Delft University of Technology

Adaptive hybrid synchronisation in uncertain Kuramoto networks with limited information

Baldi, Simone; Tao, Tian; Kosmatopoulos, Elias B.

DOI10.1049/iet-cta.2018.5465Publication date2019Document VersionAccepted author manuscriptPublished inIET Control Theory and Applications

Citation (APA)Baldi, S., Tao, T., & Kosmatopoulos, E. B. (2019). Adaptive hybrid synchronisation in uncertain Kuramotonetworks with limited information. IET Control Theory and Applications, 13(9), 1229-1238.https://doi.org/10.1049/iet-cta.2018.5465

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Adaptive hybrid synchronization inuncertain Kuramoto networks with limitedinformation

ISSN 1751-8644doi: 0000000000www.ietdl.org

Simone Baldi1∗, Tian Tao1, Elias B. Kosmatopoulos2

1Delft Center for Systems and Control, Delft University of Technology, Delft, 2628CD, The Netherlands2 Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, 67100 Greece, andInformatics & Telematics Institute, Center for Research and Technology Hellas (ITI-CERTH), Thessaloniki 57001, Greece* E-mail: s.baldi@tudelft.nl

Abstract: In this work we study adaptive synchronization in networks with Kuramoto units whose parameters are unknown andwhere measurements are quantized over the communication network (therefore information is limited). We show that, for an undi-rected connected graph, synchronization is enabled via appropriate adaptive protocols that counteract the effect of heterogeneity,uncertainty, and quantized information. In particular, to address heterogeneity and uncertainty, appropriate adaptive laws aredesigned to drive the network to frequency synchronization; to address quantized information, a dynamic quantizer is introducedand embedded into the adaptive mechanism via a zooming-based approach (therefore with hybrid dynamics). The resulting pro-tocol ends up being an adaptive hybrid synchronization strategy that can be distributed throughout the network: the quantizeris co-designed with the controller, as typical for zooming-based quantization. The proposed integrated adaptation+quantizationprotocol guarantees asymptotic synchronization to a desired frequency, which is shown via an appropriately designed distributedLyapunov function. Numerical simulations are also used to demonstrate the effectiveness of the proposed protocol.Keywords: Adaptive synchronization, Kuramoto dynamics, uncertain systems, quantization, adaptive hybrid control.

1 Introduction

Synchronization is a collective phenomenon occurring in systemsof interacting units [1–4]. Synchronization can be leaderless [5] orleader-follower [6, 7]: in the second case the network is steered insome desired and a priori known solution defined by a leader unit. Inthe 80’s Kuramoto proposed an exactly solvable model of collectivesynchronization, which became known as the Kuramoto model [8]:this model has been adopted in many fields like swarms [9], smartgrids [10, 11], among others [12].

Most synchronization models for non-evolving (or non-adaptive)Kuramoto or Kuramoto-like networks (e.g. networks of phase oscil-lators) have shown that synchronization is favored if the couplingstrength among the units is large enough and the spectrum of vari-ety of the oscillators is narrow: this last point amounts to assuminghomogeneous or almost homogeneous oscillators [13]. However,real-world networks have uncertain and heterogeneous parameterswhich might even change with time. Therefore, in place of staticcouplings, researchers have later been focusing on Kuramoto orKuramoto-like networks characterized by evolving, adapting cou-plings which vary in time according to different environmentalconditions, leading to the study of evolving (or adaptive) networks[14–16]. In [17] a simple model of adaptive Kuramoto network isgiven via mechanisms of homophily (reinforcing interactions withcorrelated units) and homeostasis (preserving the overall connectionstrength). In [18] a set of adaptive strategies for synchronization andconsensus of complex networks of dynamical systems is presented.The authors in [19] devise an adaptive scheme to achieve phase syn-chronization by suppressing the negative effect of the heterogeneityin the network, while in [20] protocols are designed to adaptivelyinteract with system dynamics and preserve the sum of all incom-ing pairwise coupling strengths. In [21] a co-evolutionary rewiringstrategy that depends only on the phase differences of neighboringoscillators is studied for Kuramoto units. However, adaptive syn-chronization of heterogeneous Kuramoto networks is usually shownnumerically but not analytically proven.

Within the scope of synchronization, this work will be focus-ing on how to achieve adaptive synchronization in uncertain andheterogeneous Kuramoto networks in the presence of networked-induced quantization. Established approaches to quantization rely ondynamic quantization mechanisms such as logarithmic quantization[22], or zooming-based hybrid control [23]. The latter mechanismtakes its name from the analogy with the zooming in digital cam-eras: when the state is outside its range region, the quantizer ‘zoomsout’ so that the state can be captured within the region. On theother hand, once the state comes close to the origin, we can ‘zoomin’ by reducing the size of the range so that the quantization reso-lution becomes finer while the region becomes smaller. Repeatingthis zooming in, we can obtain asymptotic stabilization. Recently,adaptive approaches to quantization have been proposed for stand-alone systems via different techniques, namely passification-basedadaptive control [24, 25], direct adaptive control [26–28], adap-tive backstepping [29, 30], or sliding mode [31]. However, suchapproaches have not been studied for networks of uncertain andheterogeneous units. In fact, for network systems, while severalapproaches to synchronization with quantized measurements can befound in literature [32–34], to the best of the authors’ knowledge,none of them is dealing with parametric uncertainty, which motivatesthis work.

