- Email: [email protected]
An LMI criterion for linear-state-feedback based chaos synchronization of a class of chaotic systems
Chaos, Solitons and Fractals 26 (2005) 437–443 www.elsevier.com/locate/chaos
An LMI criterion for linear-state-feedback based chaos synchronization of a class of chaotic systems Guo-Ping Jiang a b
a,*
, Wei Xing Zheng
b
Department of Electronic Engineering, Nanjing University of Posts and Telecommunications, P.O. Box 145, Nanjing, 210003, PR China School of Quantitative Methods and Mathematical Sciences, University of Western Sydney, Penrith South DC NSW 1797, Australia Accepted 7 January 2005
Abstract Based on the Lyapunov stability theory in control theory, a new sufficient condition is proposed in this paper for chaos synchronization by the linear-state-feedback approach for a class of chaotic systems. By using Schur theorem and some matrix techniques, this criterion is then transformed into the Linear Matrix Inequality (LMI) form, which can be easily verified and resolved using the MATLAB LMI Toolbox. It is shown that under the proposed criterion chaos synchronization can be achieved at an exponential convergence rate. The effectiveness of the criterion proposed herein is verified and demonstrated by the chaotic Murali–Lakshmanan–Chua system. 2005 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, chaos synchronization has attracted many researchers interests [1–9,17–21,24,25,27–30]. Chaos synchronization can be viewed from a state-observer perspective, in the sense that the response system can be regarded as the state-observer of the drive system [1,3,6,8,10,18,19,27]. In the state-observer based approach, the output (or driving signal) can be chosen to be a linear or nonlinear combination or function of the system state variables. Recently, it has been reported that there is a limitation in the traditional proportional state observer, where the output or measurement disturbance is amplified by the proportional gain [26]. To relax this limitation, the so-called integral observer has been proposed, first for linear systems, and then it has been extended to a class of nonlinear systems lately. It has been shown that the integral observer can obtain a better performance in reducing the effect of the measurement or output disturbances as compared with the traditional proportional observer [26]. Similar deficiency has also been found in [28] for chaos synchronization and chaotic secure communications based on traditional proportional observer when disturbances and noises exist in the transmission channel. On the other hand, chaos synchronization can also be considered from a dynamical control viewpoint [5,8]. This is because chaos synchronization can be regarded as a model-tracking problem, in which the response system can track the drive system asymptotically by using the linear-state-feedback approach. Similarly to the state-observer based
*
Corresponding author. Tel.: +86 25 8349 2043; fax: +86 25 8349 2042. E-mail addresses: [email protected] (G.-P. Jiang), [email protected] (W.X. Zheng).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.01.012
438
G.-P. Jiang, W.X. Zheng / Chaos, Solitons and Fractals 26 (2005) 437–443
synchronization approach, both linear-state-feedback and nonlinear-state-feedback control schemes can be used for chaos synchronization. Due to its simple configuration, easy implementation and robustness to the transmission noise in the channel, the linear-state-feedback scheme is especially attractive and has been commonly adopted for practical application. For this reason, based on the Lyapunov stability theory [12,22] and Linear quadratic optimal control theory [23], a matrix inequality condition has been developed for chaos synchronization via linear-state-feedback approach for a class of chaotic systems in our recent paper [5]. Note that, only a scalar feedback has been used through a single connection for the design of chaos synchronization, which can be easily applied to secure communication based on chaos synchronization [28]. In this paper, chaos synchronization is further studied by using the linear-state-feedback control approach. An easily verified sufficient condition, formulated in the LMI form, is developed for chaos synchronization, which is applicable to a large class of general chaotic systems. The proposed criterion can ensure chaos synchronization at an exponential convergence rate. An example is included to verify the effectiveness of the criterion proposed in the paper. The rest of this paper is organized as follows. In Section 2, an LMI criterion is presented for chaos synchronization via linear-state-feedback approach for a class of chaotic systems. In Section 3, the effectiveness of the proposed criterion is demonstrated by the chaotic Murali–Lakshmanan–Chua system [7]. Some concluding remarks are drawn in Section 4.
