Chaos Suppression and Stabilization of Generalized Liu Chaotic Control System

1. International Journal of Trend in International Open Access Journal International Open Access Journal | www.ijtsrd.com International Journal of Trend in Scientific Research and Development (IJTSRD) Research and Development (IJTSRD) www.ijtsrd.com ISSN No: 2456 ISSN No: 2456 - 6470 | Volume - 3 | Issue – 1 | Nov 1 | Nov – Dec 2018 Chaos Suppression Generalized Liu Chaotic Control System Generalized Liu Chaotic Control System Generalized Liu Chaotic Control System Chaos Suppression and Stabilization of f Yeong Yeong-Jeu Sun1, Jer-Guang Hsieh2 1Professor,2Chair Professor Electrical Engineering, I-Shou University, Kaohsiung Department of Electrical Engineering Kaohsiung, Taiwan ABSTRACT In this paper, the concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the time approach with differential inequalities, a suitable control is presented such that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily pre specified but also the critical time can be correctly estimated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Key Words: Generalized synchronization, Liu chaotic system, critical time, exponential convergence rate 1.INTRODUCTION In recent years, chaotic dynamic systems have been widely investigated by researchers; see, for instance, [1-12] and the references therein. Very often, chaos in many dynamic systems is an origin of the generation of oscillation and an origin of instability. control system, it is important to design a controller that has both good transient and steady-state response. Furthermore, suppressing the occurrence of chaos plays an important role in the controller design of a nonlinear system. In the past decades, various methodologies in control design of chaotic system have been presented variable structure control approach, approach, adaptive control approach, adaptive sliding mode control approach, back stepping control approach, and others. In this paper, the concept of generalized for nonlinear dynamic systems is introduced and the stabilizability of generalized Liu chaotic system will be investigated domain approach with differential inequality, a suitable control will be offered generalized stabilization can be achieved for a class of Liu chaotic system. Not only correctly estimated, but exponential convergence rate can be arbitrarily pre specified. Several numerical simulations will also be provided to illustrate the use of the main results. The layout of the rest of this paper is organized as follows. The problem formulation, mai controller design procedure are presented Numerical simulations are given in Section 3 the effectiveness of the developed results. conclusion remarks are drawn 2.PROBLEM FORMULATION AND MAIN RESULTS Nomenclature n ℜ the n-dimensional real space a the modulus of a complex number A the transport of the matrix x the Euclidean norm of the vector In this paper, we explore the following generalized Liu chaotic system: ( ) 1 2 2 1 1 1 t u t x a t x a t x + + = & ( ) 2 3 1 4 1 3 2 t u t x t x a t x a t x + + = & ( ) ( ) , 0 , 3 1 8 ≥ ∀ + + t t u t x a he concept of generalized stabilization for nonlinear systems is introduced and the stabilization of the generalized Liu chaotic control system is explored. Based on the time-domain approach with differential inequalities, a suitable uch that the generalized stabilization for a class of Liu chaotic system can be achieved. Meanwhile, not only the guaranteed exponential convergence rate can be arbitrarily pre- specified but also the critical time can be correctly numerical simulations are given to demonstrate the feasibility and effectiveness generalized stabilizability dynamic systems is introduced and the generalized Liu chaotic control investigated. Based on the time- domain approach with differential inequality, a