Real-time fuzzy-pid synchronization of memristor-based chaotic circuit using graphical coded algorithm in secure communication applications

In 1971, L. Chua found the fourth missing element and called memristor (memory resistor) [1]. Definition of the memristor is the relationship between two of the four basic circuit variables, such as the current i, the voltage v, the charge q, and the flux φ and it provides a functional relation between magnetic flux and charge as

$$\begin{eqnarray}&&{d}\varphi ={Mdq}\end{eqnarray} \tag{ 1 }$$

The relation of the fundamental circuit component and the I–V curve are given in figure 1 [2]. When the relation of flux-charge is analyzed for memristor, the electrical characteristic of the element is called memristance. If the relation of charge-flux is analyzed, that is described as memductance [3]. Memristance has the same unit with the resistance whereas memductance has the inverse of the resistance. As can be seen from figure 1, the I−V curve shows a non-linear characteristic, a pinched hysteresis loop, and a straight line at low frequencies, at intermediate frequencies, and at high frequencies, respectively.

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The charge-controlled memristor is called memristance and its equations are given in (2).

$$\begin{eqnarray}&&\begin{array}{l}\varphi =f(q)\\ \,\displaystyle \frac{d\varphi }{dt}=\displaystyle \frac{df\left(q\right)}{dt}\displaystyle \frac{dq}{dt}\\ \,v\left(t\right)=M(q)i(t)\\ \,M\left(q\right)=\displaystyle \frac{df\left(q\right)}{dq}\end{array}\end{eqnarray} \tag{ 2 }$$

The charge-controlled memristor is called memductance and its equations are given in (3).

$$\begin{eqnarray}&&\begin{array}{l}q=f(\varphi )\\ \,\displaystyle \frac{dq}{dt}=\displaystyle \frac{df(\varphi )}{dt}\displaystyle \frac{d\varphi }{dt}\\ \,i\left(t\right)=W(\varphi )v(t)\\ \,W\left(\varphi \right)=\displaystyle \frac{df\left(\varphi \right)}{d\varphi }\end{array}\end{eqnarray} \tag{ 3 }$$

Memristor can be used in a wide range of work areas due to its scalability, low power consumption, and good compatibility with CMOS structures. However, except for a few products produced in a laboratory environment, it still has not taken its place in the markets as a commercial product [4, 5]. Thus, many studies about memristor-based systems such as chaotic systems and analog integrated circuits are implemented only by using simulation tools like SPICE and emulator circuits [3–12]. While doing these, linear and nonlinear dopant drift models are generally used to implement circuits.

Since memristors have a natural nonlinear behavior, they can be successfully applied to chaotic circuits. Hence, many scientists have studied this topic. It would not be wrong to say that one of the main application areas of memristive systems is chaotic systems and circuits [13–22].

Since information security comes to the fore due to the rapidly developing internet technology, scientists are trying to implement secure communication applications using chaotic systems. In these studies, the master-slave synchronization of the chaotic system and secure communication applications with various control algorithms have been implemented [16–22].

In this paper, the master-slave synchronization of the memristor-based chaotic circuit was performed for a secure communication application by using LabVIEW environments. Recently, LabVIEW software and hardware such as data acquisition cards (DAQs) and c-RIO systems have been widely used to implement systems like control, communication, biomedical systems [13–19]. In the designed system, the Fuzzy-PID controller synchronizes the master-slave chaotic system. At the same time, image encryption and decryption are realized to implement secure communication applications. Various test parameters such as information entropy value, correlation coefficient value, etc. are calculated to test the result of the secure communication application.

In this study, the following sections introduce as follows: the second section consists of two parts. The first part introduces memristor-based chaotic circuits while the second introduces the dynamics of the system. In section 3, the synchronization of the memristor-based chaotic circuit with the Fuzzy-PID control method is given. Real-time applications of secure communication systems and some security analyses are implemented in section 4. In the last section, conclusions are presented.

The memristor-based chaotic circuit is shown in figure 2.

