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Chaos control for a unified chaotic system using output feedback controllers
Accepted Manuscript Chaos control for a unified chaotic system using output feedback controllers Zhiping Shen, Juntao Li PII: DOI: Reference:
S0378-4754(16)30147-1 http://dx.doi.org/10.1016/j.matcom.2016.06.011 MATCOM 4365
To appear in:
Mathematics and Computers in Simulation
Received date: 20 September 2007 Revised date: 5 October 2014 Accepted date: 12 June 2016 Please cite this article as: Z. Shen, J. Li, Chaos control for a unified chaotic system using output feedback controllers, Math. Comput. Simulation (2016), http://dx.doi.org/10.1016/j.matcom.2016.06.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Chaos control for a unified chaotic system using output feedback controllers Zhiping Shen, Juntao Li College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007,PR China.Email: [email protected];[email protected]
Abstract This paper presents a new method of controlling a unified chaotic system by using output feedback control strategy. In particular, for an arbitrarily given equilibrium point of a unified chaotic system, we design explicit and simple output feedback control laws by which the equilibrium point is globally and exponentially stabilized. Computer simulations are employed to illustrate the theoretical results. Keywords: A unified chaotic system, Output feedback control, Equilibrium point, Globally asymptotical stability 1. Introduction Chaotic system is a complex dynamical nonlinear system and its response exhibits some specific characteristics such as excessive sensitivity to initial conditions, broad Fourier transform spectrums, and irregular identities of the motion in phase space [4,10,17,18]. Also, it has been found to be useful in analyzing many problems, such as information processing, power systems collapse prevention, high-performance circuits and devices, etc. In 1963, Lorenz presented the first classical chaotic system [10] in a thirdorder autonomous system with only two multiplication-type quadratic terms but the system displays very complex dynamical behaviors. Chen and Ueta found another chaotic system in 1999, the Chen system, which is similar, but topologically non-equivalent to the Lorenz system [4]. By the definition of Vanecek and Celikovsky [17], the two systems belong to two different types: The Lorenz system satisfies the condition a12 a21 > 0 and the Chen system satisfies a12 a21 < 0, where a12 and a21 are the corresponding elements in the Preprint submitted to MATHEMATICS AND COMPUTERS IN SIMULATIONOctober 4, 2014
linear part matrix A = (aij )3×3 of the system. In 2002, L¨ u and Chen found another critical system between the Lorenz and Chen systems, bearing the name of the L¨ u system [11], which satisfies the condition of a12 a21 = 0. To bridge the gap between the Lorenz and Chen systems, L¨ u et al. introduced a unified chaotic system [12] in the same year, which contains the Lorenz and Chen systems as two extremes and the L¨ u system as a transition system between the Lorenz and Chen systems [3,12]. For quite a long period, people thought that chaos was neither predictable nor controllable. However, the OGY method [20] developed in 1990s of the last century had completely changed the situation, and then the study of chaos control began. The main goal of chaos control was to eliminate chaotic motion and to stabilize one of the system’s equilibrium points.Until now, many efficient approaches had been proposed for controlling chaos, such as state feedback control [3], variable structure control [19], generalized OGY control [20], inverse optimal control [14,15], parameter identification control [3], digital control [3], fuzzy control [1,2], adaptive control [3,5],and data sampling control [13], etc. Most of these methodologies need state vectors, while the state vectors are not all measurable in practical application,and thus they are not very useful in practice. In this paper, based on appropriate Lyapunov functions [6-9], we provide a new method to design explicit and simple output feedback controllers. For an arbitrarily given equilibrium point of a unified chaotic system, we design output feedback controllers to stabilize the equilibrium point globally, with exponential convergence, and provide the Lyaponov negative exponent estimation. This is particularly useful in practical designs. Computer simulations are presented to illustrate the theoretical results. 2. Main results Consider the unified chaotic system described by[16] x˙ 1 x˙ 2 x˙ 3 y
= = = =
(25α + 10)(x2 − x1 ) (28 − 35α)x1 + (29α − 1)x2 − x1 x3 x1 x2 − 8+α x3 3 x2
(1)
Where x1 , x2 , x3 are state variables, y is the output variable, and α ∈ [0, 1]. Obviously, system(1) becomes the original Lorenz system for α = 0; while system(1) becomes the original Chen system for α = 1. When α = 0.8, 2
system(1) becomes the critical system. In particular, system(1) bridges the gap between Lorenz system and Chen system. Moreover, system(1) is always chaotic in the whole interval α ∈ [0, 1]. It is easy to find the three equilibrium points of the system: E1 = (0, 0, 0), E2 = (β, β, γ), E3 = (−β, −β, γ).
√ where β = (8 + α)(9 − 2α) and γ = 27 − 6α. Denoting the above equilibrium points as (x01 , x02 , x03 ), the following transformation is introduced: ei = xi − x0i , (i = 1, 2, 3), ey = y − x02
(2)
and then by adding output feedback controls to system(1), the following control system can finally get yielded: e˙ 1 = (25α + 10)(e2 − e1 ) + u1 (ey ) e˙ 2 = (28 − 35α − x03 )e1 + (29α − 1)e2 −e1 e3 − x01 e3 + u2 (ey ) e3 + e1 e2 + x02 e1 + x01 e2 e˙ 3 = − 8+α 3 ey = e2
(3)
where ui (ey ) are linear control functions satisfying ui (0) = 0, (i = 1, 2). Before proceeding further, we will state a well-known Lemma as follows. Lemma 1. For any real scalar ϵ > 0 and any vectors with appropriate dimensions a and b, the following inequality holds : 2|ab| ≤ ϵa2 + ϵ−1 b2 . Theorem 1. The following control: u1 = k1 ey , k1 = −(10 + 25α), u2 = k2 ey , k2 = 1 − 29α − µ, (µ > 0).
