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TUTORIAL SESSION: Synchronization in Coupled Oscillators: Theory and Applications Exploring Synchronization in Complex Oscillator Networks Florian Dörfler and Francesco Bullo Florian Dörfler & Francesco Bullo Center for Control, Dynamical Systems, & Computation Alexandre Mauroy, Pierre Sacré, & Rodolphe J. Sepulchre University of California at Santa Barbara Murat Arcak http://motion.me.ucsb.edu 51st IEEE Conference on Decision and Control, Maui, HI 51st IEEE Conference on Decision and Control, Maui, HI A Brief History of Sync A Brief History of Sync how it all began the odd kind of sympathy is still fascinating Christiaan Huygens (1629 – 1695) physicist & mathematician engineer & horologist observed “an odd kind of sympathy ” between coupled & heterogeneous clocks watch movie online here: http://www.youtube.com/watch?v=JWToUATLGzs& list=UUJIyXclKY8FQQwaKBaawl A&index=3 [Letter to Royal Society of London, 1665] Sync of 32 metronomes at Ikeguchi Laboratory, Saitama University, 2012 Recent reviews, experiments, & analysis [M. Bennet et al. ’02, M. Kapitaniak et al. ’12] F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 2 / 38 F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 3 / 38 A Brief History of Sync Coupled Phase Oscillators a field was born Sync in mathematical biology [A. Winfree ’80, S.H. Strogatz ’03, . . . ] ∃ various models of oscillators & interactions Sync in physics and chemistry [Y. Kuramoto ’83, M. Mézard et al. ’87. . . ] Today: canonical coupled oscillator model [A. Winfree ’67, Y. Kuramoto ’75] Sync in neural networks [F.C. Hoppensteadt and E.M. Izhikevich ’00, . . . ] Coupled oscillator model: Xn θ̇i = ωi − aij sin(θi − θj ) Sync in complex networks [C.W. Wu ’07, S. Bocaletti ’08, . . . ] j=1 . . . and countless technological applications (reviewed later) !2 n oscillators with phase θi ∈ S1 !3 a23 a12 non-identical natural frequencies ωi ∈ R1 a13 elastic coupling with strength aij = aji undirected & connected graph G = (V, E, A) F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 4 / 38 Phenomenology and Challenges in Synchronization Synchronization is a trade-off: coupling vs. heterogeneity θ̇i = ωi − Xn j=1 F. Dörfler and F. Bullo (UCSB) !1 Sync in Complex Oscillator Networks CDC 2012 5 / 38 CDC 2012 7 / 38 Applications of the Coupled Oscillator Model aij sin(θi − θj ) Coupled oscillator model: Xn θ̇i = ωi − aij sin(θi − θj ) j=1 coupling small & |ωi − ωj | large ⇒ incoherence & no sync X Some related applications: centroid ✓i (t) Sync in a population of fireflies density [G.B. Ermentrout ’90, Y. Zhou et al. ’06, . . . ] coupling large & |ωi − ωj | small ⇒ coherence & frequency sync Deep-brain stimulation and neuroscience X centroid [N. Kopell et al. ’88, P.A. Tass ’03, . . . ] ✓i (t) density Sync in coupled Josephson junctions [S. Watanabe et. al ’97, K. Wiesenfeld et al. ’98, . . . ] Some central questions: (still after 45 years of work) F. Dörfler and F. Bullo (UCSB) proper notion of sync & phase transition quantify “coupling” vs. “heterogeneity” interplay of network & dynamics Sync in Complex Oscillator Networks CDC 2012 6 / 38 Countless other sync phenomena in physics, biology, chemistry, mechanics, social nets etc. [A. Winfree ’67, S.H. Strogatz ’00, J. Acebrón ’01, . . . ] F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks Example 1: AC Power Transmission Network 29 1 22 4 17 8 21 5 24 7 12 14 15 23 36 6 16 2 13 7 Di 32 3 θ1 + θ2 + 19 20 33 34 Yij 4 |Vi ||Vj ||Yij | | {z } j: · sin θi − θj aij =max power transfer power balance at node i: Pi |{z} X = power injection j inverter in microgrid = controllable AC source 2 5 power transfer on line i physics: Pi 3 ` •◦ : Pl,i < 0 and Di ≥ 0 F. Dörfler and F. Bullo (UCSB) − aij sin(θi − θj ) Droop-control [M.C. Chandorkar et. al., ’93]: X Mi θ̈i + Di θ̇i = Pm,i − aij sin(θi − θj ) j X aij sin(θi − θj ) Di θ̇i = Pl,i − Closed-loop for inverters & load `: [J.W. Simpson-Porco et. al., ’12] CDC 2012 − n 2 Di θ̇i = Pi∗ − Pi ` Di θ̇i = Pi∗ − ai` sin(θi − θ` ) X 0 = P` − a`j sin(θ` − θi ) j j Sync in Complex Oscillator Networks θn + − 1 = ai` sin(θi − θ` ) Structure-Preserving Model [A. Bergen & D. Hill ’81]: : swing eq with Pm,i > 0 θ� 11 10 |Vi | ei✓i 35 28 18 31 Pl,i |Vi | ei✓i 27 3 6 Yij 9 Pm,i 1 26 2 Yik (islanded) microgrid = autonomously managed low-voltage network 9 38 25 Pn�� 30 P2�� 37 10 P1�� 8 |Vj | ei✓j 1 Yij 39 |Vi | ei✓i Example 2: DC/AC Inverters in Microgrids 8 / 38 F. Dörfler and F. Bullo (UCSB) Example 3: Flocking, Schooling, & Vehicle Coordination Sync in Complex Oscillator Networks CDC 2012 9 / 38 Example 4: Canonical Coupled Oscillator Model DCRN Chapter 3: Robotic network models and complexity notions Network of Dubins’ vehicles r= ṙi = v e iθi sensing/comm. graph G = (V, E, A) for coordination of autonomous