear Nonlinear Systems. 1. INTRODUCTION that, under some mild assumptions, there always exists a 1. INTRODUCTION INTRODUCTION that, some mild there always 1. that, under under some mild assumptions, assumptions, there strength always exists exists region S in the parameter space (coupling γ ver-aa 1. INTRODUCTION that, under some mild assumptions, there always exists a region S in the parameter space (coupling strength γ region S in the parameter space (coupling strength γ verversus time-delay τ ), such that if (γ, τ ) ∈ S then the systems The emergence of synchronization in networks of coupled region S in the parameter space (coupling strength γ versus such (γ, S the The emergence emergence of synchronization synchronization in networks networks of sus time-delay time-delayToτττ ), ),derive such that that ifresults, (γ, τττ ))) ∈ ∈the S then then the systems systems synchronize. theirif authors assume The of in of coupled dynamical systems is a pervasive phenomenon in coupled various sus time-delay ), such that if (γ, ∈ S then the systems The emergence of synchronization in networks of coupled synchronize. To derive their results, the authors assume dynamical systems is a pervasive phenomenon in various synchronize. To derive their results, the authors assume that the individual systems are semipassive (Pogromsky dynamical systems is a pervasive phenomenon in various scientific disciplines ranging from biology, physics, and synchronize. To derive their are results, the authors assume dynamical systems isranging a pervasive phenomenon in various the individual systems semipassive (Pogromsky scientific disciplines disciplines from biology, physics, physics, and that that the individual systems are semipassive (Pogromsky et al. (1999)) with respect to the coupling variable and the scientific ranging from biology, and chemistry to social networks and technological applicathat the individual systems are semipassive (Pogromsky scientific disciplines ranging from biology, physics, and et al. (1999)) with respect to the coupling variable and the chemistry to examples social networks networks and technological technological applicaet al. al. (1999)) (1999)) with with respect to the the coupling coupling variablesystems, and the the corresponding internal dynamics are convergent chemistry to social and tions. Several of synchronous behavior inapplicascience et respect to variable and chemistry to social networks and technological applicacorresponding internal dynamics are convergent systems, tions. Several examples of synchronous behavior in science corresponding internal dynamics are convergent systems, (see Pavlov et al. (2004)). In the same spirit, here we prove tions. Several examples of synchronous behavior in science and engineering can be of found in, for instance, internal dynamics are spirit, convergent systems, tions. Several examples synchronous behaviorBlekhman in science corresponding (see et In the here and engineering engineering can be(2001), found in, for instance, instance, Blekhman (see Pavlov Pavlov et al. al.S(2004)). (2004)). In bounded the same same by spirit, here we we prove prove that the region is always a unimodal funcand be found in, for (1988), Pikovsky can et al. Strogatz (2003), Blekhman and refer- (see Pavlov et al. (2004)). In the same spirit, here we prove and engineering can be found in, for instance, Blekhman the region S is always bounded by aa unimodal func(1988),therein. Pikovsky et of al.the (2001), Strogatz (2003), and refer- that that the region S is always bounded by unimodal function ϕ(γ) defined on some set J ⊂ R; and consequently, (1988), Pikovsky et al. (2001), Strogatz (2003), and references One first technical results regarding that the region S is always bounded by a unimodal func(1988), Pikovsky et of al.the (2001), Strogatz (2003), and refer- tion ϕ(γ) defined on some set J ⊂ R; and consequently, ences therein. therein. One first technical results regarding tion ϕ(γ) ϕ(γ) defined on some set J ⊂ ⊂ R; R; and consequently, consequently, that there defined always on exists an set optimal coupling strength γ ∗∗ ences One of the first technical results regarding synchronization of coupled nonlinear systems is presented tion some J and ences therein. One of the first technical results regarding that there exists coupling γ synchronization of coupled coupled nonlinear systems isthe presented there always exists an an optimal optimal coupling strength γ ∗∗ leads always to the maximum time-delay τ ∗∗ = strength ϕ(γ ∗∗ ) that synchronization of nonlinear systems presented in Fujisaka and Yamada (1983). In this paper, is authors that that there always exists an optimal coupling strength γ synchronization of coupled nonlinear systems isthe presented that leads to the maximum time-delay τ = ϕ(γ ) that ∗ ∗ in Fujisaka and Yamada (1983). In this paper, authors that leads to the the maximum time-delay = ϕ(γ ϕ(γ ∗ )) that that can be induced to maximum the network without compromising the in Fujisaka and (1983). In paper, the show that coupled chaotic oscillators synchronize in that leads to time-delay ττ ∗ = in Fujisaka and Yamada Yamada (1983). In this thismay paper, the authors authors be induced to the network without compromising the ∗ the show that coupled chaotic oscillators may synchronize in can can be induced to the network without compromising and synchronous behavior. It follows that for γ = γ show that coupled chaotic oscillators may synchronize in spite of their high sensitivity to initial conditions. After can be induced to the network without compromising the ∗ show that coupled chaotic oscillators may synchronize in and synchronous behavior. It follows that for γ = γ ∗ ∗ spiteresult, of their their high sensitivity sensitivity toininitial initial conditions. After for and synchronous behavior. It follows follows that for for γ γ = the γ ∗ gain any τ ≤ behavior. τ ∗ the systems synchronize, i.e.,= spite of high to conditions. After this considerable interest the notion of synchroand synchronous It that γ spite of their high sensitivity to initial conditions. After for ≤ ∗ the systems synchronize, i.e., the gain this result, result, considerable interest in the the notion notion of Here, synchro∗ for any ≤ τττto the systems synchronize, i.e., the gain gain γ =any γ ∗∗ τττleads the systems best tolerance againsti.e., time-delays this considerable interest in of synchronization of general nonlinear systems arisen. we for any ≤ the synchronize, the this result, considerable interest in thehas notion of Here, synchroγ = γ leads to the best tolerance against time-delays ∗ nization of general nonlinear systems has arisen. we ∗ leads to system. γ = γ the best tolerance against time-delays of the closed-loop Finally, we analyze the effect of nization of general nonlinear systems has arisen. Here, we focus on network synchronization of identical nonlinear γ = γ leads to the best tolerance against time-delays nization general synchronization nonlinear systems arisen. nonlinear Here, we of the closed-loop system. Finally, we analyze the effect of focus on on of network of has identical of the closed-loop system. Finally, we analyze the effect of the network topology on the values of both optimal focus network synchronization of identical nonlinear systems interacting through diffusive time-delayed couof the closed-loop system. Finally, we analyze the effect of focus on interacting network synchronization of identical nonlinear the topology on the values both the optimal ∗ network ∗of systems through diffusive time-delayed couthe network topology on the values of both the optimal γ and the maximum time-delay τ , i.e., we show how systems through diffusive time-delayed couplings oninteracting networks with general topologies. Time-delayed ∗ network topology on the values∗of both the optimal systems through diffusive time-delayed cou- the γ the ττ ∗∗ ,, i.e., we how plings on oninteracting networks with general topologies. Time-delayed γ ∗∗ and and the maximum maximum time-delay i.e., we show show how eigenvalues of the time-delay corresponding Laplacian matrix plings networks with general topologies. Time-delayed couplings arise naturally for interconnected systems since the γ and the maximum time-delay τ , i.e., we show how plings on networks with general topologies. Time-delayed the eigenvalues of the corresponding Laplacian matrix ∗ ∗ couplings arise naturally for interconnected systems since the eigenvalues of the corresponding Laplacian matrix affect ϕ(γ), γ , and τ . This, in turn, gives insights on couplings arise naturally for interconnected systems since the transmission of signals is expected to take some time. the eigenvalues of the corresponding Laplacian matrix ∗ ∗ couplings arise naturally foris interconnected systems since affect ϕ(γ), γ ∗ , and τ ∗ . This, in turn, gives insights on the transmission transmission of signals signals expected to take take some faults time. affect ϕ(γ), γ ∗ the , and and τ ∗ .. This, This, in turn, turn, gives insights insights on how toϕ(γ), design network topology in order to enhance the of is expected to some time. Time-delays caused by signal transmission and/or affect γ , τ in gives on the transmission of signals is expected to take some time. how to design the network topology in order to enhance Time-delays caused by signal transmission and/or faults how to design the network topology in order to enhance robustness against time-delays. Time-delays caused signal and/or faults in the communication affect the behavior the how to design the network topology in order to enhance Time-delays caused by bychannels signal transmission transmission and/or of faults in the the communication communication channels affect the behavior of the robustness robustness against against time-delays. time-delays. in behavior of the interconnected systemschannels (e.g., in affect terms the of stability and/or against time-delays. in the communication channels affect the behavior of the robustness interconnected systems (e.g., in terms of stability and/or 2. PRELIMINARIES interconnected systems (e.g., in terms of stability and/or performance). interconnected 2. performance). systems (e.g., in terms of stability and/or 2. PRELIMINARIES PRELIMINARIES performance). 2. PRELIMINARIES performance). The results presented here follow the same research line as In this section, we introduce some important properties The results presented here follow the same research line as The results presented here research line Steur and Nijmeijer Steurthe etsame al. (2014), Murguia section, we some important properties The results presented(2010), here follow follow the research line as as In In this this section, which we introduce introduce somefor important properties and definitions are needed the subsequent reSteur and Nijmeijer Nijmeijer (2010), Steur etsame al. (2014), (2014), Murguia In this section, we introduce some important properties Steur and (2010), et al. Murguia et al. (2015a), and Murguia etSteur al. (2015b), where sufficient and definitions which are needed for the subsequent reSteur and Nijmeijer (2010), Steur et al. (2014), Murguia and definitions which are needed for the subsequent results. Throughout this paper, the following notation is et al. (2015a), and Murguia et al. (2015b), where sufficient and definitions which are needed for the subsequent reet al. and al. (2015b), sufficient conditions for synchronization interconnected Throughout paper, the is et al. (2015a), (2015a), and Murguia Murguia et et of al.diffusively (2015b), where where sufficient sults. sults. the Throughout this paper, the following following notation is used: symbol Rthis the set notation of positive >0 (R ≥0 ) denotes conditions for synchronization of diffusively interconnected sults. Throughout this paper, the following notation is conditions for synchronization of interconnected semipassive systems with time-delays are derived. In par- (nonnegative) used: )) The denotes the of >0 (R conditions for synchronization of diffusively diffusively interconnected used: the the symbol symbol Rnumbers. (R≥0 denotes the set setnorm of positive positive real R Euclidian in Rnn >0 (R ≥0 ) semipassive systems with time-delays are derived. In parused: the symbol R denotes the set of positive >0 ≥0 semipassive systems time-delays particular, in Steur and with Nijmeijer (2010),are thederived. authorsIn prove real The norm in 2 Euclidian semipassive systems time-delays par- (nonnegative) (nonnegative) real numbers. numbers. The Euclidian norm in R Rnn is denoted simply as | · |, |x| xTT x. The notation ticular, in in Steur Steur and with Nijmeijer (2010),are thederived. authorsInprove prove (nonnegative) real numbers. The Euclidian norm in R 2 = ticular, and Nijmeijer (2010), the authors is denoted simply as | · |, |x| = x x. The notation 2 T ticular, in Steur and Nijmeijer (2010), the authors prove 2 = x T x. The is denoted simply asfor|| ··the |, |x| |x| notation col(x ) standsas column vector composed of This work was supported by the National Research Foundation 1 , ..., xnsimply is denoted |, = x x. The notation col(x x stands for the column vector composed of This 1 ,, ..., n) was supported bySingapore, the National Research Foundation col(xelements ..., x ) stands for the column vector composed of the x , ..., x . This notation is also used in case This work 1 n 1 n (NRF), Prime Minister’s Office, under its National Cybercol(x , ..., x ) stands for the column vector composed of work was supported by the National Research Foundation 1 n This work was supported bySingapore, the National Research Foundation the components elements x This is used case (NRF), Prime Minister’s Office, under its National Cyberelements x x111 ,,,x..., ..., xnnn ... vectors. This notation notation is nalso also used in in case the are The n × identity matrix security R&D Programme (Award No. NRF2014NCR-NCR001-40) i (NRF), Prime Minister’s Office, Singapore, under its National Cyberthe elements x ..., x This notation is also used in case (NRF), Prime Office, Singapore, under its National Cyberthe components vectors. The × security R&D Minister’s Programme (Award No. NRF2014NCR-NCR001-40) thedenoted components xiiiorare aresimply vectors. The nconfusion ×n n identity identity matrix and administered by the National R&D Directorate. is by Inx I if non canmatrix arise. security R&D Programme Programme (AwardCybersecurity No. NRF2014NCR-NCR001-40) NRF2014NCR-NCR001-40) the components x are vectors. The n × n identity matrix security R&D (Award No. and administered by the National Cybersecurity R&D Directorate. is denoted by I or simply I if no confusion can arise. n and administered by the National Cybersecurity R&D Directorate. is denoted by I or simply I if no confusion can arise. n and administered by the National Cybersecurity R&D Directorate. is denoted by In or simply I if no confusion can arise.