2. LMI criterion of chaos synchronization Consider a chaotic system described by x_ ¼ Ax þ gðxÞ þ hðtÞ; n·1
n·n
ð1Þ n·1
where x 2 R , A 2 R , h(t) 2 R Lipschitz condition, namely,
is the external input signal, and g(x) is a continuous nonlinear function, satisfying
~k: kgðxÞ gð~ xÞk 6 qkx x
ð2Þ
Note that in (2) k Æ k denotes the Euclidean norm and q is the Lipschitz constant. Based on the linear-state-feedback approach, the response system can be constructed as follows [5]: ~_ ¼ A~ x x þ gð~ xÞ þ hðtÞ þ Bðy ~y Þ; y ¼ Kx; ~y ¼ K~ x;
ð3Þ
where y = Kx is a scalar chaotic signal, which is a linear combination of the state variables of the drive system and will be transmitted to the response system, K = [k1, k2, . . . , kn] is the feedback gain to be designed, and B = [b1, b2, . . . , bn] 0 is chosen such that (A, B) is controllable. Remark 1. It is noted that, the synchronization scheme (3) is based on the state-feedback approach, and only a scalar variable, which is the linear combination of the state variables is used for chaos synchronization, so it is convenient to apply in real applications. Remark 2. The most attractive feature of the linear-state-feedback approach is that it can reduce the influence of noise on the performance of communication systems when it is applied to chaotic secure communications [28]. In the linearstate-feedback-based chaotic secure communication system [28], one can choose a special matrix B with simple configuration, which just ensures that (A, B) is controllable, to make the message signal s easily injected into the driver. On the other hand, due to the choice of the special matrix B, the message signal s does not have to be very small, and consequently, the communication system can have a better performance in SNR. However, for the state-observer based approach, the message signal s multiplied by the state-observer gain matrix L, which, in general, is quite large to ensure the error dynamics system asymptotically stable, is injected into the driver. So a very smaller s is required to guarantee the drive system to be chaotic, leading to a poor performance in SNR. ~ ! x. Our aim is to find a suitable B and K, such that chaos synchronization is achieved, i.e., x From (1)–(3), we can get the following error dynamics system ~_ ¼ Ax þ gðxÞ þ hðtÞ ðA~ ~Þ x þ gð~ xÞ þ hðtÞ þ Bðy ~y ÞÞ ¼ Ae þ gðxÞ gð~ xÞ BKðx x e_ ¼ x_ x ¼ Ae BKe þ gðxÞ gð~ xÞ ¼ ðA BKÞe þ gðxÞ gð~ xÞ; ~ is the error vector. where e ¼ x x
ð4Þ
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439
Theorem 1. If a suitable matrix B is chosen such that (A, B) is controllable, and a suitable feedback gain K is selected such that AT P þ PA KT BT P PBK þ q2 PP þ I þ 2dP < 0;
ð5Þ
