feedback synchronization using pole-placement control

International Journal of Bifurcation and Chaos, Vol. 10, No. 11 (2000) 2611–2617 c World Scientific Publishing Company

FEEDBACK SYNCHRONIZATION USING POLE-PLACEMENT CONTROL ROBERTO TONELLI for Plasma Research, University of Maryland, College Park, MD 20742, USA INFM-Physics Department, University of Cagliari, 09100 Cagliari, Italy

∗ Institute

YING-CHENG LAI Department of Mathematics, Department of Electrical Engineering, Center for Systems Science and Engineering Research, Arizona State University, Tempe, AZ 85287, USA CELSO GREBOGI∗ Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA Received August 23, 1999; Revised November 15, 1999 Synchronization in chaotic systems has become an active area of research since the pioneering work of Pecora and Carroll. Most existing works, however, rely on a passive approach: A coupling between chaotic systems is necessary for their mutual synchronization. We describe here a feedback approach for synchronizing chaotic systems that is applicable in high dimensions. We show how two chaotic systems can be synchronized by applying small feedback perturbations to one of them. We detail our strategy to design the control based on the pole-placement method, and give numerical examples.

1. Introduction Chaotic systems are characterized by a sensitive dependence on initial conditions: two trajectories starting from slightly different initial conditions diverge exponentially in time. As such, synchronization between even identical chaotic systems becomes a highly intriguing problem. It has been recognized, however, since 1983 that synchronization can indeed occur in chaotic systems [Fujisaka & Yamada, 1983; Afraimovich et al., 1986; Pecora & Carroll, 1990; Chua et al., 1993; Heagy et al., 1994; Kocarev & Parlitz, 1996; Pecora et al., 1997]. It was Pecora and Carroll [1990] who first pointed out that synchronous chaos could be utilized for nonlinear digital communication [Parlitz et al., 1992; Cuomo & Oppenheim, 1993; Cuomo et al., 1993; Short, 1994, 1996]. Since then, synchronization in

chaotic systems has received a tremendous amount of attention and it remains to be one of the most active research areas in chaotic dynamics [Ditto & Showalter, 1997]. Most existing approaches to synchronizing chaotic systems are passive in the sense that some predesigned coupling scheme is utilized to warrant synchronization. After the coupled system is switched on so that synchronization is achieved, no external control or perturbation is necessary to keep the coupled subsystems synchronized. This approach can guarantee robust synchronization when a proper coupling scheme is employed, and it has been quite successful indeed [Pecora & Carroll, 1990; Heagy et al., 1994; Ditto & Showalter, 1997]. However, in some situations, it is difficult to devise a coupling scheme to achieve synchronization. For concreteness, say we have a chaotic system

2611

2612 R. Tonelli et al.

described by the following dynamical equations dx = F(x) , dt

(1)

where x ∈ RN and F = [F1 (x), F2 (x), . . . , FN (x)]. In the original synchronization scheme proposed and investigated by Pecora and Carroll [1990], a subset of dynamical variables, say [x1 , . . . , xk ] (k < N ), is utilized to drive a replica of the complementary subsystem of Eq. (1) dyi = Fi (x1 , . . . , xk , yk+1 , . . . , yN ), i = k+1, . . . , N, dt (2) so that synchronization between (xk+1 , . . . , xN ) and (yk+1, . . . , yN ) may be achieved lim |xi (t) − yi (t)| = 0 , i = k + 1, . . . , N .

t→∞

(3)

A necessary condition for this type of synchronization is that the Lyapunov exponents of the subsystem Eq. (2) are all negative [Pecora & Carroll, 1990]. Given a chaotic system, depending on the division of the subsystems, synchronization may or may not be achieved. The purpose of this paper is thus to investigate a feedback approach for synchronizing chaotic systems. Given two nearly identical chaotic systems, we conceive that small, time-dependent feedback control can be applied to one of them to achieve synchronization [Lai & Grebogi, 1993]. This is similar to the Ott–Grebogi–Yorke (OGY) idea of controlling chaos [Ott et al., 1990]. In particular, to synchronize two chaotic systems A and B, we imagine that some parameter of one system (say B) is externally adjustable. Assume that some state variables of both systems A and B can be measured. Based on this measurement and on our knowledge about the system (we can, for example, observe and learn the system first), when it is determined that the state variables of A and B are close, we calculate a small parameter perturbation and apply it to system B. The two systems can then be synchronized, although their trajectories are still chaotic, by applying small feedback perturbations. Under the influence of external noise, there is a finite probability that the two already synchronized trajectories may lose synchronization. However, due to ergodicity of chaotic trajectories, after a finite amount of transient time, the trajectories of A and B will get close and be synchronized again. A method based on the OGY strategy of controlling chaos [Ott et al., 1990] to design the

