c 2005 Nonlinear Phenomena in Complex Systems °
Predictive Control of Higher Dimensional Chaos A. Boukabou1 and N. Mansouri2 1
Department of Electronics, Jijel University, 98 Ouled Assa, Jijel, 18000, ALGERIA E-mail: a
[email protected] 2 Department of Electronics, Mentouri University, Route Ain-El-Bey, 25000, Constantine, ALGERIA E-mail: nor
[email protected]
(Received 01 February 2005) This paper addresses predictive control method of high order chaotic systems. Once the chaotic behaviour is identified on the bifurcation diagram, the control algorithm is then used to stabilize a desired unstable periodic orbits (UPOs) embedded in the chaotic system. The design procedure is illustrated by using the chaotic Chua and R¨ossler systems as examples, on which simulation results demonstrate the effectiveness of the proposed methodology. Key words: chaos, high order systems, bifurcation diagram, predictive control PACS numbers: 05.45.-a; 05.45.Gg; 05.45.Jn
1
Introduction
A deterministic system is said to be chaotic whenever its evolution sensitively depends on the initial conditions. Due to their critical dependence on the initial conditions, and due to the fact that, in general, these systems are unpredictable. This feature made chaos undesirable. Indeed, if it is true that a small perturbation can give rise to a very large response in the course of time, it is also true that a judicious choice of such a perturbation can direct the trajectory to wherever one wants in the chaotic system, and to produce a series of desired dynamical states. The idea of chaos control was enunciated by Ott-Grebogi-Yorke known as OGY method [1]. It is a feedback control method which uses chaos in the dynamical system to stabilize an unstable periodic orbit (UPO). When control is switched on, the controlled state converges to a desired orbits. Recently, controlling chaos has become a challenging topic in nonlinear dynamics and has been studied in many scientific and engineering fields such as biology, physics, chemistry, electrical cir258
cuit, etc., and several extension and applications of the original OGY control method have been reported [2]-[11]. A successful extension of the original OGY method is developed in [12] and [13] in which original UPOs of some high order chaotic systems were well controlled. However, there exists limitation in the OGY method. It was shown in particular that the UPOs are not stabilizable when the odd number of real eigenvalues are greater than one [14]. Moreover, the difference between the current states and the stabilized orbits is obtained by numerical calculation, that is, approximations of the orbits are used. To compensate this fault, Nakajima and Ueda [15] proposed a method based on the symmetry of periodic orbits. Chen and Yu [16] have shown some analytic sufficient conditions using the timedelay feedback control approach for both stabilization and tracking problems. Ushio and Yamamoto [19] proposed a dynamic delayed feedback controller for chaotic discrete-time systems based on the prediction of τ -time future state and the control law is calculated from the difference between the current state and the τ -time future
Nonlinear Phenomena in Complex Systems, 8:3 (2005) 258 - 265
A. Boukabou and N. Mansouri: Predictive Control of Higher Dimensional Chaos state of the chaotic system. States controlled by the predictive feedback control will converge to the stabilized fixed points since the approximations are not used in the feedback loop, which is an advantage of this method. In this paper, we develop the predictive control strategy to deal with high-dimensional chaotic systems in order to guide the chaotic trajectories to an unstable periodic orbits or unstable fixed points. Under this framework, we introduce necessary and sufficient condition for the stabilization of the controlled system in section two. Numerical results by applying the predictive method are given in section three for the two Chua and R¨ossler continuous chaotic systems. Concluding remarks are provided in the section four.
2
Feedback predictive control
x(t)
K x(t) – + x p(t)
Prediction
FIG. 1. The block diagram.
