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EP-based PID control design for chaotic synchronization with application in secure communication
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Expert Systems with Applications Expert Systems with Applications 34 (2008) 1169–1177 www.elsevier.com/locate/eswa
EP-based PID control design for chaotic synchronization with application in secure communication Hsin-Chieh Chen a, Jen-Fuh Chang b, Jun-Juh Yan b, Teh-Lu Liao b
a,*
a Department of Engineering Science, National Cheng Kung University, Tainan, 70101 Taiwan, ROC Department of Computer and Communication, Shu Te University, Yen Chau, Kaosiung County, 82445 Taiwan, ROC
Abstract In this paper, the evolutionary programming (EP)-based proportional-integral-derivative (PID) control design is presented for synchronization of chaotic systems with application in secure communication. A PID controller is developed via the EP algorithm. By using the EP algorithm, optimal control gains in PID controlled chaotic systems are derived such that a performance index of integrated absolute error (IAE) is as minimal as possible. Moreover, as an application, the proposed EP-based PID control scheme is then applied to a chaotic secure communication system. To verify the system performance, basic electronic components containing OPA, resistor and capacitor elements are used to implement the proposed PID-based chaotic secure communication system. Finally, both simulation results and the circuit experiments demonstrate the proposed PID scheme’s success in the communication application. 2006 Elsevier Ltd. All rights reserved. Keywords: Evolutionary programming (EP); PID; Chaotic systems; Secure communication
1. Introduction Evolutionary programming (EP) algorithms have been considered both availability and promising techniques for global optimization of complex functions, and also have been applied to solve difficult problems in control engineering (Cao, 1997; Chen & Huang, 2003; Fogel, 1995; Huang, Tzeng, & Ong, 2006; Kim, Min, & Han, 2006; Yan, Tsai, & Sheen, 2001). Generally, the EP algorithm for global optimization contains four parts: initialization, mutation, competition, and reproduction. Furthermore, in the work of Cao (1997) a quasi-random sequence (QRS) is used to generate the initial population for EP to avoid formulating clusters around an arbitrary local optimal. On the other hand, it is well-known that a chaotic system is a very complex, dynamic nonlinear system and its response possesses some characteristics such as excessive *
Corresponding author. Tel.: +886 62757575x63337; fax: +886 6276 6549. E-mail address: [email protected] (T.-L. Liao). 0957-4174/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2006.12.023
sensitivity to initial conditions, broad spectrums of the Fourier transform and fractal motion properties in its phase space. Moreover, the synchronization of chaotic circuits for the secure communication has received much attention in the literature (Alvarez, Herna´ndez, Mun˜oz, Montoya, & Li, 2005; Li, Li, Wen, & Soh, 2003; Li & Xu, 2004; Wen, Wang, Lin, Han, & Li, 2006; Zhou, Huang, Qi, Yang, & Xie, 2005). This is due to the fact the utilizing the chaotic circuit offers some advantages in communication systems such as broadband noise-like waveform, prediction difficulty, sensitivity to initial condition variations, etc. (Huang, Tsay, & Wu, 2005; Pecora & Carroll, 1990). Therefore, a chaotic communication system can be setup to obtain a secure communication (Liao & Tsai, 2000). In this paper, we first proposed a new EP-based PID control scheme to solve the synchronization problem of chaotic systems. A proportional-integral-derivative (PID) controller is developed via the EP algorithm. By using the EP algorithm, optimal control gains in PID controlled chaotic systems are derived such that a performance index of integrated absolute error (IAE) as minimal as possible
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and then, synchronization of the master and slave chaotic systems is achieved. Moreover, as an application, the proposed PID control scheme is then applied to a chaotic secure communication system. The proposed PID controller is used to achieve the synchronization between the transmitter and the receiver in communication systems. Thus, it can ensure that the message masked by chaotic state in the transmitter can be recovered by a chaotic state in the receiver and high performance communication can be obtained by the proposed chaotic communication system. To verify the system performance, a PID-based chaotic secure communication system, which includes two chaotic Sprott circuits (transmitter and receiver) and an EP-based PID controller, is realized by using some electronic components containing operational amplifiers (OPAs), resistors and capacitors. Finally, both simulation and experimental results are given to verify the proposed EP-based PID scheme’s success in the communication application. 