- Email: [email protected]
CHAOTIC SYNCHRONISATION FOR SECURE COMMUNICATION USING PI-OBSERVERS
CHAOTIC SYNCHRONISATION FOR SECURE COMMUNICATION USING PI-OBSERVERS P. Johnson and K. Busawon
Northumbria University, School of Computing, Engineering and Information Sciences, Ellison Building, Newcastle-Upon-Tyne, NE1 8ST, U.K. [email protected]
Abstract: In this paper, we present a chaotic synchronisation scheme for secure communication using proportional integral (PI) observers. It is shown that this proposed synchronisation scheme is able to handle measurement or message noise in the sense that the noise can be attenuated up to some reasonable level. Simulations are carried out to show the message recovery performance of the scheme. Keywords: Chaotic synchronisation, observers, secure communications, masking scheme
1. INTRODUCTION There has been a lot of interest in the problem of synchronization of chaotic systems for secure communication over the last decade. Indeed, several synchronization schemes have been developed using different techniques (see references herein). The classical observer-based chaotic synchronization scheme for secure communication is illustrated in Figure 1.
by an observer in order to produce an estimate yˆ(t) of the output y(t). This implies that a certain degree of robustness must be exhibited by the observer in generating the estimated output yˆ(t) - since it is excited by the transmitted signal y 0 (t) which obviously does not entirely coincide with the carrier signal y(t). This also implies that the message should not be of a too high amplitude compared to that of the output. The received message mr (t) is generally obtained by performing the following substraction: mr (t) = y 0 (t) − yˆ(t) = y(t) − yˆ(t) + m(t). The observer is generally designed such that lim |y(t) − yˆ(t)| → 0. As a result, the received
t→+∞
Figure 1. Classical observer-based synchronisation scheme In this scheme the chaotic transmitter generates a chaotic signal y(t) upon which a message m(t) is superimposed. The output y(t) of the transmitter acts as a message carrier. At the receiver end, the transmitted signal y 0 (t) = y(t) + m(t) is processed
message mr (t) will asymptotically converge to the transmitted message m(t). Obviously, if the estimated output yˆ(t) converges exponentially to the output y(t), then we will have a better convergence between mr (t) and m(t). Unfortunately, in many cases, it is not possible to obtain an exponential convergence of the output error to zero. This is mainly due to the fact that, in many instances, the error dynamics of the observer contain some residual terms that are dependent on
the transmitted message. This generally gives rise to a distortion in the received message. An appropriate filter might be employed, in some cases, to filter out the message from the residual terms (Morgul 1999). In other cases, either the message m(t) or the transmitted signal y 0 (t) is fedback into the chaotic drive system in order to compensate for these residual terms. These are symbolised by the dotted lines in Figure 1. The main difficulties in the design and implementation of the above scheme are: i) nonlinear observers design is, in general, not an easy task. Some hypotheses on the nonlinearities are generally required (lipschitz condition, triangular structure, persistency of inputs etc.). ii) the injection of either the message or the transmitted signal cannot be done in an arbitrary fashion. This has to be done via an input function. As a result, full cancellation of non-linearities and full compensation of any residual terms appearing in the error dynamics might not be possible. iii) the noise in the output or in the message might hamper the recovery of the message at the receiving end. In effect, if η(t) is an additive noise affecting the message m(t) or the output y(t) then the received message will be affected by the noise as follows mr (t) = y(t)− yˆ(t) + m(t) + η(t). This implies that even if yˆ(t) tracks y(t) exponentially the received message will not be exempt from the noise; i.e. mr (t) ≈ m(t) + η(t). This means that some mechanism for attenuating the effect of the noise influence in the received message has to be found. On the other hand, it is shown in (Busawon and Kabore 2001) that proportional integral observers (or in short PI-observers) have the capability of attenuating disturbances. Taking into account the above remarks, we proposed, in this paper, a PIobserver-based chaotic synchronization scheme for secure communication. This is illustrated in Figure 2.