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The state equations of this chaotic circuit are obtained using Kirchoff's laws and are presented below:

$$\begin{eqnarray}&&\begin{array}{l}{\dot{V}}_{C1}=-\displaystyle \frac{{i}_{L}\left(t\right)}{{C}_{1}}-\displaystyle \frac{{i}_{M}\left(t\right)}{{C}_{1}}\\ {\dot{V}}_{C1}=-\displaystyle \frac{{i}_{L}\left(t\right)}{{C}_{2}}\\ {\dot{i}}_{L}=-\left(\displaystyle \frac{{V}_{C1}\left(t\right)}{L}-\displaystyle \frac{{V}_{C2}\left(t\right)}{L}\right)\\ {\dot{V}}_{CM}=\displaystyle \frac{{V}_{C1}\left(t\right)}{{R}_{1}{C}_{{\rm{M}}}}\end{array}\end{eqnarray} \tag{ 4 }$$

When the equations of the memristor-based chaotic circuit developed using a flux controlled memristor model created using the nonlinear cubic property given in [17]:

$$\begin{eqnarray}&&\begin{array}{l}{\dot{V}}_{C1}=-\,\displaystyle \frac{{i}_{L}(t)}{{C}_{1}}-\displaystyle \frac{{V}_{C1}\left(t\right)\left({m}_{0}+{m}_{1}{V}_{CM}{\left(t\right)}^{2}\right)}{{C}_{1}}\\ {\dot{V}}_{C2}=\,\displaystyle \frac{{i}_{L}(t)}{{C}_{2}}\\ {\dot{i}}_{L}=-\left(\displaystyle \frac{{V}_{C1}\left(t\right)}{L}-\displaystyle \frac{{V}_{C2}\left(t\right)}{L}\right)\\ {\dot{V}}_{CM}=\displaystyle \frac{{V}_{C1}\left(t\right)}{{R}_{1}{C}_{{\rm{M}}}}\end{array}\end{eqnarray} \tag{ 5 }$$

For the set of equations given above, if V_C1 = x₁ , V_C2 = x₂ , i_L = x₃ , V_CM = u, a = 1/C₁ , b = 1/C₂ , d = 1/L and n = 1/R₁ C_M , new sets of nondimensional equations:

$$\begin{eqnarray}&&\begin{array}{l}\dot{x}1=-a.x3-a.x1.{m}_{0}-a.x1.{m}_{1}.{u}^{2}\\ \,\dot{x}2=-b.x3\\ \,\dot{x}3=-d.\left(x1-x2\right)\\ \,\dot{u}=n.x1\end{array}\end{eqnarray} \tag{ 6 }$$

In order for the circuit to show chaotic properties, the coefficients in equation (5) are a = 3.75, b = 10, d = 1, m₀ = −0.33, m₁ = 0.25 and n = −1. Also, the initial parameters of the chaotic circuit which are x₁ (0), x₂ (0) x₃ (0) and u(0) are chosen as 0.01, 0,0 and 0, respectively. State space diagrams on x₁ -x₂ , x₁ -x₃ , x₁ -x₄ , x₂ -x₃, and x₃ -x₄ of memristor based chaotic circuit are shown in figure 3.

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2.1. Lyapunov exponents of the memristor-based chaotic circuit

The Lyapunov exponent of a dynamic system is defined as a quantity that characterizes the rate of separation of infinitely close trajectories. Thus, it can be said that they mathematically indicate whether a system has time series. It is known that at least one of the Lyapunov exponents must be positive for a system to be said to exhibit chaotic behavior [15]. For this dynamical system, four Lyapunov exponents are obtained and their values are L1 = 1.7875, L2 = 2.1063, L3 = 0.00001, and L4 = −0.5057. Initial conditions of the system are selected for (0.01, 0, 0, 0). The Lyapunov exponent spectrum of the memristor-based chaotic circuit is given in figure 4.

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As can be seen, since two of the Lyapunov exponents in figure 4 have positive values, the necessary condition is fulfilled for being a chaotic system.

2.2. Bifurcation behavior analysis of the memristor-based chaotic circuit

Bifurcation diagrams are important in the analysis of dynamic systems and chaos. Bifurcation analysis is performed to determine the sensitivity of the system to the initial conditions and from where the system has moved from periodicity to chaos [23–25]. The bifurcation diagram obtained for the selected control parameters a and d in the state-equations is shown in figure 5. The range for a is between 0.9 and 1.35 and that for d is between 1.1–1.8. The diagrams are represented with two different colors which show the obtained results with different initial conditions like black (0.1, 0, 0, 0) and blue (0.1, 0, 0.01, 0). It is understood that the a and d parameters show chaotic properties in the determined ranges.

