Chaotic synchronization between oscillators using robust GPI control. Alberto Luviano-Juárez, John Cortés-Romero and Hebertt Sira-Ramırez. Departamento de ...
Chaotic synchronization between oscillators using robust GPI control Alberto Luviano-Ju´arez, John Cort´es-Romero and Hebertt Sira-Ram´ırez Departamento de Ingenier´ıa El´ectrica, Secci´on de Mecatr´onica. CINVESTAV-IPN. Apartado postal 14740, M´exico DF C.P. 07360 E-mail: aluviano{jcortes,hsira}@cinvestav.mx
Abstract— We tackle the problem of synchronization of two oscillators as a control problem. The system to control is taken as a chain of integrators control where the remainder of the dynamics is considered as an additive disturbance. This disturbance is locally approximated by a m−order time polynomial and then it is cancelled out by means of a robust Generalized Proportional Integral controller. Two examples are shown. The first one is a Duffing system as a reference and a controlled Van Der Pol as a tracker, and the second example is the synchronization of two Rossler’s systems. Some numerical results are shown.
Keywords: Chaotic oscillators, Generalized Proportional Integral Control, Nonlinear systems. I. I NTRODUCTION Synchronization and control of chaotic systems has been an active area of research which is motivated by its application in areas such as secure communication systems and power electronics systems where the main interest in the topic of synchronization arises from the possibility of encoding or masking a message using as an analog “carrier” a signal representing a state or an output of a given chaotic system. The problem is to recover the hidden, or encrypted message at the receiving end by means of an estimator system which employs one or more of the transmitted signals. The reader is referred to [1], [2], [3], [4], [5] for a more detailed content in relation to this wide topic. Many approaches have been proposed to solve this problem, from many points of view such as the ones who avoid using feedback control techniques, which are mainly based upon dynamical system theory (OttGrebory-Yorke approach [6], resonant methods). A very commonly used technique is the synchronization problem as a observation process where it is frequently to observe the some finite time derivatives of the output in order to reconstruct the non available states of the chaotic oscillator. A possible decoding process is based on the remote generation of the state estimates of the coding systems (by means of an observability law and some available outputs) and a comparison with the transmitted signals containing the message modulated states (see [5] and references therein for more details). Many of these approaches need accurate knowledge of the nonlinear dynamics of the system; hence, turns to be inapplicable if the model for the process includes uncertainties. In the sense of synchronization as a control problem, considerable efforts have been devoted to the synchronization
of chaotic systems evolving from different initial conditions, there are some passivity based-methodologies where chaotic dynamical system stabilization and some dynamic characteristics can be analyzed. Some interesting works about this approach are found in [7], [8], [9]. Other techniques which have reached a lot of development are active control [10], adaptive control [11], backstepping design [12], and sliding mode control [13] among others. In this paper, we take the synchronization problem from a control perspective. We present two cases of synchronization, one between Duffing and Van Der Pol oscillators and the other contemplates the synchronization of two Rossler’s systems. The approach is based on the robust Generalized Proportional Integral control technique (GPI) [14] which allows us to see the control problem as the control of a chain of integrators and a totally unknown additive bounded (in amplitude and frequency) disturbance. This disturbance is the addition of external and state-depending perturbations and the control process is based upon the local approximation of the disturbance function by a m−degree time polynomial. The order of the polynomial and the integrator chain to control determines the number of integral gain parameters in an easy methodology. The remainder of the paper is the following: In section II we tackle the problem of a synchronization of two different oscillators (a Duffing system as a reference and a Van Der Pol as a controlled system). In section III the synchronization of two Rossler’s systems is taken. The methodology is reported and some numerical results are shown to show the effectiveness of the controller. II. S YNCHRONIZATION OF A D UFFING SYSTEM In this section, a Duffing system will be synchronized by means of a Van Der Pol system. The Duffing system is described as follows x˙ ∗1 = x∗2 ,
x˙ ∗2 = ax∗1 − b(x∗1 )3 − cx∗2 + E cos(ωt),
(1)
y ∗ = x∗1 .
where a, b, c, E, ω are completely unknown. Now, consider the following Van Der Pol oscillator with a control input:
x˙ 1 = x2 , x˙ 2 = μ(1 − x21 )x2 − x1 + u, y = x1 .
(2)
where μ is completely unknown. Equation (2) can be expressed as x˙ 1 = x2 , x˙ 2 = ξ(t) + u, y = x1 .