Figure 2. PI-observer-based synchronisation scheme In the proposed scheme, the transmitted signal y 0 (t) is integrated at the receiver end yielding Rt x0 = t0 y 0 (τ )dτ . This integrated signal together with the transmitted signal is used to design a
PI-observer. It is shown that the proposed PIobserver has the capability of reducing the message or sensor noise up to some acceptable level. In addition, an output feedback g(y 0 , u) is derived and fedback to the chaotic drive system via an input function. It allows the cancellation of the nonlinearities involved in the error dynamics of the observer, and, it also clarifies how a nonlinear output injection is actually performed in practice. In the next section, the proposed synchronization scheme is described. An example using the Duffing oscillator is given in order to show the design procedure. It is shown that the proposed scheme has a better message recovery performance than the classical proportional observer-based (or P-observer-based) synchronization scheme in the presence of measurement and/or message noise. In Section 3, simulations are carried out to show the performance of the proposed scheme compared to P-observer-based synchronization scheme. Finally, some conclusions are given. 2. MAIN METHODOLOGY In this section, the proposed PI-observer-based chaotic synchronization scheme for secure communication, illustrated by Figure 2, is described. We shall assume that the transmitter system is composed of a chaotic system described by ¾ x˙ = Ax + Bf (y) +h (t) + Bu (1) y = Cx
where x ∈Rn , u∈R, y∈R, f is a smooth function, h is the forcing function. The matrices A, B and C are of the following form: 0 0 1 0 .. . . .. . A= . , B = . 0 1 a1 · · · an 1 ¡ ¢ C= 1 0 ··· 0 .
The function u represents an input function through which an output feedback can be applied. For simplicity, we have considered only one nonlinearity to clarify the design procedure. However, the procedure can be extended to the case where we have several nonlinearities provided each nonlinearity can be influenced by some input function. Note that many chaotic systems, if they are not already of the above form, can be brought into the above form by a change of variable. In particular the Duffing oscillator is already of the above form. 2.1 The masking scheme The transmitted signal is described as y 0 = y + m = Cx + m.
where m(t) denotes a message. Note that the following difference f (y) − f (y0 ) can always be expressed as a function of the message and the transmitted signal only. In effect, we have 0
0
0
0
f (y) − f (y ) = f (y − m) − f (y ) = g(y ,m). Consequently, to create the masking system, we apply the following feedback via the input function u(t): u = −g(y 0 ,m).
As a result the masking scheme will be defined as follows: ¾ x˙ = Ax + h (t) + B (f (y) − g(y 0 ,m)) (2) y 0 = Cx + m
¢ ¡ ¢ ¡ ¯ − KI C0 e + CT0 − Kp m ¯ − Kp C e˙ = A ¢ ¡ = Fe + CT0 − Kp m
¡Now, it remains¢ to choose Kp such that the pair ¯ − Kp C, ¯ A ° T C0 ° is observable and at the same ° time C0 − Kp ° ≈ 0. Once Kp is chosen, we ¡can then choose K¢I such that the matrix F = ¯ − Kp C ¯ − KI C0 is stable. As a result, the A error will converge asymptotically zero as t → +∞. In general, if we wish to keep the influence of the message minimal in the error dynamics, we can choose Kp as Kp = αCT0 with 0 < α < 1. In such a case, the error dynamics will reduce to
In other words, in closed-loop the masking scheme is given by: ¾ x˙ = Ax + Bf (y 0 ) + h (t) (3) y 0 = Cx + m
e˙ = Fe + (1 − α) CT0 m. ¡ ¢ ¯ C0 ¯ − Kp C, Finally, note that if α = 1, the pair A is not observable. This means that one cannot totally eliminate the message from the error dynamics.
Note that it is because the nonlinearity f (y) is directly affected by the input function that the above output injection is possible.
2.3 Performance with respect to noise
2.2 The PI-observer-based receiver design By placing an integrator at the receiving end we obtain an integrated transmitted signal defined by Z t Z t y 0 (τ ) dτ = (Cx+m) dτ = yI . x0 = 0
0
In other words, x˙ 0 = Cx + m. By combining the above equation with the transmitter system, we can create the augmented system: ¯ (t) + CT0 m ¯ aug + Bf ¯ (y 0 ) + h x˙ aug = Ax yI = C0 xaug ¯ aug + m y 0 = Cx (4) where µ
¶
µ
¶
0 C , 0n×1 A µ ¶ µ ¶ 0 ¯ (t) = ¯= 0 , h B , B h (t) ¡ ¢ ¡ ¢ ¯ = 0C . C0 = C 0 and C
xaug =
x0 x
¯ = , A
A PI-observer for such a system is given by: .
¯ + KI (yI − C0 x ¯ xaug + Bf ¯ (y 0 ) + h x ˆaug = Aˆ ˆaug ) ¡ 0 ¢ ¯ (5) xaug +Kp y − Cˆ
By defining e = xaug −ˆ xaug , the error dynamics is given by
If the message or the output was corrupted by an additive noise η(t), then the error dynamics will become ¢ ¡ ¢ ¡ ¯ − KI C0 e+ CT0 − Kp (m + η) . ¯ − Kp C e˙ = A
One can choose Kp appropriately such that the amplitude of noise and the message is attenuated up to some reasonable level. Once the proportional gain is chosen for that purpose, the integral ¡gain KI can then be¢chosen to stabilise the matrix ¯ − Kp C ¯ − KI C0 . A 2.4 Comparison with the proportional observer In this subsection, we compare the performance of the above design procedure with the classical one using the proportional observer or (P-observer) as a receiver. In this case, the masking system will be given as before by system (3). At the receiving end, the observer will be given by .
x ˆ = Aˆ x + Bf (y 0 ) + h (t) + Kp (y 0 − Cˆ x) . (6) By defining e = x − x ˆ, the error dynamics is given by e˙ = (A − Kp C) e − Kp m.