where P is a positive definite symmetric matrix, I is the identity matrix, and d is a positive constant, then the error dynamical system (4) is globally exponentially stable, implying that the coupled systems (1) and (3) are globally exponentially synchronized. Proof. Define a Lyapunov function V = eTPe, where P is a positive definite and symmetric constant matrix. Differentiating V along the error dynamical trajectory (4) and using (2) yield V_ ¼ e_ T Pe þ eT P_e ¼ ½ðA BKÞe þ gðxÞ gð~ xÞ T Pe þ eT P½ðA BKÞe þ gðxÞ gð~ xÞ xÞ T Pe 6 eT ððA BKÞT P þ PðA BKÞÞe þ 2qkek kPek: ¼ eT ððA BKÞT P þ PðA BKÞÞe þ 2½gðxÞ gð~ Since 2kek Æ qkPek 6 q2kPek2 + kek2, using (5) we further have V_ 6 eT ððA BKÞT P þ PðA BKÞÞe þ q2 kPek2 þ kek2 ¼ eT ðAT P þ PA PBK KT BT P þ q2 PP þ IÞe 6 2deT Pe ¼ 2dV < 0: Based on the Lyapunov stability theory, the error dynamical system (4) is globally exponentially stable, and hence, the two coupled systems (1) and (3) are globally exponentially synchronized. h S11 S12 , where S11 ¼ ST11 , S12 ¼ ST21 , Lemma 1 [(Schur Complements [11])]. For a given symmetric matrix S ¼ S21 S22 S22 ¼ ST22 , the condition S < 0 is equivalent to S22 < 0;
and
T S11 S12 S1 22 S12 < 0:
Using Lemma 1, the condition (5) can be easily transformed to be " # PA þ AT P þ 2ðd qÞP PBK KT BT P qP þ I < 0: ð6Þ qP þ I I 1 P 0 If (6) is multiplied by from the left-hand and right-hand sides, respectively, and letting X = P1 and 0 I W = (KP1)T, then (6) can easily be further transformed into the following LMI form (7). Theorem 2. If suitable matrices X and W are selected such that the following LMI " # AX þ XAT þ 2ðd qÞX WBT BWT qI þ X < 0; qI þ X I
ð7Þ
is satisfied, then the error dynamical system (4) with the feedback gain K = WTX1, is globally exponentially stable, implying that the coupled systems (1) and (3) are globally exponentially synchronized. Remark 3. The feasible set of X and W satisfying (7) can be easily found by using the LMI Toolbox in MATLAB software.
3. An illustrative example In this section, the chaotic Murali–Lakshmanan–Chua circuit is simulated to illustrate the above-derived criterion for chaos synchronization. The chaotic Murali–Lakshmanan–Chua circuit is described by [7] 0 1 x1 f ðx1 Þ x_ 1 0 ¼ þ þ ; ð8Þ b r x2 x_ 2 0 F sinðxtÞ where r > 0, b > 0, F > 0, x > 0 and f(Æ) is a piecewise linear function 1 f ðx1 Þ ¼ bx1 þ ða bÞðjx1 þ 1j jx1 1jÞ; 2
ð9Þ
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with a < b < 0. From (9), one easily has [4,5] f ðx1 Þ f ð~x1 Þ ¼ k x1;~x1 ðx1 ~x1 Þ;
ð10Þ
where k x1;~x1 depends on x1 and ~x1 , and varies within the interval [a, b] for t P 0. Hence, k x1;~x1 is bounded as f ðx1 Þ 0 0 1 and gðxÞ ¼ , hðtÞ ¼ . a 6 k x1;~x1 6 b < 0. From (8), we get A ¼ F sinðwtÞ b r 0 b Choose B ¼ 1 , where b1 5 0. Obviously, (A, B) is controllable. Referring to (3), the response system can be con0 structed by " # f ð~x1 Þ 0 1 0 ~x1 b1 ~x_ 1 þ ¼ ðy ~y Þ; þ þ _~x2 0 0 b r ~x2 F sinðxtÞ ð11Þ y ¼ Kx; ~y ¼ K~ x: Subtracting (11) from (8) gives b1 e1 0 1 e1 ðf ðx1 Þ f ð~x1 ÞÞ e_ 1 K ¼ þ ; 0 e_ 2 0 e2 b r e2
ð12Þ
where e1 ¼ x1 ~x1 and e2 ¼ x2 ~x2 are the errors. Furthermore, (12) can be rewritten as e_ ¼ Ae þ gðxÞ gð~ xÞ BKe; e1 f ð~x1 Þ . where e ¼ and gð~ xÞ ¼ 0 e2
ð13Þ
Consider gðxÞ gð~ xÞ ¼ where Mx;~x ¼
k x1;~x1 0
ðf ðx1 Þ f ð~x1 ÞÞ 0
¼
k x1;~x1 ðx1 ~x1 Þ 0
¼
k x1;~x1
0
0
0
e1 e2
¼ Mx;~x e;
0 . 0
Fig. 1. Chaotic signals and chaotic attractor of MLC circuit.
