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Sliding mode control for uncertain chaotic systems with input nonlinearity
Commun Nonlinear Sci Numer Simulat 17 (2012) 341–348
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Sliding mode control for uncertain chaotic systems with input nonlinearity Juntao Li a,⇑, Wenlin Li a, Qiaoping Li b a b
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, PR China The Mathematics Department of Henan Institute of Science and Technology, Xinxiang, Henan Province, PR China
a r t i c l e
i n f o
Article history: Received 9 June 2010 Received in revised form 17 January 2011 Accepted 10 April 2011 Available online 19 April 2011 Keywords: Unified chaotic system Uncertainty Input nonlinearity Sliding mode control
a b s t r a c t For the sliding mode controller of uncertain chaotic systems subject to input nonlinearity, the upper bound of the norm of uncertainties is commonly used to determine the controller parameter. However, this will cause serious chattering. In order to overcome this drawback, two new sliding mode controllers are proposed to ensure robust synchronization for a classes of chaotic systems with input nonlinearities and external uncertainty. Compared with the existing results, the proposed controllers can effectively reduce the chattering nearby sliding mode and improve the dynamic performance of the systems. Simulation results are provided to verify the proposed methods. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction As an important topic in nonlinear science, chaos synchronization has attracted much attention since it was proposed in 1990 by Pecora and Carroll. Now, chaos synchronization has been widely explored in a variety of systems including physical, chemical and ecological systems, and the application of chaos synchronization to secure communication has become a hot topic in various fields of related research [1–5]. There have existed some types of synchronization in dynamical systems, for example, generalized synchronization [6–9], phase synchronization [10,11], lag synchronization [12–14], anti-synchronization [15–17] and so on [18]. However, the control inputs are linear function or dead-zone nonlinear function. In fact, the control inputs of practical systems are frequently subject to nonlinearity as a result of physical limitations. It has been shown that input nonlinearity can cause a serious degradation of the system performance. In a worst-case scenario, system failure will occur if the controller is not well designed [19]. Therefore, the effects of input nonlinearity must be taken into account when analyzing and implementing a control scheme for the unified chaotic systems. However, a review of the published literature suggests that the problem of controlling chaotic systems subject to input nonlinearity has received relatively little attention. Synchronization between different chaotic systems with input nonlinearity has been studied in [11]. However, this method did not consider the uncertainties. By incorporating an adaptive control into sliding mode control, the chaos synchronization problem for drive-response Chua’s systems coupled with dead-zone nonlinear input has been presented. However, in general form, the concept of input nonlinearity in the presence of unknown un-modeled dynamic and uncertainties in chaos synchronization has not been studied. In this paper, we propose a new sliding mode controller to synchronize two different chaotic systems with unknown bounded uncertainties. The proposed controller is robust to the nonlinear input and guarantees the occurrence of sliding motion of the controlled chaotic systems with uncertainties. Furthermore, in the sliding mode, the investigated uncertain
⇑ Corresponding author. E-mail address: [email protected] (J. Li). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.018
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J. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 341–348
chaotic system with input nonlinearity still possesses advantages of fast response, good transient performance and insensitive to the uncertainties as the systems with linear input. The main theorem of the paper and simulation results will justify our claim. The rest of this paper is organized as follows. Section 2 presents the problem statement. Section 3 presents a new sliding mode controller. The simulations results for synchronization of Chua’s circuit and Genesio systems are given in Sections 4 and 5 concludes the paper. 2. Problem statement In this paper, we consider chaos synchronization for the unified chaotic system. The mathematical model for the unified chaotic system is given by