In this case, it can be seen that the proportional gain is multiplied by the message m(t). If a too high gain is chosen, the message recovery might be compromised. On the other hand, is a too low gain is chosen then the stability of the error dynamics might be compromised. Otherwise, whenever possible, one can use a filter in order to recover the
x˙ 0 = x1 + m
message as suggested in (Morgul 1999). Note that if the message or the output was corrupted by an additive noise η(t), then the error dynamics will be given by
x˙ 1 = x2 y0 x˙ 2 = − − y 03 + 11 cos t 4 y 0 = x1 + m
e˙ = (A − Kp C) e − Kp (m + η) . It is clear that, in this case, the proportional gain cannot handle both the noise and the stability of the matrix (A − Kp C) adequately at the same time. Consequently, this scheme is not suitable for handling noise.
3. APPLICATION USING THE DUFFING OSCILLATOR In this section, we shall apply the above PIobserver-based synchronization scheme by using the Duffing oscillator as the drive system. For comparison purposes, the classical P-observerbased synchronization scheme will also be presented. Consider the Duffing oscillator system x˙ 1 = x2 y x˙ 2 = − − y 3 + 11 cos t + u 4 y = x1 . The system is of the form (1) with µ
µ ¶ ¡ ¢ 0 A= , B= , C= 1 0 1 µ ¶ y 0 . f (y) = − − y 3 , h (t) = 11 cos t 4 01 00
¶
By setting the y 0 = x1 + m, we note that 0
f (y) − f (y ) =
µ
y y0 − 4 4
¶
¢ ¡ + y 03 − y 3
1 2 = m + 3 (y 0 ) m − 3y 0 m2 + m3 4 = g(y 0 , m) 3.1 The masking system By applying the output control u(t) = −g(y 0 , m), the masking system is given by x˙ 1 = x2 y0 x˙ 2 = − − y 03 + 11 cos t 4 3.2 PI-observer-based scheme Rt
By setting x0 = 0 y 0 (τ ) dτ = yI , we have x˙ 0 = x1 + m so that we have the following augmented system:
yI = x0 and the PI-observer is given by . x ˆ =x ˆ1 + kI0 (x0 − x ˆ0 ) + kp0 (y 0 − x ˆ1 ) .0 0 x ˆ1 = x ˆ2 + kI1 (x0 − x ˆ0 ) + kp1 (y − x ˆ1 ) (7) . y0 03 x ˆ2 = − − y + 11 cos t + kI2 (x0 − x ˆ0 ) 4 ˆ1 ) + kp2 (y 0 − x
where KpT = (kp0 , kp1 , kp2 ) is the proportional gain and KIT = (kI0 , kI1 , kI2 ) is the integral gain. The error dynamics are given by e˙ 0 = −kI0 e0 + βe1 + βm e˙ 1 = −kI1 e0 − kp1 e1 + e2 − kp1 m e˙ 2 = −kI2 e0 − kp2 e1 − kp2 m
(8)
where β = (1 − kp0 ). We choose kp1 = kp2 = 0, so that we have: e0 −kI0 β 0 β e˙ 0 e˙ 1 = −kI1 0 1 e1 + 0 m e˙ 2 e2 0 −kI2 0 0
It can be shown after some calculations that the transfer function between e1 (t) and m(t) is given by: βkI1 s + βkI2 E1 (s) =− 3 = GP I (s) 2 M (s) s + s kI0 + kI1 βs + kI2 β (9) To give GP I (s) a zero gain while maintaining stability we choose β very small and kI0 > 1. The gain kI1 and kI2 can be chosen reasonably large.