(4)
can be used to stabilize the equilibrium point (x01 , x02 , x03 ) globally, where µ is a given parameter satisfying µ > 0)
3
Proof. From (4) and (3), we get: e˙ 1 = −(25α + 10)e1 e˙ 2 = (28 − 35α − x03 )e1 − µe2 − e1 e3 − x01 e3 e3 + e1 e2 + x02 e1 + x01 e2 e˙ 3 = − 8+α 3
(5)
By solving the first equation of (5) , it is given that: e1 (t) = e1 (0)e−(25α+10)t
(6)
which implies that e1 (t) is exponentially stable. One can construct positive definite quadratic Lyapunov function as: V = e22 + e23 Differentiating V in: V˙ = = ≤
with respect to time along the solution of system(5) results
2(e2 e˙ 2 + e3 e˙ 3 ) 2[−µe22 + (28 − 35α − x03 )e1 e2 + x02 e1 e3 − e23 ] + 2|x02 ||e1 e3 | −2[µe22 + 8+α 3 +2|28 − 35α + x03 ||e1 e2 | e23 ] + 2M2 |e1 ||e3 | ≤ −2[µe22 + 8+α 3 +2|28 − 35α + x03 ||e1 ||e2 |
8+α 2 e3 ] 3
(7)
Where M2 = |x02 | = ̸ 0. Therefore, by Lemma 1, we get:
2M2 |e1 ||e3 | + 2|28 − 35α + x03 ||e1 ||e2 | ≤ M2 (ε|e3 |2 + ε−1 |e1 |2 ) + |28 − 35α + x03 |(ε|e2 |2 + ε−1 |e1 |2 ) Since (8), Eq.(7) becomes V˙ ≤ −[2µ − |28 − 35α + x03 |ε]e22 − [ 2(8+α) − M2 ε]e23 3 0 −1 2 +(|28 − 35α + x3 | + M2 )ε |e1 | Choosing ε satisfies: ε < min{
2µ 2(8 + α) , } 0 |28 − 35α + x3 | 3M2
and denoting: λ = min{2µ − |28 − 35α + x03 |ε, 2(8+α) − M2 ε}, 3 γ = (|28 − 35α + x03 | + M2 )ε−1 4
(8)
Then, we get:
V˙ ≤ −λV + γ|e1 |2
(9)
Substituting Eq.(6) into (9), we can obtain: ¯ V˙ ≤ −λV + γ|e1 (0)|2 e−2αt
(10)
Where α ¯ = 25α + 10. Additionally, integrating the above Eq.(10) from 0 to t, we get: ∫t ¯ V ≤ e−λt V0 + γ 0 |e1 (0)|2 e−λ(t−τ ) e−2ατ dτ γ 2 −λt (λ−2α)t ¯ = e−λt V0 + λ−2 |e (0)| e (e − 1) α ¯ 1 γ −λt 2 −2αt ¯ −λt = e V0 + λ−2α¯ |e1 (0)| (e −e )→0 Which implies that: lim e2 (t) = lim e3 (t) = 0,
t→∞
t→∞
and considering Eq.(6), we have: lim e1 (t) = lim e2 (t) = lim e3 (t) = 0.
t→∞
t→∞
t→∞
This completes the proof. From the theorem 1, we have the following corollary. Corollary 1: When α = 0, the equilibrium point (x01 , x02 , x03 ) is asymptotically stable by choosing the following single input and single output feedback control: u1 = −10ey , u2 = 0. Theorem 2. By choosing the following single input and single output feedback control: u1 = 0, u2 = k2 ey . (11) where
25α + 10 + ρ(28 − 35α + M3 ) + 29α − 1, 2ερ 16 + 2α 2 20 + 50α 2 0 < ρ < min{( ) ,( ) }, 3M2 M2
−k2 >
5
(12) (13)
0<ε<
50α + 20 − M2 ρ1/2 . 25α + 10 + ρ(28 − 35α + M3 )
(14)
M2 = |x02 | = ̸ 0, M3 = |x03 | , one can stabilize the equilibrium point (x01 , x02 , x03 ) globally and exponentially, with corresponding negative Lyapunov estimations. Proof. From Eqs.(3) and (11), we can obtain: e˙ 1 = (25α + 10)(e2 − e1 ) e˙ 2 = (28 − 35α − x03 )e1 + (29α − 1 + k2 )e2 −e1 e3 − x01 e3 e˙ 3 = − 8+α e3 + e1 e2 + x02 e1 + x01 e2 + e1 e2 3
(15)
Constructing a Lyapunov function: V = e21 + ρ(e22 + e23 ) The time derivative of the Lyapunov function along the trajectories of system (15) is: V˙
= −2(25α + 10)e21 + 2(25α + 10)e1 e2 +2ρ(28 − 35α)e1 e2 + 2ρx02 e1 e3 e23 −2ρx03 e1 e2 + ρ(58α − 2 + 2k2 )e22 − ρ(16+2α) 3 2 2 = −2(25α + 10)e1 − ρ(2 − 58α − 2k2 )e2 e23 + 2ρx02 e1 e3 − ρ(16+2α) 3 +2[10 + 25α + ρ(28 − 35α − x03 )]e1 e2
(16)
2[10 + 25α + ρ(28 − 35α − x03 )]|e1 ||e2 | ≤ [10 + 25α + ρ(28 − 35α − x03 )](εe21 + ε−1 e22 )
(17)
By Lemma 1:
2ρ|x02 ||e1 e3 | = 2|x02 |ρ1/4 |e1 |ρ3/4 |e3 | √ ≤ M2 ( ρe21 + ρ3/2 e23 )
(18)
From (16), (17) and (18), we have:
(19) V˙ ≤ −∆1 e21 − ∆2 e22 − ∆3 e23 √ where ∆1 = [(20 + 50α)(1 − ε) − M2 ρ − ρ(28 − 35α − x03 )ε], ∆2 = ρ[2 − √ 58α − 2k2 − (10 + 25α + ρ(28 − 35α + M3 ))ε−1 ], ∆3 = ρ[ 16+2α − M2 ρ]. 3 6
From (14), we have: √ (10 + 25α)(2 − ε) − M2 ρ − ρ(28 − 35α − x03 )ε = λ1 > 0
(20)
From (12), we get: ρ[2 − 58α − 2k2 − (10 + 25α + ρ(28 − 35α + M3 ))ε−1 ] = λ2 > 0
(21)
From (13), we can obtain: 16 + 2α √ − M 2 ρ = λ3 > 0 3
(22)
Substituting (20), (21) and (22) into (19), we can obtain: V˙ ≤ −λV, λ = min{λ1 , λ2 , λ3 },
(23)
V ≤ V (0)e−λt .