ð14Þ
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441
Since a 6 k x1;~x1 6 b < 0, letting k x1;~x1 ¼ aþb þ k x1;~x1 with ab 6 k x1;~x1 6 ab yields 2 2 2 aþb k x1;~x1 0 2 0 k x1;~x1 0 ~Þ þ ~Þ ¼ ~Þ ðx x ðx x gðxÞ gð~ xÞ ¼ ðx x 0 0 0 0 0 0 ~Þ þ gðxÞ gð~ ~ Þ þ Mx;~x ðx x ~Þ; xÞ ¼ A1 ðx x ¼ A1 ðx x
ð15Þ
where A1 ¼
aþb 2 0
0 ; 0
Mx;~x ¼
k x1;~x1 0
0 ; 0
and
gðxÞ gð~ xÞ ¼
k x1;~x1 0
0 ~ Þ: ðx x 0
Then, (13) can be rewritten as e_ ¼ Ae þ gðxÞ gð~ xÞ BKe ¼ ðA þ A1 Þe þ gðxÞ gð~ xÞ BKe ¼ Ae þ gðxÞ gð~ xÞ BKe; aþb 2 1 where A ¼ A þ A1 ¼ , and b r a b kek ¼ qkek; ~Þ 6 k x1;~x1 kek ¼ kgðxÞ gð~ xÞk ¼ Mx;~x ðx x 2 . where q ¼ ab 2
Fig. 2. Chaos synchronization of chaotic Murali–Lakshmanan–Chua circuit.
Fig. 3. Chaos synchronization of chaotic Murali–Lakshmanan–Chua circuit with the transmission noise.
ð16Þ
ð17Þ
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Remark 4. The aim of transforms in (15)–(17) is to increase the flexibility of the design. Since k x1;~x1 < k x1;~x1 , the value of q can be reduced. Consequently, the value range of the matrix K is increased. Remark 5. It is noted that, the choice of vector B is not unique, and can be selected as any other values provided that BÞ is controllable. ðA; In the following, chaos synchronization based on the chaotic Murali–Lakshmanan–Chua circuit is demonstrated. The parameters of the circuit used are r = 1.015, b = 1.0, F = 0.15, x = 0.75, a = 1.02, b = 0.55, for which the system exhibits chaotic behavior, as shown in Fig. 1. We have q = 0.235 from (17). Choose b1 = 1 and d = 0.5. Then one easily obtains K ¼ ½ 3:9368 0:5452 from (7) by using the MATLAB LMI Toolbox. According to Theorem 2, chaos synchronization is achieved, as shown in Figs. 2 and 3, where Fig. 3 is for the case with the transmission noise d(t), which is a random noise with zero mean (see Fig. 3(a)). From Fig. 3, we can see that the effect of the transmission noise on chaos synchronization can be suppressed, which shows that the scheme has strong robustness to the transmission noise.
4. Conclusions A new LMI criterion has been derived for chaos synchronization via the Lyapunov stability theory and the Linear Matrix Inequality technique. The criterion has been applied to the chaotic Murali–Lakshmanan–Chua circuit for illustration. It is noted that the scheme and criterion proposed herein are applicable to a large class of chaotic systems. Similar approach can be applied to the Chuas circuit [16], Ro¨ssler system [15], Lorenz system [14], Chens circuit [13] and so on.
Acknowledgements This work was supported in part by Jiangsu Qing Lan Project Program [JS200407], China, in part by the Key Project of Natural Science Foundation of Universities of Jiangsu Province [04KJA510092], China, and in part by a research grant from the Australian Research Council [A00102928].