We set the control input as follow:
Consider the following general continuous system of non linear autonomous differential equations described by equation (1) x(t) ˙ = f (x(t))
Continuous Chaotic Systems
259
(1)
where x ∈ Rn is the state vector and f ∈ Rn is assumed to be differentiable. Also, we assume that the system is dissipative and has a chaotic behaviour. Suppose that the chaotic system (1) has an unstable fixed point or an unstable periodic orbits x ¯, and is currently in a chaotic state. The purpose of predictive control is to assure the system asymptotically converges towards x ¯ with only extremely small applied force u(t). In order to avoid such behaviour, we design a conventional feedback controller u(t) added to the dynamical system (1) of the form x(t) ˙ = f (x(t)) + u(t) (2) u(t) ∈ Rn is determined by the difference between the predicted states and the current states. It is chosen in such a way to make the trajectory of the system (1) converge to an unstable fixed point x ¯. Figure 1 shows the block diagram of the closedloop systems.
u(t) = K(xp (t) − x(t))
(3)
where K is a gain vector, xp (t) is the predicted future state of uncontrolled chaotic systems from the current state x(t). The controlled chaotic system is then given by: x(t) ˙ = f (x(t)) + K(xp (t) − x(t))
(4)
Using a one step ahead-prediction, the predictive control (3) becomes u(t) = K(x(t) ˙ − x(t))
(5)
Thus, the controlled chaotic system becomes x(t) ˙ = f (x(t)) + K(x(t) ˙ − x(t))
(6)
The simplest way to formulate an applicable control law is to make use of the fact that the dynamics of any smooth nonlinear system is approximately linear in a small neighborhood of a fixed point. Thus, near x ¯, we can use the linear approximation for the uncontrolled system by (x(t) ˙ −x ¯) = A(x(t) − x ¯)
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(7)
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A. Boukabou and N. Mansouri: Predictive Control of Higher Dimensional Chaos
where A ∈ Rn×n is the Jacobian matrix of f (x(t)) evaluated at the fixed points x ¯, which is defined as follow ¯ ∂ x(t) ˙ ¯¯ A = Dx f (¯ x) = (8) ∂x(t) ¯x¯ we can rewrite (7) in the form δ x(t) ˙ = Aδx(t)
(9)
δx(t) = x(t) − x ¯
(10)
with The controlled system is linearized around x ¯ by δ x(t) ˙ = Aδx(t) + K(δ x(t) ˙ − δx(t)) (11)
All the fixed points x ¯i (i = 1, 2, ...) of the uncontrolled system are stabilized by the predictive control method if K satisfies: |A + K(A − I)| < I
(13)
So the controlled system will be described by: ½ f (x(t)) + u(t) if r(t) < ε x(t) ˙ = (14) f (x(t)) otherwise where ε is a positive small real number. It is noted that the dynamics of uncontrolled system has the performance that, given the properties of a chaotic attractor, any trajectory will
Numerical results
In this section, we give two numerical control examples of high-dimensional chaotic systems in order to test the proposed control methods.
Chua circuit
The Chua circuit is a nonlinear circuit with chaotic behaviour for some values of parameters. The normalized equations representing the circuit are: x˙ 1 (t) = α(x2 (t) − x1 (t) − f (x1 (t))) . x2 (t) = x1 (t) − x2 (t) + x3 (t) x˙ 3 (t) = −βx2 (t)
(12)
where I ∈ Rn×n is the Identity matrix. Gain K exists if and only if det(A − I) 6= 0, then the controlled system is exponentially stable around its fixed points x ¯. In order to apply the proposed predictive control strategy, we have to determine the correction to apply in the vicinity of the fixed point to adjust the next point so it falls on the fixed one. The vicinity of the fixed point is determined by equation (13). r(t) = |x(t) − x(t − 1)|
3
3.1
= Aδx(t) + K(Aδx(t) − δx(t)) = (A + K(A − I))δx(t).
come, after a possible long period of time, arbitrarily close to x ¯. But after the close encounter, the trajectory will rapidly move away from x ¯. The controller should assure that once the system is close to the unstable fixed point, then it will remains there, and asymptotically converges towards x ¯.
