Experiments on node-to-node pinning control of Chua’s circuits

Physica D 239 (2010) 454–464

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Physica D journal homepage: www.elsevier.com/locate/physd

Experiments on node-to-node pinning control of Chua’s circuits Maurizio Porfiri ∗ , Francesca Fiorilli Department of Mechanical and Aerospace Engineering, Polytechnic Institute of New York University, Brooklyn, NY 11201, United States

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Article history: Received 5 August 2009 Received in revised form 18 January 2010 Accepted 19 January 2010 Available online 25 January 2010 Communicated by A. Pikovsky Keywords: Pinning controllability Chaos synchronization Chua’s circuit Fast switching Global exponential stability

abstract In this paper, we study the global intermittent pinning controllability of networks of coupled chaotic oscillators. We explore the feasibility of the recently presented node-to-node pinning control strategy through experiments on Chua’s circuits. We focus on the case of two peer-to-peer coupled Chua’s circuits and we build a novel test-bed platform comprised of three inductorless Chua’s oscillators. We investigate the effect of a variety of design parameters on synchronization performance, including the coupling strength between the oscillators, the control gains, and the switching frequency of node-to-node pinning control. Experimental results demonstrate the effectiveness of this novel pinning control strategy in rapidly taming chaotic oscillator dynamics onto desired reference trajectories while minimizing the overall control effort and the number of pinned network sites. From an analytical standpoint, we present sufficient conditions for global node-to-node pinning controllability and we estimate the maximum switching period for network controllability by adapting and integrating available results on Lyapunov stability theory and partial averaging techniques. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Pinning control problems entail the design of control laws to tame the dynamics of a complex oscillator network onto a common desired reference trajectory by exerting control actions on a limited set of network oscillators, see for example [1–20]. The reference trajectory is commonly generated by a so-called reference oscillator, that acts as a master for the oscillator network, see for example [8,11]. The reference trajectory may also correspond to a fixed equilibrium of the individual oscillator, see for example [6,13]. Control inputs applied at the so-called pinned sites are spread in the whole network through diffuse coupling among the oscillators, see for example [2]. In general, pinning controllability problems can be viewed as synchronization problems on augmented oscillator networks that incorporate the reference oscillator. Such augmented networks are typically directed and weighted to account for the unidirectional nature of the pinning control, see for example [11]. Synchronization on directed weighted networks is studied in [21–25] among others. Similarly to the case of master–slave synchronization problems, see for example [26–29], pinning controllability problems can be generally cast into classical stability problems for nonlinear systems. More specifically, network synchronization can be described by the error dynamics, that represents the difference between the

∗

Corresponding author. Tel.: +1 718 260 3681; fax: +1 718-260-3532. E-mail addresses: [email protected] (M. Porfiri), [email protected] (F. Fiorilli). 0167-2789/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2010.01.012

states of the network oscillators and the state of the reference oscillator. The linear stability of the error dynamics can be assessed using the master stability function originally introduced in [30]. The master stability function has been utilized in [11,12,19,31] to design and analyze pinning control schemes. This approach is easy to be implemented, allows for analyzing a broad class of chaotic oscillators, and provides necessary and sufficient conditions for synchronization. Nevertheless, the approach can only be used to analyze local synchronization of oscillator networks, since it is based on the linearized error dynamics. Criteria for global pinning controllability of complex networks can be derived by using classical Lyapunov stability theory. Within this framework, chaotic oscillators characterized by bounded stability regions are generally excluded, see for example [32]. However, synchronization of a large variety of important chaotic oscillators can be properly tackled, including Chua’s circuits and Lorenz systems [33]. Lyapunov stability approaches have been utilized in [2,4,5,7,8,10,14–18] to provide insight into node selection for effective synchronization performance as well as to develop pinning gains’ adaptation rules. For undirected network topologies, it has been observed that pinning control effectiveness is generally maximized when pinning a large set of network nodes with a uniformly distributed gain level, see for example [12,14,19]. Nevertheless, limitations in realworld applications set stringent constraints on the simultaneous accessibility of the entire node set. In [9], we have presented a practical pinning control strategy to overcome this problem. This scheme, termed as node-to-node pinning control, consists in cyclically pinning all the network nodes at a fast-switching rate and at a uniform gain level. In other words, in a short time period, all the network nodes are individually pinned at the same gain level for

