High-gain observer for chaotic synchronization ... - Semantic Scholar
INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMS Int. J. Commun. Syst. 2005; 18:487–500 Published online 9 February 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/dac.720
High-gain observer for chaotic synchronization and secure communication Wen Yun,y Departamento de Control Automatico, CINVESTAV-IPN, A.P. 14-740, Av.IPN 2508, Me!xico D.F., 07360, Me´xico
SUMMARY An information signal can be embedded in a transmitter which produces a chaotic signal, it is inviable to the observer who has no knowledge about the transmitter, but it can be recovered if the receiver is a replica of the chaotic transmitter. In this paper, a new aspect of the chaotic communication is introduced. A highgain observer at the receiver’s end replaces the conventional chaotic system. A single parameter is adjusted so that the receiver and transmitter can be synchronized. Three chaotic systems, Duffing equation, Van der Pol oscillator and Chua’s circuit, are provided to illustrate the effectiveness of the chaotic communication. Copyright # 2005 John Wiley & Sons, Ltd. KEY WORDS:
synchronization; chaotic communication; observer
1. INTRODUCTION The general idea for transmitting information via chaotic systems is that, an information signal is embedded in the transmitter system which produces a chaotic signal, the information signal is inviable to the observer who has no knowledge about the transmitter system, it is recovered when the transmitter and the receiver are identical. Since Pecora and Carroll’s observation on the possibility of synchronizing two chaotic systems [1] (so-called drive–response configuration), several synchronization schemes have been developed. Synchronization can be classified into two types: mutual synchronization (or bidirectional coupling) [2] and master–slave synchronization (or unidirectional coupling) [1]. The chaos-based secure communications have updated their fourth generation [3]. The continuous synchronization is adopted in the first three generations while the impulsive synchronization is used in the fourth generation. Less than 94 Hz of bandwidth is needed to transmit the synchronization signal for a third-order chaotic transmitter in the fourth generation while 30 kHz bandwidth needed for transmitting the synchronization signals in the other three generations [4]. There are many applications on chaotic communication [5] and chaotic network synchronization [6]. The techniques of chaotic communication can be divided into three n
Correspondence to: Wen Yu, Departamento de Control Automatico, CINVESTAV-IPN, A.P. 14-740, Av.IPN 2508, M!exico D.F., 07360, Me´xico. y E-mail: [email protected]
Copyright # 2005 John Wiley & Sons, Ltd.
Received 12 February 2004 Revised 9 December 2004 Accepted 12 December 2004
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categories: (a) chaos masking [7], the information signal is added directly to the transmitter; (b) chaos modulation [5, 8–10], it is based on the master–slave synchronization, where the information signal is injected into the transmitter as a nonlinear filter; (c) chaos shift keying [11], the information signal is supposed to be binary, and it is mapped into the transmitter and the receiver. In these three cases, the information signal can be recovered by a receiver if the transmitter and the receiver are synchronized. In order to reach synchronization, the receiver should be a replica of the transmitter [5]. Linear and nonlinear observers in control theory literatures can be applied to design receivers. The receiver is regarded as a chaotic observer, which has two parts: a duplicated chaotic system of the transmitter and an adjustable observer gain [9]. Some modifications were made when it is difficult to obtain a replica of the synchronization. For example, the transmitter and the receiver are set up into the same chaotic structures, parameter identification methods can be used to construct the chaotic receiver [12]; when there are uncertainties in synchronization (the transmitter is not known exactly, there is noise in the transmission line, etc.), the transmitter and the receiver could be established in the same fuzzy models, fuzzy mode-based design method was applied to reach synchronization [13]; stability analysis of observer-based chaotic communication with respect to uncertainties can be found in References [14, 8]. In this paper, a novel design approach for chaotic communication is proposed, where the receiver is a pure high-gain observer. The main difference with the above methods is that the receiver is no longer a chaotic system. A single parameter is adjusted so that the transmitter and the receiver can be synchronized. Since the receiver is not a replica of the transmitter, the uncertainty of the transmitter will not affect the synchronization. The proposed communication scheme can be more robust than systems that employ chaotic systems as transmitter and receiver. But a single adjustable parameter means that the information may be recovered by the observer who has no knowledge about the transmitter, this is a big challenger to secure communication by the means of chaos. Numerical demonstrations using prototype of chaotic oscillators are also provided.
2. CHAOTIC COMMUNICATION BASED ON HIGH-GAIN OBSERVER In normal chaotic communication, the transmitter and the receiver are chaotic systems. They can be described in the form of the following nonlinear system: x’ ¼ f ðxÞ þ gðxÞu y ¼ hðxÞ
ð1Þ
where x 2 Rn is the state of the plant, u 2 R is control input, y 2 R is measurable output, f ; g and h are smooth nonlinear functions. Most of chaotic systems have uniform relative degree n; i.e. Lg hðxÞ ¼ ¼ Lg Lfn2 hðxÞ ¼ 0;
Lg Ln1 hðxÞ=0 f
So there exists a mapping x ¼ TðxÞ Copyright # 2005 John Wiley & Sons, Ltd.
ð2Þ Int. J. Commun. Syst. 2005; 18:487–500
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which can transform system (1) into the following normal form [15]: x’ i ¼ xiþ1 ; i ¼ 1 n 1 x’ n ¼ FðxÞ þ GðxÞu
ð3Þ
y ¼ x1 First, we discuss a simple case, the transmitter and the receiver are 2nd order chaotic oscillators, for example Duffing equation and Van der Pol oscillator. When n ¼ 2; (3) becomes x’ 1 ¼ x2 x’ 2 ¼ Hðx1 ; x2 ; uÞ y ¼ x1
ð4Þ
where Hðx1 ; x2 ; uÞ ¼ Fðx1 ; x2 Þ þ Gðx1 ; x2 Þu: Duffing equation describes a specific chaotic circuit [16]. It can be written as x’ 1 ¼ x2
ð5Þ
x’ 2 ¼ p1 x1 p2 x31 px2 þ q cosðotÞ þ ut
where p; p1 ; p2 ; q and o are constants. ut is control input. It is known that the solution of (5) exhibits almost periodic and chaotic behaviour. In uncontrolled case, if we select p1 ¼ 1:1; p2 ¼ 1; p ¼ 0:4; q ¼ 2:1; o ¼ 1:8; the Duffing oscillator (5) has a chaotic response as in Figure 1. The Van der Pol oscillator can be described as [17] x’ 1 ¼ x2
ð6Þ
x’ 2 ¼ a1 ½ð1 a2 x21 Þx2 a3 x1 þ ut 3
2
1
x2
0
-1
-2
-3
-4 -3
-2
-1
0 x1
1
2
3
Figure 1. Chaotic behaviour of Duffing equation with xð0Þ ¼ ½0; 0T : Copyright # 2005 John Wiley & Sons, Ltd.
