Robust chaotic control of Lorenz system by backstepping design

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Chaos, Solitons and Fractals 37 (2008) 598–608 www.elsevier.com/locate/chaos

Robust chaotic control of Lorenz system by backstepping design Chao-Chung Peng, Chieh-Li Chen

*

Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan Accepted 7 September 2006

Communicated by Prof. Ji-Huan He

Abstract This work presents a robust chaotic control strategy for the Lorenz chaos via backstepping design. Backstepping technique is a systematic tool of control law design to provide Lyapunov stability. The concept of extended system is used such that a continuous sliding mode control (SMC) effort is generated using backstepping scheme. In the proposed control algorithm, an adaptation law is applied to estimate the system parameter and the SMC offers the robustness to model uncertainties and external disturbances so that the asymptotical convergence of tracking error can be achieved. Regarding the SMC, an equivalent control algorithm is chosen based on the selection of Lyapunov stability criterion during backstepping approach. The converging rate of error state is relative to the corresponding dynamics of sliding surface. Numerical simulations demonstrate its advantages to a regulation problem and an orbit tracking problem of the Lorenz chaos. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction The chaotic behavior is a very interesting nonlinear phenomenon which has been intensively studied during the last two decades. The effect of chaotic system is usually undesirable in practice due to its sensitivity to initial conditions, unpredictable behavior and thereby restricts the operation of physical plants. Because of the difficulty of accurate prediction of a chaotic system behavior, chaos may cause system instability or degradation in performance, and it should be eliminated in many cases. Regarding the field of chaotic analysis and control, the Lorenz system is often taken as a paradigm, since it captures many of the features of chaotic dynamics. For example, the Lorenz system describes some of the unpredictable behavior which associates with the weather. It also formulates an incompressible fluid between two parallel horizontal boundaries, with the lower boundary at a higher temperature than the upper boundary. Many approaches and techniques have been proposed for the control of chaos such as OGY method [1], bang–bang control [2], optimal control [3], intelligent control base on neural network [4], feedback linearization [5], differential *

Corresponding author. Tel.: +886 6 2389121/4; fax: +886 6 2389940. E-mail address: [email protected] (C.-L. Chen).

0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.057

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geometric method [6], adaptive control [7–10], H1 control method [11], and many others [12,13]. Sliding mode control (SMC) provides an effective alternative to deal with uncertain chaotic systems. However, it is assumed that the control can be applied with infinite fast switching. In practice, it is impossible due to finite time delays and therefore causes limitation in practical system. This non-ideal switching will result in an undesirable phenomenon so-called chattering which deteriorates system performance and also cause wear and tear in mechanical devices. In order to overcome this chattering phenomenon, the concept of extended systems is used, and it has been successfully applied in controlling chaos [14,15] and physical plant [16]. The backstepping approach is one of the most popular nonlinear techniques of control design. It is capable of generating a globally asymptotically stabilizing control laws to suppress and synchronize chaotic system [17–20]. Publications regarding Lorenz chaos control, assumed that precise model parameters are available for the feedback linearization and the system is not subjected to external disturbance. However, the ideal condition can not always be provided. In this work, a control strategy was developed for chaos suppression as well as model uncertainties and disturbance. The proposed strategy belongs to an input-output control scheme which inherits advantages of adaptive control and SMC via the backstepping design procedure. The adaptation law was design for parameter estimation whereas SMC mainly focuses on the disturbance elimination. The extended model corresponding to the chaotic system is formulated to obtain a continuous control input. Simulation results verify that the proposed controller can suppress the Lorenz chaos in spite of model uncertainties and external disturbances.

2. System description for uncertain nonlinear systems Consider the following class of nonlinear systems whose trajectories are contained in a chaotic attractor: x_ ¼ f ðxÞ þ gðxÞ  u;

ð1Þ

where x(t) 2 Rn is a state vector, u 2 R is a scalar input, and f(x) and g(x) are smooth vector fields. Generally, the nonlinear mode is only an approximate description of the actual plant due to the presence of various uncertainties. Some nth order nonlinear chaotic dynamical systems with the relative degree n may be directly described or transformed by states transformation as follows:  x_ i ¼ xiþ1 ; 1 6 i 6 n  1 ð2Þ ; x ¼ ½x1 x2    xn  2 Rn ; x_ n ¼ b0 ðX ; tÞ þ DbðX ; tÞ þ dðtÞ þ uðtÞ where xðtÞ ¼ ½x1 ðtÞx2 ðtÞ    xn ðtÞ ¼ ½xðtÞ_xðtÞ    xðn1Þ ðtÞ 2 Rn is the state vector, Db(x,t) is continuous uncertainty of chaotic systems, u(t) 2 R is the control input, and d(t) is denoted as the continuous, smooth external disturbance. In general, the uncertain term Db(x,t) and disturbance term d(t) are assumed bounded, i.e. jDbðx; tÞj 6 a and jdðtÞj 6 b;