2. System description and problem formulation In order to observe the synchronization procedure in chaotic systems, we consider two single-input single output (SISO) master and slave systems described by the following differential equations: x_ m ðtÞ ¼ f ðt; xm Þ ð1Þ y m ðtÞ ¼ Cxm x_ s ðtÞ ¼ f ðt; xs Þ þ BuðtÞ ð2Þ y s ðtÞ ¼ Cxs where xm(t) = [xm1, xm2, . . ., xmn] 2 Rn and xs(t) = [xs1, xs2, . . ., xsn] 2 Rn are the state vectors of master and slave systems, respectively. f: R · Rn ! Rn is a given nonlinear vector function. ym(t) 2 R and ys(t) 2 R are the outputs of the master and slave systems, respectively. B 2 Rn·1 and C 2 R1·n. The control input u 2 R introduced to system (2) that will be derived to achieve synchronization between master and slave systems. Generally many chaotic systems can be expressed by (1). For examples, the Sprott circuit (Huang et al., 2005), modified Chua’s circuit (Yassen, 2003), Duffing-Holmes system (Tsai, Fuh, & Chang, 2002), Lorenz system (Yang, Chen, & Yau, 2002), and Lu¨ system (Lu¨ & Lu, 2003) all belong to the class defined by (1). Now let us define the state errors between the master system and slave system as e1 ¼ xm1 xs1 ; e2 ¼ xm2 xs2 ; ; en ¼ xmn xsn :
of a PID controller, with input ye(Æ) and output u(Æ), is generally given as Z t 1 d uðtÞ ¼ K p y e ðtÞ þ y ðsÞds þ T d y e ðtÞ ð4Þ Ti 0 e dt where Kp is the proportional gain, Ti is the integral time constant, and Td is the derivative time constant. We can also rewrite (4) as Z t d uðtÞ ¼ K p y e ðtÞ þ K i y e ðsÞds þ K d y e ðtÞ ð5Þ dt 0 where Ki = Kp/Ti is the integral gain and Kd = KpTd is the derivative gain. In general, for a controller design, the performance criterion or objective function can be defined according to our desired specifications. Two kinds of performance criteria usually considered in the EP algorithm are the integrated squared error (ISE) and the integrated absolute error (IAE). In this study, the IAE index is used as the objective function (OF), which is given as Z 1 OF :¼ IAE ¼ jkEðsÞkjds: ð6Þ 0
where E(t) = [e1,e2 ,. . . ,en] and kÆk is the Euclidean norm of a vector. In the following, based on using EP algorithm, we will develop a tuning method for the PID controller with optimal gain parameters to minimize the objective function score (6). 3. An evolutionary programming (EP) algorithm to solve the optimization problem Since the EP algorithm has been considered as an available and promising technique for the global optimization of complex functions, we introduce the EP algorithm to solve this problem. The EP-based PID control system consists of master and slave chaotic systems to be synchronized, a PID controller and an EP algorithm, as shown in Fig. 1. ym is the output of the master system, ys is the output of the slave system, and u is the control input generated by the PID controller as defined in (5). The PID con-
ð3Þ
One main objective of this work is to present a simple but effective PID controller based on EP algorithm to achieve the synchronization of two identical chaotic systems. The term u in (2) is a PID controller obtained via EP algorithm to guarantee the synchronization performance. The procedure to determine the PID controller u is to first define the output error signal ye = ym ys, then the continuous-form
Fig. 1. Blocks diagram of secure communication system.
H.-C. Chen et al. / Expert Systems with Applications 34 (2008) 1169–1177
troller is proposed with the optimal parameters derived from the EP algorithm such that the value of IAE in (6) is minimized as possibly.
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z ¼ ½K p ; K i ; K d is such that the performance index of IAE of system is minimized as possibly. Otherwise, return to Step III.
3.1. Optimization problem formulation Let z be the continuously differentiable matrix-valued function defined for z 2 S, where S = {z 2 R3|0 6 zi 6 zmax, zmax < 1, i = 1, 2, 3}. The optimization problem involves finding z ¼ ½K p ; K i ; K d 2 S such that the performance index of IAE of system is minimized. More accurately, the optimization problem is stated mathematically as (P1): To find z* 2 S such that Z 1 OF ¼ jkEðsÞkjds for z 2 S ð7Þ 0
is minimized. 3.2. Optimization procedures Based on the results shown by Cao (1997), the procedures of extended EP algorithm for solving the above optimization problem are described as follows:
Fig. 2. Convergence curve of IAE.
