- Email: [email protected]
Chaos control of Lorenz systems using adaptive controller with input saturation
Chaos, Solitons and Fractals 34 (2007) 1567–1574 www.elsevier.com/locate/chaos
Chaos control of Lorenz systems using adaptive controller with input saturation Her-Terng Yau a
a,*,1
, Chieh-Li Chen
b
Department of Electrical Engineering, Far-East College, No 49 Jung-Haw Road, Hsin-Shih Town, Tainan 744, Taiwan b Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan Accepted 20 April 2006
Communicated by Prof. Ji-Huan He
Abstract This paper presents an adaptive sliding mode control scheme for Lorenz chaos subject to saturating input. The state of Lorenz system can be asymptotically driven to an equilibrium point in spite of the presence of input saturation and external disturbance using the proposed control scheme. Numerical simulations demonstrate the effectiveness of its application to chaotic system control. It also shows that the settling time will be decreased, if the saturation bound of control input is relaxed. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Chaotic behaviour is an interesting nonlinear phenomenon which has been intensively studied during the last three decades. Chaotic behaviour is commonly detected in a wide variety of physical systems, such as electrical, mechanical, and thermal systems [1]. A fundamental characteristic of chaotic systems is its unpredictability. The Lorenz system is in the paradigm of chaotic system. It describes some of the unpredictable behaviour which associates with the weather. The unpredictable chaotic mode in Lorenz system can be very destructive. Thus, it is important to know when the Lorenz system will jump into a chaotic mode and if it does how to recover and control it. Many approaches have been presented for the control of Lorenz chaos, such as bang–bang control, optimal control, PID control, feedback linearization, adaptive control and variable structure control [2–13]. Most of them are based on the assumption that the actuator will not be saturated during the control process. However, actuator will saturate due to its physical limitations in practice. The presence of saturation in control input may cause serious influence on system stability and performance. Besides, the saturation of control input may cause the chaotic system been perturbed to unpredictable results due to its high sensitivity to system parameters. Therefore, the effect of actuator saturation cannot be ignored in analysis of control design and realization. Hence, the derivation of controller with input saturation is an *
1
Corresponding author. Fax: +886 6 5977570. E-mail addresses: [email protected], [email protected] (H.-T. Yau), [email protected] (C.-L. Chen). Supported by the National Science, Council of Republic of China under contract NSC-94-2212-E-269-003.
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.04.048
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important problem. To our knowledge, the control of Lorenz chaos containing input saturation has not been well discussed. In this paper, an adaptive sliding mode control scheme [17] has been proposed for chaos control of Lorenz system with input saturation. Results show that the proposed controller manipulates the chaotic behaviour in spite of external disturbance and input saturation. The paper is organized as follows; Section 2 describes the control problem of Lorenz system; the switching surface and error dynamic are derived in Section 3; an adaptive controller is proposed to control Lorenz chaos in Section 4, and resulting system performance is shown in Section 5.
2. System definition In this paper, we consider a Lorenz chaotic system described as x_ 1 ¼ rx1 þ rx2 ; x_ 2 ¼ rx1 x2 x1 x3 ; x_ 3 ¼ x1 x2 bx3 ;
ð1Þ
where x1, x2, x3 are state variables, r, r and b are real positive parameters, and Lorenz chaotic system can exhibit quite complex dynamics depending on parameter values [1]. For the elimination of chaotic behaviour and regulation of state of system (1), a control input is introduced in the differential equation for x2 giving a control process similar to that studied in [7]. In order to describe the saturation phenomenon in practical physical system, the control input is assumed to be a nonlinear function sat(u(t)). Thus the controlled Lorenz dynamic becomes x_ 1 ¼ rx1 þ rx2 ; x_ 2 ¼ rx1 x2 x1 x3 þ dðtÞ þ satðuðtÞÞ; x_ 3 ¼ x1 x2 bx3 ;
ð2Þ
where d(t) is the system external disturbance which is unknown but bounded, that is, jd(t)j 6 d, where d > 0 is given. The function sat(u(t)) is defined as 8 if uðtÞ > uH ; > < uH satðuðtÞÞ ¼ uðtÞ if uL 6 uðtÞ 6 uH ; ð3Þ > : uL if uðtÞ < uL ; where uH, uL 2 R+ are actuator limitations and they are bounded. The saturating function (3) is conveniently expressed as
where
satðuÞ ¼ bðuðtÞÞuðtÞ;
ð4Þ
8 u H > > > > uðtÞ < bðuðtÞÞ ¼ 1 > > uL > > : uðtÞ
ð5Þ
if uðtÞ > uH ; if uL 6 uðtÞ 6 uH ; if uðtÞ < uL
and such that 0 < b(u(t)) 6 1. We are interested in deriving an adaptive control law such that in the closed-loop system (2), state trajectories are regulated to a specified equilibrium point in the state space, even when the system is experiencing disturbance and input saturation. It is desired that x1 be regulated to equilibrium state x1r, where x1r is a given constant. From Eq. (1), if x1(t) = x1r, then x_ 1 ðtÞ ¼ 0 and x2(t) = x2r = x1r. For x1(t) = x2(t) = x1r, solving the differential equation for x3 results in x3 ðtÞ ¼ ebt x3 ð0Þ þ
x21r ð1 ebt Þ; b x2
which shows that x3(t) converges to x3r ¼ b1r when the time t ! 1. Therefore, for a regulated equilibrium point x1(t) = x1r , the control law seeks to steer the trajectory to the equilibrium point xr = (x1r, x1r, x3r)T, where (Æ)T denotes transposition of (Æ).