(24)
then It implies that the equilibrium point (x01 , x02 , x03 ) is globally and exponentially stable. This completes the proof. Note: When M2 = |x02 | = 0, we can obtain corresponding conclusions via the similar derivation, and thus we omit this case. 3. Numerical simulations We will show a series of numerical experiments by using the fourth-order Runge-Kutta method with step size 0.001. The control is active at t = 10 for all simulations. In the first numerical experiment, the parameter α is chosen as α = 0 to ensure the existence of chaos in the absence of control, and the control law given in Theorem 1 is used, with k1 = −10 and k2 = −10. √ we intend to √ control the chaos points E1 = (0, 0, 0), E2 = (6 2, 6 2, 27), √ to equilibrium √ and E3 = (−6 2, −6 2, 27) of the system (1). Figs.1-7 show the results. The initial conditions are x1 (0) = 8.0, x2 (0) = 3.0 and x3 (0) = −10.
7
25 20 15 10
x (t)
5
1
0 −5
−10 −15 −20 40 60
20 40
0
x2(t)
20 −20
0 −40
x3(t)
−20
Fig.1. The trajectory of the system (1) is without control with the initial point:x1 (0) = 8.0, x2 (0) = 3.0 and x3 (0) = −10. 25 20 15
x1(t)
10 5 0
−5 −10 −15 −20 40 60
20 x2(t)
40
0 20 −20
0 −40
−20
x3(t)
Fig.2. The trajectory of the system (1) with control converges to the equilibrium point:E1 = (0, 0, 0) from the initial point (8, 3 -10), the control is activated at t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 − 0) = ki x2 , (i = 1, 2) 60 x (t) 1 x2(t) x3(t0
50
40
30
20
10
0
−10
−20
−30
0
5
10
15
20
25 Time (sec)
8
30
35
40
45
50
Fig.3. The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, the control is activated at t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 − 0) = ki x2 , (i = 1, 2)
25 20 15 10 x1(t)
5 0 −5 −10 −15 −20 40 60
20 40
0
x2(t)
20 −20
0 −40
x3(t)
−20
Fig.4. The trajectory of √ the√system (1) with control converges to the equilibrium point:E2 = (6 2, 6 2, 27) from the initial point (8, 3 -10), the control is activated at √ t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 − 6 2), (i = 1, 2) 50 x1(t) x2(t) x3(t)
40
30
20
10
0
−10
−20
−30
0
5
10
15
20
25 time (sec)
30
35
40
45
50
Fig.5. The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, the √ control is activated at t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 − 6 2), (i = 1, 2)
9
25 20 15
x1(t)
10 5 0 −5 −10 −15 −20 40 60
20 x (t)
40
0 20
2
−20
0 −40
x (t) 3
−20
Fig.6. The trajectory of √ the system √ (1) with control converges to the equilibrium point:E3 = (−6 2, −6 2, 27) from the initial point (8, 3 -10), the control is activated at √ t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 + 6 2), (i = 1, 2) 60 x1(t) x2(t) x3(t)
50
40
30
20
10
0
−10
−20
−30
0
5
10
15
20
25 time (sec)
30
35
40
45
50
Fig.7. The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, the √ control is activated at t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 + 6 2), (i = 1, 2) In the second numerical experiment, the parameter α is chosen as α = 1 to ensure the existence of chaos in the absence of control, and the control law given in Theorem 2 is used, with k1 = 0 and k2 = −8. We to √ √ intend 7, 3 7, 21), control the chaos to equilibrium points E = (0, 0, 0), E = (3 1 2 √ √ and E3 = (−3 7, −3 7, 21) of the system (1). Figs.8-14 show the results. The initial conditions are x1 (0) = −10, x2 (0) = 5 and x3 (0) = 10. 10
30 20
1
x (t)
10 0
−10 −20 40 30 20
60
10 x2(t)
0
40 −10
20
−20 −30
x3(t)
0
Fig.8. The trajectory of the system (1) without control with the initial :x1 (0) = −10.0, x2 (0) = 5.0 and x3 (0) = 10.0. 30 20
x1(t)
10 0
−10 −20 −30 40 60
20 x2(t)
0
40 −20
20 −40
x3(t)
0
Fig.9. The trajectory of the system (1) with control converges to the equilibrium point:E1 = (0, 0, 0) from the initial point (8, 3 -10), the control is activated at t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 − 0) = ki x2 , (i = 1, 2) 60 x1(t) x2(t) x3(t)
50
40
30
20
10
0
−10
−20
−30
−40
0
5
10
15
20
25 time (sec)
11
30
35
40
45
50
Fig.10.The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, the control is activated at t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 − 0) = ki x2 , (i = 1, 2)
30 20
x1(t)
10 0 −10 −20 −30 40 60
20 x2(t)
0
40 −20
20 −40
x3(t)
0
