Intelligent Controller for Synchronization New Three Dimensional Chaotic System 41

I.J. Modern Education and Computer Science, 2014, 7, 40-46 Published Online July 2014 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijmecs.2014.07.06 Intelligent Controller for Synchronization New

Three Dimensional Chaotic System

Alireza Sahab Faculty of Engineering, Electrical Engineering Group, Islamic Azad University, Lahijan Branch, Iran Email: [email protected]

Masoud Taleb Ziabari Faculty of Engineering, Computer Engineering Group, Ahrar University, Rasht, Iran Email: [email protected]

Abstract—One of the most important phenomena of methods which is based on limbic system of mammalian some systems is chaos which is caused by nonlinear brain. This controller is based on emotional behaviors in dynamics. In this paper, the new 3 dimensional chaotic biological systems. Emotion is an emergent behavior in system is first investigated and then utilized an intelligent biological systems for fast decision making in complex controller based on brain emotional learning (BELBIC), environments. The advantages of this behavior cannot be this new chaotic system is synchronized. The BELBIC neglected in creature survival [2]. During the past few consists of reward signal which accepts positive values. years, the BELBIC has been used in control devices for Improper selection of the parameters causes an improper several industrial applications. The BELBIC has been behavior which may cause serious problems such as successfully employed for making decisions and instability of the system. It is needed to optimize these controlling linear and nonlinear systems such as, Brain parameters. Genetic Algorithm (GA), Cuckoo Emotional Learning Intelligent Controller (BELBIC) for Optimization Algorithm (COA), Particle Swarm the control of two benchmark nonlinear plants was Optimization Algorithm (PSO) and Imperialist applied in [3]. In [4], a problem of speed tracking of Competitive Algorithm (ICA) are used to compute the permanent magnet stepper motor has been discussed optimal parameters for the reward signal of BELBIC. based on the static PID and newly type of intelligent These algorithms can select appropriate and optimal control which mimics the emotional learning in limbic values for the parameters. These minimize the Cost system of mammalians. Also BELBIC was used to Function, so the optimal values for the parameters will be control the Locally Linear Neuro-Fuzzy Model founded. Selected cost function is defined to minimizing (LOLIMOT) of Washing Machine [5]. BELBIC was the least square errors. Cost function enforces the system applied to a Switched Reluctance Motor (SRM) to tackle errors to decay to zero rapidly. Numerical simulation will the speed and position control problem in [6]. show that this method much better, faster and more Furthermore, BELBIC was applied for real time effective than previous methods can be new 3D chaotic positioning of laboratorial overhead traveling crane in [7]. system mode to bring synchronized. In [8], BELBIC was applied to electrically heated micro- heat exchanger, which was a nonlinear plant. In [9], Index Terms—New 3D chaotic system, Synchronization, BELBIC is introduced to stabilize uncertain nonlinear BELBIC, Genetic Algorithm, Cuckoo Optimization systems via robust adaptive method. Also an intelligent Algorithm, Particle Swarm Optimization Algorithm, adaptive approach for aerospace launch vehicle control is Imperialist Competitive Algorithm, Cost Function. presented in [10]. Furthermore, the design of PID and BELBIC controllers in path tracking and controlling problem is studied in [11] and finally an intelligent I. INTRODUCTION autopilot control design for a 2-Dimensional helicopter model is studied in [12]. In parallel with industrial and technological In this study, utilizing BELBIC model introduced in improvement, control systems and their control methods [11, 12], we will design an intelligent controller to have become sophisticated. Control of new systems using synchronized two new 3D chaotic systems [13]. previous old methods has become difficult. Further, Simulation results depict that this proposed controller can considering human brain patterns and abilities in order to synchronize these chaotic systems better than Active control and solve problems has resulted in emergence of Control [19, 20]. Finally, the parameters of BELBIC are new intelligent controlling methods which utilizes human improved by swarm intelligent algorithms. brain operation patterns which are mentioned below. The rest of the paper is organized as follows. The Brain Brain Emotional Learning Based Intelligent Controller Emotional Learning Based Intelligent Controller (BELBIC) was introduced for the first time by Lucas in (BELBIC) is described in section 2. The new 3D chaotic 2004 [1]. Brain Emotional Learning Based Intelligent system is described in section 3. Synchronization Controller (BELBIC) is an example of bioinspired control