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3. Switching surface and error dynamics Let the new disturbance term be D(t) = d(t) x1x3, such that the nonlinear term is cancelled, then the dynamic equation with disturbance D(t) from (2) is given by x_ 1 r r x1 0 0 ¼ þ DðtÞ þ bðuðtÞÞ uðtÞ; ð6aÞ x_ 2 r 1 x2 1 1 ð6bÞ x_ 3 ¼ x1 x2 bx3 : Because the external disturbance jd(t)j 6 d, then D(t) satisfies jDðtÞj ¼ jdðtÞ x1 x3 j 6 jdðtÞj þ jx1 x3 j 6 d þ jx1 x3 j ¼ a: From Eq. (6a), it can be seen that the dynamics of x1(t) and x2(t) are decoupled from x3(t). Eqs. (6a) and (6b) show that x3(t) represents the internal dynamics of the control system, which will converge and stabilize when x1(t) and x2(t) con verge to x1r. z1 x1 According to the state transformation by Yang et al. [11], the new system state vector Z ¼ ¼ P is defined, z2 x2 1=r 0 where the transform matrix is P ¼ . Let the trajectory error states be e1 = z1 z1r, e1 = z1 z1r and 1 1 e3 = x3(t) x3r, where the transformed regulation system states are z1r ¼ r1 x1r and z2r = x1r + x2r = 0. Thus, the error state dynamic equations can be obtained as following: e_ 1 ¼ e2 ; e_ 2 ¼ rðr 1Þe1 ðr þ 1Þe2 þ rðr 1Þz1r þ DðtÞ þ satðuðtÞÞ ¼ f ðe1 ; e2 Þ þ DðtÞ þ bðuðtÞÞ uðtÞ;
ð7aÞ ð7bÞ
where f(e1, e2) = r(r 1)e1 (r + 1)e2 + r(r 1)z1r , and the internal dynamics of the error state is e_ 3 ¼ be3 þ gðe1 ; e2 Þ;
ð7cÞ
where g(e1, e2) = bx3r + re1e2 + re2z1r + r 2(e1 + z1r)2. Now, a sliding surface suitable for the application can be defined as sðtÞ ¼ c1 e1 ðtÞ þ e2 ðtÞ; where s(t) 2 R and c1 is a design parameter to be determined. For the existence of the sliding mode, it is necessary and sufficient that [14,15] sðtÞ ¼ c1 e1 ðtÞ þ e2 ðtÞ ¼ 0
ð8aÞ
s_ ðtÞ ¼ c1 e_ 1 ðtÞ þ e_ 2 ðtÞ ¼ 0:
ð8bÞ
and
Therefore, the following sliding mode dynamics can be obtained as e_ 1 ¼ c1 e1 ; e_ 2 ¼ rðr 1Þe1 ðr þ 1Þe2 þ rðr 1Þz1r þ DðtÞ þ satðuðtÞÞ; e_ 3 ¼ be3 þ gðe1 ; e2 Þ:
ð9aÞ ð9bÞ ð9cÞ
Obviously, if the design parameter c1 < 0, the stability of (9a) is guaranteed, that is limt!1e1(t) ! 0. Furthermore, by Eq. (8a), e2(t) is also stable, that is limt!1e2(t) ! 0. Since (e1(t), e2(t)) converge to zero, it follows that there exists a function h(t) such that jgðtÞj jgðe1 ; e2 Þj 6 hðtÞ
ð10Þ
and h(t) ! 0, as t ! 1. Thus given any h0, there exists a finite time t1 such that jgðtÞj < h0 ;
for t P t1 :
By solving (9c), gives bðtt1 Þ
e3 ðtÞ ¼ e
x3 ðt1 Þ þ
Z
t
ebðtsÞ gðsÞ ds: t1
ð11Þ
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Therefore, for t > t1 bðtt1 Þ
je3 ðtÞj 6 e
jx3 ðt1 Þj þ h0
Z
t bðtsÞ
e
bðtt1 Þ
ds ¼ e
t1
h0 h0 þ : jx3 ðt1 Þj b b
ð12Þ
Since h0 can be made arbitrarily small by taking t1 sufficiently large, and b > 0, it follows that limt!1e3(t) ! 0. Thus, the closed-loop system state (x1, x2, x3) ! xr, as t ! 1.