feedback control for synchronization was studied in [Lai & Grebogi, 1993]. The aim of this paper is to demonstrate that a well-known technique in control engineering, the pole-placement method, can be adapted to achieve robust synchronization using feedback control. Our motivation is that the method studied by Lai and Grebogi [1993] is applicable only to chaotic systems described by two-dimensional invertible maps, while the poleplacement technique is in principle applicable to higher-dimensional systems. In fact, it was demonstrated by Romeiras et al. [1992] that the poleplacement method can be effective to stabilize unstable periodic orbits in high-dimensional chaotic systems. It has been known that high-dimensional chaotic systems possess nonhyperbolic properties such as unstable dimension variability [Dawson et al., 1994; Kostelich et al., 1997; Sauer et al., 1997; Lai & Grebogi, 1999; Lai, 1999; Lai et al., 1999] that are not present in two-dimensional invertible maps. It is thus important to study methods that can be utilized to synchronize chaos in high dimensions by feedback control. The methodology outlined in this paper represents a primary step forward in this direction. The paper is organized as follows. In Sec. 2, we outline the pole-placement method and demonstrate how it can be adapted to the problem of feedback synchronization. In Sec. 3, we apply the method to two chaotic maps: the H´ enon map [Hénon, 1976] and the kicked double rotor map [Grebogi et al., 1987; Romeiras et al., 1992]. A conclusion is presented in Sec. 4.

2. Pole-Placement Technique 2.1. Review of the pole-placement method for stabilizing unstable periodic orbits Consider dynamical systems described by the following discrete-time map xn+1 = F(xn , pn ) ,

(4)

where xn ∈ RN , F is a smooth vector function, and pn is an externally adjustable parameter. Since we conceive using only small perturbations, we vary p in a small interval around some nominal value p |pn − p| ≤ δ .

(5)

Feedback Synchronization Using Pole-Placement Control 2613

If F(xn ) possesses a chaotic attractor, then a typical trajectory can be stabilized in the vicinity of any desirable unstable periodic orbit embedded in the attractor, for almost all initial conditions in the basin of attraction [Ott et al., 1990; Romeiras et al., 1992]. In particular, consider a small neighborhood of size proportional and comparable to δ of the desired periodic orbit. In this neighborhood, the dynamics is approximately linear. Since linear systems are stabilizable if the perturbations obey a standard controllability condition, the chosen unstable periodic orbit can be stabilized by feedback control, say, perturbations to the parameter p in the range: (p − δ, p + δ). The ergodic nature of the chaotic dynamics guarantees that the state trajectory enters the chosen neighborhood in its course of time evolution. The stabilizing feedback control law guarantees that the trajectory be kept in the neighborhood of the desired orbit. The formulation of the pole-placement technique for stabilizing a periodic orbit is as follows. Consider a fixed point (period-1) x∗ (p) of the map F. For pn close to p, the map can be linearized in the neighborhood of x∗ (p) xn+1 − x∗ (p) = A[xn − x∗ (p)] + B(pn − p) , (6) where A is the N × N Jacobian matrix and B is an N -dimensional vector A = Dx F(x, p) , B = Dp F(x, p) .

(7)

Note that in Eq. (7), the partial derivatives are evaluated at x = x∗ and p = p. In a small neighborhood of x∗ (p), the parameter perturbation to be applied to drive the trajectory point xn to x∗ (p) is proportional to the distance between xn and x∗ (p). Thus, we can write pn − p = −KT [xn − x∗ (p)]

(8)

where KT is a 1 × N matrix to be determined in order to stabilize the fixed point x∗ (p). Substituting Eq. (8) into Eq. (6), we obtain xn+1 − x∗ (p) = (A − BKT )[xn − x∗ (p)] .