Fig.11. The trajectory of√the √ system (1) with control converges to the equilibrium point:E2 = (3 7, 3 7, 21) from the initial point (-10, 5, 10), the control is activated at √ t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 − 3 7), (i = 1, 2) 60 x (t) 1 x (t) 2 x3(t)
50
40
30
20
10
0
−10
−20
−30
−40
0
5
10
15
20
25 time (sec)
30
35
40
45
50
Fig.12.The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, √ the control is activated at t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 − 3 7), (i = 1, 2)
12
30 20
x1(t)
10 0 −10 −20 −30 40 60
20 x (t)
0
40
2
−20
20 −40
x3(t)
0
Fig.13. The trajectory of the √ system √ (1) with control converges to the equilibrium point:E3 = (−3 7, −3 7, 21) from the initial point (-10, 5, 10), the control is activated at √ t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 + 3 7), (i = 1, 2) 60 x (t) 1 x2(t) x3(t)
50
40
30
20
10
0
−10
−20
−30
−40
0
5
10
15
20
25 time (sec)
30
35
40
45
50
Fig.14. The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, √ the control is activated at t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 + 3 7), (i = 1, 2) 4. Conclusion In this paper, output feedback controllers are presented to stabilize a unified chaotic system when the states are not all measurable. Compared with the present results, the controllers designed in this paper have many advantages, such as small feedback gain, simple structure and less conservation.
13
5. Acknowledgement The authors are grateful for the support of the National Natural Science Foundation of China under grant 61203293, the Natural Science Foundation of Guangdong Province under grant S2012010008462, and Project supported by Scientific Research Fund of Henan Normal University, China(Grant No. 01016500158). [1]W. Chang, J.B. Park, Y.H. Joo, G.R. Chen, Output feedback control of Chen’s chaotic attractor using fuzzy logic, In: Proceedings of IECON’2000, Japan, October 2000,pp. 2159-2164. [2]W. Chang, J.B. Park, Y.H. Joo, G.R. Chen, Static output feedback fuzzy controller for Chen’s chaotic systems with uncertainties, Inform. Sci. 151 (2002) 227-244. [3]G.R. Chen, J.H. L¨ u, Dynamical analysis, control and synchronization of Lorenz families, Chinese Science Press, Beijing, 2003. [4]G.R Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcat Chaos 9(1999)1465-1466. [5]X.P. Guan, Z.P. Fan, H.M. Peng , Y.Q. Wang, Adaptive control of Chen’s chaotic system, J. Phys. 3 (2001) 324-416 (in Chinese). [6]X.X. Liao, Mathematical theory and application of stability. 2nd ed, Huazhong Normal University Press, Wuhan 2001. [7]X.X. Liao, G.R. Chen, Chaos synchronization of general Lure system via time-delay feedback control, Int. J. Bifurcat. Chaos 13 (2003) 207-213. [8]X.X. Liao , G.R. Chen, On feedback-controlled synchronization of chaotic systems, Int. J. Syst. Sci. 43 (2003) 453-461. [9]X.X. Liao, P. Yu, Analysis on the global exponent synchronization of Chua’s circuit using absolute stability theory, Int. J. Bifurcat. Chaos 12 (2005) 12-15. [10]E.N. Lorenz, Deterministic nonperiodic flows, J. Atmos. Sci. 20 (1963) 130-141.
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[11]J.H L¨ u,G. Chen, A new chaotic attractor coined, Int. J. Bifurcat Chaos 12(3)(2002)659-661. [12]J.H L¨ u,G. Chen,D.Z. Cheng, S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurcat Chaos 12(2002)291726. [13]O.E. R¨ossler, An equation for continuous chaos, Phys. Lett. A 57 (1976) 397-398. [14]E.N. Sanchez, J.P. Peren, M. Martizez, G.R. Chen, Global asymptotic stability of Chen’s chaotic system via inverse optimal control, In: Proceedings of 8th IEEE Mediterranean Conference on Control and Automation University of Patars Green, July 2000,pp.17-19. [15]E.N. Sanchez, J.P. Peren, M. Martizez, G.R. Chen, Chaos stabilization: an inverse optimal control approach, Latin Am Appl. Res. Int. J. 32 (2002) 111-114. [16]C. Tao, H. Xiong, F. Hu, Two novel synchronization criterions for a unified chaotic system, chaos solit.Fract.27 (2006) 115-120. [17]A. Vanecek, S. Celikovsky, Control systems: from linear analysis to synthesis of chaos, Prentice-Hall: London 1996. [18]H.T. Yau, C.K. Chen, C.L. Chen, Sliding mode control of chaotic systems with uncertainties, Int. J. Bifurcat Chaos 10(2000)1139-1147. [19]X. Yu, Controlling Lorenz chaos, Int. J. Syst. Sci. 27 (1996) 355-359. [20]X. Yu, G.R. Chen, Y. Xia, Y. Song , Z. Gao, An invariant-manifold-based method for chaos control, IEEE Trans. Circ. Syst. 48 (2001) 930-937.