vehicles relative sensing control ui = fi (θi − θj ) for neighbors {i, j} ∈ E yields closed-loop X θ̇i = ω0 (t) − K · aij sin(θi − θj ) j φ x y (x, y) θ ve # ẋ = f (x) + · δ(t) (b) K = +1 local phase dynamics near γ with phase response curve Q(ϕ) x1 Figure 3.1 A two-wheeled vehicle (a) and four-wheeled vehicle (b). In each case, the orientation of the vehicle is indicated by the small triangle. The unicycle. The controls v and ω take value in [−1, 1] and [−1, 1], respectively. ”phase ϕ̇ = Setsync” v = (ωright + ωleft )/2 and ω = (ωright − The differential drive robot. ωleft )/2 and assume that both ωright and ωleft take value in [−1, 1]. ✏ · Q(') (t) ' ⌦ Ω + · Q(ϕ)δ(t) + O(2 ) ⇒ same phase reduction applied to interacting oscillators TheKDubins = 1 vehicle. The control v is set equal to 1 and the control ω takes value in [−1, 1]. ⇒ coord. & time transf. + averaging ⇒ θ̇i = Finally, the four-wheeled planar vehicle, composed of a front and a rear axle separated by a distance ", is described by the same dynamical system (3.1.2) with the following distinctions: (x, y) ∈ R2 is the position of the midpoint of the rear axle, θ ∈ S1 is the orientation of the rear axle, the control v is the forward linear velocity of the rear axle, and the angular velocity satisfies v ω = tan φ, where the control φ is the steering angle of the vehicle. • " ”phase balance” ⇒ 0th and 1st (odd) Fourier mode: θ̇i = ωi + Next, we generalize the notion of synchronous network introduced in Definition 1.38 and introduce a corresponding notion of robotic network. Sync in Complex Oscillator Networks ) The Reeds–Shepp car. The control v takes values in {−1, 0, 1} and the control ω takes values in [−1, 1]. [R. Sepulchre et al. ’07, D. Klein et al. ’09, L Consolini et al ’10] F. Dörfler and F. Bullo (UCSB) ✏ · (t) θ ✓ (a) x2 dynamical system with stable limit cycle γ and weak perturb. i✓i (x, y) θ̇i = ui (r , θ) with speed v and steering control ui (r , θ) P P j hij (θi − θj ) j aij sin(θi − θj ) [F.C. Hoppensteadt & E.M. Izhikevich ’00, Y. Kuramoto ’83, E. Brown et al. ’04, . . . ] Definition 3.2 (Robotic network). The physical components of a robotic F. Dörfler and F. Bullo (UCSB) 2012 10),/where 38 network S consist ofCDC a tuple (I, R, Ecmm (i) I = {1, . . . , n}, I is called the set of unique identifiers (UIDs); Sync in Complex Oscillator Networks CDC 2012 11 / 38 gait cycle, a ‘configuration’ of the limbs is defined by the joint angles at the hip, knee and ankle joints. As a person walks, the gait cycle is repeated, and the configurations of the limbs are revisited in a nearly periodic fashion. the eigenstructure of the(see Laplacian matrix As elaborated by linear dynamic systems (11) and (17) for a preview)in the time (full duplex, see Remark 2).! NThe basic mechanism of con2π and random [34] topologies. We will provide some comments # i (t+ )isis # following (and with some details GraphquantiTheory and tinuously wherecoupled whose system matrix is linearly relatedinto“Algebraic a key algebraic clocks locking (seej (t Figure Each to #̇ji(t ) − = εphase α ij ·difference f(# ) −with #6). )), (10) 0 the phase i (trespect on these important cases in the following, and we point to refTi j=1, phase j!= iits phase j, and α ijthrough Distributed of particular relevance is the null node,node and are detector-specific G, namely ith, f(·) ty that describes Synchronization”), the connectivity graph the Laplacian say the measures detector (PD)features, the erences for further details. space of matrix that isas sometimes referred asthe theadjasynchro- convex namely a nonlinear function and convex combination weights L [32]. L = I − A, A is matrix This isL,defined where to combination of phase differences "N subspace. Specifically, multiplicity of the (i.e., j=1 α ij = 1 and α ij ≥ 0),respectively (recall the discus0). It i != j and [A] cencynization matrix of the graph ([A] ij = αthe ij for ii =zero-eigenREMARK 3 N ! λ(L) = the 0 determines value whether a synchronized state is sion in the previous section). Notice that the choice of a convex is then clear that performance of mutual synchronization $# i (t ) = α ij · f(# j (t ) − # i (t )), (9) The (average) accuracy of different eventually achieved or not, while the left combination in (9) ensures that the output of the phase detector G) eigenvector depends on the network topology (connectivity graph through j=1, j!= i ∆Φ2(t) clocks is sometimes measured in parts- T T PD the v = [v1 · · · vN ]of corresponding $# i (t ) takes values in the range between minimum and the to λ(L)L.