A chaotic modulation scheme based on algebraic observability and

Chaos, Solitons and Fractals 26 (2005) 363–377 www.elsevier.com/locate/chaos

A chaotic modulation scheme based on algebraic observability and sliding mode differentiators Barbara Cannas a, Silvano Cincotti b

b,*

, Elio Usai

a

a DIEE, University of Cagliari, piazza D’Armi, I-09123 Cagliari, Italy Department of Biophysical and Electronic Engineering, University of Genoa, via Opera Pia 11a, I-16145 Genova, Italy

Accepted 20 December 2004

Abstract A chaotic communication technique for the transmission of secure information signals is presented. The proposed method allows the reconstruction of the system input (i.e., the information signal) from a scalar observable (i.e., the transmitted signal) and its derivatives. The approach is based on the concept of algebraic observability. A systematic procedure for the chaotic demodulation of the class of algebraic chaotic systems is described and discussed. The proposed procedure also allows one to directly identify a suitable ‘‘response’’ system and the ‘‘drive signal’’. Moreover, it is shown that sliding differentiators can be used to reconstruct the time derivatives of the observable, and thus the information signal is recovered at the receiving end through some simple signal-processing operations such as multiplication, addition and subtraction. This allows the estimation of the system state and of the input signal (i.e., the information recovery) in a finite time.
2005 Elsevier Ltd. All rights reserved.

1. Introduction In the last years, there has been a growing interest in chaotic communication techniques. Several methods have been proposed to scramble information signals with analogue chaotic signals at a transmitter, and subsequently to recover them from chaos at a receiver (see Fig. 1). Generally speaking, two main classes of chaotic communication schemes can be identified, i.e., non-coherent and coherent demodulations [1]. The first makes use of some statistical properties of the chaotic signal, whereas the second is based on the synchronization of chaotic systems, and thus makes use of the nonlinear dynamic equations that originate the chaotic signal. Synchronization-based coherent systems are advantageous in terms of bandwidth efficiency (in narrow band systems) and data rate (in chaotic systems). Conversely, when the synchronization between the transmitter and the receiver cannot be maintained (e.g., in presence of a strongly noisy channel) non-coherent systems may be preferable [2]. In recent years several techniques for non-coherent communication systems have been proposed that are based on chaotic spreading sequences for Multiple Access [3–6], on symbolic dynamics [7], on Chaotic Pulse Position Modulation schemes [8], etc. General approaches to non-coherent chaotic communication systems are described in [9,10]. *

Corresponding author. Tel.: +39 010 353 2080; fax: +39 010 353 2290. E-mail address: [email protected] (S. Cincotti).

0960-0779/$ - see front matter
2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.12.035

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Information signal

Transmitter (chaotic)

Public channel Receiver Retrieved information signal

Fig. 1. General layout of a chaotic communication system.

Concerning coherent communication systems, three main techniques have been proposed in the literature, i.e., chaotic masking [11,12], chaotic switching [13] and chaotic modulation [12,14,15]. Generally speaking, all these techniques strongly depend on the synchronization capabilities of the chaotic system used for the encryption process. From a security point of view, the chaotic modulation technique is, in general, more efficient than the other techniques based on chaotic encryption. The improvements in the security performances of chaotic modulation is counter parted by a more complex design of the receiver, which can usually be expressed in terms of a proper observer. Generally speaking, two main approaches to the problem of extracting a message from chaos by means of an observer can be found in the literature, i.e., the system identification approach and the inverse system one. The first considers a signal as a parameter of a non-linear system, and uses parameter identification methods to reconstruct it [16,17]. In the second case, the task is to find an asymptotic inverse mapping from the information signal to the transmitted signal [18]. The requirement for an efficient and realistic differential operator is frequently involved in chaotic synchronization and system inversion [19,20]. The main problem is to combine differentiation exactness with robustness to possible measurement errors and input noise. Recently, it has been shown that exact and robust differentiators with finite time convergence can be obtained by means of higher-order sliding mode control algorithms [21–23], and sliding differentiators have be used in two different observation schemes leading to chaos synchronization [24,25]. Starting from these results, a chaotic modulation scheme based on algebraic observability is presented and discussed. From a general point of view, the proposed approach belongs to the class of inverse system methods. In particular, the decryption problem is formulated as an observability problem, i.e., input reconstruction from measurements of an output variable under the assumption that the systemÕs structure and parameters are known. Thus, the information signal is recovered at the receiving end through some simple signal processing operations such as multiplication, addition and subtraction. This allows the estimations of the system state and of the input signal (i.e., the information recovery), in a finite time. As an extension of the Takens theorem [26] to non-autonomous systems, an inverse system represents a diffeomorphic mapping of an input time series to an output time series. A systematic procedure (i.e., independent of the choice of the drive system) is described, i.e., given a chaotic system, the proposed procedure allows one to directly identify a suitable ‘‘response’’ system and the ‘‘drive signal’’.