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An LMI criterion for linear-state-feedback based chaos synchronization of a class of chaotic systems Guo-Ping Jiang a b
a,*
, Wei Xing Zheng
b
Department of Electronic Engineering, Nanjing University of Posts and Telecommunications, P.O. Box 145, Nanjing, 210003, PR China School of Quantitative Methods and Mathematical Sciences, University of Western Sydney, Penrith South DC NSW 1797, Australia Accepted 7 January 2005
Abstract Based on the Lyapunov stability theory in control theory, a new sufficient condition is proposed in this paper for chaos synchronization by the linear-state-feedback approach for a class of chaotic systems. By using Schur theorem and some matrix techniques, this criterion is then transformed into the Linear Matrix Inequality (LMI) form, which can be easily verified and resolved using the MATLAB LMI Toolbox. It is shown that under the proposed criterion chaos synchronization can be achieved at an exponential convergence rate. The effectiveness of the criterion proposed herein is verified and demonstrated by the chaotic Murali–Lakshmanan–Chua system. 2005 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, chaos synchronization has attracted many researchers interests [1–9,17–21,24,25,27–30]. Chaos synchronization can be viewed from a state-observer perspective, in the sense that the response system can be regarded as the state-observer of the drive system [1,3,6,8,10,18,19,27]. In the state-observer based approach, the output (or driving signal) can be chosen to be a linear or nonlinear combination or function of the system state variables. Recently, it has been reported that there is a limitation in the traditional proportional state observer, where the output or measurement disturbance is amplified by the proportional gain [26]. To relax this limitation, the so-called integral observer has been proposed, first for linear systems, and then it has been extended to a class of nonlinear systems lately. It has been shown that the integral observer can obtain a better performance in reducing the effect of the measurement or output disturbances as compared with the traditional proportional observer [26]. Similar deficiency has also been found in [28] for chaos synchronization and chaotic secure communications based on traditional proportional observer when disturbances and noises exist in the transmission channel. On the other hand, chaos synchronization can also be considered from a dynamical control viewpoint [5,8]. This is because chaos synchronization can be regarded as a model-tracking problem, in which the response system can track the drive system asymptotically by using the linear-state-feedback approach. Similarly to the state-observer based
*
Corresponding author. Tel.: +86 25 8349 2043; fax: +86 25 8349 2042. E-mail addresses: [email protected] (G.-P. Jiang), [email protected] (W.X. Zheng).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.01.012
438
G.-P. Jiang, W.X. Zheng / Chaos, Solitons and Fractals 26 (2005) 437–443
synchronization approach, both linear-state-feedback and nonlinear-state-feedback control schemes can be used for chaos synchronization. Due to its simple configuration, easy implementation and robustness to the transmission noise in the channel, the linear-state-feedback scheme is especially attractive and has been commonly adopted for practical application. For this reason, based on the Lyapunov stability theory [12,22] and Linear quadratic optimal control theory [23], a matrix inequality condition has been developed for chaos synchronization via linear-state-feedback approach for a class of chaotic systems in our recent paper [5]. Note that, only a scalar feedback has been used through a single connection for the design of chaos synchronization, which can be easily applied to secure communication based on chaos synchronization [28]. In this paper, chaos synchronization is further studied by using the linear-state-feedback control approach. An easily verified sufficient condition, formulated in the LMI form, is developed for chaos synchronization, which is applicable to a large class of general chaotic systems. The proposed criterion can ensure chaos synchronization at an exponential convergence rate. An example is included to verify the effectiveness of the criterion proposed in the paper. The rest of this paper is organized as follows. In Section 2, an LMI criterion is presented for chaos synchronization via linear-state-feedback approach for a class of chaotic systems. In Section 3, the effectiveness of the proposed criterion is demonstrated by the chaotic Murali–Lakshmanan–Chua system [7]. Some concluding remarks are drawn in Section 4.