x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ; x_ 2 ¼ ð28 35aÞx1 þ ð29a 1Þx2 x1 x3 ; 8þa x3 ; x_ 3 ¼ x1 x2 3
ð1Þ
where x1, x2, x3 are state variables and parameter a 2 [0, 1]. Obviously, system (1) becomes the original Lorenz system for a = 0; while system becomes the original Chen system for a = 1. When a = 0.8, system becomes the critical system. It should be noted that system (1) bridges the gap between Lorenz system and Chen system. It should also be noted that system (1) is always chaotic in the whole interval a 2 [0, 1]. The aforementioned system (1) is represented by a precise mathematical model. It is well-known that a certain mathematical model can not always express the behavior of the real physical plant. In particular, due to the limit of physical devices and the influence of interference (such as noise, temperature, etc.), the system appears uncertainties. Let x1 be the lumped uncertainties representing the nonlinear parameter perturbations and the external disturbances. Taking the uncertainties x1 into consideration the system (1) can be represented by
x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ; x_ 2 ¼ ð28 35aÞx1 þ ð29a 1Þx2 x1 x3 þ x1 ðx; tÞ; 8þa x3 : x_ 3 ¼ x1 x2 3
ð2Þ
Taking the nonlinear input into consideration, the controlled system corresponding to (2) can be easily given by
y_ 1 ¼ ð25a þ 10Þðy2 y1 Þ; y_ 2 ¼ ð28 35aÞy1 þ ð29a 1Þy2 y1 y3 þ x2 ðy; tÞ þ uðuÞ; 8þa y_ 3 ¼ y1 y2 y3 ; 3
ð3Þ
where y1, y2, y3 are state variables, x2 represents the lumped uncertainties of the systems (3), u 2 R is the control input, u(u) is a continuous, nonlinear function u : R ? R, u(0) = 0 within the sector [a, b], i.e., 2
au2 6 uuðuÞ 6 bu ;
ð4Þ
where a and b are nonzero positive constants. For the convenience of controller design, we rewrite u(u) into the following form
uðuÞ ¼ kð1 þ hðt; uÞÞu;
ð5Þ ba . bþa
where k = (a + b)/2, jhðt; uÞj < d ¼ A nonlinear function satisfying the properties (4) and (5) is illustrated in Fig. 2. For example, nonlinear function u(u) = 0.6u + 0.3u sin u can be rewritten as
uðuÞ ¼ 0:6ð1 þ 0:5 sin uÞu; k ¼ 0:6;
hðu; tÞ ¼ 0:5 sin u:
Let the sate error be e = y x. According to (2) and (3), the following error system can be easily obtained
e_ 1 ¼ ð25a þ 10Þðe2 e1 Þ; e_ 2 ¼ ð28 35aÞe1 þ ð29a 1Þe2 y1 y3 þ x1 x3 þ xðx; y; tÞ þ uðuÞ; 8þa e_ 3 ¼ e1 e2 þ e1 x2 þ x1 e2 e3 ; 3 where x(x, y, t) = x2(y, t) x1(x, t).
ð6Þ
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The control goal considered in this paper is to design a sliding mode controller that can ensure the stability of the error system (6), i.e., limt?1e(t) = 0. 3. Switching surface design The sliding mode design method usually contains two steps involving the establishment of a sliding surface to yield desired performance and the design of a controller to ensure that the sliding mode is attained. First, we select the following switching surface:
s ¼ ce1 þ e2 ;
ð7Þ
where c is a constant: c > 1. When the system (4) operates on sliding mode s = 0, from (6) and (7) the sliding mode dynamics can be obtained as:
e_ 1 ¼ ð25a þ 10Þð1 þ cÞe1 ;
c > 1;
e2 ¼ ce1 ; e_ 3 ¼ ce21 þ ðx2 cx1 Þe1
ð8Þ
8þa e3 : 3
For the sliding mode dynamics, we have the following theorem: Theorem 1. The sliding mode dynamics (8) is asymptotically stable. Proof. Solve the first and third equation of (8) yields
e1 ðtÞ ¼ e1 ð0Þeð25aþ10Þð1þcÞt ;
ð9Þ
e2 ðtÞ ¼ ce1 : Since states of chaotic system are bounded, then jx2(t) cx1(t)j 6 M
je3 ðtÞj 6 je3 ð0Þjeð8þaÞt=3 þ
Z
t
eð8þaÞðtsÞ=3 je1 ðsÞjjx2 ðsÞ cx1 ðsÞjds
0
6 je3 ð0Þjeð8þaÞt=3 þ Mje1 ð0Þj
Z
t
eð8þaÞðtsÞ=3 eð25aþ10Þð1þcÞs ds < je3 ð0Þjeð8þaÞt=3 þ
0
3Mje1 ð0Þj ð8þaÞt=3 : e ð74a þ 22Þ
ð10Þ
Hence, the sliding mode dynamics (8) is asymptotically stable. h
4. Sliding mode controller design After establishing the appropriate switching surface (7), we will focus on designing a sliding mode controller to drive the system trajectories onto the sliding mode s = 0 even though uncertain and input nonlinearity are presented. Let us first analyze the evolution of s(t). To facilitate the derivation, we take c = 0, i.e.