Copyright © 2016, 2016 IFAC 1 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 1 Copyright © 2016 IFAC 1 Control. Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 1 10.1016/j.ifacol.2016.07.985
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Likewise, the n × m matrices composed of only ones and only zeros are denoted as 1n×m and 0n×m , respectively. The spectrum of a matrix A is denoted by spec(A). The symbol ⊗ denotes Kronecker product. Let X ⊂ Rn and Y ⊂ Rm . The space of continuous functions from X to Y is denoted by C(X , Y). If the functions are (at least) r ≥ 0 times continuously differentiable, then it is denoted by C r (X , Y). If the derivatives of a function of all orders (r = ∞) exist, the function is called smooth and if the derivatives up to a sufficiently high order exist the function is named sufficiently smooth. For simplicity of notation, we often suppress the explicit dependence of time t.
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initial condition. A sufficient condition for system (1a) to be convergent is presented in the following proposition. Proposition 1. Demidovich (1967) and Pavlov et al. (2004). If there exists a positive definite matrix P ∈ Rn×n such that all the eigenvalues λi (Q) of the symmetric matrix: T 1 ∂f ∂f P (x, u) + (x, u) Q(x, u) = P , (2) 2 ∂x ∂x are negative and separated from zero, i.e., there exists a constant c ∈ R>0 such that λi (Q) ≤ −c < 0, for all i ∈ {1, ..., n}, u ∈ U , and x ∈ Rn , then system (1a) is globally exponentially convergent. Moreover, for any pair of solutions x1 (t), x2 (t) ∈Rn of (1a), it is satisfied that: T 2 d x1 − x2 P x1 − x2 ≤ −α |x1 − x2 | with constant dt α := λmaxc (P ) and λmax (P ) being the largest eigenvalue of the symmetric matrix P .
2.1 Communication Graphs Given a set of interconnected systems, the communication topology is encoded through a communication graph. Let G = (V, E, A) denote a weighted undirected graph, where V = {v1 , v2 , ..., vk } is the set of nodes, E ⊆ V × V is the set of edges, and A ∈ Rk×k := aij is the weighted adjacency matrix with entries aij = aji ≥ 0. The set of neighbors of vi , denoted as Ei , is the set of nodes which have edges pointing to vi . Throughout this manuscript, it is assumed that the communication graph is strongly connected and simple, see Bollobas (1998). We denote the degree matrix D ∈ Rk×k := diag{d1 , ..., dk } with di := j∈Ei aij , and L := D − A, which is called the Laplacian matrix of the graph G.
3. PROBLEM STATEMENT Consider k identical nonlinear systems of the form ζ˙i = q(ζi , yi ),
(3) (4) y˙ i = a(ζi , yi ) + CBui , with i ∈ I := {1, ..., k}, state xi := col(ζi , yi ) ∈ Rn , internal state ζi ∈ Rn−m , output yi ∈ Rm , input ui ∈ Rm , sufficiently smooth functions q : Rn−m × Rm → Rn−m and a : Rn−m × Rm → Rm , matrices C ∈ Rm×n and B ∈ Rn×m , and the matrix CB ∈ Rm×m being similar to a positive definite matrix. For the sake of simplicity, it is assumed that CB = Im (results for the general case with CB being similar to a positive definite matrix can be easily derived). The system (3),(4) is assumed to be strictly C 1 -semipassive and the internal dynamics (3) is assumed to be an exponentially convergent system. Let the k systems (3),(4) interact on simple strongly connected graph through the diffusive time-delayed coupling ui (t) = γ aij (yj (t − τ ) − yi (t − τ )) , (5)
2.2 Semipassive Systems Consider the system x˙ = f (x, u), (1a) y = h(x), (1b) with state x ∈ Rn , input u ∈ Rm , output y ∈ Rm , and sufficiently smooth functions f : Rn × Rm → Rn and h : Rn → Rm . Definition 1. Pogromsky et al. (1999). System (1) is called C r -semipassive if there exists a nonnegative storage function V ∈ C r (Rn , R≥0 ) such that V˙ ≤ y T u − H(x), where the function H ∈ C(Rn , R) is nonnegative outside some ball, i.e., ∃ ϕ > 0 s.t. |x| ≥ ϕ → H(x) ≥ (|x|), for some continuous nonnegative function (·) defined for |x| ≥ ϕ. If the function H(·) is positive outside some ball, then the system (1) is said to be strictly C r -semipassive.
j∈Ei
where τ ∈ R≥0 denotes a constant time-delay, yj (t − τ ) and yi (t − τ ) are the delayed outputs of the j-th and i-th systems, γ ∈ R≥0 denotes the coupling strength, aij ≥ 0 are the weights of the interconnections, and Ei is the set of neighbors of system i. It is assumed that the graph is undirected, i.e., aij = aji . Since the coupling strength is encompassed in the constant γ, it is assumed without loss of generality that maxi∈I j∈Ei aij = 1. Note that all signals in coupling (5) are time-delayed. Such a coupling may arise, for instance, when the systems are interconnected through a centralized control law. Coupling (5) can be written in matrix form as u = −γ (L ⊗ Im ) y(t − τ ) with u := col(u1 , . . . , uk ), y := col(y1 , . . . , yk ), and Laplacian matrix L = LT ∈ Rk×k . The authors in Steur and Nijmeijer (2010) prove that the k coupled systems (3)-(5) asymptotically synchronize provided that γ is sufficiently large and the product of the coupling strength and the time-delay γτ is sufficiently small. It follows that there exists a region S in the parameter space, such that if (γ, τ ) ∈ S, the systems synchronize. In this manuscript, we go one step further by showing that this region S is actually bounded by a unimodal function ϕ : J → R≥0 , γ → ϕ(γ). Hence, there exists an optimal coupling
A (strictly) C r -semipassive system behaves like a (strictly) passive system for large |x(t)|. 2.3 Convergent Systems
Consider system (1a) and suppose f (·) is locally Lipschitz in x, u(·) is piecewise continuous in t and takes values in some compact set u ∈ U ⊆ Rm . Definition 2. System (1a) is said to be convergent if and only if for any bounded signal u(t), defined on the whole interval (−∞, +∞), there is a unique bounded globally asymptotically stable solution x ¯u (t), defined in the same interval, for which it holds that limt→∞ |x(t) − x ¯u (t)| = 0 for all initial conditions. For a convergent system, the limit solution is solely determined by the external excitation u(t) and not by the 2
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nize provided that the coupling strength γ is sufficiently large and the time-delay τ is smaller than some unimodal function ϕ(γ), see Figure 1. Hence, there exists a region S (colored area in Figure 1) such that if (γ, τ ) ∈ S, the systems synchronize. Estimates of both the constant γ and the unimodal function ϕ(γ) are derived in the proof of Theorem 1, in (A.13) and (A.15), respectively. Corollary 1. Consider k coupled systems (3)-(5) with coupling strength γ ∈ R≥0 and time-delay τ ∈ R≥0 on a simple strongly connected graph G. Suppose that the conditions stated in Theorem 1 are satisfied. Then, for every γ¯ ∈ J , there exists a maximum time-delay τ¯ ∈ R≥0 , such that if γ = γ¯ , the systems synchronize for all τ ≤ τ¯. Moreover, there exists an optimal coupling strength γ = γ ∗ ∈ J and its corresponding maximum time-delay τ ∗ := max(τ ∈ S) such that τ¯ < τ ∗ for all γ = γ ∗ .
Figure 1. Synchronization Region S.
strength γ ∗ that leads to the maximum delay τ ∗ = ϕ(γ ∗ ) that can be induced to the network without compromising the synchronous behavior. Moreover, we characterize the function ϕ(γ), the optimal γ ∗ , and the maximum timedelay τ ∗ in terms of the spectrum of the Laplacian matrix L and the vector fields q(·) and a(·).
Proof: The assertion follows from unimodality of ϕ(γ). The result stated in Corollary 1 implies that for every strongly connected graph G, there exists an optimal coupling strength γ ∗ that leads to the maximum time-delay τ ∗ = ϕ(γ ∗ ) = max(τ ∈ S) that can be induced to the network without compromising the synchronous behavior, i.e., there exists a gain γ = γ ∗ that leads to the best tolerance against time-delays of the closed-loop system, see Figure 1. In the following corollary, we characterize the effect of the network topology on the values of both the optimal γ ∗ and the maximum time-delay τ ∗ . Let G¯ := {Gs | s ∈ N } be the set of all simple strongly connected undirected graphs. For each graph Gs , consider k = ks coupled systems (3)-(5) with coupling strength γ = γs ∈ R≥0 and delay τ = τs ∈ R≥0 interacting on Gs . Assume that the conditions stated in Theorem 1 are satisfied, then from Theorem 1, there exist constants γs ∈ R>0 and unimodal functions ϕs : Js → R≥0 , s ∈ N, such that if (γs , τs ) ∈ Ss , where ¯ γs > γs , τs < ϕs (γs )}, Ss := {γs , τs ∈ R≥0 | γs < δ/2, the ks systems of each Gs asymptotically synchronize. Let γs∗ and τs∗ = max(τs ∈ Ss ) be the optimal coupling strength and the corresponding maximum time-delay of Gs and define τ¯∗ := {τs∗ | s ∈ N }, i.e., the set of all maximum ¯ and time-delays of G, ∗ τmax := max(¯ τ ∗ ). (7) Corollary 2. For s = 1, 2. Consider k = ks interconnected systems (3)-(5) with coupling strength γ = γs ∈ R≥0 and time-delay τ = τs ∈ R≥0 on a simple strongly connected graph Gs , i.e., two independent diffusively timedelayed coupled networks. Assume that G1 = G2 and the conditions stated in Theorem 1 are satisfied. Then, by Theorem 1, there exist a region Ss such that if (γs , τs ) ∈ Ss the ks systems of Gs asymptotically synchronize. Let γs∗ and τs∗ = max(τs ∈ Ss ) be the optimal coupling strength and the corresponding maximum time-delay of Gs . Additionally, let Ls ∈ Rks ×ks be the Laplacian matrix of Gs with real positive eigenvalues λj (Ls ), j = 2, . . . , ks , and λ2 (Ls ) ≤ . . . ≤ λks (Ls ). Then:
4. MAIN RESULTS Define x := col(x1 , . . . , xk ) and the synchronization manifold M := {x ∈ Rkn | xi = xj , ∀ i, j ∈ I}. The coupled systems (3)-(5) are said to synchronize if the synchronization manifold M is invariant under the closed-loop dynamics and contains an asymptotically stable subset. Clearly, if the systems are interconnected through (5), the coupling functions vanish on M; hence, the manifold M is positively invariant under the dynamics (3)-(5). In the following theorem, we give sufficient conditions for the existence of an asymptotically stable subset of the synchronization manifold. Theorem 1. Consider k coupled systems (3)-(5) with coupling strength γ ∈ R≥0 and time-delay τ ∈ R≥0 on a simple strongly connected graph G. Assume that: (H4.1) Each system (3),(4) is strictly C 1 -semipassive with input ui , output yi , radially unbounded storage function V (xi ), and the functions H(xi ) are such that there exists a constant R ∈ R>0 such that |xi | > R implies that H(xi ) − δ|yi |2 > 0 for some constant δ ∈ R>0 . Let δ¯ be the largest δ that satisfies (H4.1). Then, the solutions of the coupled systems (3)-(5) are ultimately ¯ bounded for any finite τ ∈ R>0 and γ < δ/2. In addition assume that: (H4.2) The internal dynamics (3) is exponentially convergent, i.e., there is a positive definite matrix P such that the eigenvalues of the symmetric matrix T 1 ∂q ∂q P (ζi , yi ) + (ζi , yi ) P , (6) 2 ∂ζi ∂ζi are uniformly negative and bounded away from zero for all ζi ∈ Rn−m and yi ∈ Rm . Then, there exist a constant γ ∈ R>0 and a unimodal function ϕ : J := [γ , ∞) → R≥0 , γ → ϕ(γ), where ϕ(γ ) = 0 and limγ→∞ ϕ(γ) = 0, such that if (γ, τ ) ∈ ¯ S := {γ, τ ∈ R≥0 | γ < 2δ , γ > γ , τ < ϕ(γ)}, then there exists a globally asymptotically stable subset of the synchronization manifold M.
