Synchronization of Kuramoto Oscillators with Non-identical Natural Frequencies: a Quantum Dynamical Decoupling Approach

Transcription

1 53r IEEE Conference on Decision an Control December 5-7,. Los Angeles, California, USA Synchronization of Kuramoto Oscillators with Non-ientical Natural Frequencies: a Quantum Dynamical Decoupling Approach Zhifei Zhang, Alain Sarlette an Zhihao Ling Abstract This paper proposes a metho to counter the rift associate to unknown non-ientical natural frequencies in the Kuramoto moel of couple oscillators. Inspire by the quantum ynamical ecoupling technique, it buils on a time-varying variant of the ynamics to effectively bring the oscillator phases closer to the same value. This allows effective synchronization espite arbitrarily large ifferences in natural frequencies. For two agents amitting instantaneous position exchanges, we exactly compute how the relative phase converges to a stable perioic fixe point. The latter tens to zero when the ynamics switches at a faster rate. With continuous state evolutions, using a relate ynamic controller instea of instantaneous jumps, we show with a Lyapunov function that exact phase synchronization is obtaine. We generalize the metho to multiple oscillators with instantaneous state exchanges, that can be implemente by cycling through a preefine or ranom sequence of exchanges. Simulation results illustrate the effectiveness of the algorithms. I. INTRODUCTION The Kuramoto moel of couple oscillators has attracte much interest in chemistry [], biology [], as well as physics an engineering [3], []. In a typical control application, it escribes the interactions governing the planar heaing of networke robotic systems, with the objective to achieve coorinate rotation through limite feeback between neighboring robotic agents; other applications coul involve e.g. electronic oscillators. When each agent for some reason features a heterogeneous rift ( natural frequency ) in aition to the feeback term, the coupling gain must be larger than some critical value [], [5] to prevent the oscillators from eviating away. Furthermore, even if the couple oscillators o converge to a common frequency, the resiual constant phase ifferences between oscillators of ifferent natural frequencies can significantly iffer from zero. In a control context, this spurs the introuction of some kin of integral control or other, in orer to allow oscillators with arbitrary heterogeneous rifts to synchronize towars small phase ifferences, even uner weak coupling strength. The Zhifei Zhang an Zhihao Ling are with the Key Laboratory of Avance Control an Optimization for Chemical Processes, Ministry of Eucation, East China University of Science an Technology, No.3, Meilong Roa, Shanghai 37, China. Zhifei Zhang an Alain Sarlette are with the SYSTeMS research group, Faculty of Engineering an Architecture, Ghent University, Technologiepark Zwijnaare 9, 95 Zwijnaare(Ghent), Belgium. zhifei.zhang@ugent.be, zhhling@ecust.eu.cn, alain.sarlette@ugent.be. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, an Optimization), fune by the Interuniversity Attraction Poles Programme, initiate by the Belgian State, Science Policy Office. The first author s visit to the SYSTeMS research group is supporte by a CSC scholarship, initiate by the China Scholarship Council. present paper shows how the quantum ynamical ecoupling approach [], esigne to suppress time-inepenent but unknown rift in quantum system operators, can be applie to this aim in the classical Kuramoto moel. The key point of quantum ynamical ecoupling is to iterate unitary actions that o not commute with the timeinepenent rift Hamiltonian, in orer to effectively average the rift out in the overall evolution (at least to first orer in the time between iterations). For example, rift on a single qubit which is strictly analogous to rotation aroun a constant axes on a sphere, calle the Bloch sphere can be remove by applying rotations orthogonal to the rift irection; if the rift irection is unknown, then two ecoupling irections must be