(I) Generate an initial population P0 = [p1, p2, . . ., pN] of size N by randomly initializing each 3-dimensional solution vector pi 2 S, i = 1, 2, . . ., N, according to the quasi-random sequence (QRS). (II) Calculate the fitness score (objective function) fi = f(pi) for every pi, i = 1, 2, . . ., N, where Z 1 kEðsÞkds; pi ¼ ½ K p K i K d 2 S: f ðpi Þ ¼ 0
ð8Þ (III) Mutate every pi, i = 1, . . ., N, based on the statistics to double the population size from N to 2N, and generate pi+N in the following fi piþN ;j ¼ pi;j þ N 0; b 8j ¼ 1; 2; 3 ð9Þ fR
Fig. 3. Trajectories of Kp, Ki, and Kd.
where pi,j denotes the jth element of the ith individual, N 0; b ffRi represents a Gaussian random variable with a mean zero and variance b ffRi , fR is the sum of every fitness scores, and b is a parameter to scale ffRi . (IV) Calculate the fitness score fi+N for every pi+N, i = 1, 2, . . ., N by using Eq. (5). By the stochastic competition process, pi, i = 1, 2, . . ., N, competes with pj, j = N + 1, . . ., 2N, randomly each other. If fi < fj, pi is the winner and survives; otherwise, pj is the winner and pi is replaced by pj. After the competition process, we select the N winners for the next generation and let the individual with the minimum objective function in the winners be p1. (V) If the value fR converges to a minimum value, then z* = p1 is the global optimum value and
Fig. 4. Output response using the EP-based PID controller with ½K p ; K i ; K d ¼ ½ 20 0:00455 4:8092 .
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Master Sprott circuit:
4. Simulations and experimental results In this section, we illustrate the EP-based PID controller design for synchronization of Sprott circuits, which are typical chaotic systems that have been thoroughly studied (Almeida, Alvarez, & Barajas, 2006). Now consider the following Sprott circuits:
x_ m1 ¼ xm2 x_ m2 ¼ xm3 x_ m3 ¼ 1:2xm1 xm2 0:6xm3 þ 2signðxm1 Þ y m ¼ xm1
Fig. 5. The proposed communication system.
Fig. 6. Electronic implementation of master Sprott circuit.
ð10Þ
H.-C. Chen et al. / Expert Systems with Applications 34 (2008) 1169–1177
Fig. 7. Electronic implementation of slave Sprott circuit.
Fig. 8. EP-based PID controller circuit.
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Slave Sprott circuit: x_ s1 ¼ xs2 x_ s2 ¼ xs3 þ u x_ s3 ¼ 1:2xs1 xs2 0:6xs3 þ 2signðxs1 Þ
ð11Þ
y s ¼ xs1 where x_ m and x_ s denote the derivative of xm and xs with respect to time t, respectively. The initial value conditions ½xm1 ð0Þ; xm2 ð0Þ; xm3 ð0Þ ¼ ½ 0:1 0:1 0:1 and ½xs1 ð0Þ; xs2 ð0Þ; xs3 ð0Þ ¼ ½ 1 2 1 are used in this example. We solved the optimization problem (P1) with N = 30 and b = 0.001 by using the control toolbox of Matlab and Simulink. According to the proposed EP algorithm, we generated P0 = [p1, p2, . . ., p30] according to the QRS. After a series of EP algorithm manipulations, the convergence curve of IAE value versus iteration is depicted in Fig. 2. It can be easily observed from Fig. 2 that it converges after about 80 iterations and its final value of IAE is f(z*) = 0.6726. Correspondingly, the PID control gains are z ¼ ½K p ; K i ; K d ¼ ½ 20 0:0045 4:8092 . The trajectories of Kp, Ki, and Kd during the evolutionary procedure are also shown in Fig. 3. For reference, the output response using the resulting PID control gains z* is then shown in Fig. 4. The simulation results show that the proposed PID controller via EP algorithm is viable for synchronization of chaotic systems. 5. Applications of secure communication systems and experimental results In this section, the preceding EP-based PID control synchronization scheme is applied to chaotic secure communications. Fig. 5 illustrates that the proposed communication
system consists of a transmitter and a receiver (master and slave Sprott circuit, respectively). The input message s(t) is masked by the state of the chaotic transmitter and is simultaneously transmitted to the receiver. In addition, the output signal is also transmitted to the receiver for synchronization. From the results of Example 1, it shows that the synchronization between the transmitter and the receiver can be achieved, and the input message s(t) can be completely recovered on the receiver side. In the following, we use the electronic components: OPAs, resistors and capacitors to implement the presented system. In order to speed up the dynamic response of Sprott chaotic circuit, we rescale the systems (10) and (11) by a new time scale s ¼ kt , and then we have the following systems (12) and (13), respectively x_ m1 ¼ kðxm2 Þ x_ m2 ¼ kðxm3 Þ x_ m3 ¼ k½1:2xm1 xm2 0:6xm3 þ 2signðxm1 Þ
ð12Þ
y m ¼ xm1 x_ s1 ¼ kðxs2 Þ x_ s2 ¼ kðxs3 þ uÞ x_ s3 ¼ k½1:2xs1 xs2 0:6xs3 þ 2signðxs1 Þ
ð13Þ
y s ¼ xs1 where x_ m and x_ s denote the derivative of xm and xs with respect to time s, respectively. k is a scaling factor, and 1 > k P 1. 5000. Practical circuits of the chaotic master and slave systems supplied ±15 V in Sprott circuits with k = 5000 are shown in Figs. 6 and 7, respectively. The circuit of EP-based PID controller with three gains ½K p ; K i ; K d ¼ ½ 20 0:00455 4:8092 is shown in Fig. 8. In order to demonstrate the chaotic secure communication,
Fig. 9. Audio signal input and recovered audio signal output.
