Partial Synchronization of Nonidentical Chaotic Systems Via Adaptive Control, with Applications to Modeling Coupled Nonlinear Systems

International Journal of Bifurcation and Chaos, Vol. 12, No. 3 (2002) 561–570 c World Scientiﬁc Publishing Company

PARTIAL SYNCHRONIZATION OF NONIDENTICAL CHAOTIC SYSTEMS VIA ADAPTIVE CONTROL, WITH APPLICATIONS TO MODELING COUPLED NONLINEAR SYSTEMS

DAVID J. WAGG Department of Mechanical Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, UK [email protected]

Received February 23, 2001; Revised May 25, 2001

We consider the coupling of two nonidentical dynamical systems using an adaptive feedback linearization controller to achieve partial synchronization between the two systems. In addition we consider the case where an additional feedback signal exists between the two systems, which leads to bidirectional coupling. We demonstrate the stability of the adaptive controller, and use the example of coupling a Chua system with a Lorenz system, both exhibiting chaotic motion, as an example of the coupling technique. A feedback linearization controller is used to show the diﬀerence between unidirectional and bidirectional coupling. We observe that the adaptive controller converges to the feedback linearization controller in the steady state for the Chua– Lorenz example. Finally we comment on how this type of partial synchronization technique can be applied to modeling systems of coupled nonlinear subsystems. We show how such modeling can be achieved where the dynamics of one system is known only via experimental time series measurements.

Keywords: Synchronization; adaptive; feedback linearization.

1. Introduction partial synchronization using an adaptive synchronization technique. The problem of synchronizing two identical dynam- A case of particular interest is when an addi- ical systems has been studied by many authors, for tional feedback signal exists between the two sys- example, Ashwin et al. [1994], Kozlov and Shalfeev tems such that the coupling is bidirectional and [1996], Ashwin [1998], and Yang and Duan [1998], the two systems interact dynamically, giving rise following the work of Pecora and Carroll [1990]. to a complex dynamical behavior. This has appli- When this is achieved using adaptive control type cations to dynamic substructuring, where systems methods, the process is referred to as adaptive syn- are modeled by coupling a set of interacting sub- chronization [John & Amritkar, 1994; Boccaletti structures together [Ohayon et al., 1997; Wagg & et al., 1997; Fradkov & Markov, 1997; Dedieu & Stoten, 2001]. We demonstrate this concept us- Ogorzalek, 1997]. More recently, the concept of ing both a feedback linearization controller and an partial synchronization between two or more sim- adaptive feedback linearization controller. ilar chaotic systems has been studied [Hasler, 1998; In addition, we demonstrate how the adaptive Yanchuk et al., 2001]. In this paper we consider controller can be designed when coupling single and coupling two nonidentical dynamical systems via multiple variables from each of the nonidentical

561 562 D. J. Wagg nonlinear systems. We show that this type of adaptive controller is stable for such a coupled system. ff 11 12 This is demonstrated using the example of coupling a Lorenz system with a Chua system; for similar examples see [Di Benardo, 1996]. In this example, we observe that the steady state adaptive controller ff converges to the feedback linearization controller. 21 22 Finally we discuss applications to modeling dynamical systems composed of a set of coupled non-

linear dynamical systems. We discuss how partial Signals from the two synchronization can be used to achieve this type of ff systems coupled via 11 22 modeling. We also discuss how the concepts of syn- partial synchronization chronizing dynamical systems [Ashwin, 1998] can be used to monitor the performance of the controller Synchronization f f producing the coupling and hence the modeling pro- 12 21 cess itself, using the Lorenz Chua system as an Fig. 1. Schematic representation of a system formed by par- example. tial synchronization.