λ
λ
(a) If λk21 (L1 ) = λk22 (L2 ) > 1, then τ1∗ = τ2∗ . In addition, λ2 (L1 ) ≥ λ2 (L2 ) implies γ1∗ ≤ γ2∗ .
The proof of Theorem 1 can be found in the appendix. Theorem 1 implies that the solutions of (3)-(5) are ultimately bounded and the systems asymptotically synchro-
λ
λ
∗ . In (b) If λk21 (L1 ) = λk22 (L2 ) = 1, then τ1∗ = τ2∗ = τmax ∗ ∗ addition, λ2 (L1 ) ≥ λ2 (L2 ) implies γ1 ≤ γ2 .
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Figure 2. Synchronization Regions Ss , s = 1, 2. Left: Case (a) in Corollary 2. Right: Case (b) in Corollary 2.
Figure 4. Synchronization regions Ss computed for different topologies Gs , s = 1, . . . , 7. 2 2 . More+25z2i storage function V (z1i , z2i , yi ) := 12 yi2 +σz1i over, the (z1i , z2i )-dynamics (the internal dynamics) is exponentially convergent (in the sense of Definition 2), i.e., it satisfies the Demidovich condition (6) (with P = I2 ); hence, assumption (H4.2) in Theorem 1 is satisfied. This particular experiment is taken from Neefs et al. (2010), where a detailed experimental study is presented. B. Bounded Solutions and Synchronization. It can be easily verified that the function H(xi ) obtained from the V (z1i , z2i , yi ) defined above satisfies the boundedness assumption (H4.1) for arbitrary large coupling strength γ, see Neefs et al. (2010). Therefore, by Theorem 1, the solutions of the coupled systems (8),(5) with γ = γs and τ = τs are ultimately bounded for any finite coupling strength γs and time-delay τs . Finally, given that all the graphs in Figure 3 are strongly connected and (H4.1) and (H4.2) are satisfied, then, by Theorem 1, there exist regions Ss , s = 1, . . . , 7, (as depicted in Figure 1), such that if (γs , τs ) ∈ Ss , the systems synchronize. C. Simulation Results. In Figure 4, we show results obtained through extensive computer simulations. We depict the synchronization regions Ss , s = 1, . . . , 7 for each network topology Gs . These regions are clearly bounded by unimodal functions; and therefore, for each network, there exists an optimal coupling strength γs∗ and its corresponding maximum time-delay τs∗ that can be induced to the network without compromising the synchronous behavior. These maximum time-delays are strongly influenced by the network topology, (see the proofs of Theorem 1 and Corollary 2). In Table 1, we show the numerical values of the optimal coupling strengths, the maximum time-delays, and the quotients λks /λ2 . These values match with the theoretical predictions given in Corollary 2.
Figure 3. Network topologies. λ
77
λ
(c) If λk21 (L1 ) > λk22 (L2 ), then τ1∗ < τ2∗ . In addition, λ2 (L1 ) ≥ λ2 (L2 ) implies γ1∗ ≤ γ2∗ . The proof of Corollary 2 can be found in the appendix. The result stated in Corollary 2 amounts to the following. The effect of the network topology on the maximum timedelay τ ∗ is solely determined by the quotient ( λλk2 ). Case (a) implies that any two strongly connected networks with the same quotient have the same maximum time-delay τ ∗ . Additionally, in this case, the value of the optimal coupling strength γ ∗ is also determined by λ2 , i.e., if the two networks have the same quotient, then the larger the λ2 the smaller the γ ∗ , and vice versa. Case (b) implies that networks with quotient equal to one have the best tolerance against time-delays, i.e., if λλk2 = 1, then τ ∗ = ∗ ∗ with τmax as defined in (7). Case (c) implies that the τmax larger the quotient λλk2 the smaller the τ ∗ , and vice versa. See Figure 2. 5. SIMULATION EXPERIMENT A. Network Topology, Strict Semipassivity, and Convergence. Consider a network of ks , s ∈ {1, . . . , 7}, systems coupled according to the graphs Gs depicted in Figure 3. For each network, the weights of the interconnections are set to aij = 1/ks if {i, j} ∈ Es and aij = 0 otherwise. The networks are strongly connected and undirected. Each system in the networks is assumed to be a Hindmarsh-Rose neuron, see Hindmarsh and Rose (1984), of the form z˙1i = 1 − 5yi2 − z1i , z˙2i = 0.005(4yi + 6.472 − z2i ), (8) y˙ i = −yi3 + 3yi2 + z1i − z2i + 3.25 + ui , with output yi ∈ R, internal states zi1 , zi2 ∈ R, state xi = col(zi1 , zi2 , yi ) ∈ R3 , input ui ∈ R, and i ∈ I = {1, . . . , ks }. In Steur et al. (2009), the authors prove that the Hindmarsh-Rose neuron is strictly C 1 -semipassive with
6. CONCLUSIONS We have presented a result on network synchronization of coupled nonlinear systems in the case when the coupling functions are subject to constant time-delays. Using the notions of semipassivity and convergent systems, we have proved that, under some mild assumptions, there always exists a region S in the parameter space (coupling strength γ versus time-delay τ ), such that if γ, τ ∈ S, the systems synchronize. We proved that this region S is always bounded by a unimodal function ϕ(γ); and consequently, that there always exists an optimal coupling strength γ ∗ which leads to the maximum time-delay τ ∗ = ϕ(γ ∗ ) that the network can tolerate without breaking the synchrony. 4
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Table 1. Simulation results: Optimal coupling strength γs∗ , maximum time-delay τs∗ , eigenvalues λks and λ2 of the corresponding Laplacian matrices Ls , and their quotient (λks /λ2 ). G1 G2 G3 G4 G5 G6 G7
γs∗ 2.00 5.70 1.95 10.6 3.85 3.75 1.90
τs∗ [ms] 4.25 1.10 4.25 0.33 2.55 2.50 4.15
λks 1 1 1 0.8536 1 1 1
λ2 1 1 3
1 0.1464 1 2 1 2
1
strongly connected and undirected. Then, the Laplacian matrix is symmetric and its eigenvalues are real. Moreover, the matrix L has an algebraically simple eigenvalue λ1 = 0 and 1k×1 is the corresponding ˜ = spec(L)\{0}, then from Gerschgorin’s eigenvector. Since spec(L) ˜ are disc theorem, it can be concluded that the eigenvalues of L ˜ has eigenvalues λ2 , ..., λk ∈ R>0 positive real, i.e., the matrix L with 0 < λ2 ≤ · · · ≤ λk . Coupling (5) can be written in matrix form u = −γ (L ⊗ Im ) y(t − τ ) where u = col(u1 , ..., uk ) ∈ Rkm . Denote u ˜ = col((u1 − u2 ), ..., (u1 − uk )), it follows that
λks /λ2 1 3 1 5.8306 2 2 1
˜ ⊗ Im )˜ u ˜(t) = −γ(L y (t − τ ),
˙ ˜ y1 , ζ1 ), ζ˜ = q˜(˜ y , ζ, ˜ y1 , ζ1 ) − γ(L ˜ ⊗ Im )˜ y (t − τ ), y˜˙ = a ˜(˜ y , ζ,
In Corollary 2, we have provided tools for selecting the network topology in order to enhance robustness against time-delays of the coupled systems. Case (b) in Corollary 2 implies that networks with quotient (λk /λ2 ) equal to 1 have the best tolerance against time-delays. This is the case for all-to-all networks.
−τ
˜ y1 , ζ1 ) − γ(L ˜ ⊗ Im )˜ ˜ ⊗ Im ) y˜˙ = a(˜ y , ζ, y + γ(L
A.1 Proof of Theorem 1 : First, we prove ultimate boundedness of the closed-loop system (3)-(5). By assumption, each system (3),(4) is strictly C 1 -semipassive with input ui , output yi , and radially unbounded function V (xi ). Define the functional: W (xt (θ)) :=
k
νi V (xi ) + γ
i=1
aij
j∈Ei
−τ
substitution of (A.5) in (A.6) yields
|yi (t + s)|2 ds ,
˜ ⊗ Im ) + γ(L
i=1
˙ ≤ ν (−H(xi )+2γ |yi |2 ), see the proof of Theorem 1 in Steur W i=1 i and Nijmeijer (2010) for details. By assumption (H4.1), there exist positive constants R, δ ∈ R>0 such that for |xi | > R it is satisfied that H(xi )−δ|yi |2 > 0. Let δ¯ be the largest δ that satisfies (H4.1) for ¯ and for suffiarbitrarily large R < ∞. Then, for γ satisfying γ ≤ δ/2 ˙ < 0. Then, by the same arguments ciently large |x|, it follows that W presented in the proof of Corollary 2 in Steur and Nijmeijer (2010), it can be concluded that the solutions of the closed-loop system (3)-(5) ¯ are ultimately bounded for any finite τ ≥ 0 and γ ≤ δ/2. Next, we prove the asymptotic stability of the synchronization manifold M. Let ζ = col(ζ1 , . . . , ζk ) ∈ Rk(n−m) and y = col(y1 , . . . , yk ) ∈ Rkm . Define M := (1k−1 − Ik−1 ) ∈ R(k−1)×k and introduce the new coordinates ζ˜ = (M ⊗ In−m )ζ and y˜ = (M ⊗ Im )y. Note that y˜ = ζ˜ = 0 implies that the systems are synchronized. Assumption (H4.2), Proposition 1, smoothness of the vector fields, and boundedness of the solutions imply the existence of a positive ˜ such that definite function V2 : R(k−1)(n−m) → R≥0 , ζ˜ → V2 (ζ)
˜ = M
1 0k−1 1k−1 −Ik−1
˜ LM ˜ −1 = →M
0 ∗ ˜ 0k−1 L
,
0 −τ
y˜(t − τ + s)ds
˜ y1 , ζ1 )(t + s)ds. a(˜ y , ζ,
y |2 + γ y¯T (Λ ⊗ Im ) V˙ 3 ≤ −γλ2 |¯
(A.7)
(A.8) 0 −τ 0
y¯(t − τ + s)ds
(A.9)
¯ y1 , ζ1 )(t + s)ds, a ¯(¯ y , ζ, −τ
0
¯ y1 , ζ1 )(t + s)ds a ¯(¯ y , ζ, −τ
¯ y1 , ζ1 ) − γ 2 y¯T (Λ2 ⊗ Im ) ¯(¯ y , ζ, + y¯ a T
0 −τ
y¯(t − τ + s)ds. (A.10)
Ultimate boundedness of the solutions and smoothness of the func¯ y1 , ζ1 ) ≤ c1 |¯ ¯ for some tion a(·) imply that y¯T a ¯(¯ y , ζ, y |2 + c2 |¯ y | |ζ| ¯ y¯) := positive constants c1 , c2 ∈ R>0 . Let the function V1 (ζ, ¯ + V3 (¯ V2 (ζ) y ) be a Lyapunov-Razumikhin function such that if ¯ ¯ + θ), y¯(t + θ)) for θ ∈ [−2τ, 0] and some V1 (ζ(t), y¯(t)) > κ 2 V1 (ζ(t constant κ > 1, then
(A.1)
¯ y1 , ζ1 ), and a ¯ y1 , ζ1 ) := ¯ y1 , ζ1 ) := q˜((U −1 ⊗Im )¯ y , ζ, ¯(¯ y , ζ, where q¯(¯ y , ζ, ¯ y1 , ζ1 ). Notice that y¯ = ζ¯ = 0 implies (U ⊗ Im )˜ a((U −1 ⊗ Im )¯ y , ζ, that the systems are synchronized because U is nonsingular. Since stability is invariant under change of coordinates and U = 1, from (A.1), it follows that there exists a positive definite function ¯ such that V¯˙ 2 (ζ, ¯ y¯) ≤ −α|ζ| ¯2 + V¯2 : R(k−1)(n−m) → R≥0 , ζ¯ → V¯2 (ζ) ¯ |¯ c0 |ζ| y | for some α, c0 ∈ R>0 . Define V3 (¯ y ) := 12 y¯T y¯, then
for some constants α, c0 ∈ R>0 , see Pogromsky et al. (1999) for further details. Notice that
y˜˙ (t + s)ds, (A.6)
−τ
−τ
− γ (Λ ⊗ Im ) y¯(t) + γ(Λ ⊗ Im )
i
˜ |˜ ˜ y˜) ≤ −α|ζ| ˜ 2 + c0 |ζ| y| , V˙ 2 (ζ,
0
¯ y1 , ζ1 ) − γ 2 (Λ2 ⊗ Im ) y¯˙ = a ¯(¯ y , ζ,
sumption, where yiτ := yi (t − τ ).Using Young’s inequality, the fact that ν T L = 0, and maxi∈I aij = 1, it follows that j∈E
k
¯ y1 , ζ1 ), ζ¯˙ = q¯(¯ y , ζ,
i
j∈Ei
0
˜ is nonsingular and symmetric, then there exists a The matrix L ˜ −1 = Λ, matrix U ∈ R(k−1)×(k−1) such that U = 1 and U LU where Λ denotes a diagonal matrix with the nonzero eigenvalues of L on its diagonal. Introduce the change of coordinates y¯ = (U ⊗Im )˜ y ˜ In the new coordinates, the and for consistency of notation ζ¯ = ζ. closed-loop system takes the form
where x = col(x1 , ..., xk ), xt (θ) = x(t + θ) ∈ C, θ ∈ [−τ, 0], C being the Banach space of continuous functions mapping the interval [−τ, 0] into Rkn , and the constants νi denoting the entries of the left eigenvector corresponding to the simple zero eigenvalue of the Laplacian matrix L, i.e., ν = (ν1 , ..., νk )T and ν T L = ν T (D − A) = 0. Note that L is singular by construction and, since it is assumed that the graph is strongly connected, the zero eigenvalue is simple. Using the Perron-Frobenius theorem, it can be proved that the vector ν has strictly positive real entries, see Bollobas (1998). Then, 2 k ˙ = W νi (y T ui − H(xi ) + γ aij (|yi |2 − y τ )) by asi
˜ y1 , ζ1 ) − γ(L ˜ ⊗ Im )˜ ˜ 2 ⊗ Im ) y˜˙ = a(˜ y , ζ, y − γ 2 (L
0
(A.4) (A.5)
˜ y1 , ζ1 ) = col(a(y1 , ζ1 )−a(y1 − y˜1 , ζ1 − ζ˜1 ), . . . , a(y1 , ζ1 )− with a ˜(˜ y , ζ, ˜ y1 , ζ1 ) = col(q(y1 , ζ1 ) − q(y1 − a(y1 − y˜k−1 , ζ1 − ζ˜k−1 )) and q˜(˜ y , ζ, y˜1 , ζ1 − ζ˜1 ), . . . , q(y1 , ζ1 ) − q(y1 − y˜k−1 , ζ1 − ζ˜k−1 )). Using Leibniz’s rule and continuity of the solutions, y˜(t − τ ) can be written as 0 y˜(t − τ ) = y˜(t) − y˜˙ (t + s)ds. Then, (A.5) can be written as
Appendix A. PROOFS
(A.3)
˜ as in (A.2). Then, the closed-loop system can be written as with L
2 y |2 V˙ 1 ≤ −α ζ¯ + κτ λ2k γ 2 + c1 (1 + κγτ λk ) − γλ2 |¯
(A.2)
y| . + (c0 + c2 (1 + κγτ λk )) ζ¯ |¯
(A.11)
The constant κ can be arbitrarily close to one as long as it is greater than one. Then, for simplicity of notation, we take κ on the boundary
where 0k−1 = 0(k−1)×1 , 1k−1 = 1(k−1)×1 , and L denotes the Laplacian matrix. By assumption, the communication graph is
5
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κ = 1 for the rest of the proof. It can be easily verified that (A.11) is negative definite if
with
γ :=
λ2 γ − γ
− λk λ k γ +
λ2 c¯1 γτ − k (γτ )2 > 0, c¯2 2¯ c2
τ
(A.12)
c¯1 γλk
±
c¯2 +
c¯1 γλk
2
+
2¯ c2 (λ2 γ − γ ) . λ2k γ 2
c¯1 ϕ(γ) := − c¯2 + γλk
+
c¯2 +
c¯1 γλk
2
+
c¯1 +
λ 2¯ c2 γ λk 2
+
c¯2
λ
λ
2¯ c2 γ + 4¯ c2 c¯1 γ λk + 2¯ c2 γ λk 2
2
2
.