H.-C. Chen et al. / Expert Systems with Applications 34 (2008) 1169–1177
we have masked the input message s(t) (audio signal from an MP3 player) by the state xs2 in the transmitter, and the masked signal is then sent to the receiver. The chaotic masking circuit and the recovery circuit are shown in Fig. 9. A photograph of the proposed secure communication system is shown in Fig. 10. 5.1. Experimental results Fig. 11a–f shows the experimental of synchronization between state xm and state xs. Fig. 12a shows the audio sig-
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nal masked by xm2. From this figure, we can observe that the masked signal becomes chaotic and is different from the original signal. It is not easy to recovery the original signal from the transmitted signal except the synchronization between the transmitter and the receiver can be obtained. Fig. 12b shows the recovered audio signal can be completely recovered at the receiver side by the PIDbased control for chaotic synchronization. The experimental results demonstrate that the secure communication is obtained through the proposed EP-based PID control scheme.
Fig. 10. A photograph of the proposed secure communication system.
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Fig. 11. Experimental results of synchronization between state xm with state xs. (a) The state xm1 versus xs1 before PID controller turned on. (b) The state xm1 versus xs1 after PID controller turned on. (c) The state xm2 versus xs2 before PID controller turned on. (d) The state xm2 versus xs2 after PID controller turned on. (e) The state xm3 versus xs3 before PID controller turned on. (f) The state xm3 versus xs3 after PID controller turned on.
Fig. 12. Experimental results of audio signal process. (a). Channel 1: original audio signal; Channel 2 masked audio signal. (b). Channel 1: original audio signal; Channel 2 recovered audio signal.
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6. Conclusions Based on the evolutionary programming algorithm, a simple and successful PID controller has been proposed for synchronizing two chaotic systems. An EP algorithm is derived such that three PID controller gains can be directly obtained by solving specified optimization problems as defined above. Moreover, the proposed EP-based PID scheme is then successfully applied to a secure communication system. The message masked by the chaotic state at the transmitter can be recovered by the chaotic state at the receiver. Furthermore, some basic electronic circuits are used to implement the EP-based secure communication system. The simulation and experimental results verify that the methods are correct and practical. Compared with the existing reports for chaotic synchronization, the proposed EP-based PID controller is not only effective but also simple in design to implement. References Almeida, D. I. R., Alvarez, J., & Barajas, J. G. (2006). Robust synchronization of Sprott circuits using sliding mode control. Chaos, Solitons & Fractals, 30(1), 11–18. Alvarez, G., Herna´ndez, L., Mun˜oz, J., Montoya, F., & Li, S. (2005). Security analysis of communication system based on the synchronization of different order chaotic systems. Physics Letters A, 345(4–6), 245–250. Cao, Y. J. (1997). Eigenvalue optimization problems via evolutionary programming. Electronics Letters, 33, 642–643. Chen, M. C., & Huang, S. H. (2003). Credit scoring and rejected instances reassigning through evolutionary computation techniques. Expert Systems with Applications, 24(4), 433–441. Fogel, D. B. (1995). Evolutionary computation: toward a new philosophy of machine intelligence. IEEE Press.