2. Input reconstruction by differential algebra Let us consider a non-linear time invariant dynamic system described by 8 < dx ¼ f ½xðtÞ; uðtÞ dt : y ¼ h½xðtÞ

ð1Þ

where x is the n-dimensional state variable vector, u is a scalar input, y a scalar output, and f and h, are algebraic function vectors in x. We assume that only one scalar output is measured as y(t) (i.e., the state is not directly available). The goal to be attained is the reconstruction of the system input u by using only the information contained in y. Definition 1. An algebraic observer for the input u of system (1) is a polynomial P = p(u, y, y(1), . . . , y(m)), with m 6 n [27]. The input of system (1) can be reconstructed if a smooth function g exists such that u = g(y, y(1), . . . , y(m)). The possibility that function g may exists is strictly linked to the ‘‘observability’’ concept. In the case of algebraic systems, the most natural way is to refer to ‘‘algebraic observability’’ [28,29]. The algebraic observability property can be easily tested within the differential algebra context by resorting to the concept of ‘‘Characteristic Set’’ associated to the dynamic equations. The Characteristic Set was introduced by Ritt [30] in

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1950, and since 1990 it has been widely used for the study of dynamic systems [28,29]. In order to define the Characteristic Set, we need to introduce some concepts of Differential Algebra. A detailed description can be found in [30]. Remark 1. The peculiarity of the Characteristic Set is that it summarizes all the information contained in the differential equations defining a dynamic system. If one chooses the ranking of the variables and their derivatives ð1Þ

ð1Þ

ð2Þ

ð2Þ

ðnÞ

ðnÞ

u < uð1Þ < uð2Þ <    < y < x1 < x2 <    y ð1Þ < x1 < x2    < y ð2Þ < x1 < x2    < y ðnÞ < x1 < x2 <    for a system of the form (1), the Characteristic Set exhibits n + 1 differential polynomials, that is: (1) an Input/Output (I/O) relation that is a differential polynomial in u, y and their derivatives, and is denoted by k(u, y); (2) n differential polynomials, triangular with respect to the state components, and denoted by the n-dimensional vector K(u, x, y). Property 1. The input is algebraically observable if at least one of the following equivalent relations is verified: (i) derivatives of the input u do not appear in the Characteristic Set; (ii) k(u, y) is of order n in the output y. Proposition 1. A necessary condition for the finite time global reconstruction of the input of system (1) is that the I/O relation is an algebraic observer of the input. If the I/O relation is not an algebraic observer, it contains some derivatives of the input, hence the input is the solution of a differential equation whose initial conditions are unknown. Thus, Remark 1 involves that no further information is available, and that the input admits infinite solutions. Conversely, if the I/O relation is an algebraic observer, the number of solutions is finite. Proposition 2. The input function can be globally recovered in a finite time iff it appears in the I/O relation with order zero and degree one. It is worth noting that, under the hypothesis of Proposition 2, the input u is characterized by an intriguing property, i.e., the global recovery in finite time holds for a non-derivable input. Several other definitions of input reconstruction can be obtained by direct extensions of Propositions 1 and 2 (e.g., local, asymptotical and partial) but they are not reported here for the sake of compactness. 3. Sliding mode approach to differentiation The availability of efficient devices or algorithms that produce the derivatives of a measured signal is a common need in system theory and in practical applications. Indeed, many observation and output feedback control problems can be solved under the assumption that the derivatives of a signal (e.g., the system output) can be evaluated [31–33]. Most differentiation devices are based on either numerical interpolation techniques [32], or linear high-gain filters [33,34], or non-linear filters [21–23,35], or proper electrical circuits [36]. Nevertheless, the first three techniques seem to be more attracting as they can be implemented in discrete time as computer routines. The main problem is to combine differentiation exactness with robustness to possible measurement errors and input noise. Because of their characteristics, even in the case of noisy-free signals, both interpolation techniques and high-gain differentiators are affected by a residual error. On the contrary, differentiation devices base on higher-order sliding modes are, to the best of the authorÕs knowledge, the only class of differentiators that gives, in principle, an exact and robust estimate of the derivatives of a sufficiently smooth signal in a finite time [21]. When the noise is taken into account, a precise evaluation of the accuracy of the derivative estimate is, in general, not easily known a priori. In particular, for the high-gain differentiator it is very dependent on the noise spectrum, while for the interpolation and higher-order sliding mode differentiation techniques it is possible to evaluate their accuracy. In fact, if d is the magnitude of the noise affecting the measured data, and L is a Lipshitz constant for the signal derivative, it was proven that both interpolation techniques with a proper regularization procedure [37] and sliding differentiators based on second-order sliding modes [21,22] guarantee that the estimation error of the differentiator is bounded, i.e.,