concatenate. There are two possible ways to implement these rotations. Bang-Bang ecoupling [] inserts instantaneous pulses of π rotation at iscrete times, which simplifies the computation of the resulting evolution. As a more realistic moel of experimental settings, Eulerian ecoupling [7] as boune Hamiltonian controls to the existing ynamics, esigne such that in absence of rift the system woul evolve smoothly through various states corresponing to Cayley graph vertices of a so-calle ecoupling group of transformations. When large ecoupling groups are neee, a ranom switching sequence can be more effective than eterministic cycling [8]. This is interesting for our istribute network application, where we can thus avantageously ispose of a supervisor. In the present paper, we show that quantum ynamical ecoupling can be aapte to synchronize multiple heterogeneous oscillators into a state with arbitrarily small phase ifferences. We esign both Bang-Bang an Eulerian ecoupling strategies to synchronize two oscillators. For the multi-agent case, we consier both eterministic an ranom Bang-Bang ecoupling an show their capacity to effectively average out rift, both in simulations an analytically uner an averaging approximation. In future work, we will combine the present results towars Eulerian ecoupling for multiple agents. The open-loop control nature of quantum ynamical ecoupling coul be a practical avantage, in settings where the active controller is meant to correct rifts in a network with existing Kuramoto-like coupling. Inee, also in our non-quantum multi-agent aaptation, the jump actions require by Bang-Bang ecoupling might be implementable without knowing the system states, by physically exchanging e.g. subsystem connections to external influences. Our aaptation of Eulerian ecoupling however introuces a new ynamical variable in the controller, which must be upate on basis of state measurements an remins //$3. IEEE 585

2 a kin of integral controller. The structure of the paper is as follows. In Section II, we recall the notions of Kuramoto moel an quantum ynamical ecoupling metho, an show how they can be linke. In section III, we present synchronization of two oscillators with Bang-Bang ecoupling an Eulerian ecoupling. In section IV, Bang-Bang ecoupling algorithms for multi-agent synchronization are analyze an simulate, both with eterministic an ranom ecoupling paths. A. Kuramoto Moel II. GENERAL PRINCIPLE The Kuramoto moel [] escribes a set of N oscillators whose phases attract each other accoring to: θ k = ω k + K N N sin(θ j θ k ), k =,,..., N, () j= where K is the coupling strength an ω k enotes the natural frequency of oscillator k. Variants have been stuie where each oscillator k is not couple to all others, but only to a subset N k specifie by some interconnection graph. A solution θ (t) of () is sai to be phase-locke or frequency synchronize if an only if θ j θ k =, j, k,,..., N. () Oscillators are further sai to be exactly synchronize or phase synchronize if they are phase-locke with equal value, i.e. θj = θk for all j, k. The critical conition inicates that oscillators which evolve accoring to () can achieve frequency synchronization only if K K critical = max j,k ω j ω k /, or equivalently κ = K j, k. (3) ω j ω k If κ <, then () allows no relative equilibrium solution ( t (θ k θ j ) = for all j, k) an the fast oscillator(s) will overtake the slow oscillator(s) repeately. However, even when κ > an the oscillators get phase-locke, this might happen with very ifferent phase values. This is the stanar eviation from equilibrium necessary to compensate rifts in systems with proportional feeback. In applications where the actual goal is phase synchronization, this eviation can be etrimental an eicate control techniques shoul be ae to compensate the rift. B. Dynamical Decoupling A close quantum system evolves as t x = i(h + H c )x, () where x C n is the wave function, an we efine the Hermitian matrices H = H, H c = Hc C n n as the rift an control Hamiltonians respectively. Here enotes complex conjugate