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Huang, C. K., Tsay, S. C., & Wu, Y. R. (2005). Implementation of chaotic secure communication systems based on OPA circuits. Chaos, Solitons and Fractals, 23, 589–600. Huang, J. J., Tzeng, G. H., & Ong, C. S. (2006). Optimal fuzzy multicriteria expansion of competence sets using multi-objectives evolutionary algorithms. Expert Systems with Applications, 30(4), 739–745. Kim, M. J., Min, S. H., & Han, I. (2006). An evolutionary approach to the combination of multiple classifiers to predict a stock price index. Expert Systems with Applications, 31(2), 241–247. Liao, T. L., & Tsai, S. H. (2000). Adaptive synchronization of chaotic systems and its application to secure communications. Chaos, Solitons and Fractals, 11(9), 1387–1396. Li, Z., Li, K., Wen, C., & Soh, Y. C. (2003). A new chaotic secure communication system. IEEE Transactions on Communications, 51(8), 1306–1312. Li, Z., & Xu, D. (2004). A secure communication scheme using projective chaos synchronization. Chaos, Solitons & Fractal, 22(2), 477–481. Lu¨, J., & Lu, J. (2003). Controlling uncertain Lu¨ system using linear feedback. Chaos, Solitons & Fractal, 17(1), 127–133. Pecora, L. M., & Carroll, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters, 64, 821–824. Tsai, H. H., Fuh, C. C., & Chang, C. N. (2002). A robust controller for chaotic systems under external excitation. Chaos, Solitons & Fractal, 14(4), 627–632. Wen, G., Wang, Q. G., Lin, C., Han, X., & Li, G. (2006). Synthesis for robust synchronization of chaotic systems under output feedback control with multiple random delays. Chaos, Solitons & Fractals, 29(5), 1142–1146. Yang, S. K., Chen, C. L., & Yau, H. T. (2002). Control of chaos in Lorenz system. Chaos, Solitons & Fractal, 13(4), 767–780. Yan, J. J., Tsai, S. H., & Sheen, I. E. (2001). Stability analysis of largescale time-varying systems via evolutionary programming algorithm. Journal of Dynamic Systems, Measurement, and Control, 123, 293–296. Yassen, M. T. (2003). Adaptive control and synchronization of a modified Chua’s circuit system. Applied Mathematics and Computation, 135, 113–128. Zhou, J., Huang, H. B., Qi, G. X., Yang, P., & Xie, X. (2005). Communication with spatial periodic chaos synchronization. Physics Letters A, 335(2–3), 191–196.
Expert Systems with Applications Expert Systems with Applications 34 (2008) 1169–1177 www.elsevier.com/locate/eswa
EP-based PID control design for chaotic synchronization with application in secure communication Hsin-Chieh Chen a, Jen-Fuh Chang b, Jun-Juh Yan b, Teh-Lu Liao b
a,*
a Department of Engineering Science, National Cheng Kung University, Tainan, 70101 Taiwan, ROC Department of Computer and Communication, Shu Te University, Yen Chau, Kaosiung County, 82445 Taiwan, ROC
Abstract In this paper, the evolutionary programming (EP)-based proportional-integral-derivative (PID) control design is presented for synchronization of chaotic systems with application in secure communication. A PID controller is developed via the EP algorithm. By using the EP algorithm, optimal control gains in PID controlled chaotic systems are derived such that a performance index of integrated absolute error (IAE) is as minimal as possible. Moreover, as an application, the proposed EP-based PID control scheme is then applied to a chaotic secure communication system. To verify the system performance, basic electronic components containing OPA, resistor and capacitor elements are used to implement the proposed PID-based chaotic secure communication system. Finally, both simulation results and the circuit experiments demonstrate the proposed PID scheme’s success in the communication application. 2006 Elsevier Ltd. All rights reserved. Keywords: Evolutionary programming (EP); PID; Chaotic systems; Secure communication
1. Introduction Evolutionary programming (EP) algorithms have been considered both availability and promising techniques for global optimization of complex functions, and also have been applied to solve difficult problems in control engineering (Cao, 1997; Chen & Huang, 2003; Fogel, 1995; Huang, Tzeng, & Ong, 2006; Kim, Min, & Han, 2006; Yan, Tsai, & Sheen, 2001). Generally, the EP algorithm for global optimization contains four parts: initialization, mutation, competition, and reproduction. Furthermore, in the work of Cao (1997) a quasi-random sequence (QRS) is used to generate the initial population for EP to avoid formulating clusters around an arbitrary local optimal. On the other hand, it is well-known that a chaotic system is a very complex, dynamic nonlinear system and its response possesses some characteristics such as excessive *
Corresponding author. Tel.: +886 62757575x63337; fax: +886 6276 6549. E-mail address: [email protected] (T.-L. Liao). 0957-4174/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2006.12.023
