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IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 9, SEPTEMBER 2013
One-to-Many Chaotic Synchronization with Application in Wireless Sensor Network Leisheng Jin, Yan Zhang, and Lijie Li, Senior Member, IEEE
Abstract—Chaotic synchronization between one and multiple systems has been innovatively postulated in this work for the purpose of achieving secure communication in the wireless sensor networks. The chaotic modulating signals for both the base station that acts as the commander and all sensor nodes are modelled based on the time-delay Lorenz system. Synchronization between the base and sensor nodes has been achieved by using the open-plus-closed-loop method. Identification of each sensor is viable using a selection matrix that is combined with the coupling equation. Realization of such system is also discussed. Index Terms—One-to-many chaotic synchronization, wireless sensor network.
I. I NTRODUCTION
W
IRELESS sensor network (WSN) has attracted much attention from both academic and industrial sectors as it has been ubiquitously used in surveillance [1], industrial process control [2], structural health monitoring [3], and even in healthcare [4]. A WSN is usually constructed by many sensors that detect physical and environmental conditions based on various transduction technologies, and a base station that receives and processes data from all of those sensors. Though the WSN technology is a well-established area, there are many challenges such as energy sources for sensors, production of miniature sensors, and communication between sensors and the base station. Particularly in terms of data transferring in a WSN, quite often a more secure communication is required. Previously modulating data by a chaotic signal was regarded as a more secure approach [5]. A lot of efforts have been devoted in this development in the past. For example, two chaotic coding and de-coding methods based on artificial neural networks were reported in [6]. Chaotic synchronization was used in discrete-time systems connected by bandlimited channels [7]. The bit error rate performance for differential chaos shift keying chaotic communication was evaluated by using the non-central F distribution of the decision variable in the reference [8]. In order to establish a communication between two chaotic signals from both the base and one sensor node, chaotic synchronization is essential and has to be carefully researched. Several chaotic synchronization methods have been investigated including the open-plus-closed-loop (OPCL) method that was first proposed by Jackson and Grosu [9]. An open loop control is also called non-feedback control,
Manuscript received May 20, 2013. The associate editor coordinating the review of this letter and approving it for publication was M. Leeson. L. Jin and L. Li are with the College of Engineering, Swansea University, Swansea, SA2 8PP, UK (e-mail:
[email protected]). L. Li is also with Wuhan University of Technology, P. R. China. L. Li is the corresponding author. Y. Zhang is with the Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences. Digital Object Identifier 10.1109/LCOMM.2013.081313.131169
which is a type of control that does not use feedback to determine the output; its advantage is low cost and simplicity. In contrast, closed loop control, also called feedback control, means that the output is determined by both the input and feedback, which can have a more accurate, more adaptive control. OPCL combines the advantages of open loop and closed loop controls. In recent years, since the OPCL method has been invented, it has been widely used to control complex nonlinear systems [10], [11]. This method has been predominately used between a single system and another single system. However, one-to-many chaotic synchronization has not been mentioned previously. In this paper, a new concept of synchronizing one chaotic system with many chaotic systems has been proposed. Mathematical model of this one-to-many chaotic synchronization has been constructed based on the OPCL method. This technique provides a platform where much secured communication in WSN can be realized. II. M ODEL C ONSTRUCTION Considering a WSN schematically sketched in Figure 1, the chaotic signal is added to both the sensors and the base station. Due to the nature of the chaos, chaotic signals for every sensor and the base are out of phase; even they are generated by the same function. That is because a very little change in the initial setting or some other factors will lead to complete different chaotic signals. Hence synchronization