M. Porfiri, F. Fiorilli / Physica D 239 (2010) 454–464

an equal fraction of time. Under fast-switching conditions, that is, for switching rates considerably faster than the individual oscillator dynamics, the node-to-node pinning scheme is virtually equivalent to simultaneously pinning all the network nodes with a homogeneous gain level. In [9], the analysis of the node-to-node pinning strategy is based on the master stability function and is thus limited to local controllability. Further, estimates of the minimum switching period required to guarantee network pinning controllability are not reported and the performance assessment of the scheme is limited to computer simulations. The aim of this paper is threefold: (i) to validate node-tonode pinning control through experiments on coupled nonlinear circuits; (ii) to extend node-to-node pinning control to the analysis of global pinning controllability problems; and (iii) to provide quantitative bounds for the lowest switching rate that enables global pinning controllability. Experiments are conducted on a platform of Chua’s circuits comprising two peer-to-peer coupled oscillators and a reference oscillator. The novelty of the platform is in the full state time varying coupling of the three circuits. Available experimental results on networks of coupled Chua’s circuits consider only static voltage coupling among the oscillators [34]. The performance of node-to-node pinning control is studied by parametrically varying the coupling between the peer oscillators and the switching rate/gain of the intermittent pinning control. Experiments on static pinning are also conducted to better elucidate the advantages of node-to-node pinning control. In such experiments, the coupling among the oscillators, the number of pinned sites, and the overall pinning control gain are varied. In addition, node-to-node pinning control is compared to a different intermittent control strategy, in which network nodes are evenly pinned at any instant of time. The analytical treatment of node-tonode pinning control is based on the recent results on intermittent master–slave synchronization presented in [28] and on the criteria for global pinning controllability reported in [8]. We adapt the quadratic Lyapunov function of [8] for the case in which all the network nodes are simultaneously pinned at the same gain level. We follow the line of argument in [28] to show that, under fastswitching conditions, such Lyapunov function decreases at every switching event in case of node-to-node pinning control, which in turn implies that the error dynamics asymptotically approaches zero. We adopt a standard notation throughout the paper. We indicate the Euclidean norm in Rn and the corresponding induced norm with k · k. The identity matrix in Rn×n is indicated as In . We indicate with sym(A) the symmetric part of the matrix A ∈ Rn×n , that is, sym(A) = 12 (A + AT ). The eigenvalues of A are called {λi (A)}ni=1 and, if A is symmetric, λmin (A) is the smallest eigenvalue of A. Superscript T means matrix transposition and ⊗ indicates Kronecker product. 2. Problem statement We consider a network of N oscillators, and we assume that the network is unity weighted and undirected. The network topology is described by a unity weighted undirected graph G = (V , E ), where the node set V comprises all the network oscillators and the edge set E includes all the unordered pairs (i, j) of interconnected nodes. In other words, the pair (i, j) indicates a communication link between the ith and the jth node of the graph G. The Laplacian matrix L is the algebraic description of the graph G, see for example [35], and it is defined as the difference between the degree matrix D and the adjacency matrix A, that is, L = D − A.

(1)

The degree matrix D is a diagonal matrix whose ith diagonal element di equals the connectivity of the ith node. The entries of the adjacency matrix aij are 1 if (i, j) ∈ E and 0 otherwise.

455

The Laplacian matrix is symmetric, positive semi-definite, and zero row-sum, see for example [35]. This implies that λmin (L) = 0 and 1N ∈ Null(L)

(2)

where 1N is the vector comprised of all ones. The time evolution of the ith oscillator is described by x˙ i (t ) = f (xi (t )) − σ

N X

lij h(xj (t )) + ui (t )

(3)

j =1

xi (t0 ) = xi0 ,

i = 1, . . . , N .

Here, xi (t ) ∈ Rn is the n-dimensional state vector corresponding to the ith oscillator and xi0 indicates the initial condition at the initial time t = t0 . The function f : Rn −→ Rn is a nonlinear function of the state xi and describes the characteristic individual dynamics of an oscillator. The function h : Rn −→ Rn describes inner coupling among oscillators in the network, and it is called inner linking function. Coupling among oscillators is weighted by the parameter σ > 0. The scalar quantity lij is the (ij)th element of the graph Laplacian matrix L. The vector function ui (t ) is the external control input acting on the ith oscillator. Note that in (3), all the oscillators are assumed to be identical; the effect of parameter mismatch on synchronization is experimentally studied for example in [36]. Following [26], we assume that the function f is globally Lipschitz with Lipschitz constant α . Therefore, the matrix Fξ,eξ defined by f (ξ) − f (e ξ) = Fξ,eξ (ξ − e ξ)

(4)

satisfies

kFξ,eξ k ≤ α,

∀ξ, e ξ ∈ Rn .

(5)

Note that the Lipschitz constant can be selected as the smallest value of α in (5). Within pinning control strategies, a nonzero control input is applied only to a restricted number of nodes. We refer to P (t ) as the set of network nodes pinned at time t, and to p(t ) as the cardinality of this set. The goal of pinning control is to drive each oscillator dynamics onto a common reference trajectory s(t ), whose evolution satisfies s˙ (t ) = f (s(t )).

