Complexity
Volume 2017 (2017), Article ID 1892618, 19 pages
https://doi.org/10.1155/2017/1892618

Research Article

Fractional Order Memristor No Equilibrium Chaotic System with Its Adaptive Sliding Mode Synchronization and Genetically Optimized Fractional Order PID Synchronization

Karthikeyan Rajagopal, Laarem Guessas, Anitha Karthikeyan, Ashokkumar Srinivasan, and Girma Adam

Centre for Non-Linear Dynamics, Defense University, Addis Ababa, Ethiopia

Correspondence should be addressed to Karthikeyan Rajagopal; rkarthiekeyan@gmail.com

Received 12 October 2016; Revised 21 November 2016; Accepted 22 December 2016; Published 26 March 2017

Academic Editor: Ahmed G. Radwan

Copyright © 2017 Karthikeyan Rajagopal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper introduces a fractional order memristor no equilibrium (FOMNE) chaotic system and investigates its adaptive sliding mode synchronization. Firstly the dynamic properties of the integer order memristor no equilibrium system are analyzed. The fractional order memristor no equilibrium system is then derived from the integer order model. Lyapunov exponents and bifurcation with fractional order are investigated. An adaptive sliding mode control algorithm is derived to globally synchronize the identical fractional order memristor systems and genetically optimized fractional order PID controllers are designed and used to synchronize the FOMNE systems. Finally the fractional order memristor no equilibrium system is realized using FPGA.

1. Introduction

Chaotic systems are a special case of nonlinear systems which can be categorized as chaotic if the system possesses at least one positive Lyapunov exponent and hyperchaotic if the system possesses two or more positive Lyapunov exponents. Lyapunov exponents and fractal dimension studies are important in defining the complexity of chaotic and hyperchaotic systems [1, 2]. Chaotic systems found significant importance after the discovery of a 3D weather model by Lorenz [3].

By early 21st century many researchers have announced different chaotic systems such as Chen system [4] Liu system [5], Sundarapandian system [6], Sundarapandian system [7], and Pham system [8]. Chaotic systems with no equilibrium are of great interest in chaos literature. If sum of all the Lyapunov exponents is zero then the system is a conservative system [9].

Memristors popularly known as the fourth fundamental circuit element were characterized as a nonlinear and low power device was proposed by Chua [10, 11]. In 2008 Hewlett-Packard [HP] engineers announced the first physical realization of memristors. In memristor literatures several models had been presented such as linear and nonlinear ion drift model and threshold adaptive memristor model [12–14]. A Murali-Lakshmanan-Chua’s circuit with a piecewise linear active flux controlled memristors with hyperchaotic behavior was investigated by Ishaq Ahamed and Lakshmanan [15]. A memristor based hyperchaotic complex Lu system and its adaptive synchronization were studied by Wang et al. [16].

Synchronization of chaotic systems is of great importance when one chaotic oscillator drives the other. Because of high sensitivity to initial conditions two identical chaotic systems may have exponentially diverging state trajectories. Many methods have been proposed in the literature such as active control method [17, 18], adaptive control method [19, 20], extended back stepping control [21, 22], sliding mode control [23, 24], and adaptive sliding mode [25–27].

Fractional calculus [28–31] has fully emerged into a mathematical field with applications in nonlinear controls, electrical and mechanical controls, and so forth. Fraction chaotic systems have been investigated by many researchers [28–35]. Fractional order controllers [24–27, 36–38] are more effective compared to its integer order models especially in chaos control and synchronization. Fractional order systems with no equilibrium are announced and investigated by Li and Chen [39]. For the numerical simulation of fractional order chaotic system Ivo [40] explained a methodology with a register memory component.

For chaos to exist there should be at least one unstable equilibrium point. Recently many researchers have announced chaotic systems with no equilibrium or one equilibrium [32–34]. Chaotic systems with no equilibrium exhibit hidden attractions as the orbit of attractions does not intersect with any equilibrium points [35, 41]. An algorithmic search methodology was developed by Jafari et al. [42] to discover chaotic flows with no equilibrium.

PID controllers are objects of steady effort for improvements of their quality and robustness. One of the possibilities to improve PID controllers is to use the fractional order controllers (FOPID) with noninteger derivation and integration parts; they generalize the integer order PID controller, used and verified their effectiveness in [38, 40, 43–45], and add more flexibility to control design with accuracy for the real world processes. But finding appropriate parameters values for the FOPID controller is still a difficult task, so in practice control engineers still often use trial and error for the tuning process.