(3)
with ξ(t) a smooth bounded low frequency unknown function. By approximating this function locally by a time valued polynomial, we have: ξ(t) = p0 + p1 t + p2 t2 + ... + pm tm + H.O.T where H.O.T are High Order Terms and pi ∈ R. These high order terms can be negligible for a sufficiently high order approximation as in a truncated Taylor approximation. We have: y¨ =
m i=0
Remark 1: It is necessary to include m+2 different integral gain parameters to be able to manipulate all the terms of the closed-loop tracking error characteristic polynomial. The details will be shown in the rest of the section. Applying the control law, the closed loop system dynamics satisfies the following integro-differential equation
e¨y + km+3 e˙ y + km+2 ey + km+1 ey + km (3) (m+1) ey + . . . + k1 ey + k0 + km−1 n km+3 pi ti + km+3 e˙ y (0)
ey + (m+2)
ey =
i=0
Now, taking the (m+2) time derivative of last equation we have: e(m+4) + km+3 e(m+3) + km+2 e(m+2) + k(m+1) e(m+1) + y y y y with a characteristic polynomial defined by
∗
x2 − x∗2
Let ey y − y be the tracking error in x1 , ex2 and let u∗ be a feedforward input such that y¨∗ = u∗ under no disturbance instances. The input error is given by eu u − u∗ = u − y¨∗ . Since y¨∗ is unknown, then, it can be included into ξ(t) and instead of using eu , u will be taken. The output tracking error is governed by: m
pi t i
i=0
By integrating once the tracking error dynamics, the following relation is obtained: e˙ y = e˙ y (0) +
(5)
+ km−1 e(m−1) + . . . + k1 e˙ y + k0 ey = 0 km e(m) y y
pi ti + u,
e¨y = u +
ey − u = − km+3 e˙ y − km+2 ey − km+1 ey − km (3) (m+1) (m+2) ey − . . . − k1 ey − k0 ey − km−1
u+
m
pi t
P (s) =sm+4 + km+3 sm+3 + km+2 sm+2 + k(m+1) sm+1 + + km sm + km−1 sm−1 + . . . + k1 s + k0 = 0 By selecting km+3 , . . . k0 such that P (s) has all its roots with a negative real part, the tracking error will converge asymptotically to zero as time elapses. Since e˙ y = ex2 , then if ey → 0, implies ex2 → 0 and the synchronization problem is solved. Remark 2: Note that the proposed controller can be written, in terms of Laplace transforms in the following manner. Substituting (4) in (5) and taking the Laplace transform we have km+1 km+3 u + ey km+2 + + u=− s s km+1 km k1 k0 + + + . . . + + s2 s3 sm+1 sm+2
i
i=0
The integral reconstructor of e˙ y [15] is defined as follows: (4) e˙ y u
u=−
which leads to the following relation: e˙ y + e˙ y (0) + e˙ y =
Working out u, the controller is defined by
m
pi t i .
i=0
Let us propose the following feedback controller:
km+2 sm+2 + km+1 sm+1 + . . . + k1 s + k0 sm+1 (s + km+3 )
ey (s)
This transfer function is proper, as presented in some lead compensators, and it contains the minimal quantity of terms to construct the characteristic polynomial of the trajectory tracking error.
A. Numerical results
2
To illustrate the methodology, a numerical simulation with both systems with different initial conditions was done. The parameters for the Duffing system were a = 1, b = 1, c = 1.5, E = 1, ω = 1, and the initial conditions were given by x∗1 (0) = −2, x∗2 (0) = −.5. The Van Der Pol oscillator parameters were μ = 1, and the initial conditions were x1 (0) = −.5, x∗2 (0) = −1. The disturbance ξ(t) was locally approximated by a fourth order time polynomial, then the proposed controller was defined by the following law:
Reference Tracking
1 0 −1 −2 −3 0
2
4
6
8
ey − k0
(6)
16
18
20
0 −5 −10 0
2
4
Fig. 1.
14
Reference Tracking
ey (5)
12
5
6
8
e˙ y − k6 ey − k5 ey − k4 u = −k7 (3) (4) − k3 ey − k2 ey − k1
10 Time (s)
10 Time (s)
12
14
16
18
20
Simulation results for the GPI control.
ey 2 error x
1
or in Laplace notation: k6 s6 + k5 s5 + k4 s4 + k3 s3 + k2 s2 + k1 s + k0 u= ey (s) s6 + k7 s4 where the pole placement was such that the characteristic polynomial of the tracking error was set to be equal to P (s) = (s + 2ζωn + ωn2 )4 . In this case the controller gain parameters were defined by:
0 −2 0
2
4
6
8
10 Time (s)
12
14
16
18
20
10 error x
2
5 0 −5 0
2
4
6
8
10 Time (s)
12
14
16
50
18
20
Control input
0 −50 0
2
4
6
8
k7 =8ζωn k6 =(24ζ 2 + 4)ωn2
Fig. 2.