=As0elaborated (v L = 0)inyields the eigenstructure the Laplacian matrix the the ε (s) per-million (PPM) by calculating the f (# j (t ) −with # i (trespect )). Finally, steady-state frequency and phases of the Graph clocksTheory (see (12), maximum # iphase (t ) is differences # j (t ) − of following (and with some details in “Algebraic and (14),where the phase difference to the average (absolute value of) the clock ) ij(9)are and (19)Synchronization”), for a preview). is phase fed to adetector-specific loop filter ε(s), features, whose output j, and f(·)$# Distributed of particular relevance is the null nodedifference and i (t α ∆Φ3(t) error after one second. There exists a s2(t) whichweights a final in the discussion we synchrohave limited drives the voltage controlled (VCO), updates the L, remark, space ofAs matrix that is sometimes referred above, to as the namely a nonlinear function andoscillator convex combination PD VCO ε (s) "N clear trade-off between accuracy and the subspace. scope to time-invarying deterministic topologies, but (i.e., local phase as1 and α ij ≥ 0),respectively (recall the discusα ij = nization Specifically, the and multiplicity of the zero-eigenj=1 power consumption. For instance, accuanalysis can be extended to both time-varying [31], λ(L) = 0 determines valuethe whether a synchronized state is [33] sion in the previous section). Notice that the choice of aT1convex 2 racies of typical clocks range between N ! 2π s (t) and random [34] topologies. We will provide some comments eventually achieved or not, while the left eigenvector combination in (9) ensures that the output of the phase detector 3 VCO #̇ i (t ) = + ε0 α ij · f(# j (t ) − # i (t )), (10) aroundT 10−4 and 10−11 PPM with coron1 ·these cases in pointthe to ref- $# i (t ) takes values v = [v · · vN ]important = 0)weyields λ(L)following, = 0 (v T Land range between the minimum and the corresponding to the Ti in the j=1, j!= i 1 responding power consumptions on the erences frequency for furtherand details. f (# (t ) − # (t )). steady-state phases of the clocks (see (12), (14), maximum of phase differences Finally, the j i T3 order of 1µW and hundreds of and (19) for a preview). difference $# i (t ) (9) is fed to a loop filter ε(s), whose output megawatts, respectively [12]. 3 AsREMARK a final remark, in the discussion above, we have limited drives the voltage controlled oscillator∆Φ (VCO), which updates the 1(t) PD ε (s) The (average) accuracy different topologies, but the scope to time-invarying andofdeterministic local phase as CONTINUOUSLY ∆Φ2(t) clocks iscan sometimes measured in partsthe analysis be extended to both time-varying [31], [33] PD COUPLED ANALOG CLOCKS ε (s) N ! 2π per-million by calculating the some comments and random [34] (PPM) topologies. We will provide #̇ i (t ) = + ε0 α ij · f(# (t ) − # i (t )), (10) In this section, we study the problem of s1j(t) average (absolute the clock on these important casesvalue in theof) following, and we point to refTi VCO j=1, j!= i distributed synchronization of coupled ∆Φ3(t) error onedetails. second. There exists a erences forafter further s2(t) 1 PD analog clocks. The interest of such probVCO ε (s) clear trade-off between accuracy and T1 lem for wireless communications is relatpower3consumption. For instance, accuREMARK 1 ed to applications such as, e.g., T2 of typical clocks between The racies (average) accuracy of range different s (t) cooperative beamforming or frequency [FIG6] 3 Block VCO diagram of N = 3 continuously coupled oscillators ∆Φ2(t) (PD: phase detector; VCO: 10−4 andmeasured 10−11 PPM around with corclocks is sometimes in partsPD division multiple access in ad hoc net- voltage controlled oscillator). ε (s) 1 responding power on the per-million (PPM) by consumptions calculating the T3 order of 1µW and average (absolute value of) hundreds the clock of ∆Φ3(t)MAGAZINE [87] SEPTEMBER 2008 [12].exists a errormegawatts, after one respectively second. There IEEE SIGNAL PROCESSING s (t) PD VCO ε (s) ∆Φ12(t) clear trade-off between accuracy and PD ε (s) t CONTINUOUSLY power consumption. For instance, accu1 T2 COUPLED CLOCKS racies of typicalANALOG clocks range between s3(t) −11study VCO In this 10−4section, around and 10we PPM the withproblem cor- of s1(t) VCO distributed synchronization of coupled 1 responding power consumptions on the T3 1 analog of suchofprob1µW The order of clocks. and interest hundreds T1 lem for respectively wireless communications is relatmegawatts, [12]. t b t PD ∆Φ1(t) ε (s) ed to applications such as, e.g., cooperative beamforming or frequency [FIG6] Block diagram of N = 3 continuously coupled oscillators (PD: phase detector; VCO: CONTINUOUSLY division multipleCLOCKS access in ad hoc net- voltage controlled oscillator). COUPLED ANALOG t B In this section, we study the problem of s1(t) VCO distributed synchronization of coupled IEEE SIGNAL PROCESSING MAGAZINE [87] SEPTEMBER 2008 1 analog clocks. The interest of such probT1 lem for wireless communications is related to applications such as, e.g., cooperative beamforming or frequency [FIG6] Block diagram of N = 3 continuously