2. LMI criterion of chaos synchronization Consider a chaotic system described by x_ ¼ Ax þ gðxÞ þ hðtÞ; n·1
n·n
ð1Þ n·1
where x 2 R , A 2 R , h(t) 2 R Lipschitz condition, namely,
is the external input signal, and g(x) is a continuous nonlinear function, satisfying
~k: kgðxÞ gð~ xÞk 6 qkx x
ð2Þ
Note that in (2) k Æ k denotes the Euclidean norm and q is the Lipschitz constant. Based on the linear-state-feedback approach, the response system can be constructed as follows [5]: ~_ ¼ A~ x x þ gð~ xÞ þ hðtÞ þ Bðy ~y Þ; y ¼ Kx; ~y ¼ K~ x;
ð3Þ
where y = Kx is a scalar chaotic signal, which is a linear combination of the state variables of the drive system and will be transmitted to the response system, K = [k1, k2, . . . , kn] is the feedback gain to be designed, and B = [b1, b2, . . . , bn] 0 is chosen such that (A, B) is controllable. Remark 1. It is noted that, the synchronization scheme (3) is based on the state-feedback approach, and only a scalar variable, which is the linear combination of the state variables is used for chaos synchronization, so it is convenient to apply in real applications. Remark 2. The most attractive feature of the linear-state-feedback approach is that it can reduce the influence of noise on the performance of communication systems when it is applied to chaotic secure communications [28]. In the linearstate-feedback-based chaotic secure communication system [28], one can choose a special matrix B with simple configuration, which just ensures that (A, B) is controllable, to make the message signal s easily injected into the driver. On the other hand, due to the choice of the special matrix B, the message signal s does not have to be very small, and consequently, the communication system can have a better performance in SNR. However, for the state-observer based approach, the message signal s multiplied by the state-observer gain matrix L, which, in general, is quite large to ensure the error dynamics system asymptotically stable, is injected into the driver. So a very smaller s is required to guarantee the drive system to be chaotic, leading to a poor performance in SNR. ~ ! x. Our aim is to find a suitable B and K, such that chaos synchronization is achieved, i.e., x From (1)–(3), we can get the following error dynamics system ~_ ¼ Ax þ gðxÞ þ hðtÞ ðA~ ~Þ x þ gð~ xÞ þ hðtÞ þ Bðy ~y ÞÞ ¼ Ae þ gðxÞ gð~ xÞ BKðx x e_ ¼ x_ x ¼ Ae BKe þ gðxÞ gð~ xÞ ¼ ðA BKÞe þ gðxÞ gð~ xÞ; ~ is the error vector. where e ¼ x x
ð4Þ
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439
Theorem 1. If a suitable matrix B is chosen such that (A, B) is controllable, and a suitable feedback gain K is selected such that AT P þ PA KT BT P PBK þ q2 PP þ I þ 2dP < 0;
ð5Þ