transpose. Quantum Dynamical Decoupling (DD) is a metho to esign H c in open loop in orer to minimize the effect of an unknown H on any state x []. It esigns H c such that, (i) for H = the system woul unergo a perioic motion of ecoupling perio T c, an (ii) the possible effects of a nonzero constant H on the state at T c, T c,... are average out (as much as possible) over this perioic motion. Explicitly, let U c (t) enote the propagator associate with () when H =. DD ientifies a iscrete group G = {g j } SU(n), j = {,..., G } an a corresponing representation P (g) such that P (g j ) H P (g j ) = λ I (5) G j a multiple of the ientity matrix, for all expecte isturbances H. It then ivies T c into M G subintervals for some integer M, an it implements controls H c such that U c (t) P (g j ) over M subintervals for each j, with U c (T c ) = I the ientity. This open-loop control rotates the effective impact of the rift H, such that its associate rift propagator U(t) satisfies t U(t) = i Hr (t) U(t) () with H r (t) = U c (t) H U c (t). The effective Hamiltonian H eff over T c is efine as: U(T c ) = e ihefftc = e i[ H + H +...]T c, (7) where the H, H,... are Magnus expansion terms of orer,,... In particular, H = Tc H T (u) r u. (8) c The general goal of DD is to get H eff as close as possible to a multiple of the ientity matrix. Inee then the effect of H at every t = T c, T c,... is suppresse since U(t) = exp iλt with λ R oes not influence outputs in quantum ynamics [9]. Achieving this ieal H eff at the first orer of the Magnus expansion (8) is irectly linke to (5). Bang-Bang Decoupling (BBD) lets strong pulses in H c implement iscrete jumps, such that U c (t) = P (g l ) for t [l, (l + ) ) an l =,,..., G, so T c = G. The corresponing first-orer effective error Hamiltonian H is exactly (5). In general the iscrete U c (t) facilitate computation an hence esign of the corresponing H eff, but in practice they require very strong control Hamiltonians. Eulerian Decoupling (ED) uses smooth control Hamiltonians of lower amplitue, an thereby achieves U c (l ) = P (g k ), g k G only at iscrete time steps [7]. A symmetric canceling of H at all times can then nevertheless be ensure by constructing the sequence f : (,,..., M G ) G : l g k as an Eulerian path through a Cayley graph constructe from M generators f,..., f M of G. Inee, uner the reasonable assumption that the transition associate to a given generator f α is each time implemente with the same control profile Hc α (t) of uration, the resulting 58

3 first-orer effective error Hamiltonian writes H = ( M ) P (g j ) Uc α (s) H Uc α (s) s P (g j ) G M = G g j G α= P (g j ) H P (g j ). (9) g j G Both these methos aim at canceling efficient mainly for small T c. C. Dynamical Decoupling on Kuramoto H, an hence are We can write the Kuramoto moel in the complex plane: t [eiθ,, e iθ N ] = (H + H K ) [ie iθ,, ie iθ N ], () where H = iag(ω,..., ω N ) an H K = iag(f,..., f N ) with f k = K N k j N k sin(θ j θ k ). This highlights a ynamical structure similar to a quantum system. Remarkably, also the objectives match, since a rift Hamiltonian proportional to the ientity matrix is perfectly transparent for phase synchronization in the Kuramoto moel. A ifference is that the state in () is restricte to {x C N : x k = for k =,..., N}, while quantum DD exploits the freeom to move in {x C N : x = }. In this paper, we replace the effect of H c by either (BBD) exchanging among subsystems the states or the causes of their natural frequencies, or (EDD) aapting H K with a ynamic controller that as the missing egrees of freeom to (). Another novelty w.r.t. stanar quantum DD is that we here have to analyze the DD effect on H K, a Hamiltonian which we want to keep. An important point is that H K is state-epenent, unlike H, an hence it might be ifferently affecte by the time-varying ecoupling ynamics. III. TWO-AGENT SYNCHRONIZATION While our ultimate goal is to work with multiple agents, we present as a preliminary step how to synchronize two oscillators uner BBD an ED because this situation allows