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S0378-4754(16)30147-1 http://dx.doi.org/10.1016/j.matcom.2016.06.011 MATCOM 4365
To appear in:
Mathematics and Computers in Simulation
Received date: 20 September 2007 Revised date: 5 October 2014 Accepted date: 12 June 2016 Please cite this article as: Z. Shen, J. Li, Chaos control for a unified chaotic system using output feedback controllers, Math. Comput. Simulation (2016), http://dx.doi.org/10.1016/j.matcom.2016.06.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Click here to view linked References
Chaos control for a unified chaotic system using output feedback controllers Zhiping Shen, Juntao Li College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007,PR China.Email: [email protected];[email protected]
Abstract This paper presents a new method of controlling a unified chaotic system by using output feedback control strategy. In particular, for an arbitrarily given equilibrium point of a unified chaotic system, we design explicit and simple output feedback control laws by which the equilibrium point is globally and exponentially stabilized. Computer simulations are employed to illustrate the theoretical results. Keywords: A unified chaotic system, Output feedback control, Equilibrium point, Globally asymptotical stability 1. Introduction Chaotic system is a complex dynamical nonlinear system and its response exhibits some specific characteristics such as excessive sensitivity to initial conditions, broad Fourier transform spectrums, and irregular identities of the motion in phase space [4,10,17,18]. Also, it has been found to be useful in analyzing many problems, such as information processing, power systems collapse prevention, high-performance circuits and devices, etc. In 1963, Lorenz presented the first classical chaotic system [10] in a thirdorder autonomous system with only two multiplication-type quadratic terms but the system displays very complex dynamical behaviors. Chen and Ueta found another chaotic system in 1999, the Chen system, which is similar, but topologically non-equivalent to the Lorenz system [4]. By the definition of Vanecek and Celikovsky [17], the two systems belong to two different types: The Lorenz system satisfies the condition a12 a21 > 0 and the Chen system satisfies a12 a21 < 0, where a12 and a21 are the corresponding elements in the Preprint submitted to MATHEMATICS AND COMPUTERS IN SIMULATIONOctober 4, 2014
linear part matrix A = (aij )3×3 of the system. In 2002, L¨ u and Chen found another critical system between the Lorenz and Chen systems, bearing the name of the L¨ u system [11], which satisfies the condition of a12 a21 = 0. To bridge the gap between the Lorenz and Chen systems, L¨ u et al. introduced a unified chaotic system [12] in the same year, which contains the Lorenz and Chen systems as two extremes and the L¨ u system as a transition system between the Lorenz and Chen systems [3,12]. For quite a long period, people thought that chaos was neither predictable nor controllable. However, the OGY method [20] developed in 1990s of the last century had completely changed the situation, and then the study of chaos control began. The main goal of chaos control was to eliminate chaotic motion and to stabilize one of the system’s equilibrium points.Until now, many efficient approaches had been proposed for controlling chaos, such as state feedback control [3], variable structure control [19], generalized OGY control [20], inverse optimal control [14,15], parameter identification control [3], digital control [3], fuzzy control [1,2], adaptive control [3,5],and data sampling control [13], etc. Most of these methodologies need state vectors, while the state vectors are not all measurable in practical application,and thus they are not very useful in practice. In this paper, based on appropriate Lyapunov functions [6-9], we provide a new method to design explicit and simple output feedback controllers. For an arbitrarily given equilibrium point of a unified chaotic system, we design output feedback controllers to stabilize the equilibrium point globally, with exponential convergence, and provide the Lyaponov negative exponent estimation. This is particularly useful in practical designs. Computer simulations are presented to illustrate the theoretical results. 2. Main results Consider the unified chaotic system described by[16] x˙ 1 x˙ 2 x˙ 3 y
= = = =
(25α + 10)(x2 − x1 ) (28 − 35α)x1 + (29α − 1)x2 − x1 x3 x1 x2 − 8+α x3 3 x2
(1)
Where x1 , x2 , x3 are state variables, y is the output variable, and α ∈ [0, 1]. Obviously, system(1) becomes the original Lorenz system for α = 0; while system(1) becomes the original Chen system for α = 1. When α = 0.8, 2
system(1) becomes the critical system. In particular, system(1) bridges the gap between Lorenz system and Chen system. Moreover, system(1) is always chaotic in the whole interval α ∈ [0, 1]. It is easy to find the three equilibrium points of the system: E1 = (0, 0, 0), E2 = (β, β, γ), E3 = (−β, −β, γ).
√ where β = (8 + α)(9 − 2α) and γ = 27 − 6α. Denoting the above equilibrium points as (x01 , x02 , x03 ), the following transformation is introduced: ei = xi − x0i , (i = 1, 2, 3), ey = y − x02
(2)
and then by adding output feedback controls to system(1), the following control system can finally get yielded: e˙ 1 = (25α + 10)(e2 − e1 ) + u1 (ey ) e˙ 2 = (28 − 35α − x03 )e1 + (29α − 1)e2 −e1 e3 − x01 e3 + u2 (ey ) e3 + e1 e2 + x02 e1 + x01 e2 e˙ 3 = − 8+α 3 ey = e2
(3)
where ui (ey ) are linear control functions satisfying ui (0) = 0, (i = 1, 2). Before proceeding further, we will state a well-known Lemma as follows. Lemma 1. For any real scalar ϵ > 0 and any vectors with appropriate dimensions a and b, the following inequality holds : 2|ab| ≤ ϵa2 + ϵ−1 b2 . Theorem 1. The following control: u1 = k1 ey , k1 = −(10 + 25α), u2 = k2 ey , k2 = 1 − 29α − µ, (µ > 0).