Motivated by the above, in this paper we announce a novel integer order memristor no equilibrium chaotic system. We then derive a fractional order model of the proposed novel system. The dynamic properties of both integer order and fractional order novel systems are investigated. A fractional order adaptive sliding mode control and genetically optimized fractional order PID are proposed to synchronize the identical novel fraction order systems. Finally the proposed fractional order system is implemented in FPGA.

2. Memristor No Equilibrium Chaotic System

In this section we introduce a novel 4D memristor no equilibrium chaotic system (MNECS) with three parameters derived from the Sundarapandian system [46] by including a fourth state which is a combination of state feedback and the flux controlled memristor [47, 48] as described by

and the parameter values are , , and and is the memductance of a flux controlled memristor characterized by the cubic nonlinearity ,  , and [47].

Figure 1 shows the state portrait of system (1).

(a)

(b)

Figure 1: 3D state portrait of the novel memristor no equilibrium system ( plane; plane).

For numerical simulations, we take the initial values of the hyperchaotic system (1) as

3. Properties of the 4D Memristor No Equilibrium Chaotic System

3.1. Equilibrium Points

The equilibrium points of system (1) can be found by solving

From (3) it can be clearly seen that . As    and    the only solution is . It can be seen that and make ; we conclude that . Hence system (1) has no equilibrium points.

3.2. Lyapunov Exponents and Kaplan-Yorke Dimension

The Jacobian matrix of the novel system (1) is calculated as

The eigenvalues of the Jacobian matrix at equilibrium are , , , and . The Lyapunov exponents of the system are , , , and . The existence of positive Lyapunov exponents confirms the chaotic behavior of system (1). The sum of all the Lyapunov exponents is negative confirming that the system is dissipative.

The Kaplan-Yorke dimension of a chaotic system is defined as

where is the maximum integer such that .

3.3. Bifurcation, Bicoherence, and Poincare Map

In order to understand the dynamical behavior of the MNECS, the bifurcation plots are derived for three cases as follows.

Case 1. Fix , , , and and vary between .

Case 2. Fix , , , and and vary between .

Case 3. Fix , , , and and vary between .

As discussed in [49], the transient behaviors occurring in memristor based nonlinear systems may result in longer simulation times to reach steady states. Hence we used the ode45 solver for numerical simulations. Figures 2(a), 2(b), and 2(c) show the bifurcation plots of the system for the parameters , , and , respectively. The variation in Lyapunov exponents with reference to the parameters , , and is also derived and compared with the bifurcation plots. Figure 3 shows the variation of the Lyapunov exponents with the parameters. As it is observed from Figures 2 and 3, for the values of , the bifurcation plots show denser points confirming the existence of a stable positive Lyapunov exponent.

(a)

(b)

(c)

Figure 2: (a) Bifurcation plot versus . (b) Bifurcation plot versus . (c) Bifurcation plot versus .

Figure 3: Positive Lyapunov exponent for variation in parameters ().

The regular power spectra investigations cannot be used to analyze nonlinear systems as it does not have the required phase information. Hence researchers have adopted bicoherence or bispectrum analysis to analyze the period doubling sequence occurring in chaotic systems. Bicoherence contour of the novel system (1) as shown in Figure 4 shows the coupling between Fourier modes and is centered about the dominant frequencies composing the limit cycles [50]. As the system exhibits positive Lyapunov exponents, the chaotic behavior of the system increases resulting in period doubling and motions at additional frequencies because of the quadratic interactions between the dominant frequencies and itself. In Figure 4, the dominant frequencies are shown in a circled area which contributes to the bispectrum as indicated.

Figure 4: Contour of bicoherence for the novel system (1).

The autobispectrum with the Fourier coefficients can be derived as

where is the radian frequency and is the Fourier coefficients of the time series. The normalized magnitude spectrum of the bispectrum known as the squared bicoherence is given by

where and are the power spectra at and .