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pffiffiffiffiffiffi ky_  ^y_ k 6 k diff dL

ð2Þ

where kdiff is a proper constant depending on the chosen differentiation algorithm. In the case of sliding differentiators this theoretical result is, in some sense, corrupted when the real sliding behaviour is considered that is, an equivalent delay s affects the ideal instantaneous switching. In practice, the estimation error due to the delay sp isffiffiffidefined by the size of the boundary layer of a real second-order _ 6 k sd Ls). Therefore, the accuracy of a 2-SM differentiator in the pressliding mode (2-SM) behaviour [38] (i.e., jy_  ^yj ence of both measurement noise and switching delay cannot be better than the maximum error, that is,  pffiffiffiffiffiffi pffiffiffi  _ 6 k M max k sn dL; k sd Ls ð3Þ jy_  ^yj This means that, if the available signal is very noisy, it is non-sense to use high quality devices that guarantee a very small equivalent delay. The above relationships allow an almost direct extension of the results if a 2-SM algorithm is implemented in discrete time. In this case, the parameter s represents the sampling period, and the only additional requirement is that the controller parameters are properly tuned [22,39]. A systematic comparison of different differentiators is beyond the scope of this paper, in which we explore the possibility of using a sliding differentiator, based on higher-order sliding modes to process the transmitted signal in order to estimate its derivatives. In Section 2 it has been shown that the proposed method to invert a nth order dynamic system needs the derivatives of the measurable output up to the nth one. They could be estimated by the cascade implementation of n simple differentiators based on second-order sliding modes, but a better accuracy can be obtained by a recursive scheme, using a basic first-order 2-SM differentiator where the higher derivatives estimates are fed forward to improve accuracy [23,40]. The scheme based on the sub-optimal algorithm [40] requires a higher computational burden and greater care in setting the parameters than the scheme based on the super-twisting algorithm [23], therefore the latter is considered for the proposed application to signal recovery in chaotic systems. 3.1. Derivatives estimation by higher-order sliding modes Consider a smooth signal y(t) to be differentiated, and assume that its rth derivative is Lipshitz, i.e., a real positive constant L exists such that jy ðrþ1Þ j 6 L

ð4Þ

The rth-order sliding differentiator (r-SMD) is represented by the following set of differential equations whose solution exists in the FilippovÕs sense such that the corresponding motion is characterised by a (r + 1)-SM on the surface y  z0 = 0 [23], rþ1

z_ 0 ¼ C0 jy  z0 jrþ2 signðy  z0 Þ þ z1 r

z_ 1 ¼ C1 j_z0  z1 jrþ1 signð_z0  z1 Þ þ z2 .. . rþ1i

z_ i ¼ Ci j_zi1  zi jrþ2i signð_zi1  zi Þ þ ziþ1 ...

ð5Þ

z_ r ¼ Cr j_zr2  zr1 j1=2 signð_zr1  zr Þ þ zr z_ rþ1 ¼ Crþ1 signð_zr1  zr Þ where constants Ci (i = 0, 2, . . . , r + 1) are proper positive constants to be defined on the basis of the Lipshitz constant L. In particular, if some C i (i = 0, 2, . . . , r + 1) are effective for a Lipshitz constant L = 1, then L1=ðriþ1Þ C i (i = 0, 2, . . ., r + 1) work for any L > 0 [23]. 3.2. Robustness issues of higher-order sliding differentiators It has been shown that such a multiple differentiator is exact (i.e., it assures the perfect estimation of the first r-derivatives of the available signal y in a finite time) and robust (i.e., it has regularity properties with respect to the measure-