In this work we study adaptive synchronization in undirectedconnected networks with Kuramoto units whose parameters areunknown and where information is quantized over the communi-cation network. Synchronization is enabled via appropriate adaptivehybrid protocols in the presence of both heterogeneous and uncertaindynamics. To address quantized information a dynamic quantizeris introduced and embedded into the adaptive mechanism via azooming-based approach. the quantizer is co-designed with the con-troller, i.e. the quantizer parameters are part of the design process.This is the typical approach used both in logarithmic and zooming-based quantization [22, 23]. The main contributions of this work arethe following: (a) proving asymptotic synchronization of Kuramotounits in an adaptive way and with quantized information; (b) adopt-ing a Lyapunov function that allows to distribute the dynamic

IET Research Journals, pp. 1–10c© The Institution of Engineering and Technology 2015 1

quantization strategy using only local information. The main ben-efit of the proposed integrated adaptation+quantization protocol isto guarantee asymptotic synchronization, whereas in the absenceof any dynamic quantization mechanism, one would obtain at mostbounded synchronization error.

The rest of the paper is organized as follows. Sect. 2 providesthe problem formulation. Synchronization protocols are given inSects. 3 and 4 without and with quantization, respectively. Numericalexamples are in Sect. 5, and Sect. 6 concludes the work.

Notation: The notation in this paper is standard. The transpose ofa matrix or of a vector is indicated with XT and xT respectively.A vector signal x ∈ Rn is said to belong to L∞ class (x ∈ L∞),if max

t≥0

∥∥x(t)∥∥ < ∞, ∀t ≥ 0. A time-invariant undirected commu-

nication graph of order N is completely defined by the pair G =(V ,E ), where V = 1, . . . ,N is a finite nonempty set of nodes, andE ⊆ V × V is a set of corresponding non-ordered pair of nodes,called edges. The adjacency matrix of a weighted undirected graphK = [ki j] is defined as kii = 0 and ki j = k ji > 0 if (i, j) ∈ E , wherei 6= j. The adjacency matrix A = [ai j] of an unweighted undirectedgraph is defined as aii = 0 and ai j = a ji = 1 if (i, j) ∈ E , wherei 6= j. The Laplacian matrix of the unweighted graph is defined asL = [li j], where lii = ∑ j ai j and li j = −ai j, if i 6= j. An undirectedgraph G is said to be connected if, taken any arbitrary pair of nodes(i, j) where i, j ∈ V , there is a path that leads from i to j.

2 Problem formulation

The following network of heterogeneous coupled oscillators withunknown dynamics is considered in this work

miθi +diθi = τi−N

∑j=1

ki j sin(θi−θ j),

i ∈ V , 1,2, . . . ,N

(1)

where the time index t may be omitted whenever obvious. The mean-ing of the parameters in (1) can be examined via the mechanicalanalogy of mass points in Figure 1. After neglecting any collision,

Fig. 1: Mechanical analogy of a network of three coupled oscilla-tors.

each point, or unit, will move on the circle describing an angle (orphase, by analogy) θi and an angular velocity (or frequency, by anal-ogy) θi, under the effect of an external driving torque τi, an elasticrestoring torque ki j sin(θi−θ j) (with ki j = k ji), and a viscous damp-ing torque diθi that is opposite to the direction of motion. All inertialcoefficients mi, damping coefficients di and stiffness coefficients ki jhave positive but unknown value. The external driving torque hastwo components

τi = ωi +ui, i ∈ V (2)

where ui is the actual control torque and ωi is a term proportional tothe natural angular velocity (or natural frequency) of the unit i, thatis the angular velocity it would have if there were no couplings. Letus define for convenience the state xi = θi. Then, (1) can be rewritten

as

xi =−di

mixi +

1mi

ui +ωi−N

∑j=1

ki j sin(θi−θ j)

(3)

The following connectivity assumption is made.

Assumption 1. The graph G of the network of Kuramoto oscilla-tors (1) is undirected and connected. In addition, let us assume thatthe network through which the oscillators can exchange informa-tion coincides with the network through which the oscillators arephysically coupled.

The following assumption on the uncertainty set is made.

Assumption 2. Upper bounds for the positive constants mi, di andki j, call them mi, di and ki j are known. Let us call the resultinguncertainty set Θi for compactness.

Remark 1. Assumption 1 is a standard connectivity assumptionrequired for convergence of consensus dynamics. Assumption 2 isrequired to obtain a bound to the increasing rate of the tracking errorduring the zooming in phase, as it will be explained in Section 4.

Quantized information is now introduced. Let z ∈ Rn be the vari-able being quantized. The uniform static quantizer is described bya function g : Rn → Q, where Q ⊂ Rn. The finite set of values isdefined as z ∈ Rn : q(z) = i, i ∈ Q. With these considerations inmind, we define the following dynamic quantizer:

qµ (z) = µq(

zµ

)(4)

where µ > 0. Note that the dynamic quantizer (4) satisfies thefollowing condition:

µ

∥∥∥∥q(

zµ

)∥∥∥∥≤ µM (5)

where µM represents the quantization range and M > 0 is the quan-tization range of the static quantizer g(z). The quantization range Mand quantization error ∆ will be adjusted by using a hybrid controlpolicy defined in Section 4.

We are now ready to formulate the synchronization problem.

Problem 1. [Adaptive frequency synchronization] Consider a net-work of unknown oscillators (1) satisfying Assumptions 1 and 2.Find a distributed adaptive strategy (i.e. exploiting only measure-ments from neighbors) for the control input ui such that the networkfrequency synchronizes to an a priori defined frequency x0, i.e. xi−x0→ 0, ∀i. In addition, assuming that information is quantized overthe communication network according to (4), find a distributed adap-tive strategy for the control input ui and for the dynamic parameterµi such that xi− x0→ 0, ∀i.