(6)

The control input u (t ) is a state-feedback control, that is, u (t ) = ki (t )[h(s(t )) − h(xi (t ))], where ki is a nonnegative function. For ease of presentation, we limit our analysis to a linear statefeedback control law, meaning that the inner linking function is in the form h(x) = Hx, where H is a constant matrix. We group the nonnegative feedback gains ki (t ) into the positive semi-definite diagonal matrix K(t ). We consider the case of intermittent pinning control, that is, we assume that the gain matrix K is a periodic time function of period T that switches among a finite set of constant matrices K1 , . . . , Kr , where r is an integer number. We refer to T as the switching period and we call δi the period fraction during which K(t ) = Ki for i = 1, . . . , r. Note that static pinning control is a special case of intermittent pinning control. We define the error vector e(t ) ∈ RnN as i

e(t ) = 1N ⊗ s(t ) − x(t ).

i

(7)

The {(i − 1)n + 1, . . . , in} elements of this vector represent the components of the error between the state of the ith and the reference oscillators, that is, ei (t ) = s(t ) − xi (t ). Following [8], we tackle the global controllability problem by analyzing the stability of the error dynamics about the origin. Through standard manipulation and using (2)–(4) and (7), we express the error

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dynamics as follows





e˙ (t ) = Fs(t ),s(t )−e(t ) − (σ L + K(t )) ⊗ H e(t )

(8)

where Fs(t ),s(t )−e(t ) is a block diagonal matrix, whose ith block equals Fs(t ),s(t )−ei (t ) . We associate to system (8) a partially averaged system, see [28], where the time average is taken with respect to the sole switching dynamics. The partially averaged system is described by e˙ (t ) = Fs(t ),s(t )−e(t ) e(t ) − (σ L + K) ⊗ He(t ).

(9)

The feedback gain matrix for the partially averaged system K is constant and equals the time average of the time varying matrix K(t ) over a period T , that is, K=

1 T

Z

(l+1)T

K(τ )dτ =

r X

lT

δi Ki

(10)

where l is an arbitrary integer. In case of node-to-node pinning control, r = N, δi = 1/N, and Ki = C π i π Ti , where π i is the column vector comprised of all zeros except of a one at the ith entry and C measures the overall cost of network pinning. In addition, the matrix K is equal to (C /N )IN . In what follows, we present a general claim on intermittent pinning controllability of complex networks and we later specialize it to node-to-node pinning control. 3. Intermittent global pinning controllability We provide sufficient conditions to guarantee global exponential stability of the error dynamics (8) by extending and integrating results from [8,28]. In particular, we enforce global pinning controllability of the oscillator network by imposing that the error dynamics of the partially averaged system in (9) is globally exponentially stable and that the switching period T is sufficiently small as compared to the oscillator dynamics. We denote with T ∗ an upper bound to the switching period that guarantees that the stability properties of the static pinning control are inherited by the intermittent pinning control. Loosely speaking, if T < T ∗ , the oscillator network is virtually pinned by a constant coupling matrix K, that is indeed capable of synchronizing the network dynamics onto the reference trajectory. A sufficient condition for global pinning controllability of (9) is presented in [8]. More specifically, Corollary 5 in [8], herein reported for completeness as Theorem 1, allows for a proper selection of the control gain matrix K as a function of the underlying network topology and the coupling σ . Theorem 1. The partially averaged system (9) is globally exponentially stable if, for some Q ∈ Rn×n symmetric and positive definite, symQH is a positive definite matrix and (11)

where α is the Lipschitz constant in (5). In addition, the error dynamics exponentially approaches zero. Theorem 1 is based on the quadratic Lyapunov function V (e) = e (IN ⊗ Q)e T

for t > t0 .

φ(T ) = 2T 2 L∗ 2 kIN ⊗ Qk exp(L∗ T )(1 + exp(L∗ T )).

(16)

The constant L∗ is a bound for the matrix measure of Fs(t ),x(t ) − (σ L + K(t )) ⊗ H, that can be estimated as

L∗ = α + max µ(−(σ L + Ki ) ⊗ H)

(17)

where the matrix measure of a matrix A is defined by

µ(A) = max λi (symA).

(18)

i=1,...,n

The quantity L∗ is a bound for the norm of Fs(t ),x(t ) −(σ L + K(t ))⊗ H, that can be computed as

L∗ = α + max k(σ L + Ki ) ⊗ Hk.

(19)

i=1,...,r

Global exponential stability of the error dynamics of (8), that is, global pinning controllability of the oscillator network under intermittent coupling is possible if the right hand side of (15) is negative. The maximum switching period is estimated as the smallest nonzero solution of

φ(T ) − w T = 0.

(20) ∗

r i i=1

Note that the estimate of T depends on the duty cycles {δ } only through the average gain matrix K that, in turn, controls the rate of decay w in (13). Note that the switching time can also be infinitely large if (20) does not have a nonzero solution. Less conservative estimates may be potentially obtained by further exploiting the spectral properties of the gain matrices {Ki }ri=1 using commutators, as suggested in [37] for linear systems. The above discussion can be condensed in the following claim. Proposition 1. The system (8) is globally exponentially stable if the hypotheses of Theorem 1 are satisfied and T < T ∗ , where T ∗ is the smallest nonzero solution of (20). In case of node-to-node pinning control, inequality (11) reduces to

C N

>

αkQk λmin (symQH)

(21)

and all the constants appearing in (16) should be computed by using Ki = C π i π Ti . 4. Case study: Pinning control of two peer-to-peer coupled Chua’s oscillators In this section, we specialize the results illustrated in Section 3 to Chua’s oscillators. We consider two peer-to-peer coupled Chua’s oscillators controlled by a third one that acts as a reference. The Laplacian matrix of the network is



1 −1

 −1

.