The Poincare map is usually constructed by plotting the value of the state variable for every time it passes through a particular plane. Plotting the value every time it passes through the plane where is changing from negative to positive is the normal practice done when studying most of the chaotic attractors. In the case of the proposed novel system (1), the regular method of plotting zero crossing of the plane is of no interest because of the nature of the system. Hence we plotted the zero crossing of system (1) with reference to the plane. Figure 5 shows the Poincare sections in the 3D state portrait of system (1). indicates the negative to positive crossing and indicates the positive to negative crossing in the plane.

Figure 5: Poincare sections in the 3D state portrait of system (1) ( plane).

4. Fractional Order Memristor No Equilibrium Chaotic (FOMNE) System

In this section we derive the fractional order model of the memristor no equilibrium system (1). There are three commonly used definitions of the fractional order differential operator, Grunwald-Letnikov, Riemann-Liouville, and Caputo [28–31].

In this section, we will study the dynamical behavior of fractional order system derived from novel system (1) with the Grunwald-Letnikov (GL) definition, which is defined as

where and are limits of the fractional order equation, is generalized difference, is the step size, and is the fractional order of the differential equation.

For numerical calculations the above equation is modified as

Theoretically fractional order differential equations use infinite memory. Hence when we want to numerically calculate or simulate the fractional order equations we have to use finite memory principal, where is the memory length and is the time sampling.

The binomial coefficients required for the numerical simulation are calculated as

Using (10), the fractional order model of system (1) is derived as

where , , , , , , , , and are the fractional orders of the system, and is the memductance of a flux controller memristor characterized by the nonlinearity ,  , and [46]. For numerical simulations, we take the initial values of the hyperchaotic system (1) as , , , and . Figure 6 shows the 3D state portrait of the fractional order system (13) for .

(a)

(b)

Figure 6: 3D state portrait of the FOMNE system ( plane; plane).

5. Stability Analysis of FOMNE System

As the FOMNE system has no defined equilibrium points, direct calculation of eigenvalues is impossible and hence we consider the origin as the only time bound trivial solution and use it for stability analysis of the system.

5.1. Commensurate Order

For commensurate FOMNE system of order , the system is stable and exhibits chaotic oscillations if , where is the Jacobian matrix at the equilibrium and are the eigenvalues of the FOMNE system where . As seen from the FOMNE system, the eigenvalues should remain in the unstable region and the necessary condition for the FOMNE system to be stable is . As the eigenvalues of the system are , , , and , it is clearly seen that the value is a saddle point and remains in the unstable region contributing to the existence of chaotic oscillations.

5.2. Incommensurate Order

The necessary condition for the FOMNE system to exhibit chaotic oscillations in the incommensurate case is , where is the LCM of the fractional orders. If , , , and , then . The characteristic equation of the system evaluated at the equilibrium is and by substituting the values of and the fractional orders , the characteristic equation is . The approximated solution of the characteristic equation is whose argument is zero which is the minimum argument and hence the stability necessary condition becomes which solves for and hence the FOMNE system is stable and chaos exists in the incommensurate system.

6. Dynamic Analysis of the Fractional Order Memristor No Equilibrium System

Most of the dynamic properties of the integer order system (1) like the Lyapunov exponents, eigenvalues, and bifurcation with parameters are preserved in the fractional order if , where The most important analysis of interest when investigating a fractional order system is the bifurcation with fractional order. The resistance of the memristor increases from the initial value until it reaches its maximum in a certain time period which is called the saturation time [51]. The fractional order parameter can be used to control the saturation time from a part of a second up to several minutes under the same input voltage [51, 52]. The saturation time can be controlled through the fractional order where it can be less than 1 sec when up to higher values when . This specific character of the fractional order memristor is useful in determining the largest positive Lyapunov exponent of the system. Figure 7 shows the 3D state portrait ( plane) of the FOMNE system for various fractional orders . The largest positive Lyapunov exponent () of the FOMNE system appears when against its largest integer order Lyapunov exponent (). Hence fractional order chaos suppression/control and synchronization prove efficient than the integer order controls as the systems show the largest positive Lyapunov exponent in fractional order close to 1. It can also be seen that, as the fractional order decreases, the FOMNE system starts losing its largest positive Lyapunov exponent. When the only positive Lyapunov exponent of the system becomes negative and thus the chaotic oscillations in the system disappear.

Figure 7: 3D state portrait ( plane) of FOMNE system for various fractional orders.