ð3Þ

where a, b are positive scalar. The control task is to force the system to track an n-dimensional desired vector xd(t), where xd ðtÞ ¼ ðn1Þ ½xd1 ðtÞ; xd2 ðtÞ;    ; xdn ðtÞ ¼ ½xd ðtÞ; x_ d ðtÞ;    ; xd , which belong to a class of continuous functions on [t0, 1]. Define the tracking error as ðn1Þ

eðtÞ ¼ xd ðtÞ  xðtÞ ¼ ½xd ðtÞ  xðtÞ_xd ðtÞ  x_ ðtÞ    xd

ðtÞ  xðn1Þ ðtÞ ¼ ½eðtÞ_eðtÞ    eðn1Þ ðtÞ ¼ ½e1 ðtÞe2 ðtÞ    en ðtÞ ð4Þ

The control goal considered in this paper is that for any given set point or target orbit, an integral type backstepping sliding mode control law is designed, such that the resulting state response of tracking error vector satisfies limt!1 keðtÞk ¼ limt!1 kxd ðtÞ  xðtÞk ! 0:

ð5Þ

where k Æ k is the Euclidean norm of a vector.

3. Backstepping design for the Lorenz chaotic system In 1963, Lorenz proposed a simple model which describes the unpredictable behavior of the weather. The dynamic of Lorenz’s equations can be written as

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50

x3

50

40

40

x1, x2, x3

x3

30

30 20 10 (10,10,10)

0 40

20 10 0 -10

20

x2

0 -20 -40

-20

-10

0

x1

10

20

-20 -30 0

a

x1

x2 5

10

15 Time (s)

20

25

30

b

Fig. 1. Chaotic behavior of uncontrolled Lorenz system. (a) The butterfly effect and (b) states response.

x_ 1 ¼ rx1 þ rx2 x_ 2 ¼ rx1  x2  x1 x3 x_ 3 ¼ x1 x2  bx3

ð6Þ

where r and r are relative to the Prandtl number and Rayleigh number, respectively. The parameter b belongs to a geometric factor. Depending on these parameter values, the Lorenz system performs complex dynamics as shown in Fig. 1a and b. With the initial condition was placed at (10, 10, 10) and the values are taken as r = 10, r = 28 and b = 8/3, the dynamic behavior (6) acts a two-lobed pattern so called the butterfly effect. 3.1. Lorenz system standardization with control input and disturbance Consider the following chaotic system with external disturbance d. x_ 1 ¼ rx1 þ rx2 x_ 2 ¼ rx1  x2  x1 x3 þ u þ d x_ 3 ¼ x1 x2  bx3

ð7Þ

where u 2 R1 is a scalar control input and d is denoted as the bounded disturbance, that is jdj 6 b. One of the control purposes is to stabilize the system to a specific state, and the other task is to force the system response to the desired orbit by u in the direction of x2 even when the system is experiencing parameter uncertainty and disturbance. Let the control input u = u1 + u2 and u1 = x1(t)x3(t) such that one can obtain the following decoupled system x_ 1 ¼ rx1 þ rx2 x_ 2 ¼ rx1  x2 þ u2 þ d

ð8Þ

x_ 3 ¼ x1 x2  bx3

ð9Þ

and

Eq. (9) shows that x3(t) represents the internal dynamic of the system, which will converge when x1(t) and x2(t) converge to a specific point. In order to derive the integral type backstepping sliding mode control algorithm, system (8) is reformulated into an extended form as x_ 1 ¼ rx1 þ rx2 x_ 2 ¼ x3 x_ 3 ¼ rrx1 þ rrx2  x3 þ u_ 2  d_

ð10Þ

C.-C. Peng, C.-L. Chen / Chaos, Solitons and Fractals 37 (2008) 598–608

Define

601

2

3 2 3 x1 x1c 6 7 7 1 6 xc ¼ 4 x2c 5 ¼ T 4 x2 5 x3c x3

ð11Þ

where xc is the new state vector and the transformation matrix is defined by 2 3 ^ 0 0 r 6 7 ^ 1 05 T ¼ 4r ^ 1 0 r