(A.20)
λ
2
(A.20) that networks with equal λk have the same maximum time2 delay. Moreover, from (A.19), it is clear that if two networks have the λk same quotient λ , then the value of γ ∗ is solely determined by λ2 , 2 the larger the λ2 the smaller the γ ∗ , and vice versa. Case (b): From λ (A.20), it is clear that τ ∗ has its maximum value at λk = 1. The part 2 ∗ regarding γ follows from the same arguments of case (a). Case (c): λk Clearly, from (A.20), the larger the quotient λ the smaller the τ ∗ , 2 and vice versa. Moreover, from (A.19), it is clear that the larger the λ quotient λk the smaller the γ ∗ for a fixed λ2 , and for a fixed quotient, 2 the larger the λ2 the smaller the γ ∗ and the assertion follows.
(A.14)
The time-delay τ is nonnegative by definition. Hence, in order to satisfy (A.14), it is sufficient to consider the possible positive values of the right-hand side of (A.14), i.e., the positive square root. Then, inequality (A.14) amounts to τ < ϕ(γ) with
Next, we analyze the cases stated in Corollary 2. Case (a): The maximum time-delay in (A.20) relies on the constants c¯1 , c¯2 , γ ∈ λ R>0 and the quotient λk . The constants c¯1 , c¯2 , and γ do not depend 2 on the network topology (they depend on the vectorfields q(·) and λ a(·)), then the effect of network is solely determined by λk . From 2 Gerschgorin’s disc theorem, it can be concluded that the eigenvalues λ2 and λk of any strongly connected undirected graph are positive λ real, i.e., 0 < λ2 ≤ λk , and therefore λk ≥ 1. Then, it is clear from
All the constants in (A.12) are positive by construction and γ and τ are nonnegative by definition. Then, a necessary condition for (A.12) to be satisfied is λ2 γ > γ . After some straightforward computations (A.12) can be rewritten as follows
= λk λ2
2αc1 + c0 c2 + c22 (c0 + c2 )2 2α , c¯2 := 2 . (A.13) + c1 , c¯1 := 4α c22 c2
τ < − c¯2 +
∗
79
2¯ c2 (λ2 γ − γ ) . (A.15) λ2k γ 2
We are only interested in possible values of γ, τ ∈ R≥0 such that (A.15) is satisfied. Then, we restrict the function ϕ(γ) to the set
J := [ λγ , ∞). Next, we prove that the function ϕ : J → R≥0 is 2 unimodal. The function ϕ(·) is continuous and real-valued on J .
REFERENCES
ϕ( λγ ) 2
= 0, and Moreover, it is strictly positive on the interior of J , limγ→∞ ϕ(γ) = 0. The function ϕ(·) is differentiable on J ; then, we can compute its local extrema by computing its critical points. It is ∂ϕ(γ) easy to verify that ∂γ = 0 only for γ = γ ∗ and γ = γ ˜ with γ =
γ ˜ =
∗
λ2 1+ λ2 + 2λk c¯1
λ2 λ2 + 2λk c¯1
1+
γ + λ2
γ
λ2
−
2¯ c21 c¯2 γ (λ22 + 2λ2 λk c¯1 + 2λ2k c¯2 γ ) c¯2 λ2 (λ2 + 2λk c¯1 )
Blekhman, I. (1988). Synchronization in science and technology. ASME, New York. Bollobas, B. (1998). Modern Graph Theory. Springer-Verlag, New York. Demidovich, B. (1967). Lectures on Stability Theory. Moscow. In Russian. Fujisaka, H. and Yamada, T. (1983). Stability theory of synchronized motion in coupled-oscillator systems. Progress of Theoretical Physics, 69, 32–37. Hindmarsh, J.L. and Rose, R.M. (1984). A model of neuronal bursting using three coupled first order differential equations. Proceedings of the Royal Society of London B: Biological Sciences, 221, 87–102. Murguia, C., Fey, R.H.B., and Nijmeijer, H. (2015a). Network synchronization of time-delayed coupled nonlinear systems using predictor-based diffusive dynamic couplings. Chaos, 25, 023108. Murguia, C., Fey, R.H.B., and Nijmeijer, H. (2015b). Network synchronization using invariant-manifold-based diffusive dynamic couplings with time-delay. Automatica, 57, 34 – 44. Neefs, P., Steur, E., and Nijmeijer, H. (2010). Network complexity and synchronous behavior an experimental approach. Int. J. of Neural Systems, 20, 233–247. Pavlov, A., Pogromsky, A., van de Wouw, N., and Nijmeijer, H. (2004). Convergent dynamics, a tribute to Boris Pavlovich Demidovich. Syst. Control Lett., 52, 257. Pikovsky, A., Rosenblum, M., and Kurths, J. (2001). Synchronization: A universal concept in nonlinear Science. Cambridge University Press, Cambridge. Pogromsky, A., Glad, T., and Nijmeijer, H. (1999). On diffusion driven oscillations in coupled dynamical systems. Int. J. Bif. Chaos, 9, 629–644. Steur, E. and Nijmeijer, H. (2010). Synchronization in networks of diffusively time-delay coupled (semi-)passive systems. IEEE Transactions on Circuits and Systems-I, 58(6), 1358–1371. Steur, E., Tyukin, I., and Nijmeijer, H. (2009). Semi-passivity and synchronization of diffusively coupled neuronal oscillators. Physica D, 238, 2119–2128. Steur, E., Michiels, W., Huijberts, H., and Nijmeijer, H. (2014). Networks of diffusively time-delay coupled systems: Conditions for synchronization and its relation to the network topology. Physica D: Nonlinear Phenomena, 277, 22 – 39. Strogatz, S. (2003). SYNC. The emerging science of spontaneous Order. Hyperion, New York.
,
(A.16) 2¯ c21 c¯2 γ (λ22
+ 2λ2 λk c¯1 +
2λ2k c¯2 γ )
c¯2 λ2 (λ2 + 2λk c¯1 )
.
(A.17)
˜ are the critical points of ϕ(γ) and ϕ(γ ∗ ) and Then, γ = γ ∗ and γ = γ ϕ(˜ γ ) are the corresponding extrema. Notice that γ ∗ > ∗ γ belongs to the interior of J . Rewrite (A.17) as 2¯ c2 γ − c¯21
γ ˜=
c¯2 (λ2 + c¯1 λk ) + c¯1
γ ; λ2
+2λ2 λk c ¯1 +2λ2 c ¯ γ ) c ¯2 (λ2 2 k 2 2γ
.