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To realize the synchronization of the memristor-based chaotic circuit with the Fuzzy-PID control method, it will be necessary to obtain the error dynamics of the outputs at different initial conditions. In equation (6) the model of the Master system, in (7) the models of the system consisting of the Slave system and the control function are given.

Master system:

$$\begin{eqnarray}&&\begin{array}{l}{\dot{x}}_{m1}=-a{x}_{m3}-a{x}_{m1}{m}_{0}-a{x}_{m1}{m}_{1}{{x}_{m4}}^{2}\\ \,{\dot{x}}_{m2}=-b{x}_{m3}\\ \,{\dot{x}}_{m3}=-d\left({x}_{m1}-{x}_{m2}\right)\\ \,{\dot{x}}_{m4}={\rm{n}}{x}_{m1}\end{array}\end{eqnarray} \tag{ 7 }$$

Slave system:

$$\begin{eqnarray}&&\begin{array}{l}{\dot{x}}_{s1}=-a{x}_{s3}-a{x}_{s1}{m}_{0}-a{x}_{s1}{m}_{1}{{x}_{s4}}^{2}+\mu (t)\\ \,{\dot{x}}_{s2}=-b{{\rm{x}}}_{{\rm{s}}3}\,\\ \,{\dot{x}}_{s3}=-d\left({x}_{s1}-{x}_{s2}\right)\\ \,{\dot{x}}_{s4}=n{x}_{s1}\end{array}\end{eqnarray} \tag{ 8 }$$

Error:

$$\begin{eqnarray}&&\begin{array}{l}{e}_{1}={x}_{m1}-{x}_{s1}\\ \,{e}_{2}={x}_{m2}-{x}_{s2}\\ \,{e}_{3}={x}_{m3}-{x}_{s3}\\ \,{e}_{4}={x}_{m4}-{x}_{s4}\end{array}\end{eqnarray} \tag{ 9 }$$

where x_m , x_s , c(t), and e are respectively master system, slave system, control function, and error. The coefficients of master and slave system are a = 3.75, b = 10, d = 1, m0 = −0.33, m1 = 0.25 and n = −1. The equation of the control function is given in (9). It is calculated by the Fuzzy-PID algorithm. The fuzzy system determines the Kp, Ki, and Kd parameters depending on the error and the change in the error. Thus, control function μ (t) is obtained to provide synchronization performance.

The Fuzzy-PID control:

$$\begin{eqnarray}&&\mu \left(t\right)={K}_{p}{e}_{1}+{K}_{i}\displaystyle \int {e}_{1}+{K}_{d}e{^\circ }_{1}\end{eqnarray} \tag{ 10 }$$

Figure 6 shows the communication system general block diagram. In this system, the control input of the PID system is defined as the μ function.

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The fuzzy controller has two inputs and three outputs. The inputs are error and change in error whereas the outputs are Kp, Ki, and Kd. The membership functions of inputs and outputs are shown in figures 7 and 8.

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One of the rule tables of the fuzzy system is given in table 1.

Linguistic variables in membership functions represent negative, zero, and positive for the inputs while they represent small, medium, and big for the outputs.

Simulation and real-time synchronization of the memristor-based chaotic system are implemented in the LabVIEW environment. The master-slave synchronization system performed with the Fuzzy-PID controller using the eight differential equations of the chaotic system is shown in figure 9.

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In the block diagram, first of all, the eight-state equations of the master and slave system are created. At the same time, the image was imported into the system and the encrypted image was obtained with the help of the image encryption algorithm by using the signals from the state equations.

During the specified time, the algorithm transmits the encrypted image. At the end of the specified time, the Fuzzy-PID controller is activated and the slave system follows the master system. The real image is obtained again with the developed decryption algorithm.

According to the fuzzy controller algorithm, the PID control parameters were obtained as [Kp Ki Kd] = [8.33 62.8 27.9]. After the designed system is run, master and slave systems' variables and errors between them are obtained as figure 10. The designed Fuzzy-PID synchronization was activated at 10 s. After that time, the error between the master and slave system quickly decreased to zero.

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The appearance of the real-time experimental setup and the real-time state-space diagrams are shown in figures 11 and 12.

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As can be seen from figure 10, it is understood that xm which is the master system's first variable and xs which is the slave system's first variable has different values up to the 10th second and the error between them is large. After the 10th second, it is seen that xm and xs have the same values when the controller is activated and the error goes to zero. This situation has also been achieved in real-time applications. In addition, 5 different state-space diagrams obtained from the simulation are also obtained in real-time applications. During this application, since the analog input voltage range of the DAQ is 0–10 V, scaling processing has been carried out in the algorithm so that it does not exceed this value.