A result is now given which is instrumental to solving the problemabove.

Proposition 1. [Homogeneization] There exists a family of scalarsk∗i ∈ R and l∗i > 0 such that

− di

mi+

1mi

k∗i = 0

l∗i1mi

= 1(6)

Furthermore, there exists an ideal controller

u∗i = k∗i θi− l∗iN

∑j=1

ai j(θi− θ j)+ c∗i +N

∑j=1

g∗i jai j sin(θi−θ j) (7)

IET Research Journals, pp. 1–102 c© The Institution of Engineering and Technology 2015

with c∗i =−ωi, g∗i j = ki j which leads to the following dynamics

xi =−N

∑j=1

ai j(xi− x j) (8)

Proof: The proof directly follows from applying the control input(7) to unit (3), and using (6).

Remark 2. Conditions (6) basically correspond to matching theuncertain dynamics to first-order consensus dynamics (8) for whichleaderless synchronization (xi− x j → 0, ∀i, j) has been well studiedin literature [35]. It is not difficult to show that the controller

u∗i = k∗i θi− l∗iN

∑j=1

ai j(θi− θ j)− l∗i (θi− θ0)

+ c∗i +N

∑j=1

g∗i jai j sin(θi−θ j)

(9)

where θ0 is constant and a priori given, leads to leader-followersynchronization dynamics

xi =−N

∑j=0

ai j(xi− x j) (10)

where ai0 = 1 ∀i by definition, which guarantees convergence of allθi to θ0 [35]. The connectivity assumption (Assumption 1) is instru-mental in guaranteeing convergence of the dynamics (10), whichturn out to be leader-follower consensus dynamics [36, 37].

Remark 3. It is worth mentioning that, since mi, di, ωi, ki j areunknown, the ideal control (7) cannot be implemented to solve Prob-lem 1. Therefore, some adaptation mechanisms must be devised toestimate the unknown ideal gains in Proposition 1 by exploiting onlymeasurements from neighbors. This is shown in the next section.

3 Distributed adaptive synchronization

The following synchronizing protocol is proposed

ui(t) =ki(t)θi− li(t)N

∑j=1

ai j(θi(t)− θ j(t))− li(t)(θi− θ0)

+ ci(t)+N

∑j=1

gi j(t)ai j sin(θi(t)−θ j(t))

(11)

where ki, li, ci, gi j , are the (time-dependent) estimates of k∗i , l∗i , c∗i ,g∗i j , respectively. The following synchronization result holds.

Theorem 1. Define the errors

ei =N

∑j=0

ai j(θi− θ j), e = [e1,e2, . . . ,eN ]T . (12)

Under Assumption 1, the heterogeneous Kuramoto network (3),controlled using the synchronizing protocol (11) and the following

adaptive laws

ki = PΘi

−γ

N

∑j=0

ai j(ei− e j)

θi

li = PΘi

γ

N

∑j=0

ai j(ei− e j)

ei

ci = PΘi

−γ

N

∑j=0

ai j(ei− e j)

gi j = PΘi

−γ

N

∑j=0

ai j(ei− e j)

sin(θi−θ j)

(13)

where PΘi represent parameter projection in the uncertainty set Θi,and γ > 0 is the adaptive gain, reaches frequency synchronization(θi− θ0→ 0, ∀i).

Proof: The closed-loop network formed by (3) and (11) is

xi =(− di

mi+

1mi

ki)xi− li1mi

N

∑j=1

ai j(xi− x j)− li1mi

(xi− x0)

+1mi

(ci +ωi)+1mi

N

∑j=1

ai j(gi j− ki j)sin(θi−θ j)

(14)

which can be rewritten as a function of the estimation errors,

xi =1mi

ki(t)xi− (li(t)1mi−1)

N

∑j=1

ai j(xi− x j)+1mi

ci(t)−

(li(t)1mi−1)(xi− x0)+

1mi

N

∑j=1

gi j(t)ai j sin(θi−θ j)

(15)

where ki(t) = ki(t)−k∗i , li(t) = li(t)− l∗i , ci(t) = ci(t)−c∗i , gi j(t) =gi j(t)−g∗i j. By defining for compactness

Bk(t) = diag(1

m1k1(t), . . . ,

1mN

kN(t))

Bl(t) = diag(−l1(t)1

m1, . . . ,−lN(t)

1mN

)

Bc(t) = diag(1

m1c1(t), . . . ,

1mN

cN(t))

Bg(t) = diag(1

m1

N

∑j=1

g1 j(t)a1 j sin(θ1−θ j), . . .

. . . ,1

mN

N

∑j=1

gN j(t)aN j sin(θN −θ j))

(16)

the closed-loop for the overall network can be written as

x = Bk(t)x+(−IN +Bl(t))e+Bc(t)+Bg(t) (17)

where x = [x1,x2, . . . ,xN ]T . Recalling that the synchronization error

is e = IN(x − x0) + L x, where x0 = [x0,x0, . . . ,x0]T , the error

IET Research Journals, pp. 1–10c© The Institution of Engineering and Technology 2015 3

dynamics are

e =− (IN +L )e+

+(IN +L )(Bk(t)x+Bl(t)e+Bc(t)+Bg(t)).(18)

The adaptive laws (13) arise from considering the Lyapunov functioncandidate

V =N

∑i=1

e2i (t)+

N

∑i=1

k2i (t)γ

−1

|l∗i |+

N

∑i=1

l2i (t)γ

−1

|l∗i |

+N

∑i=1

c2i (t)γ

−1

|l∗i |+

N

∑i=1

N

∑j=1

g2i j(t)γ

−1

|l∗i |.