coupled oscillators (PD: phase detector; VCO: division multiple access in ad hoc net- voltage controlled oscillator). Example 5: Other technological applications Particle filtering to estimate limit cycles [A. Tilton & P. Mehta et al. ’12] Fig. 1: Single gait cycle. Clock synchronization over networks [Y. Hong & A. Scaglione ’05, O. Simeone Mathematically, an idealized gait cycle is represented as a limit cycle (periodic orbit) in the phase space comprising et al. ’08, Y. Wang & F. Doyle et al. ’12] of joint angles and their velocities. An idealized gait cycle may be obtained by numerically solving the Euler-Lagrange Central pattern generators and equations of motion; cf., [3]. robotic locomotion [J. Nakanishi et al. In this paper, we take a phenomenological approach: We use q to denote the state of the gait cycle at time t. The state ’04, S. Aoi et al. ’05, L. Righetti et al. ’06] evolves on the circle [0, 2p], modeled here as a stochastic differential equation (sde): Decentralized maximum likelihood q̇ = w + s Ḃ , mod 2p, estimation [S. Barbarossa et al. ’07] where {Ḃ } is a white noise process, and w, s (1) are parameters that represent natural frequency of the gait cycle and standard deviation of the process noise, respectively. Carrier sync without phase-locked theThe phenomenological model (1) is useful because we loops [M. Rahman et al. ’11] are interested, after all, only in estimating the state qt of the gait cycle. The model can be rigorously justified by employing a normal form reduction starting from the limit F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 12 / 38 cycle in the phase space; cf., [1]. For an idealized gait cycle, the normal form reduction procedure establishes one-one correspondence between the values of the phase angle qt and the configuration of the joint angles, and their velocities (for homogenous coupling aij = K /n) in the phase space. A small process noise is included in the phenomenological P the small model (1) iψ Define the order parameter (centroid) by re to account = n1 fornj=1 e iθi cycle-to-cycle , then variability. B. Observation Model IEEE SIGNAL PROCESSING MAGAZINE [87] SEPTEMBER 2008 Order Parameter θ̇i = ωi − K Xn sin(θi −θj ) j=1 n Intuition: synchronization = K small & |ωi − ωj | large Figure 2 depicts the Ankle Foot Orthosis (AFO) device a pneumatic θ̇i = ωi −The Krdevice sin(θincludes ⇔used in the experiments. i − ψ) power source, a rotary actuator at the ankle, and is equipped with three sensors. The two force sensors on the bottom of the foot are referred to as the iψ heel sensor and the toe sensor. entrainmentUnder by normal meanwalking field conditions, re the toe and heel sensors are approximately binary valued. For example, the heel X X re K large & |ωi − ωj | small i r ei density density ⇒ analysis based on concepts from statistical mechanics & cont. limit: [Y. Kuramoto ’75, G.B. Ermentrout ’85, J.D. Crawford ’94, S.H. Strogatz ’00, J.A. Acebrón et al. ’05, E.A. Martens et al. ’09, H. Yin et al. ’12, . . . ] F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 13 / 38 up to 12Nm at 120psi; D) Sensors: two force sensors under the heel and toe, and potentiometer at the ankle joint [8]. Outline sensor outputs one value when the foot is in contact with the ground, and another value when it is not. In this paper, we model the sensor response of each of the sensors with an indicator function: 1 ⇢ + hm q 2 (fm1 , fm2 ) hm (q ) = hm otherwise, 2 where f 1 , f 2 , h+ , and h for m = 1, 2 are determined from Introduction and motivation Synchronization notions, metrics, & basic insights m m m m the experimental data (see Sec. III). Using the corresponding sensor’s response function hm3(qt ), the observation model for each sensor is, Phase synchronization and more basic insights Ytm = hm (qt ) + smẆtm , Synchronization in complete networks 4 {Ẇ m } is an independent white noise processes, and where t sm is the standard deviation for m = 1, 2. The noise term has been included to account for the presence of sensor noise. 5 C. Filtering Problem Synchronization in sparse networks The objective of the filtering problem is to estimate the posterior distribution of qt given the history of observations 6 Yt := s (Ys1 ,Ys2 : s t). We denote the posterior distribution by p⇤ (q ,t), so that for any measurable set A ⇢ [0, 2p], Open problems and research directions Z A p⇤ (q ,t) dq = P[qt 2 A|Yt ]. The posterior p⇤ (q ,t) represents the ‘belief F. Dörfler and F. distribution Bullo (UCSB) Sync in Complex Oscillator Networks state’ of the process qt given the history of observations. Using the posterior, the estimate (conditional mean) is obtained as, SynchronizationZ 2pNotions & Metrics q̂t := E[qt |Yt ] = 0 q p⇤ (q ,t) dq . In the remainder ofsync: this paper, we describe a coupled 1) frequency θ̇i (t) = θ̇j (t) ∀ i, j