L=

(13)

The state of a single Chua’s oscillator comprises three components, x1 , x2 , and x3 , and its dynamics is given by, see for example [38,39],

that is, V˙ (e(t )) ≤ −wke(t )k2

(15)

where k is any integer such that kT ≥ t0 and the function φ is given by

(12)

whose decay is faster than

w = −2αkQk + 2λmin (symQH)λmin (σ L + K)

V (e((k + 1)T )) − V (e(kT )) ≤ (φ(T ) − w T )ke(kT )k2

i=1,...,r

i=1

αkQk λmin (σ L + K) > λmin (symQH)

We use the Lyapunov function of the partially averaged system in (9) to enforce the global exponential stability of the error dynamics of the switched system (8). In particular, by following the line of proof of Theorem 2 in [28], the increment of (12) over a switching period along the trajectory of the switched system (8) can be expressed as

( (14)

1

x˙ 1 (t ) = a(x2 (t ) − x1 (t ) − g (x1 (t ))) x˙ 2 (t ) = x1 (t ) − x2 (t ) + x3 (t ) x˙ 3 (t ) = −bx2 (t )

(22)

(23)

M. Porfiri, F. Fiorilli / Physica D 239 (2010) 454–464

457

a

Fig. 1. Level curves for the minimum eigenvalue λmin (σ L + K) as a function of C and (k1 − k2 )/C for σ = 0.021 and σ = 2.1.

b

c

Fig. 2. Inductorless realization of Chua’s circuit. The inductor is synthesized by using Antoniou’s circuit.

where a > 0 and b > 0. The nonlinear function g (x1 ) is given by g (x1 ) = m1 x1 +

1 2

(m0 − m1 )(|x1 + 1| − |x1 − 1|)

(24)

where m0 and m1 are negative quantities. Further, we define the function ωη,e η ∈ R such that η , ∀η,e g (η) − g (e η) = ωη,eη (η − e η) ∀η,

e η ∈ R.

(25)

The scalar quantity ωη,e η is bounded by m0 ≤ ωη,e η ≤ m1 , see [26]. The subscript indicates the dependence of ωη,e η. η on both η and e We assume that the oscillators are equally coupled through all their states, that is, we set H = I3 . This assumption allows for a direct application of Theorem 1, since symQH = Q > 0. Thus, upon coupling, the dynamics of a single network oscillator becomes

x˙ i (t ) = a(xi (t ) − xi (t ) − g (xi (t ))) 1 2 1 1   2  X  j   −σ lij x1 (t ) − ki (t )(xi1 (t ) − s1 (t ))     j =1   2  X j x˙ i2 (t ) = xi1 (t ) − xi2 (t ) + xi3 (t ) − σ lij x2 (t )   j=1   i   − k ( t )( x ( t ) − s ( t )) i 2 2    2 X   j i i  lij x (t ) − ki (t )(xi (t ) − s3 (t )). x˙ (t ) = −bx (t ) − σ 3

2

3

j=1

3

Fig. 3. (a) Circuit design of two peer-to-peer coupled Chua’s circuits, (b) circuit design of pinning control, and (c) controlled peer-to-peer coupled Chua’s circuits.

Superscripts i and j belong to the set {1, 2}, while s1 , s2 , and s3 represent the components of the reference oscillator state, that is, s = [s1 , s2 , s3 ]T . From Theorem 1, global pinning controllability of the coupled oscillators is possible if the partially averaged feedback gain matrix K and the coupling σ satisfy

λmin (σ L + K) > (26)

αkQk λmin (Q)

(27)

for some Q = QT > 0, see Theorem 1. For ease of illustration, we set Q = I3 and we analyze λmin (σ L + K). The parameter λmin (σ L + K) can be viewed as a performance index of the pinning control strategy that is described by the time averaged gain matrix K, see also [9]. Clearly, in static pinning, the average gain matrix equals the actual gain matrix.