s ¼ e2 :
ð11Þ
From (6) and (11), it follows that
s_ ¼ e_ 2 ¼ ð28 35aÞe1 þ ð29a 1Þe2 y1 y3 þ x1 x3 þ x þ uðuÞ ¼ F þ x þ uðuÞ;
ð12Þ
where F = (28 35a)e1 + (29a 1)e2 y1y3 + x1x3. Assumption 1. Suppose jx1(x, t)j + jx2(y, t)j 6 l. According to the traditional approach [1,2], the states of the error system (6) can be driven to the sliding surface if we select u = 1g sgn s, where 1 > 1a, g = jFj + l. However, this will incur a serious chattering near by the sliding surface when switching jFj is large. In fact, when sF < 0, it can be easily obtained that
ss_ ¼ sðF þ x þ uðuÞÞ 6 jFjjsj þ ljsj þ suðuÞ 6 gjsj þ suðuÞ:
ð13Þ
If we take u = 1g sgn s, then
suðuÞ ¼ suð1g sgn sÞ ¼
jsj sgn s
1g
1guð1g sgn sÞ:
Note that au2 6 uu(u) 6 bu2, uu(u) 6 au2
ð14Þ
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1g sgn suð1g sgn sÞ 6 að1g sgn sÞ2 ¼ að1gÞ2 :
ð15Þ
Hence, substituting (15) into (14) yields
suðuÞ 6 a1gjsj < gjsj ¼ ðjFj þ lÞjsj:
ð16Þ
From (13) and (16), we have
ss_ 6 gjsj þ suðuÞ 6 gjsj ðjFj þ lÞjsj 6 ðjFj þ l þ gÞjsj: Hence, a serious chattering will occur when switching gain jFj is large. In order to reduce the chattering, we design the following controller:
u¼
F þ q sgn s ; kð1 þ dÞ
ð17Þ
subject to
qP
2djFj þ ð1 þ dÞðl þ eÞ ; 1d
ð18Þ
where e is small positive constant which is much smaller than jFj. Theorem 2. If the control input u is designed as (17) subject to (18), then the state trajectory of the error system (6) will reach sliding mode s = 0 in finite time. Proof. From (5) and (12), we have
ss_ ¼ sðF þ x þ uðuÞÞ ¼ s½F þ x þ kð1 þ hÞu 6 sF þ ljsj þ ksð1 þ hÞu:
ð19Þ
Substituting (17) into (19) yields
ss_ 6 sF þ ljsj þ ksð1 þ hÞu ¼ sF þ ljsj sð1 þ hÞ ¼ ljsj qjsj sðh dÞ
F þ q sgn s : 1þd
F þ q sgn s F þ q sgn s ¼ sF þ ljsj sð1 þ d d þ hÞ 1þd 1þd
Note that jhj 6 d. Hence,
2dðjFj þ qÞ 1d 2djFj þ lð1 þ dÞ ð1 dÞq 2djFj lð1 þ dÞ jsj ¼ jsj ¼ jsj: ss_ 6 l q þ q 1þd 1þd 1þd 1þd
ð20Þ
Substituting (18) into (20) yields
ð1 dÞq þ 2djFj þ lð1 þ dÞ 2djFj ð1 þ dÞðl þ eÞ þ 2djFj þ lð1 þ dÞ ð1 þ dÞe djsj jsj 6 jsj 6 jsj ¼ ejsj 1þd 1þd dt 1þd 6 e:
ss_ 6
ð21Þ
So the state trajectory of the error system will reach sliding mode s = 0 in finite time. Note that e is much smaller than jFj + l + g. Hence, the chattering nearby sliding mode is largely reduced by using the controller (17) subject to (18). In order to improve the dynamic performance of the arrival process, we can take:
u¼
qP
F þ qs þ q sgn s ; cð1 þ dÞ
q>0
2dðjFjÞ þ ð1 þ dÞðl þ eÞ : 1d
ð22Þ
ð23Þ
In this case, we get
djsj jsj e; 6 q dt
¼ q
ð1 dÞ q: 1þd
This implies that controller (22) subject to (23) not only can speed up the rate of arriving sliding surface but also can improve the dynamic performance of the motion arriving sliding surface.
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J. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 341–348
Fig. 1a. Lorenz chaotic attractor.
20 15 10
x1
5 0 −5 −10 −15 −20
0
5
10
15
20
25
30
35
40
45
50
45
50
Fig. 1b. The state (x1) graph of Lorenz system.
30 20
x2
10 0 −10 −20 −30
0
5
10
15
20
25
30
35
40
Fig. 1c. The state (x2) graph of Lorenz system.
5. Simulation results In this section, we apply the simulation results of uncertain the Lorenz chaotic system to illustrate the effectiveness of the proposed approaches. Here we consider the uncertain Lorenz chaotic system (2) with uncertainty x1 = 0.5x2 sin x1 and the corresponding controlled system (3) with x2 = 0.3y2 cos y1 and u(u) = 0.6u + 0.3u sin u. Select the initial value [x1, x2, x3] = [3, 1, 2], [y1, y2, y3] = [2, 2, 1], the switching surface
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J. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 341–348 50 40
x3
30 20 10 0 −10 0
5
10
15
20
25
30
35
40
45
50
Fig. 1d. The state (x3) graph of Lorenz system.
ϕ (u )
ku
bu O
au
u
Fig. 2. Nonlinear function u(u).