7. Adaptive Fractional Order Sliding Mode Synchronization

In this section we investigate the adaptive fractional order sliding mode synchronization of identical systems. Let us define a generalized fractional order master system as

where is the fractional order of the master system and generalized slave system with adaptive controller as

where is the adaptive controller for synchronizing the nonidentical systems and is the fractional order of the slave system.

Let us define the synchronization error as

The sliding surface for the integral sliding mode control [26, 27, 36] is defined as

The fractional first derivative of the sliding surface is derived as

The fractional order error dynamics are defined as

Using (14) and (15) in (19),

Let us define the adaptive controller as

where , , and are positive gain values, and are parameter estimates of master and slave systems, and is the sliding surface.

Using (21) in (20), the error dynamics simplify to

The Lyapunov candidate function to analyze the stability of the controller is defined as

The Lyapunov first derivate is derived as

By definition of fractional calculus [29, 30],

Using (25) in (24),

Finding the sign of the Lyapunov first derivative using (26) seems difficult and hence we use the modified fractional order Lyapunov method defined by Rajagopal et al. [37] as

Using (27), (20), and (18) in (24),

Let us define the parameter estimate laws as

Using (29) in (28)

as and are all positive and is negative definite. Using Barbalat’s lemma [53], we conclude that as .

7.1. Adaptive Sliding Mode Synchronization of Identical FOMNE Systems

For synchronization let us define the master system as

and the slave system with the adaptive controller is defined as

where , , and are the unknown system parameters and , , , and are the adaptive controllers.

The synchronization error dynamics are defined as

where , , , and are the synchronization errors given by

and , , and are slave and master systems, respectively.

Using (31) and (32) in (33), we simplify error dynamics as

The parameter estimation errors are defined as

The first derivatives of (36) are

Let us define the integral sliding surface as

where are the states, , is the sliding surface, and is the synchronization errors. The first derivative of the sliding surface (38) can be derived as

Let us define the Lyapunov candidate function as

The first derivative of the Lyapunov candidate function (40) is

By definition of fractional calculus [29, 30],

Applying (42) in (41)

As observed in (43), analysis of the Lyapunov first derivative is difficult; hence we use the modified fractional order Lyapunov method defined by Rajagopal et al. [37] as

Applying (44) in (41),

After some mathematical simplifications,

Let us define the adaptive controllers as

And parameter estimation adaptive laws can be defined as

Using the adaptive controllers (47) and the parameter estimation laws (48) in (46),

as , , , and and , , , and are all positive and is negative definite. Using Barbalat’s lemma [53], we conclude that as . For numerical simulations we take the initial conditions as , , , , , , , , and , , and . Figure 8 shows the time history of the synchronization errors. Figure 9 shows the adapted parameters.

Figure 8: Time history of synchronization errors ().

Figure 9: Estimated Parameters ().

8. Genetically Optimized Fractional Order PID (FOPID) Controller for the Synchronization of FOMNE Systems

In this section we investigate the synchronization of FOMNE systems using genetically optimized (GA) FOPID controllers () [38, 44, 45] implemented in feedback loops given by

where is the fractional order PID action control for and , are the fractional order differential and integral operators, is the error signal, and , , and are the proportional, integral, and the derivative gains to synchronize the fractional order model of the memristor no equilibrium system (1). We use genetic algorithm [54, 55] to optimize the controller gains such that the error is minimized. Matlab tools are used for numerical simulations with the following constraints defined.

Variable bounds matrix of the proportional, integral, and the derivative gains = ; population size = 80, GA. Genetically, bigger population size give better approximation; number of generations = 100; selection function = stochastic uniform; crossover fraction = 0.8; mutation function = Gaussian; stopping criteria = error performance criterion; length of the chromosome = 12 (decimal codage).

Tuning of PID controllers involves the selection of , , and gains for better control performance which is defined with reference to the required performance index. There are two important performance indices ISE (Integrated Squared Error) and IAE (Integrated Absolute Error) as given in the following:

In this paper we use IAE as the objective function and the fitness functions is as given below

We synchronize the master system (31) and the slave system (32) using the GA optimized PID controllers with parameters known (Case  1) and parameters unknown (Case  2).

Case 1 (parameters known). In this section we assume that the parameters of the slave system are known. We take the same master system defined by (31) and the slave system (32) is modified with FOPID controllers as given below.