where P is a positive definite symmetric matrix, I is the identity matrix, and d is a positive constant, then the error dynamical system (4) is globally exponentially stable, implying that the coupled systems (1) and (3) are globally exponentially synchronized. Proof. Define a Lyapunov function V = eTPe, where P is a positive definite and symmetric constant matrix. Differentiating V along the error dynamical trajectory (4) and using (2) yield V_ ¼ e_ T Pe þ eT P_e ¼ ½ðA BKÞe þ gðxÞ gð~ xÞ T Pe þ eT P½ðA BKÞe þ gðxÞ gð~ xÞ xÞ T Pe 6 eT ððA BKÞT P þ PðA BKÞÞe þ 2qkek kPek: ¼ eT ððA BKÞT P þ PðA BKÞÞe þ 2½gðxÞ gð~ Since 2kek Æ qkPek 6 q2kPek2 + kek2, using (5) we further have V_ 6 eT ððA BKÞT P þ PðA BKÞÞe þ q2 kPek2 þ kek2 ¼ eT ðAT P þ PA PBK KT BT P þ q2 PP þ IÞe 6 2deT Pe ¼ 2dV < 0: Based on the Lyapunov stability theory, the error dynamical system (4) is globally exponentially stable, and hence, the two coupled systems (1) and (3) are globally exponentially synchronized. h S11 S12 , where S11 ¼ ST11 , S12 ¼ ST21 , Lemma 1 [(Schur Complements [11])]. For a given symmetric matrix S ¼ S21 S22 S22 ¼ ST22 , the condition S < 0 is equivalent to S22 < 0;
and
T S11 S12 S1 22 S12 < 0:
Using Lemma 1, the condition (5) can be easily transformed to be " # PA þ AT P þ 2ðd qÞP PBK KT BT P qP þ I < 0: ð6Þ qP þ I I 1 P 0 If (6) is multiplied by from the left-hand and right-hand sides, respectively, and letting X = P1 and 0 I W = (KP1)T, then (6) can easily be further transformed into the following LMI form (7). Theorem 2. If suitable matrices X and W are selected such that the following LMI " # AX þ XAT þ 2ðd qÞX WBT BWT qI þ X < 0; qI þ X I
ð7Þ
is satisfied, then the error dynamical system (4) with the feedback gain K = WTX1, is globally exponentially stable, implying that the coupled systems (1) and (3) are globally exponentially synchronized. Remark 3. The feasible set of X and W satisfying (7) can be easily found by using the LMI Toolbox in MATLAB software.
3. An illustrative example In this section, the chaotic Murali–Lakshmanan–Chua circuit is simulated to illustrate the above-derived criterion for chaos synchronization. The chaotic Murali–Lakshmanan–Chua circuit is described by [7] 0 1 x1 f ðx1 Þ x_ 1 0 ¼ þ þ ; ð8Þ b r x2 x_ 2 0 F sinðxtÞ where r > 0, b > 0, F > 0, x > 0 and f(Æ) is a piecewise linear function 1 f ðx1 Þ ¼ bx1 þ ða bÞðjx1 þ 1j jx1 1jÞ; 2
ð9Þ
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with a < b < 0. From (9), one easily has [4,5] f ðx1 Þ f ð~x1 Þ ¼ k x1;~x1 ðx1 ~x1 Þ;
ð10Þ
where k x1;~x1 depends on x1 and ~x1 , and varies within the interval [a, b] for t P 0. Hence, k x1;~x1 is bounded as f ðx1 Þ 0 0 1 and gðxÞ ¼ , hðtÞ ¼ . a 6 k x1;~x1 6 b < 0. From (8), we get A ¼ F sinðwtÞ b r 0 b Choose B ¼ 1 , where b1 5 0. Obviously, (A, B) is controllable. Referring to (3), the response system can be con0 structed by " # f ð~x1 Þ 0 1 0 ~x1 b1 ~x_ 1 þ ¼ ðy ~y Þ; þ þ _~x2 0 0 b r ~x2 F sinðxtÞ ð11Þ y ¼ Kx; ~y ¼ K~ x: Subtracting (11) from (8) gives b1 e1 0 1 e1 ðf ðx1 Þ f ð~x1 ÞÞ e_ 1 K ¼ þ ; 0 e_ 2 0 e2 b r e2
ð12Þ
where e1 ¼ x1 ~x1 and e2 ¼ x2 ~x2 are the errors. Furthermore, (12) can be rewritten as e_ ¼ Ae þ gðxÞ gð~ xÞ BKe; e1 f ð~x1 Þ . where e ¼ and gð~ xÞ ¼ 0 e2
ð13Þ
Consider gðxÞ gð~ xÞ ¼ where Mx;~x ¼
k x1;~x1 0
ðf ðx1 Þ f ð~x1 ÞÞ 0
¼
k x1;~x1 ðx1 ~x1 Þ 0
¼
k x1;~x1
0
0
0
e1 e2
¼ Mx;~x e;
0 . 0
Fig. 1. Chaotic signals and chaotic attractor of MLC circuit.