ð12Þ

Then, dynamic Eq. (10) becomes 2 3 2 32 3 2 3 x_ 1c x1c 0 r=^ r 0 0 6 7 6 76 7 6 7 _ ð^ r  rÞ 1 4 x_ 2c 5 ¼ 4 0 54 x2c 5 þ 4 0 5ðu_ 2 þ dÞ ^ð1 þ r ^Þ ð^ x_ 3c x3c 0 rð^ r þ rÞ  r r þ 1Þ 1

ð13Þ

^þr ~, where r ^ is the nominal value and r ~ denotes the parameter uncertainty. If the exact value of r was Note that r ¼ r known, then (13) can be simplified as 2 3 2 32 3 2 3 x_ 1c x1c 0 1 0 0 6 7 6 76 7 6 7 _ _ x x ð14Þ ¼ 0 0 1 þ 4 2c 5 4 54 2c 5 4 0 5ðu_ 2 þ dÞ x_ 3c x3c 0 rðr  1Þ ðr þ 1Þ 1 Eq. (14) is of controllable canonical form. However, the exact parameter value can not always be known previously. Therefore, (13) is considered in the following design procedure. Transform (13) in the form of error equations, one can obtain ~s e_ 1c ¼ e2c þ r ~ðx2dc  e2c Þ e_ 2c ¼ e3c þ r

ð15:aÞ ð15:bÞ

^ ð1 þ r ^Þe2c  ð^ e_ 3c ¼ ½rð^ r þ rÞ  r r þ 1Þe3c þ g  u_ 2  d_ ^Þ  rð^ ~ =^ where s ¼ ðe2c  x2dc Þ=^ r, g ¼ ½^ rð1 þ r r þ rÞx2dc þ ð^ r þ 1Þx3dc þ x_ 3dc and jr rj  1, xdc = [x1dcx2dcx3dc] transformed desired vector, i.e., xdc = T1xd

ð15:cÞ T

is the

3.2. Backstepping sliding mode controller design with adaptation law In this section, the backstepping design technique is applied to obtain control law of error system (15). The design procedure is divided into three steps shown as follows. Step.1 From (15), treat the system state e2c as an independent input and let e2c ¼ /1 ðe1c Þ ¼ k 1 e1c ;

k 1 > 0:

ð16Þ

where /1(e1c) is defined as a desired virtual stabilizing algorithm for subsystem (15.a). Select a Lyapunov function V 1 ¼ e21c =2, and then V_ 1 ¼ e1c e_ 1c ¼ e1c ½/1 ðe1c Þ þ r ~s ¼ k 1 e21c þ r ~se1c

ð17Þ

~=^ rÞe21c 6 0 with j~ r=^ rj < 1. It Consider regulation problem (x2dc ¼ x_ 1dc ¼ 0), (17) can be represented as V_ 1 ¼ k 1 ð1 þ r infers that the asymptotical convergence of e1c can be achieved without the contribution of adaptation law even in ~. However, it is not the case for tracking problem. It can be seen that (17) can not guarthe presence of uncertainty r ~se1c . In order to overcome this problem, an antee the asymptotical stability due to the existence of uncertain term r adaptation algorithm is introduced as follows. Let ^_ ¼ lse1c r

ð18Þ

where l is an adaptation gain which relative to the convergent rate of parameter estimation. Then, choose a Lyapunov function as 1 1 2 ~ ; V 1 ¼ e21c þ r 2 2l

ð19Þ

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and the derivative of (19) is ~_ =l ¼ e1c ½/1 ðe1c Þ þ r ~r ~s þ r ~ðlse1c Þ=l ¼ k 1 e21c V_ 1 ¼ e1c e_ 1c þ r

ð20Þ

~ are bounded. Also, the uniform continuity of From (19) and (20), it implies that V1(t) < V1(0), and thereby the e1c and r V_ 1 can be checked by ~s ~s ¼ 2k 21 e21c  2k 1 e1c r V€ 1 ¼ 2k 1 e1c e_ 1c ¼ 2k 1 e1c ½/1 ðe1c Þ þ r Therefore, V€ 1 is bounded and according to the Barbalat’s lemma [21], (which states that if the differentiable function f(t) has a finite limit as t ! 1, and f€ ðtÞ is bounded, then f_ ðtÞ ! 0 as t ! 1), V_ 1 is uniform continuous and it indicates that ~ðtÞ ! 0 for tracking task. V_ 1 ! 0 as t ! 1. In other words, lim e1c ðtÞ ! 0 and it implies that lim r t!1

t!1

Step.2: In practice, e2c may be different from /1(e1c) for all time. Therefore, define a new error variable z1 = e2c  /1 (e1c). The dynamic of the subsystem (15.a) can be represented as ~s: e_ 1c ¼ /1 ðe1c Þ þ z1 þ r