(19)

Then we have

2eT e =−2eT (IN +L )e+

+2eT (IN +L )(Bkx+Ble+Bc +Bg)(20)

and

V ≤−2N

∑i=1

σ(IN +L )e2i

+2N

∑i=1

ki(t)|l∗i |

γ−1 ˙ki(t)+ xi

N

∑j=0

ai j(ei− e j)

+2N

∑i=1

li(t)|l∗i |

γ−1 ˙li(t)− ei

N

∑j=0

ai j(ei− e j)

+2N

∑i=1

ci(t)|l∗i |

γ−1 ˙ci(t)+

N

∑j=0

ai j(ei− e j)

+2

N

∑i=1

N

∑j=1

gi j(t)|l∗i |

γ−1 ˙gi j(t)+ sin(θi−θ j)

N

∑j=0

ai j(ei− e j)

(21)

where σ(·) is a real number representing the smallest eigenvalue ofa square symmetric matrix. Using (13) we have

V ≤−2N

∑i=1

σ(IN +L )e2i (22)

which is negative semi-definite. Using standard Lyapunov argumentswe can prove boundedness of all closed-loop signals and conver-gence of e to 0. In fact, since V > 0 and V ≤ 0, it follows that V (t) hasa limit, i.e. limt→∞ V (e(t),Ω(t)) =V∞ < ∞, where we have collectedall parametric errors in Ω. The finite limit implies V , e, Ω ∈L∞. Inaddition, since V is uniformly continuous in time (this is satisfiedbecause V is finite), the Barbalat’s lemma implies V → 0 as t → ∞

and hence e→ 0, from which we derive xi→ x0, ∀i. This concludesthe proof.

Remark 4. Theorem 1 provides a leader-follower synchroniza-tion protocol driving the synchronization error to zero. In standardleader-follower synchronization protocols, the desired reference isavailable to a single node (or to a few nodes) in the team acting asleader(s). However, traditionally, the literature on Kuramoto oscil-lators has considered the reference to be available to all nodes: seefor example the relevant works [38, 39]. Here, the pacemaker pro-vides a reference frequency to all nodes: such a setting is justified inview of the fact that Kuramoto models are often used in smart gridapplications [40, 41]. In this scenario, θ0 represents the globally-known 50 or 60 Hz to be followed by all systems. In line with such

literature, also in the proposed protocol the knowledge of θ0 is avail-able to all systems in the network: nevertheless, using a distributedobserver, it is not difficult to extend the proposed approach to thesetting where the desired reference is available to a single or to afew nodes. This issue was studied by some of the authors in [16].With a similar smart grid application in mind, we have been focus-ing on frequency synchronization, rather than on full-state (phaseand frequency) synchronization. The proposed protocol requires themeasurements of both θi and θi. This might sound demanding, butmany management algorithms in smart grids (frequency regulation,optimal power flow, etc.) require both θi and θi. In addition, becauseof the widespread use of phasor measurement units (PMUs), it isnowadays quite straightforward to measure θi and θi in smart grids.A PMU is a device which measures the electrical waves on anelectricity grid.

Remark 5. Note that the parameter projection PΘi [·] in (13) is notnecessary for stability. As a matter of fact, stability can be achievedevn without any projection PΘi [·] (see for example the full-state syn-chronization strategy in [16]). However, since parameter projectionis fundamental for stability in the presence of quantized informationas explained in the next section, then parameter projection has beenintroduced in Theorem 1 as well.

Remark 6. In order to implement (13), and in particular the term∑

Nj=1 ai j(ei − e j), it is required to communicate among neighbors

the extra variable ei. In fact, the variable ei can also be thoughtto be local information for the following rationale. Communicationof extra local variables is often at the core of many synchroniza-tion protocols: for example, synchronization based on distributedobserver [42, 43] requires communication of extra local variablesrepresenting the observer states. Therefore, communication of ei ishomologous to communication of any other auxiliary variable. It hasto be noted that, in our setting, communication of the local vari-able ei is equivalent to communicating xi to the neighbors of theneighbors (2-hop communication). This apparently more complexcommunication architecture actually brings a crucial useful feature:the Lyapunov function V in (19) ends up being block-diagonal (nosynchronization error cross terms among units are present, and sim-ilarly no parameter estimation error cross terms among units arepresent). This reminds of the typical block-diagonal design in con-sensus problems, V = e′(I⊗P)e [35, 44], and it is thus consistentwith such literature: in the next section we will see that such a fea-ture is also crucial to implement a dynamic quantization strategythat does not require the complete knowledge of L .

4 Distributed adaptive synchronization withquantized information

Being the systems connected through a network, information (fre-quency and phase) should be exchanged among neighbors, andtherefore quantized. Therefore, system i will received some quan-tized information from its neighbors j. In line with the work byLiberzon and Ishii [22, 23] and most literature stemming from there,a dynamic quantizer will be codesigned together with the controller,i.e. the quantizer parameters will also be part of the design. Weconsider the following quantization setting:

• Instead of quantizing the frequency θ j itself, we assume that eachsystem j quantizes the difference with respect to θ0, i.e. qµ (θ j− θ0)which, upon synchronization, tends to zero;• Instead of quantizing the phase θ j itself, we assume that each sys-tem j sends a correction factor with respect to the phase increment,according to the relation

θ j(t) = b j(t)+(θ0 +a j(t))t (23)

IET Research Journals, pp. 1–104 c© The Institution of Engineering and Technology 2015

where qµ (b j(t)) and qµ (a j(t)) are the packages to be sent (y-intercept and incremental slope correction of the line) which, uponsynchronization, tend to a constant and to zero, respectively.