3.3 P-observer-based scheme In this case, the receiver is given by . x ˆ1 = x ˆ2 + kp1 (y 0 − x ˆ1 ) 0 . y 3 x ˆ2 = − − (y 0 ) + 11 cos t + kp2 (y 0 − x ˆ1 ) 4 (10) and the error dynamics is given by ½ e˙ 1 = −kp1 e1 + e2 − kp1 m (11) e˙ 2 = −kp2 e1 − kp2 m. In this case, it can be shown that the transfer function between e1 (t) and m(t) is given by: kp1 s + kp2 E1 (s) =− 2 = GP (s) M (s) s + kp1 s + kp2
(12)
Here again to give GP (s) a zero gain, we should choose k1 and k2 very small. This means that
the poles of the P-observer should be placed very closed to the origin. Remark 2. It is difficult to carry out a clear cut comparison between GP (s) and GP I (s). However, by observing GP (s) and GP I (s), it can be noticed that the gain kI0 provides an additional degree of freedom to improve the stability of the transfer function GP I (s). Indeed, if we set βkI1 = kp1 and βkI2 = kp2 , then we have GP I (s) = −
kp1 s + kp2 . s3 + s2 kI0 + kp1 s + kp2
In that case, the numerator of both transfer functions GP (s) and GP I (s) are the same. However, the gain kI0 can be used adjust the poles of GP I (s) while the poles of GP (s) are already fixed by kp1 and kp2 . 3.4 Simulations results:
10% of the message amplitude was employed and the following gains were used kI0 = 10, kI1 = kI2 = 20, kp1 = kp2 = 2. It can be seen that the message recovery performance of the PI is very good the presence of noise. It should be stressed that the above comparison is not perfect since the PI-observer is of order 3 while the Pobserver is of order 2. The comparison could also have been made by fixing two of the poles of both observers at similar locations. However, the general conclusions obtained above would still be valid in such a case. It is clear that there are better choice of gains of both observers. For example when kI0 = 20, kI1 = kI2 = 10 it can be shown that the PI-observer would perform better. Finally, while there might be better choice of gains for both observers than the ones presented above it can nevertheless be concluded that the performance of the PI-observer in the presence of noise is better than that of the P-observer. 8
Simulation studies of the above PI-observer and P-observer synchronization schemes are carried out to show their message recovery performance both in the presence and absence of message and/or measurement noise. The following message was used: m(t) = sin πt. We have chosen kp0 = 0.9 so that β = 0.1. To make the comparison with the P-observer relevant, we shall choose kp1 = kp2 = βkI1 = βkI2 since we are mainly concerned in making the numerators of the transfer functions GP (s) and GP I (s) defined previously as small as possible. Figure 3 and 4 shows the output of the chaotic drive and the transmitted message. It can be seen that both signals look similar and the message is not discernible from the transmited signal. Figure 5 and Figure 6 show the original message m(t) and the received message mr (t) (in dotted lines and denoted as ’mr’ in the figure legend) by the PI-observer and the P-observer respectively when a fairly low gain was applied. These set of simulation were carried out for the following choice of gain: kI0 = 10, kI1 = kI2 = 10, the poles of kp1 = kp2 = 1. That is, we have √ the P-observer located at − 12 ± 12 i 3 and that of the PI-observer at −9. 9093 and −4. 5366 × 10−2 ± 0. 31442i. In this case, the poles of the Pobserver are close to the imaginary axis. It can be observed that both the PI and the P-observer has good message recovery performance. Even though not shown here, if a high gain was used, the Pobserver will not perform well since the message will be very distorted and its amplitude will be considerably reduced compared to that obtained with the PI-observer. Figure 7 and Figure 8 show the original message and the received message by the PI-observer and the P-observer respectively in the presence of noise. A random zero-mean Gaussian noise of
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Fig. 4. The transmitted signal y 0 (t)
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Fig. 8. m(t), mr (t) with P-observer (noise)
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Cuomo, K. M. and Oppenheim, A. V., 1993, “Circuit implementation of synchronized chaos with applications to communications, ” Phys. Rev. Lett. 71, 65-68.
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REFERENCES Busawon, K. and Kabore P., Disturbance attenuation using proportional integral observers, Int. J. Control, 74, No.6, 2001, 618-627.
Fig. 6. m(t) and mr (t) with P-observer
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Chua LO, Kocarev LJ, Eckert K, Itoh M., Experimental chaos synchronization in Chua circuit, Int. J. Bifurcation and Chaos, 1992; 2:705-8.
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Kocarev, L. and Parlitz, U., 1995, “General approach for chaotic synchronization with applications to communication,” Phys. Rev. Lett. 75, 5028-5031.
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In this paper, we have presented a chaotic synchronization scheme for secure communication using proportional integral observers. We have shown that this proposed synchronization scheme is capable of attenuating measurement and/or message noise. As a result, the proposed synchronization scheme has a better message recovery performance in the presence of noise compared to the classical proportional observer-based chaotic synchronization scheme.
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4. CONCLUSIONS
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Fig. 7. m(t), mr (t) with PI-observer (noise)
Kolumban G., Kennedy MP, Chua LO, The role of synchronization in digital communications using chaos, IEEE Trans. Circuits Systems I: Fundamental Theory Appl. 2000. Morgul, O., 1999, “Necessary condition for observerbased chaos synchronization,” Phys. Rev. Lett. 82, No.1, 169-176.