ð21Þ

One can find that V_ 1 < 0 if zl in (21) equals to zero. Further, consider the dynamic of zl ~ðx2dc  e2c Þ þ k 1 e_ 1c ¼ e3c þ r ~ðx2dc  e2c Þ þ k 1 ½/1 ðe1c Þ þ z1 þ r ~s z_ 1 ¼ e_ 2c  /_ 1 ðe1c Þ ¼ e3c þ r ~½ðx2dc  e2c Þ þ k 1 s ¼ e3c þ k 1 ½/1 ðe1c Þ þ z1  þ r ~½ðx2dc  e2c Þ þ k 1 ðe2c  x2dc Þ=^ r ¼ e3c þ k 1 ½/1 ðe1c Þ þ z1  þ r ^ ¼ e3c þ k 1 ½/1 ðe1c Þ þ z1 ; for k 1 ¼ r

ð22Þ

In a similar manner, take the state e3c as an independent input of the form /2(e1c,z1c) as the following, e3c ¼ /2 ðe1c ; z1 Þ ¼ e1c  ðk 1 þ k 2 Þz1  k 1 /1 ðe1c Þ; k 2 2 R; R > 0:

ð23Þ

Select a Lyapunov function of the subsystem (e1c, z1) in the form of 1 V 2 ¼ V 1 þ z21 : 2

ð24Þ

Based on the assumption (23), ~_ =l þ z1 z_ 1 ¼ e1c ½/1 ðe1c Þ þ z1 þ r ~r ~s  r ~se1c þ z1 ½e3c þ k 1 ð/1 ðe1c Þ þ z1 Þ V_ 2 ¼ e1c e_ 1c þ r ~s  r ~se1c þ z1 ½/2 ðe1c ; z1 Þ þ k 1 ð/1 ðe1c Þ þ z1 Þ ¼ e1c ½/1 ðe1c Þ þ z1 þ r ~s  r ~se1c þ z1 ½e1c  ðk 1 þ k 2 Þz1  k 1 /1 ðe1c Þ þ k 1 ð/1 ðe1c Þ þ z1 Þ ¼ k 1 e21c  k 2 z21 6 0 ¼ e1c ½/1 ðe1c Þ þ z1 þ r

ð25Þ

According to Barbalat’s lemma, the subsystem (e1c, z1) is asymptotically stable.

50 50 40

30

30

x1, x2, x3

x3

x3

40

20 10

(10,10,10)

0 40

20 10 0 - 10

20

x2

0

- 20 - 40

-20

a

-10

0

x1

10

20

x1 x2

- 20 - 30

0

5

10

15 Time (s)

20

25

30

b

Fig. 2. Chaotic behavior of uncontrolled Lorenz system subjected to disturbance. (a) The butterfly effect and (b) states response.

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603

Step.3 Following steps 1–2, one can find that if (23) came into effect, then the desired behavior of the subsystem (e1c, z1) can be achieved. In other words, the asymptotically stability of the error system can be guaranteed. Thus, according to the requirement of Lyapunov stability (25), a sliding manifold is selected as follows s ¼ e3c  /2 ðe1c ; z1 Þ ¼ e3c þ e1c þ ðk 1 þ k 2 Þz1 þ k 1 /1 ðe1c Þ ¼ e3c þ e1c þ ðk 1 þ k 2 Þðe2c  /1 ðe1c ÞÞ þ k 1 /1 ðe1c Þ ¼ e3c þ e1c þ ðk 1 þ k 2 Þe2c  k 2 /1 ðe1c Þ ¼ e3c þ ðk 1 þ k 2 Þe2c þ ð1 þ k 1 k 2 Þe1c

ð26Þ

Eq. (26) shows that once the sliding mode occurs (i.e., s = 0), the settling time of the error responses can be determine by assigning the corresponding eigen-values in the sliding surface. _ the corresponding equivalent control force u_ 2eq can be obtained by In the absence of uncertainty and disturbance d, s_ ¼ 0, that is ^ð^r  1Þe2c  ð^ u_ 2eq ¼ r r þ 1Þe3c þ ð1 þ k 1 k 2 Þe2c þ ðk 1 þ k 2 Þe3c þ ^ g:

ð27Þ

^ð1  ^rÞx2dc þ ð^ r þ 1Þx3dc þ x_ 3dc . where ^g ¼ r Regarding the enhancement of the system robustness such that system states stay on the sliding surface even in the presence of model uncertainties and external disturbances, a switching control action is integrated as follows 80

x3

States Response

60 40 20 0

x1 x2

-20 -40

0

2

4

6

SMC 8

10 12 Time (s)

14

16

18

20

Fig. 3. States responses for regulation problem. The control input is active at t = 10.