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ment noise and the switching delays) [23]. In fact, when a sampled noisy-free signal is differentiated with sampling period s the following estimation errors bounds result jz0  yj 6 k d 0 ðLÞsrþ1 ;

_ 6 k d 1 ðLÞsr ; jz1  yj

;

jzr  y ðrÞ j 6 k d r ðLÞs

ð6Þ

while, if only a measurement noise with amplitude d is taken into account, the estimation accuracy is defined by the following inequalities: rþ1i

jzi  y ðiÞ j 6 k ni ðLÞd rþ1 ;

i ¼ 0; 1; 2; . . . ; r

ð7Þ

Exact evaluation of constants kdi and kni (i = 0, 1, . . . , r) is not easy, but their upper bounds can be estimated by computer simulations taking into account the homogeneity properties of Eq. (5). In fact, if coefficients k ni (i = 0, 1, . . . , r) are evaluated by computer simulation for L = 1, then the following inequalities are valid for any L > 0 [23] rþ1i

jzi  y ðiÞ j 6 k ni Li=ðrþ1Þ d rþ1 ;

i ¼ 0; 1; 2; . . . ; r

Extending inequality (3) to the rth-order differentiator (5), it can be claimed that the differentiator output variables zi(i = 0, 1, . . . , r) are the estimates of the derivatives of the available signal y, affected by a bounded estimation error, i.e.,  rþ1i  zi ¼ y ðiÞ  ei L; d rþ1 ; srþ1i   ð8Þ rþ1i jei j 6 k M i max k ni Li=ðrþ1Þ d rþ1 ; k d i ðLÞsrþ1i ; i ¼ 0; 1; 2; . . . ; r This relationship can be used to over-estimate a noise-to-signal ratio and therefore the effectiveness of the method in the presence of noisy sampled signals. 4. Reconstruction and demodulation of signals Let us consider the reference dynamic system described by Eq. (1) and let us assume that the input is algebraically observable. Thus, there exists an algebraic relation between the input u and the output y with its first m 6 n derivatives, i.e., nðy; y ð1Þ ; . . . ; y ðmÞ ; uÞ ¼ 0

ð9Þ

Eq. (9) represents the I/O relation of the system. For a globally recoverable input, Eq. (9) can be rewritten as u ¼ gðy; y ð1Þ ; . . . ; y ðmÞ Þ

ð10Þ

Thus, in the case of global reconstruction, a m-order differentiator allows one to determine the input to the system uniquely by starting from the output measures and independently of the initial conditions [24,25]. Conversely, in the case of local reconstruction, the state of the system and the input signal are determined uniquely, provided that the initial conditions of the reconstructor are close to the initial conditions of the system [24,25]. The block-scheme depicted in Fig. 2 is the reference scheme proposed for the input recovery by the algebraic observability approach in the case of global reconstruction. It is worth noting that, in both cases, a m-order differentiator is mandatory. Therefore, it is possible to conclude that the general feature of the observability-based approach is the requirement for a differential operator. y

u

y(1)

x& = f ( x,u )

y

y( 2 ) g ( y, y (1) , y (2) ,..., y ( m ) )

y = h (x)

u$

y( m–1) x t =0 = x0

y( m )

Fig. 2. The reference scheme for input recovery in the case of global reconstruction.