Within this synchronization setting it is possible to reconstruct witharbitrary precision all signals, provided that the quantization rangeof (4) can be made arbitrarily small. When information is quantizedaccording to (4), the closed-loop system (14) modifies into

xi =(− di

mi+

1mi

ki)xi− li1mi

N

∑j=1

ai j(xi− x j)− li1mi

(xi− x0)

+1mi

(ci +ωi)+1mi

N

∑j=1

ai j(gi j− ki j)sin(θi−θ j)−µili1mi

∆zi

(24)where ∆z is defined as the quantization error, and in case of no sat-uration∗ it holds

∥∥∆zi

∥∥≤ ∆. Note that µi represents the quantizationrange for system i.

In view of (24), the evolution of the synchronization error can bewritten as:

e =− (IN +L )e+

+(IN +L )(Bk(t)x+Bl(t)e+Bc(t)+Bg(t)+Blµ (t))(25)

where we have used the same variables defined in (26), and inaddition

Blµ (t) = diag((−1−µ1 l1(t))1

m1∆z1 , . . . ,(−1−µN lN(t))

1mN

∆zN )

(26)

In order to analyze the stability of the closed-loop system (25), itis natural to consider the same Lyapunov function defined in (19). Inaddition, let us consider the same adaptation laws as in (13). Afterrepeating similar steps as in the proof of Theorem 1, we arrive at theexpression

V ≤−2N

∑i=1

σ(IN +L )ei(ei +(1+µi li)1mi

∆zi). (27)

Because all control gains are bounded due to the projection strategyin (13), we can define ρi ∈ R≥ 0 such that:

ρi = maxt≥0

γ−1 k2

i (t)+ l2i (t)+ c2

i (t)+∑Nj=1 g2

i j(t)

|l∗i |

(28)

and

ρ =N

∑i=1

ρi. (29)

In addition, because of (19) we have

eT e≤V ≤ eT e+ρ. (30)

The most interesting observation is that, because of the separablestructure of V = ∑

Ni=1 Vi (Vi represent the terms in (30) related only

to system i), it is possible to find ‘decentralized’ versions of the

∗Saturation occurs when the signal z exceeds the maximum quantized level.

In case of no saturation (‖z‖ ≤ µM), it holds∥∥qµ (z)− z

∥∥= µ∥∥∆zi

∥∥≤ µ∆.

inequality (30), i.e.

e2i ≤Vi ≤ e2

i +ρi, ∀i ∈ V . (31)

The time derivative of V in (27) in case of no saturation can beexpressed as

V ≤−2N

∑i=1

σ(IN +L )ei(ei +(1+ li)1mi

µi∆)

≤−2N

∑i=1|ei|σ(IN +L )

(|ei|−

σ(IN +L )

σ(IN +L )max

mi∈Θ,i∈V

∥∥∥∥∥1+limi

∥∥∥∥∥︸ ︷︷ ︸R

µi∆

)

=⇒ V ≤−2N

∑i=1|ei|σ(IN +L )(|ei|−µiR∆) (32)

where R is bounded, in view of Assumption 2. The last inequalitybasically reveals that the time derivative of V can also be separatedinto N inequalities, i.e. V = ∑

Ni=1 Vi (Vi represent the terms in (32)

related only to system i). According to (5), the condition for nosaturation is satisfied if the following holds:

|ei| ≤ µiM. (33)

We define, for each system i, the following regions:

B1i(µi) :=

ei : |ei| ≤ µiM

I1i(µi) :=

ei : e2i ≤ µ

2i M2

B2i(µi) :=

ei : |ei| ≤ µiR∆

I2i(µi) =

ei : e2

i ≤ µ2i R2

∆2.

(34)

Note that, when

M > R∆

then B2(µi)⊂I2(µi)⊂I1(µi)⊂B1(µi).

4.1 Main Result

Using the previously explained design, we can obtain an inte-grated mechanism where appropriate adaptive laws interact witha zooming-based quantizer. The following stability result can bederived.

Theorem 2. Under Assumptions 1 and 2, consider the heteroge-neous Kuramoto network (3), controlled using the synchronizingprotocol∗

ui(t) =ki(t)θi− li(t)N

∑j=0

ai j(θi(t)−qµ (θ j(t)))

+ ci(t)+N

∑j=1

gi j(t)ai j sin(θi(t)−qµ (θ j(t)))

(35)

∗In the next and the following equation we use, with some abuse of nota-

tion, the symbols qµ (θ j), qµ (θ j) and qµ (e j) to indicate variable quantized

as indicated at the beginning of the section.

IET Research Journals, pp. 1–10c© The Institution of Engineering and Technology 2015 5

and the following adaptive laws

ki = PΘi

−γ

N

∑j=0

ai j(qµ (ei− e j))

θi

li = PΘi

γ

N

∑j=0

ai j(qµ (ei− e j))

qµ (ei)

ci = PΘi

−γ

N

∑j=0

ai j(qµ (ei− e j))

gi j = PΘi

−γ

N

∑j=0

ai j(qµ (ei− e j))

sin(θi−qµ (θ j))

(36)

where PΘi represent parameter projection in the uncertainty set Θi,and γ > 0 is the adaptive gain. If the following holds

M > R∆ (37)

then there exists an error-based hybrid quantized feedback controlpolicy that reaches frequency synchronization (θi− θ0→ 0, ∀i).