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Table 1 Values of the circuit components. Nominal data are followed by experimental data in curly braces, the first two lines refer to the two peer-to-peer coupled circuits and the third one refers to the reference circuit. Tolerance on resistances is 5% of the nominal value. Tolerance on the capacitances is 20% of the nominal value. The deviation with respect to nominal values are partially caused by parallel/series realizations. Chua’s circuit

Chua’s diode RC 1 = 220 

 1.613 k 1.615 k 1.610 k  9.90 nF 9.90 nF 9.92 nF  97.05 nF 97.40 nF 97.40 nF  18.80 mH 19.10 mH 19.10 mH

R = 2 k

C1 = 10 nF

C2 = 100 nF

L = 15 mH

RC 2 = 220 

RC 3 = 2.2 k

RC 4 = 22 k

RC 5 = 22 k

RC 6 = 3.3 k

Table 2 Values of circuits’ components for the peer-to-peer coupling. Peer-to-peer coupling

σ

R1

R2

CT

0.021 2.1

750 k 7.5 k

76 k 760 

5 µF 50 nF

Antoniou’s circuit

 221.00  220.50  220.00   220.00 220.05  220.10   2.260 k 2.190 k 2.195 k  21.200 k 21.140 k 21.406 k  21.120 k 22.200 k 21.900 k  3.270 k 3.270 k 3.350 k

RA1 = 100 

RA2 = 1 k

RA3 = 1 k

RA4 = 1.5 k

CA = 100 nF

 97.00  98.00  97.00   992.00  992.00  989.00   981.00  982.00  990.00   1.480 k 1.480 k 1.490 k  132.30 nF 133.05 nF 132.00 nF

strength σ and equals C /2. We remark that in case of node-to-node pinning control, each circuit is pinned for the same time duration at the gain level C . Also, Fig. 1 and (28) show that the performance index is bounded above by σ that is asymptotically attained as C goes to infinity when a single network node is pinned. 4.1. Hardware implementation

Table 3 Values of circuits’ components for pinning controlling both the network oscillators at the same gain level. When statically pinning a single oscillator or in case of nodeto-node pinning control, the resistances Rx1 and Rx2 and the capacitance Cx3 are divided by two and the resistance R3 is doubled. Pinning control

C

Rx1

Rx2

Cx3

R3

1.2 2 15 30 42 217 310 1000

25 k 15 k 2 k 1 k 750  150  100  30 

2.5 k 1.5  200  100  75  15  10  3

0.2 µF 0.1 µF 14 nF 7 nF 5.3 nF 1 nF 0.7 nF 0.2 nF

70  120  900  1.8 k 2 k 13 k 18 k 60 k

Fig. 1 shows the minimum eigenvalue λmin (σ L+K) as a function of the overall control action C and the skewness of the control gains, defined as (k1 − k2 )/C , for two different values of the coupling σ . Such eigenvalue can be readily computed from (22) as



v u  2 u 2σ 2σ C λmin (σ L + K) = 1 + −t + 2

C

C

k1 − k2

C

!2

Chua’s circuits are easily implementable and widely studied chaotic oscillators, see for example [38–45]. Their simplest implementation scheme includes four passive elements (one resistor, two capacitors, and one inductor) and a nonlinear active component called Chua’s diode. In this work, the nonlinear Chua’s diode is constructed by following the synthesis proposed in [38] and adopted in [39], see Fig. 2. The dimensional state components V1 and V2 (voltages) and I (current) in Fig. 2 are related to the corresponding nondimensional variables x1 , x2 , and x3 by x1 = V1 /V0 , x2 = V2 /V0 , and x3 = IR/V0 , where V0 is the breakpoint voltage of the I–V characteristic of Chua’s diode and R is the resistance illustrated in Fig. 2. The constants m0 and m1 , that quantify the nonlinear behavior of Chua’s diode, can be directly determined from its components. More specifically, see for example [38], we have

 m0 = −R

   .(28)

The gains k1 and k2 are varied in the representative range 0.01 and 1000 and the parameters σ = 0.021 and σ = 2.1 are investigated. This parameters’ choice is experimentally studied in what follows. Fig. 1 can be used to select numerical values for the time averaged pinning gains to enforce synchronization under fast-switching conditions or, in case of static pinning, to select the values of the time constant pinning gains. Fig. 1 shows that the performance index λmin (σ L + K) increases as σ and C increase. Such performance index is maximized when the control action is uniformly distributed throughout the network. In this case, the performance index is independent of the oscillator coupling

RC 3

 m1 = −R

1 1 RC 3

− +

1



RC 4 1 RC 6



(29)

.

We realize our test-bed platform using an inductorless Chua’s circuit obtained by simulating the inductor through Antoniou’s circuit, see [46]. The RC -synthesis of the inductor allows a fine tuning of the oscillator dynamics and improves the circuit robustness. Another advantage of this design is the availability of a circuit node whose voltage can be used to directly estimate x3 , see [39]. The simulated inductance of Antoniou’s circuit is related to the circuit components by L = RA1 RA3 RA4 CA /RA2 where the meaning of the symbols is illustrated in Fig. 2. The time constant for nondimensionalizing time is selected as τ0 = C2 R. The dimensionless parameters in Eq. (23) are a = C2 /C1 and b = C2 R/(L/R).

M. Porfiri, F. Fiorilli / Physica D 239 (2010) 454–464

Fig. 4. Trajectory of the three circuits in the V1 –V2 plane from experimental data.