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Such an efficient and realistic differential operator has often been involved in other approaches to chaotic synchronization. As an example, Itoh et al. have studied the reconstruction of chaotic attractors for a subclass of LurÕe systems by using an output and its derivatives [19]. A major problem associated with these methods concerns the errors that may be incurred in evaluation of the derivatives. Indeed, in practical implementations, the use of differentiation may generate high-frequency noise. As discussed in the previous section, such differentiators may be realized with sliding mode techniques [21–23]. The smoothness and robustness properties of higher-order sliding modes provide accurate finite time estimates of the first and higher derivatives of a smooth signal in the ideal case. Finally, it should be remarked that the proposed approach cannot be adopted by high gain observers, as they are approximate, and not robust, differentiators. The proposed approach (based on algebraic observability and sliding mode differentiators) to chaotic modulation recovery has been tested making reference to chaotic and hyperchaotic systems proposed in literature. 4.1. ChuaÕs circuit The great simplicity and considerable robustness have made the ChuaÕs circuit a paradigm to generate chaotic signals [41]. Concerning secure communication approaches based on chaotic systems, most of the papers fulfilled in recent Special Issues devoted to chaos synchronization [42] and non-coherent chaotic communications [9] make reference to ChuaÕs circuit. ChuaÕ s circuit is shown in Fig. 3. In the proposed chaotic communication system, ChuaÕ s circuit is used as the transmitter. A signal u(t) (i.e., the information signal) is given as input to ChuaÕs circuit in the transmitter. The signal u(t) can be introduced by a current source in parallel with the capacitor C1 or the capacitor C2 or by a voltage source in series with the inductor L. The signal y is the output of the transmitter, it is transmitted over a public channel to the receiver and used to synchronize the receiver so as to recover the information signal. The signal y may be either voltage vc1 , voltage vc2 , or current iL. Care should be exercised in choosing u(t) as this signal must guarantee the chaotic nature of the dynamics of the forced ChuaÕs circuit. Moreover, y should show an invariant ‘‘look’’, as compared with an autonomous ChuaÕs circuit. The configurations that allow the input reconstruction for ChuaÕs transmitter are characterized by  y ¼ v c1 • i.e., the voltage across the capacitor C1 is transmitted and u is injected through a voltage generator in series u ¼ u3 with the inductor L,  y ¼ iL i.e., the current through the inductor L is transmitted and u is injected through a current generator in par• u ¼ u1 allel with the capacitor C1. Let us consider the iL-drive configuration. In this case, the dynamics of ChuaÕs transmitter is given by the following state equation: 8 dx1 > > ¼ acx1 þ ax2  ax31 þ u1 > > > ds > < dx2 ¼ x1  x2 þ x3 > ds ð11Þ > > > > dx 3 > : ¼ bx2 ds y ¼ x3 ig R C2

C1

vC

vC

vg

2

1

L

iL

NR Fig. 3. ChuaÕ s circuit.

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The parameters used in the numerical simulation were fixed at a = 10, b = 16 and c = 0.143 that guarantee a chaotic behaviour for ChuaÕs circuit [43]. The transmitterÕs initial state was fixed at 3 2 3 2 0:1 x1 7 6 7 6 4 x2 5 ¼ 4 0:1 5 0:1 x3 and the receiverÕs initial state was fixed at 2 3 2 3 x1 0 6 7 6 7 4 x2 5 ¼ 4 0 5 x3

0

Rather a direct inspection of the dynamic equations (11) allows one to determine the algebraic observer for ChuaÕs circuit, i.e., 2 3 1 0 0 0 2 3 6 2 3 1 72 3 x3 0 0 0 7 y  6 0 6 7 3 b 6x 7 6 6 y_ 7 6 0 7 7 1 6 27 6 7 6 7 76 1 1 € ð12Þ  a y þ ð_ y þ y Þ 6 7 ¼ 6 1 6 7 6 7   0 7 4 x1 5 6 405 b 74 €y 5 b b 6 7 v 4 u1 1 b þ aðc  1Þ ac þ 1 15 y ac    b b b which ensures that the information signal u1 can be globally recovered in a finite time (see Proposition 2). Furthermore, Eqs. (8) and (12) allow for an a priori evaluation of the demodulation error. It is worth noting that, in the case of the reconstruction scheme proposed by Huijberts et al. for an information signal injected in the 3rd equation of Eq. (11), two outputs have been considered, i.e., x2 and x3 [17]. The properties of the observer scheme described by Eq. (12) have been verified making reference to both sinusoidal and random trapezoidal signals. In the first case considered, the input signal u1(t) was a sinusoidal wave of amplitude 0.12 and period 1 (normalized quantities). The forced ChuaÕs circuit was characterized by a double-scroll behaviour, as shown in Fig. 4a. Thus, it is possible to conclude that the input signal does not modify the qualitative chaotic features of the system. The state component x3 was transmitted over an ideal communication channel. At the receiving end, a 3-SMD was used to reconstruct the 1st, 2nd and 3rd derivatives of the transmitted signal. Such quantities were utilized to drive the algebraic observer described by Eq. (12). The reconstructed state components of ChuaÕs circuit are shown in Fig. 4b and c, together with the approximation errors. It is worth noting that a high precision was obtained within a finite time. Moreover, Fig. 4d presents the demodulated input signal. As clearly shown, the approximation error was within 104; this value points out the effectiveness of the proposed approach. In the case of a random trapezoidal input, the signal u1(t) was a wave of amplitude 0.1, period equal to 2, and rising (descending) time equal to 0.2 at instant 0.3 (0.7). Also in this case, the forced ChuaÕs circuit was characterized by a double-scroll behaviour, as shown in Fig. 5a. The reconstructed state components of ChuaÕs circuit are shown in Fig. 5b and c, together with the approximation errors, showing that, also in this case, a high precision was obtained within a finite time. Finally, Fig. 5d depicts the demodulated input signal and the approximation error. As is shown, the approximation error was within 104. The effectiveness of this approximation is further confirmed by the fact that a trapezoidal signal is not derivable. However, in accordance with Proposition 2, the algebraic observer defined by Eq. (12) guarantees that, also in this case, the input signal can be globally recovered in finite time. 4.2. The Ro¨ssler system In 1976, Otto Ro¨ssler developed the simplest continuous dynamical system he could judge capable of generating chaotic solutions [44]. The resulting state equations are 8 dx1 > > ¼ x2  x3 > > > dt > < dx2 ð13Þ ¼ x1 þ ax2 > dt > > > > > : dx3 ¼ b þ x3 ðx1  cÞ dt where a, b and c are parameters [44].