therefore,
(A.18)
The denominator of (A.18) is strictly positive, then the sign of γ ˜ is solely determined by its numerator. Substitution of (A.13) in the numerator of (A.18) yields 2¯ c2 γ − c¯21 = −4αc1 (c0 c2 + αc1 ) /c22 , which is strictly negative. It follows that γ ˜ is strictly negative; in consequence, γ ˜∈ / J . Then, the function ϕ(·) has a unique extremum on J and it is given by ϕ(γ ∗ ). Moreover, given that ϕ(γ ) = 0, limγ→∞ ϕ(γ) = 0, φ(γ) is strictly positive on the interior of J , and ϕ(γ ∗ ) is the unique extremum on J , it follows that ϕ(γ ∗ ) is a unique local maximum on J ; therefore, it can be concluded that the function ϕ(·) is a unimodal function. Hence, (A.11) is negative definite if λ2 γ > γ and τ < ϕ(γ). Finally, ultimate boundedness of the solutions and the Lyapunov-Razumikhin theorem imply that the set {ζ¯ = y¯ = 0} is a global attractor for λ2 γ > γ and τ < ϕ(γ). A.2 Proof of Corollary 2 From Theorem 1, the optimal coupling strength γ ∗ for any strongly connected graph is given by (A.16). Moreover, the corresponding maximum time-delay τ ∗ is given by ϕ(γ ∗ ) with unimodal function ϕ(·) given in (A.15). After some simple algebra, γ ∗ and τ ∗ = ϕ(γ ∗ ) can be rewritten as γ γ = λ2 ∗
1+
1 λ
1 + 2¯ c1 λk 2
+
1+
1 λ
(1 + 2¯ c1 λk )2 2
+
2 c¯21 − c¯2 γ
λ
c¯2 γ (1 + 2¯ c1 λk )
,
2
(A.19)
6
ScienceDirect IFAC-PapersOnLine 49-14 (2016) 074–079
Robust Network Network Synchronization Synchronization of of Robust Robust Network Synchronization Time-Delayed Coupled Coupled Systems Systems of Time-Delayed Time-Delayed Coupled Systems
∗∗ Carlos Murguia ∗∗ Justin Ruths ∗∗ Henk Nijmeijer ∗∗ Carlos Murguia ∗ Justin Ruths ∗ ∗ Henk Nijmeijer ∗∗ ∗∗ ∗ Carlos Murguia Justin Ruths Henk Nijmeijer Carlos Murguia Justin Ruths Henk Nijmeijer ∗ Singapore University of Technology and Design, Engineering ∗ ∗ Singapore University of Technology and Design, Engineering ∗ Singapore University of Center Technology and Design, Design, Engineering Systems and Design (ESD), for Research in Cyber Security Singapore University of Technology and Engineering Systems and Design (ESD), Center for Research in Cyber Security Systems and Design (ESD), Center for Research in Cyber Security (iTrust), e-mails: murguia [email protected] and Systems (iTrust), and Design (ESD), Center for Research in Cyber Security murguia (iTrust), e-mails: e-mails: murguia [email protected] [email protected] and and [email protected] (iTrust), e-mails: murguia [email protected] and [email protected] [email protected] ∗∗ [email protected] ∗∗ Eindhoven University of Technology, Mechanical Engineering Eindhoven University of Mechanical ∗∗ ∗∗ Eindhoven University e-mail: of Technology, Technology, Mechanical Engineering Engineering Department, [email protected] Eindhoven University of Technology, Mechanical Engineering Department, e-mail: [email protected] Department, e-mail: [email protected] Department, e-mail: [email protected] Abstract: We address the problem of controlled synchronization in networks of time-delayed Abstract: We the of synchronization in of Abstract: We address address the problem problem of controlled controlled synchronization in networks networks of time-delayed time-delayed coupled nonlinear systems. In particular, we prove that, under some mild conditions, there Abstract: We address the problem of controlled synchronization in networks of time-delayed coupled nonlinear systems. In particular, we prove that, under some mild conditions, there coupled nonlinear systems. In particular, we prove that, under some mild conditions, there always exists a unimodal region in the parameter space (coupling strength γ versus time-delay coupled nonlinear systems. In particular, we prove that, under some mild conditions, there always parameter space (coupling strength versus always exists unimodal region intothe the parameter space (coupling strength γ γWe versus time-delay τ ), suchexists that a ifunimodal γ and τ region belongin this region, the systems synchronize. showtime-delay how this always exists aaif unimodal region in the parameter space (coupling strength γ versus time-delay ττunimodal ), such that γ and τ belong to this region, the systems synchronize. We show how this ), such that if γ and τ belong to this region, the systems synchronize. We show how this region scales with the network topology, which, in turn, provides useful insights of τunimodal ), such that if γscales and with τ belong to this region, thewhich, systems synchronize. We showinsights how this region the network topology, in turn, provides useful of unimodal region scales withtopology the network network topology,robustness which, in in against turn, provides provides usefulThe insights of how to design thescales network to maximize time-delays. results unimodal region with the topology, which, turn, useful insights of how to the network to maximize against time-delays. The how to design design by thecomputer network topology topology to of maximize robustness againstneural time-delays. The results results are illustrated simulations coupled robustness Hindmarsh-Rose chaotic oscillators. how to design the network topology to maximize robustness against time-delays. The results are illustrated by computer simulations of coupled Hindmarsh-Rose neural chaotic oscillators. are illustrated by computer simulations of coupled Hindmarsh-Rose neural chaotic oscillators. are illustrated by computer simulations of coupled Hindmarsh-Rose neural chaotic oscillators. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Robustness, Delays, Synchronization, Control of Networks, Nonlinear Systems. Keywords: Robustness, Delays, Delays, Synchronization, Control Control of Networks, Networks, Nonlinear Systems. Systems. Keywords: Keywords: Robustness, Robustness, Delays, Synchronization, Synchronization, Control of of Networks, Nonlinear Nonlinear Systems. 1. INTRODUCTION that, under some mild assumptions, there always exists a 1. INTRODUCTION INTRODUCTION that, some mild there always 1. that, under under some mild assumptions, assumptions, there strength always exists exists region S in the parameter space (coupling γ ver-aa 1. INTRODUCTION that, under some mild assumptions, there always exists a region S in the parameter space (coupling strength γ region S in the parameter space (coupling strength γ verversus time-delay τ ), such that if (γ, τ ) ∈ S then the systems The emergence of synchronization in networks of coupled region S in the parameter space (coupling strength γ versus such (γ, S the The emergence emergence of synchronization synchronization in networks networks of sus time-delay time-delayToτττ ), ),derive such that that ifresults, (γ, τττ ))) ∈ ∈the S then then the systems systems synchronize. theirif authors assume The of in of coupled dynamical systems is a pervasive phenomenon in coupled various sus time-delay ), such that if (γ, ∈ S then the systems The emergence of synchronization in networks of coupled synchronize. To derive their results, the authors assume dynamical systems is a pervasive phenomenon in various synchronize. To derive their results, the authors assume that the individual systems are semipassive (Pogromsky dynamical systems is a pervasive phenomenon in various scientific disciplines ranging from biology, physics, and synchronize. To derive their are results, the authors assume dynamical systems isranging a pervasive phenomenon in various the individual systems semipassive (Pogromsky scientific disciplines disciplines from biology, physics, physics, and that that the individual systems are semipassive (Pogromsky et al. (1999)) with respect to the coupling variable and the scientific ranging from biology, and chemistry to social networks and technological applicathat the individual systems are semipassive (Pogromsky scientific disciplines ranging from biology, physics, and et al. (1999)) with respect to the coupling variable and the chemistry to examples social networks networks and technological technological applicaet al. al. (1999)) (1999)) with with respect to the the coupling coupling variablesystems, and the the corresponding internal dynamics are convergent chemistry to social and tions. Several of synchronous behavior inapplicascience et respect to variable and chemistry to social networks and technological applicacorresponding internal dynamics are convergent systems, tions. Several examples of synchronous behavior in science corresponding internal dynamics are convergent systems, (see Pavlov et al. (2004)). In the same spirit, here we prove tions. Several examples of synchronous behavior in science and engineering can be of found in, for instance, internal dynamics are spirit, convergent systems, tions. Several examples synchronous behaviorBlekhman in science corresponding (see et In the here and engineering engineering can be(2001), found in, for instance, instance, Blekhman (see Pavlov Pavlov et al. al.S(2004)). (2004)). In bounded the same same by spirit, here we we prove prove that the region is always a unimodal funcand be found in, for (1988), Pikovsky can et al. Strogatz (2003), Blekhman and refer- (see Pavlov et al. (2004)). In the same spirit, here we prove and engineering can be found in, for instance, Blekhman the region S is always bounded by aa unimodal func(1988),therein. Pikovsky et of al.the (2001), Strogatz (2003), and refer- that that the region S is always bounded by unimodal function ϕ(γ) defined on some set J ⊂ R; and consequently, (1988), Pikovsky et al. (2001), Strogatz (2003), and references One first technical results regarding that the region S is always bounded by a unimodal func(1988), Pikovsky et of al.the (2001), Strogatz (2003), and refer- tion ϕ(γ) defined on some set J ⊂ R; and consequently, ences therein. therein. One first technical results regarding tion ϕ(γ) ϕ(γ) defined on some set J ⊂ ⊂ R; R; and consequently, consequently, that there defined always on exists an set optimal coupling strength γ ∗∗ ences One of the first technical results regarding synchronization of coupled nonlinear systems is presented tion some J and ences therein. One of the first technical results regarding that there exists coupling γ synchronization of coupled coupled nonlinear systems isthe presented there always exists an an optimal optimal coupling strength γ ∗∗ leads always to the maximum time-delay τ ∗∗ = strength ϕ(γ ∗∗ ) that synchronization of nonlinear systems presented in Fujisaka and Yamada (1983). In this paper, is authors that that there always exists an optimal coupling strength γ synchronization of coupled nonlinear systems isthe presented that leads to the maximum time-delay τ = ϕ(γ ) that ∗ ∗ in Fujisaka and Yamada (1983). In this paper, authors that leads to the the maximum time-delay = ϕ(γ ϕ(γ ∗ )) that that can be induced to maximum the network without compromising the in Fujisaka and (1983). In paper, the show that coupled chaotic oscillators synchronize in that leads to time-delay ττ ∗ = in Fujisaka and Yamada Yamada (1983). In this thismay paper, the authors authors be induced to the network without compromising the ∗ the show that coupled chaotic oscillators may synchronize in can can be induced to the network without compromising and synchronous behavior. It follows that for γ = γ show that coupled chaotic oscillators may synchronize in spite of their high sensitivity to initial conditions. After can be induced to the network without compromising the ∗ show that coupled chaotic oscillators may synchronize in and synchronous behavior. It follows that for γ = γ ∗ ∗ spiteresult, of their their high sensitivity sensitivity toininitial initial conditions. After for and synchronous behavior. It follows follows that for for γ γ = the γ ∗ gain any τ ≤ behavior. τ ∗ the systems synchronize, i.e.,= spite of high to conditions. After this considerable interest the notion of synchroand synchronous It that γ spite of their high sensitivity to initial conditions. After for ≤ ∗ the systems synchronize, i.e., the gain this result, result, considerable interest in the the notion notion of Here, synchro∗ for any ≤ τττto the systems synchronize, i.e., the gain gain γ =any γ ∗∗ τττleads the systems best tolerance againsti.e., time-delays this considerable interest in of synchronization of general nonlinear systems arisen. we for any ≤ the synchronize, the this result, considerable interest in thehas notion of Here, synchroγ = γ leads to the best tolerance against time-delays ∗ nization of general nonlinear systems has arisen. we ∗ leads to system. γ = γ the best tolerance against time-delays of the closed-loop Finally, we analyze the effect of nization of general nonlinear systems has arisen. Here, we focus on network synchronization of identical nonlinear γ = γ leads to the best tolerance against time-delays nization general synchronization nonlinear systems arisen. nonlinear Here, we of the closed-loop system. Finally, we analyze the effect of focus on on of network of has identical of the closed-loop system. Finally, we analyze the effect of the network topology on the values of both optimal focus network synchronization of identical nonlinear systems interacting through diffusive time-delayed couof the closed-loop system. Finally, we analyze the effect of focus on interacting network synchronization of identical nonlinear the topology on the values both the optimal ∗ network ∗of systems through diffusive time-delayed couthe network topology on the values of both the optimal γ and the maximum time-delay τ , i.e., we show how systems through diffusive time-delayed couplings oninteracting networks with general topologies. Time-delayed ∗ network topology on the values∗of both the optimal systems through diffusive time-delayed cou- the γ the ττ ∗∗ ,, i.e., we how plings on oninteracting networks with general topologies. Time-delayed γ ∗∗ and and the maximum maximum time-delay i.e., we show show how eigenvalues of the time-delay corresponding Laplacian matrix plings networks with general topologies. Time-delayed couplings arise naturally for interconnected systems since the γ and the maximum time-delay τ , i.e., we show how plings on networks with general topologies. Time-delayed the eigenvalues of the corresponding Laplacian matrix ∗ ∗ couplings arise naturally for interconnected systems since the eigenvalues of the corresponding Laplacian matrix affect ϕ(γ), γ , and τ . This, in turn, gives insights on couplings arise naturally for interconnected systems since the transmission of signals is expected to take some time. the eigenvalues of the corresponding Laplacian matrix ∗ ∗ couplings arise naturally foris interconnected systems since affect ϕ(γ), γ ∗ , and τ ∗ . This, in turn, gives insights on the transmission transmission of signals signals expected to take take some faults time. affect ϕ(γ), γ ∗ the , and and τ ∗ .. This, This, in turn, turn, gives insights insights on how toϕ(γ), design network topology in order to enhance the of is expected to some time. Time-delays caused by signal transmission and/or affect γ , τ in gives on the transmission of signals is expected to take some time. how to design the network topology in order to enhance Time-delays caused by signal transmission and/or faults how to design the network topology in order to enhance robustness against time-delays. Time-delays caused signal and/or faults in the communication affect the behavior the how to design the network topology in order to enhance Time-delays caused by bychannels signal transmission transmission and/or of faults in the the communication communication channels affect the behavior of the robustness robustness against against time-delays. time-delays. in behavior of the interconnected systemschannels (e.g., in affect terms the of stability and/or against time-delays. in the communication channels affect the behavior of the robustness interconnected systems (e.g., in terms of stability and/or 2. PRELIMINARIES interconnected systems (e.g., in terms of stability and/or performance). interconnected 2. performance). systems (e.g., in terms of stability and/or 2. PRELIMINARIES PRELIMINARIES performance). 2. PRELIMINARIES performance). The results presented here follow the same research line as In this section, we introduce some important properties The results presented here follow the same research line as The results presented here research line Steur and Nijmeijer Steurthe etsame al. (2014), Murguia section, we some important properties The results presented(2010), here follow follow the research line as as In In this this section, which we introduce introduce somefor important properties and definitions are needed the subsequent reSteur and Nijmeijer Nijmeijer (2010), Steur etsame al. (2014), (2014), Murguia In this section, we introduce some important properties Steur and (2010), et al. Murguia et al. (2015a), and Murguia etSteur al. (2015b), where sufficient and definitions which are needed for the subsequent reSteur and Nijmeijer (2010), Steur et al. (2014), Murguia and definitions which are needed for the subsequent results. Throughout this paper, the following notation is et al. (2015a), and Murguia et al. (2015b), where sufficient and definitions which are needed for the subsequent reet al. and al. (2015b), sufficient conditions for synchronization interconnected Throughout paper, the is et al. (2015a), (2015a), and Murguia Murguia et et of al.diffusively (2015b), where where sufficient sults. sults. the Throughout this paper, the following following notation is used: symbol Rthis the set notation of positive >0 (R ≥0 ) denotes conditions for synchronization of diffusively interconnected sults. Throughout this paper, the following notation is conditions for synchronization of interconnected semipassive systems with time-delays are derived. In par- (nonnegative) used: )) The denotes the of >0 (R conditions for synchronization of diffusively diffusively interconnected used: the the symbol symbol Rnumbers. (R≥0 denotes the set setnorm of positive positive real R Euclidian in Rnn >0 (R ≥0 ) semipassive systems with time-delays are derived. In parused: the symbol R denotes the set of positive >0 ≥0 semipassive systems time-delays particular, in Steur and with Nijmeijer (2010),are thederived. authorsIn prove real The norm in 2 Euclidian semipassive systems time-delays par- (nonnegative) (nonnegative) real numbers. numbers. The Euclidian norm in R Rnn is denoted simply as | · |, |x| xTT x. The notation ticular, in in Steur Steur and with Nijmeijer (2010),are thederived. authorsInprove prove (nonnegative) real numbers. The Euclidian norm in R 2 = ticular, and Nijmeijer (2010), the authors is denoted simply as | · |, |x| = x x. The notation 2 T ticular, in Steur and Nijmeijer (2010), the authors prove 2 = x T x. The is denoted simply asfor|| ··the |, |x| |x| notation col(x ) standsas column vector composed of This work was supported by the National Research Foundation 1 , ..., xnsimply is denoted |, = x x. The notation col(x x stands for the column vector composed of This 1 ,, ..., n) was supported bySingapore, the National Research Foundation col(xelements ..., x ) stands for the column vector composed of the x , ..., x . This notation is also used in case This work 1 n 1 n (NRF), Prime Minister’s Office, under its National Cybercol(x , ..., x ) stands for the column vector composed of work was supported by the National Research Foundation 1 n This work was supported bySingapore, the National Research Foundation the components elements x This is used case (NRF), Prime Minister’s Office, under its National Cyberelements x x111 ,,,x..., ..., xnnn ... vectors. This notation notation is nalso also used in in case the are The n × identity matrix security R&D Programme (Award No. NRF2014NCR-NCR001-40) i (NRF), Prime Minister’s Office, Singapore, under its National Cyberthe elements x ..., x This notation is also used in case (NRF), Prime Office, Singapore, under its National Cyberthe components vectors. The × security R&D Minister’s Programme (Award No. NRF2014NCR-NCR001-40) thedenoted components xiiiorare aresimply vectors. The nconfusion ×n n identity identity matrix and administered by the National R&D Directorate. is by Inx I if non canmatrix arise. security R&D Programme Programme (AwardCybersecurity No. NRF2014NCR-NCR001-40) NRF2014NCR-NCR001-40) the components x are vectors. The n × n identity matrix security R&D (Award No. and administered by the National Cybersecurity R&D Directorate. is denoted by I or simply I if no confusion can arise. n and administered by the National Cybersecurity R&D Directorate. is denoted by I or simply I if no confusion can arise. n and administered by the National Cybersecurity R&D Directorate. is denoted by In or simply I if no confusion can arise.