2. Partial Synchronization for To achieve partial synchronization, we need to Nonidentical Systems synchronize the dynamics of f12 and f21.Thuswe add a controller, to the coupled system, such that We consider two nonidentical dynamical systems, Eq.(2)canbewrittenas one with state variable x ∈Rp, and the second, q with state variable y ∈R, with governing equa- x˙n(t)=f11(xn,xs,t) tions of the general form x˙s(t)=f12(xn,xs,t)+g(u, t) (3) y˙ (t)=f (y ,y,t) x˙(t)=f1(x, t) , s 21 n s (1) y˙ (t)=f (y ,y,t) y˙(t)=f2(y, t). n 22 n s · In general, we consider that the dynamics of the two where u is the control signal, and g( ) represents systems are nonlinear and that there is no cross cou- the controller function. In this form, the dynam- pling between the two sets of state variables. We de- ics of f21 can be thought of as the reference model [Landau, 1979], which we want f +g(u, t) to repli- ﬁne a coordinate subset of x, x ∈Rn, and similarly 12 s cate and f represents the plant. y ∈Rn, which represent the coordinates which re- 12 s So, in the formulation of Eq. (3), a part of sys- quire synchronization to achieve coupling between tem 1 will be forced to behave like part of system 2. the two systems. So, we will consider the class of However, for bidirectional coupling, system 1 will systems for which Eq. (1) can be expressed as also have an inﬂuence on the behavior of systems 2. In this case, an additional feedback signal between x˙ n(t)=f11(xn,xs,t) f1 and f2 can be used to represent the coupling. x˙s(t)=f12(xn,xs,t) (2) We represent it by adding a coupling function to y˙s(t)=f21(yn,ys,t) f21, such that

y˙n(t)=f22(yn,ys,t) x˙ n(t)=f11(xn,xs,t) where xn = {xi ∈ x : xi ∈/ xs} and xi denotes the x˙s(t)=f12(xn,xs,t)+g(u, t) { ∈ ∈ (4) ith element of x, and likewise yn = yi y : yi / y˙s(t)=f21(yn,ys,t)+c(xn,xs,t) ys}.Thenifxs→ysas t →∞we say that the sys- y˙n(t)=f22(yn,ys,t) tem is partially synchronized. When such partial synchronization occurs a coupled system is formed In the case where f1 is a physical system and f2 which is shown schematically in Fig. 1. The case is an analytical model the dynamics of f11 can be where xs = x and ys = y is the standard synchro- assumed to be unknown, and c(xn,xs,t)would nization problem [Pecora & Carroll, 1990]. typically be a recorded time series from f11.The Partial Synchronization of Nonidentical Systems 563 functions f22 and f21 must be known explicitly, so therefore e is scalar in this case. Then we can write that they can be computed numerically, and the the error dynamics as structure of f12 must be known. Knowledge of spe- cific parameter values is not required, as the adap- e˙(t)=−λe + L − g(u, t) , (8) tive controller can be applied without this information. If c = 0 the coupling between the two systems where L =∆f+λe+c,andλ>0. This type of for- (via partial synchronization) is effectively unidirec- mulation is possible with a wide variety of both lin- tional, whereas if c =6 0 the coupling is bidirectional; ear and nonlinear systems [Di Benardo, 1996], and examples will be discussed in Sec. 3.1, 4.1 and 5.1. this requirement is therefore not overly restrictive. We note also that the analysis in this section is for It is clear from Eq. (8) that (L − g1(u, t)) → 0, and autonomous systems, however it is possible to ap- λ>0 will stabilize the required equilibrium, e =0. ply this analysis to some nonautonomous systems Therefore L is the feedback linearization controller [Wagg & Stoten, 2001] which we briefly discuss in for the system [Di Benardo, 1996]. Sec. 5.1. For the class of systems considered in this work, L can be expressed as L = k∗ξ, where k∗ represents 2.1. Controller design a set of (constant) parameters, and ξ the vector of coupling variables. For such systems we use an To design a controller for the system we first reduce adaptive controller which has essentially the same Eq. (4) to the form form as L, g(u, t)=u=k(t)ξ, where k(t)isthe adaptive gain. Using these definitions enables us to x˙ s(t)=f12(d1(t),xs,t)+g(u, t) (5) express Eq. (8) as y˙s(t)=f21(d2(t),ys,t)+c(t) − where the dynamics of xn and yn are now rep- e˙(t)= λe + φ(t)ξ(t) , (9) resented by the functions d and d respectively, 1 2 ∗ which we assume act as disturbances. Then we can where φ(t)=k −k(t) is the parameter error. We formulate the error dynamics for the system such then need to find an expression for k(t)whichstabi- that lizes the system such that φ(t) → 0ast→∞.This we can achieve by choosing a Lyapunov function of − e˙(t)=f21(d2(t),ys,t) f12(d1(t),xs,t) the form − e2 φφT +c(t) g(u, t)(6)V (t)= + , (10) 2 2γ where the error, e = ys − xs.Thiscanthenbe expressed as where γ is the controller gain. Then the derivative of V with respect to time is e˙(t)=∆f(t)+c(t)−g(u, t)(7) − 1 where ∆f(t)=f21 f12. For effective performance V˙ (t)=e(−λe + φ(t)ξ(t)) + φφ˙T , (11) of the controller, we require that the equilibrium, γ e = 0 is stable. From Eq. (7) we see that the con- such that choosing φ˙ T = −γeξ,resultsinV˙ =−λe2 troller has to compensate for the difference between which implies that the controller is Lyapunov stable f and f ,∆f(t) and the additional feedback sig- 12 21 for λ>0. As k∗ is constant, φ˙T = −k˙ T = −γeξ, nal c(t). such that the adaptive gain becomes In this formulation there are two additional dis- Z t turbances, d1, d2. These functions are not external kT = γ eξdt (12) disturbances in the ordinary sense, but signals from t=t some other part of the coupled system. As a result, [Sastry & Bodson, 1989]. Thus k(t) → k∗ as φ → 0 the controller must compensate for the influence of and e → 0. Note: providing φ → 0, the final adap- these additional signals. tive gain values correspond to the unknown set of system parameters k∗. In general k(t) → k∗ pro- 3. Single Variable Coupling vided the adaptive controller has a persistently ex- citing signal (see for example [Sastry, 1999]. From Let us first consider the case where only a single qualitative examination of our numerical simula- coordinate of f1 and f2 is to be synchronized, and tions in this paper this is nearly always the case. 564 D. J. Wagg