Northumbria University, School of Computing, Engineering and Information Sciences, Ellison Building, Newcastle-Upon-Tyne, NE1 8ST, U.K. [email protected]
Abstract: In this paper, we present a chaotic synchronisation scheme for secure communication using proportional integral (PI) observers. It is shown that this proposed synchronisation scheme is able to handle measurement or message noise in the sense that the noise can be attenuated up to some reasonable level. Simulations are carried out to show the message recovery performance of the scheme. Keywords: Chaotic synchronisation, observers, secure communications, masking scheme
1. INTRODUCTION There has been a lot of interest in the problem of synchronization of chaotic systems for secure communication over the last decade. Indeed, several synchronization schemes have been developed using different techniques (see references herein). The classical observer-based chaotic synchronization scheme for secure communication is illustrated in Figure 1.
by an observer in order to produce an estimate yˆ(t) of the output y(t). This implies that a certain degree of robustness must be exhibited by the observer in generating the estimated output yˆ(t) - since it is excited by the transmitted signal y 0 (t) which obviously does not entirely coincide with the carrier signal y(t). This also implies that the message should not be of a too high amplitude compared to that of the output. The received message mr (t) is generally obtained by performing the following substraction: mr (t) = y 0 (t) − yˆ(t) = y(t) − yˆ(t) + m(t). The observer is generally designed such that lim |y(t) − yˆ(t)| → 0. As a result, the received
t→+∞
Figure 1. Classical observer-based synchronisation scheme In this scheme the chaotic transmitter generates a chaotic signal y(t) upon which a message m(t) is superimposed. The output y(t) of the transmitter acts as a message carrier. At the receiver end, the transmitted signal y 0 (t) = y(t) + m(t) is processed
message mr (t) will asymptotically converge to the transmitted message m(t). Obviously, if the estimated output yˆ(t) converges exponentially to the output y(t), then we will have a better convergence between mr (t) and m(t). Unfortunately, in many cases, it is not possible to obtain an exponential convergence of the output error to zero. This is mainly due to the fact that, in many instances, the error dynamics of the observer contain some residual terms that are dependent on
the transmitted message. This generally gives rise to a distortion in the received message. An appropriate filter might be employed, in some cases, to filter out the message from the residual terms (Morgul 1999). In other cases, either the message m(t) or the transmitted signal y 0 (t) is fedback into the chaotic drive system in order to compensate for these residual terms. These are symbolised by the dotted lines in Figure 1. The main difficulties in the design and implementation of the above scheme are: i) nonlinear observers design is, in general, not an easy task. Some hypotheses on the nonlinearities are generally required (lipschitz condition, triangular structure, persistency of inputs etc.). ii) the injection of either the message or the transmitted signal cannot be done in an arbitrary fashion. This has to be done via an input function. As a result, full cancellation of non-linearities and full compensation of any residual terms appearing in the error dynamics might not be possible. iii) the noise in the output or in the message might hamper the recovery of the message at the receiving end. In effect, if η(t) is an additive noise affecting the message m(t) or the output y(t) then the received message will be affected by the noise as follows mr (t) = y(t)− yˆ(t) + m(t) + η(t). This implies that even if yˆ(t) tracks y(t) exponentially the received message will not be exempt from the noise; i.e. mr (t) ≈ m(t) + η(t). This means that some mechanism for attenuating the effect of the noise influence in the received message has to be found. On the other hand, it is shown in (Busawon and Kabore 2001) that proportional integral observers (or in short PI-observers) have the capability of attenuating disturbances. Taking into account the above remarks, we proposed, in this paper, a PIobserver-based chaotic synchronization scheme for secure communication. This is illustrated in Figure 2.
Figure 2. PI-observer-based synchronisation scheme In the proposed scheme, the transmitted signal y 0 (t) is integrated at the receiver end yielding Rt x0 = t0 y 0 (τ )dτ . This integrated signal together with the transmitted signal is used to design a
PI-observer. It is shown that the proposed PIobserver has the capability of reducing the message or sensor noise up to some acceptable level. In addition, an output feedback g(y 0 , u) is derived and fedback to the chaotic drive system via an input function. It allows the cancellation of the nonlinearities involved in the error dynamics of the observer, and, it also clarifies how a nonlinear output injection is actually performed in practice. In the next section, the proposed synchronization scheme is described. An example using the Duffing oscillator is given in order to show the design procedure. It is shown that the proposed scheme has a better message recovery performance than the classical proportional observer-based (or P-observer-based) synchronization scheme in the presence of measurement and/or message noise. In Section 3, simulations are carried out to show the performance of the proposed scheme compared to P-observer-based synchronization scheme. Finally, some conclusions are given. 2. MAIN METHODOLOGY In this section, the proposed PI-observer-based chaotic synchronization scheme for secure communication, illustrated by Figure 2, is described. We shall assume that the transmitter system is composed of a chaotic system described by ¾ x˙ = Ax + Bf (y) +h (t) + Bu (1) y = Cx
where x ∈Rn , u∈R, y∈R, f is a smooth function, h is the forcing function. The matrices A, B and C are of the following form: 0 0 1 0 .. . . .. . A= . , B = . 0 1 a1 · · · an 1 ¡ ¢ C= 1 0 ··· 0 .