(15)
where α = 10, β = 14.87, m0 = −1.27, m1 = −0.68 and f (x1 (t)) = m1 x1 (t)
(16)
1 + m0 −m (|x1 (t) + 1| − |x1 (t) − 1|) 2
represents the nonlinear element of the circuit. This system has three equilibrium points: 0 (17) x ¯1 = 0 0 and x ¯2 =
m1 −m0 m1 +1
0
m0 −m1 m1 +1
3
x ¯
=
m0 −m1 m1 +1
0
1.8437 = , 0 −1.8437
=
m1 −m0 m1 +1
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−1.8437 0 1.8437
(18)
A. Boukabou and N. Mansouri: Predictive Control of Higher Dimensional Chaos where x ¯2 , x ¯3 are two symmetric equilibrium points. Bifurcation analysis is investigated by changing the values of the parameter α from 0 to 10.5. The resulting bifurcation diagram is shown in figure 2.
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For simplicity, the control law is applied only to the state variable x1 (t) by u1 (t) = k1 (x˙ 1 (t) − x1 (t)) u(t) = (20) u (t) = 0 2 u3 (t) = 0 Luckily, computer simulations have shown that these gains indeed yield a successful controller. The system under predictive control is then given by:
3.5
3
2.5
x˙ 1 (t) = α(x2 (t) − x1 (t) − f (x1 (t))) 2
x1
+k1 (α(x2 (t) − x1 (t) − f (x1 (t)) − x1 (t)), .
x2 (t) = x1 (t) − x2 (t) + x3 (t),
1.5
x˙ 3 (t) = −βx2 (t).
1
The system state variable x1 (t) is linearized around the fixed point x ¯i1 (i = 1, 2, 3) by:
0.5
0 6
(21)
7.5
α
9
10.5
FIG. 2. Bifurcation diagram of Chua system.
One can observe a period-doubling route to chaos. Before the period doubling, all the trajectories are stabilized on a fixed point for a value of α < 7.731. At this value starts a stable 2cycle periodic orbits followed by a stable 4-cycle at α = 7.897 and so on until α = 7.940, at this value, the period-doubling regime eventually leads to a chaotic regime. In order to control the Chua system to one of the three unstable equilibrium points, we have to determine the correction which will be applied to the current state of the chaotic system. For this purpose, we determine the control input u(t) defined by equation (5). In the Chua systems, the predictive control u is applied as follow: k1 (x˙ 1 (t) − x1 (t)) u(t) = k2 (x˙ 2 (t) − x2 (t)) . (19) k3 (x˙ 3 (t) − x3 (t))
δ x˙ 1 (t) =
∂ x˙ 1 (t) δx1 (t) ∂x1 (t)
(22)
The state variable x1 (t) is linearized around the first equilibrium points x ¯11 by δ x˙ 1 (t) = (α(−1−m0 )+k1 (α(−1−m0 )−1))δx1 (t) (23) 1 The predictive feedback control stabilize x ¯1 if k1 satisfy the inequality |α(−1 − m0 ) + k1 (α(−1 − m0 ) − 1)| < 1 (24) =⇒ −2.176 < k1 < −1 (25) ¡ ¢ By choosing K = −1.5, 0, 0 and starting from the initial conditions (x1 (0), x2 (0), x3 (0)) = (−0.1, −0.1, −0.1), results of application the predictive control are shown in the figure 3. The control is activated for t > 30. The control input u(t) takes a non zero values, the states variables are close to x ¯1 and the system trajectory stabilize on the unstable fixed point x ¯1 . In the same way, x1 (t) is linearized around x ¯21 and x ¯31 by δ x˙ 1 (t) = (α(−1−m1 )+k1 (α(−1−m1 )−1))δx1 (t) (26)
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FIG. 3. Controlled Chua chaos on x ¯ .
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FIG. 4. Controlled Chua chaos on x ¯2 .
|α(−1 − m1 ) + k1 (α(−1 − m1 ) − 1)| < 1 (27)
X
5
R¨ ossler system
In this simulations studies, the predictive feedback control is applied to the R¨ossler system given by x˙ 1 (t) = −(x2 (t) + x3 (t)) x˙ 2 (t) = x1 (t) + ax2 (t) x˙ 3 (t) = b + (x1 (t) − c)x3 (t)
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The feedback predictive control suppresses chaotic behaviour of the Chua system to its two unstable equilibriums x ¯21 and to x ¯31 respectively.