(4)
can be used to stabilize the equilibrium point (x01 , x02 , x03 ) globally, where µ is a given parameter satisfying µ > 0)
3
Proof. From (4) and (3), we get: e˙ 1 = −(25α + 10)e1 e˙ 2 = (28 − 35α − x03 )e1 − µe2 − e1 e3 − x01 e3 e3 + e1 e2 + x02 e1 + x01 e2 e˙ 3 = − 8+α 3
(5)
By solving the first equation of (5) , it is given that: e1 (t) = e1 (0)e−(25α+10)t
(6)
which implies that e1 (t) is exponentially stable. One can construct positive definite quadratic Lyapunov function as: V = e22 + e23 Differentiating V in: V˙ = = ≤
with respect to time along the solution of system(5) results
2(e2 e˙ 2 + e3 e˙ 3 ) 2[−µe22 + (28 − 35α − x03 )e1 e2 + x02 e1 e3 − e23 ] + 2|x02 ||e1 e3 | −2[µe22 + 8+α 3 +2|28 − 35α + x03 ||e1 e2 | e23 ] + 2M2 |e1 ||e3 | ≤ −2[µe22 + 8+α 3 +2|28 − 35α + x03 ||e1 ||e2 |
8+α 2 e3 ] 3
(7)
Where M2 = |x02 | = ̸ 0. Therefore, by Lemma 1, we get:
2M2 |e1 ||e3 | + 2|28 − 35α + x03 ||e1 ||e2 | ≤ M2 (ε|e3 |2 + ε−1 |e1 |2 ) + |28 − 35α + x03 |(ε|e2 |2 + ε−1 |e1 |2 ) Since (8), Eq.(7) becomes V˙ ≤ −[2µ − |28 − 35α + x03 |ε]e22 − [ 2(8+α) − M2 ε]e23 3 0 −1 2 +(|28 − 35α + x3 | + M2 )ε |e1 | Choosing ε satisfies: ε < min{
2µ 2(8 + α) , } 0 |28 − 35α + x3 | 3M2
and denoting: λ = min{2µ − |28 − 35α + x03 |ε, 2(8+α) − M2 ε}, 3 γ = (|28 − 35α + x03 | + M2 )ε−1 4
(8)
Then, we get:
V˙ ≤ −λV + γ|e1 |2
(9)
Substituting Eq.(6) into (9), we can obtain: ¯ V˙ ≤ −λV + γ|e1 (0)|2 e−2αt
(10)
Where α ¯ = 25α + 10. Additionally, integrating the above Eq.(10) from 0 to t, we get: ∫t ¯ V ≤ e−λt V0 + γ 0 |e1 (0)|2 e−λ(t−τ ) e−2ατ dτ γ 2 −λt (λ−2α)t ¯ = e−λt V0 + λ−2 |e (0)| e (e − 1) α ¯ 1 γ −λt 2 −2αt ¯ −λt = e V0 + λ−2α¯ |e1 (0)| (e −e )→0 Which implies that: lim e2 (t) = lim e3 (t) = 0,
t→∞
t→∞
and considering Eq.(6), we have: lim e1 (t) = lim e2 (t) = lim e3 (t) = 0.
t→∞
t→∞
t→∞
This completes the proof. From the theorem 1, we have the following corollary. Corollary 1: When α = 0, the equilibrium point (x01 , x02 , x03 ) is asymptotically stable by choosing the following single input and single output feedback control: u1 = −10ey , u2 = 0. Theorem 2. By choosing the following single input and single output feedback control: u1 = 0, u2 = k2 ey . (11) where
25α + 10 + ρ(28 − 35α + M3 ) + 29α − 1, 2ερ 16 + 2α 2 20 + 50α 2 0 < ρ < min{( ) ,( ) }, 3M2 M2
−k2 >
5
(12) (13)
0<ε<
50α + 20 − M2 ρ1/2 . 25α + 10 + ρ(28 − 35α + M3 )
(14)
M2 = |x02 | = ̸ 0, M3 = |x03 | , one can stabilize the equilibrium point (x01 , x02 , x03 ) globally and exponentially, with corresponding negative Lyapunov estimations. Proof. From Eqs.(3) and (11), we can obtain: e˙ 1 = (25α + 10)(e2 − e1 ) e˙ 2 = (28 − 35α − x03 )e1 + (29α − 1 + k2 )e2 −e1 e3 − x01 e3 e˙ 3 = − 8+α e3 + e1 e2 + x02 e1 + x01 e2 + e1 e2 3
(15)
Constructing a Lyapunov function: V = e21 + ρ(e22 + e23 ) The time derivative of the Lyapunov function along the trajectories of system (15) is: V˙
= −2(25α + 10)e21 + 2(25α + 10)e1 e2 +2ρ(28 − 35α)e1 e2 + 2ρx02 e1 e3 e23 −2ρx03 e1 e2 + ρ(58α − 2 + 2k2 )e22 − ρ(16+2α) 3 2 2 = −2(25α + 10)e1 − ρ(2 − 58α − 2k2 )e2 e23 + 2ρx02 e1 e3 − ρ(16+2α) 3 +2[10 + 25α + ρ(28 − 35α − x03 )]e1 e2
(16)
2[10 + 25α + ρ(28 − 35α − x03 )]|e1 ||e2 | ≤ [10 + 25α + ρ(28 − 35α − x03 )](εe21 + ε−1 e22 )
(17)
By Lemma 1:
2ρ|x02 ||e1 e3 | = 2|x02 |ρ1/4 |e1 |ρ3/4 |e3 | √ ≤ M2 ( ρe21 + ρ3/2 e23 )
(18)
From (16), (17) and (18), we have:
(19) V˙ ≤ −∆1 e21 − ∆2 e22 − ∆3 e23 √ where ∆1 = [(20 + 50α)(1 − ε) − M2 ρ − ρ(28 − 35α − x03 )ε], ∆2 = ρ[2 − √ 58α − 2k2 − (10 + 25α + ρ(28 − 35α + M3 ))ε−1 ], ∆3 = ρ[ 16+2α − M2 ρ]. 3 6
From (14), we have: √ (10 + 25α)(2 − ε) − M2 ρ − ρ(28 − 35α − x03 )ε = λ1 > 0
(20)
From (12), we get: ρ[2 − 58α − 2k2 − (10 + 25α + ρ(28 − 35α + M3 ))ε−1 ] = λ2 > 0
(21)
From (13), we can obtain: 16 + 2α √ − M 2 ρ = λ3 > 0 3
(22)
Substituting (20), (21) and (22) into (19), we can obtain: V˙ ≤ −λV, λ = min{λ1 , λ2 , λ3 },
(23)
V ≤ V (0)e−λt .