(9)

If the matrix KT is chosen such that the new matrix A−BKT is asymptotically stable, the desirable fixed point x∗ (p) can be stabilized. The pole-placement method solves the following problem: determine KT so that the eigenvalues

of the matrix A − BKT have specific values known as regulator poles [Ogata, 1990]. The pole placement problem has a unique solution if and only if the N × N matrix C = (B, AB, A2 B, . . . , An−1 B)

(10)

is of rank N , where C is the controllability matrix. The solution to the problem is then given by [Ogata, 1990] KT = (αN − aN , . . . , α1 − a1 )T−1 ,

(11)

where T = CW and aN −1   aN −2   W =  ...    a1 1 

aN −2 aN −3 .. . 1 0

··· ··· ··· ···

a1 1 .. . 0 0

1 0   ..  .   0 0 

(12)

Here {a1 , . . . , aN } and {α1 , . . . , αN } are the coefficients of the characteristic polynomials of A and (A − BKT ), respectively ksI − Ak = sN + a1 sN −1 + · · · + aN ,

ksI − (A − BKT )k = sN + α1 sN −1 + · · · + αN . (13) It was demonstrated by Romeiras et al. [1992] that the pole-placement method is equivalent to the OGY-method [Ott et al., 1990] when the eigenvalues in the unstable directions are set to zero and those in other directions are left unchanged. There are in fact many choices for the eigenvalues for stabilizing the fixed point. Combining Eqs. (5) and (8), we obtain |KT [xn − x∗ (p)]| ≤ δ .

(14)

This defines a slab in RN of width 2δkKT k. The control is activated only when the trajectory falls in the slab, otherwise no control is applied and the control parameter is left at its nominal value p. Ergodicity of chaotic trajectories on the attractor ensures that a typical trajectory will enter the slab at some time so that the fixed point can be stabilized. Extension of the pole-placement method to unstable periodic orbits of higher periods is straightforward [Romeiras et al., 1992].

2.2. Pole-placement method for feedback synchronization We now demonstrate that the pole-placement method discussed in Sec. 2.1 can be adapted for


synchronizing chaotic trajectories from two nearly identical systems. Consider the following chaotic maps on a Poincaré surface of section

KT such that the matrix (A − BKT ) is asymptotically stable under the condition

A : xn+1 = F(xn , p) , B : yn+1 = F(yn , pn ) ,

The solution is given by Eq. (11). In principle, the method so formulated does not require knowledge of the system equations. Because of ergodicity of chaotic trajectories, the stability properties (eigenvalues and eigendirections) of many points on the attractor can be pre-learned. In practice, this may be difficult.

(15)

where x ∈ RN , y ∈ RN , p is the nominal parameter value, and pn is the time-varying parameter. The idea is that system A remains unperturbed, while system B is perturbed slightly [Eq. (5)] so that a chaotic trajectory from B can follow a trajectory from A to achieve chaotic synchronization between A and B [Lai & Grebogi, 1993]. Say we start the two systems from different initial conditions. The resulting trajectories are uncorrelated because they are chaotic. To achieve synchronization, we wait until both trajectories come close enough to activate control to one of the systems. When a close encounter, which is guaranteed by ergodicity of chaotic systems, occurs, the dynamics from system B can be linearized in the neighborhood of the target trajectory {xn } yn+1 −xn+1 (p) = A[yn −xn (p)]+B(pn −p) ,

(16)

where A = Dx F and B = Dp F are the Jacobian matrix of F and its parameter-derivative vector, respectively, evaluated with respect to the target trajectory {xn } and the nominal parameter p. Consider the following distance: zn = yn − xn . Synchronization between x and y corresponds to the fixed point z = 0, which can then be stabilized by employing the pole-placement method. In the linearized neighborhood of xn , the perturbation to be applied is proportional to the distance |yn − xn (p)| pn − p = −KT [yn − xn (p)] .