between two new 3D chaotic systems by BELBIC is proposed in section 4. Parameters of BELBIC are (1) improved by Evolutionary Algorithms in section 5. The conclusion of this study is provided in section 6. In which , is the input sensor and are two states that are depended on input sensor. Index represents the Th sensor and its related states. These two equations will II. BRAIN LEARNING BASED INTELLIGENT CONTROLLER be updated by following equations [11, 12]. (BELBIC) In this method, emotional factors like excitement and ( ∑ ) anxiety are the roots of learning. Here, the roots of ( ∑ ∑ ( )) (2) anxiety are assumed as some stimulants and the control system should react in the way that reduces the anxiety of In which α are training coefficients and is the the system caused by these stimulants. The Brain reward signal. Amygdala acts as an actuator and Emotional Learning (BEL) is divided into two parts, very orbitofrontal corex acts as a preventer. Therefore the roughly corresponding to the amygdala and the control signal of BELBIC is: orbitofrontal cortex, respectively. The amygdaloid part receives inputs from the thalamus and from cortical areas, ∑ ∑ (3) while the orbital part receives inputs from the cortical areas and the amygdala only. The system also receives This paper uses the continuous form of BELBIC. In reinforcing (REW) signal. The emotional learning model continuous form the BELBIC states are updated by in amygdala and orbitofrontal corex is illustrated in Fig 1. following equations.

̇ ( ) ̇ ( ) (4)

A BELBIC controller has to be designed to synchronize two chaotic systems. For traction force sensory inputs are considered.

(5)

is the error between the master system and the slave Fig. 1. Scheme of BELBIC structure [1] system. The structure of the control system is illustrated in Fig 2. BELBIC has some input sensors that can be chosen by designer. Each input sensor has two different states that can be described as.

Sensory Input

Master Chaotic - e u Slave Chaotic BELBIC System System + Reward Signal Builder

Fig. 2. Control system configuration using BELBIC

The reward signal will be obtained the reward function. (6) This function has a great role in BELBIC. The designer must define a reward function that has its maximum and are the positive parameters of the reward values in the most desired regions. In this study, the function. The reward function for this BELBIC controller reward function is chosen as a linear function of system is as Fig 3. error.

̇ Rewi ̇ (8) ̇ k1iei+k2i k2i 20 x ei y 15 z

Fig. 3. Reward Function 0 Trajectory of States -5

III. SYSTEM DESCRIPTION -10

Recently, Dadras and Momeni proposed the three- -15 0 5 10 15 20 25 30 35 40 45 50 dimensional autonomous chaotic system that generating Time (sec) two, three and four-scroll attractors [13]. The system is described by: Fig. 5. State trajectory of the 3D chaotic attractors (7).

And the response system is presented as follows ̇ ̇ (7) ̇ ( ) ̇ ̇ ( ) (9)

̇ ( ) Here are the state variables and are the positive constant parameters. When Where ( ) ( ) and ( ) are control functions to , the system (7) is chaotic witn the be determined to achieve the synchronization between Lyapunov exponents . two systems (8) and (9). Define state errors between The corresponding phase portraits are depicted in Fig 4 system (8) and (9) as follows and the state trajectory of the system (7) is displayed in Fig 5.

(10)

We obtain the following error dynamical system by 10 subtracting the driving system (8) from the response 5 system (9).

0 ̇ ( ) ( )

z State -5 ̇ ( ) ( ) (11) -10 ̇ ( ) ( ) -15 10 5 20 Thus, the error system (11) to be controlled with 0 15 control inputs ( ) ( ) and ( ) as functions of error y State 10 -5 5 x State states and . When system (11) is stabilized by -10 0 control inputs ( ) ( ) and ( ), and will

converage to zeroes as time tends to infinity. Which Fig. 4. Phase portraits of the 3D chaotic attractors (7) in (a) the space, (b) the z space, (c) the space, (d) the space. implies that system (8) and (9) are synchronized. To achieve this purpose, Input sensory of BELBIC is chosen as (12).

IV. SYNCHRONIZATION BETWEEN TWO NEW 3D CHAOTIC

SYSTEM { (12)

In this section, the BELBIC is applied to synchronize between two new 3D chaotic systems. Suppose the The reward function’s parameters for the BELBIC driving system takes the following froms: controller are as follows.