(24)
then It implies that the equilibrium point (x01 , x02 , x03 ) is globally and exponentially stable. This completes the proof. Note: When M2 = |x02 | = 0, we can obtain corresponding conclusions via the similar derivation, and thus we omit this case. 3. Numerical simulations We will show a series of numerical experiments by using the fourth-order Runge-Kutta method with step size 0.001. The control is active at t = 10 for all simulations. In the first numerical experiment, the parameter α is chosen as α = 0 to ensure the existence of chaos in the absence of control, and the control law given in Theorem 1 is used, with k1 = −10 and k2 = −10. √ we intend to √ control the chaos points E1 = (0, 0, 0), E2 = (6 2, 6 2, 27), √ to equilibrium √ and E3 = (−6 2, −6 2, 27) of the system (1). Figs.1-7 show the results. The initial conditions are x1 (0) = 8.0, x2 (0) = 3.0 and x3 (0) = −10.
7
25 20 15 10
x (t)
5
1
0 −5
−10 −15 −20 40 60
20 40
0
x2(t)
20 −20
0 −40
x3(t)
−20
Fig.1. The trajectory of the system (1) is without control with the initial point:x1 (0) = 8.0, x2 (0) = 3.0 and x3 (0) = −10. 25 20 15
x1(t)
10 5 0
−5 −10 −15 −20 40 60
20 x2(t)
40
0 20 −20
0 −40
−20
x3(t)
Fig.2. The trajectory of the system (1) with control converges to the equilibrium point:E1 = (0, 0, 0) from the initial point (8, 3 -10), the control is activated at t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 − 0) = ki x2 , (i = 1, 2) 60 x (t) 1 x2(t) x3(t0
50
40
30
20
10
0
−10
−20
−30
0
5
10
15
20
25 Time (sec)
8
30
35
40
45
50
Fig.3. The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, the control is activated at t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 − 0) = ki x2 , (i = 1, 2)
25 20 15 10 x1(t)
5 0 −5 −10 −15 −20 40 60
20 40
0
x2(t)
20 −20
0 −40
x3(t)
−20
Fig.4. The trajectory of √ the√system (1) with control converges to the equilibrium point:E2 = (6 2, 6 2, 27) from the initial point (8, 3 -10), the control is activated at √ t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 − 6 2), (i = 1, 2) 50 x1(t) x2(t) x3(t)
40
30
20
10
0
−10
−20
−30
0
5
10
15
20
25 time (sec)
30
35
40
45
50
Fig.5. The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, the √ control is activated at t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 − 6 2), (i = 1, 2)
9
25 20 15
x1(t)
10 5 0 −5 −10 −15 −20 40 60
20 x (t)
40
0 20
2
−20
0 −40
x (t) 3
−20
Fig.6. The trajectory of √ the system √ (1) with control converges to the equilibrium point:E3 = (−6 2, −6 2, 27) from the initial point (8, 3 -10), the control is activated at √ t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 + 6 2), (i = 1, 2) 60 x1(t) x2(t) x3(t)
50
40
30
20
10
0
−10
−20
−30
0
5
10
15
20
25 time (sec)
30
35
40
45
50
Fig.7. The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, the √ control is activated at t = 10, k1 = −10, k2 = −10, ui = ki ey = ki (x2 + 6 2), (i = 1, 2) In the second numerical experiment, the parameter α is chosen as α = 1 to ensure the existence of chaos in the absence of control, and the control law given in Theorem 2 is used, with k1 = 0 and k2 = −8. We to √ √ intend 7, 3 7, 21), control the chaos to equilibrium points E = (0, 0, 0), E = (3 1 2 √ √ and E3 = (−3 7, −3 7, 21) of the system (1). Figs.8-14 show the results. The initial conditions are x1 (0) = −10, x2 (0) = 5 and x3 (0) = 10. 10
30 20
1
x (t)
10 0
−10 −20 40 30 20
60
10 x2(t)
0
40 −10
20
−20 −30
x3(t)
0
Fig.8. The trajectory of the system (1) without control with the initial :x1 (0) = −10.0, x2 (0) = 5.0 and x3 (0) = 10.0. 30 20
x1(t)
10 0
−10 −20 −30 40 60
20 x2(t)
0
40 −20
20 −40
x3(t)
0
Fig.9. The trajectory of the system (1) with control converges to the equilibrium point:E1 = (0, 0, 0) from the initial point (8, 3 -10), the control is activated at t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 − 0) = ki x2 , (i = 1, 2) 60 x1(t) x2(t) x3(t)
50
40
30
20
10
0
−10
−20
−30
−40
0
5
10
15
20
25 time (sec)
11
30
35
40
45
50
Fig.10.The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, the control is activated at t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 − 0) = ki x2 , (i = 1, 2)
30 20
x1(t)
10 0 −10 −20 −30 40 60
20 x2(t)
0
40 −20
20 −40
x3(t)
0
Fig.11. The trajectory of√the √ system (1) with control converges to the equilibrium point:E2 = (3 7, 3 7, 21) from the initial point (-10, 5, 10), the control is activated at √ t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 − 3 7), (i = 1, 2) 60 x (t) 1 x (t) 2 x3(t)