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x1 of the transmitter x1 at the receiver

3 0.6

2

x1

0.3

x

3

0.0

1

-0.3

0

-0.6

-1 0

0.6 0.3 0.0

0.00 -0.3

0.05

x2

0.10

x1

-0.6

20

40

60

80

100

Time

2

Error (10-8)

-0.15 -0.10 -0.05

0 -2

50

100

Time (b)

(a) 0.3 0.25

x2 of the transmitter x2 at the receiver

0.20 0.15 0.10

0.1

0.05

u1

x2

u1 of the transmitter u1 at the receiver

0.2

0.00

0.0

-0.05 -0.10

-0.1

-0.15 20

40

60

80

0

100

20

40

60

80

100

Time

Time

Error (10-4)

Error (10-12)

0

-2

2 0 -2 50

100

Time (c)

0 2 25

50

75

100

Time (d)

Fig. 4. ChuaÕs circuit (y = x3 and sinusoidal input u = u3): (a) the double scroll attractor of the forced ChuaÕs circuit; (b) x2 and x2 reconstructed; (c) x1 and x1 reconstructed; (d) u1 and u1 reconstructed.

The configurations that allow the input reconstruction for the Ro¨ssler transmitter are characterized by • u = u2 and y = x3; • u = u3 and y = x2. Let us consider the x2-drive configuration. In this case, the dynamics of the Rossler system is determined by the state equations 8 dx1 > > > dt ¼ x2  x3 > > > < dx2 ¼ x1 þ ax2 > dt > > > > > : dx3 ¼ b þ x3 ðx1  cÞ þ u3 dt y ¼ x2

ð14Þ

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371

4

x1 of the transmitter x1 at the receiver

3

0.8 0.6

2

x1

0.4 0.2

1

0.0

x3

-0.2

0

-0.4 -0.6

-1

0.6 0.4 0.2 0.0 -0.2 x1 -0.4

0.00

x2

0.05 0.10

-0.6

0

Error (10-8)

-0.8 -0.15 -0.10 -0.05

20

40

60

80

100

Time

2 0 -2

50

100

Time (b)

(a) 0.2

0.25 0.20

x2 of the transmitter

u1 of the transmitter

x2 at the receiver

u1 at the receiver

0.15

u1

x2

0.10 0.05

0.1

0.00 -0.05 -0.10 -0.15 0

20

40

60

80

100

0.0 0

Error (10-4 )

Error (10-12)

Time 2 0 -2 100

50

Time (c)

2 1 0 -1 -2 20

20

40

60

80

100

Time

40

60

80

100

Time (d)

Fig. 5. ChuaÕs circuit (y = x3 and random trapezoidal wave input u = u3): (a) the double scroll attractor of the forced ChuaÕs circuit; (b) x2 and x2 reconstructed; (c) x1 and x1 reconstructed; (d) u1 and u1 reconstructed.