a etaile analytic stuy. For two oscillators, let θ = θ θ, ω = ω ω, the incremental ynamics can be erive as: θ = K sin(θ) + ω. () No equilibrium of θ exists for < κ = (K)/ω <. For κ >, there is a stable fixe point θ = arcsin(/κ) with cos(θ ) >, an an associate unstable fixe point θ = arcsin(/κ) with cos(θ ) <. When κ tens to from above, these two fixe points converge towars π/, in fact far from phase synchronization θ =. For N = in (), ecoupling is achieve with a twoelements group G = {e, g}, where P (e) is the ientity matrix an P (g) = [, ;, ]. This inee generates effective rifts H = iag[ω, ω ] an H = P (g) H P (g) = iag[ω, ω ], whose sum H is proportional to the ientity. A. Bang-Bang Decoupling For BBD of two agents, instantaneous switches P (g) at k T c /, k =,,... can be implemente by exchanging positions or natural frequency(-inucing fiels). For the sake of analysis, these situations are equivalent to changing the sign of relative phase: θ(t + ) = θ(t ), t = T c /, T c, 3T c /, T c,... () where θ(t ) an θ(t+) are respectively the states just before an just after a switch at t. Since BDD is a perioic operation, we analyze the solution of the system (),() at perioic times θ(k T c ). Proposition : The relative phase θ(k T c ) between N = oscillators with non-ientical frequencies, evolving uner (), () an starting from any point on the circle, except one unstable perioic equilibrium, converges to ( ( )) a θ = arctan ( at ) ω t + ω, (3) where a = ω (K), t = tan(a T c /) for < κ, an a = (K) ω, t = tanh(a T c /) for κ >. Proof: When ( < κ, the general ) solution of () is θ(t) = arctan K a tan(a(c t)/) ω, where C epens on initial value θ(). After the variable change x = ω tan(θ/), this general solution writes x(t) = K a tan(a(c t)/) = K a tan(ac/) tan(at/) +tan(ac/) tan(at/). Using this expression over intervals (, T c /) an (T c /, T c ), with a switch at the en of each perio, a few calculations yiel x(t c+ ) = C C C = f(x()) with constants C 3+x() 3 < C < < C respectively efine by C = m m, C = m, C 3 = m m + where m = K a/ tan(at c /), m + = K+a/ tan(at c /), m = a K (+/ tan (at c /)). Figure illustrates the corresponing iscrete-time map f( ). Since C 3 < C <, the line y = x can only cross the upperleft curve of y = f(x). Therefore x((k +)T c ) = f(x(kt c )) ( ) has two equilibria x, = a a t ± t + ω. The negative one x correspons to (3) an it is locally stable since f (x ) < (see Fig. ); it irectly attracts all points x (, x ). Points above x = C 3 > get mappe to negative values an then converge to x, while points in (x, C 3 ] get mappe to increasingly larger values until they excee C 3 an join the former evolution. In conclusion, all points except x converge to x, as announce. The calculations for κ > are similar an therefore not etaile here. Remark: For fast swapping agents, that is T c, we have the approximation x t ω a 8 T cω an θ = arctan( ω x ) T cω. Thus faster switching rives the final relative phase closer to zero, an this inepenently of K in particular even for κ <. This remins integral control in linear systems, where the en equilibrium oes not epen on the proportional feeback strength. 587

4 (x,y ) y = x This is the moifie control law that we will apply. Going back to the ynamics of iniviual agents, we write θ = (K sin(θ θ ) + ω ) Ω cot( γ) cos(θ θ ) y (x,y ) y = C x = C 3 x Fig.. Function y = f(x) escribing the iscrete evolution x((k + ) T c+ ) = f(x(k T c+ )) of x = ω tan(θ/) uner (), (). B. Eulerian Decoupling Bang-Bang ecoupling, requiring instantaneous phase exchanges, has its limitations when applie to physical systems. We therefore propose a synchronization algorithm with continuous ecoupling, inspire by EDD. This continuous evolution requires to introuce one more egree of freeom, which will be a ynamic variable of the controller. The smallest setting with EDD is a qubit [7], whose state evolves on the sphere (so-calle Bloch sphere ) S R 3. We hence consier that θ = θ θ evolves on the equator of a sphere, an an auxiliary variable is use to moel ynamics that is require outsie of the