sensitivity to initial conditions, broad spectrums of the Fourier transform and fractal motion properties in its phase space. Moreover, the synchronization of chaotic circuits for the secure communication has received much attention in the literature (Alvarez, Herna´ndez, Mun˜oz, Montoya, & Li, 2005; Li, Li, Wen, & Soh, 2003; Li & Xu, 2004; Wen, Wang, Lin, Han, & Li, 2006; Zhou, Huang, Qi, Yang, & Xie, 2005). This is due to the fact the utilizing the chaotic circuit offers some advantages in communication systems such as broadband noise-like waveform, prediction difficulty, sensitivity to initial condition variations, etc. (Huang, Tsay, & Wu, 2005; Pecora & Carroll, 1990). Therefore, a chaotic communication system can be setup to obtain a secure communication (Liao & Tsai, 2000). In this paper, we first proposed a new EP-based PID control scheme to solve the synchronization problem of chaotic systems. A proportional-integral-derivative (PID) controller is developed via the EP algorithm. By using the EP algorithm, optimal control gains in PID controlled chaotic systems are derived such that a performance index of integrated absolute error (IAE) as minimal as possible
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and then, synchronization of the master and slave chaotic systems is achieved. Moreover, as an application, the proposed PID control scheme is then applied to a chaotic secure communication system. The proposed PID controller is used to achieve the synchronization between the transmitter and the receiver in communication systems. Thus, it can ensure that the message masked by chaotic state in the transmitter can be recovered by a chaotic state in the receiver and high performance communication can be obtained by the proposed chaotic communication system. To verify the system performance, a PID-based chaotic secure communication system, which includes two chaotic Sprott circuits (transmitter and receiver) and an EP-based PID controller, is realized by using some electronic components containing operational amplifiers (OPAs), resistors and capacitors. Finally, both simulation and experimental results are given to verify the proposed EP-based PID scheme’s success in the communication application. 2. System description and problem formulation In order to observe the synchronization procedure in chaotic systems, we consider two single-input single output (SISO) master and slave systems described by the following differential equations: x_ m ðtÞ ¼ f ðt; xm Þ ð1Þ y m ðtÞ ¼ Cxm x_ s ðtÞ ¼ f ðt; xs Þ þ BuðtÞ ð2Þ y s ðtÞ ¼ Cxs where xm(t) = [xm1, xm2, . . ., xmn] 2 Rn and xs(t) = [xs1, xs2, . . ., xsn] 2 Rn are the state vectors of master and slave systems, respectively. f: R · Rn ! Rn is a given nonlinear vector function. ym(t) 2 R and ys(t) 2 R are the outputs of the master and slave systems, respectively. B 2 Rn·1 and C 2 R1·n. The control input u 2 R introduced to system (2) that will be derived to achieve synchronization between master and slave systems. Generally many chaotic systems can be expressed by (1). For examples, the Sprott circuit (Huang et al., 2005), modified Chua’s circuit (Yassen, 2003), Duffing-Holmes system (Tsai, Fuh, & Chang, 2002), Lorenz system (Yang, Chen, & Yau, 2002), and Lu¨ system (Lu¨ & Lu, 2003) all belong to the class defined by (1). Now let us define the state errors between the master system and slave system as e1 ¼ xm1 xs1 ; e2 ¼ xm2 xs2 ; ; en ¼ xmn xsn :
of a PID controller, with input ye(Æ) and output u(Æ), is generally given as Z t 1 d uðtÞ ¼ K p y e ðtÞ þ y ðsÞds þ T d y e ðtÞ ð4Þ Ti 0 e dt where Kp is the proportional gain, Ti is the integral time constant, and Td is the derivative time constant. We can also rewrite (4) as Z t d uðtÞ ¼ K p y e ðtÞ þ K i y e ðsÞds þ K d y e ðtÞ ð5Þ dt 0 where Ki = Kp/Ti is the integral gain and Kd = KpTd is the derivative gain. In general, for a controller design, the performance criterion or objective function can be defined according to our desired specifications. Two kinds of performance criteria usually considered in the EP algorithm are the integrated squared error (ISE) and the integrated absolute error (IAE). In this study, the IAE index is used as the objective function (OF), which is given as Z 1 OF :¼ IAE ¼ jkEðsÞkjds: ð6Þ 0
where E(t) = [e1,e2 ,. . . ,en] and kÆk is the Euclidean norm of a vector. In the following, based on using EP algorithm, we will develop a tuning method for the PID controller with optimal gain parameters to minimize the objective function score (6). 3. An evolutionary programming (EP) algorithm to solve the optimization problem Since the EP algorithm has been considered as an available and promising technique for the global optimization of complex functions, we introduce the EP algorithm to solve this problem. The EP-based PID control system consists of master and slave chaotic systems to be synchronized, a PID controller and an EP algorithm, as shown in Fig. 1. ym is the output of the master system, ys is the output of the slave system, and u is the control input generated by the PID controller as defined in (5). The PID con-
ð3Þ
One main objective of this work is to present a simple but effective PID controller based on EP algorithm to achieve the synchronization of two identical chaotic systems. The term u in (2) is a PID controller obtained via EP algorithm to guarantee the synchronization performance. The procedure to determine the PID controller u is to first define the output error signal ye = ym ys, then the continuous-form
Fig. 1. Blocks diagram of secure communication system.
H.-C. Chen et al. / Expert Systems with Applications 34 (2008) 1169–1177
troller is proposed with the optimal parameters derived from the EP algorithm such that the value of IAE in (6) is minimized as possibly.
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z ¼ ½K p ; K i ; K d is such that the performance index of IAE of system is minimized as possibly. Otherwise, return to Step III.