25 { (13) 20

15 u 1 u 10 2 u The parameters and are equal to 1 and 3 3 5 respectively. We take the reward gains as for . For drive and response systems, we take 0 -5 initial conditions ( ( ) ( ) ( )) ( ) Signals Control -10 and ( ( ) ( ) ( )) ( ) . After using BELBIC, we compare the results with the results -15 obtained by Active Control [19]. Synchronization errors -20 -25 ( ) of BELBIC and Active Control [19] in new 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 3D chaotic systems are shown in fig 6 and fig 7. The time Time (sec) response of the control inputs ( ) of BELBIC Fig. 8. The time response of the control inputs ( ) via BELBIC. and Active Control [19] to the synchronize new 3D chaotic systems are shown in fig 8 and fig 9. The time 2500 response of ( ) states for drive system (8) and response system (9) via BELBIC is shown in fig 10. 2000 u 1 u 2 u 1500 3 30 e x e 1000 y 20 e

z Signals Control 500

-500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (sec) Trajectory of Errors -10

Fig. 9. The time response of the control inputs ( ) via Active -20 Control [19].

-30 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 45 x Time (sec) 1 40 x 2 Fig. 6. Synchronization errors ( ) in drive system (8) and 35

response system (9) via BELBIC. 30

40 20

e x State x 15 e y 20 e 10 z 5 0 0

-5 -20 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec)

Trajectory of Errors -40 30 y 1 y 2 -60 20

-80 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time (sec)

Fig. 7. Synchronization errors ( ) in drive system (8) and y State

response system (9) via Active Control [19]. -10

-20

-30 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec)

Table 3. Particle Swarm Optimization Algorithm Parameters. 40 z 1 z Parameters Values 2 30 Size population 80 20 Maximum iterations 30 Initial and Final value of the global best 10 2 and 2 z State acceleration factor 0 1 and Initial and Final value of the inertia factor 0.99 -10 Search interval [1 10]

-20 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec) Table 4. Imperialist Competitive Algorithm Parameters.

Fig. 10. The time response of signals ( ) for drive system (8) and Parameters Values response system (9) via BELBIC.

Number of Initial Countries 80 Number of Decades 30 V. IMPROVING THE PARAMETERS OF CONTROLLER Number of Initial Imperialists 8 The reward signal is very important for BELBIC. The system responses differently to each value. It is necessary Revolution Rate 0.3 to select proper parameters to obtain a good response because the improper selection of parameters leads to Search interval [1 10] inappropriate responses or even may lead to instability of the system. The Genetic Algorithm [14,15], Cuckoo The optimal parameters of reward signals using genetic Optimization Algorithm [16], Particle Swarm algorithm, cuckoo optimization algorithm, particle swarm Optimization Algorithm [17] and Imperialist Competitive optimization algorithm and imperialist competitive Algorithm [18] are used to search the optimal parameter algorithm are listed in table. 5. ( ) in order to guarantee the stability of the systems by ensuring negativity of the Lyapunov function and having Table 5. Optimal Parameters of Reward Signals. a suitable time response. The reward signals in the equation (13) are optimized by the Cost Function in the 9.46 1.09 equation 14. The selected cost function is defined to GA 3.55 1.26 6.01 8.02 3.46 9.49 minimize the least square errors. The cost function 7.19 1.53 enforces the synchronization errors to decay to zero PSO 3.79 6.27 5.64 3.97 2.92 8.11 rapidly. 8.51 1.28 COA 10 7.54 10 8.58 1.66 10

( ) √∑ (14) 10 1 ICA 4.03 1 5.89 10 1 10

Table 1. Genetic Algorithm Parameters. Synchronization errors ( ) in new 3D chaotic Parameters Values systems are shown in order Fig 11 to Fig 13. Size population 80 30 Maximum of generation 30 GA PSO 25 Prob.crossover 0.75 ICA COA Prob.mutation 0.001 20 Search interval [1 10]

x e Table 2. Cuckoo Optimization Algorithm Parameters. 10 Parameters Values 5 Size clusters 2

Maximum number of 0 80 cuckoo -5 Size initial population 5 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Maximum iterations of Time (sec) 30 cuckoo Fig. 11. Synchronization error ( ) in drive system (8) and response Search interval [1 10] system (9).

controller is faster, more optimal and more efficient than 5 active control.

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Generalized Projective Synchronization of Three-Scroll

How to cite this paper: Alireza Sahab, Masoud Taleb Ziabari,"Intelligent Controller for Synchronization New Three Dimensional Chaotic System", IJMECS, vol.6, no.7, pp.40-46, 2014.DOI: 10.5815/ijmecs.2014.07.06