The parameters used in the numerical simulation were a = 0.2, b = 0.2, and c = 6, which guarantee a chaotic behaviour for the Ro¨ssler transmitter [44]. The transmitterÕs initial state was 2

3 2 3 x1 3:9 6 7 6 7 4 x2 5 ¼ 4 3:2 5 x3 0:03

ð15Þ

and the receiverÕs initial state was 3 2 3 0 x1 6 7 6 7 4 x2 5 ¼ 4 0 5 0 x3 2

ð16Þ

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From rather a direct inspection of the dynamic equations, it follows: 3 2 a x1 6x 7 6 1 6 27 6 6 7¼6 4 x3 5 4 1 2

c

u3

1 0 a

0 0 1

0 0 0

ac  1

ac

1

32 y 3 2 3 0 7 6 7 76 _ y 6 7 76 7 6 0 7 76 7  6 7ððy  a_y þ €y Þ ðay  y_ Þ þ bÞ 54 €y 5 4 0 5 v

ð17Þ

1

y

thus ensuring that the information signal u3 can be entirely recovered with a demodulation error that can be evaluated a priori by means of Eqs. (8) and (17). It is worth noting that, in this case, the subsystem (x1x2) is not stable driven by x3, thus the master slave approach [45] and the scheme inspired by the method proposed in [16] cannot be used. The properties of the observer scheme described by Eq. (17) have been verified making reference to both sinusoidal and random trapezoidal signals. In the first case considered, the input signal u1(t) was a sinusoidal wave of amplitude

x1 of the transmitter

40

25

x1 at the receiver 20

20

x3

-20

10

-40

5

5

-10

-5 -5

5

-10

10

0

15

0

0

x2

10

Error (10-10)

0 -15

0

x1

15

x1

25

50

75

100

125

150

175

Time

1.0 0.5 0.0 -0.5 -1.0 50

100

150

Time

(a)

(b)

30

0.30

25

0.15 0.10

x3

u1

15 10

0.05 0.00

-0.05

00

-0.10

0 25

50

75

100

125

150

175

-0.15

0

40

Time -5 0 5 25

50

75

100

60

Time Error (10-3)

Error (10-7)

u1 at the receiver

0.20

20

5

u1 of the transmitter

0.25

x3 at the receiver

u1

x3 at the transmitter

125

150

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Fig. 6. Rossler system (y = x2 and sinusoidal input u = u3): (a) attractor of the forced Rossler system; (b) x1 and x1 reconstructed; (c) x3 and x3 reconstructed; (d) u3 and u3 reconstructed.

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0.12 and period 1. The qualitative features of the Ro¨ssler system are not significantly altered by the input signal, as shown in Fig. 6. The state component x2 was transmitted over an ideal communication channel. At the receiving end, a 3-SMD was employed to reconstruct the 1st, 2nd and 3rd derivatives of the transmitted signal and used as input to the algebraic observer described by Eq. (17). The reconstructed state components of the Ro¨ssler system are shown in Fig. 6b and c, together with the approximation errors. As expected, also in this case a high precision was obtained within a finite time. Moreover, Fig. 6d depicts the demodulated input signal, showing that the approximation error was constrained within 103. In the case of a random trapezoidal input, the input signal u1(t) was a wave of amplitude 0.1, period equal to 2, and rising (descending) time equal to 0.2 at instant 0.3 (0.7). The reconstructed state components of the Ro¨ssler system are given in Fig. 7b and c, together with the approximation errors. As clearly shown, a high precision was obtained within a finite time. Finally, Fig. 7d presents the demodulated signal, together with the approximation error. It is worth noting that the approximation error was within 104.

x 1 of the transmitter x 1 at the receiver

40

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(d)

Fig. 7. Rossler system (y = x2 and random trapezoidal wave input u = u3): (a) attractor of the forced Rossler system; (b) x1 and x1 reconstructed; (c) x3 and x3 reconstructed; (d) u3 and u3 reconstructed.

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4.3. The Lorenz system The coupled system of differential equations proposed by Lorenz is [46] 8 dx1 > > ¼ rðx2  x1 Þ > > > dt > < dx2 ¼ qx1  x2  x1 x3 > dt > > > dx > > : 3 ¼ bx3 þ x1 x2 dt