The test-bed platform implemented in this study comprises three oscillators. Two circuits are mutually coupled through a full state diagonal coupling and a third circuit acts as a reference oscillator by driving the peer-to-peer coupled oscillator network. Consistently with (26), we use superscript s, 1, and 2 to identify voltages and currents in the reference, first, and second oscillator, respectively. We refer to Fig. 3 for the meaning of symbols introduced below. The bidirectional coupling between the voltages V11 and V12 is realized through the resistance R1 . Fig. 3(a) illustrates the peerto-peer coupled oscillators in absence of the reference oscillator.

459

The coupling scheme for the second state components V21 and V22 follows a similar architecture, through the resistance R2 , see Fig. 3(a). The coupling on the current state variables is different. A transformer with unitary transformation ratio R = 1 : 1 connects the two peer-to-peer coupled circuits. We note that transformers may as well be synthesized using active circuits similar to Antoniou’s circuit, see for example [47]. The voltage across the primary winding has opposite sign with respect to the voltage across the secondary winding, while currents flow in the same direction. The secondary winding is in parallel connection with a capacitance CT . Fig. 3(b) and (c) display the coupling between the reference oscillator and the peer-to-peer coupled circuits. An operational amplifier in voltage follower configuration along with resistance Rx1 is used for directionally coupling the voltage V1s to the voltages V11 or V12 . A switching device is used to generate intermittent directional coupling. Note that static coupling is simply obtained by short-circuiting the ports of the switching devices. The directional coupling for the second state components is similar and thus does not require modification of the pristine circuit architecture, see Fig. 3(c). In contrast, when the third states of the peer-to-peer coupled circuits are driven by the reference oscillator, the pristine circuit architecture is significantly modified as illustrated in Fig. 3(c). In particular, pinning control is obtained by including the resistance R3 on the circuit branch of I 1 (I 2 ) and connecting a terminal of the transformer (transformer and capacitor CT ) to the reference oscillator voltage Vout . The voltage Vout is the output voltage of an integrator connecting Antoniou’s circuit of the reference oscillator to the peer-to-peer coupled

Fig. 5. Experimental results for the uncontrolled network. Time evolution of the two voltage states of the peer-to-peer coupled circuits for (a) σ = 0.021 and (b) σ = 2.1. The black and the grey lines refer to the network oscillators. Synchronization graphs in the V11 − V12 and in the V21 − V22 space for (c) σ = 0.021 and (d) σ = 2.1.

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reported in Tables 1–3. TL082 operational amplifiers biased at ±8.5 V, DG411DJ switches, and a Midcom 671–8014 transformer are used in the experimental platform. The average value of the dimensionless system parameters are calculated from the hardware components in Table 1. In particular, substituting values from the reference circuit hardware components, see Table 1, we obtain a = 9.82, b = 13.22, m0 = −1.2141, and m1 = −0.6583. The mutual coupling σ is adjusted through proper selection of the resistances R1 , R2 , and the capacitance CT . These parameters must satisfy the following constraints

σ = aR/R1 = R/R2 = C2 R/CT RA4

Fig. 6. Synchronization error E for different values of the pinning control gain C and number of pinned nodes for σ = 0.021 (black line) and σ = 2.1 (red line) in the case of static pinning control. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

network, see Fig. 3(b). A similar circuit synthesis is used in [9] to simulate master–slave coupling of two Chua’s oscillators. A switching device alternates the connection from the integrator output terminal Vout with the transformer winding ending, see Fig. 3(c). Values of the circuits’ components are selected for this study following the implementation proposed in [28,38,39] and are

(30)

in order to evenly couple the dimensionless state variables of the oscillators, that is, H = I3 in Eq. (8). The pinning gains are adjusted by selecting the resistances Rx1 , Rx2 , and RA4 and the capacitance Cx3 . In case of node-to-node pinning control, these quantities satisfy

C = aR/Rx1 = R/Rx2 = C2 R/(Cx3 RA4 ) = C2 R/(L/R3 ).

(31)

When statically pinning a single network site, the switching devices do not operate and these component values are unaltered. When simultaneously pinning both the network sites, the switching devices are bypassed. In this case, the circuit components are replaced in order to maintain the same total control gain unaltered. In case of node-to-node pinning control, switches are driven in a synchronous way by the FG7002C LG function generator. The driving signal is a square wave with duty cycles δ1 = δ2 = 0.5 and adjustable frequency f .

a

b

c

d

Fig. 7. Experimental results in case the network is equally pinned at both its nodes at C = 42. Time evolution of the two voltage states of the three circuits for (a) σ = 0.021 and (b) σ = 2.1. The black and the grey lines refer to the network oscillators and the red line is related to the reference circuit. Synchronization graphs in the V11 − V12 − V1s and in the V21 − V22 − V2s space for (c) σ = 0.021 and (d) σ = 2.1.