3.1. Optimization problem formulation Let z be the continuously differentiable matrix-valued function defined for z 2 S, where S = {z 2 R3|0 6 zi 6 zmax, zmax < 1, i = 1, 2, 3}. The optimization problem involves finding z ¼ ½K p ; K i ; K d 2 S such that the performance index of IAE of system is minimized. More accurately, the optimization problem is stated mathematically as (P1): To find z* 2 S such that Z 1 OF ¼ jkEðsÞkjds for z 2 S ð7Þ 0
is minimized. 3.2. Optimization procedures Based on the results shown by Cao (1997), the procedures of extended EP algorithm for solving the above optimization problem are described as follows:
Fig. 2. Convergence curve of IAE.
(I) Generate an initial population P0 = [p1, p2, . . ., pN] of size N by randomly initializing each 3-dimensional solution vector pi 2 S, i = 1, 2, . . ., N, according to the quasi-random sequence (QRS). (II) Calculate the fitness score (objective function) fi = f(pi) for every pi, i = 1, 2, . . ., N, where Z 1 kEðsÞkds; pi ¼ ½ K p K i K d 2 S: f ðpi Þ ¼ 0
ð8Þ (III) Mutate every pi, i = 1, . . ., N, based on the statistics to double the population size from N to 2N, and generate pi+N in the following fi piþN ;j ¼ pi;j þ N 0; b 8j ¼ 1; 2; 3 ð9Þ fR
Fig. 3. Trajectories of Kp, Ki, and Kd.
where pi,j denotes the jth element of the ith individual, N 0; b ffRi represents a Gaussian random variable with a mean zero and variance b ffRi , fR is the sum of every fitness scores, and b is a parameter to scale ffRi . (IV) Calculate the fitness score fi+N for every pi+N, i = 1, 2, . . ., N by using Eq. (5). By the stochastic competition process, pi, i = 1, 2, . . ., N, competes with pj, j = N + 1, . . ., 2N, randomly each other. If fi < fj, pi is the winner and survives; otherwise, pj is the winner and pi is replaced by pj. After the competition process, we select the N winners for the next generation and let the individual with the minimum objective function in the winners be p1. (V) If the value fR converges to a minimum value, then z* = p1 is the global optimum value and
Fig. 4. Output response using the EP-based PID controller with ½K p ; K i ; K d ¼ ½ 20 0:00455 4:8092 .
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Master Sprott circuit:
4. Simulations and experimental results In this section, we illustrate the EP-based PID controller design for synchronization of Sprott circuits, which are typical chaotic systems that have been thoroughly studied (Almeida, Alvarez, & Barajas, 2006). Now consider the following Sprott circuits:
x_ m1 ¼ xm2 x_ m2 ¼ xm3 x_ m3 ¼ 1:2xm1 xm2 0:6xm3 þ 2signðxm1 Þ y m ¼ xm1
Fig. 5. The proposed communication system.
Fig. 6. Electronic implementation of master Sprott circuit.
ð10Þ
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Fig. 7. Electronic implementation of slave Sprott circuit.
Fig. 8. EP-based PID controller circuit.
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Slave Sprott circuit: x_ s1 ¼ xs2 x_ s2 ¼ xs3 þ u x_ s3 ¼ 1:2xs1 xs2 0:6xs3 þ 2signðxs1 Þ
ð11Þ
y s ¼ xs1 where x_ m and x_ s denote the derivative of xm and xs with respect to time t, respectively. The initial value conditions ½xm1 ð0Þ; xm2 ð0Þ; xm3 ð0Þ ¼ ½ 0:1 0:1 0:1 and ½xs1 ð0Þ; xs2 ð0Þ; xs3 ð0Þ ¼ ½ 1 2 1 are used in this example. We solved the optimization problem (P1) with N = 30 and b = 0.001 by using the control toolbox of Matlab and Simulink. According to the proposed EP algorithm, we generated P0 = [p1, p2, . . ., p30] according to the QRS. After a series of EP algorithm manipulations, the convergence curve of IAE value versus iteration is depicted in Fig. 2. It can be easily observed from Fig. 2 that it converges after about 80 iterations and its final value of IAE is f(z*) = 0.6726. Correspondingly, the PID control gains are z ¼ ½K p ; K i ; K d ¼ ½ 20 0:0045 4:8092 . The trajectories of Kp, Ki, and Kd during the evolutionary procedure are also shown in Fig. 3. For reference, the output response using the resulting PID control gains z* is then shown in Fig. 4. The simulation results show that the proposed PID controller via EP algorithm is viable for synchronization of chaotic systems. 5. Applications of secure communication systems and experimental results In this section, the preceding EP-based PID control synchronization scheme is applied to chaotic secure communications. Fig. 5 illustrates that the proposed communication