100

500

0

400

-300 -400

12

14 16 18 20

-500

200 100 0 - 100 - 200

-600

- 300

SMC

-700 -800

SMC

300 1.5 1 0.5 0 -0.5 -1

-200

Sliding Surface

Control Input

-100

- 400

0

2

4

6

8

10 12 Time (s)

a

14

16

18

20

- 500

0

2

4

6

8

10 12 Time (s)

14

b

Fig. 4. Response of the proposed control algorithm. (a) Control input and (b) sliding surface.

16

18

20

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C.-C. Peng, C.-L. Chen / Chaos, Solitons and Fractals 37 (2008) 598–608

u_ 2 ¼ u_ 2eq þ wsgnðsÞ

ð28Þ

_ Choose a Lyapunov function Vs = s /2. Then, the reaching condition can be guaranteed for w > jDbðec ; tÞj þ jdðtÞj, ~r þ r ~ðk 1 þ k 2 Þðxdc2  e2c Þ þ r ~ð1 þ k 1 k 2 Þs þ ~ where jDbðec ; tÞj ¼ j^ rð~ r þ ~rÞ þ r gj is bounded (i.e., jDb(ec,t)j 6 a). Hence,   _ _ _ V_ s ¼ s_s ¼ s½wsgnðsÞ  Dbðec ; tÞ  dðtÞ 6 s½wsgnðsÞ þ jDbðec ; tÞj þ jdðtÞj 6 jsj w  jDbðec ; tÞj  jdðtÞj 6 0; 2

ð29Þ Eq. (29) confirms the existence of sliding mode dynamics. Therefore, the system is globally asymptotically stable. From (27) and (28), an integral type control effort for the Lorenz chaotic system can be represented as Z t   u_ 2eq þ wsgnðsÞ dt u ¼ u1 þ u2 ¼ x1 x3 þ ¼ x1 x3 þ

0

Z

t

^ ð^r  1Þe2c  ð^ ½r r þ 1Þe3c þ ð1 þ k 1 k 2 Þe2c þ ðk 1 þ k 2 Þe3c þ ^ g þ wsgnðsÞdt:

ð30Þ

0

x3

x2 x1 Fig. 5. Trajectory of controlled Lorenz system.

20

25

x1 xd1

20 Tracking Response of x2

Tracking Response of x1

15 10 5 0 -5 -10

with adaptation law

SMC -15 -20

x2 xd2

15 10 5 0 -5 -10 -15

with adaptation law

SMC

-20 -25

0

5

10

15 Time (s)

a

20

25

30

0

5

10

15 Time (s)

b

Fig. 6. Tracking response. (a) x1(t) and (b)x2(t). The control input is active at t = 5.