As can be seen in figure 11, the developed algorithm has a user interface (GUI). In this GUI, both the chaotic signals and the encryption/decryption process can be seen instantly. In the encryption process of the developed algorithm, firstly, pseudo-random numbers were generated and then applied to the master and slave system's every chaotic variable. Image encryption was performed for 10 s. During this time, the chaotic signals of eight variables taken from the master and slave systems were sampled separately, and the image encryption process was carried out by performing an XOR operation between these scaled eight variables and the real image data. The decryption process starts after the Fuzzy-PID synchronization process takes place after 10 s. Also, this process is performed for XOR processing between the encrypted variables and slave systems' variables for 40 s [26, 27].

Three images used in the literature were used to test the performance of the image encryption process for the secure communication system. Histogram graphs and real-time results for three images in the encryption/decryption process are shown between figures 13–18 [28].

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When the histograms of three encrypted images are examined, it is seen that the image information is effectively hidden because of being distributed uniformly. The information entropy values of three encrypted images are calculated by equation (10) and the results are given in table 2 [29].

$$\begin{eqnarray}&&H\left(s\right)=\displaystyle \sum _{i=0}^{{2}^{k}-1}P({s}_{i})\left({\mathrm{log}}_{2}\displaystyle \frac{1}{Psi}\right)\end{eqnarray} \tag{ 11 }$$

As can be seen from table 2, information entropy values are very close to the ideal value of 8 for three encrypted images. In addition, correlation coefficient (CC), peak signal to noise ratio (PSNR), and structural similarity index (SSIM) are examined for every three images in table 3.

The values of similarity analysis for three images in table 3 show that there are no similarities between encrypted and decrypted images as [30–32].

In the final test of similarity analysis, the correlation coefficients obtained from randomly selected pixel pairs in the horizontal, vertical, and diagonal directions from the three encrypted images are given in table 4. It can be seen that the designed encryption algorithm reduces the correlation coefficient of the image. These values seem to be more appropriate when compared with the studies done in [33, 34].

Number of pixels change rate (NPCR) and unified average changing intensity (UACI) which show the resistant to differential attacks were employed to test the sensitivity of the developed encryption algorithm. The equations of NPCR and UACI are given in (11). For a good encryption process, NPCR value is greater than 99.6% and UACI value is greater than or equal to 30% [23, 29, 35, 36]. The calculated NPRC and UACI values for three encrypted images are given in table 5.

$$\begin{eqnarray}&&\begin{array}{l}NPCR=\,\displaystyle \frac{\displaystyle {\sum }_{i=1}^{M}\displaystyle {\sum }_{j=1}^{N}D(i,\,j)}{M\,x\,N}\times \,100 \% \\ \,UACI=\displaystyle \frac{1}{M\,x\,N}\,\left(\displaystyle {\sum }_{i=1}^{M}\displaystyle {\sum }_{j=1}^{N}\displaystyle \frac{\left|{C}_{1}\left(i,\,j\right)-{C}_{2}(i,\,j)\right|}{255}\right)\times \,100 \% \\ \,D\left(i,\,j\right)=\left\{\begin{array}{c}\,0,\,{C}_{1}\left(i,\,j\right)={C}_{2}\left(i,\,j\right),\\ \,1,\,{C}_{1}\left(i,\,j\right)\cong {C}_{2}\left(i,\,j\right),\end{array}\right.\end{array}\end{eqnarray} \tag{ 12 }$$

According to the tests results to see performance analysis for the designed encryption/decryption algorithm, it can be said that the memristor-based chaotic system has good encryption and decryption properties.

In the article study, real-time Fuzzy-PID synchronization of memristor-based chaotic circuits was implemented by using the LabVIEW environment. A secure communication simulation and image encryption process have been successfully performed with real-time Fuzzy-PID synchronization. Histogram analysis, correlation and entropy coefficient, UACI, NPCR were calculated to show the performance of the encryption algorithm and the reliability of the encrypted images. It has been seen that the results of these tests are among the values accepted for similarity and security analysis in the image encryption and decryption process in the literature. Thus, it can be said that the proposed system can be used secure communication process where privacy is important.

The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.

The authors declare that they have no conflict of interest.

This study was not funded by any project.

All data generated or analyzed during this study are included in this published article.