Proof: We distinguish two phases, namely the zooming-out andzooming-in phases. In the zooming-out phase µi is chosen sothat ei ∈ B1i(µi) and thus boundedness can be guaranteed. Dur-ing zooming-in phase the objective is to shrink the smaller regionI2i(µi) by reducing the dynamic quantizer parameter µi so thatstate-tracking properties can be concluded. The two phases areexplained in the following.

Zooming-out phase: Let µi(0) = 1. If∣∣ei(0)

∣∣ > M we have sat-uration. In this case we make µi(t) increase in a piecewise fashionto dominate the growth factor of ei, which can be seen by (24) that

equals

∣∣∣∣∣emaxmi ,di∈Θ

∣∣∣− di+kimi

∣∣∣∣∣∣∣∣, where maxmi,di∈Θ

∣∣∣− di+kimi

∣∣∣ is bounded in

view of Assumption 2. There will be a time instant t0 > 0 and abounded µi(t0) at which ei(t0) ∈I1i(µi(t0))∩B1i(µi(t0)). Becauseei(t0) ∈B1i(µi(t0)) and B2i(µi)⊂B1i(µi) from (37), we get from(32) Vi ≤ 0. Note that this does not imply that ei will decrease, sinceVi comprises also the parametric estimation errors. Therefore, fort > t0 we might have two cases: either e2

i is decreasing, in whichcase there is no saturation and we go to zooming-in phase; or e2

i isincreasing, in which case we keep increasing µi(t) at the same rate.For this second case, because µi(t) is updated continuously at muchhigher rate compared to the growth of ei(t) to avoid saturation, wecan assume that ∀t ≥ t0, ei(t) ∈B1i(µi(t)).

Zooming-in phase: Let t ′ be a time instant such that t ≥ t ′ ≥ t0,and ei(t) ∈ B1i(µi(t ′)). Then it is true that Vi ≤ 0 as long as ei /∈B2i(µi(t ′)). One can see from (34) that B2i(µi)⊂I2i(µi). Thus, attime t with t ≥ t ′, when ei(t) ∈I2i(µi(t ′)), µ(t) is updated

µi(t) =R∆

M︸︷︷︸Ω

µi(t ′). (38)

Obviously Ω < 1 due to (37). Thus, zooming-in event occurs, andone can see that I1i(µi(t)) = I2i(µi(t ′)), where µi(t ′) is the valueof µi that prevents saturation. After the zooming-in event one mighthave two cases: either the tracking error increases tending to violateei ∈B1i(µi(t)), in which case a new zooming-out phase is activated;or the tracking error keeps decreasing in which case a new zooming-in will eventually be triggered.

Combined behavior: Let us now look at the combined behavior ofzooming-in and zooming-out phases for the function V and the sets

B1(µ) :=

e : |e| ≤ NµM

I1(µ) :=

e : eT e≤ N2µ

2M2

B2(µ) :=

e : |e| ≤ NµR∆

I2(µ) =

e : eT e≤ N2

µ2R2

∆2,

(39)

where µ is taken as the maximum µi among all systems i in thenetwork, i.e. µ = maxi µi. Now, provided that the same designparameters M, ∆ and R are chosen for all units, including the sameinitial condition µi(0) = 1, it is possible to extend the stability argu-ment to the whole Lyapunov function V . In fact, for t ≥ t0 (taken asthe maximum among all systems), at both zooming-in and zooming-out phases it holds V ≤ 0 =⇒ V (t)≤V (t0), thus V is upper-boundedby V (t0) and lower-bounded by 0. Because V is bounded and V islower bounded by V (t0), because it holds V ≤ 0 ∀t ≥ t0, we canconclude using the Barbalat’s lemma that lim

t→∞V (t) = 0.

The following relation from (32) holds:

limt→∞

V (t)≤−2 limt→∞

∥∥e(t)∥∥σ(IN +L )

(∥∥e(t)∥∥− µ(t)NR∆

)=⇒

0≤−2 limt→∞

∥∥e(t)∥∥σ(IN +L )

(∥∥e(t)∥∥− µ(t)NR∆

).

(40)The above relation is true when

limt→∞

∥∥e(t)∥∥= 0 or lim

t→∞

∥∥e(t)∥∥− µ(t)NR∆≤ 0. (41)

The second relation implies that e ∈ B2(µ). However whene ∈ I2(µ), and because I2(µ) ⊃ B2(µ), µ is decreasing as in(38) because zooming-in occurs, and consequently e /∈B2(µ). Asa consequence lim

t→∞µ(t) = 0 and by (41) we conclude lim

t→∞

∥∥e(t)∥∥ =

0.

A state flow diagram of the adaptive hybrid control strategy, withrules for zooming in/out, is shown in Fig. 2.

Fig. 2: Error dependent adaptive hybrid control strategy, with rulesfor zooming in/out.

Some remarks follow.

Remark 7. The peculiar feature and contribution of the zoomingprocedure of Theorem 2 is to adopt a Lyapunov function (19) that

IET Research Journals, pp. 1–106 c© The Institution of Engineering and Technology 2015

allows distributing the quantization strategy throughout the network.In other words, thanks to the diagonal structure of V in (19) and tothe diagonal structure of V in (32) each system can implement itsown zooming strategy only by using measurements from neighbors.