oscillator particle filter to approximate the filtering task. ⇔recall θ̇i (t) ωsyncfilter ∀ icomprises ∈ {1, .of. .N,stochastic n} Here, we that= a particle processes {qti : 1 i N}, where the value qti 2 [0, 2p] is the state for the ith particle at time t. For any measurable set 2) phase sync: θi (t) = formed θj (t) by∀ the i, jparticle A ⇢ [0, 2p], the empirical distribution population defined ⇔ ris = 1 by, CDC 2012 2) X ✓i = \ei 3) X 1 N Â 1{qti 2 A}. N i=1 r = 0 balancing: r ei = 0 p(N) (A,t) := 3) phase (e.g., splay state = uniform spacing on S1 ) 12 / 38 4) 4) arc invariance: all angles in Arcn (γ) (closed arc of length γ) for γ ∈ [0, 2π] 5) phase cohesiveness: all angles in ¯ G (γ) = θ ∈ Tn : max{i,j}∈E |θi − θj | ≤ γ ∆ for some γ ∈ [0, π/2[ F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks 5) CDC 2012 14 / 38 Geometric & Algebraic Insights I Geometric & Algebraic Insights II symmetries Jacobian is Laplacian Coupled oscillator model: Xn θ̇i = f (θ) = ωi − aij sin(θi −θj ) ✓1 = ✓2 Coupled oscillator model: Xn θ̇i = f (θ) = ωi − aij sin(θi −θj ) 12 ✓⇤ j=1 |✓1 θ̇i (t) = i=1 n P ! ωi = i=1 n P ✓2 | ⇡/2 i=1 ωsync ⇒ sync frequ. ωsync = ωavg = n P 1 n k=2 L(θ∗ ) = ωi i=1 wlog: assume ωavg = 0 ⇒ frequency sync = equilibrium manifold [θ∗ ] = θ ∈ Tn : θ∗ + ϕ1n , f (θ∗ ) = 0 , ϕ ∈ [0, 2π] Sync in Complex Oscillator Networks ✓⇤ CDC 2012 15 / 38 Geometric & Algebraic Insights II |✓1 ✓2 | ⇡/2 negative Jacobian −∂f /∂θ evaluated at θ∗ ∈ Tn is given by Pn ∗ ∗ ∗ ∗ ⇒ transf. to rot. frame with freq. ωavg ⇔ ωsync = 0 ⇔ ωi 7→ ωi − ωavg F. Dörfler and F. Bullo (UCSB) 12 j=1 vector field f (θ) possesses rotational symmetry: f (θ∗ ) = f (θ∗ + ϕ1n ) n P ✓1 = ✓2 a1k cos(θ1 − θk ) . . . −an1 cos(θn∗ − θ1∗ ) −a12 cos(θ1 − θ2 ) . . ... . . . . ∗ −an,n−1 cos(θn∗ − θn−1 ) ... −a1n cos(θ1∗ − θn∗ ) n−1 P k=1 . . . ain cos(θn∗ − θk∗ ) = Laplacian matrix of graph (V, E, Ã) with weights ãij = aij cos(θi∗ −θj∗ ) ⇒ all weights ãij > 0 for {i, j} ∈ E ⇔ max{i,j}∈E |θi∗ − θj∗ | < π/2 ⇒ algebraic graph theory: L(θ∗ ) is p.s.d. and ker(L(θ∗ )) = span(1n ) F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 16 / 38 Outline Jacobian is Laplacian Coupled oscillator model: Xn θ̇i = f (θ) = ωi − aij sin(θi −θj ) ✓1 = ✓2 12 ✓⇤ j=1 |✓1 ✓2 | ⇡/2 Lemma [C. Tavora and O.J.M. Smith ’72] If there exists an equilibrium manifold [θ∗ ] in ∆G (π/2) = θ ∈ Tn : max{i,j}∈E |θi − θj | < π/2 , then [θ∗ ] is 1 locally exponentially stable (modulo symmetry), and 2 ¯ G (π/2) (modulo symmetry). unique in ∆ F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 17 / 38 1 Introduction and motivation 2 Synchronization notions, metrics, & basic insights 3 Phase synchronization and more basic insights 4 Synchronization in complete networks 5 Synchronization in sparse networks 6 Open problems and research directions F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 17 / 38 Phase Synchronization Phase Synchronization a forced gradient system main result θ̇i = ωi − Xn j=1 aij sin(θi − θj ) {phase sync} = {θ ∈ Tn : θi = θj ∀ i, j} θ̇i = ωi − Xn j=1 aij sin(θi − θj ) {phase sync} = {θ ∈ Tn : θi = θj ∀ i, j} Theorem: [P. Monzon et al. ’06, Sepulchre et al. ’07] Classic intuition [P. Monzon et al. ’06, Sepulchre et al. ’07]: Coupled oscillator model is forced gradient flow θ̇i = ωi − ∇i U(θ) , P where U(θ) = {i,j}∈E aij 1 − cos(θi − θj ) (spring potential) assume that ωi = 0 ∀ i ∈ {1, . . . , n} ⇒ gradient flow θ̇ = −∇U(θ) The following statements are equivalent: 1 For all {i, j} ∈ {1, . . . , n}, we have that ωi = ωj ; and 2 There exists a locally exp. stable phase synchronization manifold. Proof of “⇒”: wlog in rot. frame: ωi = ωj = 0 ⇒ follow previous args ⇒ global convergence to critical points {∇U(θ) = 0} ⊇ {phase sync} Proof of “⇐”: phase sync’d solutions satisfy θi = θj & θ̇i = θ̇j ⇒ ωi = ωj ⇒ previous Jacobian arguments: {phase sync} is local minimum & stable Remark: “almost global phase sync” for certain topologies (trees, cmplt., short cycles) [P. Monzon, E.A. Canale et al. ’06-’10, A. Sarlette ’09] F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 18 / 38 Phase Synchronization F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 19 / 38 Outline further insights when all ωi = 0 θ̇i = ωi − Xn j=1 aij sin(θi − θj ) {phase sync} = {θ ∈ Tn : θi = θj ∀ i, j} Convexity simplifies life: 1 Introduction and motivation 2 Synchronization notions, metrics, & basic insights 3 Phase synchronization and more basic insights 4 Synchronization in complete networks 5 Synchronization in sparse networks 6 Open problems and research directions if all oscillators in open semicircle Arcn (π) ⇒ convex hull maxi,j∈{1,...,n} |θi (t) − θj (t)| is contracting max |✓i (t) i,j ✓j (t)| [L. Moreau ’04, Z. Lin et al. ’08] Phase balancing: inverse gradient flow (ascent) θ̇ = +∇U(θ) ⇒ phase balancing for circulant graphs [L. Scardovi et al. ’07, Sepulchre et al. ’07] F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks X r ei = 0 CDC 2012 20 / 38 F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 20 / 38 in aof Complete – Synchronization Kron reduction graphs& Homogeneous Graph brief review 3 7 2 1 3 9 3 4 5 1 4 1 0 2 0 3 3 3 1 5 2 3 1 6 1 3 1 1 2 1 9 3 3 4 8 2 9 3 4 1 7 2 1 2 4 1 2 6 1 4 1 5 1 0 2 3 3 4 Necessary conditions: θ̇i = θ̇j !sync 3 2 0 3 4 ∀i, j ⇒ 5 2 3 1 6 1 3 1 1 1 9 3 2 2 5 7 5 2 8 3 6 1 3 9 4 9 2 3 1 6 3 6 1 3 9 2 3 2 0 8 2 6 2 7 3 1 8 2 2 1 0 3 2 5 0 2 3 1 3 1 1 1 3 2 1 1 5 0 1 7 1 4 1 3 1 2 6 9 2 4 [D. AeyelsY etreduced al. ’04, R.E.