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461

a

4.2. Experimental results We measure the first two state components of each circuit. We collect data by using six analog input channels from a National-Instruments PCI − 6229 acquisition board. The sampling frequency is equal to 25 kHz and the capacity of each channel is 5000 samples. All measurements reported in the following figures are dimensional. In particular, voltages are expressed in volts (V) and time in milliseconds (ms). Fig. 4 displays the chaotic dynamics of the uncoupled Chua’s circuits. From Fig. 4, we can appreciate the similarity of the systems, that is made possible by using the above described circuit implementation. The synchronization performance is quantified by the dimensional error measure E E =

1 t2 − t1

Z

t2

(|V11 (t ) − V1s (t )| + |V12 (t ) − V1s (t )|

t1

+ |V21 (t ) − V2s (t )| + |V22 (t ) − V2s (t )|)dt

(32)

where t2 − t1 is the measurement duration (0.2 s). Note that the selected error measure accounts only for the voltage states of Chua’s circuits. We report experimental results for two different cases corresponding to a value of coupling strength σ equal to 2.1 and 0.021. These two values are selected to analyze pinning control in case of oscillator networks that synchronize or do not synchronize in absence of external control actions. More specifically, for σ = 2.1, the peer-to-peer coupled oscillators synchronize without any external control actions, whereas synchronization is not observed for σ = 0.021 as illustrated in Fig. 5. The Lipschitz constant as defined in Eq. (5) is computed to be equal to α = 16.5. Therefore, from condition (21), the peerto-peer coupled circuits are controllable through node-to-node pinning if C > 33. For the selected values of the parameter σ , the sufficient condition in (11) with Q = I3 does not imply pinning controllability by statically pinning a single Chua’s circuit. In other words, the network is controllable through a single site if the site location rapidly switches over time, but static pinning of a single node may not guarantee network controllability. This is further illustrated in Fig. 1, where the network performance in the range 0.01 ≤ C ≤ 1000 is reported and the critical value 16.5 is depicted as a separate level curve. Fig. 1 also shows that for given values of the pinning gains, higher values of the mutual coupling σ imply larger value of the performance index. The characteristic constants of the system L∗ and L∗ computed using Eqs. (17) and (19) do not show considerable variations with respect to the selected values of the coupling strength. More specifically, L∗ = 16.48 and L∗ = 58.51 in case of σ = 0.021, and L∗ = 14.50 and L∗ = 60.70 in case of σ = 2.1. The switching rate needed for enforcing node-to-node pinning controllability can be estimated from Eq. (20). In this case, the constant w is given by w = −2α + 2C /N, see Eq. (13). For an overall control gain C = 42, the constant w is equal to 9. Therefore, from Eq. (20), we find T ∗ = 0.60 × 10−3 for σ = 2.1 and T ∗ = 0.65 × 10−3 for σ = 0.021. These values correspond to a switching frequency of approximately 10 MHz, since the characteristic time τ0 is equal to 0.1568 ms. We first analyze the synchronization performance in case of static pinning control. Experimental data in Fig. 6 are obtained using a constant feedback gain matrix. Fig. 6 shows that the synchronization error is generally smaller for strongly coupled oscillators and that the error is minimized when statically pinning both the oscillators at the same value of control gain. In this case, the synchronization error is relatively independent of the oscillator coupling. On the other hand, σ is an important parameter when pinning a single network site. In particular, when pinning a

b

Fig. 8. Experimental results for node-to-node pinning control at C = 42; the switching frequency is equal to 150 kHz. Synchronization graph in the V11 − V12 − V1s and in the V21 − V22 − V2s space for (a) σ = 0.021 and (b) σ = 2.1.

single network site, the synchronization error drastically decreases as the oscillator coupling increases while holding the total pinning gain constant. The weakly coupled oscillators do not exhibit synchronization even for considerably large values of C when pinned at a single network site. In case σ = 2.1, the synchronization performance tends to improve as C increases above 200 and is in general comparable with the synchronization performance for any studied value of C . These results are in agreement with the general trend of the performance index reported in Fig. 1, that is independent of σ when both oscillators are pinned and improves as σ increases when a single oscillator network is pinned. Fig. 7(a) and (b) show the time trace of the voltage signals for a subset of the cases illustrated in Fig. 6. More specifically, they show the time evolution of the voltage signals in case the network is equally pinned at both its nodes with overall gain C = 42. The time traces of the signals for the two different values of σ are similar and illustrate satisfactory synchronization, corresponding to E = 0.615 V at σ = 0.021 and E = 0.606 V at σ = 2.1. Fig. 7(c) and (d) show the synchronization plot in the three dimensional voltage spaces for the two studied inner coupling conditions. The spatial extent of these plots provide a qualitative measure of the synchronization performance. In particular, in case of perfectly synchronized oscillators, these plots degenerate to simple lines. Similar evidence has been reported in [34] for a network of three Chua’s oscillators coupled through their first state. In the following, we report experimental results on the nodeto-node pinning control strategy to assess its effectiveness as

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a

b

Fig. 9. Experimental results for node-to-node pinning control at C = 42; the switching frequency is equal to 25 kHz. Synchronization graph in the V11 − V12 − V1s and in the V21 − V22 − V2s space for (a) σ = 0.021 and (b) σ = 2.1.