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4 3 2
x2
1 0 -1 -2 -3 -4 -3
-2
-1
0 x1
1
2
3
Figure 2. Chaotic behaviour of Van der Pol oscillator with xð0Þ ¼ ½1; 1T :
In uncontrolled case, if we select a1 ¼ 1:5; a2 ¼ 1; a3 ¼ 1; the Van der Pol oscillator (6) has a chaotic response as in Figure 2. In this paper chaos modulation [14, 8, 9, 18] is used for communication, where the information signal s is embedded into the chaotic transmitter. The transmitter is a slight modification of the normal chaotic systems (4) as follows: x’ 1 ¼ x2 þ l1 s x’ 2 ¼ Hðy; x2 Þ þ l2 s
ð7Þ
y ¼ x1 þ s where the output y ¼ x1 þ s is chaotic masking, l1 s and l2 s are chaotic modulation, l1 and l2 are constants. A normal observer-based receiver is x’# 1 ¼ x# 2 þ l1 ðy y#Þ x’# 2 ¼ Hðy#; x# 2 Þ þ l2 ðy y#Þ y# ¼ x# 1
ð8Þ
where x# 1 ; x# 2 and y# are the states in the receiver side. The information signal is recovered as s# ¼ y y#: It can be seen that the receiver (8) is a replica of the transmitter (7). A drawback of such scheme is that it is difficult to choose l1 and l2 such that s# ! s: Riccati inequality (LMI tools) [8, 9] and observability matrix [14] can be used to obtain them. Copyright # 2005 John Wiley & Sons, Ltd.
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In this paper we discuss a new observer-based receiver. The high-gain observer-based receiver is x’# 1 ¼ x# 2 þ
k1 ðy y#Þ e
k2 ðy y#Þ e2 y# ¼ x# 1
ð9Þ
x’# 2 ¼
where e is a small positive parameter 05e51; k1 and k2 are positive constants. We select k1 and k2 such that A is stable, i.e. the roots of detðmI AÞ ¼ m2 þ k1 m þ k2 are in open left-hand side. The schematic diagram of the chaotic communication based on high gain observer is shown in Figure 3. The receiver (9) proposed in this paper is very easy to be applied and more robust compared with the other receivers in chaotic communication. For example in References [8, 9, 14]
Chaotic dynamic
1 s
+ +
1 s
+
x2 +
l2
y
+ +
l1
s Chaotic transmitter
1 s
+ +
k2 ε +
1 s xˆ1
+
xˆ2 +
yˆ
k1 ε2 sˆ No-identical receiver
Figure 3. Chaotic communication with high gain observer-based receiver. Copyright # 2005 John Wiley & Sons, Ltd.
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the receiver is x’# 1 ¼ x# 2 þ l1 ðy y#Þ x’# 2 ¼ a1 ½ð1 a2 x# 21 Þx# 2 a3 x# 1 þ l2 ðy y#Þ y# ¼ x# 1
ð10Þ
where l1 and l2 are the solution of a Riccati inequality. Any uncertainty in the transmitter side, for example the parameters a1 ; a2 and a3 are not known exactly, will affect the recovery accuracy of the information signal. Let us define the synchronization error as x* ¼ x x#
ð11Þ
where x ¼ ½x1 ; x2 T ; x# ¼ ½x# 1 ; x# 2 T : Since s# ¼ y y# ¼ x* 1 þ s by (7) and (9) the synchronization error can be formed as k1 k1 x* 1 s þ l1 s e e k k2 2 x’* 2 ¼ 2 x* 1 þ Hðy; x2 Þ 2 s þ l2 s e e
x’* 1 ¼ x* 2
ð12Þ
Let us define a new pair of variables z*1 ¼ x* 1 z*2 ¼ ex* 2
ð13Þ
In this paper the synchronization parameters are selected as l1 ¼
k1 ; e
l2 ¼
k2 e2
ð14Þ
(12) becomes ez’*1 ¼ z*2 k1 z*1 ez’*2 ¼ k2 z*1 þ e2 Hðz*1 ; z*2 ; sÞ
ð15Þ
ez’* ¼ Az* þ e2 BHðz*1 ; z*2 ; sÞ
ð16Þ
In the matrix form, (15) is
where
" z* ¼
½z*T1 ; z*T2 T ;
A¼
k1
1
k2
0
# ;
B¼
" # 0 1
ð17Þ
Next theorem gives the upper bound of the signal recovery error. Theorem 1 The high gain observer-based receiver (9) can recover the information signal s which is embedded in the chaotic transmitter (7), the signal recovery error s* ¼ s s# converges to Copyright # 2005 John Wiley & Sons, Ltd.