Fig. 3a. The trajectories of the synchronization errors (e1, e2, e3) under controller (25) subject to (26).
s ¼ e1 þ e2 ¼ 0
ð24Þ
and take the sliding mode controller
u¼
F þ q sgn s ; kð1 þ dÞ
ð25Þ
subject to
qP
2dðjFjÞ þ ð1 þ dÞðl2 þ l1 þ eÞ ; 1d
ð26Þ
where F = 28e1 e2 y1y3 + x1x3, k = 0.6, d = 0.5, e = 0.1, l1 = 0.5, l2 = 0.3. The simulation results are shown in Figs. 3a–4.
J. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 341–348
Fig. 3b. The evolution of x1, y1.
Fig. 3c. The evolution of x2, y2.
Fig. 3d. The evolution of x3, y3.
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J. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 341–348
Fig. 4. The evolution of control law (25).
Fig. 3a gives the trajectories of the synchronization errors. It shows that all the synchronization errors convergence to the origin under controller (25) subject to (26) even when the system is undergoing uncertainty. Figs. 3a–3d give the trajectories of states variables x1,y1, x2,y2, x3,y3, respectively. It shows that the states variables have improved dynamic performance and the chattering is effectively reduced (compared with Figs. 1a, 1b, 1c, 1d, 2, 3a, 3b, 3c, 3d). Fig. 4 give the trajectories of the control input. Simulation results illustrate the applicability and the effectiveness of the proposed approach. 6. Conclusions In this paper, a sliding mode control strategy for uncertain chaotic systems subject to input nonlinearity is proposed. It has been shown that the proposed controller has the ability to eliminate the model uncertainties and to reduce the chattering on the sliding surface. In addition, computational simplicity of the proposed method is another prominent feature that should be emphasized. Acknowledgements This work was supported by National Natural Science Foundation of China (Nos. 60850004, 60774003), Foundation of Henan Educational Committee (2011B120005), Youth Science Foundation of Henan Normal University (2010qk01). References [1] Nana B, Woafo P, Domngang S. Chaotic synchronization with experimental application to secure communications. Commun Nonlinear Sci Numer Simul 2009;14(5):2266–76. [2] Grzybowski JMV, Rafikov M, Balthazar JM. Synchronization of the unified chaotic system and application in secure communication. Commun Nonlinear Sci Numer Simul 2009;14(6):2793–806. [3] Salarieh Hassan, Alasty Aria. Adaptive synchronization of two chaotic systems with stochastic unknown parameters. Commun Nonlinear Sci Numer Simul 2009;14(2):508–19. [4] Bowong Samuel. Adaptive synchronization between two different chaotic dynamical systems. Commun Nonlinear Sci Numer Simul 2007;12(6):976–85. [5] Fang JQ, Hong Y, Chen G. A switching manifold approach to chaos synchronization. Phys Rev E 1999;59:2523. [6] Lu J, Zhang S. Controlling Chen’s chaotic attractor using backstepping design based on parameters identification. Phys Lett A 2001;286:145–9. [7] Chen F, Zhang W. LMI criteria for robust chaos synchronization of a class of chaotic systems. Nonlinear Anal TMA 2007;67:3384–93. [8] Suykens JAK, Curran PF, Vandewalle J. Robust nonlinear synchronization of chaotic Lur’e systems. IEEE Trans Circuits Syst I 1997;44(10):891–904. [9] Yau, Her-Terng. Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Solitons Fract 2004;22:341–7. [10] Jui-Sheng Lin, Jun-Juh Yan. Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller. Nonlinear Anal: RWA 2009;10(2):1151–9. [11] Yau Her-Terng, Yan Jun-Juh. Chaos synchronization of different chaotic systems subjected to input nonlinearity. Appl Math Comput 2008;197(2, 1):775–88. [12] Hung Y-C, Yan J-J, Liao T-L. Projective synchronization of Chua’s chaotic systems with dead-zone in the control input. Math Comput Simul 2008;77(4):374–82. [13] Khalil HK. Nonlinear systems. New York: Macmillan; 1992. [14] Slotine JJE, Li W. Applied nonlinear control. Upper Saddle River, NJ: Prentice-Hall; 1991. [15] Chua LO, Lin GN. Canonical realization of Chua’s circuit family. IEEE Trans Circuits Syst 1990;37(1):885–902. [16] Genesio R, Tesi A. A harmonic balance method for the analysis of chaotic dynamics in nonlinear systems. Automatica 1992;28:531–48. [17] Li Zhi, Shi Songjiao. Robust adaptive synchronization of Rossler and Chen chaotic systems via slide technique. Phys Lett A 2003;311:389–95. [18] Hunga Y-C, Yan b J-J, Liao T-L. Projective synchronization of Chua’s chaotic systems with dead-zone in the control input. Math Comput Simul 2008;77:374–82. [19] Hsu KC. Adaptive variable structure control design for uncertain time-delay systems with nonlinear input. Dyn Contr. 1998;8:341–54.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Sliding mode control for uncertain chaotic systems with input nonlinearity Juntao Li a,⇑, Wenlin Li a, Qiaoping Li b a b