4. Adaptive controller design Before proceeding to the adaptive sliding mode control design, the Barbalat lemma is provided: Lemma 1 (Barbalat lemma [16]). If w : R ! R is a uniformly continuous function for t P 0 and if the limit of the integral Z t jwðkÞj dk ð13Þ lim t!1
0
exists and is finite, then lim wðtÞ ¼ 0:
ð14Þ
t!1
We choose a control law of the form uðtÞ ¼ n^cðtÞgsignðsðtÞÞ;
n > 1;
ð15Þ
where g = jc1e2j + jf(e1, e2)j + a. The adaptive law is ^c_ ðtÞ ¼ ng^c3 ðtÞjsðtÞj;
^cð0Þ ¼ ^c0 ;
ð16Þ
where ^c0 is the bounded positive initial value of ^cðtÞ. The adaptive law can be also rewritten in the integral form as Z t ^cðtÞ ¼ ^c0 þ n^c3 ðkÞgðkÞjsðkÞj dk: 0
Based on the control law (15), the reaching condition sðtÞ_sðtÞ < 0 is achieved using the following theorem, that is, the proposed scheme (15) will derive the system (7) with saturating input onto the sliding mode s(t) = 0. Remark 1. Since from theoretical point of view, s(t) will not be exactly equal to zero in finite time, thus the adaptive parameter ^c will increase (even if s(t) is a very small number) until s(t) = 0. A simple way for overcoming this disadvantage is to modify the adaptive law (16) by ‘‘dead-zone’’ technique [16] as qksk; ksk P d; ^c_ ðtÞ ¼ ð17Þ 0; ksk < d; where d is a small positive constant. Theorem 1. Consider the error dynamics system (7a) and (7b) with saturating input. The hitting condition of the sliding mode is satisfied, if the control u(t) is given by (15). Proof. Letting the Lyapunov function of the system be V ¼ 12 ½s2 ðtÞ þ ~c2 ðtÞ, then its derivative with respect to time is V_ ¼ sðtÞ_sðtÞ þ ~cðtÞ~c_ ðtÞ ¼ s½c1 e_ 1 þ e_ 2 þ ~c~c_ ¼ s½c1 e2 þ f ðe1 ; e2 Þ þ DðtÞ þ bðuðtÞÞ uðtÞ þ ~c ~c_ 6 jsjðjc1 e2 j þ jf ðe1 ; e2 Þj þ aÞ þ sbðuðtÞÞuðtÞ þ ~c~c_ 6 jsjg n^cgbðuðtÞÞjsðtÞj þ ~c~c_ :
ð18Þ
Since 0 < b(u) 2 R 6 1, according the density property of real number, there always exists a constant c > 0 satisfying 0 < c < bðuÞ 6 1;
8uðtÞ; t 2 ½0; 1Þ:
ð19Þ
Combined with (19), Eq. (18) yields V_ ðtÞ 6 gjsðtÞj nc^cðtÞgjsðtÞj þ ~cðtÞ~c_ ðtÞ:
ð20Þ
Now define the adaptation error as ~cðtÞ ¼ ^c1 ðtÞ c:
ð21Þ
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Since c is an existing constant then c_ ¼ 0. Thus the following expression holds: ~c_ ðtÞ ¼ ^c_ 1 ðtÞ c_ ¼ ^c_ 1 ðtÞ ¼ ^c2 ðtÞ^c_ ðtÞ:
ð22Þ
Inserting (16), (21) and (22) into the right hand of inequality (20) this yields V_ ðtÞ 6 gjsðtÞj þ ð^c1 ðtÞ cÞ n^cðtÞgjsðtÞj ngjsðtÞj ~cðtÞ^c2 ðtÞ^c_ ðtÞ 6 ðn 1ÞgjsðtÞj: |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
ð23Þ
~c
Therefore, if n > 1, then V_ < 0. Now if we define w(t) = (n 1)gkS(t)k, and integrate the above Eq. (23) from zero to t, it yields Z t Z t Z t wðkÞ dk ) V ð0Þ P V ðtÞ þ wðkÞ dk P wðkÞ dk: ð24Þ V ðtÞ 6 V ð0Þ 0
0
0
Taking the limit as t ! 1 on both sides of (24) gives Z t wðkÞ dk: 1 > V ð0Þ P lim t!1
ð25Þ
0
According to Barbalat lemma (see Lemma 1), It obtains lim wðtÞ ¼ lim ðn 1ÞgjsðtÞj ! 0:
t!1
ð26Þ
t!1
Since n > 1, g > 0, (26) implies s(t) ! 0 as t ! 1. Hence the proof is complete.
