Sliding mode control for chaotic systems based on LMI

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Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1410–1417 www.elsevier.com/locate/cnsns

Sliding mode control for chaotic systems based on LMI q Hua Wang *, Zheng-zhi Han, Qi-yue Xie, Wei Zhang School of Electronic, Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Received 21 April 2007; received in revised form 10 December 2007; accepted 10 December 2007 Available online 23 December 2007

Abstract This paper investigates the chaos control problem for a general class of chaotic systems. A feedback controller is established to guarantee asymptotical stability of the chaotic systems based on the sliding mode control theory. A new reaching law is introduced to solve the chattering problem that is produced by traditional sliding mode control. A dynamic compensator is designed to improve the performance of the closed-loop system in sliding mode, and its parameter is obtained from a linear matrix inequality (LMI). Simulation results for the well known Chua’s circuit and Lorenz chaotic system are provided to illustrate the effectiveness of the proposed scheme. Ó 2007 Elsevier B.V. All rights reserved. PACS: 05.45.Gg; 05.45.Ac Keywords: Chaos control; Sliding mode; Dynamic compensator; Linear matrix inequality (LMI)

1. Introduction Dynamic chaos has aroused considerable interest in many areas of science and technology due to its powerful applications in chemical reactions, power converters, biological systems, information processing, secure communication, etc. For quite a long period, many reports concluded that chaos was neither predictable nor controllable. However, the OGY method developed by Ott et al. [1] has reopened this research topic, and the study of chaos control begun. Many approaches have been proposed such as feedback linearization [2,3], differential geometric method [4] and backstepping [5], etc. There have been many reports on Chua’s circuit and Lorenz system. In [6] the authors studied chaos synchronization using control based on tridiagonal structure. In [7], the authors proposed a method for the synchronization of the fractional Lorenz system, Chen system and Chua’s circuit. For designing a robust controller of uncertain systems, sliding mode control is frequently adopted due to its inherent advantages of easy realization, rapid response, better transient performance, and insensitivity to variations in plant

q *

Supported by the National Natural Science Foundation of China (Grant No. 60674024). Corresponding author. Tel.: +86 21 34203655; fax: +86 21 62932083. E-mail address: [email protected] (H. Wang).

1007-5704/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2007.12.006

H. Wang et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1410–1417

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parameters or external disturbances. Usually, the designing process is divided into two steps. The first step is to design a sliding surface and ensure that the sliding mode equation is asymptotically stable. The second step is to design a controller that can drive the system states to the sliding surface in a finite time [8]. Recently, the sliding mode variable structure design technique has become one of the most popular methods in chaotic systems [9–12]. In [9], the authors presented the sliding mode controller, and in [10–12], the authors prosed the sliding mode control for robust chaos suppression. Compared with these methods, the design presented in this paper analyses the stability as well as the reaching time. In general, eliminating or reducing the chattering phenomenon is a difficult problem in sliding mode control. A saturation function method is proposed but it is not easy to proceed the stability analysis. In this paper, we adopt a new reaching law and deduce a continuous controller that can reduce the chattering phenomenon effectively. The sliding surface parameters are obtained by solving a linear matrix inequality (LMI). It can be solved easily by Matlab’s LMI toolbox. To overcome the difficulty of dealing with the nonlinear items in the chaotic systems, we add a dynamic compensator to improve the stability quality successfully. 2. System description In this paper, we mainly consider chaos control for Chua’s circuit and the Lorenz chaotic system. The mathematical model for the controlled Chua’s circuit is given by 8 > < x_ 1 ðtÞ ¼ x1 þ x2 þ x3 ; ð1Þ x_ 2 ðtÞ ¼ bx1 þ u1 ; > : x_ 3 ðtÞ ¼ aðx1 x3 gðx3 ÞÞ þ u2 ; where gðx3 Þ ¼ nx3 þ 12 ðm nÞðjx3 þ 1j jx3 1jÞ. When a ¼ 40; b ¼ 93:333; m ¼ 1:139; n ¼ 0:711 and u1 ¼ 0; u2 ¼ 0 the system has a chaotic attractor as shown in Fig. 1. The mathematical model for the controlled Lorenz system is 8 > < x_ 1 ðtÞ ¼ aðx2 x1 Þ; ð2Þ x_ 2 ðtÞ ¼ rx1 x2 x1 x3 þ u1 ; > : x_ 3 ðtÞ ¼ bx3 þ x1 x2 þ u2 : It has a chaotic attractor when a ¼ 10; b ¼ 83 ; r ¼ 28 and u1 ¼ 0; u2 ¼ 0 as shown in Fig. 2 and u ¼ ðu1 ; u2 Þ is the controller to be designed. These two chaotic systems can be unified as the following description: 0 0 þ u; ð3Þ x_ ðtÞ ¼ Ax þ f ðxÞ þ Bu ¼ Ax þ I f2 ðxÞ T

where x 2 R3 ; u ¼ ðu1 ; u2 Þ 2 R2 . A; I and 0 are matrices and vector with appropriate dimensions. It is direct to check when

x1

5

0

−5 −0.4

−0.2

−5 0

x2

0 0.2

0.4

5

x3

Fig. 1. Chua’s circuit chaotic attractor ðu ¼ 0Þ.