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HIGH-GAIN OBSERVER
493
De ¼ f*s j jj*sjj42e2 CT g
ð18Þ
the following residual set: where CT ¼ sup jjBHjj jjPjj t2½0;T
P is a solution of the following Lyapunov equation: AT P þ PA ¼ I
ð19Þ
Proof Due to the fact that A is stable, there exists a positive definite matrix P such that the Lyapunov equation (19) is established. Consider following Lyapunov function: Vðz*Þ ¼ ez*T Pz* The derivation along the solutions of (15) is V’ ¼ ez’*T Pz* þ ez*T Pz’* ¼ ½Az* þ e2 BHðz*1 ; z*2 ; sÞT Pz* þ z*T P½Az* þ e2 BHðz*1 ; z*2 ; sÞ ¼ z*T ðAT P þ PAÞz* þ 2e2 ½BHðz*1 ; z*2 ; sÞT Pz* 4 z*T ðAT P þ PAÞz* þ 2e2 jjBHðz*1 ; z*2 ; sÞjj jjPjj jz*j
ð20Þ
For any chaotic communication system, the solution of the transmitter (7) in t 2 ½0; T is bounded, so jjHðz*1 ; z*2 ; sÞjj is bounded for any finite time T: We conclude that jjBHðz*1 ; z*2 ; sÞjj jjPjj is bounded. From (19) we have V’ 4 jjz*jj2 þ K% ðeÞ jjz*jj where K% ðeÞ ¼ 2e2 CT ;
CT ¼ sup jjBHðz*1 ; z*2 ; sÞjj jjPjj t2½0;T
It is noted that if jjz*ðtÞjj > K% ðeÞ
ð21Þ
then V’ i 50; 8t 2 ½0; T: So the total time during which jjz*ðtÞjj > K% ðeÞ is finite. Let Tk denotes the time interval during jjz*ðtÞjj > K% ðeÞ *
*
If only finite times that z*ðtÞ stay outside the ball of radius K% ðeÞ and then reenter, z*ðtÞ will eventually stay inside of this ball. If z*ðtÞ leave the ball infinite times, since the total time z*ðtÞ leave the ball is finite, 1 X
Tk 51
k¼1
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so lim Tk ¼ 0
ð22Þ
k!1
Now z*ðtÞ is bounded via an invariant set argument. From (16) z’*ðtÞ is also bounded. Let jjz*k ðtÞjj denote the largest tracking error during the Tk interval. Then (22) and bounded z’*ðtÞ imply that lim ½jjz*k ðtÞjj K% ðeÞ ¼ 0 k!1
1 So jjz*k ðtÞjj will convergence to K% ðeÞ: Because s* ¼ ½1; 0 x* ¼ ½1; 0 0 So jj*sjj converges to the ball of radius K% ðeÞ:
0 1 e
z*; and e51: &
The design procedure for the chaotic communication is as follows: k1 1 1. Choose two positive constants k1 and k2 such that A ¼ is stable, solve k2 0 Lyapunov equation (19) to get P: 2. Select e according to the accuracy requirement, jj*sjj42e2 supðjjHjj jjPjjÞ: 3. Select the synchronization parameters l1 and l2 in the transmitter, such that l1 ¼ k1 =e; l2 ¼ k2 =e2 : Remark 2 Since CT is bounded, we can select e arbitrary small (the gain of the receiver (9) becomes bigger) in order to make the observer error small enough. So j#s sj can be arbitrary small when e ! 0: Remark 3 Although we have restricted ourselves to the case of second-order chaotic system, the observer construction and convergence analysis can be extended to n-dimensional case. The chaotic transmitter is x’ j ¼ xjþ1 þ lj s; j ¼ 1; . . . ; n 2 x’ n1 ¼ xn2 þ ln1 s x’ n ¼ Hðx; sÞ þ ln s y ¼ x1 þ s the high-gain observer-based receiver is constructed as kj x’# j ¼ x# jþ1 þ j ðy y#Þ; j ¼ 1; . . . ; n 2 e kn1 x’# n1 ¼ x# n2 þ n1 ðy y#Þ e ’x# n ¼ kn ðy y#Þ en
ð23Þ
n1 where the constants kj are chosen such that the polynomial mn þ kn1 þ þ k1 ¼ 0 has all n m its roots in the open left-hand side of the complex plane. As the second-order case, it can be proved that the synchronization error x* converges to any accuracy by selecting sufficiently small
Copyright # 2005 John Wiley & Sons, Ltd.
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HIGH-GAIN OBSERVER
values of observer gain e: The synchronization parameter is chosen as k1 kn T L ¼ ½l1 . . . ln T ¼ n e e Remark 4 Some chaotic systems do not have the normal form as (3), we cannot apply high gain observer directly on the receiver, for example Chua’s circuit C1 x’ 1 ¼ Gðx2 x1 Þ gðx1 Þ þ u C2 x’ 2 ¼ Gðx1 x2 Þ þ x3
ð24Þ
Lx’ 3 ¼ x2 y ¼ x3
where gðx1 Þ ¼ m0 x1 þ 12ðm1 m0 Þ½jx1 þ Bp j þ jx1 Bp j; x1 ; x2 ; x3 denote the voltages across C1 ; C2 and L: It is known that with C1 ¼ 19; C2 ¼ 1; L ¼ 17; G ¼ 0:7; m0 ¼ 0:5; m1 ¼ 1:5; Bp ¼ 1 the circuit displays double scroll as Figure 4. If we make transformation x ¼ TðxÞ as x 1 ¼ x3 x2 ¼ Lx2 LG L x3 ¼ ðx2 x1 Þ x3 C2 C2
ð25Þ
the Chua’s circuit becomes the normal form x’ 1 ¼ x2 x’ 2 ¼ x3 x’ 3 ¼ f ðx1 ; x2 ; x3 Þ þ gu y ¼ x1 15 10
x2
5 0 -5 -10 -15 4 2
10 5
0 x3
-2
-5 -4
0 x1
-10
Figure 4. Chaotic behaviour of Chua’s circuit with initial condition ½0; 0; 1T : Copyright # 2005 John Wiley & Sons, Ltd.
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where f ðx1 ; x2 ; x3 Þ ¼ G=C2 fx3 ðG=C1 Þ½2x2 ð1=GLÞx1 ðC2 =GÞx3 ð1=C1 Þgðx1 ð1=GLÞx1 ðC2 =GÞx3 Þg ð1=C2 LÞx1 ; g ¼ G=C1 C2 : Now high-gain observer-based receiver (23) can be applied.