is an only method that can match two chaotic signals together to achieve precise data transferring. In this work, chaotic signal generation and modulation parts are ignored, and it is mainly focused on the synchronization between the base and sensor nodes. It is worth noting that the synchronization between the base and sensors cannot be simultaneously established, obtaining data from all sensor nodes can be done using a scanning procedure, which is schematically shown in Figure 2. In the synchronization model, the time-delay Lorenz system is used for chaotic signals to be added to both the base and sensors. The Lorenz system is a system of ordinary differential equations, which has chaotic solutions for certain parameter values and initial conditions. It has been widely seen in simplified models for lasers [12], electric circuits [13], and chemical reactions [14]. Lorenz system has clear advantages such as symmetry and invariance, thus it has attracted massive interests from the researchers [15]. In particular, the coupled time-delay systems have potential applications in secure communication, cryptography and controlling because of its intrinsic nature of generating high-dimensional chaotic signal and the experimental realization of these systems [16], [17], [18], [19], [20]. A typical three-dimensional time-delay Lorenz system x˙ = F (x), x ∈ R3 used for describing the chaotic
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JIN et al.: ONE-TO-MANY CHAOTIC SYNCHRONIZATION WITH APPLICATION IN WIRELESS SENSOR NETWORK
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signals being coded to the base station can be mathematically expressed as x˙1 = −σx1 + σx2 x˙2 = rx1 − x2 − x1 (t − τ ) · x3 (t − τ ) x˙3 = −bx3 + x1 (t − τ ) · x2 (t − τ )
(1)
where x1 , x2 , and x3 , are the system states; t is time; τ is the delayed time; σ, r, and b are system parameters. For the sensor nodes, the same time-delay Lorenz systems are used for describing the chaotic signals being coded to the sensor signals. In the real cases, usually there is parameter mismatch between chaotic signals in sensors, therefore each sensor system is written as y˙ = F (y) + ΔF (y), y ∈ R3 . The three-dimensional form is y˙i1 = (σ + Δσi ) · (yi2 − yi1 ) y˙i2 = (r + Δri ) · yi1 − yi2 − yi1 (t − τ ) · yi3 (t − τ )
(2)
y˙i3 = −(b + Δbi ) · yi3 + yi1 (t − τ ) · yi2 (t − τ )
Fig. 1. Schematic sketch of a wireless sensor network, blue dots represent sensor nodes, and bidirectional arrows stand for the communication between sensor nodes and the base station.
where yi1 , yi2 , and yi3 , are the system states; t is time; τ is time delay; σ, r, and b are system parameters; Δσ, Δr, and Δb are designated as parameter mismatch. Assuming there are N sensors in a WSN, i = 1, 2, 3, . . . N . i represents each sensor node. In order to realize synchronization between the base station and each sensor node, according to the OPCL method, equation (1) should be driven by a coupling scheme D(x, αyi ) and the driven system is given by: x˙ = F (x) + D(x, αyi ), where D(x, αyi ) is defined as: D(x, αyi ) =C · (αyi − F (αyi )
⎡ x − αy 1 i1 ⎢ x1τ − αyi1τ ⎢ x − αy ⎢ 2 i2 + A[(H − JF (αyi )] ⎢ ⎢ x2τ − αyi2τ ⎣ x − αy 3 i3 x3τ − αyi3τ
⎤ ⎥ ⎥ ⎥ ⎥) ⎥ ⎦
(3)
Equations (1) and (2) represent two chaotic systems that are added on the base station and the sensor. OPCL method has been applied on the base station to achieve complete synchronization between the base and the sensor. Equation (3) is the OPCL synchronizer that will be added to equation (1). In equation (3), C is a matching matrix which can be used to identify and switch the synchronization between each node and base station. H is an arbitrary constant Hurwitz matrix (N × N ) whose eigenvalues all have negative real parts, and J = ∂/∂(αy) is the Jacobian matrix. In the chaotic system the variables that contain time delay can be treated as independent variables, which are described in the 6 × 1 matrix in the equation (3). yi1τ , yi2τ , and yi3τ represent the yi1 (t−τ ), yi2 (t−τ ), and yi3 (t−τ ) respectively. The matrix 1 1 0 0 0 0 A = 0 0 1 1 0 0 . The matching matrix C is an 0 0 0 0 1 1 n0 × m matrix, m = n0 × N . n0 is the dimension of the system, so in this case, n0 = 3 for this three dimensional Lorenz system. N is the number of sensor nodes. It is worth noting that the matrix C keeps changing for every sensor node. It functions like a single pole, multi throw switch
Fig. 2. Synchronization between the base and individual sensor node can be achieved by using an identification matrix C, which essentially scans the synchronizer from the first sensor node to number N node.