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, PR China The Mathematics Department of Henan Institute of Science and Technology, Xinxiang, Henan Province, PR China
a r t i c l e
i n f o
Article history: Received 9 June 2010 Received in revised form 17 January 2011 Accepted 10 April 2011 Available online 19 April 2011 Keywords: Unified chaotic system Uncertainty Input nonlinearity Sliding mode control
a b s t r a c t For the sliding mode controller of uncertain chaotic systems subject to input nonlinearity, the upper bound of the norm of uncertainties is commonly used to determine the controller parameter. However, this will cause serious chattering. In order to overcome this drawback, two new sliding mode controllers are proposed to ensure robust synchronization for a classes of chaotic systems with input nonlinearities and external uncertainty. Compared with the existing results, the proposed controllers can effectively reduce the chattering nearby sliding mode and improve the dynamic performance of the systems. Simulation results are provided to verify the proposed methods. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction As an important topic in nonlinear science, chaos synchronization has attracted much attention since it was proposed in 1990 by Pecora and Carroll. Now, chaos synchronization has been widely explored in a variety of systems including physical, chemical and ecological systems, and the application of chaos synchronization to secure communication has become a hot topic in various fields of related research [1–5]. There have existed some types of synchronization in dynamical systems, for example, generalized synchronization [6–9], phase synchronization [10,11], lag synchronization [12–14], anti-synchronization [15–17] and so on [18]. However, the control inputs are linear function or dead-zone nonlinear function. In fact, the control inputs of practical systems are frequently subject to nonlinearity as a result of physical limitations. It has been shown that input nonlinearity can cause a serious degradation of the system performance. In a worst-case scenario, system failure will occur if the controller is not well designed [19]. Therefore, the effects of input nonlinearity must be taken into account when analyzing and implementing a control scheme for the unified chaotic systems. However, a review of the published literature suggests that the problem of controlling chaotic systems subject to input nonlinearity has received relatively little attention. Synchronization between different chaotic systems with input nonlinearity has been studied in [11]. However, this method did not consider the uncertainties. By incorporating an adaptive control into sliding mode control, the chaos synchronization problem for drive-response Chua’s systems coupled with dead-zone nonlinear input has been presented. However, in general form, the concept of input nonlinearity in the presence of unknown un-modeled dynamic and uncertainties in chaos synchronization has not been studied. In this paper, we propose a new sliding mode controller to synchronize two different chaotic systems with unknown bounded uncertainties. The proposed controller is robust to the nonlinear input and guarantees the occurrence of sliding motion of the controlled chaotic systems with uncertainties. Furthermore, in the sliding mode, the investigated uncertain
⇑ Corresponding author. E-mail address: [email protected] (J. Li). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.018
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J. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 341–348
chaotic system with input nonlinearity still possesses advantages of fast response, good transient performance and insensitive to the uncertainties as the systems with linear input. The main theorem of the paper and simulation results will justify our claim. The rest of this paper is organized as follows. Section 2 presents the problem statement. Section 3 presents a new sliding mode controller. The simulations results for synchronization of Chua’s circuit and Genesio systems are given in Sections 4 and 5 concludes the paper. 2. Problem statement In this paper, we consider chaos synchronization for the unified chaotic system. The mathematical model for the unified chaotic system is given by