Remark 8. Even though the adaptive hybrid synchronization pro-tocol in Theorem 2 is distributed, it requires some global knowledgein order to obtain the parameter R in (32). In particular, someglobal knowledge of the uncertainty set Θi for all units is requiredin order to obtain maxmi∈Θ,i∈V

∥∥∥1+ limi

∥∥∥: this global knowledge canbe obtained after assuming that all units have the same uncertaintyset. In addition, the knowledge of σ(IN +L ) and of σ(IN +L ) isrequired in order to obtain R: it is important to remark that appro-priate upper and lower bound for σ(IN +L ) and for σ(IN +L ) canbe obtained by simply knowing the number of units in the network, orthe maximum number of neighbors for each unit. In fact, conserva-tive bounds in this direction can be found in [45–47]. In other words,it is not necessary to know the exact topology of the network in orderto estimate σ(IN +L ) and σ(IN +L ), and therefore to estimate R.

Remark 9. Differently from the non-adaptive zooming procedure[23, 48], where a unique zooming-in occurs, in the adaptive casewe might have multiple zooming-in and zooming-out phases. Thisis because the Lyapunov function (19) is quadratic in both the syn-chronization and the parameter estimation error. Therefore, V ≤ 0might be due not only to a decreasing synchronization error (eventu-ally leading to zooming-in), but also to decreasing parameter errorcombined with increasing synchronization error (eventually lead-ing to a new zooming-out). Therefore, convergence of µ may be notmonotonic. This is illustrated by the numerical example in the nextsection.

Remark 10. From the dynamics in (24) we can see that the quanti-zation of the phase θi does not contribute to stability or instability ofthe adaptive loop: in fact, being the phase always bounded in [0,2π],the quantization of θi simply acts as a bounded disturbance that canbe handled by parameter projection. It is the quantization of θi thatmust be carefully handled in order not to lead to instability. As amatter of fact, if one is interested only in bounded synchronization (inplace of asymptotic synchronization), one can have a static quantizerfor θi and a dynamic quantizer for θi: using similar steps as in theproof of Theorem 2, one would be able to conclude global uniformlyultimately boundedness of the synchronization error, where the ulti-mate bound would depend on the precision of the static quantizer forθi.

5 Numerical example

Simulations using the adaptive synchronization protocol of Theorem1 and the adaptive hybrid synchronization protocol of Theorem 2 arecarried out in the following, considering the weighted graph shownin Figure 3. The parameters and initial conditions for each heteroge-neous Kuramoto unit (3) are reported in Table 1. Please recall thatthe unit parameters are unknown to the designer, i.e. the values ofTable 1 are used for simulations but not for control design.0

1 2

3 4

5 6

3

3

3

45

5

1

2

Fig. 3: The Kuramoto undirected weighted graph. The weights ki jare indicated on the links.

Table 1 Parameters and initial conditions for the Kuramoto units.mi di ωi θi(0) θi(0)

unit #1 1.1 0.1 5 0 0.6unit #2 1.3 0.15 10 π 0.5unit #3 1.2 0.2 15 π/2 0.4unit #4 1.8 0.21 20 (5/4)π 0.3unit #5 1.5 0.25 25 π/4 0.2unit #6 1 0.3 30 (3/2)π 0.1

The reference model is chosen as an integrator with initial con-dition xm(0) = 1 (representing the frequency to be followed). Theadaptive gain is taken as γ = 1. The projection bounds are chosenas: ki ∈ [0.05, 4], li ∈ [0.8, 2], ci ∈ [−32, −2], and gi j ∈ [0, 6]. Theestimated control gains are initialized as ki(0) = 0.2, li(0) = 1.5,ci(0)=−10, gi j(0)= 0. The parameters for the quantizer are M = 10and ∆ = 0.01: in addition, the parameter R has been calculated as79.2 (resulting in Ω = 0.0792), and the dynamic ranges are ini-tialized as µi(0) = 1. Clearly, condition (37) is satisfied, thereforeTheorem 2 is applicable.

0 5 10 15 20 250

1

2

3

4

5

x1, ... , x

6

0 5 10 15 20 25

time

-40

-30

-20

-10

0

10

u1, ... ,u

6

Fig. 4: Adaptive synchronization: synchronization of the frequen-cies of each unit i to the reference frequency, and control inputsui.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

k1, ... ,k

6

0 5 10 15 20 25

time

0.5

1

1.5

2

l 1, ... ,l

6

Fig. 5: Adaptive synchronization: adaptive gains ki and li.

The adaptive synchronization resulting from Theorem 1 is shownin Figure 4. Synchronization is achieved and it can be noted that

IET Research Journals, pp. 1–10c© The Institution of Engineering and Technology 2015 7

0 5 10 15 20 25

time

-40

-30

-20

-10

0

c1, ... ,c

6

0 5 10 15 20 25

time

0

2

4

6

g1

1, ... , g

66

Fig. 6: Adaptive synchronization: adaptive gains ci and gi j.