(17)

Substituting Eq. (17) into Eq. (16), we obtain yn+1 − xn+1 (p) = (A − BKT )[yn − xn (p)] . (18) A difference between stabilizing a fixed point and synchronizing chaotic trajectories is that in the latter case, the stability matrix (A − BKT ) changes at every time step. In principle, the stability of a time-variant system cannot be decided based on eigenvalues of the stability matrix only. One should use criteria such as requiring that the norm of the stability matrix be less than unity. Here for our synchronization problem, however, it suffices to choose

|KT [yn − xn (p0 )]| ≤ δ .

(19)

3. Numerical Examples 3.1. The H´ enon map The Hénon map is given by [Hénon, 1976]: x y

!

→

x′ y′

!

a − x2 + 0.3y x

=

!

,

(20)

where a is the control parameter. Consider two such Hénon maps: one with fixed parameter value at a = 1.4 and another with a varying in the range: an ∈ [1.39, 1.41]. The quantities required for computing the time-dependent parameter perturbations are as follows A = Dx F(x, a) =

−2x∗ 1 1 0

B = Da F(x, a) =

W=

2x∗ 1

1 0

!

,

!

T=

KT = ( −λ1 − λ2 − 2x∗

,

!

,

(21)

,

1 −2x∗ 0 1

C = (B AB) =

!

0.3 0

0 1 1 0

!

(22) ,

λ1 λ2 + 0.3 ) ,

(23) (24)

where λ1 and λ2 are the regulator poles, i.e. the eigenvalues of (A−BKT ), and x∗ stands for the notion that all quantities are evaluated with respect to the target trajectory. The perturbation to be applied is given by ∆an = ∆xn (λ1 + λ2 + 2x∗ ) − (λ1 λ2 + 0.3)∆yn

(25)

where ∆xn and ∆yn are the differences in the trajectory coordinates at time n. To achieve synchronization, the eigenvalues of the matrix (A − BKT ) have to be chosen so that it is asymptotically


2.0

6

x

1.0

4

0.0 2

−2.0 2300

∆ x1

−1.0

2400

2500

2600

t

−2

Fig. 1. Feedback synchronization between two chaotic Hénon maps. Shown is the time evolution of the x-variables before and after control.

stable. There is a great deal of freedom to do this, in so far as the regulator poles are inside the unit circle. For instance, we can set to zero the eigenvalue in the unstable direction and leave the stable ones unchanged. We start two Hénon systems with different initial conditions: (x1 , y1 ) = (0.5, −0.8) and (x2 , y2 ) = (0.001, 0.001). At time step 2443, these two trajectories are close to each other within a circle of radius of 0.01. Parameter perturbations computed using Eq. (25) are then turned on. Figure 1 shows the x variables of the two trajectories, before and after control. We see that synchronization is achieved: although the two trajectories are still chaotic, they now evolve synchronously, as can be seen in Fig. 2. In the presence of noise, synchronization can be lost and the trajectories can go uncorrelated again. However, after a transient time, due to ergodicity, the two trajectories will be close enough again so that synchronization can be restored. The average transient time τ to achieve control can be shown to scale with the maximum allowed parameter perturbation δ as: τ ∼ δ−γ , where γ is determined by the stable and unstable Lyapunov numbers (λs and λu ) of the trajectory on the chaotic attractor [Lai & Grebogi, 1993] γ =1−

ln λu , ln |λs |

0

(26)

if the controlling neighborhood is chosen to be a circle.

−4 −6 0

500

1000

Time

1500

2000

Fig. 2. For feedback synchronization of two chaotic Hénon maps, the difference in the x-variables as a function of time.

3.2. The kicked double rotor map The double rotor map [Grebogi et al., 1987; Romeiras et al., 1992] models a physical system consisting of two thin, massless connected rods that are capable of rotating relative to each other, as shown in Fig. 3. The first rod of length l1 pivots about P1 (fixed), and the second rod of length 2l2 , pivots about P2 which moves. A mass m1 is attached at P2 , and the two masses m2 /2 are attached to each end of the second rod. The end of the second rod (P3 ) receives vertical periodic impulse kicks of period T and strength f0 . The motion is in the horizontal plane so that gravity can be neglected. The double rotor is also subject to friction at P1 and P2 which is proportional to the angular velocity dθ1 (t)/dt and dθ2 (t)/dt − dθ1 (t)/dt with proportional constants ν1 and ν2 , respectively. Due to the periodic forcing, the set of differential equations describing the double rotor can be reduced to the following four-dimensional map by using the stroboscopic sectioning technique Xn+1 Yn+1

!