offered such that the stabilization can be achieved for a class of system. Not only the critical time can be , but exponential convergence rate can be arbitrarily pre- . Several numerical simulations will also be provided to illustrate the use of the main results. also also the guaranteed The layout of the rest of this paper is organized as The problem formulation, main result, and controller design procedure are presented in Section 2. given in Section 3 to show developed results. Finally, Generalized synchronization, Liu chaotic system, critical time, exponential convergence rate tic dynamic systems have been ; see, for instance, ] and the references therein. Very often, chaos in an origin of the generation an origin of instability. For a chaotic control system, it is important to design a controller in Section 4. ROBLEM FORMULATION AND MAIN dimensional real space the modulus of a complex number a the transport of the matrix A the Euclidean norm of the vector state response. Furthermore, suppressing the occurrence of chaos T oller design of a ∈ ℜ n x the following generalized In the past decades, various methodologies in control presented, such as time-domain ( ) ( ) ( ) 2 ( ) ( ) ( ) ( ) ( ) ( ), (1a) (1b) ( ), approach, adaptive control approach, adaptive sliding ( ) t stepping control = + + x & t a x t a x t a x 3 5 1 6 2 7 3 (1c) @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1112

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 ( ) [ 30 20 10 x x x = of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 [ ] ( ) 0 ( ) ] , + − 1 p q T 0 0 x x x where , with , with and α = p > > > ∈ : q , 0 , 0 , a b p q N (1d) 1 2 3 − 2 1 p T In convergence rate and the guaranteed critical time given by b and  this case, the pre- guaranteed critical time are -specified exponential where ( ) ( ) : = t u [ x 10 the parameters of the system. The original system is a special case of system (1) with 10 2 1 = − = a a , , 40 a a 4 a . It is well known that the system (1) without any control (i.e., ( ) 0 = t u ) displays chaotic behavior for certain values of the parameters [1]. The paper is to search a novel control for the system (1) such that the generalized stability of the controlled system can be guaranteed. In this concept of generalized introduced. Motivated by time-domain approach with differential inequality, a suitable control be established. Our goal is to design a control such that the generalized stabilization of system achieved. Let us introduce a definition which will be used in subsequent main results. 1There exist two positive numbers k that ( ) 0 , ≥ ∀ ⋅ ≤ t e k t x 2There exists a positive number ( ) t t x ≥ ∀ = , 0 . Definition 1 The system (1) is said to realize the stabilization, provided that there exist a suitable control u such that the conditons (i) and (ii) are satisfied. In this case, the positive number the exponential convergence rate and number ct is called the critical time. Now we present the main result for the stabilization of the system (1) via approach with differential inequalities. Theorem 1 The system (1) realizes the generalized under the following control ( ) 2 2 1 1 1 ax t x a t x b a t u − − + − = ( ) ( ), 2 t ax − ( ) ( ) 3 1 8 t ax t x a − − [ ] ( ) t ( ) t ] ( ) t is the state vector, is the system is the state vector, T = ∈ is the system control, ∈ b , . The original Liu chaotic ℜ 3 : x ( ) 1 t u x 20 t x ( ) x ( ) 3 x 1 t is the initial value, and ai, 2 t 3 [ T ∈ ℜ 3 u u  a 2 ]T   indicate ℜ x b + ln 30   [ ( )] 0 a ( ) 0 ( ) ( 1 2 α − 1     + + 2 3 2 1 2 2 2 3 ) 0 x x x is a special case of system (1) with ( ) b (3) = = , 0 u t , and , tc ) α − − b b 3= = , 1 − = a = = 5 . 