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Likewise, the n × m matrices composed of only ones and only zeros are denoted as 1n×m and 0n×m , respectively. The spectrum of a matrix A is denoted by spec(A). The symbol ⊗ denotes Kronecker product. Let X ⊂ Rn and Y ⊂ Rm . The space of continuous functions from X to Y is denoted by C(X , Y). If the functions are (at least) r ≥ 0 times continuously differentiable, then it is denoted by C r (X , Y). If the derivatives of a function of all orders (r = ∞) exist, the function is called smooth and if the derivatives up to a sufficiently high order exist the function is named sufficiently smooth. For simplicity of notation, we often suppress the explicit dependence of time t.
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initial condition. A sufficient condition for system (1a) to be convergent is presented in the following proposition. Proposition 1. Demidovich (1967) and Pavlov et al. (2004). If there exists a positive definite matrix P ∈ Rn×n such that all the eigenvalues λi (Q) of the symmetric matrix: T 1 ∂f ∂f P (x, u) + (x, u) Q(x, u) = P , (2) 2 ∂x ∂x are negative and separated from zero, i.e., there exists a constant c ∈ R>0 such that λi (Q) ≤ −c < 0, for all i ∈ {1, ..., n}, u ∈ U , and x ∈ Rn , then system (1a) is globally exponentially convergent. Moreover, for any pair of solutions x1 (t), x2 (t) ∈Rn of (1a), it is satisfied that: T 2 d x1 − x2 P x1 − x2 ≤ −α |x1 − x2 | with constant dt α := λmaxc (P ) and λmax (P ) being the largest eigenvalue of the symmetric matrix P .
2.1 Communication Graphs Given a set of interconnected systems, the communication topology is encoded through a communication graph. Let G = (V, E, A) denote a weighted undirected graph, where V = {v1 , v2 , ..., vk } is the set of nodes, E ⊆ V × V is the set of edges, and A ∈ Rk×k := aij is the weighted adjacency matrix with entries aij = aji ≥ 0. The set of neighbors of vi , denoted as Ei , is the set of nodes which have edges pointing to vi . Throughout this manuscript, it is assumed that the communication graph is strongly connected and simple, see Bollobas (1998). We denote the degree matrix D ∈ Rk×k := diag{d1 , ..., dk } with di := j∈Ei aij , and L := D − A, which is called the Laplacian matrix of the graph G.
3. PROBLEM STATEMENT Consider k identical nonlinear systems of the form ζ˙i = q(ζi , yi ),
(3) (4) y˙ i = a(ζi , yi ) + CBui , with i ∈ I := {1, ..., k}, state xi := col(ζi , yi ) ∈ Rn , internal state ζi ∈ Rn−m , output yi ∈ Rm , input ui ∈ Rm , sufficiently smooth functions q : Rn−m × Rm → Rn−m and a : Rn−m × Rm → Rm , matrices C ∈ Rm×n and B ∈ Rn×m , and the matrix CB ∈ Rm×m being similar to a positive definite matrix. For the sake of simplicity, it is assumed that CB = Im (results for the general case with CB being similar to a positive definite matrix can be easily derived). The system (3),(4) is assumed to be strictly C 1 -semipassive and the internal dynamics (3) is assumed to be an exponentially convergent system. Let the k systems (3),(4) interact on simple strongly connected graph through the diffusive time-delayed coupling ui (t) = γ aij (yj (t − τ ) − yi (t − τ )) , (5)
2.2 Semipassive Systems Consider the system x˙ = f (x, u), (1a) y = h(x), (1b) with state x ∈ Rn , input u ∈ Rm , output y ∈ Rm , and sufficiently smooth functions f : Rn × Rm → Rn and h : Rn → Rm . Definition 1. Pogromsky et al. (1999). System (1) is called C r -semipassive if there exists a nonnegative storage function V ∈ C r (Rn , R≥0 ) such that V˙ ≤ y T u − H(x), where the function H ∈ C(Rn , R) is nonnegative outside some ball, i.e., ∃ ϕ > 0 s.t. |x| ≥ ϕ → H(x) ≥ (|x|), for some continuous nonnegative function (·) defined for |x| ≥ ϕ. If the function H(·) is positive outside some ball, then the system (1) is said to be strictly C r -semipassive.
j∈Ei
where τ ∈ R≥0 denotes a constant time-delay, yj (t − τ ) and yi (t − τ ) are the delayed outputs of the j-th and i-th systems, γ ∈ R≥0 denotes the coupling strength, aij ≥ 0 are the weights of the interconnections, and Ei is the set of neighbors of system i. It is assumed that the graph is undirected, i.e., aij = aji . Since the coupling strength is encompassed in the constant γ, it is assumed without loss of generality that maxi∈I j∈Ei aij = 1. Note that all signals in coupling (5) are time-delayed. Such a coupling may arise, for instance, when the systems are interconnected through a centralized control law. Coupling (5) can be written in matrix form as u = −γ (L ⊗ Im ) y(t − τ ) with u := col(u1 , . . . , uk ), y := col(y1 , . . . , yk ), and Laplacian matrix L = LT ∈ Rk×k . The authors in Steur and Nijmeijer (2010) prove that the k coupled systems (3)-(5) asymptotically synchronize provided that γ is sufficiently large and the product of the coupling strength and the time-delay γτ is sufficiently small. It follows that there exists a region S in the parameter space, such that if (γ, τ ) ∈ S, the systems synchronize. In this manuscript, we go one step further by showing that this region S is actually bounded by a unimodal function ϕ : J → R≥0 , γ → ϕ(γ). Hence, there exists an optimal coupling
A (strictly) C r -semipassive system behaves like a (strictly) passive system for large |x(t)|. 2.3 Convergent Systems
Consider system (1a) and suppose f (·) is locally Lipschitz in x, u(·) is piecewise continuous in t and takes values in some compact set u ∈ U ⊆ Rm . Definition 2. System (1a) is said to be convergent if and only if for any bounded signal u(t), defined on the whole interval (−∞, +∞), there is a unique bounded globally asymptotically stable solution x ¯u (t), defined in the same interval, for which it holds that limt→∞ |x(t) − x ¯u (t)| = 0 for all initial conditions. For a convergent system, the limit solution is solely determined by the external excitation u(t) and not by the 2
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nize provided that the coupling strength γ is sufficiently large and the time-delay τ is smaller than some unimodal function ϕ(γ), see Figure 1. Hence, there exists a region S (colored area in Figure 1) such that if (γ, τ ) ∈ S, the systems synchronize. Estimates of both the constant γ and the unimodal function ϕ(γ) are derived in the proof of Theorem 1, in (A.13) and (A.15), respectively. Corollary 1. Consider k coupled systems (3)-(5) with coupling strength γ ∈ R≥0 and time-delay τ ∈ R≥0 on a simple strongly connected graph G. Suppose that the conditions stated in Theorem 1 are satisfied. Then, for every γ¯ ∈ J , there exists a maximum time-delay τ¯ ∈ R≥0 , such that if γ = γ¯ , the systems synchronize for all τ ≤ τ¯. Moreover, there exists an optimal coupling strength γ = γ ∗ ∈ J and its corresponding maximum time-delay τ ∗ := max(τ ∈ S) such that τ¯ < τ ∗ for all γ = γ ∗ .
Figure 1. Synchronization Region S.
strength γ ∗ that leads to the maximum delay τ ∗ = ϕ(γ ∗ ) that can be induced to the network without compromising the synchronous behavior. Moreover, we characterize the function ϕ(γ), the optimal γ ∗ , and the maximum timedelay τ ∗ in terms of the spectrum of the Laplacian matrix L and the vector fields q(·) and a(·).