each unit has different control inputs ui that reach different steady-state values: this is due to heterogeneity, since each unit might needto converge to different steady-state values in order to track the ref-erence frequency. The adaptive gains are reported in Figure 5 andFigure 6, which can be shown to be projected inside their bounds.It can be noted that the weights gi j are not necessarily symmetricgi j 6= g ji: this is because, in order to compensate for heterogeneity,the adaptive laws might converge to different values. The adaptivesynchronization resulting from Theorem 2 is shown in Figure 7. Syn-chronization is achieved and, to some extent, convergence is similarto the previous results. The initial transient is slightly different, as itcan be noted from the different control inputs ui. The adaptive gainsare reported in Figure 8 and Figure 9. It is important to note thatthe gains do not necessarily converge to the same values as in theprevious simulation: this is a well-know result in adaptive controlapproaches, since, in the absence of a persistency of excitation con-dition, convergence of the tracking error to zero might be achievedwithout the need to converge to the actual parameters (cf. [24–30]and references therein). In other words, adaptive synchronizationdoes not require ki, li, ci and gi j to converge to k∗i , l∗i , c∗i and g∗i j.Finally, Figure 10 reports how the dynamic range µi for each unitevolves with time: after starting from 1, the different µi’s converge,with different trends, quite fast to small values, which in turn guar-antees high quantization precision and asymptotic tracking. Overall,the proposed protocols show the capability to synchronize, in a dis-tributed way, uncertain and heterogeneous units, even in the presenceof limited (quantized) information.

By selecting the parameters for the quantizer to be M = 10 and∆ = 0.1, we now have Ω = 0.792. This means that the zooming-in is smaller than in the previous case (with ∆ = 0.01 we obtainedΩ = 0.0792). Figure 11 and Figure 12 show the new synchroniza-tion and dynamic range (adaptive gains are not reported for lackof space): it can be seen that asymptotic synchronization is stillreached, but the dynamic range takes a bit more time to decreasetowards zero. Finally, in order to clarify the benefits of the pro-posed method, we provide the synchronization with a static quantizer(with no zooming-based strategy). The tracking error can be seenin Figure 13, where it is clear from the zoomed part that the staticquantization error will lead to bounded synchronization, in place ofasymptotic synchronization.

6 Conclusions

In this work we have studied adaptive synchronization in net-works with Kuramoto units whose parameters are unknown andwhere information is quantized over the communication network.In the presence of both heterogeneous and uncertain dynamics,synchronization has been enabled via appropriate adaptive proto-cols that counteract the effect of heterogeneity and uncertainty. In

0 5 10 15 20 25

x1, ... , x

6

0

1

2

3

4

time

0 5 10 15 20 25

u1, ... ,u

6

-40

-30

-20

-10

0

10

Fig. 7: Adaptive hybrid synchronization: synchronization of the fre-quencies of each unit i to the reference frequency, and control inputsui.

0 5 10 15 20 25

k1, ... ,k

6

0

0.1

0.2

0.3

0.4

time

0 5 10 15 20 25

l 1, ... ,l

6

1.2

1.4

1.6

1.8

2

Fig. 8: Adaptive hybrid synchronization: adaptive gains ki and li.

time

0 5 10 15 20 25

c1, ... ,c

6

-30

-25

-20

-15

-10

-5

time

0 5 10 15 20 25

g1

1, ... , g

66

0

2

4

6

Fig. 9: Adaptive hybrid synchronization: adaptive gains ci and gi j.

particular, to address quantized information, a dynamic quantizerwas introduced and embedded into the adaptive mechanism via azooming-based approach. The resulting protocol ended up beingan adaptive hybrid control strategy, for which synchronization to a

IET Research Journals, pp. 1–108 c© The Institution of Engineering and Technology 2015

time

0 5 10 15 20 25

µ1, ... ,µ

6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 10: Adaptive hybrid synchronization: dynamic range µi foreach unit i.

0 5 10 15 20 25

x1, ... , x

6

0

1

2

3

4

time

0 5 10 15 20 25

u1, ... ,u

6

-40

-30

-20

-10

0

10

Fig. 11: Adaptive hybrid synchronization with ∆ = 0.1: synchro-nization of the frequencies of each unit i to the reference frequency,and control inputs ui.

time

0 5 10 15 20 25

µ1, ... ,µ

6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 12: Adaptive hybrid synchronization with ∆ = 0.1: dynamicrange µi for each unit i.

desired frequency was shown via Lyapunov analysis. Finally, numer-ical examples have demonstrated the effectiveness of the proposedprotocol.

0 5 10 15 20 25

x1, ... , x

6

0

1

2

3

4

5

time

0 5 10 15 20 25

u1, ... ,u

6

-40

-30

-20

-10

0

10

5 10 15 20 25

0.6

0.8

1

1.2

Fig. 13: Static quantizer synchronization: synchronization of thefrequencies of each unit i to the reference frequency, and controlinputs ui.

Future work might include the following aspects: in smart grids,which are a typical application field of Kuramoto networks, it isoften of practical interest to optimize the power flow: this impliessatisfying some constraints on the phase differences between unit[49, 50]. This aspect has not been addressed in this work and wouldbe of interest for future work. Also, some practical input constraintslike input saturation or discrete inputs are of interest in view of:the limits of generation or load shedding, or some possibly discretegeneration or load shedding actions [51]. An adaptive approach inthis direction has been recently investigated in [52], even though theextension to network systems is open. Other networked-induced con-straints like sampled information have been studied in [53], althoughadaptivity to parametric uncertainty has not been covered. All theseaspects are of extreme interest for future work.

Acknowledgments

Mr. Ilario Azzollini and Dr. Shuai Yuan are gratefully acknowledgedfor useful discussions on the topics of this work. The research lead-ing to these results has been partially funded by the European Com-mission H2020-SEC-2016-2017-1, Border Security: autonomoussystems and control systems, under contract #740593 (ROBOR-DER) and H2020-ICT-2014-1, FIRE+ (Future Internet Research &Experimentation), under contract #645220 (RAWFIE).

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