Finally, there is an extra eﬀect on the stabil- The reference model is f21, and therefore the control ity of the partially synchronized systems due to the signal must be applied to f12 such that signals d1, d2 and c. For Lyapunov stability these signals must remain bounded. As they are depen- f12 = −δx2 + u. (17) dent on state variables, they can only become un- bounded if the system becomes unstable. There- Thus we are coupling the two systems by control- fore providing the system reaches a stable state ling f12 to follow f21. Sointhiscasewecanthink with d1, d2 and c bounded the system will remain of the Lorenz system as the master or forcing sys- stable. tem and the Chua as a slaved system, such that the coupling is unidirectional. We can introduce bidirectional coupling by adding a coupling function, 3.1. Example of coupling Chua c(t), to the Lorenz system, such that the reference and Lorenz systems f21 can be written as We now consider an example of coupling a Chua − − system with a Lorenz system. In this example (for f21 = σ(y1 y2)+c(t) (18) an example of adaptive control using similar sys- where c is set to zero in the unidirectional case. tems see [Stoten & Di Bernardo, 1996]), we use a Chua system deﬁned as 3.1.1. Feedback linearization controller

x˙ 1 = α1(x2 − x1)+α2x1 To demonstrate the diﬀerence between unidirec- −α3(|x1 +1|−|x1−1|) tional, and bidirectional coupling, we ﬁrst use a (13) x˙ =x −x +x controller based on feedback linearization (see for 2 1 2 3 example [Di Benardo, 1996]). To design such a con- − x˙3 = δx2 troller we need to know ∆f explicitly, which in this example is and a Lorenz system

∆f = f21 −f12 = −σy1 +σy2 +c+δx2 −u. (19) y˙1 = −σ(y1 − y2) − − y˙2 = ry1 − y2 − y1y3 (14) The error variable e = ys xs = y1 x3,sothat Eq. (19) can be expressed as y˙3 = y1y2 − by3 ∆f = f − f = −λe + σ(y − x )+c+δx − u, To ensure that both systems are chaotic, we se- 21 12 2 3 2 (20) lect the parameter values: α1 = 10, α2 =0.68, where λ = σ in this case. So, in this example as α3 =0.59, δ = −14.87, σ = 10, r =28andb=8/3. Initial conditions for the system were selected as σ>0, we can write x1(0) = 1.1, x2(0) = 1.0, x3(0) = 7.0, y1(0) = −1.1, ∆f = f21 − f12 = −σe + L − u, (21) y2(0) = −1.0andy1(0) = −5.0. This choice of parameters and initial conditions is arbitrary: control can be applied for any parameter values. which is equivalent to the right-hand side of Eq. (9). Now let us consider the case when we wish to Thus for feedback linearization we set u = L = σ(y2 − x3)+c+δx2. couple (i.e. synchronize) x3 and y1. Thus, we deﬁne T A numerical simulation of the unidirectional xn =[x1,x2] ,xs =x3,ys =y1 and yn =[y2,y3]. Then (master-slave) system, c = 0, is shown in Fig. 2. ! Here the response of x3 from the Chua system is shown as a solid line, while the response y of the α1(x2 −x1)+α2x1−α3(|x1+1|−|x1−1|) 1 f11 = Lorenz system is shown as a dashed line. The con- x −x +x 1 2 3 troller is initially turned oﬀ u = 0, and the responses (15) of the two systems are unsynchronized. Then at t = 12 the feedback linearization controller is turned and ! on. The two selected coordinates from the systems ry − y − y y 1 2 1 3 quickly synchronize, with the Chua x3 coordinate f22 = . (16) y1y2 − by3 slaved to the Lorenz y1 coordinate. Partial Synchronization of Nonidentical Systems 565