= Mirollo et al. ’05, M. Verwoerd et al. ’08] Ynetwork /Y interior Some properties of theY Kron reduction process: network 5 2 2 2 4 1 2 6 4 3 7 2 1 7 1 7 5 7 1 3 2 2 1 7 3 5 8 4 5 7 3 2 9 2 8 1 8 8 8 2 6 2 7 3 6 4 9 3 4 9 2 5 2 8 3 3 5 One appropriate sync notion: 3 3 2 9 2 8 2 3 1 6 1 9 6 1 8 8 2 6 2 7 3 1 5 reduced = Ynetwork Ynetwork /Y interior Implicit equations for existence of sync’d fixed points 1 9 θ ∈ Arcn (γ) 0 3 5 2 5 0 2 0 3 7 0 3 2 2 3 3 θ̇i = ωavg ∀ i 8 1 3 1 3 1 0 1 4 1 3 1 1 2 9 2 1 2 4 1 0 3 7 2 2 1 7 1 2 6 8 3 5 4 2 8 5 7 2 1 9 3 3 4 K> ωmax −ωmin 2 · n Well-posedness: Symmetric & irreducible (loopy) Laplaciann−1 matrices [N. Chopra et al. ’09, A. Jadbabaie et al. ’04, J.L. van Hemmen et al. ’93] Some properties of the Kron reduction process: can be reduced and are closed under Kron reduction 1 frequency sync: θ̇i = ωavg Pn with ωavg = n1 i=1 ωi 2 1 8 8 1 arc invariance: θ ∈ Arcn (γ) for small γ ∈ [0, π/2[ 3 9 1 2 7 3 2 9 6 1 8 2 6 6 K Xn sin(θi − θj ) θ̇i = ωi − j=1 n 9 2 5 0 3 6 3 3 6 8 0 frequency sync: arc invariance: 7 Detour – Kron reduction of graphs 1 1 K Xn θ̇i = ωi − sin(θi − θj ) j=1 n Y 7 Classic Kuramoto model of coupled oscillators: 1 recall definitions 3 9 Synchronization in a Complete & Homogeneous Graph Detour Numerous results on sync conditions & bifurcations 1 2 Topological Well-posedness: Symmetric & irreducible (loopy) Laplacian matrices properties: Sufficient conditions, e.g., K > k(. . . , ωi −ωj , . . . )k2 , ∞ ·f (n, γ) can be reduced and are closed Kron reduction interior networkunder connected reduced network complete 2 at leastChopra one node in interior network features a self-loop Topological properties: et al. ’09, G. Schmidt et al. ’09, F. Dörfler and F. Bullo ’09, S.J. Chung [J.L. van Hemmen et al. ’93, A. Jadbabaie et al. ’04, F. de Smet et al. ’07, N. all nodes reduced feature and J.J.in Slotine ’10, A.network Franci et al. ’10, S.Y.self-loops Ha et al. ’10, . . . ] interior network connected reduced network complete CDC 2012 3 21 / 38 and F. Bullo (UCSB) Sync in Complex Oscillator Networks Algebraic properties: self-loops in interior network . . . at least one node F.inDörfler interior network features a self-loop all nodes indecrease reduced mutual networkcoupling feature in self-loops reduced network [A. Jadbabaie et al. ’04, P. Monzon et al. ’06, Sepulchre et al. ’07, F. de Smet et al. ’07, N. Chopra et al. ’09, A. Franci et al. ’10, S.Y. Ha et al. ’10, D. Aeyels et al. ’04, , J.L. van Hemmen et al. ’93, R.E. Mirollo et al. ’05, M. Verwoerd et al. ’08, . . . ] F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks Synchronization in a Complete & Homogeneous Graph main result 3 K Xn θ̇i = ωi − sin(θi − θj ) j=1 n Theorem [F. Dörfler & F. Bullo ’11] frequency sync: arc invariance: 2 3 Synchronization in a Complete & Homogeneous Graph θ̇i =decrease ωavg ∀ Florian i mutual in reduced network 1 Arc invariance: Dörflercoupling (UCSB) Power Networks Synchronization Candidacy 31 / 3 θ(t) in γ arc ⇔ arc-length Advancement V (θ(t)) is tonon-increasing θ ∈increase Arcn (γ) self-loops in reduced network Power Networks Synchronization ⇔ Florian Dörfler (UCSB) V (✓(t)) Coupling dominates heterogeneity, i.e., K > Kcritical , ωmax − ωmin ; ∃ γmax ∈ ]π/2, π] s.t. all Kuramoto models with ωi ∈ [ωmin , ωmax ] and θ(0) ∈ Arcn (γmax ) achieve exponential frequency sync; and 2 K n Moreover, we have Kcritical /K = sin(γmin ) = sin(γmax ) Sync in Complex Oscillator Networks CDC 2012 23 / 38 Advancement to Candidacy ! D + V (θ(t)) ≤ 0 31 / 36 Frequency synchronization ⇔ consensus protocol in Rn where aij (t) = and practical phase synchronization: from γmax arc → γmin arc V (θ(t)) = maxi,j∈{1,...,n} |θi (t) − θj (t)| true if K sin(γ) ≥ Kcritical ∃ γmin ∈ [0, π/2[ s.t. all Kuramoto models with ωi ∈ [ωmin , ωmax ] feature a locally exp. stable equilibrium manifold in Arcn (γmin ). F. Dörfler and F. Bullo (UCSB) 22 / 38 mainself-loops proof ideas inininterior increase self-loops reducednetwork network . . . Algebraic properties: The following statements are equivalent: 1 CDC 2012 3 Xn d θ̇i = − aij (t)(θ̇i − θ̇j ) , j=1 dt cos(θi (t) − θj (t)) > 0 for all t ≥ T Necessity: all results exact for bipolar distribution