3. NUMERICAL SIMULATION We use three different chaotic systems as transmitters, the information signal s is embedded in the output y: (a) Duffing equation 1 x’ 1 ¼ x2 þ s t 1 ð26Þ x’ 2 ¼ 1:1y y3 0:4x2 þ 2:1 cosð1:8tÞ þ 2 s t y ¼ x1 þ s; xð0Þ ¼ ½0; 0T
(b) Van der Pol oscillator 1 x’ 1 ¼ x2 þ s t 1 s t2 xð0Þ ¼ ½2; 1T
ð27Þ
x’ 2 ¼ 1:5½ð1 x21 Þx2 x1 þ y ¼ x1 þ s;
(c) Chua’s circuit, we use the following parameters: C1 ¼ 19; C2 ¼ 1; L ¼ 17; G ¼ 0:7; m0 ¼ 0:5; m1 ¼ 1:5; Bp ¼ 1: By transformation (25), the Chua’s circuit (24) can be written as x’ 1 ¼ x2 x’ 2 ¼ x3 31 310 22 22 220 x3 x1 x3 x2 x1 0:7x2 7x1 þ 22gðx1 ; x2 ; x3 Þ x’ 3 ¼ 4:9 7 4:9 7 7 y ¼ x2 ; xð0Þ ¼ ½1; 0; 7T 1 1 10 1 1 10 where gðx1 ; x2 ; x3 Þ ¼ j x3 x2 x1 þ 1j j x3 x2 x1 1j: We use x2 and 4:9 7 7 4:9 7 7 x3 as the transmitter 1 x’ 2 ¼ x3 þ s t 31 310 22 22 220 1 x3 x1 x3 y x1 0:7y 7x1 þ 22gðx1 ; y; x3 Þ þ 2 s x’ 3 ¼ 4:9 7 4:9 7 7 t y ¼ x2 þ s
ð28Þ
where x1 satisfies x’ 1 ¼ x2 : Copyright # 2005 John Wiley & Sons, Ltd.
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Now the high-gain-based receiver. We choose k1 ¼ k2 ¼ 1; the eigenvalues of we design k1 1 A¼ are 0:5000 0:8660i; so A is stable. The solution of (19) is k2 0 " # 1 12 T P¼P ¼ >0 12 32 If we require the maximum signal recovery error is less than 2%, we can select e ¼ 0:01: For (26)–(28) we can estimate supðCT Þ ¼ 100; according to Theorem 1, j#s sj42CT 104 ¼ 2 102 : The high-gain observer-based receiver is x’# 1 ¼ x# 2 þ 102 ðy y#Þ x’# 2 ¼ 104 ðy y#Þ y# ¼ x# 1 ;
ð29Þ
x# ð0Þ ¼ ½1; 1T
The synchronization parameters are l1 ¼ 1=e ¼ 102 ; l2 ¼ 1=e2 ¼ 104 : The information signal s is chosen as sinusoidal signal with frequency of 100 Hz as in References [8, 9], i.e. s ¼ 0:05 sinð200ptÞ Figures 5–7 show the communication process with three different chaotic transmitters and one receiver, here the waveform of the transmitted signal y is shown in subplot (a), the convergence behavior of s s# is shown in subplots (b). After transient process ðt > 0:1 sÞ; the maximum relative error is defined as
The output of transmitter (y)
emax ¼ 4 2 0 -2
0
2
4
6
8
10
Time (s)
(a) Information recovery error ( s − sˆ )
maxðjs s#jÞ maxðjsjÞ
0.5 0 -0.5 -1
0
0.05
(b)
0.1
0.15
0.2 0.25 Time (s)
0.3
0.35
0.4
0.45
Figure 5. Duffing equation for chaotic communication. Copyright # 2005 John Wiley & Sons, Ltd.
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The output of transmitter (y)
4 2 0 -2 -4
0
2
4
6
8
10
Time (s)
Information recovery error ( s − sˆ )
(a)
0.5 0 -0.5 -1
0
0.05
0.1
0.15
0.2 0.25 Time (s)
(b)
0.3
0.35
0.4
0.45
Figure 6. Van der Pol oscillator for chaotic communication.
The output of transmitter (y)
4 2 0 -2 -4 0
Information recovery error ( s − sˆ )
(a)
2
4
Time (s)
6
8
10
1 0.5 0
-0.5 0
0.05
0.1
(b)
0.15
0.2 0.25 Time (s)
0.3
0.35
0.4
0.45
Figure 7. Chua’s circuit for chaotic communication.
For Duffing oscillator, emax ffi 1:55%: For Van der Pol oscillator, emax ffi 2%: For Chua’s circuit, emax ffi 0:915%: Although the relative errors are different, they are acceptable for signal communication. It is interesting to see that one receiver (29) can recover the information signal from three different chaotic transmitters. Copyright # 2005 John Wiley & Sons, Ltd.
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0.3 0.25
Signal error ( s − sˆ )
0.2 0.15
Model-based observer
0.1
P high-gain observer (l1 = 0.9 × 10 2 , l 2 = 0.9 × 104 )
0.05 0 -0.05 -0.1
0
0.5
1
1.5
2 2.5 Time (s)
3
3.5
4
Figure 8. Signal errors for different types of receivers.
Model-based observer requires complete information of the transmitter. We use linear observer [9] to compare with our results, see Figure 8. We find that model-based observer gives the best performance, but if the transmitter is unknown or partly known, this kind of receiver does not work. Another advantage of model-based receiver is that it can be applied to any chaotic transmitter as in (1), but high-gain observer is only suitable for the chaotic system which has the normal form as in (3).
4. CONCLUSION In this paper, we propose a novel chaotic communication approach, where the receiver is a highgain observer. The main difference with normal chaos modulation in communication is that the receiver is no longer a chaotic system, only one parameter L ¼ f ðki ; eÞ can synchronize the transmitter and the receiver. The proposed scheme can be more robust and less secure than communications that employ chaotic systems as transmitter and receiver. Although the communication single error can be arbitrary small by selecting a proper observer gain in the receiver, large observer gain will also enlarge transmission noise. High-gain observer-based receiver cannot work as well as the normal receivers, but it may impose security risk on the current secure communication system using chaotic communication technique when the transmitter is in the form of (3) or it can be transformed into this form. To the best of our knowledge, this kind of receiver has not yet been applied in real application. But we hope this paper will encourage the research effort in the real chaotic communication field. For example, in real application we should avoid to use a transmitter which has the form of (3). Copyright # 2005 John Wiley & Sons, Ltd.
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AUTHOR’S BIOGRAPHY
Wen Yu received the BS degree from Tsinghua University, Beijing, China in 1990 and the MS and PhD degrees, both in Electrical Engineering, from Northeastern University, Shenyang, China, in 1992 and 1995, respectively. From 1995 to 1996, he served as a Lecturer in the Department of Automatic Control at Northeastern University, Shenyang, China. In 1996, he joined CINVESTAV-IPN, M!exico, where he is a professor in the Departamento de Control Autom!atico. He has held a research position with the Instituto Mexicano del Petro! leo, from December 2002 to November 2003. His research interests include adaptive control, neural networks, and fuzzy control.