h
5. An Illustrative example In this section, simulation results are presented with initial condition (x1, x2, x3) = (10, 10, 10). For the uncontrolled Lorenz system, the parameters are r = 15, b ¼ 123, and r = 35, and the regulating equilibrium point is xr = (11.66, 11.66, 34)T. Let c1 = 10 to result in a stable sliding mode and the saturating input is defined as 8 if uðtÞ > uH ; > < uH ¼ 10 if uL 6 uðtÞ 6 uH ; satðuðtÞÞ ¼ uðtÞ ð27Þ > : uL ¼ 10 if uðtÞ < uL : Base on (16), parameter of adaptive control law is n = 1.5 to satisfy the condition (23). The external disturbance d(t) is assumed to be 0.5 cos(3pt), then jd(t)j 6 0.5 = d. Combining Eqs. (2), (15) and (27) gives the simulated closed-loop system. The simulation results with uH = uL = 10 are shown in Figs. 1–5. Fig. 1 shows the time response of sliding surface s(t). Fig. 2(a)–(c) represents respectively the time responses of error states of e1, e2 and e3, and the phase plane behaviour is shown in Fig. 3. Fig. 4 shows the state response for the controlled Lorenz system. Otherwise, the simulation results with uH = uL = 5 are shown in Figs. 5 and 6. Compare Fig. 4 with Fig. 5, it shows that the settling time will be decreased, if the saturation bound of control input is relaxed.
6 4
s(t)
2 0 –2 –4 –6 –8 0
control in action 1
2
3
4
5
6
7
8
9
Time(sec) Fig. 1. Time responses of sliding surface with uH = uL = 10.
10
H.-T. Yau, C.-L. Chen / Chaos, Solitons and Fractals 34 (2007) 1567–1574 1 0.5
e1(t)
0 –0.5 –1 –1.5 –2 control in action –2.5
1
0
2
6
5
4
3
(a)
8
7
9
10
9
10
9
10
Time(sec) 20 15 10
e2(t)
5 0 –5 –10 –15
control in action 0
1
2
3
(b)
4
5
6
7
8
Time(sec) 30 20
e3(t)
10 0 –10 –20 control in action –30
0
1
2
3
4
5
6
7
8
Time(sec)
(c)
Fig. 2. Time response of error state (a) e1(t), (b) e2(t), (c) e3 (t) with uH = uL = 10.
15 A
10
e3(t)
1572
5 B 0 –5
–10 5 0.4
0
e2(t)
0.2 0
5
–0.2 10
–0.4
e1(t)
Fig. 3. The phase plane of error states with uH = uL = 10; the controller is turned at t = 5 (the point A).
x1, x2, x3
H.-T. Yau, C.-L. Chen / Chaos, Solitons and Fractals 34 (2007) 1567–1574 60 50 40 30 20 10 0 –10 –20 –30 –40
1573
x3
x1
x2 0
1
2
3
control in action
4
5
6
7
8
9
10
Time(sec)
x1, x2, x3
Fig. 4. The state time responses of Lorenz system with uH = uL = 10.
60 50 40 30 20 10 0 –10 –20 –30 –40
0
2
4
6
8
10
Time(sec) Fig. 5. The state time responses of Lorenz system with uH = uL = 5.
20
e3(t)
10
A
0
–10 –20 10 0.5
5 0
e2(t)
0
–5 –10
–0.5
e1(t)
Fig. 6. The phase plane of error states with uH = uL = 5; the controller is turned at t = 5 (the point A).
6. Conclusions In this paper, an adaptive sliding mode control method for the Lorenz chaos with input saturation is proposed. Based on the Lyapunov stability theory, the adaptive sliding mode controller can stabilize Lorenz chaos even in the existence of system external disturbance. Simulation results show that the proposed controller is able to control Lorenz chaos and the settling time will decrease if the saturation bound is relaxed.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Nayfeh AH. Applied nonlinear dynamics. New York: John Wiley & Sons, Inc.; 1994. Vincent TL, Yu J. Control of a chaotic system. Dynam Control 1991;1:35–52. Luce R, Kernevez JP. Controllability of Lorenz equation. Int Ser Numer Math 1991;97:257. Yeap TH, Ahmed NU. Feedback control of chaotic systems. Dynam Control 1994;4:97–114. Gallegos JA. Nonlinear regulation of a Lorenz system by feedback linearization techniques. Dynam Control 1994;4:277–98. Fuh CC, Tung PC. Controlling chaos using differential geometric method. Phys Rev Lett 1995;75:2952–5. Zeng Y, Singh SN. Adaptive control of chaos in lorenz system. Dynam Control 1997;7:143–54. Ge SS, Wang C, Lee TH. Adaptive backstepping control of a class of chaotic systems. Int J Bifurcat Chaos 2000;5:1149–56. Yu X. Controlling Lorenz chaos. Int J Syst Sci 1996;27:355–9. Tian YP. Controlling chaos using invariant manifold. Int J Control 1999;72:258–66. Yang SK, Chen CL, Yau HT. Control of chaos in Lorenz system. Chaos, Solitons & Fractals 2002;13:767–80. Yau HT, Chen CK, Chen CL. Sliding mode control of chaotic system with uncertainties. Int J Bifurcat Chaos 2000;10:1139–47. Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific; 1998. Utkin VI. Sliding mode and their applications in variable structure systems. Moscow: Mir Editors; 1978. Utkin VI. Sliding mode in control optimization. Berlin: Springer-Verlag; 1992. Slotine JE, Li W. Applied nonlinear control. Englewood Cliffs, New Jersey: Prentice-Hall; 1991. Yau H-T, Yan J-J. Memoryless adaptive variable structure control of time-delayed systems with unknown bounded uncertainties. Int J Nonlinear Sci Numer Simul 2005;6(4):249–54.