=

MYn + Xn LYn + G(Xn+1 )

!

,

(27)


l1 cos x1∗ 0

f H(X∗ ) = 0 I

0 l2 cos x2∗

!

,

BT = (0, 0, l1 sin x1∗ /I, l2 sin x2∗ /I) , . . . . C = (B..AB..A2 B.. . . . ..An−1 B) , 

a3

a  W= 2  a1

1

Fig. 3. Schematic representation of the kicked double rotor system.

where X = (x1 , x2 )T , Y = (y 1 , y 2 )T , x1 and x2 are the angular positions of the rods at the instant of the kth kick, and y 1 and y 2 are the angular velocities of the rods immediately after the kth kick. L and M are constant 2 × 2 matrices defined by, L=

2 X

Wj eλj T ,

M=

a b b d

1 ν1 a= 1+ 2 ∆

Wj

eλj T

,

!

,

W2 =

d −b −b a

1 ν1 d= 1− 2 ∆

1 λ1,2 = − (ν1 + ν2 ± ∆) , 2

∆=

−1

λj

j=1

j=1

W1 =

2 X

,

!

,

b=−

q

,

ν2 , ∆

ν12 + 4ν22 .

∆f0 = |KT [yn − xn (f0 )]| ≤ δ ,

(31)

where xn and yn are four-dimensional vectors representing the dynamical variables of the two systems at time n. We choose again to set to zero the eigenvalues in the unstable direction and leave the stable ones unchanged. We start the two systems with the initial conditions: x0 = (0, 0, −3, −1) and y0 = (1, −0.5, −2, −0.5). At the time step 296 the two trajectories are close enough so that control perturbation is turned on. Figure 4 shows the

(28)

4.0 !

,

(29)

where c1 = f0 l1 /I, c2 = f0 l2 /I, and I = (m1 + m2 )l12 = m2 l22 . For illustrative purposes√we fix ν = T = I = m1 = m2 = l2 = 1 and l2 = 1/ 2. To apply the pole-placement technique, we choose f0 as the control parameter so it can be varied about its nominal value. Let (X∗ , Y∗ ) be a fixed point to be stabilized. The quantities required in the application of the pole-placement technique are as follows, A=

1 0  , 0 0

We consider two kicked double rotors. The first one is the target system, where we fixed the parameter value f0 = 6.9, just after the bifurcation cascade so that the double rotor is chaotic. In the second one, f0 is the adjustable parameter to be varied inside the slab

I2 M H(X∗ ) L + H(X∗ )M

!

,

2.0

∆x

c1 sin x1 c2 sin x2

a1 1 0 0

KT = (α4 − a4 , α3 − a3 , α2 − a2 , α1 − a1 )T−1 . (30)

The function G(X) is given by, G(X) =

a2 a1 1 0

T = CW ,



0.0 −2.0 −4.0

2000

2400

t

2800

Fig. 4. Feedback synchronization between two kicked double rotor systems.


transition to the synchronized state when control is activated. This example illustrates that feedback synchronization can also be realized in highdimensional maps.

4. Conclusion The principal idea of this paper is that chaotic systems can be synchronized by taking a feedback approach. Specifically, instead of using passive coupling between two chaotic systems, feedback control can be applied to one of them to synchronize them. Technically, we demonstrate that the pole-placement method used widely in control engineering can be adapted readily for computing the feedback control for synchronization. The idea of feedback synchronization may be appealing because it provides an alternative method to the important problem of realizing synchronous chaos.

Acknowledgments Y. C. Lai was supported by AFOSR under Grant No. F49620-98-1-0400 and by NSF under Grant No. PHY-9722156. C. Grebogi was supported by ONR (Physics) and by an NSF-CNPq joint grant.

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