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H. Wang et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1410–1417 50 40

x3

30 20 10 0 −20

−10

50 0

10

x1

20

0

−50

x2

Fig. 2. Lorenz chaotic attractor ðu ¼ 0Þ.

0

1

B A ¼ @ b a

1 0 0

1

1

C 0 A; a

f 2 ðxÞ ¼ a

0 nx3 þ 12 ðm nÞðjx3 þ 1j jx3 1jÞ

;

system (3) will denote the Chua’s circuit (1). When 0 1 a a 0 x1 x3 B C A ¼ @ r 1 0 A; f 2 ðxÞ ¼ ; x1 x2 0 0 b system (3) will represent the Lorenz chaotic system (2). 3. Main results The sliding mode design method contains usually 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. This section follows such a line and consists of two subsections. Firstly, we establish a sliding surface which contains a dynamic compensator. Secondly, we adopt a new reaching law and design a continuous controller to ensure the states of the chaotic system reach the sliding surface in a finite time. 3.1. Sliding surface design In this section, our aim is to design an appropriate sliding surface for the chaotic system (3). The existence of the nonlinear item f2 ðxÞ makes the design of the sliding surface difficult. Thus, we introduce a dynamic compensator z_ ¼ Kx z;

ð4Þ

where z 2 R2 is the state of the compensator. K 2 R23 is a matrix to be designed later by solving a linear matrix inequality (LMI). Consider the following sliding surface: s ¼ Cx þ z;

ð5Þ

where C ¼ ½C 1 ; I; C 1 2 R21 , I is the 2 2 identity matrix and C 1 will be decided later. Differentiating the sliding surface along the solution of the system (3) we obtain s_ ¼ C x_ þ z_ ¼ CAx þ Cf ðxÞ þ CBu þ Kx z:

ð6Þ

From the definitions of C and B we have CB ¼ I. Let s_ ¼ 0. Then we can get the equivalent control ueq ueq ¼ CAx Cf ðxÞ Kx þ z:

ð7Þ


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Substitute (7) into (3) and it yields x_ ¼ Ax þ f ðxÞ þ BðCAx Cf ðxÞ Kx þ zÞ ¼ ðA BCA BKÞx þ Bz:

ð8Þ

On the sliding surface s ¼ 0, we have z ¼ Cx. System (7) can be written as x_ ¼ ðA BCA BK BCÞx ¼ ½A BðK þ C þ CAÞx:

ð9Þ

From the variable structure theory [13] we know that the sliding mode dynamic equation is in a lower dimension. Also we know that we have introduced a dynamic compensator z_ ¼ Kx z. This means the real sliding mode dynamic equation will have a dimension 3. Then the differential equation (9) is the dynamic sliding mode equation of the system composed by (3), (4) and (7). From (9) we can see that if ðA; BÞ is controllable, then there always exists a matrix M such that A þ BM < 0. The eigenvalues of the matrix A þ BM can be arbitrarily assigned and we can adopt the pole assignment method to decide the matrices K; C in (9). Nowadays, the LMI method has attracted many interests because the LMI can be solved easily by the Matlab’s toolbox. In the following we also try to obtain the matrices K; C from the solution of a LMI. Theorem 1. The sliding mode dynamic equation (9) is asymptotically stable if there exist general matrixes K and C such that the following LMI is feasible: 0 1 a13 ða11 þ 1Þc21 k 21 2a11 a12 ða11 þ 1Þc11 k 11 B C 2c11 a12 2k 12 2 c11 a13 k 13 c21 a12 k 22 A < 0; ð10Þ X¼@

2c21 a13 2k 23 2

where aij ; cij ; k ij ði; j ¼ 1; 2Þ are the elements of the matrix A, C and K, respectively. Proof. Consider V ðxÞ ¼ 12 xT x as the candidate Lyapunov function of the system (9). The time derivation of V ðxÞ along the solution of (9) is V_ ðxÞ ¼ xT ðA BCA BK BC þ AT AT C T BT K T BT C T BT Þx ¼ xT Xx:

ð11Þ

If X < 0 then V_ ðxÞ < 0. So the sliding mode equation (9) is asymptotically stable if the LMI (10) is feasible. h 4. Controller design We design a feedback control in this subsection. Let we start with a Lemma that introduced a new reaching law. Lemma 1 [14]. When we adopt the following reaching law: s_ ðtÞ ¼ ðl þ gkska1 Þs;

0 < a < 1; l; g > 0;

ð12Þ

the system state x(t) will reach the sliding surface fxjs ¼ 0g in a finite time t: t ¼

1 lksð0Þk1a þ g ln : lð1 aÞ g

ð13Þ

Proof. Left multiply Eq. (12) by sT and it leads to sT s_ ¼ ðl þ gkska1 Þksk2 :

ð14Þ

On the other hand 2

sT s_ ¼

1 dðsT sÞ 1 dksk dksk ¼ ksk ¼ : 2 dt 2 dt dt

ð15Þ

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It follows that dksk a ¼ lksk gksk dt

ð16Þ

from simple calculation we can get dt ¼

dksk kska dksk 1 dksk1a ¼ : a ¼ 1a lksk þ gksk 1 a lksk1a þ g lksk þ g

ð17Þ

To get the reaching time we can integral (17) from 0 to t. Let sðtÞ ¼ 0 then we can obtain the reaching time t as follows: Z sðtÞ 1a 1a 1 dksk 1 1 lksð0Þk þ g sðtÞ 1a t ¼ lnðlksk ln : ð18Þ þ gÞj ¼ ¼ sð0Þ 1 a sð0Þ lksk1a þ g lð1 aÞ lð1 aÞ g From (12), when t P t; sðtÞ 0. It proves the conclusion.

h

Theorem 2. Let the sliding surface be s ¼ Cx þ z and the feedback control be u ¼ CAx Cf Kx þ z ðl þ gkska1 Þs;

0 < a < 1; l; g > 0:

ð19Þ

Then the state starts from any initial point will move toward the switching surface and reach the sliding surface s ¼ 0 in a finite time. As a result, the closed-loop composed with system (3), (4) and (19) is globally asymptotically stable with this controller. Proof. Let us consider the candidate Lyapunov function: 1 V ðtÞ ¼ sT ðtÞsðtÞ: 2

ð20Þ

Its derivative along the trajectory of (3) is V_ ðtÞ ¼ sT ½CAxðtÞ þ Cf ðxÞ þ CBu þ z_ ¼ sT ½CAxðtÞ þ Cf ðxÞ CAx Cf Kx þ z ðl þ gkska1 Þs þ Kx z ¼ sT ðl þ gkska1 Þs 2

a

¼ lksk gksk :

ð21Þ

From Lemma 1, the state of the chaotic system (3) will move toward the sliding surface and reach the sliding surface s ¼ 0 in a finite time. Once the state reach the sliding surface s ¼ 0 it will retain on the sliding surface s ¼ 0. When the state is on the sliding surface the dynamic behavior of the system can be represented by the differential equation (9). Combined with Theorem 1, the sliding mode dynamic equation (9) is asymptotically stable. So the chaotic system (3) is asymptotically stable with the controller (19). This completes the proof. h 5. Numerical simulation In this section, we apply the fourth-order Runge–Kutta integration method to obtain the solutions of the examples. The models for the Chua’s circuit and Lorenz system are also used in examples of [9] where Jamal and Ammar proposed sliding mode control for Chua’s circuit and Lorenz system. In all the simulation process, the initial value of the chaotic system is ðx1 ð0Þ; x2 ð0Þ; x3 ð0ÞÞ ¼ ð1; 0; 1Þ. The initial state of the compensator is ðz1 ð0Þ; z2 ð0ÞÞ ¼ ð1; 2Þ. The parameters of the controller is l ¼ 1, g ¼ 2, a ¼ 12. 5.1. Chua’s circuit For Chua’s circuit, we can obtain through Matlab’s LMI toolbox that c11 ¼ 71:4856, c21 ¼ 137:9351, k 11 ¼ 1, k 12 ¼ 70:9894, k 13 ¼ 33:2247, k 21 ¼ 1, k 22 ¼ 33:2247, k 23 ¼ 138:4314. Fig. 3 shows the state of Chua’s circuit converges to zero quickly by using the proposed sliding mode control (19). Fig. 4 shows time

H. Wang et al. / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1410–1417 1

x1 x2 x3

0.5

x1,x2,x3

1415

0 −0.5 −1 −1.5 −2

0

2

4

6

8

10

time(s) Fig. 3. State response of the Chua’s circuit under the sliding mode controller.