The function u represents an input function through which an output feedback can be applied. For simplicity, we have considered only one nonlinearity to clarify the design procedure. However, the procedure can be extended to the case where we have several nonlinearities provided each nonlinearity can be influenced by some input function. Note that many chaotic systems, if they are not already of the above form, can be brought into the above form by a change of variable. In particular the Duffing oscillator is already of the above form. 2.1 The masking scheme The transmitted signal is described as y 0 = y + m = Cx + m.
where m(t) denotes a message. Note that the following difference f (y) − f (y0 ) can always be expressed as a function of the message and the transmitted signal only. In effect, we have 0
0
0
0
f (y) − f (y ) = f (y − m) − f (y ) = g(y ,m). Consequently, to create the masking system, we apply the following feedback via the input function u(t): u = −g(y 0 ,m).
As a result the masking scheme will be defined as follows: ¾ x˙ = Ax + h (t) + B (f (y) − g(y 0 ,m)) (2) y 0 = Cx + m
¢ ¡ ¢ ¡ ¯ − KI C0 e + CT0 − Kp m ¯ − Kp C e˙ = A ¢ ¡ = Fe + CT0 − Kp m
¡Now, it remains¢ to choose Kp such that the pair ¯ − Kp C, ¯ A ° T C0 ° is observable and at the same ° time C0 − Kp ° ≈ 0. Once Kp is chosen, we ¡can then choose K¢I such that the matrix F = ¯ − Kp C ¯ − KI C0 is stable. As a result, the A error will converge asymptotically zero as t → +∞. In general, if we wish to keep the influence of the message minimal in the error dynamics, we can choose Kp as Kp = αCT0 with 0 < α < 1. In such a case, the error dynamics will reduce to
In other words, in closed-loop the masking scheme is given by: ¾ x˙ = Ax + Bf (y 0 ) + h (t) (3) y 0 = Cx + m
e˙ = Fe + (1 − α) CT0 m. ¡ ¢ ¯ C0 ¯ − Kp C, Finally, note that if α = 1, the pair A is not observable. This means that one cannot totally eliminate the message from the error dynamics.
Note that it is because the nonlinearity f (y) is directly affected by the input function that the above output injection is possible.
2.3 Performance with respect to noise
2.2 The PI-observer-based receiver design By placing an integrator at the receiving end we obtain an integrated transmitted signal defined by Z t Z t y 0 (τ ) dτ = (Cx+m) dτ = yI . x0 = 0
0
In other words, x˙ 0 = Cx + m. By combining the above equation with the transmitter system, we can create the augmented system: ¯ (t) + CT0 m ¯ aug + Bf ¯ (y 0 ) + h x˙ aug = Ax yI = C0 xaug ¯ aug + m y 0 = Cx (4) where µ
¶
µ
¶
0 C , 0n×1 A µ ¶ µ ¶ 0 ¯ (t) = ¯= 0 , h B , B h (t) ¡ ¢ ¡ ¢ ¯ = 0C . C0 = C 0 and C
xaug =
x0 x
¯ = , A
A PI-observer for such a system is given by: .
¯ + KI (yI − C0 x ¯ xaug + Bf ¯ (y 0 ) + h x ˆaug = Aˆ ˆaug ) ¡ 0 ¢ ¯ (5) xaug +Kp y − Cˆ
By defining e = xaug −ˆ xaug , the error dynamics is given by
If the message or the output was corrupted by an additive noise η(t), then the error dynamics will become ¢ ¡ ¢ ¡ ¯ − KI C0 e+ CT0 − Kp (m + η) . ¯ − Kp C e˙ = A
One can choose Kp appropriately such that the amplitude of noise and the message is attenuated up to some reasonable level. Once the proportional gain is chosen for that purpose, the integral ¡gain KI can then be¢chosen to stabilise the matrix ¯ − Kp C ¯ − KI C0 . A 2.4 Comparison with the proportional observer In this subsection, we compare the performance of the above design procedure with the classical one using the proportional observer or (P-observer) as a receiver. In this case, the masking system will be given as before by system (3). At the receiving end, the observer will be given by .
x ˆ = Aˆ x + Bf (y 0 ) + h (t) + Kp (y 0 − Cˆ x) . (6) By defining e = x − x ˆ, the error dynamics is given by e˙ = (A − Kp C) e − Kp m.