ð18Þ

where the dimensionless parameters r, r and b are assumed to be positive. In the case of the Lorenz system, the input can be recovered if it is injected into the 3rd of Eqs. (18) and the x1 component is transmitted. 4.4. The Ro¨ssler hyperchaotic system When multiple chaotic interactions occur in a multidimensional phase space, a system can become hyperchaotic. Many real world systems may show this behaviour, but relatively few of them have been studied in detail. Hyperchaos requires a minimum four-dimensional system with two or more positive Lyapunov exponents. The first hyperchaotic system that has been studied is the Ro¨ssler hyperchaotic system [47]. Its dynamics is given by the state equations 8 dx1 > > ¼ x2  x3 > > dt > > > > dx2 > > ¼ x1 þ ax2 þ x4 < dt ð19Þ > dx > > 3 ¼ c þ x3 x1 > > dt > > > > dx4 > : ¼ dx3 þ ex4 dt where a, b, c and d are parameters. In this case, the input can be recovered if it is injected: Case A: into the 2nd of Eqs. (19) and the state component x4 is transmitted; Case B: into the 4th of Eqs. (19) and the state component x3 is transmitted.

4.5. Coupled Chua’s circuits Let us consider two bi-directionally coupled, identical ChuaÕs circuits [48,49]. The mutual coupling is realized by connecting a voltage-defined non-linear resistor between the two capacitors c1 in the two ChuaÕs circuits. Moreover, in each ChuaÕs circuit is introduced a non-linear voltage controlled current generator (VCCS) driven by the voltage vc1 of the other ChuaÕs circuit [48,49]. The dynamics of the system is described by the following set of six dimensionless differential equations: 8 dx1 a > > ¼ acx1 þ ay 1  ax31  ax32  K ðx1  x2 Þ3 > > > 2 ds > > > > du1 > > > > ds ¼ x1  y 1 þ z1 > > > > dz1 > > ¼ by 1 < ds ð20Þ > dx a > > 2 ¼ acx2 þ ay 2  ax32  ax31 þ K ðx1  x2 Þ3 > > ds 2 > > > > du > 2 > ¼ x 2  y 2 þ z2 > > > ds > > > > dz > : 2 ¼ by 2 ds

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In this case, the input can be recovered if it is: Case A: injected into the 6th of Eqs. (20) and the state component z1 is transmitted; Case B: injected into the 3rd of Eqs. (20) and the state component z2 is transmitted.

4.6. Sprott’s systems SprottÕs systems [50] are 19 examples of chaotic flows with quadratic non-linearities of the form 3 3 X X dx ¼aþ bi xi þ cij xi xj dt i¼1 j¼1

ð21Þ

where x = (x, y, z) is a real three-dimensional variable and a, b, and c are real three-dimensional coefficient vectors. The 19 distinct cases are characterized by a simple algebraic representation, i.e., either five terms and two non-linearities or six terms and one non-linearity. It is worth noting that SprottÕs systems are somehow simpler than the Lorenz and Ro¨ssler systems, which are characterized by seven terms and either two quadratic non-linearities or one quadratic non-linearity, respectively [50]. The analysis of the I/O relations for the 19 systems allows one to conclude that there exists at least one configuration for the input reconstruction for each system.

5. Conclusions In this paper, an algebraic observability approach to chaotic modulation has been presented. The proposed approach can be applied to a wide class of chaotic systems, i.e., algebraic chaotic and hyperchaotic systems. A systematic procedure for the chaotic demodulation of the class of algebraic chaotic systems has been described and discussed. The proposed procedure assumes a measurable chaotic signal and allows one to directly identify a suitable ‘‘response’’ system for the ‘‘drive signal’’. In short, the proposed receiver reconstructs the chaotic attractor and the input, starting from the knowledge of the transmitter dynamics. The receiver realization, based on the algebraic concept concerning the Characteristic Set equations, requires several differential operators. An efficient and realistic differential operator is frequently required in practical applications in order to reduce the errors in evaluating the derivatives. Indeed, in practical implementations, the use of differentiation can generate high frequency noise. In this paper, the problem of an efficient calculus of the time derivatives has been faced by a sliding approach. Sliding mode differentiators are the only class of differentiators that gives, in principle, exact and robust estimates of the derivatives of a sufficiently smooth signal in a finite time. Results show that, in the case of algebraic observability, the sliding differentiator allows the input recovery in a finite time, within prefixed boundary on the estimate error. Examples have pointed out the efficiency of the proposed approach (even for higher-order derivatives) and its interesting properties for hardware implementations.

Acknowledgments Authors would like to thank Dr. A. Levant for useful discussions and suggestions. This work is partially supported by the University of Genova, the University of Cagliari and the Italian Ministry of Education, University and Research (MIUR) under the grant FIRB 2001.

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