linking the base station with each sensor. The function of matrix C is to identify and chose the sensor node that is expected to synchronize. It is an n0 × m matrix multiplied to D function. For example if we want to synchronize the sensor node i with the base, the value of Cn,n0 ·i−n0 +n = 1 and all other elements in the matrix should be 0, where n = 1, 2, ...n0 . III. C ASE S TUDY A synchronization example is given between one sensor node and the base station, if for example the second sensor node is targeted, i = 2, then the matrix C is 0 0 0 1 0 0 ... C = 0 0 0 0 1 0 ... (4) 0 0 0 0 0 1 ... The successful synchronization between this second sensor node and the base station can be established by implementing the OPCL synchronizer to the base system described in equation (1), which is the combination of the equations (3) and (4). It should be noted that the matrix H in equation (3)
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IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 9, SEPTEMBER 2013
In addition, stability analysis for this case has been conducted by defining the error e = x1 − y21 .and then writing the F (x) using Taylor series expansion ⎡ x − αy ⎤ 1
i1
⎢ x1τ − αyi1τ ⎢ x − αy ⎢ 2 i2 F (x) = F (αyi ) + JF (αyi ) ⎢ ⎢ x2τ − αyi2τ ⎣ x − αy 3 i3 x3τ − αyi3τ
Fig. 3. Calculated results show that the complete synchronization (CS) between the base system and the sensor node 2 has been reached. Fig. 3(a). The time series of x1 and y21 are plotted. Figs. 3(b) and (c) show the calculated results of x1 vs. y21 , and the e vs. t respectively.
Fig. 4. Calculation results of the projection of attractor in x1 − x2 plane is plotted in Fig. 4(a). Fig. 4(b) shows the projection of attractor in the second sensor node.
has eigenvalues all with negative real parts. According to the RH criterion, H matrix can be chosen as ⎡ ⎤ −2.7 0 0 0 0 0 0 0 0 ⎥ ⎢ 0 −3.1 0 ⎢ ⎥ 0 −10 0 0 0 ⎥ ⎢ 0 (5) H =⎢ ⎥ 0 0 0 0 −1 ⎥ ⎢ 0 ⎣ ⎦ 0 0 0 0 −7.5 0 0 0 0 2.8 0 −0.01 Following the above procedure, the chaotic signal of the base station has been modified by the OPCL synchronizer in the following form x˙1 = − σx1 + σx2 + αΔσ(y22 − y21 )(−2.7 + σ)(x1 − αy21 ) − σ(x2 − αy22 ) − 3.1(x1τ − αy21τ ) x˙2 =rx1 − x2 − x1 x3 + αΔry1 + αy23τ (x1τ − αy21τ ) + (αy21τ − 1)(x3τ − αy23τ ) − r(x1 − αy21 ) − 9(x2 − αy22 ) x˙3 = − bx3 + x1 x2 + αΔby23 − αy22τ (x1τ − αy21τ ) + (2.8 − αy22τ ) + (b − 7.5)(x3 − αy23 ) − 0.01(x3τ − αy23τ )
⎥ ⎥ ⎥ ⎥+··· ⎥ ⎦
(7)
The dynamic error e˙ = He can be derived by keeping the first order of equation (7) and substituting it into equation (3). If H becomes a Hurwitz matrix whose eigenvalues all have negative real parts, Stable synchronization can be obtained [9]. In Figure 3 a, it is seen that in the time domain chaotic signals of the second sensor and the modified base station are completely overlapped after a very short time period. Figure 3 b shows that these two signals are exactly in phase, and Figure 3 c displays a zero dynamic error. In this case study, the parameters τ = 0.01, σ = 10, r = 28, b = 8/3 for the purpose of guaranteeing all the signals remain chaotic. Here for simplicity, mismatched parameters Δσ and Δb are set to zero; only one mismatched parameter Δr is set to 1 for the sake of better demonstration. The attractors for these two systems are also calculated and show in the Figure 4. It is seen that attractors of the chaotic signals in both the sensor and the base station have the exactly same shape, indicating a complete synchronization. Synchronization between the base station and other sensor node can also be established by