Proof: The assertion follows from unimodality of ϕ(γ). The result stated in Corollary 1 implies that for every strongly connected graph G, there exists an optimal coupling strength γ ∗ that leads to the maximum time-delay τ ∗ = ϕ(γ ∗ ) = max(τ ∈ S) that can be induced to the network without compromising the synchronous behavior, i.e., there exists a gain γ = γ ∗ that leads to the best tolerance against time-delays of the closed-loop system, see Figure 1. In the following corollary, we characterize the effect of the network topology on the values of both the optimal γ ∗ and the maximum time-delay τ ∗ . Let G¯ := {Gs | s ∈ N } be the set of all simple strongly connected undirected graphs. For each graph Gs , consider k = ks coupled systems (3)-(5) with coupling strength γ = γs ∈ R≥0 and delay τ = τs ∈ R≥0 interacting on Gs . Assume that the conditions stated in Theorem 1 are satisfied, then from Theorem 1, there exist constants γs ∈ R>0 and unimodal functions ϕs : Js → R≥0 , s ∈ N, such that if (γs , τs ) ∈ Ss , where ¯ γs > γs , τs < ϕs (γs )}, Ss := {γs , τs ∈ R≥0 | γs < δ/2, the ks systems of each Gs asymptotically synchronize. Let γs∗ and τs∗ = max(τs ∈ Ss ) be the optimal coupling strength and the corresponding maximum time-delay of Gs and define τ¯∗ := {τs∗ | s ∈ N }, i.e., the set of all maximum ¯ and time-delays of G, ∗ τmax := max(¯ τ ∗ ). (7) Corollary 2. For s = 1, 2. Consider k = ks interconnected systems (3)-(5) with coupling strength γ = γs ∈ R≥0 and time-delay τ = τs ∈ R≥0 on a simple strongly connected graph Gs , i.e., two independent diffusively timedelayed coupled networks. Assume that G1 = G2 and the conditions stated in Theorem 1 are satisfied. Then, by Theorem 1, there exist a region Ss such that if (γs , τs ) ∈ Ss the ks systems of Gs asymptotically synchronize. Let γs∗ and τs∗ = max(τs ∈ Ss ) be the optimal coupling strength and the corresponding maximum time-delay of Gs . Additionally, let Ls ∈ Rks ×ks be the Laplacian matrix of Gs with real positive eigenvalues λj (Ls ), j = 2, . . . , ks , and λ2 (Ls ) ≤ . . . ≤ λks (Ls ). Then:
4. MAIN RESULTS Define x := col(x1 , . . . , xk ) and the synchronization manifold M := {x ∈ Rkn | xi = xj , ∀ i, j ∈ I}. The coupled systems (3)-(5) are said to synchronize if the synchronization manifold M is invariant under the closed-loop dynamics and contains an asymptotically stable subset. Clearly, if the systems are interconnected through (5), the coupling functions vanish on M; hence, the manifold M is positively invariant under the dynamics (3)-(5). In the following theorem, we give sufficient conditions for the existence of an asymptotically stable subset of the synchronization manifold. Theorem 1. Consider k coupled systems (3)-(5) with coupling strength γ ∈ R≥0 and time-delay τ ∈ R≥0 on a simple strongly connected graph G. Assume that: (H4.1) Each system (3),(4) is strictly C 1 -semipassive with input ui , output yi , radially unbounded storage function V (xi ), and the functions H(xi ) are such that there exists a constant R ∈ R>0 such that |xi | > R implies that H(xi ) − δ|yi |2 > 0 for some constant δ ∈ R>0 . Let δ¯ be the largest δ that satisfies (H4.1). Then, the solutions of the coupled systems (3)-(5) are ultimately ¯ bounded for any finite τ ∈ R>0 and γ < δ/2. In addition assume that: (H4.2) The internal dynamics (3) is exponentially convergent, i.e., there is a positive definite matrix P such that the eigenvalues of the symmetric matrix T 1 ∂q ∂q P (ζi , yi ) + (ζi , yi ) P , (6) 2 ∂ζi ∂ζi are uniformly negative and bounded away from zero for all ζi ∈ Rn−m and yi ∈ Rm . Then, there exist a constant γ ∈ R>0 and a unimodal function ϕ : J := [γ , ∞) → R≥0 , γ → ϕ(γ), where ϕ(γ ) = 0 and limγ→∞ ϕ(γ) = 0, such that if (γ, τ ) ∈ ¯ S := {γ, τ ∈ R≥0 | γ < 2δ , γ > γ , τ < ϕ(γ)}, then there exists a globally asymptotically stable subset of the synchronization manifold M.
λ
λ
(a) If λk21 (L1 ) = λk22 (L2 ) > 1, then τ1∗ = τ2∗ . In addition, λ2 (L1 ) ≥ λ2 (L2 ) implies γ1∗ ≤ γ2∗ .
The proof of Theorem 1 can be found in the appendix. Theorem 1 implies that the solutions of (3)-(5) are ultimately bounded and the systems asymptotically synchro-
λ
λ
∗ . In (b) If λk21 (L1 ) = λk22 (L2 ) = 1, then τ1∗ = τ2∗ = τmax ∗ ∗ addition, λ2 (L1 ) ≥ λ2 (L2 ) implies γ1 ≤ γ2 .
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Figure 2. Synchronization Regions Ss , s = 1, 2. Left: Case (a) in Corollary 2. Right: Case (b) in Corollary 2.
Figure 4. Synchronization regions Ss computed for different topologies Gs , s = 1, . . . , 7. 2 2 . More+25z2i storage function V (z1i , z2i , yi ) := 12 yi2 +σz1i over, the (z1i , z2i )-dynamics (the internal dynamics) is exponentially convergent (in the sense of Definition 2), i.e., it satisfies the Demidovich condition (6) (with P = I2 ); hence, assumption (H4.2) in Theorem 1 is satisfied. This particular experiment is taken from Neefs et al. (2010), where a detailed experimental study is presented. B. Bounded Solutions and Synchronization. It can be easily verified that the function H(xi ) obtained from the V (z1i , z2i , yi ) defined above satisfies the boundedness assumption (H4.1) for arbitrary large coupling strength γ, see Neefs et al. (2010). Therefore, by Theorem 1, the solutions of the coupled systems (8),(5) with γ = γs and τ = τs are ultimately bounded for any finite coupling strength γs and time-delay τs . Finally, given that all the graphs in Figure 3 are strongly connected and (H4.1) and (H4.2) are satisfied, then, by Theorem 1, there exist regions Ss , s = 1, . . . , 7, (as depicted in Figure 1), such that if (γs , τs ) ∈ Ss , the systems synchronize. C. Simulation Results. In Figure 4, we show results obtained through extensive computer simulations. We depict the synchronization regions Ss , s = 1, . . . , 7 for each network topology Gs . These regions are clearly bounded by unimodal functions; and therefore, for each network, there exists an optimal coupling strength γs∗ and its corresponding maximum time-delay τs∗ that can be induced to the network without compromising the synchronous behavior. These maximum time-delays are strongly influenced by the network topology, (see the proofs of Theorem 1 and Corollary 2). In Table 1, we show the numerical values of the optimal coupling strengths, the maximum time-delays, and the quotients λks /λ2 . These values match with the theoretical predictions given in Corollary 2.
Figure 3. Network topologies. λ
77
λ
(c) If λk21 (L1 ) > λk22 (L2 ), then τ1∗ < τ2∗ . In addition, λ2 (L1 ) ≥ λ2 (L2 ) implies γ1∗ ≤ γ2∗ . The proof of Corollary 2 can be found in the appendix. The result stated in Corollary 2 amounts to the following. The effect of the network topology on the maximum timedelay τ ∗ is solely determined by the quotient ( λλk2 ). Case (a) implies that any two strongly connected networks with the same quotient have the same maximum time-delay τ ∗ . Additionally, in this case, the value of the optimal coupling strength γ ∗ is also determined by λ2 , i.e., if the two networks have the same quotient, then the larger the λ2 the smaller the γ ∗ , and vice versa. Case (b) implies that networks with quotient equal to one have the best tolerance against time-delays, i.e., if λλk2 = 1, then τ ∗ = ∗ ∗ with τmax as defined in (7). Case (c) implies that the τmax larger the quotient λλk2 the smaller the τ ∗ , and vice versa. See Figure 2. 5. SIMULATION EXPERIMENT A. Network Topology, Strict Semipassivity, and Convergence. Consider a network of ks , s ∈ {1, . . . , 7}, systems coupled according to the graphs Gs depicted in Figure 3. For each network, the weights of the interconnections are set to aij = 1/ks if {i, j} ∈ Es and aij = 0 otherwise. The networks are strongly connected and undirected. Each system in the networks is assumed to be a Hindmarsh-Rose neuron, see Hindmarsh and Rose (1984), of the form z˙1i = 1 − 5yi2 − z1i , z˙2i = 0.005(4yi + 6.472 − z2i ), (8) y˙ i = −yi3 + 3yi2 + z1i − z2i + 3.25 + ui , with output yi ∈ R, internal states zi1 , zi2 ∈ R, state xi = col(zi1 , zi2 , yi ) ∈ R3 , input ui ∈ R, and i ∈ I = {1, . . . , ks }. In Steur et al. (2009), the authors prove that the Hindmarsh-Rose neuron is strictly C 1 -semipassive with
6. CONCLUSIONS We have presented a result on network synchronization of coupled nonlinear systems in the case when the coupling functions are subject to constant time-delays. Using the notions of semipassivity and convergent systems, we have proved that, under some mild assumptions, there always exists a region S in the parameter space (coupling strength γ versus time-delay τ ), such that if γ, τ ∈ S, the systems synchronize. We proved that this region S is always bounded by a unimodal function ϕ(γ); and consequently, that there always exists an optimal coupling strength γ ∗ which leads to the maximum time-delay τ ∗ = ϕ(γ ∗ ) that the network can tolerate without breaking the synchrony. 4
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Table 1. Simulation results: Optimal coupling strength γs∗ , maximum time-delay τs∗ , eigenvalues λks and λ2 of the corresponding Laplacian matrices Ls , and their quotient (λks /λ2 ). G1 G2 G3 G4 G5 G6 G7
γs∗ 2.00 5.70 1.95 10.6 3.85 3.75 1.90
τs∗ [ms] 4.25 1.10 4.25 0.33 2.55 2.50 4.15
λks 1 1 1 0.8536 1 1 1
λ2 1 1 3
1 0.1464 1 2 1 2
1
strongly connected and undirected. Then, the Laplacian matrix is symmetric and its eigenvalues are real. Moreover, the matrix L has an algebraically simple eigenvalue λ1 = 0 and 1k×1 is the corresponding ˜ = spec(L)\{0}, then from Gerschgorin’s eigenvector. Since spec(L) ˜ are disc theorem, it can be concluded that the eigenvalues of L ˜ has eigenvalues λ2 , ..., λk ∈ R>0 positive real, i.e., the matrix L with 0 < λ2 ≤ · · · ≤ λk . Coupling (5) can be written in matrix form u = −γ (L ⊗ Im ) y(t − τ ) where u = col(u1 , ..., uk ) ∈ Rkm . Denote u ˜ = col((u1 − u2 ), ..., (u1 − uk )), it follows that
λks /λ2 1 3 1 5.8306 2 2 1
˜ ⊗ Im )˜ u ˜(t) = −γ(L y (t − τ ),
˙ ˜ y1 , ζ1 ), ζ˜ = q˜(˜ y , ζ, ˜ y1 , ζ1 ) − γ(L ˜ ⊗ Im )˜ y (t − τ ), y˜˙ = a ˜(˜ y , ζ,
In Corollary 2, we have provided tools for selecting the network topology in order to enhance robustness against time-delays of the coupled systems. Case (b) in Corollary 2 implies that networks with quotient (λk /λ2 ) equal to 1 have the best tolerance against time-delays. This is the case for all-to-all networks.