equator. Let p R 3 a vector on the unit sphere, enote z, x the unit vectors pointing respectively perpenicular to the equator an on the equator at θ =, an the vector prouct in R 3, such that for y = z x the vectors (x, y, z) form a right-oriente orthonormal frame. Inspire by quantum EDD on the Bloch sphere, we let the rift be a rotation aroun z an apply EDD as a constant rotation aroun x: p = (Ωx + ( K sin θ + ω)z) p. () t In this moel, θ is the (azimuth) angle between the (z, p) plane an the x axis. It is not efine when p is parallel to z, but then anyways the corresponing factor z p = so it oes not matter. Let γ [, π] be the (polar) angle between p an z. Then p = [sin γ cos θ, sin γ sin θ, cos γ] T an p/t = [(K sin θ ω) sin γ sin θ, (K sin θ ω) sin γ cos θ Ω cos γ, Ω sin γ sin θ] T. Let c be the circle through p an perpenicular to z, an c the great circle through p an z (black circles on Fig.); the unit vectors tangent to c an c are given by v = [ sin θ, cos θ, ] T an v = [cos γ cos θ, cos γ sin θ, sin γ] T. Projecting p/t onto irections v an v yiels θ = sin γ v p t = (K sin θ ω) Ω cot γ cos θ, (5) γ = v p t = Ω sin θ. θ = (K sin(θ θ ) + ω ) Ω cot γ cos(θ θ ) with γ an auxiliary variable, share by both agents. This ynamics features a jump in θ for the rare trajectories which pass through one of the poles. For K =, the ynamics p/t = (Ωx + ωz) p implements a constant rotation rate about the axis n = [Ω,, ω] T, for which the unit tangent vector is v 3 (blue circle on Fig.). We now show that the effect ae with K > is to ecrease the angle φ between the vectors p an n, until p coincies with n which means that θ =. Base on this observation, the convergence of (5) can be summarize as: Proposition : The control law (5) for synchronizing oscillators an countering rift will rive the system asymptotically to the equilibrium θ =,γ = arctan(ω/ω), for any starting point except (θ() = π, γ() = arctan(ω/ω)) which is an unstable equilibrium. Proof: We propose the Lyapunov function V = cos φ. Projecting p/t onto the irection of n gives: n t p = Ω ω Ω ω K p sin θ = K p sin γ sin θ. sin γ cos θ sin γ sin θ cos γ This inicates that the angle between n an p/t is less than or equal to 9, thus φ. Since φ π, we then have V = sin φ φ. Accoring to LaSalle s invariance principle, the system converges to the largest invariant set where V =. In this conition, sin(φ) = correspons to the global maximum an minimum of V, i.e. the unstable initial point an the target equilibrium. Excluing the poles sin(γ) =, the conition V = itself alreay implies that sin(θ) =, i.e. the invariance set must satisfy y p =. Now keeping this conition invariant requires y p/t = Ω sin γ cos θ = sin γ sin θ ω K p sin θ cos γ = sin γ cos θ (ω K p sin θ) Ω cos γ. This conition in conjunction with sin γ = requires Ω = which oes not hol, so the poles cannot belong to the invariant set. In conjunction with sin θ = it reuces to the minimum (target equilibrium) an maximum (exclue unstable starting point) of V, for θ = an θ = π respectively. Remark: In (5), the erivative of θ tens to infinity when γ approaches or π. Although this is not a major problem for the analysis an practical workarouns may be imagine, a more pruent engineer might want to explicitly avoi this singularity. First an foremost, let us clarify that the poles are no accumulation point so in practice almost all trajectories will avoi passing through them. Knowing a boun on the 588

5 Fig.. n φ v v 3 p v v Directions on the sphere for Eulerian ecoupling rift ω ω, constructions which reuce the set of initial states to V (t = ) < V (p poles) woul more explicitly ensure that the poles are inaccessible in the future evolution. A better such set is obtaine with a small moification of our control law: replace the ecoupling Ωx by Ωx ω z, such that the rift to counter along z becomes (ω + ω ), possibly larger but certainly positive. Then if we restrict θ() to [ π/ + δ, π/ δ], δ > an select γ() = ɛ for small enough ɛ >, we can ensure that V (t = ) < V (p poles). This is ifferent from the analogy with integral