scanning the C matrix. The OPCL coupling scheme is robust even when the system is in the presence of noise [11], [21]. Here, the synchronization between the base station and sensor nodes under Gaussian white noise [22] has been further studied. The white noise that is present in common communication systems ξ(t) was generated using a Matlab function ξ(t) = wgn(t, 1, p), where p specifies the power of ξ in decibels relative to a watt. We used 0 for p, meaning that the power of the generated real white Gaussian noise in decibel is 0dBW . In this case, a noise signal with signal-to-noise ratio (SNR) of 13dB has been added to signals entering both the base station and sensor node. In this simulation, the noise ξ(t) has been added on the left side of the equations (2) and (6). The ξ(t) is characterized by < ξ(t) >= 0, which is shown in the Figure 5 a. The y-axis displays values of the noise vector generated. Also the time series of x1 and y21 under the white noise has been calculated and plot out in Figure 5 b. OPCL algorithm has been applied to the systems with noise, and it is shown from the simulation results (Figure 5 c) that the one-to-many system described in this work can achieve complete chaotic synchronization. IV. C ONCLUSION
(6)
In the case of α = 1, equations (2) and (6) representing the chaotic systems of the modified base station have been solved, which results in a complete synchronization (CS). Hence the secured communication between the base and the number 2 sensor has been established.
In summary, a new idea of achieving secured communication between the base station and sensor nodes in a WSN has been proposed based on a well known OPCL chaotic synchronization method. Time delay Lorenz system has been used to form the chaotic signals that are to be added to sensors and the base station. In this method, the chaotic signal in sensor nodes will be unchanged, but the chaotic signal of the
JIN et al.: ONE-TO-MANY CHAOTIC SYNCHRONIZATION WITH APPLICATION IN WIRELESS SENSOR NETWORK
Fig. 5. Calculated results show that the complete synchronization (CS) between the base system and the sensor node 2 has been reached under the Gaussian white noise. Fig. 5a shows Gaussian white noise signal; y-axis of Fig. 5a shows the amplitude of the column vector containing real white Gaussian noise of power 0 dBW. The calculated results for time series of x1 and y21 is shown in Fig. 5b, x1 vs. y21 is shown in Fig. 5c.
base will be modified using the OPCL synchronizer. In the mathematic expression of the synchronizer, the only changing item is the matrix C that is used for identifying each sensor node. A case study exhibiting the synchronization between one of the sensors and the base has been presented, and it is shown from the results that complete synchronization can be reached. ACKNOWLEDGMENT The authors would like to thank the UK Leverhulme Trust, and College of Engineering, Swansea University for support. R EFERENCES [1] M. Ali, A. Bohm, and M. Jonsson, “Wireless sensor networks for surveillance applications—a comparative survey of MAC protocols,” in Proc. 2008 International Conference on Wireless and Mobile Communications. [2] G. Zhao, “Wireless sensor networks for industrial process monitoring and control: a survey,” Network Protocols and Algorithms, vol. 3, no. 1, pp. 46–63, 2001. [3] X. Hu, B. Wang, and H. Ji, “A wireless sensor network-based structural health monitoring system for highway bridges,” Computer-Aided Civil and Infrastructure Engineering, vol. 28, no. 3, pp. 193–209, 2013.
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