Copyright # 2005 John Wiley & Sons, Ltd.
Int. J. Commun. Syst. 2005; 18:487–500
High-gain observer for chaotic synchronization and secure communication Wen Yun,y Departamento de Control Automatico, CINVESTAV-IPN, A.P. 14-740, Av.IPN 2508, Me!xico D.F., 07360, Me´xico
SUMMARY An information signal can be embedded in a transmitter which produces a chaotic signal, it is inviable to the observer who has no knowledge about the transmitter, but it can be recovered if the receiver is a replica of the chaotic transmitter. In this paper, a new aspect of the chaotic communication is introduced. A highgain observer at the receiver’s end replaces the conventional chaotic system. A single parameter is adjusted so that the receiver and transmitter can be synchronized. Three chaotic systems, Duffing equation, Van der Pol oscillator and Chua’s circuit, are provided to illustrate the effectiveness of the chaotic communication. Copyright # 2005 John Wiley & Sons, Ltd. KEY WORDS:
synchronization; chaotic communication; observer
1. INTRODUCTION The general idea for transmitting information via chaotic systems is that, an information signal is embedded in the transmitter system which produces a chaotic signal, the information signal is inviable to the observer who has no knowledge about the transmitter system, it is recovered when the transmitter and the receiver are identical. Since Pecora and Carroll’s observation on the possibility of synchronizing two chaotic systems [1] (so-called drive–response configuration), several synchronization schemes have been developed. Synchronization can be classified into two types: mutual synchronization (or bidirectional coupling) [2] and master–slave synchronization (or unidirectional coupling) [1]. The chaos-based secure communications have updated their fourth generation [3]. The continuous synchronization is adopted in the first three generations while the impulsive synchronization is used in the fourth generation. Less than 94 Hz of bandwidth is needed to transmit the synchronization signal for a third-order chaotic transmitter in the fourth generation while 30 kHz bandwidth needed for transmitting the synchronization signals in the other three generations [4]. There are many applications on chaotic communication [5] and chaotic network synchronization [6]. The techniques of chaotic communication can be divided into three n
Correspondence to: Wen Yu, Departamento de Control Automatico, CINVESTAV-IPN, A.P. 14-740, Av.IPN 2508, M!exico D.F., 07360, Me´xico. y E-mail: [email protected]
Copyright # 2005 John Wiley & Sons, Ltd.
Received 12 February 2004 Revised 9 December 2004 Accepted 12 December 2004
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categories: (a) chaos masking [7], the information signal is added directly to the transmitter; (b) chaos modulation [5, 8–10], it is based on the master–slave synchronization, where the information signal is injected into the transmitter as a nonlinear filter; (c) chaos shift keying [11], the information signal is supposed to be binary, and it is mapped into the transmitter and the receiver. In these three cases, the information signal can be recovered by a receiver if the transmitter and the receiver are synchronized. In order to reach synchronization, the receiver should be a replica of the transmitter [5]. Linear and nonlinear observers in control theory literatures can be applied to design receivers. The receiver is regarded as a chaotic observer, which has two parts: a duplicated chaotic system of the transmitter and an adjustable observer gain [9]. Some modifications were made when it is difficult to obtain a replica of the synchronization. For example, the transmitter and the receiver are set up into the same chaotic structures, parameter identification methods can be used to construct the chaotic receiver [12]; when there are uncertainties in synchronization (the transmitter is not known exactly, there is noise in the transmission line, etc.), the transmitter and the receiver could be established in the same fuzzy models, fuzzy mode-based design method was applied to reach synchronization [13]; stability analysis of observer-based chaotic communication with respect to uncertainties can be found in References [14, 8]. In this paper, a novel design approach for chaotic communication is proposed, where the receiver is a pure high-gain observer. The main difference with the above methods is that the receiver is no longer a chaotic system. A single parameter is adjusted so that the transmitter and the receiver can be synchronized. Since the receiver is not a replica of the transmitter, the uncertainty of the transmitter will not affect the synchronization. The proposed communication scheme can be more robust than systems that employ chaotic systems as transmitter and receiver. But a single adjustable parameter means that the information may be recovered by the observer who has no knowledge about the transmitter, this is a big challenger to secure communication by the means of chaos. Numerical demonstrations using prototype of chaotic oscillators are also provided.
2. CHAOTIC COMMUNICATION BASED ON HIGH-GAIN OBSERVER In normal chaotic communication, the transmitter and the receiver are chaotic systems. They can be described in the form of the following nonlinear system: x’ ¼ f ðxÞ þ gðxÞu y ¼ hðxÞ
ð1Þ
where x 2 Rn is the state of the plant, u 2 R is control input, y 2 R is measurable output, f ; g and h are smooth nonlinear functions. Most of chaotic systems have uniform relative degree n; i.e. Lg hðxÞ ¼ ¼ Lg Lfn2 hðxÞ ¼ 0;
Lg Ln1 hðxÞ=0 f
So there exists a mapping x ¼ TðxÞ Copyright # 2005 John Wiley & Sons, Ltd.
ð2Þ Int. J. Commun. Syst. 2005; 18:487–500
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which can transform system (1) into the following normal form [15]: x’ i ¼ xiþ1 ; i ¼ 1 n 1 x’ n ¼ FðxÞ þ GðxÞu
ð3Þ
y ¼ x1 First, we discuss a simple case, the transmitter and the receiver are 2nd order chaotic oscillators, for example Duffing equation and Van der Pol oscillator. When n ¼ 2; (3) becomes x’ 1 ¼ x2 x’ 2 ¼ Hðx1 ; x2 ; uÞ y ¼ x1
ð4Þ
where Hðx1 ; x2 ; uÞ ¼ Fðx1 ; x2 Þ þ Gðx1 ; x2 Þu: Duffing equation describes a specific chaotic circuit [16]. It can be written as x’ 1 ¼ x2
ð5Þ
x’ 2 ¼ p1 x1 p2 x31 px2 þ q cosðotÞ þ ut
where p; p1 ; p2 ; q and o are constants. ut is control input. It is known that the solution of (5) exhibits almost periodic and chaotic behaviour. In uncontrolled case, if we select p1 ¼ 1:1; p2 ¼ 1; p ¼ 0:4; q ¼ 2:1; o ¼ 1:8; the Duffing oscillator (5) has a chaotic response as in Figure 1. The Van der Pol oscillator can be described as [17] x’ 1 ¼ x2
ð6Þ
x’ 2 ¼ a1 ½ð1 a2 x21 Þx2 a3 x1 þ ut 3
2
1
x2
0
-1
-2
-3
-4 -3
-2
-1
0 x1
1
2
3
Figure 1. Chaotic behaviour of Duffing equation with xð0Þ ¼ ½0; 0T : Copyright # 2005 John Wiley & Sons, Ltd.