control, where we woul set γ() = π/ such that the initial value of the integral term associate to θ is. IV. MULTI-AGENT SYNCHRONIZATION The extension to multiple oscillators involves a new issue besies the selection of an appropriate ecoupling group, namely how to implement the ecoupling actions over the network. In this section, we first present a strategy where a supervisor woul coorinate the ecoupling actions, then a strategy with ranom pairwise permutations. We here consier Bang-Bang ecoupling only, an leave for future work an explicit aaptation of Eulerian ecoupling with possibly several auxiliary control variables. A. Deterministic Bang-Bang ecoupling Suppose that there exists a close Hamiltonian path, which passes exactly once through all the noes of the network, an number the agents accoring to their orer on this path. Then the cyclic permutation group generate by P π : P π [x, x,..., x N ] = [x,..., x N, x ] can be implemente to counter the rift ue to iffering natural frequencies. Inee, since Pπ iag(ω, ω,..., ω N )P π = iag(ω,..., ω N, ω ), applying N iterations of P π over T c yiels an effective N N k= P k rift π H Pπ k = ( N N k= ω k) I. We call this metho Cyclic Deterministic Bang-Bang Decoupling (CDBBD). If we cannot guarantee that the network has such a close path, but we only know that it is connecte, then it contains a spanning tree tr(g) with N eges. Let P k enote the swap of the two agents linke by ege k of this spanning tree, for k =,,..., N. A recursive argument easily shows that N k= P k = P π a cyclic permutation. Hence N repetitions of a cycle of swaps on the N eges of the spanning tree has the same effect as CDBBD, only with an enlarge ecoupling perio covering N(N ) swaps. We call this approach Swapping Deterministic Bang- Bang Decoupling (SDBBD). B. Ranom Bang-Bang ecoupling We can ispose of the nee for a supervisor coorinating the swaps, by letting the ecoupling actions take place ranomly throughout the network. Such ranom ecoupling has alreay been investigate in the quantum context [8] with positive results. During a given time interval [, T c ], we apply ecoupling actions at n (more or less) evenly space instants. At the k-th instant, (not necessarily) inepenently of previous choices, a set of compatible graph eges whose agents will be swappe is selecte ranomly from a (more or less) uniform istribution. This can be one in a ecentralize way. We enote the corresponing permutation P (k). The effective Hamiltonian over T is then to first orer H = n n j= k= j P(k) H P (k). The probability for this expression to be ɛ-close to λi converges to as n increases to, provie the network is connecte []. We can finally give the following formal synchronization result for multiple oscillators uner Bang-Bang Decoupling. Proposition 3: Consier N agents with ynamics () an applying BBD as escribe above. The following properties hol for the behavior of the system, in the averaging approximation as T c. Assume the interconnection graph is connecte. We have: Uner SDBBD or CDBBD, if the agents start within a semicircle, then the phases converge to exact synchronization for any K an ω,..., ω N. Uner RBBD, for any δ (, ) an ε >, there exists M > such that for any n > M, if the agents remain within a semicircle of each other, then the phase is riven into the neighborhoo of exact synchronization e iθj e iθ k ε, j, k =,,..., N, with probability p δ. Proof: For SDBBD an CDBBD, averaging uner ecoupling gives the Kuramoto moel with equal natural frequencies an a graph that varies in time accoring to the swapping of agents. It is known that this ynamics converges to phase synchronization, for any time-varying graph, provie the initial conitions are within half a circle (see e.g. []). For RBBD we have a similar situation, with ecoupling now inucing a ranomly varying graph. Any graph sequence woul rive the system with equal natural frequencies to exact phase synchronization, when starting within a semicircle. Arguments on ranom ynamical ecoupling [8], [] can be invoke to boun the effective rift that remains after averaging. C. Simulations We simulate synchronization uner ynamical ecoupling with N = 3 agents with three kins of switching, CDBBD, 589