20

25

30

C.-C. Peng, C.-L. Chen / Chaos, Solitons and Fractals 37 (2008) 598–608

605

4. Numerical simulations In this section, numerical studies of the Lorenz system subjected to disturbance (7) are carried out with initial conditions at (x1(0),x2(0),x3(0)) = (10,10,10) as shown in Fig. 2(a)–(b). 4.1. Regulation problem Consider a regulation problem for the specific point, xd = 0 (i.e. the equilibrium point (0, 0, 0)). The disturbance term ^ with r ^ð0Þ ¼ 9 and k2 = 15 are applied for all is selected as d(t) = 0.5sin(0.5xt) with x = 2p. The control gains k 1 ¼ r cases. The nominal value of r is performed by ^r ¼ 26 and w = 150 is chosen to satisfy the inequality condition (26). Using the proposed control algorithm, the simulation results of regulation problem by SMC with l = 0 are shown in Figs. 3 and 4, where the control input is active at t = 10. Fig. 3 shows that irregular state responses converge to the set point asymptotically even in the presence of model uncertainties and disturbances. Fig. 4(a) and (b) are the corresponding responses of control input and sliding surface, respectively. 4.2. Tracking problem However, regarding the tracking problem, asymptotically stability can not be guaranteed if there exists uncertainty in r (as mentioned in (17)). Therefore, the adaptation law must acts in the tracking task. Let the desired vector to be _ €kðtÞT ; where k ¼ asinðxtÞ with a = 0.4. The control gains are selected as the same values used in xd ¼ Txdc ¼ T½kðtÞkðtÞ regulation problem. The proposed SMC acts at t = 5 and the adaptation law acts at t = 15 with l = 200. Fig. 5 shows the trajectory of the controlled Lorenz system has been forced to the assigned orbit. Further, from Fig. 6(a)–(b), it can be seen that states x1(t) and x2(t) track the desired value while the control acts. The corresponding phase portrait of x1(t)  x2(t) is illustrated in Fig. 7. It indicates that system trajectory approach to the desired orbit and steer to it when adaptation scheme is applied. Examining the tracking performance, from Fig. 8(a), one can find that the asymptotical convergence can not be reached (before t = 15). However, the tracking errors converge to zero owing to the effort of adaptation law (after t = 15). In other words, the exact parameter value (r = 10) has been estimated accurately as shown in Fig. 9. From (9), the internal dynamics belongs to a stable system, but it is subjected to a perturbation by x1(t)x2(t). However, the states x1(t) and x2(t) have been controlled to track a bounded orbit such that the time response of x3(t) is also bounded (which can be proven by bound-input bound-output) as illustrated in Fig. 10. The oscillated boundary of the internal state is dominated by the perturbation term x1(t)x2(t), which means that the larger the jx1(t)x2(t)j, the larger the boundary jx3(t)j. As illustrated in Fig. 11(a), the proposed synthesized robust controller does not induce chattering behavior. Fig. 11(b) shows that the sliding mode dynamics is suppressed to zero as the SMC is activated.

x2

x1 Fig. 7. Phase portrait at x1  x2 plane with control input (30).

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C.-C. Peng, C.-L. Chen / Chaos, Solitons and Fractals 37 (2008) 598–608

30

30 1.5

20 15

0.5

10

0

5

- 0.5

25 with adaptation law

SMC

5

20

15

10

25

Tracking Response of x2

1

30

0 -5 -10

20

with adaptation law

SMC

15 10 5 0 -5 -10 -15

-15 -20 5

10

15 Time (s)

20

25

30

-20 -25 0

5

10

15 Time (s)

a

b

Fig. 8. Tracking performance. (a) e1(t) and (b)e2(t). The control input is active at t = 5.

10.5 with adaptation law 10 Estimation of σ

0

9.5

9

8.5

0

5

10

15 Time (s)

20

25

30

Fig. 9. Estimation behavior of the adaptive law.

45 40 State Response of x3

Tracking Response of x1

25

35 30 25 20 15 10 5 0

0

5

10

15 Time (s)

20

25

Fig. 10. Time response of the internal state x3(t).

30

20

25

30

C.-C. Peng, C.-L. Chen / Chaos, Solitons and Fractals 37 (2008) 598–608

200

500 with adaptation law

SMC

150

400

Sliding Surface

50 0

– 50

with adaptation law

SMC

300

100

Control Input

607

200 100 0 – 100 – 200

– 100 – 150

– 300 0

5

10

15 Time (s)

a

20

25

30

– 400 0

5

10

15 Time (s)

20

25

30

b

Fig. 11. Response of the proposed control algorithm. (a) Control input and (b) sliding surface.

5. Conclusions In this paper, a robust control scheme for perturbed Lorenz chaos is developed. Based on the Lyapunov stability theory, the controller design was integrated with an adaptation algorithm and the SMC via utilizing backstepping design scheme. The SMC offers the robustness to model uncertainties and external disturbances. The adaptation law estimates the system parameter such that the asymptotical convergence of tracking error can be guaranteed. Moreover, under the proposed control algorithm, the convergent rate of the error responses can be determined by selecting the corresponding eigenvalues in the sliding surface that is generated by Lyapunov stability requirements. The advantages of this method can be summarized as follows: (a) it is a systematic procedure for chaos suppression; (b) it can be applied to a variety of chaotic systems whether it contains uncertainties or not; (c) the proposed controller is robust to external disturbance; (d) it is a chattering free SMC law.

Acknowledgement Part of the work was supported by the National Science Council, Taiwan, under the grant No. NSC94-2212-E006062.

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