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4 3 2
x2
1 0 -1 -2 -3 -4 -3
-2
-1
0 x1
1
2
3
Figure 2. Chaotic behaviour of Van der Pol oscillator with xð0Þ ¼ ½1; 1T :
In uncontrolled case, if we select a1 ¼ 1:5; a2 ¼ 1; a3 ¼ 1; the Van der Pol oscillator (6) has a chaotic response as in Figure 2. In this paper chaos modulation [14, 8, 9, 18] is used for communication, where the information signal s is embedded into the chaotic transmitter. The transmitter is a slight modification of the normal chaotic systems (4) as follows: x’ 1 ¼ x2 þ l1 s x’ 2 ¼ Hðy; x2 Þ þ l2 s
ð7Þ
y ¼ x1 þ s where the output y ¼ x1 þ s is chaotic masking, l1 s and l2 s are chaotic modulation, l1 and l2 are constants. A normal observer-based receiver is x’# 1 ¼ x# 2 þ l1 ðy y#Þ x’# 2 ¼ Hðy#; x# 2 Þ þ l2 ðy y#Þ y# ¼ x# 1
ð8Þ
where x# 1 ; x# 2 and y# are the states in the receiver side. The information signal is recovered as s# ¼ y y#: It can be seen that the receiver (8) is a replica of the transmitter (7). A drawback of such scheme is that it is difficult to choose l1 and l2 such that s# ! s: Riccati inequality (LMI tools) [8, 9] and observability matrix [14] can be used to obtain them. Copyright # 2005 John Wiley & Sons, Ltd.
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In this paper we discuss a new observer-based receiver. The high-gain observer-based receiver is x’# 1 ¼ x# 2 þ
k1 ðy y#Þ e
k2 ðy y#Þ e2 y# ¼ x# 1
ð9Þ
x’# 2 ¼
where e is a small positive parameter 05e51; k1 and k2 are positive constants. We select k1 and k2 such that A is stable, i.e. the roots of detðmI AÞ ¼ m2 þ k1 m þ k2 are in open left-hand side. The schematic diagram of the chaotic communication based on high gain observer is shown in Figure 3. The receiver (9) proposed in this paper is very easy to be applied and more robust compared with the other receivers in chaotic communication. For example in References [8, 9, 14]
Chaotic dynamic
1 s
+ +
1 s
+
x2 +
l2
y
+ +
l1
s Chaotic transmitter
1 s
+ +
k2 ε +
1 s xˆ1
+
xˆ2 +
yˆ
k1 ε2 sˆ No-identical receiver
Figure 3. Chaotic communication with high gain observer-based receiver. Copyright # 2005 John Wiley & Sons, Ltd.
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the receiver is x’# 1 ¼ x# 2 þ l1 ðy y#Þ x’# 2 ¼ a1 ½ð1 a2 x# 21 Þx# 2 a3 x# 1 þ l2 ðy y#Þ y# ¼ x# 1
ð10Þ
where l1 and l2 are the solution of a Riccati inequality. Any uncertainty in the transmitter side, for example the parameters a1 ; a2 and a3 are not known exactly, will affect the recovery accuracy of the information signal. Let us define the synchronization error as x* ¼ x x#
ð11Þ
where x ¼ ½x1 ; x2 T ; x# ¼ ½x# 1 ; x# 2 T : Since s# ¼ y y# ¼ x* 1 þ s by (7) and (9) the synchronization error can be formed as k1 k1 x* 1 s þ l1 s e e k k2 2 x’* 2 ¼ 2 x* 1 þ Hðy; x2 Þ 2 s þ l2 s e e
x’* 1 ¼ x* 2
ð12Þ
Let us define a new pair of variables z*1 ¼ x* 1 z*2 ¼ ex* 2
ð13Þ
In this paper the synchronization parameters are selected as l1 ¼
k1 ; e
l2 ¼
k2 e2
ð14Þ
(12) becomes ez’*1 ¼ z*2 k1 z*1 ez’*2 ¼ k2 z*1 þ e2 Hðz*1 ; z*2 ; sÞ
ð15Þ
ez’* ¼ Az* þ e2 BHðz*1 ; z*2 ; sÞ
ð16Þ
In the matrix form, (15) is
where
" z* ¼
½z*T1 ; z*T2 T ;
A¼
k1
1
k2
0
# ;
B¼
" # 0 1
ð17Þ
Next theorem gives the upper bound of the signal recovery error. Theorem 1 The high gain observer-based receiver (9) can recover the information signal s which is embedded in the chaotic transmitter (7), the signal recovery error s* ¼ s s# converges to Copyright # 2005 John Wiley & Sons, Ltd.