Chaos control of Lorenz systems using adaptive controller with input saturation Her-Terng Yau a
a,*,1
, Chieh-Li Chen
b
Department of Electrical Engineering, Far-East College, No 49 Jung-Haw Road, Hsin-Shih Town, Tainan 744, Taiwan b Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, Taiwan Accepted 20 April 2006
Communicated by Prof. Ji-Huan He
Abstract This paper presents an adaptive sliding mode control scheme for Lorenz chaos subject to saturating input. The state of Lorenz system can be asymptotically driven to an equilibrium point in spite of the presence of input saturation and external disturbance using the proposed control scheme. Numerical simulations demonstrate the effectiveness of its application to chaotic system control. It also shows that the settling time will be decreased, if the saturation bound of control input is relaxed. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Chaotic behaviour is an interesting nonlinear phenomenon which has been intensively studied during the last three decades. Chaotic behaviour is commonly detected in a wide variety of physical systems, such as electrical, mechanical, and thermal systems [1]. A fundamental characteristic of chaotic systems is its unpredictability. The Lorenz system is in the paradigm of chaotic system. It describes some of the unpredictable behaviour which associates with the weather. The unpredictable chaotic mode in Lorenz system can be very destructive. Thus, it is important to know when the Lorenz system will jump into a chaotic mode and if it does how to recover and control it. Many approaches have been presented for the control of Lorenz chaos, such as bang–bang control, optimal control, PID control, feedback linearization, adaptive control and variable structure control [2–13]. Most of them are based on the assumption that the actuator will not be saturated during the control process. However, actuator will saturate due to its physical limitations in practice. The presence of saturation in control input may cause serious influence on system stability and performance. Besides, the saturation of control input may cause the chaotic system been perturbed to unpredictable results due to its high sensitivity to system parameters. Therefore, the effect of actuator saturation cannot be ignored in analysis of control design and realization. Hence, the derivation of controller with input saturation is an *
1
Corresponding author. Fax: +886 6 5977570. E-mail addresses: [email protected], [email protected] (H.-T. Yau), [email protected] (C.-L. Chen). Supported by the National Science, Council of Republic of China under contract NSC-94-2212-E-269-003.
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.04.048
1568
H.-T. Yau, C.-L. Chen / Chaos, Solitons and Fractals 34 (2007) 1567–1574
important problem. To our knowledge, the control of Lorenz chaos containing input saturation has not been well discussed. In this paper, an adaptive sliding mode control scheme [17] has been proposed for chaos control of Lorenz system with input saturation. Results show that the proposed controller manipulates the chaotic behaviour in spite of external disturbance and input saturation. The paper is organized as follows; Section 2 describes the control problem of Lorenz system; the switching surface and error dynamic are derived in Section 3; an adaptive controller is proposed to control Lorenz chaos in Section 4, and resulting system performance is shown in Section 5.