k5 =24ζωn3 + 32ζ 3 ωn3
10 Time (s)
12
14
16
18
20
Tracking errors and control input.
k4 =48ζ 2 ωn4 + 6ωn4 + 16ζ 4 ωn4 k3 =32ζ 3 ωn5 + 24ζωn5 2
k2 =24ζ ωn6 k1 =8ζωn7 k0 =ωn8
+
of an integrator chain control with an additive disturbance. In this case model has not this form as in the first example and the control can not be applied directly. Then a coordinate transformation must be implemented. Since system (7) has a relative degree value of 3, a local coordinate transformation can be applied. For this system we have
4ωn6
where for this instance, ζ = 2, ωn = 1. Figure 1 shows the tracking process for each state. In figure 1 we can find that the controller can reduce the error in a short time despite the fact that the controller does not have any information about the system parameters and there are state depending disturbances. Also, this figure shows the control input. III. S YNCHRONIZATION OF A ROSSLER ’ S SYSTEM Let y ∗ be a unknown oscillator output. We have the following Rossler’s chaotic system:
y = x2
y =h(x) = x2 . The new coordinates are given by: z1 = φ1 (x) =h(x) = x2 , z2 = φ2 (x) =Lf h(x) = x1 + ax2 , z3 = φ3 (x) =L2f h(x) = ax1 + (a2 − 1)x2 − x3 . where the inverse transformation is
x˙ 1 = −(x2 + x3 ), x˙ 2 = x1 + ax2 , x˙ 3 = b + x1 x3 − cx3 + u.
⎞ ⎛ ⎞ 0 −(x2 + x3 ), ⎠ + ⎝ 0 ⎠ u, x1 + ax2 x˙ =f (x) + g(x)u = ⎝ b + x1 x3 − cx3 1 ⎛
(6) (7)
where a, b, c are unknown quantities. To apply the proposed methodology, it is necessary to have a system of the form
x1 = φ−1 1 (z) =z2 − az1 ,
x2 = φ−1 2 (z) =z1 ,
x3 = φ−1 3 (z) = − z1 + az2 − z3 .
and the normal form [16] is obtained as follows
(8)
and the relation between the each original state and its reconstructor is
z˙1 =z2 , z˙2 =z3 , z˙3 =L3f h(φ−1 (z))
+
ey (0) + e¨y =¨
Lg L2f h(φ−1 (z))u
m
pi t i + e¨
i=0
= − b + az3 − z2 + c(−z1 + az2 − z3 ) − (z2 − az1 )(−z1 + az2 − z3 ) − u.
ey (0) + e˙ y =e˙ y (0) + t¨
m
y =z1
pi t i + e˙
i=0
As in the first example, the problem gets the following structure z˙1 =z2 , z˙2 =z3 , z˙3 =ξ(t) − u y =z1 ξ(t) = − b + az3 − z2 + (c + az1 − z2 )(−z1 + az2 − z3 ). which reduces the problem of the robust control of a chain of three integrators. As in the first example, we have y (3) =ξ(t) − u (y ∗ )(3) =ξ(t) − u∗ We denote the tracking error as ey y − y ∗ and the control error as eu u − u∗ . Since u∗ is unknown, let us define ξ2 (t) = (y ∗ )(3) − ξ(t). The tracking error dynamics has the form: e(3) y = ξ2 (t) − u
(9)
The disturbance ξ2 (t) can be locally approximated as a time polynomial leading to ξ2 (t) =
m i=0
where for a sufficiently high order approximation the H.O.T are small enough to be discarded without affecting the process. In this case, two integral reconstructors are needed, one of e˙ y and another for e¨y . Taking a time integration in (9) m e¨y = e¨y (0) + pi t i − u and by taking another time integration to obtain e˙ y m i e˙ y = e˙ y (0) + t¨ ey (0) + pi t − u i=0
The integral reconstructors are defined by
ey + (11)
By implementing the control law, the closed loop system dynamics satisfies the following relation
e(3) y
+ km+5 e¨y + km+4 e˙ y + km+3 ey + km+2 ey + (3) (m+2) ey + . . . + k1 ey + ey + km + km+1 (m+3) m ey = km+5 pi ti + e¨y (0) + + k0 + km+4
n
i=0
pi t + t¨ ey (0) + e˙ y (0) i
(12)
i=0
Taking the (m + 3) time derivative of (12), we have: e(m+6) + km+5 e(m+5) + km+4 e(m+4) + km+3 e(m+3) + y y y y + k0 ey = 0
(13)
The characteristic polynomial of the tracking error is then P (s) =sm+6 + km+5 sm+5 + km+4 sm+4 + k(m+3) sm+3 + + km+2 sm+2 + km+1 sm+1 + . . . + k1 s + k0 = 0 (14)
km+4 km+2 km+5 u − 2 u + ey km+3 + + s s s km+1 km k1 k0 + + + . . . + + s2 s3 sm+2 sm+3
u=−
u u
By selecting the gain parameters ki , i = 0 . . . n + 5 such that P (s) satisfies the Hurwitz condition, the tracking error and the tracking error of its three time derivatives will converge asymptotically to zero as time elapses. Let express u in terms of Laplace transform, by substituting (10) in (11) and take the Laplace transform.