ωi ∈ {ωmin , ωmax } F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 24 / 38 Synchronization in a Complete & Homogeneous Graph Synchronization in a Complete & Homogeneous Graph robustness and extensions scaling & statistical analysis Switching natural frequencies: dwell-time assumption 10 3 2 1 0 0 5 10 5 0 −5 −10 0 5 10 γmax 2.5 1.5 γmin 1 0.5 0 5 10 t t[s] 6 4 θ̇(t)θ̇(t) [rad/s] θ(t)θ(t) [rad] Slowly time-varying: kω̈(t) − ω̈avg (t)k∞ sufficiently small 3 2 1.5 1 0.5 0 0 5 10 0 −2 −4 0 F. Dörfler and F. Bullo (UCSB) X 3 2 t t[s] 5 K Xn sin(θi − θj ) j=1 n Cont. limit predicts largest Kcritical = 2 for bipolar distribution & smallest Kcritical = 4/π for uniform distribution [Y. Kuramoto ’75, G.B. Ermentrout ’85] 2 t t[s] 2.5 θ̇i = ωi − Kuramoto model with ωi ∈ [−1, 1]: 3 t t [s] 2 X Kcritical θ̇(t) θ̇(t) [rad/s] θ(t)θ(t) [rad] 4 (θ (t) [rad] [rad] VV(θ(t)) 5 (θ (t) [rad] [rad] VV(θ(t)) 1 10 γmax 2.5 4/π 2 1.5 γmin 1 0.5 t t[s] Sync in Complex Oscillator Networks 0 5 n 10 t t[s] CDC 2012 25 / 38 Outline necessary bound (◦), sufficient & tight bound (), & exact & implicit bound (♦) F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 26 / 38 Primer on Algebraic Graph Theory Undirected graph G = (V, E, A) with weight aij > 0 on edge {i, j} 1 Introduction and motivation 2 Synchronization notions, metrics, & basic insights 3 Phase synchronization and more basic insights 4 Synchronization in complete networks adjacency matrix A = [aij ] ∈ Rn×n (induces the graph) P degree matrix D ∈ Rn×n is diagonal with dii = nj=1 aij Laplacian matrix L = D − A ∈ Rn×n , L = LT ≥ 0 Notions of connectivity topological: connectivity, path lengths, degree, etc. 5 Synchronization in sparse networks 6 Open problems and research directions spectral: 2nd smallest eigenvalue of L is “algebraic connectivity” λ2 (L) Notions of heterogeneity kωkE,∞ = max{i,j}∈E |ωi − ωj |, F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 26 / 38 F. Dörfler and F. Bullo (UCSB) kωkE,2 = Sync in Complex Oscillator Networks P {i,j}∈E |ωi − ωj |2 CDC 2012 1/2 27 / 38 Synchronization in Sparse Networks Synchronization in Sparse Networks a brief review I a brief review II θ̇i = ωi − 1 Xn j=1 Assume connectivity & P ωavg = n1 ni=1 ωi = 0 aij sin(θi − θj ) necessary condition: Pn j=1 aij ≥ |ωi | ⇐ θ̇i = ωi − sync 3 Assume connectivity & P ωavg = n1 ni=1 ωi = 0 aij sin(θi − θj ) sufficient condition II: λ2 (L) > λcritical , kωkE,2 ⇒ sync Proof idea inspired by [A. Jadbabaie et al. ’04]: fixed point theorem with Proof idea: θ̇i = 0 has no solution if condition is not true sufficient condition I: j=1 [F. Dörfler and F. Bullo ’11] [C. Tavora and O.J.M. Smith ’72] 2 Xn λ2 (L) > λcritical , kωkEcmplt ,2 incremental 2-norms; condition implies kθ∗ kE,2 ≤ λcritical /λ2 (L) ⇒ sync ⇒ ∃ similar conditions with diff. metrics on coupling & heterogeneity [F. Dörfler and F. Bullo ’09] Proof idea: analogous Lyapunov proof with V (θ) = P i<j |θi − θj |2 ; condition also implies θ∗ ∈ Arcn (λcritical /λ2 (L)) ⇒ evtl. too strong! F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 28 / 38 F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks Synchronization in Sparse Networks A Nearly Exact Synchronization Condition problems . . . a “back of the envelope calculation” Problems: the sharpest general nec. & suff. conditions known to date Pn j=1 aij < |ωi | , λ2 (L) > kωkEcmplt ,2 , and λ2 (L) > kωkE,2 have a large gap and are conservative ! 2 3 j=1 (??) j=1 Unique solution (modulo symmetry) of (??) is δ ∗ = L† ω conservative bounding of trigs & network interactions conditions θ∗ ∈ Arcn λλcritical or kθ∗ kE,2 ≤ λλcritical are too strong (L) 2 2 (L) ⇒ Solution ansatz for (?): analysis with 2-norm is conservative ωi = Open problem: quantify “coupling/connectivity” vs. “heterogeneity” [S. Strogatz ’00 & ’01, J. Acebrón et al. ’00, A. Arenas et al. ’08, S. Boccaletti et al. ’06] F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 29 / 38 ¯ G (γ), then it is unique and stable Recall: if ∃ equilibrium [θ∗ ] ∈ ∆ Xn ωi = aij sin(θi − θj ) (?) Consider linear “small-angle” approximation of (?) : Xn ωi = aij (δi − δj ) ⇔ ω = Lδ Why? 1 CDC 2012 30 / 38 Xn j=1 θi∗ − θj∗ = arcsin(δi∗ − δj∗ ) aij sin(θi − θj ) = ⇒ Theorem: (for a tree) F. Dörfler and F. Bullo (UCSB) Xn j=1 (for a tree) aij sin arcsin(δi∗ − δj∗ ) = ωi X ¯ G (γ) ⇔ L† ω ∃ [θ∗ ] ∈ ∆ ≤ sin(γ) E,∞ Sync in Complex Oscillator Networks CDC 2012 31 / 38 A Nearly Exact Synchronization Condition A Nearly Exact Synchronization Condition comments Theorem [F. Dörfler, M. Chertkov, and F. Bullo ’12] Statistical correctness through Monte Carlo simulations: construct nominal randomized graph topologies, weights, & natural frequencies Under one of following assumptions: 1) graph is either tree, homogeneous, or short cycle (n ∈ {3, 4}) 2) natural frequencies: L† ω is bipolar, small, or symmetric (for cycles) Intuition: the condition † L ω ≤ sin(γ) E,∞ where γ < π/2 0 0 eigenvectors of L .. . 