20 behavior is not the same as the master slave exam- 15 ple. This can be seen in Fig. 3 from the deviation of the synchronized system from the Lorenz system 10 after t = 12. 5 3.1.2. Adaptive feedback linearization 0 control Amplitude -5 Now we consider the same synchronization problem -10 using an adaptive controller. To achieve this we have to express L, the feedback linearization con- -15 troller, as a product of an unknown parameter vec- ∗ -20 tor, k and a coupling variable vector, ξ.Thus 0 2 4 6 8 10 12 14 16 18 20 Time L=σ(y2−x3)+δx2 + c Fig. 2. Output from master slave system. Solid line x,   − − dashed line y. Initial conditions x(0) = 1.1, x(0) = 1.0,  y2 − x3  x (0)=1.1, y (0)=1.1, y (0) = 1.0andy(0)=7.0. Con- = {σ, δ, β} x (22) trol started at t =12sec.  2  c

20 so that the coupling variable vector is ξ = {(y2−x3), T ∗ x2,c} and the parameter vector is k = {σ, δ, β = 15 1},andβis a dummy parameter variable. 10 The response of the system is shown in Fig. 4,

5 where again the controller was initiated at time t = 12. The evolution of the adaptive gains k = 0 T {k1,k2,k3} which were all initiated at zero, is

Amplitude -5 shown in Fig. 5 where a controller gain of γ = 100 in Eq. (12) was found sufficient to achieve fast -10 adaption. We can see from Fig. 5 that during -15 the first 8 seconds of adaption gains vary signifi-

-20 cantly. However, at time t = 1000, the adaptive 0 2 4 6 8 10 12 14 16 18 20 gains are approximately constant with values close Time T to k = {k1 ≈ 10, k2 ≈ 15, k3 ≈ 1} .Thuswe Fig. 3. Output from system with bidirectional coupling c = x. Solid line x, dashed line y, chained dotted line y for the unidirectional c = 0 case: Note before t =12the 20 dashed and chain dotted lines are identical. Initial condi- 15 tions x(0) = −1.1, x(0) = −1.0, x(0) = 1.1, y(0)=1.1, y(0) = 1.0andy(0) = 7.0. Control started at t =12sec. 10

In Fig. 3 we show a simulation for the bidirec- 0 tional case, when c = x3. Here the response of x3 Amplitude from the Chua system is shown as a solid line, while -5 the response y1 of the Lorenz system is shown as a -10 dashed line. In addition we have plotted the output from the Lorenz system (chain dotted line) for -15 the c = 0 case as a comparison. Again the con- -20 trol is turned on at t = 12, and the two systems 0 2 4 6 8 10 12 14 16 18 20 Time quickly become synchronized. This time however Fig. 4. Output from adaptive control system. Solid line x , the dynamics are not slaved to the Lorenz system. dashed line y. Initial conditions x(0) = −1.1, x(0) = −1.0, The bidirectional coupling produces interaction be- x(0)=1.1, y(0)=1.1, y(0) = 1.0andy(0)=7.0. Con- tween the two systems, such that the dynamical trol gain γ = 100. Control started at t =12sec. 566 D. J. Wagg

