Design of adaptive feedback control for new 3D chaotic system and its digital implementation on FPGA

Background/Objectives : In this research work, digital circuit implementation on FPGA of an adaptive feedback control methodology for a new 3 – D chaotic system is proposed Methods/Statistical analysis: The chaos synchronization is achieved using adaptive feedback control method. The new adaptive controllers are designed to achieve the chaos synchronization for the identical new chaotic system. The FPGA implementation of chaos synchronization using numerical methods induces artificial suppression in the chaotic system or chaotic behavior can be dead in very short-time. In this research work, the FPGA implementation of chaos synchronization is achieved with the help of automatic code generator like System generator in Matlab simulink. The adaptive feedback control for identical new chaotic system is coded with VHDL with 32 bit fixed point number, 12 for the entire and 20 for the fraction. Findings: In this paper, we designed a new 3D chaotic system and its chaotic behavior is verified using Lyapunov exponents, stability analysis and Poincare map. The complete synchronization for proposed chaotic system is achieved using adaptive feedback control methodology. The digital circuit realization of adaptive feedback control for the synchronization of identical chaotic system based on FPGA is achieved for the various applications of digital information systems. Simulation results and FPGA outputs illustrate the effectiveness of our proposed method. Novelty/Applications: The digital implementation of adaptive feedback control has many engineering applications such as digital data transmission, digital modulation, video encryption, digital cryptosystem etc.


Introduction
Chaotic systems are nonlinear dynamical systems which are highly sensitive to initial conditions. This sensitivity is popularly known as the butterfly effect. Chaos is an interesting nonlinear phenomenon and has been studied well in the last three decades. Chaos theory has wide applications in several fields like oscillators (1,2) , image encryption (3,4) , chemical reactors (5,6) , secure communications (7,8) , biological systems (9,10) , etc. In this paper a new 3D chaotic system is introduced and its basic properties such as Lyapunnov exponents, stability analysis, Poincare map are studied. The new chaotic system introduced in this paper consists of four nonlinear terms and five parameters unstable at all equilibrium points.
Still the chaos synchronization is a nontrivial task interesting impact on chaos based engineering applications such as secure communications etc. In last two decades many researchers presented the chaos synchronization using master -slave methodology (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29) . In this methodology, particular chaotic system is considered as master system and another or same chaotic system can be considered as the slave system. The chaos synchronization can be achieved if the output of the slave system tracks the output of the master system asymptotically. In the last two decades, various schemes have been successfully applied for chaos synchronization such as OGY method, active feedback control method, adaptive feedback control method, time-delay method, back stepping design method, projective synchronization, sampled-data synchronization method etc. The literature review on chaos synchronization pinpoints that compared to any other method adaptive feedback control method is simple, convenient and efficient methodology to implement the chaos synchronization. In this research work, the chaos synchronization for chaotic systems is considered with unknown parameters for master and slave systems based on Lyapunon stability theorem. The synchronization of chaotic systems with unknown parameters is achieved in this paper by designing adaptive controllers and parameters updated law.
The digital circuit implementation of proposed adaptive feedback control method for chaotic system is very important for the digital information system such as digital communication system (30,31) , true random number generator (32) , and pseudorandom number generator (33) , secure image transmission (34) , etc. Recently, many researchers focused on the Field Programmable Gate Array (FPGA) implementation of chaotic systems using many numerical methods such as Runge -Kutta, Euler algorithm, trigonometric polynomials and Adomian decomposition method etc. In (35) , novel 5D hyperchaotic system is implemented in FPGA using Runge -Kutta (RK-4) algorithm of Verilog in which chaotic system is designed with FPGA by dividing the whole system into three modules, such as RK-4 solving module, data selector module, and numerical conversion module. In (36) , S.T.Kingni et al. presented the FPGA implementation of an autonomous Josephson junction snap oscillator using a forward Euler algorithm of Verilog. In (37) , the chaotic oscillators are implemented in FPGA using numerical method based on trigonometric polynomials. In (38) , FPGA implementation of chaotic oscillator is presented by Karthikeyan et al. using Adomian decomposition method (ADM). The literature review on FPGA implementation of chaotic system using numerical methods indicating that if one does not choose the correct time-step, numerical methods may induce artificial chaos suppression or can engender the appearance of spurious solutions. In the worst case, chaotic behavior can be dead in the very short-time. Recently, many researchers presented the design of chaotic system with FPGA using automatic code generator in Matlab simulink such as DSP builder, system generator tool etc. Karthikeyan et al. (39) proposed the FPGA implementation of chaotic system using an automatic code generator tool Xilinx system generator toolbox in Matlab simulink. With the help of this code generator any one can generate Verilog / VHDL program for chaotic system which can be downloaded directly in to FPGA. The design of chaotic system with FPGA using this method can be very simple and does not require any numerical for HDL program.
In this research work, the proposed adaptive feedback control scheme for identical new chaotic system is implemented in FPGA using Matlab simulink and Xilinx system generator tools. First, the proposed feedback control scheme is designed using Xilinx block set tools in Matlab simulink and corresponding VHDL code is downloaded from the system generator design. Then VHDL code is synthesized using Xilinx software environment. As they synthesize result, RTL schematic diagram and source consumed by adaptive feedback control scheme is obtained. The experimental results indicate that our proposed methodology is very simple and suitable for the digital implementation of adaptive feedback control scheme for the synchronization of identical chaotic systems.