i=0
e˙ y −
e¨y + km+4 e˙ y + km+3 ey + km+2 u =km+5 (3) ey + . . . + ey + km + km+1 (m+2) (m+3) ey + k0 ey + k1
+ km+1 e(m+1) + km e(m) + . . . + k1 e˙ y + + km+2 e(m+2) y y y
pi ti + H.O.T
e¨y −
Let us propose the following feedback controller:
(10)
Working out u, the controller is defined by u(s) =
km+3 sm+3 + km+2 sm+2 + . . . + k1 s + k0 sm+1 (s2 + km+5 s + km+4 )
ey (s)
u =k8 ey + e¨y + k7 e˙ y + k6 ey + k5 ey + k4 (3) (4) (5) ey + k2 ey + k2 ey + + k3 (6) ey + k0
0
20
25 Time (s)
30
35
40
0
45
Reference Tracking x 15
50
20
25 Time (s)
30
35
40
2
45
50
20 0 −20 0
Reference Tracking x 5
10
15
35
40
45
50
5
10
15
20
25 Time (s)
30
35
40
5
10
15
20
25 Time (s)
30
35
40
45
40
control 45 50
20
25 Time (s)
30
35
40
45
error x2 −0.2 0
45
50
4 2 0
error x
3
−2 0
Fig. 4.
50
Tracking error signals.
−40
10
10
30
−20
Reference Tracking x1
5
25 Time (s)
0
10
−10 0
20
20
And the pole placement was such that the characteristic polynomial of the tracking error was set to be equal to 2 2 ) (s+p). In Pd (s) = (s2 +2ζ1 ωn1 s+ω1n2 )2 (s2 +2ζ2 ωn2 s+ωn2 this case the controller gain parameters are obtain by matching Pd (s) with (14) using the fact that m = 3. For this instance, ζ1 = 2, ω1 = 1, ζ2 = 2.35, ω2 = 1, p = 1. This set of values were tuned such that |u(t)| ≤ 100. Figure 3 shows the tracking process for each state, the tracking error convergence can be observed at figure 4. Figure 5 shows the control input behavior where at first there is a slight peak but when the error is closer to zero, the control signal reduces in order to cancel out the disturbance given by ξ2 (t).
15
15
40
or in Laplace notation: k6 s6 + k5 s5 + k4 s4 + k3 s3 + k2 s2 + k1 s + k0 u= ey (s) s6 + k8 s5 + k7 s4
10
1
10
0
A numerical simulation with two Rossler’s systems with different initial conditions was done. The system parameters were a = .2, b = .2, c = 5, and the initial conditions for the reference system were given by x∗1 (0) = .5, x∗2 = 0, x∗3 (0) = 1. while the initial conditions for the tracking system were x1 (0) = .4, x2 (0) = .2, x3 (0) = −.3. The disturbance ξ2 (t) was locally approximated by a third order time polynomial (m=3), then the proposed controller was defined by the following law:
5
error x 5
0.2
A. Numerical Results
−10 0
0 −0.2 −0.4 −0.6 −0.8 0
2
−60 −80 −100 0
5
10
15
20
Fig. 5.
25 Time (s)
30
35
Control signal.
IV. C ONCLUDING R EMARKS In this work, the robust GPI control were proven to be quite suitable for robust synchronization of chaotic systems and the fundamental ideas of the controller approach can be used to advantage in robustness enhancing tasks, as demonstrated in this article. For the case of disturbances of low frequency and a bounded amplitude, the presented polynomial approximation has a good performance reducing control objectives to controlling a chain of integrators. Here we use integral compensation for disturbance rejection or unmodeled dynamics of the rest of the dynamics in each example. Simulation results show a satisfactory tracking process and the control gain parameters can be adjusted in order to extend the result in an experimental framework. Finally, this problem can be extended to a perturbation compensation based upon an observer (by the same design principle of the GPI control) in order to cancel out the disturbance with the observer information and using a lower order control law.
50
R EFERENCES Fig. 3.
Simulation results for the GPI control.
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