x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ; x_ 2 ¼ ð28 35aÞx1 þ ð29a 1Þx2 x1 x3 ; 8þa x3 ; x_ 3 ¼ x1 x2 3
ð1Þ
where x1, x2, x3 are state variables and parameter a 2 [0, 1]. Obviously, system (1) becomes the original Lorenz system for a = 0; while system becomes the original Chen system for a = 1. When a = 0.8, system becomes the critical system. It should be noted that system (1) bridges the gap between Lorenz system and Chen system. It should also be noted that system (1) is always chaotic in the whole interval a 2 [0, 1]. The aforementioned system (1) is represented by a precise mathematical model. It is well-known that a certain mathematical model can not always express the behavior of the real physical plant. In particular, due to the limit of physical devices and the influence of interference (such as noise, temperature, etc.), the system appears uncertainties. Let x1 be the lumped uncertainties representing the nonlinear parameter perturbations and the external disturbances. Taking the uncertainties x1 into consideration the system (1) can be represented by
x_ 1 ¼ ð25a þ 10Þðx2 x1 Þ; x_ 2 ¼ ð28 35aÞx1 þ ð29a 1Þx2 x1 x3 þ x1 ðx; tÞ; 8þa x3 : x_ 3 ¼ x1 x2 3
ð2Þ
Taking the nonlinear input into consideration, the controlled system corresponding to (2) can be easily given by
y_ 1 ¼ ð25a þ 10Þðy2 y1 Þ; y_ 2 ¼ ð28 35aÞy1 þ ð29a 1Þy2 y1 y3 þ x2 ðy; tÞ þ uðuÞ; 8þa y_ 3 ¼ y1 y2 y3 ; 3
ð3Þ
where y1, y2, y3 are state variables, x2 represents the lumped uncertainties of the systems (3), u 2 R is the control input, u(u) is a continuous, nonlinear function u : R ? R, u(0) = 0 within the sector [a, b], i.e., 2
au2 6 uuðuÞ 6 bu ;
ð4Þ
where a and b are nonzero positive constants. For the convenience of controller design, we rewrite u(u) into the following form
uðuÞ ¼ kð1 þ hðt; uÞÞu;
ð5Þ ba . bþa
where k = (a + b)/2, jhðt; uÞj < d ¼ A nonlinear function satisfying the properties (4) and (5) is illustrated in Fig. 2. For example, nonlinear function u(u) = 0.6u + 0.3u sin u can be rewritten as
uðuÞ ¼ 0:6ð1 þ 0:5 sin uÞu; k ¼ 0:6;
hðu; tÞ ¼ 0:5 sin u:
Let the sate error be e = y x. According to (2) and (3), the following error system can be easily obtained
e_ 1 ¼ ð25a þ 10Þðe2 e1 Þ; e_ 2 ¼ ð28 35aÞe1 þ ð29a 1Þe2 y1 y3 þ x1 x3 þ xðx; y; tÞ þ uðuÞ; 8þa e_ 3 ¼ e1 e2 þ e1 x2 þ x1 e2 e3 ; 3 where x(x, y, t) = x2(y, t) x1(x, t).
ð6Þ
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The control goal considered in this paper is to design a sliding mode controller that can ensure the stability of the error system (6), i.e., limt?1e(t) = 0. 3. Switching surface design The sliding mode design method usually contains two steps involving the establishment of a sliding surface to yield desired performance and the design of a controller to ensure that the sliding mode is attained. First, we select the following switching surface:
s ¼ ce1 þ e2 ;
ð7Þ
where c is a constant: c > 1. When the system (4) operates on sliding mode s = 0, from (6) and (7) the sliding mode dynamics can be obtained as:
e_ 1 ¼ ð25a þ 10Þð1 þ cÞe1 ;
c > 1;
e2 ¼ ce1 ; e_ 3 ¼ ce21 þ ðx2 cx1 Þe1
ð8Þ
8þa e3 : 3
For the sliding mode dynamics, we have the following theorem: Theorem 1. The sliding mode dynamics (8) is asymptotically stable. Proof. Solve the first and third equation of (8) yields
e1 ðtÞ ¼ e1 ð0Þeð25aþ10Þð1þcÞt ;
ð9Þ
e2 ðtÞ ¼ ce1 : Since states of chaotic system are bounded, then jx2(t) cx1(t)j 6 M
je3 ðtÞj 6 je3 ð0Þjeð8þaÞt=3 þ
Z
t
eð8þaÞðtsÞ=3 je1 ðsÞjjx2 ðsÞ cx1 ðsÞjds
0
6 je3 ð0Þjeð8þaÞt=3 þ Mje1 ð0Þj
Z
t
eð8þaÞðtsÞ=3 eð25aþ10Þð1þcÞs ds < je3 ð0Þjeð8þaÞt=3 þ
0
3Mje1 ð0Þj ð8þaÞt=3 : e ð74a þ 22Þ
ð10Þ
Hence, the sliding mode dynamics (8) is asymptotically stable. h
4. Sliding mode controller design After establishing the appropriate switching surface (7), we will focus on designing a sliding mode controller to drive the system trajectories onto the sliding mode s = 0 even though uncertain and input nonlinearity are presented. Let us first analyze the evolution of s(t). To facilitate the derivation, we take c = 0, i.e.