100 ˙ T − ˙ suchP that choosing φi = γieiξ,resultsinV = N− 2 80 1 λiei which implies that the controller is T T Lyapunov stable. Therefore φ˙ = −k˙ = −γieiξ. 60 i

40 4.1. Example of multivariable

20 coupling Gain

0 We now discuss an example of coupling more than a single variable again using the Chua and Lorenz -20 systems as an example. In this example, two vari- -40 ables from each system are coupled simultaneously. To demonstrate this, we select x2 and y1 as the -60 0 2 4 6 8 10 12 14 16 18 20 first pair of variables, and x3 and y2 as the sec- Time ond pair of variables for coupling. This means that Fig. 5. Output from adaptive control system. Solid line x, we now have two error variables, e1 = y1 − x2 and dashed line y. Initial conditions x(0) = −1.1, x(0) = −1.0, e2 = y2 − x3. Unidirectional coupling only is con- x(0)=1.1, y(0)=1.1, y(0) = 1.0andy(0)=7.0. Con- trol gain γ = 100. Control started at t =12sec. sidered, such that c = 0. In this case the coupling functions are ! x1 − x2 + x3 + u1 see that k1 → σ, k2 → δ and k3 → β.Thusvia f = (26) 12 − the relation u = k(t)ξ(t) we see that the steady δx2 + u2 state adaptive controller is the same as the feed- and ! back linearization controller, and that in effect we −σ(y1 − y2) have identified the system parameters. f21 = . (27) ry1 − y2 − y1y3 Therefore 4. Multivariable Coupling ∆f = f21 − f12 For multivariable coupling between nonlinear sys- ! −σ(y − y ) − x + x − x − u tems controller design is more difficult. Here, we = 1 2 1 2 3 1 , (28) take the approach of analyzing the stability of ry1 − y2 − y1y3 + δx2 − u2 the system in a partially decentralized form. This means that the linear error coordinates are decou- whichcanbeexpressedas pled, but nonlinear coupling exists. ∆f=f21 − f12 Thus for a system of N error equations, the ! ith equation can be written in a similar form to −σe + σy +(1−σ)x −x −x −u = 1 2 2 3 1 1 , Eq. (9) as −e2 +ry1 + δx2 − x3 − y1y3 − u2

e˙i(t)=−λiei+φi(t)ξ(t), (23) (29) such that we can write where φi(t)isan(1×m) parameter error vector, × and ξ is the (m 1) coordinate coupling vector for e˙1(t)=−σe1 + L1 − u1 , all N error states. We choose a Lyapunov function (30) e˙2(t)=−e2+L2−u2 of the form ! where the feedback linearization controllers are XN e2 φ φT V (t)= i + i i , (24) 2 2γ L1 = σy2 +(1−σ)x2 −x3 −x1 i=1 i (31) L2 =ry1 + δx2 − x3 − y1y3 where γi is the controller gain. Then the derivative of V with respect to time is For adaptive feedback linearization we write each L = k∗ξ such that in this case XN i i 1 T V˙ (t)= e(−λe +φ(t)ξ(t)) + φ φ˙ , ∗ i i i i γ i i k =[0,σ,(1 − σ), −1, −1, 0] 1 i 1 (32) ∗ − − (25) k2 =[r, 0,δ, 1,0, 1] Partial Synchronization of Nonidentical Systems 567

T ∗ and ξ =[y1,y2,x2,x3,x1,y1y2] . Then finally converged to k2. This behavior occurs because of we can express Eq. (30) in the required format of our choice of λi values in this example; Eq. (30). Eq. (23) by substituting ui = ki(t)ξ, giving For the x2 , y 1 synchronization λ1 = σ = 10, but for the x , y synchronization λ = 1. Considering the − 3 2 1 e˙1(t)= σe1 + φ1(t)ξ(t), − − (33) convergence when L1 u1 =0andL2 u2 =0, − e˙2(t)= e2+φ2(t)ξ(t) e1=exp(−10t) while e2 =exp(−t). Therefore the error convergence of e will be greater than that Then for this system stabilizing controllers can 1 T of e2. beR applied using the gain vectors given by ki = γ t e ξdt. The results of simulating this example i t=t i are shown in Fig. 6. As with the previous examples, 5. Applications to Modeling the control is started at time t = 12. From Fig. 6(a) Coupled Dynamical Systems we see that x2 becomes synchronized with y1 very quickly after the control starts. However, the x3, y2 Many real life dynamical systems are composed of synchronization takes significantly longer; approxi- two or more coupled systems giving rise to highly mately 2 seconds. We also find that after 1000 sec- complex dynamics. Partial synchronization tech- ≈ ∗ onds k1 k1, but that k2 has not completely niques can be applied to modeling such systems in two main ways:

20 1. To model systems composed of a set of coupled 15 nonlinear subsystems, where the structure of the

10 individual component systems is known, but the nature of the coupling is unknown. 5 2. To model systems composed of a set of coupled

0 nonlinear subsystems, where information from

Amplitude one (or more) of the subsystems is known only -5 in the form of a recorded time series. -10

-15 The first approach can be used to synchronize two variables from the subsystems to effect cou- -20 0 2 4 6 8 10 12 14 16 18 20 pling, without having explicit knowledge of the form Time ofthecouplingitself.Thesecondmethodhaspo- (a) tential uses for systems where time series data is taken from an experimental source. For example, in techniques which have a numerical and experi- 25 mental component to the modeling [Oomens et al., 20 1993; Donea et al., 1996; Wagg & Stoten, 2001] 15 these two modeling methods can be approached 10 using the coupling techniques described in Sec. 2. 5 0 5.1. Example: modeling a system Amplitude -5 of two coupled nonlinear -10 dynamical systems -15 Consider the problem of modeling the dynamics of -20 a complex dynamical system governed by the state -25 0 2 4 6 8 10 12 14 16 18 20 equation Time z˙(t)=f(z, t), (34) (b) where we have only partial knowledge of the form of Fig. 6. Multivariable coupling (synchronization) for Chua– f(z, t). We will consider the problem where f(z, t) Lorenz system using feedback linearization. (a) x solid and y dashed, (b) x solid and y dashed. Control started at is composed of two parts, one for which the dynam- t =12sec. ics is known explicitly, f2, and the other, f1 where 568 D. J. Wagg the dynamics can be divided into; a part where 5.1.1. Numerical–experimental example the structure is known, f12, and a part where the dynamics are known only via time series measure- Wagg and Stoten [2001] considered a numerical– experimental example where f11 and f12 are phys- ments f11. This is the situation in some numerical– experimental applications, where a physical system ical systems but f12 is (approximately) linear. A is acted upon by some experimental apparatus and force signal, F (t), between f11 and f12 is recorded this is coupled with a numerical model [Wagg & experimentally which represents the coupling be- Stoten, 2001]. So in this case the coordinates of tween the two functions. For this system Eq. (36) can be written in the form the apparatus would correspond to xs,andthedy- namics of xn would be known only implicitly from experimental measurements of the physical system. x˙ n(t)=f11 To create a model of f(z, t) the coordinates xs x˙ s(t)=Axs + Bu → (37) and ys need to be synchronized such that e 0. y˙s(t)=Amys+Bmyn+CmF(t) Thus if partial synchronization can be achieved then y˙ (t)=A y +B y +C r(t) Eq. (34) can be written (using Eq. (2)) as n z n z s z         where A, B, A , B , C , A , B and C are x˙n(t) f11(xn,xs,t) m m m z z z constant matrices and f11 is an unknown nonlinear z˙(t)= x˙s(t) = f12(xn,xs,t) =f(z, t).     function. In this example only part of the system y˙ (t) f22(yn,ys,t) n is nonlinear, but the development of the combined (35) model for the system uses a similar approach to that As before, this is achieved by using an adaptive described here for nonlinear systems. We note also control algorithm to ensure that f12 tracks f21 that this system has bidirectional coupling from the → i.e. f12 f21. Thus the coupled systems form a application of F (t) and is nonautonomous via the single combined model of the overall system. In this forcing signal r(t). Further details of this can be process we will effectively reconstruct the dynam- found in [Wagg & Stoten, 2001]. ics of the experimental system [Broomhead & King, 1986; Maybhate & Amritkar, 1999], while simulating the dynamics known explicitly, and thus recon- 5.2. How effective is this struct the overall dynamics of the system f(z, t). modeling process? Let us consider the case where the dynamics of In order to measure the degree of synchronization x , represented by f are unknown in a explicit n 11 between the coordinates x and y , we monitor the form but are known implicitly from a set of ex- s s error vector e = y − x . For effective modeling we perimental measurements in the form of time se- s s require that synchronization occurs within a certain ries v(t)andw(t) such that x = h (v(t)) and n 1 time limitation e → ε as t → t , where ε is small. x˙ = h (w(t)). Here h (·) are correlation functions s n 2 j The effectiveness of the synchronization process can which provide a relationship between the experi- be viewed geometrically by considering the phase mental measurements and state variables. In this space for the coupled system E = {(x, y) ∈Rp×q}. caseEq.(3),canbewrittenas Then Σ = {(x, y) ∈Rp×q :e=0}represents the synchronized subspace [Ashwin, 1998] for the cou- x˙n(t)=h2(w(t)) pled system. The dynamics which are restricted to x˙ s(t)=f12(h1(v(t)),xs,t)+g(u, t) (36) the manifold Σ correspond to that of the coupled y˙s(t)=f21(yn,ys,t) system, Eq. (34). Furthermore, out of subspace dy-