200

u1 u2

150

u1,u2

100 50 0 −50 −100 −150

0

2

4

6

8

10

time(s) Fig. 4. Controller response for the Chua’s circuit.

response of the controller and we can see that the controller is continuous and it has reduced the chattering phenomenon effectively. 5.2. Lorenz system Through Matlab’s LMI toolbox we can obtain the parameters of the controller (19) for Lorenz system are c11 ¼ 0:0497, c21 ¼ 0, k 11 ¼ 10:4476, k 12 ¼ 8:9989, k 13 ¼ 0, k 21 ¼ 0, k 22 ¼ 0, k 23 ¼ 9:4963. In [9], Jamal and 1

x1 x2 x3

0.8 0.6

x1,x2,x3

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

0.5

1

1.5

2

2.5

3

time(s) Fig. 5. Time responses of the Lorenz chaotic system with Jamal’s controller.

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Ammar proposed sliding mode control for Chua’s circuit and Lorenz system. Here we use Jamal’s controller and our controller to control the Lorenz system respectively and the results are shown in Figs. 5–8. Fig. 5 shows the state response of the controlled Lorenz system with Jamal’s controller and Fig. 6 shows the state response of the controlled Lorenz system with our controller. From the figures, we can see the state converge to zero quickly with our controller (19). Fig. 7 shows the time response of the control input of Jamal’s and 1

x1 x2 x3

0.8 0.6

x1,x2,x3

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

0.5

3

2.5

2

1.5

1

time(s) Fig. 6. Time responses of the Lorenz chaotic system with our controller.

2 0 −2 −4

u

−6 −8 −10 −12 −14 −16 −18 0

2

4

6

8

10

time(s) Fig. 7. Time response of Jamal’s controller for the Lorenz chaotic system.

20

u1 u2

10

u1,u2

0 −10 −20 −30 −40 −50

0

2

4

6

8

time(s) Fig. 8. Time response of our controller for the Lorenz chaotic system.

10


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Fig. 8 shows the time response of our controller. From Fig. 7, we can see there are serious high frequency chattering in Jamal’s control signal. Since we have adopted the new reaching law we can see from Fig. 8 the chattering phenomenon has been reduced effectively. 6. Conclusion In this paper, sliding mode method is applied to control a class of chaotic systems. Though we mainly consider Chua’s circuit and the Lorenz chaotic system in this paper, it can be seen that this sliding mode control method can be extended to other chaotic systems such as Ro¨ssler sytem, Liu chaotic system, Chen system, Lu¨ system, etc. The parameters of the sliding surface and the controller can be selected by solving a linear matrix inequality. Also, the sliding mode equation is simplified and the stability quality is improved by the dynamic compensator. Simulation results verifies the effectiveness of the proposed method. References [1] Ott E, Grebogi C, Yorke JA. Using chaos to direct trajectories to targets. Phys Rev Lett 1990;65(26):3215–8. [2] Gallegos JA. Nonlinear regulation of a Lorenz system by feedback linearization techniques. Dyn Control 1994;4:277–98. [3] Yassen MT. Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos Solitons Fract 2005;26:913–20. [4] Fuh CC, Tung PC. Controlling chaos using differential geometric method. Phys Rev Lett 1995;75:2952–5. [5] Li Guo-Hui. Projective synchronization of chaotic system using backstepping control. Chaos Solitons Fract 2006;29:490–4. [6] Liu B et al. Synchronizing chaotic systems using control based on tridiagonal structure. Chaos Solitons Fract 2007. doi:10.1016/ j.chaos.2007.06.099. [7] Li CP, Yan JP. The synchronization of three fractional differential systems. Chaos Solitons Fract 2007;32:751–7. [8] Utkin VI. Sliding mode and their application in variable structure systems. Moscow: Mir Editors; 1978. [9] Nazzal Jamal M, Natsheh Ammar N. Chaos control using sliding-mode theory. Chaos Solitons Fract 2007;33:695–702. [10] Liao Teh-Lu, Yan Jun-Juh, Hou Yi-You. Nonlinear Anal 2007. doi:10.1016/j.na.2007.04.036. [11] Chang J-F et al. Controlling chaos of the family of Rossler systems using sliding mode control. Chaos Solitons Fract 2006. doi:10.1016/j.chaos.2006.09.051. [12] Chiang TY, Hung ML, Yan JJ, Yang YS, Chang JF. Sliding mode control for uncertain unified chaotic systems with input nonlinearity. Chaos Solitons Fract 2007;34:437–42. [13] Gao Weibing. Foundation of variable structure control [M]. Beijing: Science Press; 1993. [14] Li Wen-lin, Wang Hua. Robust sliding-mode control for uncertain time-delay systems. Dyn Contin Discr Impuls Syst—Ser A-Math Anal I 2006;Part 1 Suppl(13):121–5.