system consists of a transmitter and a receiver (master and slave Sprott circuit, respectively). The input message s(t) is masked by the state of the chaotic transmitter and is simultaneously transmitted to the receiver. In addition, the output signal is also transmitted to the receiver for synchronization. From the results of Example 1, it shows that the synchronization between the transmitter and the receiver can be achieved, and the input message s(t) can be completely recovered on the receiver side. In the following, we use the electronic components: OPAs, resistors and capacitors to implement the presented system. In order to speed up the dynamic response of Sprott chaotic circuit, we rescale the systems (10) and (11) by a new time scale s ¼ kt , and then we have the following systems (12) and (13), respectively x_ m1 ¼ kðxm2 Þ x_ m2 ¼ kðxm3 Þ x_ m3 ¼ k½1:2xm1 xm2 0:6xm3 þ 2signðxm1 Þ
ð12Þ
y m ¼ xm1 x_ s1 ¼ kðxs2 Þ x_ s2 ¼ kðxs3 þ uÞ x_ s3 ¼ k½1:2xs1 xs2 0:6xs3 þ 2signðxs1 Þ
ð13Þ
y s ¼ xs1 where x_ m and x_ s denote the derivative of xm and xs with respect to time s, respectively. k is a scaling factor, and 1 > k P 1. 5000. Practical circuits of the chaotic master and slave systems supplied ±15 V in Sprott circuits with k = 5000 are shown in Figs. 6 and 7, respectively. The circuit of EP-based PID controller with three gains ½K p ; K i ; K d ¼ ½ 20 0:00455 4:8092 is shown in Fig. 8. In order to demonstrate the chaotic secure communication,
Fig. 9. Audio signal input and recovered audio signal output.
H.-C. Chen et al. / Expert Systems with Applications 34 (2008) 1169–1177
we have masked the input message s(t) (audio signal from an MP3 player) by the state xs2 in the transmitter, and the masked signal is then sent to the receiver. The chaotic masking circuit and the recovery circuit are shown in Fig. 9. A photograph of the proposed secure communication system is shown in Fig. 10. 5.1. Experimental results Fig. 11a–f shows the experimental of synchronization between state xm and state xs. Fig. 12a shows the audio sig-
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nal masked by xm2. From this figure, we can observe that the masked signal becomes chaotic and is different from the original signal. It is not easy to recovery the original signal from the transmitted signal except the synchronization between the transmitter and the receiver can be obtained. Fig. 12b shows the recovered audio signal can be completely recovered at the receiver side by the PIDbased control for chaotic synchronization. The experimental results demonstrate that the secure communication is obtained through the proposed EP-based PID control scheme.
Fig. 10. A photograph of the proposed secure communication system.
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Fig. 11. Experimental results of synchronization between state xm with state xs. (a) The state xm1 versus xs1 before PID controller turned on. (b) The state xm1 versus xs1 after PID controller turned on. (c) The state xm2 versus xs2 before PID controller turned on. (d) The state xm2 versus xs2 after PID controller turned on. (e) The state xm3 versus xs3 before PID controller turned on. (f) The state xm3 versus xs3 after PID controller turned on.
Fig. 12. Experimental results of audio signal process. (a). Channel 1: original audio signal; Channel 2 masked audio signal. (b). Channel 1: original audio signal; Channel 2 recovered audio signal.
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6. Conclusions Based on the evolutionary programming algorithm, a simple and successful PID controller has been proposed for synchronizing two chaotic systems. An EP algorithm is derived such that three PID controller gains can be directly obtained by solving specified optimization problems as defined above. Moreover, the proposed EP-based PID scheme is then successfully applied to a secure communication system. The message masked by the chaotic state at the transmitter can be recovered by the chaotic state at the receiver. Furthermore, some basic electronic circuits are used to implement the EP-based secure communication system. The simulation and experimental results verify that the methods are correct and practical. Compared with the existing reports for chaotic synchronization, the proposed EP-based PID controller is not only effective but also simple in design to implement. References Almeida, D. I. R., Alvarez, J., & Barajas, J. G. (2006). Robust synchronization of Sprott circuits using sliding mode control. Chaos, Solitons & Fractals, 30(1), 11–18. Alvarez, G., Herna´ndez, L., Mun˜oz, J., Montoya, F., & Li, S. (2005). Security analysis of communication system based on the synchronization of different order chaotic systems. Physics Letters A, 345(4–6), 245–250. Cao, Y. J. (1997). Eigenvalue optimization problems via evolutionary programming. Electronics Letters, 33, 642–643. Chen, M. C., & Huang, S. H. (2003). Credit scoring and rejected instances reassigning through evolutionary computation techniques. Expert Systems with Applications, 24(4), 433–441. Fogel, D. B. (1995). Evolutionary computation: toward a new philosophy of machine intelligence. IEEE Press.
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