6 SDBBD an RBBD. The network for CDBBD is all-toall, while for SDBBD an RBBD there are only two eges as agents an 3 are not irectly couple. We normalize K = an take rift vector ω = [, 3.5, ] T, initial phase vector θ = [, π/3, π/3 ] T. The continuous lines show the phase evolutions without ynamical ecoupling, which clearly iverge with this parameter choice. Figure 3 shows the evolution of phases uner CDBBD, applying P π : (,, 3) (, 3, ) every.5s for a ecoupling perio T c =.75s. We observe that after convergence, the relative phases actually follow an evolution with a shorter perio T c /3. Qualitatively similar results are visible on Fig. for SDBBD. For ranom permutation we increase the ecoupling spee an apply a ranom permutation every.5s. Figure 5 shows one typical ranom run, illustrating how with high probability the rift effect is reuce an the relative phases converge to a close neighborhoo of exact synchronization. Further simulations (not shown) confirm that switching faster make the phases converge closer to each other. phase (π) agent without DD agent without DD agent 3 without DD agent with CDBBD agent with CDBBD agent 3 with CDBBD time (s) Fig. 3. Phase evolutions with CDBBD, applying P π : (,, 3) (, 3, ) every.5s, all-to-all coupling. phase (π) 8 agent without DD agent without DD agent 3 without DD agent with SDBBD agent with SDBBD agent 3 with SDBBD time (s) Fig.. Phase evolutions with SDBBD, applying alternatively P : (,, 3) (,, 3) an P : (,, 3) (, 3, ) every.5s; Kuramoto coupling among oscillators (, ) an (, 3). T c T c phase (π) agent without DD agent without DD agent 3 without DD agent with RBBD agent with RBBD agent 3 with RBBD time (s) Fig. 5. Phase evolutions with RBBD, applying P : (,, 3) (,, 3) or P : (,, 3) (, 3, ) ranomly every.5s; Kuramoto coupling among oscillators (, ) an (, 3). V. CONCLUSION This paper proposes a metho to counter the effect of nonientical rift frequencies in coorination control of couple oscillators, by aing a time-varying control signal inspire by quantum ynamical ecoupling. Convergence properties are analyze in both two-agent an multi-agent situations. In particular, the oscillators can be brought closer to exact synchronization by increasing the spee of the time-varying ynamics. A irect extension of this work woul be to aapt the continuous control version, known as Eulerian ecoupling, to the multi-agent case. Future work coul focus on eveloping ynamical ecoupling solutions for other geometric configuration spaces ubiquitous in robotics, like SE() an SO(3), an possibly proviing a general theory. REFERENCES [] Y. Kuramoto. Chemical Oscillations, Waves, an Turbulence. Springer, 98. [] A. T. Winfree. The Geometry of Biological Time. Springer, 98. [3] R. Sepulchre, D. A. Paley, an N. E. Leonar. Stabilization of planar collective motion: all-to-all communication. IEEE Trans. Autom. Control, 5(5):8-8, 7. [] A. Jababaie, N. Motee, an M. Barahona. On the stability of the Kuramoto moel of couple nonlinear oscillators. Proc. Amer. Control Conference, pages 9-3,. [5] S.H. Strogatz. From Kuramoto to Crawfor: exploring the onset of synchronization in populations of couple nonlinear oscillators. Physica D, 3:,. [] L. Viola, E. Knill, an S. Lloy. Dynamical ecoupling of open quantum systems. Physical Review Letters, 8():7-, 999. [7] L. Viola an E. Knill. Robust ynamical ecoupling of quantum systems with boune controls. Physical Review Letters, 9(3), 3. [8] L.F. Santos an L. Viola. Dynamical control of qubit coherence: ranom versus eterministic schemes. Physical Review A, 7(): 33, 5. [9] J. J. Sakurai. Moern Quantum Mechanics. Aison-Wesley, New York, 7. [] L. Mazzarella, A. Sarlette an F. Ticozzi. From consensus to robust ranomize algorithms: a symmetrization approach. SIAM J. Control an Optimization, 3, submitte. [] L. Moreau. Stability of multi-agent systems with time-epenent communication links. IEEE Trans. Autom. Control, 5():9-8, 5. T 59