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HIGH-GAIN OBSERVER
493
De ¼ f*s j jj*sjj42e2 CT g
ð18Þ
the following residual set: where CT ¼ sup jjBHjj jjPjj t2½0;T
P is a solution of the following Lyapunov equation: AT P þ PA ¼ I
ð19Þ
Proof Due to the fact that A is stable, there exists a positive definite matrix P such that the Lyapunov equation (19) is established. Consider following Lyapunov function: Vðz*Þ ¼ ez*T Pz* The derivation along the solutions of (15) is V’ ¼ ez’*T Pz* þ ez*T Pz’* ¼ ½Az* þ e2 BHðz*1 ; z*2 ; sÞT Pz* þ z*T P½Az* þ e2 BHðz*1 ; z*2 ; sÞ ¼ z*T ðAT P þ PAÞz* þ 2e2 ½BHðz*1 ; z*2 ; sÞT Pz* 4 z*T ðAT P þ PAÞz* þ 2e2 jjBHðz*1 ; z*2 ; sÞjj jjPjj jz*j
ð20Þ
For any chaotic communication system, the solution of the transmitter (7) in t 2 ½0; T is bounded, so jjHðz*1 ; z*2 ; sÞjj is bounded for any finite time T: We conclude that jjBHðz*1 ; z*2 ; sÞjj jjPjj is bounded. From (19) we have V’ 4 jjz*jj2 þ K% ðeÞ jjz*jj where K% ðeÞ ¼ 2e2 CT ;
CT ¼ sup jjBHðz*1 ; z*2 ; sÞjj jjPjj t2½0;T
It is noted that if jjz*ðtÞjj > K% ðeÞ
ð21Þ
then V’ i 50; 8t 2 ½0; T: So the total time during which jjz*ðtÞjj > K% ðeÞ is finite. Let Tk denotes the time interval during jjz*ðtÞjj > K% ðeÞ *
*
If only finite times that z*ðtÞ stay outside the ball of radius K% ðeÞ and then reenter, z*ðtÞ will eventually stay inside of this ball. If z*ðtÞ leave the ball infinite times, since the total time z*ðtÞ leave the ball is finite, 1 X
Tk 51
k¼1
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so lim Tk ¼ 0
ð22Þ
k!1
Now z*ðtÞ is bounded via an invariant set argument. From (16) z’*ðtÞ is also bounded. Let jjz*k ðtÞjj denote the largest tracking error during the Tk interval. Then (22) and bounded z’*ðtÞ imply that lim ½jjz*k ðtÞjj K% ðeÞ ¼ 0 k!1
1 So jjz*k ðtÞjj will convergence to K% ðeÞ: Because s* ¼ ½1; 0 x* ¼ ½1; 0 0 So jj*sjj converges to the ball of radius K% ðeÞ:
0 1 e
z*; and e51: &
The design procedure for the chaotic communication is as follows: k1 1 1. Choose two positive constants k1 and k2 such that A ¼ is stable, solve k2 0 Lyapunov equation (19) to get P: 2. Select e according to the accuracy requirement, jj*sjj42e2 supðjjHjj jjPjjÞ: 3. Select the synchronization parameters l1 and l2 in the transmitter, such that l1 ¼ k1 =e; l2 ¼ k2 =e2 : Remark 2 Since CT is bounded, we can select e arbitrary small (the gain of the receiver (9) becomes bigger) in order to make the observer error small enough. So j#s sj can be arbitrary small when e ! 0: Remark 3 Although we have restricted ourselves to the case of second-order chaotic system, the observer construction and convergence analysis can be extended to n-dimensional case. The chaotic transmitter is x’ j ¼ xjþ1 þ lj s; j ¼ 1; . . . ; n 2 x’ n1 ¼ xn2 þ ln1 s x’ n ¼ Hðx; sÞ þ ln s y ¼ x1 þ s the high-gain observer-based receiver is constructed as kj x’# j ¼ x# jþ1 þ j ðy y#Þ; j ¼ 1; . . . ; n 2 e kn1 x’# n1 ¼ x# n2 þ n1 ðy y#Þ e ’x# n ¼ kn ðy y#Þ en
ð23Þ
n1 where the constants kj are chosen such that the polynomial mn þ kn1 þ þ k1 ¼ 0 has all n m its roots in the open left-hand side of the complex plane. As the second-order case, it can be proved that the synchronization error x* converges to any accuracy by selecting sufficiently small
Copyright # 2005 John Wiley & Sons, Ltd.
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HIGH-GAIN OBSERVER
values of observer gain e: The synchronization parameter is chosen as k1 kn T L ¼ ½l1 . . . ln T ¼ n e e Remark 4 Some chaotic systems do not have the normal form as (3), we cannot apply high gain observer directly on the receiver, for example Chua’s circuit C1 x’ 1 ¼ Gðx2 x1 Þ gðx1 Þ þ u C2 x’ 2 ¼ Gðx1 x2 Þ þ x3
ð24Þ
Lx’ 3 ¼ x2 y ¼ x3
where gðx1 Þ ¼ m0 x1 þ 12ðm1 m0 Þ½jx1 þ Bp j þ jx1 Bp j; x1 ; x2 ; x3 denote the voltages across C1 ; C2 and L: It is known that with C1 ¼ 19; C2 ¼ 1; L ¼ 17; G ¼ 0:7; m0 ¼ 0:5; m1 ¼ 1:5; Bp ¼ 1 the circuit displays double scroll as Figure 4. If we make transformation x ¼ TðxÞ as x 1 ¼ x3 x2 ¼ Lx2 LG L x3 ¼ ðx2 x1 Þ x3 C2 C2
ð25Þ
the Chua’s circuit becomes the normal form x’ 1 ¼ x2 x’ 2 ¼ x3 x’ 3 ¼ f ðx1 ; x2 ; x3 Þ þ gu y ¼ x1 15 10
x2
5 0 -5 -10 -15 4 2
10 5
0 x3
-2
-5 -4
0 x1
-10
Figure 4. Chaotic behaviour of Chua’s circuit with initial condition ½0; 0; 1T : Copyright # 2005 John Wiley & Sons, Ltd.
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where f ðx1 ; x2 ; x3 Þ ¼ G=C2 fx3 ðG=C1 Þ½2x2 ð1=GLÞx1 ðC2 =GÞx3 ð1=C1 Þgðx1 ð1=GLÞx1 ðC2 =GÞx3 Þg ð1=C2 LÞx1 ; g ¼ G=C1 C2 : Now high-gain observer-based receiver (23) can be applied.