2. System definition In this paper, we consider a Lorenz chaotic system described as x_ 1 ¼ rx1 þ rx2 ; x_ 2 ¼ rx1 x2 x1 x3 ; x_ 3 ¼ x1 x2 bx3 ;
ð1Þ
where x1, x2, x3 are state variables, r, r and b are real positive parameters, and Lorenz chaotic system can exhibit quite complex dynamics depending on parameter values [1]. For the elimination of chaotic behaviour and regulation of state of system (1), a control input is introduced in the differential equation for x2 giving a control process similar to that studied in [7]. In order to describe the saturation phenomenon in practical physical system, the control input is assumed to be a nonlinear function sat(u(t)). Thus the controlled Lorenz dynamic becomes x_ 1 ¼ rx1 þ rx2 ; x_ 2 ¼ rx1 x2 x1 x3 þ dðtÞ þ satðuðtÞÞ; x_ 3 ¼ x1 x2 bx3 ;
ð2Þ
where d(t) is the system external disturbance which is unknown but bounded, that is, jd(t)j 6 d, where d > 0 is given. The function sat(u(t)) is defined as 8 if uðtÞ > uH ; > < uH satðuðtÞÞ ¼ uðtÞ if uL 6 uðtÞ 6 uH ; ð3Þ > : uL if uðtÞ < uL ; where uH, uL 2 R+ are actuator limitations and they are bounded. The saturating function (3) is conveniently expressed as
where
satðuÞ ¼ bðuðtÞÞuðtÞ;
ð4Þ
8 u H > > > > uðtÞ < bðuðtÞÞ ¼ 1 > > uL > > : uðtÞ
ð5Þ
if uðtÞ > uH ; if uL 6 uðtÞ 6 uH ; if uðtÞ < uL
and such that 0 < b(u(t)) 6 1. We are interested in deriving an adaptive control law such that in the closed-loop system (2), state trajectories are regulated to a specified equilibrium point in the state space, even when the system is experiencing disturbance and input saturation. It is desired that x1 be regulated to equilibrium state x1r, where x1r is a given constant. From Eq. (1), if x1(t) = x1r, then x_ 1 ðtÞ ¼ 0 and x2(t) = x2r = x1r. For x1(t) = x2(t) = x1r, solving the differential equation for x3 results in x3 ðtÞ ¼ ebt x3 ð0Þ þ
x21r ð1 ebt Þ; b x2
which shows that x3(t) converges to x3r ¼ b1r when the time t ! 1. Therefore, for a regulated equilibrium point x1(t) = x1r , the control law seeks to steer the trajectory to the equilibrium point xr = (x1r, x1r, x3r)T, where (Æ)T denotes transposition of (Æ).
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3. Switching surface and error dynamics Let the new disturbance term be D(t) = d(t) x1x3, such that the nonlinear term is cancelled, then the dynamic equation with disturbance D(t) from (2) is given by x_ 1 r r x1 0 0 ¼ þ DðtÞ þ bðuðtÞÞ uðtÞ; ð6aÞ x_ 2 r 1 x2 1 1 ð6bÞ x_ 3 ¼ x1 x2 bx3 : Because the external disturbance jd(t)j 6 d, then D(t) satisfies jDðtÞj ¼ jdðtÞ x1 x3 j 6 jdðtÞj þ jx1 x3 j 6 d þ jx1 x3 j ¼ a: From Eq. (6a), it can be seen that the dynamics of x1(t) and x2(t) are decoupled from x3(t). Eqs. (6a) and (6b) show that x3(t) represents the internal dynamics of the control system, which will converge and stabilize when x1(t) and x2(t) con verge to x1r. z1 x1 According to the state transformation by Yang et al. [11], the new system state vector Z ¼ ¼ P is defined, z2 x2 1=r 0 where the transform matrix is P ¼ . Let the trajectory error states be e1 = z1 z1r, e1 = z1 z1r and 1 1 e3 = x3(t) x3r, where the transformed regulation system states are z1r ¼ r1 x1r and z2r = x1r + x2r = 0. Thus, the error state dynamic equations can be obtained as following: e_ 1 ¼ e2 ; e_ 2 ¼ rðr 1Þe1 ðr þ 1Þe2 þ rðr 1Þz1r þ DðtÞ þ satðuðtÞÞ ¼ f ðe1 ; e2 Þ þ DðtÞ þ bðuðtÞÞ uðtÞ;
ð7aÞ ð7bÞ
where f(e1, e2) = r(r 1)e1 (r + 1)e2 + r(r 1)z1r , and the internal dynamics of the error state is e_ 3 ¼ be3 þ gðe1 ; e2 Þ;
ð7cÞ
where g(e1, e2) = bx3r + re1e2 + re2z1r + r 2(e1 + z1r)2. Now, a sliding surface suitable for the application can be defined as sðtÞ ¼ c1 e1 ðtÞ þ e2 ðtÞ; where s(t) 2 R and c1 is a design parameter to be determined. For the existence of the sliding mode, it is necessary and sufficient that [14,15] sðtÞ ¼ c1 e1 ðtÞ þ e2 ðtÞ ¼ 0
ð8aÞ
s_ ðtÞ ¼ c1 e_ 1 ðtÞ þ e_ 2 ðtÞ ¼ 0:
ð8bÞ
and
Therefore, the following sliding mode dynamics can be obtained as e_ 1 ¼ c1 e1 ; e_ 2 ¼ rðr 1Þe1 ðr þ 1Þe2 þ rðr 1Þz1r þ DðtÞ þ satðuðtÞÞ; e_ 3 ¼ be3 þ gðe1 ; e2 Þ:
ð9aÞ ð9bÞ ð9cÞ
Obviously, if the design parameter c1 < 0, the stability of (9a) is guaranteed, that is limt!1e1(t) ! 0. Furthermore, by Eq. (8a), e2(t) is also stable, that is limt!1e2(t) ! 0. Since (e1(t), e2(t)) converge to zero, it follows that there exists a function h(t) such that jgðtÞj jgðe1 ; e2 Þj 6 hðtÞ
ð10Þ
and h(t) ! 0, as t ! 1. Thus given any h0, there exists a finite time t1 such that jgðtÞj < h0 ;
for t P t1 :
By solving (9c), gives bðtt1 Þ
e3 ðtÞ ¼ e
x3 ðt1 Þ þ
Z
t
ebðtsÞ gðsÞ ds: t1
ð11Þ
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Therefore, for t > t1 bðtt1 Þ
je3 ðtÞj 6 e
jx3 ðt1 Þj þ h0
Z
t bðtsÞ
e
bðtt1 Þ
ds ¼ e
t1
h0 h0 þ : jx3 ðt1 Þj b b
ð12Þ
Since h0 can be made arbitrarily small by taking t1 sufficiently large, and b > 0, it follows that limt!1e3(t) ! 0. Thus, the closed-loop system state (x1, x2, x3) ! xr, as t ! 1.