0
−5 0 1
Y
=⇒ −1 < k1 < −0.523 (28) ¡ ¢ By choosing K = −0.96, 0, 0 and starting from the same initial conditions, results of application the predictive control to stabilize the two symmetric unstable fixed points x ¯21 and x ¯31 are shown in the figure 4 and figure 5 respectively. The control is activated for t > 45 and for t > 50 respectively.
0
−0.01 0
time (s)
FIG. 5. Controlled Chua chaos on x ¯3 .
(29)
Nonlinear Phenomena in Complex Systems Vol. 8, No. 3, 2005
A. Boukabou and N. Mansouri: Predictive Control of Higher Dimensional Chaos where a, b and c are the system parameters. For a = 0.398, b = 2 and c = 4 , the R¨ossler system has a chaotic behaviour. The unstable fixed points of the system are given by: √ c2 −4ab c 3.790 2 + √2 2 −4ab c x ¯1 = − 2a = −9.522 , −√ c 2a 2 c 9.522 + c −4ab 2a
√ c2 −4ab c − 2 √2 2 −4ab c − 2a +√ c 2a c2 −4ab c 2a − 2a
=
2
x ¯
2a
(30)
0.210 = −0.527 . 0.527
263
The predictive control is applied only to the state variable x2 (t) as follow: u1 (t) = 0 u(t) = u (t) = k2 (x˙ 2 (t) − x2 (t)) 2 u3 (t) = 0 u1 (t) = 0 = u (t) = k2 (x1 (t) + ax2 (t) − x2 (t)) 2 u3 (t) = 0 (31) The controlled chaotic system will be given by the following formula:
Bifurcation diagram of the R¨ossler system has been carried out by simulations by varying the parameter c from 0 to 8. Figure 6 represents the trajectory of the variable x2 obtained when it goes through maxima, as a function of the parameter c at every iteration. √ Note that there is no solution for c < 2 a b
x˙ 1 (t) = −(x2 (t) + x3 (t)), x˙ 2 (t) = x1 (t) + ax2 (t) + k2 (x1 (t) + ax2 (t) − x2 (t)), x˙ 3 (t) = b + (x1 (t) − c)x3 (t). x2 (t) is linearized around x ¯i2 (i = 1, 2) by δ x˙ 2 (t) =
18 16
∂ x˙ 2 (t) δx2 (t) ∂x2 (t)
=⇒ δ x˙ 2 (t) = (a + k2 (a − 1))δx2 (t)
14
(32)
(33) (34)
The system is stabilized around the fixed points x ¯i2 by the predictive feedback control if k2 satisfy the inequality
12
x2
10 8
|a + k2 (a − 1)| < 1
(35)
6
Once the system is linearized around the fixed point and in order to maintain the controlled system, gain must satisfy the condition:
4 2 0 1
2
3
4
5
6
7
8
c
FIG. 6. Bifurcation diagram of R¨ossler system.
There is a stable fixed point for c < 3.44, the stable 2-cycle periodic orbit appears at c = 3.44, also through c > 3.89 where appears chaos.
|0.398 − 0.602k2 | < 1 =⇒ −1 < k2 < 2.32 (36) ¡ ¢ For gain K = 0, −0.98, 0 and starting from (x1 (0), x2 (0), x3 (0)) = (−3, 2, 1), figure 7 and figure 8 represents the behaviour of the system when control is applied for t > 40 and t > 60 respectively.
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Conclusion
This paper has dealt with controller design problems of high dimensional chaotic systems based on predictive feedback control method where we do not have to calculate target unstable periodic orbits beforehand. We show necessary and sufficient conditions for the stabilization of fixed points. The proposed control method overcomes the limitation of the delay feedback control and the stability condition is guaranteed in the same way as in the original OGY method. Application of the control law to two high-dimensional chaotic systems demonstrate its efficiency.
0
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References
time (s)
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FIG. 7. Controlled R¨ossler chaos on x ¯1 .
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