s ¼ e2 :
ð11Þ
From (6) and (11), it follows that
s_ ¼ e_ 2 ¼ ð28 35aÞe1 þ ð29a 1Þe2 y1 y3 þ x1 x3 þ x þ uðuÞ ¼ F þ x þ uðuÞ;
ð12Þ
where F = (28 35a)e1 + (29a 1)e2 y1y3 + x1x3. Assumption 1. Suppose jx1(x, t)j + jx2(y, t)j 6 l. According to the traditional approach [1,2], the states of the error system (6) can be driven to the sliding surface if we select u = 1g sgn s, where 1 > 1a, g = jFj + l. However, this will incur a serious chattering near by the sliding surface when switching jFj is large. In fact, when sF < 0, it can be easily obtained that
ss_ ¼ sðF þ x þ uðuÞÞ 6 jFjjsj þ ljsj þ suðuÞ 6 gjsj þ suðuÞ:
ð13Þ
If we take u = 1g sgn s, then
suðuÞ ¼ suð1g sgn sÞ ¼
jsj sgn s
1g
1guð1g sgn sÞ:
Note that au2 6 uu(u) 6 bu2, uu(u) 6 au2
ð14Þ
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1g sgn suð1g sgn sÞ 6 að1g sgn sÞ2 ¼ að1gÞ2 :
ð15Þ
Hence, substituting (15) into (14) yields
suðuÞ 6 a1gjsj < gjsj ¼ ðjFj þ lÞjsj:
ð16Þ
From (13) and (16), we have
ss_ 6 gjsj þ suðuÞ 6 gjsj ðjFj þ lÞjsj 6 ðjFj þ l þ gÞjsj: Hence, a serious chattering will occur when switching gain jFj is large. In order to reduce the chattering, we design the following controller:
u¼
F þ q sgn s ; kð1 þ dÞ
ð17Þ
subject to
qP
2djFj þ ð1 þ dÞðl þ eÞ ; 1d
ð18Þ
where e is small positive constant which is much smaller than jFj. Theorem 2. If the control input u is designed as (17) subject to (18), then the state trajectory of the error system (6) will reach sliding mode s = 0 in finite time. Proof. From (5) and (12), we have
ss_ ¼ sðF þ x þ uðuÞÞ ¼ s½F þ x þ kð1 þ hÞu 6 sF þ ljsj þ ksð1 þ hÞu:
ð19Þ
Substituting (17) into (19) yields
ss_ 6 sF þ ljsj þ ksð1 þ hÞu ¼ sF þ ljsj sð1 þ hÞ ¼ ljsj qjsj sðh dÞ
F þ q sgn s : 1þd
F þ q sgn s F þ q sgn s ¼ sF þ ljsj sð1 þ d d þ hÞ 1þd 1þd
Note that jhj 6 d. Hence,
2dðjFj þ qÞ 1d 2djFj þ lð1 þ dÞ ð1 dÞq 2djFj lð1 þ dÞ jsj ¼ jsj ¼ jsj: ss_ 6 l q þ q 1þd 1þd 1þd 1þd
ð20Þ
Substituting (18) into (20) yields
ð1 dÞq þ 2djFj þ lð1 þ dÞ 2djFj ð1 þ dÞðl þ eÞ þ 2djFj þ lð1 þ dÞ ð1 þ dÞe djsj jsj 6 jsj 6 jsj ¼ ejsj 1þd 1þd dt 1þd 6 e:
ss_ 6
ð21Þ
So the state trajectory of the error system will reach sliding mode s = 0 in finite time. Note that e is much smaller than jFj + l + g. Hence, the chattering nearby sliding mode is largely reduced by using the controller (17) subject to (18). In order to improve the dynamic performance of the arrival process, we can take:
u¼
qP
F þ qs þ q sgn s ; cð1 þ dÞ
q>0
2dðjFjÞ þ ð1 þ dÞðl þ eÞ : 1d
ð22Þ
ð23Þ
In this case, we get
djsj jsj e; 6 q dt
¼ q
ð1 dÞ q: 1þd
This implies that controller (22) subject to (23) not only can speed up the rate of arriving sliding surface but also can improve the dynamic performance of the motion arriving sliding surface.
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Fig. 1a. Lorenz chaotic attractor.
20 15 10
x1
5 0 −5 −10 −15 −20
0
5
10
15
20
25
30
35
40
45
50
45
50
Fig. 1b. The state (x1) graph of Lorenz system.
30 20
x2
10 0 −10 −20 −30
0
5
10
15
20
25
30
35
40
Fig. 1c. The state (x2) graph of Lorenz system.
5. Simulation results In this section, we apply the simulation results of uncertain the Lorenz chaotic system to illustrate the effectiveness of the proposed approaches. Here we consider the uncertain Lorenz chaotic system (2) with uncertainty x1 = 0.5x2 sin x1 and the corresponding controlled system (3) with x2 = 0.3y2 cos y1 and u(u) = 0.6u + 0.3u sin u. Select the initial value [x1, x2, x3] = [3, 1, 2], [y1, y2, y3] = [2, 2, 1], the switching surface
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x3
30 20 10 0 −10 0
5
10
15
20
25
30
35
40
45
50
Fig. 1d. The state (x3) graph of Lorenz system.
ϕ (u )
ku
bu O
au
u
Fig. 2. Nonlinear function u(u).