In this case, it can be seen that the proportional gain is multiplied by the message m(t). If a too high gain is chosen, the message recovery might be compromised. On the other hand, is a too low gain is chosen then the stability of the error dynamics might be compromised. Otherwise, whenever possible, one can use a filter in order to recover the
x˙ 0 = x1 + m
message as suggested in (Morgul 1999). Note that if the message or the output was corrupted by an additive noise η(t), then the error dynamics will be given by
x˙ 1 = x2 y0 x˙ 2 = − − y 03 + 11 cos t 4 y 0 = x1 + m
e˙ = (A − Kp C) e − Kp (m + η) . It is clear that, in this case, the proportional gain cannot handle both the noise and the stability of the matrix (A − Kp C) adequately at the same time. Consequently, this scheme is not suitable for handling noise.
3. APPLICATION USING THE DUFFING OSCILLATOR In this section, we shall apply the above PIobserver-based synchronization scheme by using the Duffing oscillator as the drive system. For comparison purposes, the classical P-observerbased synchronization scheme will also be presented. Consider the Duffing oscillator system x˙ 1 = x2 y x˙ 2 = − − y 3 + 11 cos t + u 4 y = x1 . The system is of the form (1) with µ
µ ¶ ¡ ¢ 0 A= , B= , C= 1 0 1 µ ¶ y 0 . f (y) = − − y 3 , h (t) = 11 cos t 4 01 00
¶
By setting the y 0 = x1 + m, we note that 0
f (y) − f (y ) =
µ
y y0 − 4 4
¶
¢ ¡ + y 03 − y 3
1 2 = m + 3 (y 0 ) m − 3y 0 m2 + m3 4 = g(y 0 , m) 3.1 The masking system By applying the output control u(t) = −g(y 0 , m), the masking system is given by x˙ 1 = x2 y0 x˙ 2 = − − y 03 + 11 cos t 4 3.2 PI-observer-based scheme Rt
By setting x0 = 0 y 0 (τ ) dτ = yI , we have x˙ 0 = x1 + m so that we have the following augmented system:
yI = x0 and the PI-observer is given by . x ˆ =x ˆ1 + kI0 (x0 − x ˆ0 ) + kp0 (y 0 − x ˆ1 ) .0 0 x ˆ1 = x ˆ2 + kI1 (x0 − x ˆ0 ) + kp1 (y − x ˆ1 ) (7) . y0 03 x ˆ2 = − − y + 11 cos t + kI2 (x0 − x ˆ0 ) 4 ˆ1 ) + kp2 (y 0 − x
where KpT = (kp0 , kp1 , kp2 ) is the proportional gain and KIT = (kI0 , kI1 , kI2 ) is the integral gain. The error dynamics are given by e˙ 0 = −kI0 e0 + βe1 + βm e˙ 1 = −kI1 e0 − kp1 e1 + e2 − kp1 m e˙ 2 = −kI2 e0 − kp2 e1 − kp2 m
(8)
where β = (1 − kp0 ). We choose kp1 = kp2 = 0, so that we have: e0 −kI0 β 0 β e˙ 0 e˙ 1 = −kI1 0 1 e1 + 0 m e˙ 2 e2 0 −kI2 0 0
It can be shown after some calculations that the transfer function between e1 (t) and m(t) is given by: βkI1 s + βkI2 E1 (s) =− 3 = GP I (s) 2 M (s) s + s kI0 + kI1 βs + kI2 β (9) To give GP I (s) a zero gain while maintaining stability we choose β very small and kI0 > 1. The gain kI1 and kI2 can be chosen reasonably large.