3. NUMERICAL SIMULATION We use three different chaotic systems as transmitters, the information signal s is embedded in the output y: (a) Duffing equation 1 x’ 1 ¼ x2 þ s t 1 ð26Þ x’ 2 ¼ 1:1y y3 0:4x2 þ 2:1 cosð1:8tÞ þ 2 s t y ¼ x1 þ s; xð0Þ ¼ ½0; 0T
(b) Van der Pol oscillator 1 x’ 1 ¼ x2 þ s t 1 s t2 xð0Þ ¼ ½2; 1T
ð27Þ
x’ 2 ¼ 1:5½ð1 x21 Þx2 x1 þ y ¼ x1 þ s;
(c) Chua’s circuit, we use the following parameters: C1 ¼ 19; C2 ¼ 1; L ¼ 17; G ¼ 0:7; m0 ¼ 0:5; m1 ¼ 1:5; Bp ¼ 1: By transformation (25), the Chua’s circuit (24) can be written as x’ 1 ¼ x2 x’ 2 ¼ x3 31 310 22 22 220 x3 x1 x3 x2 x1 0:7x2 7x1 þ 22gðx1 ; x2 ; x3 Þ x’ 3 ¼ 4:9 7 4:9 7 7 y ¼ x2 ; xð0Þ ¼ ½1; 0; 7T 1 1 10 1 1 10 where gðx1 ; x2 ; x3 Þ ¼ j x3 x2 x1 þ 1j j x3 x2 x1 1j: We use x2 and 4:9 7 7 4:9 7 7 x3 as the transmitter 1 x’ 2 ¼ x3 þ s t 31 310 22 22 220 1 x3 x1 x3 y x1 0:7y 7x1 þ 22gðx1 ; y; x3 Þ þ 2 s x’ 3 ¼ 4:9 7 4:9 7 7 t y ¼ x2 þ s
ð28Þ
where x1 satisfies x’ 1 ¼ x2 : Copyright # 2005 John Wiley & Sons, Ltd.
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HIGH-GAIN OBSERVER
Now the high-gain-based receiver. We choose k1 ¼ k2 ¼ 1; the eigenvalues of we design k1 1 A¼ are 0:5000 0:8660i; so A is stable. The solution of (19) is k2 0 " # 1 12 T P¼P ¼ >0 12 32 If we require the maximum signal recovery error is less than 2%, we can select e ¼ 0:01: For (26)–(28) we can estimate supðCT Þ ¼ 100; according to Theorem 1, j#s sj42CT 104 ¼ 2 102 : The high-gain observer-based receiver is x’# 1 ¼ x# 2 þ 102 ðy y#Þ x’# 2 ¼ 104 ðy y#Þ y# ¼ x# 1 ;
ð29Þ
x# ð0Þ ¼ ½1; 1T
The synchronization parameters are l1 ¼ 1=e ¼ 102 ; l2 ¼ 1=e2 ¼ 104 : The information signal s is chosen as sinusoidal signal with frequency of 100 Hz as in References [8, 9], i.e. s ¼ 0:05 sinð200ptÞ Figures 5–7 show the communication process with three different chaotic transmitters and one receiver, here the waveform of the transmitted signal y is shown in subplot (a), the convergence behavior of s s# is shown in subplots (b). After transient process ðt > 0:1 sÞ; the maximum relative error is defined as
The output of transmitter (y)
emax ¼ 4 2 0 -2
0
2
4
6
8
10
Time (s)
(a) Information recovery error ( s − sˆ )
maxðjs s#jÞ maxðjsjÞ
0.5 0 -0.5 -1
0
0.05
(b)
0.1
0.15
0.2 0.25 Time (s)
0.3
0.35
0.4
0.45
Figure 5. Duffing equation for chaotic communication. Copyright # 2005 John Wiley & Sons, Ltd.
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The output of transmitter (y)
4 2 0 -2 -4
0
2
4
6
8
10
Time (s)
Information recovery error ( s − sˆ )
(a)
0.5 0 -0.5 -1
0
0.05
0.1
0.15
0.2 0.25 Time (s)
(b)
0.3
0.35
0.4
0.45
Figure 6. Van der Pol oscillator for chaotic communication.
The output of transmitter (y)
4 2 0 -2 -4 0
Information recovery error ( s − sˆ )
(a)
2
4
Time (s)
6
8
10
1 0.5 0
-0.5 0
0.05
0.1
(b)
0.15
0.2 0.25 Time (s)
0.3
0.35
0.4
0.45
Figure 7. Chua’s circuit for chaotic communication.
For Duffing oscillator, emax ffi 1:55%: For Van der Pol oscillator, emax ffi 2%: For Chua’s circuit, emax ffi 0:915%: Although the relative errors are different, they are acceptable for signal communication. It is interesting to see that one receiver (29) can recover the information signal from three different chaotic transmitters. Copyright # 2005 John Wiley & Sons, Ltd.
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0.3 0.25
Signal error ( s − sˆ )
0.2 0.15
Model-based observer
0.1
P high-gain observer (l1 = 0.9 × 10 2 , l 2 = 0.9 × 104 )
0.05 0 -0.05 -0.1
0
0.5
1
1.5
2 2.5 Time (s)
3
3.5
4
Figure 8. Signal errors for different types of receivers.
Model-based observer requires complete information of the transmitter. We use linear observer [9] to compare with our results, see Figure 8. We find that model-based observer gives the best performance, but if the transmitter is unknown or partly known, this kind of receiver does not work. Another advantage of model-based receiver is that it can be applied to any chaotic transmitter as in (1), but high-gain observer is only suitable for the chaotic system which has the normal form as in (3).
4. CONCLUSION In this paper, we propose a novel chaotic communication approach, where the receiver is a highgain observer. The main difference with normal chaos modulation in communication is that the receiver is no longer a chaotic system, only one parameter L ¼ f ðki ; eÞ can synchronize the transmitter and the receiver. The proposed scheme can be more robust and less secure than communications that employ chaotic systems as transmitter and receiver. Although the communication single error can be arbitrary small by selecting a proper observer gain in the receiver, large observer gain will also enlarge transmission noise. High-gain observer-based receiver cannot work as well as the normal receivers, but it may impose security risk on the current secure communication system using chaotic communication technique when the transmitter is in the form of (3) or it can be transformed into this form. To the best of our knowledge, this kind of receiver has not yet been applied in real application. But we hope this paper will encourage the research effort in the real chaotic communication field. For example, in real application we should avoid to use a transmitter which has the form of (3). Copyright # 2005 John Wiley & Sons, Ltd.
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AUTHOR’S BIOGRAPHY
Wen Yu received the BS degree from Tsinghua University, Beijing, China in 1990 and the MS and PhD degrees, both in Electrical Engineering, from Northeastern University, Shenyang, China, in 1992 and 1995, respectively. From 1995 to 1996, he served as a Lecturer in the Department of Automatic Control at Northeastern University, Shenyang, China. In 1996, he joined CINVESTAV-IPN, M!exico, where he is a professor in the Departamento de Control Autom!atico. He has held a research position with the Instituto Mexicano del Petro! leo, from December 2002 to November 2003. His research interests include adaptive control, neural networks, and fuzzy control.
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