The initial conditions and the parameters are taken as

The synchronization error dynamics are defined as,

Table 1 shows the FOPID gain values of the controllers achieved using genetic algorithm for known parameter values. We get the best solutions tracked over generations for the complete synchronization of the FOMNE systems.
Figure 10 shows the time history of the synchronization errors using genetically optimized fractional order PID controllers; Figure 11 shows the synchronized states and Figure 12 shows the time history of the fractional order PID controllers.

Table 1: FOPID controller gain values optimized with GA.

FOPID controller

Figure 10: Time history of synchronization errors ().

Figure 11: State synchronization ().

Figure 12: Time history of PID controllers , , , and .

Case 2 (parameters unknown). In this section we assume that the parameters of the slave system are unknown. We take the same master system defined by (31) and the slave system (32) is modified with FOPID controllers as given below.

where , , and are the unknown system parameters of the slave system. The initial conditions are taken as

The synchronization error dynamics are defined as

and the parameter update laws are defined as

Table 2 shows the FOPID gain values achieved using the genetic algorithm optimization technique for unknown parameters.
Figure 13 shows the time history of the synchronization errors using genetically optimized fractional order PID controllers, Figure 14 shows the synchronized states, and Figure 15 shows the time history of fractional order PID controllers. Figure 16 shows the time history of the parameter estimates.
As can be seen from Figures 8, 10, and 13, adaptive GA optimized PID control converges much faster ( = 40 mS) than adaptive sliding mode control ( = 190 mS). Figure 17 shows the comparison of synchronization errors using adaptive sliding mode control (ASMC) and GA optimized PID control (GAPID).

Table 2: FOPID controller gain values optimized with GA.

FOPID controller

Figure 13: Time history of synchronization errors ().

Figure 14: State synchronization ().

Figure 15: Time history of PID controllers , , , and .

Figure 16: Time history of parameter estimates.

Figure 17: Comparison of synchronization errors (ASMC and GAFOPID).

9. FPGA Implementation of the FOMNE System

In this section we discuss the implementation of the novel hyperchaotic system in FPGA using the Xilinx (Vivado) system generator toolbox in Simulink. Firstly we configure the available built-in blocks of the system generator toolbox. The Add/Sub blocks are configured with zero latency and 32/16-bit fixed point settings. The output of the block is configured to rounded quantization in order to reduce the bit latency. For the multiplier block a latency of 3 is configured and the other settings are the same as in Add/Sub block. Next we will have to design the fractional order integrator which is not a readily available block in the system generator. Hence we implement the integrators using the mathematical relation discussed in (11) and (12) and the value of is taken as 0.001 and the initial conditions are fed into the forward register. Figure 18 shows the Xilinx schematics of the FOMNE system (13) implemented in Kintex 7 (device = 7k160t; package = fbg484 S) and Figure 19 shows the Xilinx Kintex 7 schematics of the fractional order integrator. Figure 20(a) shows the power consumed by the FOMNE system and Figure 20(b) shows the power consumption for variation in fractional orders. As it can be seen from Figure 20(b), the FOMNE system consumes maximum power when its fractional order compared to the integer order 1 thus confirming that the fractional order memristor system exhibits largest Lyapunov exponent compared to the integer order. Figure 21 shows the 2D state portraits of the FOMNE system using Xilinx system generator.

Figure 18: Xilinx schematics of the FOMNE system implemented in Kintex 7 (device = 7k160t; package = fbg484 S).

Figure 19: Xilinx schematics of the Fractional order integrator implemented in Kintex 7 (device = 7k160t; package = fbg484 S).

(a)

(b)

Figure 20: (a) Power utilized and (b) power utilization versus fractional order of FOMNE system.

Figure 21: 2D state portraits of the FOMENS system implemented using Xilinx system generator ( plane; plane).

10. Conclusion

In this paper we have announced a novel 4D no equilibrium memristor chaotic system. The dynamic properties of the proposed system are investigated to prove the chaotic behavior of the system. The fractional order model of the 4D no equilibrium memristor chaotic system is derived from its integer model. Fractional order bifurcation property of the FOMNE system is investigated and it is seen that largest Lyapunov exponent of the system exists when the fractional order is close to 1. The identical FOMNE systems are synchronized using adaptive sliding mode controllers and genetically optimized PID controllers. Numerical simulations are done to validate the theoretical results. Finally the proposed FOMNE system is implemented in FPGA to show that the system is hardware realizable.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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