y˙n(t)=f22(yn,ys,t) namics correspond to failure in the synchronization (control) process. Thus we can use the synchroniza- This set of equations can be used to form a cou- tion subspace to monitor the performance of the pled model for the overall system, by substituting controller, and hence the effectiveness or accuracy h1 = d1 and yn = d2, we obtain Eq. (5), and for v(t), of the modeling process. w(t) bounded, the stability proof follows. In addi- If we define the phase space of the overall sys- k tion, when f12 and f21 are synchronized, Eq. (36) tem we are trying to model as G = {z ∈R },then can be reduced to the form of Eq. (35) to provide a the combined model is a close approximation of the combined system model. overall system if dim Σ ≈ dim G. In other words Partial Synchronization of Nonidentical Systems 569 the combined system, f1 and f2, has higher dimen- used from time t = 0, and the figure shows trajec- sional dynamics than the modeled system, f(z, t), tories computed from t =40tot= 50. Lorenz-like but by synchronizing the required set of coordinates dynamics can be observed, however these dynamics we reduce the dynamics to the subspace Σ which are in fact restricted to the synchronization sub- is an approximation of the overall dynamics in G. space, which can be seen from viewing the x3, y1 Thus we can qualitatively identify the dynamics of plane; Fig. 8. Qualitatively, we observe no out of the overall system by examining the dynamics in subspace dynamics, indicating a high level of ac- thehypersurfaceΣ. curacy in the coupling and hence modeling process, which can be expected from feedback lineariza- 5.2.1. Chua–Lorenz example tion control. Similar results can be obtained using adaptive feedback linearization, however in this case This can be demonstrated using the example from some out of subspace dynamics will occur during { ∈R3} Sec. 3.1. Let E = (x3,y1,y3) be a subset the transient adaption phase. of the complete phase space, which we can use as a visualization aid. The evolution of the feedback coupled system in this space is shown in Fig. 7. 6. Conclusions In this example feedback linearization control was We have considered how nonidentical nonlinear dynamical systems can be coupled using partial synchronization with the inclusion of additional feed- y1 20 back coupling. For applications where two different 15 dynamical systems require coupling, a partial syn- 10 5 chronization method can be used where one part of 0 the system is included using only a recorded time -5 -10 series. We have demonstrated how both unidirec- -15 tional and bidirectional coupling can be simulated 15 in such a modeling process using a feedback lin- 10 5 earization controller. We have also demonstrated 15 0 20 x3 25 -5 using the example of a Chua system coupled with a 30 -10 y3 35 40 Lorenz system, how an adaptive feedback lineariza- Fig. 7. Output from feedback coupled system plotted in the tion controller can be used to effect such coupling. space E.Datafromt=40tot= 50 shown. Initial condi- The use of an adaptive controller is significant, in tions x(0) = −1.1, x(0) = −1.0, x(0) = 1.1, y(0)=1.1, that it can be used to couple systems without ex- y(0) = 1.0andy(0) = 7.0. plicit knowledge of the plant parameters, although a knowledge of the structure of the plant is re-

20 quired. In the steady state, we observed that the adaptive controller converged to the exact formula- 15 tion of the feedback linearization controller. Finally

10 we have discussed how the coupling techniques can be applied to modeling numerical–experimental and 5 other coupled systems.