a function of the switching frequency. Experimental data are collected by controlling the network at switching frequencies equal to 150 kHz, 25 kHz, and 1 kHz, see Figs. 8–10 respectively. We note that the period of the lowest switching frequency is comparable with the characteristic time τ0 of the circuit dynamics, while the other switching frequencies are considerably faster than the oscillator dynamics. Under fast-switching conditions, node-to-node pinning control is expected to yield synchronization performance comparable to that displayed in Fig. 7. Experimental data in Figs. 8–10 show that the circuits well synchronize for relatively low values of the switching frequency for both the studied inner coupling values. In particular, satisfactory synchronization performance is observed at a switching frequency of 25 kHz, see Fig. 9. This corresponds to a switching period of 40 µs that is approximately equal to 0.25 τ0 , that is, T ' 0.25. This result is in line with numerical evidence shown in [9], where performance of the network remains satisfactory in case of relatively low switching rates. We note that theoretical predictions yield a severely conservative estimate of three order of magnitude difference. As expected, the synchronization performance at different switching frequencies does not significantly change with respect to the coupling strength between the peer-to-peer coupled oscillators even if static pinning of a single oscillator network does not yield network synchronization as shown in Fig. 6. In particular, for a switching frequency of 150 kHz the error E is 0.69 V for σ = 0.021 and 0.65 V for σ = 2.1; for a switching frequency

a

b

Fig. 10. Experimental results for node-to-node pinning control at C = 42; the switching frequency is equal to 1 kHz. Synchronization graph in the V11 − V12 − V1s and in the V21 − V22 − V2s space for (a) σ = 0.021 and (b) σ = 2.1.

of 25 kHz, the error E is 0.74 V for σ = 0.021 and 0.73 V for σ = 2.1; and for the switching frequency of 1 kHz, the error E is 2.41 V for σ = 0.021 and 1.8 V for σ = 2.1 Nevertheless, we note that, in case of strong coupling, the synchronization plots have a more prominent extension in the plane of the peer-topeer coupled oscillators than in the direction along the reference oscillator. In other words, the synchronization plot is characterized by a strip-like shape, indicating that the synchronization dynamics in the network is strongly dominated by the pinning control action. For the large coupling values, the synchronization plot presents a better defined geometry. Such plots may be potentially used to garner additional synchronization features and define different error measures. As a further evidence of the effectiveness of node-to-node pinning control, we report experiments on an alternative intermittent pinning control scheme. Within this strategy, in a time period, the peer-to-peer coupled oscillators are equally pinned at the same control gain and then left free for the same time duration. We note that the time average gain matrix of this binary pinning scheme is equal to the time average gain matrix of the node-to-node pinning control. Nevertheless, the synchronization performance of this binary strategy for a switching frequency of 25 kHz, see Fig. 11, is slightly lower than the performance of node-to-node pinning control in the same conditions, see Fig. 9. 5. Conclusions In this paper, we have presented a thorough experimental study of pinning controllability of coupled Chua’s chaotic circuits.

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a

463

Acknowledgements This research was supported by the National Science Foundation under Grant No. CMMI-0745753. The authors would like to thank Mr. Matteo Aureli for his help with the figures and his careful review of the manuscript and Ms. Nicole Abaid and Mr. Pietro De Lellis for the further review of the manuscript. The authors would also like to acknowledge the anonymous reviewers for their valuable feedback that has helped the authors to improve on the quality and presentation of the manuscript. References

b

Fig. 11. Experimental results in the case of binary pinning control at C = 42; the switching frequency is equal to 25 kHz. Synchronization graph in the V11 − V12 − V1s and in the V21 − V22 − V2s space for (a) σ = 0.021 and (b) σ = 2.1.

We have analyzed static and node-to-node pinning control through an in-house developed experimental platform comprised of three Chua’s circuits coupled through all their states. The parameters experimentally varied in this work are: the total pinning control gain, the number of pinned circuits in case of static pinning control, the connection strength between the oscillators, and the switching frequency for node-to-node pinning control. In addition, results on binary switching as a different intermittent pinning control strategy have been presented along with experimental results on synchronization of coupled Chua’s circuits without pinning control. The experimental contribution of this work is not only the considerable amount of data collected for synchronization and pinning control of Chua’s circuits but also the experimental platform per se. To the best of our knowledge, the design of networks of Chua’s circuits intermittently coupled through all their states has never been explored in the literature and available experimental results are limited to static partial state coupling. The proposed circuit synthesis can be potentially used to explore different synchronization and control strategies for Chua’s oscillators. Experimental results are supported by analytical findings and, vice versa, theoretical predictions on node-to-node pinning control are validated through experiments. We have analyzed global pinning controllability of an oscillator network through node-tonode pinning control. In addition, we have established sufficient conditions on the overall control gain and maximum switching period for global pinning controllability of an oscillator network in terms of the network topology, coupling strength among the oscillators, and properties of the individual oscillator dynamics.

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