4. Adaptive controller design Before proceeding to the adaptive sliding mode control design, the Barbalat lemma is provided: Lemma 1 (Barbalat lemma [16]). If w : R ! R is a uniformly continuous function for t P 0 and if the limit of the integral Z t jwðkÞj dk ð13Þ lim t!1
0
exists and is finite, then lim wðtÞ ¼ 0:
ð14Þ
t!1
We choose a control law of the form uðtÞ ¼ n^cðtÞgsignðsðtÞÞ;
n > 1;
ð15Þ
where g = jc1e2j + jf(e1, e2)j + a. The adaptive law is ^c_ ðtÞ ¼ ng^c3 ðtÞjsðtÞj;
^cð0Þ ¼ ^c0 ;
ð16Þ
where ^c0 is the bounded positive initial value of ^cðtÞ. The adaptive law can be also rewritten in the integral form as Z t ^cðtÞ ¼ ^c0 þ n^c3 ðkÞgðkÞjsðkÞj dk: 0
Based on the control law (15), the reaching condition sðtÞ_sðtÞ < 0 is achieved using the following theorem, that is, the proposed scheme (15) will derive the system (7) with saturating input onto the sliding mode s(t) = 0. Remark 1. Since from theoretical point of view, s(t) will not be exactly equal to zero in finite time, thus the adaptive parameter ^c will increase (even if s(t) is a very small number) until s(t) = 0. A simple way for overcoming this disadvantage is to modify the adaptive law (16) by ‘‘dead-zone’’ technique [16] as qksk; ksk P d; ^c_ ðtÞ ¼ ð17Þ 0; ksk < d; where d is a small positive constant. Theorem 1. Consider the error dynamics system (7a) and (7b) with saturating input. The hitting condition of the sliding mode is satisfied, if the control u(t) is given by (15). Proof. Letting the Lyapunov function of the system be V ¼ 12 ½s2 ðtÞ þ ~c2 ðtÞ, then its derivative with respect to time is V_ ¼ sðtÞ_sðtÞ þ ~cðtÞ~c_ ðtÞ ¼ s½c1 e_ 1 þ e_ 2 þ ~c~c_ ¼ s½c1 e2 þ f ðe1 ; e2 Þ þ DðtÞ þ bðuðtÞÞ uðtÞ þ ~c ~c_ 6 jsjðjc1 e2 j þ jf ðe1 ; e2 Þj þ aÞ þ sbðuðtÞÞuðtÞ þ ~c~c_ 6 jsjg n^cgbðuðtÞÞjsðtÞj þ ~c~c_ :
ð18Þ
Since 0 < b(u) 2 R 6 1, according the density property of real number, there always exists a constant c > 0 satisfying 0 < c < bðuÞ 6 1;
8uðtÞ; t 2 ½0; 1Þ:
ð19Þ
Combined with (19), Eq. (18) yields V_ ðtÞ 6 gjsðtÞj nc^cðtÞgjsðtÞj þ ~cðtÞ~c_ ðtÞ:
ð20Þ
Now define the adaptation error as ~cðtÞ ¼ ^c1 ðtÞ c:
ð21Þ
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Since c is an existing constant then c_ ¼ 0. Thus the following expression holds: ~c_ ðtÞ ¼ ^c_ 1 ðtÞ c_ ¼ ^c_ 1 ðtÞ ¼ ^c2 ðtÞ^c_ ðtÞ:
ð22Þ
Inserting (16), (21) and (22) into the right hand of inequality (20) this yields V_ ðtÞ 6 gjsðtÞj þ ð^c1 ðtÞ cÞ n^cðtÞgjsðtÞj ngjsðtÞj ~cðtÞ^c2 ðtÞ^c_ ðtÞ 6 ðn 1ÞgjsðtÞj: |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
ð23Þ
~c
Therefore, if n > 1, then V_ < 0. Now if we define w(t) = (n 1)gkS(t)k, and integrate the above Eq. (23) from zero to t, it yields Z t Z t Z t wðkÞ dk ) V ð0Þ P V ðtÞ þ wðkÞ dk P wðkÞ dk: ð24Þ V ðtÞ 6 V ð0Þ 0
0
0
Taking the limit as t ! 1 on both sides of (24) gives Z t wðkÞ dk: 1 > V ð0Þ P lim t!1
ð25Þ
0
According to Barbalat lemma (see Lemma 1), It obtains lim wðtÞ ¼ lim ðn 1ÞgjsðtÞj ! 0:
t!1
ð26Þ
t!1
Since n > 1, g > 0, (26) implies s(t) ! 0 as t ! 1. Hence the proof is complete.