Design of 3D Chaotic System
The new 3 D chaotic system is described as given in the Equation. 1,  Here x 1 , x 2 , and x 3 are the state variables of new chaotic system (1) a, b, c, d and p are positive constant parameters. The system is chaotic when a = 2.86, b = 9, c =10, d =0.5 and p = 4. The Lyapunov exponents for the new system (1) are calculated as LE 1 = 0.755939, LE 2 = 0.060754 and LE 3 = -12.105507. The corresponding Lyapunov exponents dimension is given as, Equation 2 indicates that the attractor of new chaotic system (1) is a strange attractor with fractal dimension. The dynamics of Lyapunov exponents of the new chaotic system (1) is shown in Figure 1. The maximal Lyapunov Exponent (MLE) of proposed chaotic system (1) is, LE 1 = 0.755939 > 0 indicates that the proposed new system (1) has a chaotic behavior for the parameter values a = 2.86, b = 9, c =10, d =0.5 and p = 4. Since the sum of the Lyapunov Exponents is negative, the new system (1) is dissipative. The new system (1) is found to be invariant when coordinates are transformed as, S: ( . This indicates that the new system (1) has symmetry about the x 2 -axis. The phase portraits of new 3D chaotic system are shown in Figure 2 (a -d). .
The Eigen values obtained is shown in Table 1 and the system (1) is unstable at all equilibrium points S 1 ,S 2 , S 3 , and S 4 . The Poincare map for new chaotic system (1) for a = 2.86 is shown in Figure 3. The Poincaré map is a useful tool for analyzing the dynamical characteristics of chaotic system. In the chaotic case, the phase portrait of new chaotic system (1) is very dense in the sense that the trajectories of the motion are very close to each other. It can be only indicative of the minima and maxima of the motion. Any other characterization of the motion is difficult to be interpreted. So, one way to capture the qualitative features of the strange attractor is to obtain the Poincaré map.