50
40
30
20
10
0
−10
−20
−30
−40
0
5
10
15
20
25 time (sec)
30
35
40
45
50
Fig.12.The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, √ the control is activated at t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 − 3 7), (i = 1, 2)
12
30 20
x1(t)
10 0 −10 −20 −30 40 60
20 x (t)
0
40
2
−20
20 −40
x3(t)
0
Fig.13. The trajectory of the √ system √ (1) with control converges to the equilibrium point:E3 = (−3 7, −3 7, 21) from the initial point (-10, 5, 10), the control is activated at √ t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 + 3 7), (i = 1, 2) 60 x (t) 1 x2(t) x3(t)
50
40
30
20
10
0
−10
−20
−30
−40
0
5
10
15
20
25 time (sec)
30
35
40
45
50
Fig.14. The time responses for the states x1 (t), x2 (t) and x3 (t) of the control system (1) before and after control activation with time, √ the control is activated at t = 10, k1 = 0, k2 = −8, ui = ki ey = ki (x2 + 3 7), (i = 1, 2) 4. Conclusion In this paper, output feedback controllers are presented to stabilize a unified chaotic system when the states are not all measurable. Compared with the present results, the controllers designed in this paper have many advantages, such as small feedback gain, simple structure and less conservation.
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5. Acknowledgement The authors are grateful for the support of the National Natural Science Foundation of China under grant 61203293, the Natural Science Foundation of Guangdong Province under grant S2012010008462, and Project supported by Scientific Research Fund of Henan Normal University, China(Grant No. 01016500158). [1]W. Chang, J.B. Park, Y.H. Joo, G.R. Chen, Output feedback control of Chen’s chaotic attractor using fuzzy logic, In: Proceedings of IECON’2000, Japan, October 2000,pp. 2159-2164. [2]W. Chang, J.B. Park, Y.H. Joo, G.R. Chen, Static output feedback fuzzy controller for Chen’s chaotic systems with uncertainties, Inform. Sci. 151 (2002) 227-244. [3]G.R. Chen, J.H. L¨ u, Dynamical analysis, control and synchronization of Lorenz families, Chinese Science Press, Beijing, 2003. [4]G.R Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcat Chaos 9(1999)1465-1466. [5]X.P. Guan, Z.P. Fan, H.M. Peng , Y.Q. Wang, Adaptive control of Chen’s chaotic system, J. Phys. 3 (2001) 324-416 (in Chinese). [6]X.X. Liao, Mathematical theory and application of stability. 2nd ed, Huazhong Normal University Press, Wuhan 2001. [7]X.X. Liao, G.R. Chen, Chaos synchronization of general Lure system via time-delay feedback control, Int. J. Bifurcat. Chaos 13 (2003) 207-213. [8]X.X. Liao , G.R. Chen, On feedback-controlled synchronization of chaotic systems, Int. J. Syst. Sci. 43 (2003) 453-461. [9]X.X. Liao, P. Yu, Analysis on the global exponent synchronization of Chua’s circuit using absolute stability theory, Int. J. Bifurcat. Chaos 12 (2005) 12-15. [10]E.N. Lorenz, Deterministic nonperiodic flows, J. Atmos. Sci. 20 (1963) 130-141.
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[11]J.H L¨ u,G. Chen, A new chaotic attractor coined, Int. J. Bifurcat Chaos 12(3)(2002)659-661. [12]J.H L¨ u,G. Chen,D.Z. Cheng, S. Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurcat Chaos 12(2002)291726. [13]O.E. R¨ossler, An equation for continuous chaos, Phys. Lett. A 57 (1976) 397-398. [14]E.N. Sanchez, J.P. Peren, M. Martizez, G.R. Chen, Global asymptotic stability of Chen’s chaotic system via inverse optimal control, In: Proceedings of 8th IEEE Mediterranean Conference on Control and Automation University of Patars Green, July 2000,pp.17-19. [15]E.N. Sanchez, J.P. Peren, M. Martizez, G.R. Chen, Chaos stabilization: an inverse optimal control approach, Latin Am Appl. Res. Int. J. 32 (2002) 111-114. [16]C. Tao, H. Xiong, F. Hu, Two novel synchronization criterions for a unified chaotic system, chaos solit.Fract.27 (2006) 115-120. [17]A. Vanecek, S. Celikovsky, Control systems: from linear analysis to synthesis of chaos, Prentice-Hall: London 1996. [18]H.T. Yau, C.K. Chen, C.L. Chen, Sliding mode control of chaotic systems with uncertainties, Int. J. Bifurcat Chaos 10(2000)1139-1147. [19]X. Yu, Controlling Lorenz chaos, Int. J. Syst. Sci. 27 (1996) 355-359. [20]X. Yu, G.R. Chen, Y. Xia, Y. Song , Z. Gao, An invariant-manifold-based method for chaos control, IEEE Trans. Circ. Syst. 48 (2001) 930-937.
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