y1 0

-5 References Ashwin, P., Buescu, J. & Stewart, I. [1994] “Bubbling of -10 attractors and synchronisation of chaotic oscillators,”

-15 Phys. Lett. A193, 126–139. Ashwin, P. [1998] “Non-linear dynamics, loss of syn- -20 chronization and symmetry breaking,” Proc. Inst. -20 -15 -10 -5 0 5 10 15 20 x3 Mech. Engin., Part G: J. Aerosp. Engin. 212(3), Fig. 8. Synchronization subspace from feedback coupled 183–187. system. Data from t =40tot= 50 shown. Initial con- Boccaletti, S., Farini, A. & Arecchi, F. T. [1997] “Adap- ditions x(0) = −1.1, x(0) = −1.0, x(0) = 1.1, y(0) = 1.1, tive synchronization of chaos for secure communica- y(0) = 1.0andy(0) = 7.0. tion,” Phys. Rev. E55(5), 4979–4981. 570 D. J. Wagg

Broomhead, D. S. & King, G. P. [1986] “Extracting qual- Ohayon, R., Sampaio, R. & Soize, C. [1997] “Dynamic itative dynamics from experimental data,” Physica substructuring of damped structures using singular D20(2 & 3), 217–236. value decomposition,” Trans. ASME 64, 292–298. Dedieu, H. & Ogorzalek, M. J. [1997] “Identifiability and Oomens, C. W. J., Ratingen, M. R., Janssen, J. D., identification of chaotic systems based on adaptive Kok, J. J. & Hendriks, M. A. N. [1993] “A numerical- synchronization,” IEE Trans. Circuits Syst. 44(10), experimental method for a mechanical characteriza- 948–962. tion of biological materials,” J. Biomech. 26(4 & 5), Di Benardo, M. [1996] “An adaptive approach to the con- 617–621. trol and synchronization of continuous-time chaotic Pecora, L. M. & Carroll, T. L. [1990] “Synchronization systems,” Int. J. Bifurcation and Chaos 6(3), 557–568. in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824. Donea, J., Magonette, P., Negro, P., Pegon, P., Pinto, Sastry, S. & Bodson, M. [1989] Adaptive Control: Stabil- A. & Verzeletti, G. [1996] “Pseudodynamic capabili- ity, Convergence and Robustness (Prentice-Hall, NJ). ties of the ELSA laboratory for earthquake testing of Sastry, S. [1999] Nonlinear Systems: Analysis, Stability large structures,” Earthq. Spectra 12(1), 163–180. and Control (Springer-Verlag, NY). Fradkov, A. L. & Markov, A. Y. [1997] “Adaptive syn- Stoten, D. P. & Di Bernardo, M. [1996] “Application of chronization of chaotic systems based on speed gradi- the minimal control synthesis algorithm to the con- ent method and passification,” IEEE Trans. Circuits trol and synchronization of chaotic systems,” Int. J. Syst. 44(10), 905–912. Contr. 65(6), 925–938. Hasler, M. [1998] “Simple example of partial synchro- Wagg, D. J. & Stoten, D. P. [2001] “Substructuring of nization of chaotic systems,” Phys. Rev. E58(5), dynamical systems via the adaptive minimal control 6843–6846. synthesis algorithm,” Earthq. Engin. Struct. Dyn. 30, John, J. K. & Amritkar, R. E. [1994] “Synchronization 865–877. of unstable orbits using adaptive control,” Phys. Rev. Yanchuk, S., Maistrenko, Y. & Mosekilde, E. [2001] “Par- E49(6), 4843–4848. tial synchronization and clustering in a system of dif- Kozlov, A. K. & Shalfeev, V. D. [1996] “Exact synchro- fusively coupled chaotic oscillators,” Math. Comput. nization of mismatched chaotic systems,” Int. J. Bi- Simulation 54, 491–508. furcation and Chaos 6(3), 569–580. Yang, S. S. & Duan, C. K. [1998] “Generalized synchro- Landau, Y. D. [1979] Adaptive Control: The Model Ref- nization in chaotic systems,” Chaos Solit. Fract. 9(10), erence Approach (Marcel Dekker, NY). 1703–1707. Maybhate, A. & Amritkar, R. E. [1999] “Use of synchronization and adaptive control in parameter estimation from a time series,” Phys. Rev. E59(1), 284–293.