ð14Þ
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441
Since a 6 k x1;~x1 6 b < 0, letting k x1;~x1 ¼ aþb þ k x1;~x1 with ab 6 k x1;~x1 6 ab yields 2 2 2 aþb k x1;~x1 0 2 0 k x1;~x1 0 ~Þ þ ~Þ ¼ ~Þ ðx x ðx x gðxÞ gð~ xÞ ¼ ðx x 0 0 0 0 0 0 ~Þ þ gðxÞ gð~ ~ Þ þ Mx;~x ðx x ~Þ; xÞ ¼ A1 ðx x ¼ A1 ðx x
ð15Þ
where A1 ¼
aþb 2 0
0 ; 0
Mx;~x ¼
k x1;~x1 0
0 ; 0
and
gðxÞ gð~ xÞ ¼
k x1;~x1 0
0 ~ Þ: ðx x 0
Then, (13) can be rewritten as e_ ¼ Ae þ gðxÞ gð~ xÞ BKe ¼ ðA þ A1 Þe þ gðxÞ gð~ xÞ BKe ¼ Ae þ gðxÞ gð~ xÞ BKe; aþb 2 1 where A ¼ A þ A1 ¼ , and b r a b kek ¼ qkek; ~Þ 6 k x1;~x1 kek ¼ kgðxÞ gð~ xÞk ¼ Mx;~x ðx x 2 . where q ¼ ab 2
Fig. 2. Chaos synchronization of chaotic Murali–Lakshmanan–Chua circuit.
Fig. 3. Chaos synchronization of chaotic Murali–Lakshmanan–Chua circuit with the transmission noise.
ð16Þ
ð17Þ
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Remark 4. The aim of transforms in (15)–(17) is to increase the flexibility of the design. Since k x1;~x1 < k x1;~x1 , the value of q can be reduced. Consequently, the value range of the matrix K is increased. Remark 5. It is noted that, the choice of vector B is not unique, and can be selected as any other values provided that BÞ is controllable. ðA; In the following, chaos synchronization based on the chaotic Murali–Lakshmanan–Chua circuit is demonstrated. The parameters of the circuit used are r = 1.015, b = 1.0, F = 0.15, x = 0.75, a = 1.02, b = 0.55, for which the system exhibits chaotic behavior, as shown in Fig. 1. We have q = 0.235 from (17). Choose b1 = 1 and d = 0.5. Then one easily obtains K ¼ ½ 3:9368 0:5452 from (7) by using the MATLAB LMI Toolbox. According to Theorem 2, chaos synchronization is achieved, as shown in Figs. 2 and 3, where Fig. 3 is for the case with the transmission noise d(t), which is a random noise with zero mean (see Fig. 3(a)). From Fig. 3, we can see that the effect of the transmission noise on chaos synchronization can be suppressed, which shows that the scheme has strong robustness to the transmission noise.
4. Conclusions A new LMI criterion has been derived for chaos synchronization via the Lyapunov stability theory and the Linear Matrix Inequality technique. The criterion has been applied to the chaotic Murali–Lakshmanan–Chua circuit for illustration. It is noted that the scheme and criterion proposed herein are applicable to a large class of chaotic systems. Similar approach can be applied to the Chuas circuit [16], Ro¨ssler system [15], Lorenz system [14], Chens circuit [13] and so on.
Acknowledgements This work was supported in part by Jiangsu Qing Lan Project Program [JS200407], China, in part by the Key Project of Natural Science Foundation of Universities of Jiangsu Province [04KJA510092], China, and in part by a research grant from the Australian Research Council [A00102928].
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