Fig. 3a. The trajectories of the synchronization errors (e1, e2, e3) under controller (25) subject to (26).
s ¼ e1 þ e2 ¼ 0
ð24Þ
and take the sliding mode controller
u¼
F þ q sgn s ; kð1 þ dÞ
ð25Þ
subject to
qP
2dðjFjÞ þ ð1 þ dÞðl2 þ l1 þ eÞ ; 1d
ð26Þ
where F = 28e1 e2 y1y3 + x1x3, k = 0.6, d = 0.5, e = 0.1, l1 = 0.5, l2 = 0.3. The simulation results are shown in Figs. 3a–4.
J. Li et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 341–348
Fig. 3b. The evolution of x1, y1.
Fig. 3c. The evolution of x2, y2.
Fig. 3d. The evolution of x3, y3.
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Fig. 4. The evolution of control law (25).
Fig. 3a gives the trajectories of the synchronization errors. It shows that all the synchronization errors convergence to the origin under controller (25) subject to (26) even when the system is undergoing uncertainty. Figs. 3a–3d give the trajectories of states variables x1,y1, x2,y2, x3,y3, respectively. It shows that the states variables have improved dynamic performance and the chattering is effectively reduced (compared with Figs. 1a, 1b, 1c, 1d, 2, 3a, 3b, 3c, 3d). Fig. 4 give the trajectories of the control input. Simulation results illustrate the applicability and the effectiveness of the proposed approach. 6. Conclusions In this paper, a sliding mode control strategy for uncertain chaotic systems subject to input nonlinearity is proposed. It has been shown that the proposed controller has the ability to eliminate the model uncertainties and to reduce the chattering on the sliding surface. In addition, computational simplicity of the proposed method is another prominent feature that should be emphasized. Acknowledgements This work was supported by National Natural Science Foundation of China (Nos. 60850004, 60774003), Foundation of Henan Educational Committee (2011B120005), Youth Science Foundation of Henan Normal University (2010qk01). References [1] Nana B, Woafo P, Domngang S. Chaotic synchronization with experimental application to secure communications. Commun Nonlinear Sci Numer Simul 2009;14(5):2266–76. [2] Grzybowski JMV, Rafikov M, Balthazar JM. Synchronization of the unified chaotic system and application in secure communication. Commun Nonlinear Sci Numer Simul 2009;14(6):2793–806. [3] Salarieh Hassan, Alasty Aria. Adaptive synchronization of two chaotic systems with stochastic unknown parameters. Commun Nonlinear Sci Numer Simul 2009;14(2):508–19. [4] Bowong Samuel. Adaptive synchronization between two different chaotic dynamical systems. Commun Nonlinear Sci Numer Simul 2007;12(6):976–85. [5] Fang JQ, Hong Y, Chen G. A switching manifold approach to chaos synchronization. Phys Rev E 1999;59:2523. [6] Lu J, Zhang S. Controlling Chen’s chaotic attractor using backstepping design based on parameters identification. Phys Lett A 2001;286:145–9. [7] Chen F, Zhang W. LMI criteria for robust chaos synchronization of a class of chaotic systems. Nonlinear Anal TMA 2007;67:3384–93. [8] Suykens JAK, Curran PF, Vandewalle J. Robust nonlinear synchronization of chaotic Lur’e systems. IEEE Trans Circuits Syst I 1997;44(10):891–904. [9] Yau, Her-Terng. Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Solitons Fract 2004;22:341–7. [10] Jui-Sheng Lin, Jun-Juh Yan. Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller. Nonlinear Anal: RWA 2009;10(2):1151–9. [11] Yau Her-Terng, Yan Jun-Juh. Chaos synchronization of different chaotic systems subjected to input nonlinearity. Appl Math Comput 2008;197(2, 1):775–88. [12] Hung Y-C, Yan J-J, Liao T-L. Projective synchronization of Chua’s chaotic systems with dead-zone in the control input. Math Comput Simul 2008;77(4):374–82. [13] Khalil HK. Nonlinear systems. New York: Macmillan; 1992. [14] Slotine JJE, Li W. Applied nonlinear control. Upper Saddle River, NJ: Prentice-Hall; 1991. [15] Chua LO, Lin GN. Canonical realization of Chua’s circuit family. IEEE Trans Circuits Syst 1990;37(1):885–902. [16] Genesio R, Tesi A. A harmonic balance method for the analysis of chaotic dynamics in nonlinear systems. Automatica 1992;28:531–48. [17] Li Zhi, Shi Songjiao. Robust adaptive synchronization of Rossler and Chen chaotic systems via slide technique. Phys Lett A 2003;311:389–95. [18] Hunga Y-C, Yan b J-J, Liao T-L. Projective synchronization of Chua’s chaotic systems with dead-zone in the control input. Math Comput Simul 2008;77:374–82. [19] Hsu KC. Adaptive variable structure control design for uncertain time-delay systems with nonlinear input. Dyn Contr. 1998;8:341–54.