Adaptive Feedback Control for New -D Chaotic System
In this section, based on the adaptive feedback control theory, the complete synchronization between two identical new chaotic systems is achieved using master -slave formalization. According to that Equation 1 is considered as a master chaotic system and the slave chaotic system is given by the Equation 4.  ẏ 1 = ay 1 − y 2 y 3 + b + u 1 y 2 = −cy 2 + y 1 y 3 + dy 2 3 + u 2 y 3 = −py 3 + y 1 y 2 + y 2 y 3 + u 3 Here y 1 , y 2 , y 3 are the state variables of slave system, a, b, c, d, and p are positive constant parameters and u 1 , u 2 , and u 3 are control functions. Our goal is to determine the control function from adaptive feedback control method. The complete synchronization error are defined as,  The synchronization of error dynamics is obtained as,  The Controller u is derived for the adaptive feedback control functions such that the error drives to zero. The adaptive feedback controller is given in Equation 7.
Here a, c, d and p are the estimate values of the unknown parameters a, b, c, d and p respectively. Then by substituting Equation 7 in Equation 6, the error dynamics is obtained as in Equation 8. , Let us choose the parameter update law as,  The Equation 12 indicates that by Lyapunov stability theory, it is determined that the synchronization error e i , i = 1, 2, 3 and the parameter estimation error e a , e c , e d , e p decay to zero exponentially with time. Thus the adaptive controlled slave chaotic system is given as. 

Digital Circuit Realization on FPGA of Adaptive Feedback Control for New 3 -D Chaotic System
In this section, the digital circuit realization of proposed adaptive feedback control methodology is achieved in FPGA Virtex4 xc4vfx100-12ff1152. The new 3 -D chaotic system corresponding to Equation 1, adaptive controllers corresponding to Equation 7 and parameter update laws corresponding to Equation13 are constructed using MATLAB Simulink and Xilinx system generator block sets. Figure 4 represents the FPGA implementation of new 3 D chaotic system using Xilinx block set. Figure 5 (a -c) represents the FPGA implementation of adaptive controllers u 1 , u 2 , and u 3 using Xilinx block set respectively. Figure 6 represents the FPGA implementation of parameter update law in Xilinx block set. Figure 7 represents the FPGA implementation of proposed adaptive feedback control for new 3D chaotic system using Xilinx System Generator blocks. The master subsystem contains the new master chaotic generator ( Figure 4), the slave subsystem contains the block diagram of slave system given in Equation 4. The parameter update law subsystem contains the Figure 6 and the control commands subsystem contains the Figure 5. This FPGA implementation is adopted with a fixed point and with a representation of the real data on 32 bits (12Q20), 12 for the entire and 20 for the fraction.  The initial conditions for master chaotic system are chosen as x 1 (0) = 5, x 2 (0) = 10 and x 3 (0) = 30 and the initial conditions are feed up in the integrator block in Figure 4. The initial conditions for the state variables of slave system can be chosen as y 1 (0) = 20, y 2 (0) = 12 and y 3 (0) = 30. We adopt this FPGA implementation of adaptive feedback control method with a fixed point and with a representation of the real data on 32 bits (12Q20), 12 for the entire and 20 for the fraction. Figure 8 represents the time evolutions of the master and controlled slave system variables. Figure 9 represents the time history of error signals which reaches zero when slave system is controlled by master chaotic system. Figure 10 represents the time history of adaptive controller signals which also reaches zero when slave system is controlled by master chaotic system. After that, VHDL code for completely controlled 3 D chaotic systems is downloaded from the MATLAB Simulink design shown in Figure 7. Then the VHDL code is simulated using Xilinx software and the source consumed by the controlled 3 D chaotic system is obtained as given in Table 2. Also, RTL schematic diagram for controlled 3 D chaotic system is generated as shown in Figure 11.

Conclusion
In this paper, we proposed a new chaotic system and its properties are analyzed in detail. The synchronization of new identical chaotic system is derived using master -slave adaptive feedback control methodology. The experimental results and numerical simulations proved that the proposed feedback methodology is efficient and convenient for the chaos synchronization. Further, the digital implementation of proposed adaptive feedback control scheme is achieved on FPGA using automatic VHDL code generator such as Xilinx system generator. The Matlab simulation results and Xilinx simulation results are indicating that the proposed methodology is very simple, effective and convenient for the digital implementation of adaptive feedback control method for chaotic systems.