3.3 P-observer-based scheme In this case, the receiver is given by . x ˆ1 = x ˆ2 + kp1 (y 0 − x ˆ1 ) 0 . y 3 x ˆ2 = − − (y 0 ) + 11 cos t + kp2 (y 0 − x ˆ1 ) 4 (10) and the error dynamics is given by ½ e˙ 1 = −kp1 e1 + e2 − kp1 m (11) e˙ 2 = −kp2 e1 − kp2 m. In this case, it can be shown that the transfer function between e1 (t) and m(t) is given by: kp1 s + kp2 E1 (s) =− 2 = GP (s) M (s) s + kp1 s + kp2
(12)
Here again to give GP (s) a zero gain, we should choose k1 and k2 very small. This means that
the poles of the P-observer should be placed very closed to the origin. Remark 2. It is difficult to carry out a clear cut comparison between GP (s) and GP I (s). However, by observing GP (s) and GP I (s), it can be noticed that the gain kI0 provides an additional degree of freedom to improve the stability of the transfer function GP I (s). Indeed, if we set βkI1 = kp1 and βkI2 = kp2 , then we have GP I (s) = −
kp1 s + kp2 . s3 + s2 kI0 + kp1 s + kp2
In that case, the numerator of both transfer functions GP (s) and GP I (s) are the same. However, the gain kI0 can be used adjust the poles of GP I (s) while the poles of GP (s) are already fixed by kp1 and kp2 . 3.4 Simulations results:
10% of the message amplitude was employed and the following gains were used kI0 = 10, kI1 = kI2 = 20, kp1 = kp2 = 2. It can be seen that the message recovery performance of the PI is very good the presence of noise. It should be stressed that the above comparison is not perfect since the PI-observer is of order 3 while the Pobserver is of order 2. The comparison could also have been made by fixing two of the poles of both observers at similar locations. However, the general conclusions obtained above would still be valid in such a case. It is clear that there are better choice of gains of both observers. For example when kI0 = 20, kI1 = kI2 = 10 it can be shown that the PI-observer would perform better. Finally, while there might be better choice of gains for both observers than the ones presented above it can nevertheless be concluded that the performance of the PI-observer in the presence of noise is better than that of the P-observer. 8
Simulation studies of the above PI-observer and P-observer synchronization schemes are carried out to show their message recovery performance both in the presence and absence of message and/or measurement noise. The following message was used: m(t) = sin πt. We have chosen kp0 = 0.9 so that β = 0.1. To make the comparison with the P-observer relevant, we shall choose kp1 = kp2 = βkI1 = βkI2 since we are mainly concerned in making the numerators of the transfer functions GP (s) and GP I (s) defined previously as small as possible. Figure 3 and 4 shows the output of the chaotic drive and the transmitted message. It can be seen that both signals look similar and the message is not discernible from the transmited signal. Figure 5 and Figure 6 show the original message m(t) and the received message mr (t) (in dotted lines and denoted as ’mr’ in the figure legend) by the PI-observer and the P-observer respectively when a fairly low gain was applied. These set of simulation were carried out for the following choice of gain: kI0 = 10, kI1 = kI2 = 10, the poles of kp1 = kp2 = 1. That is, we have √ the P-observer located at − 12 ± 12 i 3 and that of the PI-observer at −9. 9093 and −4. 5366 × 10−2 ± 0. 31442i. In this case, the poles of the Pobserver are close to the imaginary axis. It can be observed that both the PI and the P-observer has good message recovery performance. Even though not shown here, if a high gain was used, the Pobserver will not perform well since the message will be very distorted and its amplitude will be considerably reduced compared to that obtained with the PI-observer. Figure 7 and Figure 8 show the original message and the received message by the PI-observer and the P-observer respectively in the presence of noise. A random zero-mean Gaussian noise of
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Fig. 4. The transmitted signal y 0 (t)
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Fig. 8. m(t), mr (t) with P-observer (noise)
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Cuomo, K. M. and Oppenheim, A. V., 1993, “Circuit implementation of synchronized chaos with applications to communications, ” Phys. Rev. Lett. 71, 65-68.
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REFERENCES Busawon, K. and Kabore P., Disturbance attenuation using proportional integral observers, Int. J. Control, 74, No.6, 2001, 618-627.
Fig. 6. m(t) and mr (t) with P-observer
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Chua LO, Kocarev LJ, Eckert K, Itoh M., Experimental chaos synchronization in Chua circuit, Int. J. Bifurcation and Chaos, 1992; 2:705-8.
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Kocarev, L. and Parlitz, U., 1995, “General approach for chaotic synchronization with applications to communication,” Phys. Rev. Lett. 75, 5028-5031.
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In this paper, we have presented a chaotic synchronization scheme for secure communication using proportional integral observers. We have shown that this proposed synchronization scheme is capable of attenuating measurement and/or message noise. As a result, the proposed synchronization scheme has a better message recovery performance in the presence of noise compared to the classical proportional observer-based chaotic synchronization scheme.
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4. CONCLUSIONS
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Fig. 7. m(t), mr (t) with PI-observer (noise)
Kolumban G., Kennedy MP, Chua LO, The role of synchronization in digital communications using chaos, IEEE Trans. Circuits Systems I: Fundamental Theory Appl. 2000. Morgul, O., 1999, “Necessary condition for observerbased chaos synchronization,” Phys. Rev. Lett. 82, No.1, 169-176.