h
5. An Illustrative example In this section, simulation results are presented with initial condition (x1, x2, x3) = (10, 10, 10). For the uncontrolled Lorenz system, the parameters are r = 15, b ¼ 123, and r = 35, and the regulating equilibrium point is xr = (11.66, 11.66, 34)T. Let c1 = 10 to result in a stable sliding mode and the saturating input is defined as 8 if uðtÞ > uH ; > < uH ¼ 10 if uL 6 uðtÞ 6 uH ; satðuðtÞÞ ¼ uðtÞ ð27Þ > : uL ¼ 10 if uðtÞ < uL : Base on (16), parameter of adaptive control law is n = 1.5 to satisfy the condition (23). The external disturbance d(t) is assumed to be 0.5 cos(3pt), then jd(t)j 6 0.5 = d. Combining Eqs. (2), (15) and (27) gives the simulated closed-loop system. The simulation results with uH = uL = 10 are shown in Figs. 1–5. Fig. 1 shows the time response of sliding surface s(t). Fig. 2(a)–(c) represents respectively the time responses of error states of e1, e2 and e3, and the phase plane behaviour is shown in Fig. 3. Fig. 4 shows the state response for the controlled Lorenz system. Otherwise, the simulation results with uH = uL = 5 are shown in Figs. 5 and 6. Compare Fig. 4 with Fig. 5, it shows that the settling time will be decreased, if the saturation bound of control input is relaxed.
6 4
s(t)
2 0 –2 –4 –6 –8 0
control in action 1
2
3
4
5
6
7
8
9
Time(sec) Fig. 1. Time responses of sliding surface with uH = uL = 10.
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H.-T. Yau, C.-L. Chen / Chaos, Solitons and Fractals 34 (2007) 1567–1574 1 0.5
e1(t)
0 –0.5 –1 –1.5 –2 control in action –2.5
1
0
2
6
5
4
3
(a)
8
7
9
10
9
10
9
10
Time(sec) 20 15 10
e2(t)
5 0 –5 –10 –15
control in action 0
1
2
3
(b)
4
5
6
7
8
Time(sec) 30 20
e3(t)
10 0 –10 –20 control in action –30
0
1
2
3
4
5
6
7
8
Time(sec)
(c)
Fig. 2. Time response of error state (a) e1(t), (b) e2(t), (c) e3 (t) with uH = uL = 10.
15 A
10
e3(t)
1572
5 B 0 –5
–10 5 0.4
0
e2(t)
0.2 0
5
–0.2 10
–0.4
e1(t)
Fig. 3. The phase plane of error states with uH = uL = 10; the controller is turned at t = 5 (the point A).
x1, x2, x3
H.-T. Yau, C.-L. Chen / Chaos, Solitons and Fractals 34 (2007) 1567–1574 60 50 40 30 20 10 0 –10 –20 –30 –40
1573
x3
x1
x2 0
1
2
3
control in action
4
5
6
7
8
9
10
Time(sec)
x1, x2, x3
Fig. 4. The state time responses of Lorenz system with uH = uL = 10.
60 50 40 30 20 10 0 –10 –20 –30 –40
0
2
4
6
8
10
Time(sec) Fig. 5. The state time responses of Lorenz system with uH = uL = 5.
20
e3(t)
10
A
0
–10 –20 10 0.5
5 0
e2(t)
0
–5 –10
–0.5
e1(t)
Fig. 6. The phase plane of error states with uH = uL = 5; the controller is turned at t = 5 (the point A).
6. Conclusions In this paper, an adaptive sliding mode control method for the Lorenz chaos with input saturation is proposed. Based on the Lyapunov stability theory, the adaptive sliding mode controller can stabilize Lorenz chaos even in the existence of system external disturbance. Simulation results show that the proposed controller is able to control Lorenz chaos and the settling time will decrease if the saturation bound is relaxed.
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