−τ
˜ y1 , ζ1 ) − γ(L ˜ ⊗ Im )˜ ˜ ⊗ Im ) y˜˙ = a(˜ y , ζ, y + γ(L
A.1 Proof of Theorem 1 : First, we prove ultimate boundedness of the closed-loop system (3)-(5). By assumption, each system (3),(4) is strictly C 1 -semipassive with input ui , output yi , and radially unbounded function V (xi ). Define the functional: W (xt (θ)) :=
k
νi V (xi ) + γ
i=1
aij
j∈Ei
−τ
substitution of (A.5) in (A.6) yields
|yi (t + s)|2 ds ,
˜ ⊗ Im ) + γ(L
i=1
˙ ≤ ν (−H(xi )+2γ |yi |2 ), see the proof of Theorem 1 in Steur W i=1 i and Nijmeijer (2010) for details. By assumption (H4.1), there exist positive constants R, δ ∈ R>0 such that for |xi | > R it is satisfied that H(xi )−δ|yi |2 > 0. Let δ¯ be the largest δ that satisfies (H4.1) for ¯ and for suffiarbitrarily large R < ∞. Then, for γ satisfying γ ≤ δ/2 ˙ < 0. Then, by the same arguments ciently large |x|, it follows that W presented in the proof of Corollary 2 in Steur and Nijmeijer (2010), it can be concluded that the solutions of the closed-loop system (3)-(5) ¯ are ultimately bounded for any finite τ ≥ 0 and γ ≤ δ/2. Next, we prove the asymptotic stability of the synchronization manifold M. Let ζ = col(ζ1 , . . . , ζk ) ∈ Rk(n−m) and y = col(y1 , . . . , yk ) ∈ Rkm . Define M := (1k−1 − Ik−1 ) ∈ R(k−1)×k and introduce the new coordinates ζ˜ = (M ⊗ In−m )ζ and y˜ = (M ⊗ Im )y. Note that y˜ = ζ˜ = 0 implies that the systems are synchronized. Assumption (H4.2), Proposition 1, smoothness of the vector fields, and boundedness of the solutions imply the existence of a positive ˜ such that definite function V2 : R(k−1)(n−m) → R≥0 , ζ˜ → V2 (ζ)
˜ = M
1 0k−1 1k−1 −Ik−1
˜ LM ˜ −1 = →M
0 ∗ ˜ 0k−1 L
,
0 −τ
y˜(t − τ + s)ds
˜ y1 , ζ1 )(t + s)ds. a(˜ y , ζ,
y |2 + γ y¯T (Λ ⊗ Im ) V˙ 3 ≤ −γλ2 |¯
(A.7)
(A.8) 0 −τ 0
y¯(t − τ + s)ds
(A.9)
¯ y1 , ζ1 )(t + s)ds, a ¯(¯ y , ζ, −τ
0
¯ y1 , ζ1 )(t + s)ds a ¯(¯ y , ζ, −τ
¯ y1 , ζ1 ) − γ 2 y¯T (Λ2 ⊗ Im ) ¯(¯ y , ζ, + y¯ a T
0 −τ
y¯(t − τ + s)ds. (A.10)
Ultimate boundedness of the solutions and smoothness of the func¯ y1 , ζ1 ) ≤ c1 |¯ ¯ for some tion a(·) imply that y¯T a ¯(¯ y , ζ, y |2 + c2 |¯ y | |ζ| ¯ y¯) := positive constants c1 , c2 ∈ R>0 . Let the function V1 (ζ, ¯ + V3 (¯ V2 (ζ) y ) be a Lyapunov-Razumikhin function such that if ¯ ¯ + θ), y¯(t + θ)) for θ ∈ [−2τ, 0] and some V1 (ζ(t), y¯(t)) > κ 2 V1 (ζ(t constant κ > 1, then
(A.1)
¯ y1 , ζ1 ), and a ¯ y1 , ζ1 ) := ¯ y1 , ζ1 ) := q˜((U −1 ⊗Im )¯ y , ζ, ¯(¯ y , ζ, where q¯(¯ y , ζ, ¯ y1 , ζ1 ). Notice that y¯ = ζ¯ = 0 implies (U ⊗ Im )˜ a((U −1 ⊗ Im )¯ y , ζ, that the systems are synchronized because U is nonsingular. Since stability is invariant under change of coordinates and U = 1, from (A.1), it follows that there exists a positive definite function ¯ such that V¯˙ 2 (ζ, ¯ y¯) ≤ −α|ζ| ¯2 + V¯2 : R(k−1)(n−m) → R≥0 , ζ¯ → V¯2 (ζ) ¯ |¯ c0 |ζ| y | for some α, c0 ∈ R>0 . Define V3 (¯ y ) := 12 y¯T y¯, then
for some constants α, c0 ∈ R>0 , see Pogromsky et al. (1999) for further details. Notice that
y˜˙ (t + s)ds, (A.6)
−τ
−τ
− γ (Λ ⊗ Im ) y¯(t) + γ(Λ ⊗ Im )
i
˜ |˜ ˜ y˜) ≤ −α|ζ| ˜ 2 + c0 |ζ| y| , V˙ 2 (ζ,
0
¯ y1 , ζ1 ) − γ 2 (Λ2 ⊗ Im ) y¯˙ = a ¯(¯ y , ζ,
sumption, where yiτ := yi (t − τ ).Using Young’s inequality, the fact that ν T L = 0, and maxi∈I aij = 1, it follows that j∈E
k
¯ y1 , ζ1 ), ζ¯˙ = q¯(¯ y , ζ,
i
j∈Ei
0
˜ is nonsingular and symmetric, then there exists a The matrix L ˜ −1 = Λ, matrix U ∈ R(k−1)×(k−1) such that U = 1 and U LU where Λ denotes a diagonal matrix with the nonzero eigenvalues of L on its diagonal. Introduce the change of coordinates y¯ = (U ⊗Im )˜ y ˜ In the new coordinates, the and for consistency of notation ζ¯ = ζ. closed-loop system takes the form
where x = col(x1 , ..., xk ), xt (θ) = x(t + θ) ∈ C, θ ∈ [−τ, 0], C being the Banach space of continuous functions mapping the interval [−τ, 0] into Rkn , and the constants νi denoting the entries of the left eigenvector corresponding to the simple zero eigenvalue of the Laplacian matrix L, i.e., ν = (ν1 , ..., νk )T and ν T L = ν T (D − A) = 0. Note that L is singular by construction and, since it is assumed that the graph is strongly connected, the zero eigenvalue is simple. Using the Perron-Frobenius theorem, it can be proved that the vector ν has strictly positive real entries, see Bollobas (1998). Then, 2 k ˙ = W νi (y T ui − H(xi ) + γ aij (|yi |2 − y τ )) by asi
˜ y1 , ζ1 ) − γ(L ˜ ⊗ Im )˜ ˜ 2 ⊗ Im ) y˜˙ = a(˜ y , ζ, y − γ 2 (L
0
(A.4) (A.5)
˜ y1 , ζ1 ) = col(a(y1 , ζ1 )−a(y1 − y˜1 , ζ1 − ζ˜1 ), . . . , a(y1 , ζ1 )− with a ˜(˜ y , ζ, ˜ y1 , ζ1 ) = col(q(y1 , ζ1 ) − q(y1 − a(y1 − y˜k−1 , ζ1 − ζ˜k−1 )) and q˜(˜ y , ζ, y˜1 , ζ1 − ζ˜1 ), . . . , q(y1 , ζ1 ) − q(y1 − y˜k−1 , ζ1 − ζ˜k−1 )). Using Leibniz’s rule and continuity of the solutions, y˜(t − τ ) can be written as 0 y˜(t − τ ) = y˜(t) − y˜˙ (t + s)ds. Then, (A.5) can be written as
Appendix A. PROOFS
(A.3)
˜ as in (A.2). Then, the closed-loop system can be written as with L
2 y |2 V˙ 1 ≤ −α ζ¯ + κτ λ2k γ 2 + c1 (1 + κγτ λk ) − γλ2 |¯
(A.2)
y| . + (c0 + c2 (1 + κγτ λk )) ζ¯ |¯
(A.11)
The constant κ can be arbitrarily close to one as long as it is greater than one. Then, for simplicity of notation, we take κ on the boundary
where 0k−1 = 0(k−1)×1 , 1k−1 = 1(k−1)×1 , and L denotes the Laplacian matrix. By assumption, the communication graph is
5
IFAC PSYCO 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Carlos Murguia et al. / IFAC-PapersOnLine 49-14 (2016) 074–079
κ = 1 for the rest of the proof. It can be easily verified that (A.11) is negative definite if
with
γ :=
λ2 γ − γ
− λk λ k γ +
λ2 c¯1 γτ − k (γτ )2 > 0, c¯2 2¯ c2
τ
(A.12)
c¯1 γλk
±
c¯2 +
c¯1 γλk
2
+
2¯ c2 (λ2 γ − γ ) . λ2k γ 2
c¯1 ϕ(γ) := − c¯2 + γλk
+
c¯2 +
c¯1 γλk
2
+
c¯1 +
λ 2¯ c2 γ λk 2
+
c¯2
λ
λ
2¯ c2 γ + 4¯ c2 c¯1 γ λk + 2¯ c2 γ λk 2
2
2
.
(A.20)
λ
2
(A.20) that networks with equal λk have the same maximum time2 delay. Moreover, from (A.19), it is clear that if two networks have the λk same quotient λ , then the value of γ ∗ is solely determined by λ2 , 2 the larger the λ2 the smaller the γ ∗ , and vice versa. Case (b): From λ (A.20), it is clear that τ ∗ has its maximum value at λk = 1. The part 2 ∗ regarding γ follows from the same arguments of case (a). Case (c): λk Clearly, from (A.20), the larger the quotient λ the smaller the τ ∗ , 2 and vice versa. Moreover, from (A.19), it is clear that the larger the λ quotient λk the smaller the γ ∗ for a fixed λ2 , and for a fixed quotient, 2 the larger the λ2 the smaller the γ ∗ and the assertion follows.
(A.14)
The time-delay τ is nonnegative by definition. Hence, in order to satisfy (A.14), it is sufficient to consider the possible positive values of the right-hand side of (A.14), i.e., the positive square root. Then, inequality (A.14) amounts to τ < ϕ(γ) with
Next, we analyze the cases stated in Corollary 2. Case (a): The maximum time-delay in (A.20) relies on the constants c¯1 , c¯2 , γ ∈ λ R>0 and the quotient λk . The constants c¯1 , c¯2 , and γ do not depend 2 on the network topology (they depend on the vectorfields q(·) and λ a(·)), then the effect of network is solely determined by λk . From 2 Gerschgorin’s disc theorem, it can be concluded that the eigenvalues λ2 and λk of any strongly connected undirected graph are positive λ real, i.e., 0 < λ2 ≤ λk , and therefore λk ≥ 1. Then, it is clear from
All the constants in (A.12) are positive by construction and γ and τ are nonnegative by definition. Then, a necessary condition for (A.12) to be satisfied is λ2 γ > γ . After some straightforward computations (A.12) can be rewritten as follows
= λk λ2
2αc1 + c0 c2 + c22 (c0 + c2 )2 2α , c¯2 := 2 . (A.13) + c1 , c¯1 := 4α c22 c2
τ < − c¯2 +
∗
79
2¯ c2 (λ2 γ − γ ) . (A.15) λ2k γ 2
We are only interested in possible values of γ, τ ∈ R≥0 such that (A.15) is satisfied. Then, we restrict the function ϕ(γ) to the set
J := [ λγ , ∞). Next, we prove that the function ϕ : J → R≥0 is 2 unimodal. The function ϕ(·) is continuous and real-valued on J .
REFERENCES
ϕ( λγ ) 2
= 0, and Moreover, it is strictly positive on the interior of J , limγ→∞ ϕ(γ) = 0. The function ϕ(·) is differentiable on J ; then, we can compute its local extrema by computing its critical points. It is ∂ϕ(γ) easy to verify that ∂γ = 0 only for γ = γ ∗ and γ = γ ˜ with γ =
γ ˜ =
∗
λ2 1+ λ2 + 2λk c¯1
λ2 λ2 + 2λk c¯1
1+
γ + λ2
γ
λ2
−
2¯ c21 c¯2 γ (λ22 + 2λ2 λk c¯1 + 2λ2k c¯2 γ ) c¯2 λ2 (λ2 + 2λk c¯1 )
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,
(A.16) 2¯ c21 c¯2 γ (λ22
+ 2λ2 λk c¯1 +
2λ2k c¯2 γ )
c¯2 λ2 (λ2 + 2λk c¯1 )
.
(A.17)
˜ are the critical points of ϕ(γ) and ϕ(γ ∗ ) and Then, γ = γ ∗ and γ = γ ϕ(˜ γ ) are the corresponding extrema. Notice that γ ∗ > ∗ γ belongs to the interior of J . Rewrite (A.17) as 2¯ c2 γ − c¯21
γ ˜=
c¯2 (λ2 + c¯1 λk ) + c¯1
γ ; λ2
+2λ2 λk c ¯1 +2λ2 c ¯ γ ) c ¯2 (λ2 2 k 2 2γ
.
therefore,
(A.18)
The denominator of (A.18) is strictly positive, then the sign of γ ˜ is solely determined by its numerator. Substitution of (A.13) in the numerator of (A.18) yields 2¯ c2 γ − c¯21 = −4αc1 (c0 c2 + αc1 ) /c22 , which is strictly negative. It follows that γ ˜ is strictly negative; in consequence, γ ˜∈ / J . Then, the function ϕ(·) has a unique extremum on J and it is given by ϕ(γ ∗ ). Moreover, given that ϕ(γ ) = 0, limγ→∞ ϕ(γ) = 0, φ(γ) is strictly positive on the interior of J , and ϕ(γ ∗ ) is the unique extremum on J , it follows that ϕ(γ ∗ ) is a unique local maximum on J ; therefore, it can be concluded that the function ϕ(·) is a unimodal function. Hence, (A.11) is negative definite if λ2 γ > γ and τ < ϕ(γ). Finally, ultimate boundedness of the solutions and the Lyapunov-Razumikhin theorem imply that the set {ζ¯ = y¯ = 0} is a global attractor for λ2 γ > γ and τ < ϕ(γ). A.2 Proof of Corollary 2 From Theorem 1, the optimal coupling strength γ ∗ for any strongly connected graph is given by (A.16). Moreover, the corresponding maximum time-delay τ ∗ is given by ϕ(γ ∗ ) with unimodal function ϕ(·) given in (A.15). After some simple algebra, γ ∗ and τ ∗ = ϕ(γ ∗ ) can be rewritten as γ γ = λ2 ∗
1+
1 λ
1 + 2¯ c1 λk 2
+
1+
1 λ
(1 + 2¯ c1 λk )2 2
+
2 c¯21 − c¯2 γ
λ
c¯2 γ (1 + 2¯ c1 λk )
,
2
(A.19)
6