0 ⇒ ∃ a unique & locally exponentially stable equilibrium manifold in ¯ G (γ) = θ ∈ Tn | max{i,j}∈E |θi − θj | ≤ γ . ∆ F. Dörfler and F. Bullo (UCSB) with high accuracy Possibly thin sets of degenerate counter-examples for large cycles 3) arbitrary one-connected combinations of 1) and 2) If ⇒ sync “for almost all graphs G (V, E, A) & ω ” Sync in Complex Oscillator Networks ... 0 .. . ... 0 1 λ2 (L) .. . ... † L ω ≤ sin(γ) E,∞ ... ... .. . 0 is equivalent to 0 0 T eigenvectors of L ω 0 1 λn (L) E,∞ ⇒ includes previous conditions on λ2 (L) and degree (≈ λn (L)) CDC 2012 32 / 38 F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 A Nearly Exact Synchronization Condition A Nearly Exact Synchronization Condition statistical analysis for power networks statistical analysis for complex networks Randomized power network test cases θ̇i = ωi − K · Comparison with exact Kcritical for with 50 % randomized loads and 33 % randomized generation F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 E,1 0.9 0.9 K/ L† ! K K n = 10 n = 20 0.25 0.5 pp 0.75 0.75 0.25 1 0.5 pp 0.75 1 (d) Student Version of MATLAB Student Version of MATLAB 0.9 K/ L† ! K K 0.5 pp p 0.7 1 (e) 0.9 0.25 0.4 1 n = 40 1 0.75 1 0.75 0.25 0.5 pp 0.75 1 (f ) Student Version of MATLAB F. Dörfler and F. Bullo (UCSB) n = 80 0.9 n = 160 0.75 0.1 0.4 p 0.7 1 ⇒ condition L† ω E,∞ ≤ sin(γ) is extremely accurate for γ = π/2 Student Version of MATLAB 34 / 38 0.75 0.1 1 E,1 0.75 0.75 ⇒ condition L† ω E,∞ ≤ sin(γ) is extremely accurate for γ ≤ 25◦ (c) E,1 0.9 K/ L† ! 0.12889 rad 0.16622 rad 0.22309 rad 0.1643 rad 0.16821 rad 0.20295 rad 0.24593 rad 0.23524 rad 0.43204 rad 0.25144 rad E,1 rad rad rad rad rad rad rad rad rad rad − 1 K/ L† ! 4.1218 · 10−5 2.7995 · 10−4 1.7089 · 10−3 2.6140 · 10−4 6.6355 · 10−5 2.0630 · 10−2 2.6076 · 10−3 5.9959 · 10−4 5.2618 · 10−4 4.2183 · 10−3 max {i,j}∈E sin(θi − θj ) j=1 aij (b) (a) θj∗ | E,1 − arcsin(kL ωkE,∞ ) |θi∗ K/ L† ! true true true true true true true true true true ≤γ † E,1 always always always always always always always always always always − {i,j}∈E Pn 33 / 38 Small World Network 1 1 K/ L ! 9 bus system IEEE 14 bus system IEEE RTS 24 IEEE 30 bus system New England 39 IEEE 57 bus system IEEE RTS 96 IEEE 118 bus system IEEE 300 bus system Polish 2383 bus system (winter peak 1999/2000) max {i,j}∈E θj∗ | Random Geometric Graph Erdös-Rényi Graph Phase cohesiveness: ! bipolar ⇒ |θi∗ Accuracy of condition: max |θi∗ − θj∗ | † Correctness of condition: kL† ωkE,∞ ≤ sin(γ) ! uniform Randomized test case (1000 instances) ≤ sin(γ) Student Version of MATLAB Sync in Complex Oscillator Networks Student Version of MATLAB CDC 2012 35 / 38 Outline Exciting Open Problems and Research Directions 1 Introduction and motivation 1 Q: What about networks of second-order oscillators ? Xn Mi θ̈i + Di θ̇i = ωi − aij sin(θi − θj ) j=1 2 Synchronization notions, metrics, & basic insights 3 Phase synchronization and more basic insights 4 Synchronization in complete networks Apps: mechanics, synchronous generators, Josephson junctions, . . . Problems: kinetic energy is a mixed blessing for transient dynamics 2 e.g., directed graphs: aij 6= aji or phase shifts: aij sin(θi − θj − ϕij ) Synchronization in sparse networks 5 Apps: sync protocols, lossy circuits, phase/time-delays, flocking, . . . Problems: algebraic & geometric symmetries are broken Open problems and research directions 6 3 F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 35 / 38 Exciting Open Problems and Research Directions 4 Q: What about asymmetric interactions ? Q: How to derive sharper results for heterogeneous networks ? F. Dörfler and F. Bullo (UCSB) CDC 2012 36 / 38 Conclusions Q: What about the transient dynamics beyond Arcn (π), general equilibria beyond ∆G (π/2), or the basin of attraction ? Coupled oscillator model: Apps: phase balancing, volatile power networks, flocking, . . . 5 Sync in Complex Oscillator Networks Xn Problems: lack of analysis tools (only for simple cases), chaos, . . . θ̇i = ωi − Q: Beyond continuous, sinusoidal, and diffusive coupling ? X θ̇i ∈ ωi − fij (θi , θj ) , θ ∈ C ⊂ Tn {i,j}∈E X θi+ ∈ θi + gij (θi , θj ) , θ ∈ D ⊂ Tn history: from Huygens’ clocks to power grids j=1 aij sin(θi − θj ) applications in sciences, biology, & technology !2 synchronization phenomenology {i,j}∈E !3 a23 a12 Apps: impulsive coupling, relaxation oscillators, neuroscience, . . . a13 network aspects & heterogeneity Problems: lack of analysis tools, coping with heterogeneity, . . . available analysis tools & results 6 !1 Q: Does anything extend from phase to state space oscillators ? F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 37 / 38 F. Dörfler and F. Bullo (UCSB) Sync in Complex Oscillator Networks CDC 2012 38 / 38