2 − , 0 a a respectively Proof. From (1)-(2), the feedback can be performed ( ) , 1 1 1 − − = x a bx x & ( ) , 2 2 2 − − = x a bx x & ( ) , 3 3 3 − − = x a bx x & Let ( ) x t x t x W = . The time derivative of feedback-controlled system is given by 2 2 x x x x W & & ⋅ + ⋅ = 2 2 1 2 2 x b x b − ⋅ − ⋅ − = ( 1 2 2 x a W b , 2 2 ⋅ − ⋅ − ≤ W a W b It follows that ( ( . 0 , 1 2 ≥ ∀ − − ≤ t a α Define ( ) , : ∀ = t t x W t Q From (6) and (7), it can be readily obtained that ( 1 2 1 2 − − ≤ − + a bQ Q α α & It is easy to deduce that ( ( ) ( ) ( . 0 , 1 2 ≥ ∀ − − ≤ t e a α It follows that ( ) 0 Q t Q e − ⋅ = 4 5 6 7 It is well known that the system (1) without ) displays chaotic behavior for 8= feedback-controlled system he aim of this for the system (1) of the feedback- In this paper, the α − (4a) (4a) (4a) 2 1 α − 2 1 α − 2 1 stabilization stabilization domain approach with suitable control strategy will a control such stabilization of system (1) can be will will be be ( ) ( ) ( ) t (5) T ( ( ) t ) along the trajectories of is given by W x Let us introduce a definition which will be used in & 1 1 2 2 α 2 1 x ) ≥ t α 2 ⋅ − a a⋅ 2 2 2 a 2x k and b , such α 2 α 2 2 ∀ = − ⋅ − + x . − t b α . 0 ct such that ct ) ( ) α & − − α − α + − α 1 1 2 1 W W bW (6) ) The system (1) is said to realize the generalized there exist a suitable such that the conditons (i) and (ii) are In this case, the positive number b is called convergence rate and the positive ( ( ) ) − α (7) 1 ≥ . 0 ), it can be readily obtained that . 0 ) ( ), ∀ ≥ t the generalized (1) via time-domain ( ) ( ) t ( ) ( ] ) Q & α α − − α ⋅ + ⋅ − 2 1 2 1 bt bt 2 1 e t e b Q [ d ( ) α − = ⋅ 2 1 bt e Q t dt ) ( ) generalized stabilization α − 2 1 bt ( ) ( ) ( ) 1 − α ( ) 2 ( ) t ( ), t (2a) α ( ) t − 2 1 1 ( ) ( ) x t = − − − u t a x t a x bx [ ] t d ( ) 2 3 1 4 1 3 2 (2b) ∫ − α ⋅ 2 1 bt e Q t dt 2 dt ( ) ( ( ), ) ( ) 3 x ( ) ( ) 0 = − − − + ( ) u t a x t a x t a b t − α 2 1 bt 3 5 1 6 2 α 7 (2c) − 2 1 @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1113

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 results, it is seen that the dynamic system of ( achieves the generalized stabilization under the control law of (9). 4.CONCLUSION In this paper, the concept of generalized for nonlinear systems has been introduced and the stabilization of generalized Liu chaotic has been studied. Based on the time with differential inequalities, a s been presented such that the generalized for a class of Liu chaotic system Besides, not only the guaranteed exponential convergence rate can be arbitrarily pre also the critical time can be correctly estimated. Finally, some numerical simulations have been offered to show the feasibility and effectiveness of the obtained results. ACKNOWLEDGEMENT The authors thank the Ministry of Science and Technology of Republic of China for supporting this work under grants MOST 106 MOST 106-2813-C-214-025-E E-214-030. REFERENCES 1.D. Su, W. Bao, J. Liu, and simulation of the fractional chaotic system and its synchronization,” Journal of the Franklin Institute vol. 355, pp. 9072-9084, 201 2.B. Wang, F.C. Zou, and memritive chaotic system and the application in digital watermark,” Optik, vol. 1 2018. 3.R. Lan, J. He, S. Wang, T. Gu, “Integrated chaotic encryption,” Signal Processing 145, 2018. 4.Y. Wang and H. Yu, “Fuzzy synchronization of chaotic systems via intermittent control Solitons & Fractals, vol. 106 5.Q. Lai, A. Akgul, M. Varan, J. Kengne, Erguzel, “Dynamic analysis and synchronization control of an unusual chaotic system with exponential term and coexisting attractors Chinese Journal of Physics 2851, 2018. 6.T.L Le, C.M. Lin, and T.T. type-2 fuzzy brain emotional learning control 2 fuzzy brain emotional learning control dynamic system of (1) stabilization under the t ( ) ( ) ∫ − α ≤ − − α 2 1 bt 2 1 a e dt 0 − ( ) a ( ) − α = , 1 − ∀ ≥ 2 1 bt . 0 e t b Consequently, we have ( ) 0  t Hence, from (6), (7), and (8), we have (i) If , 0 t ≤ ≤ generalized stabilization for nonlinear systems has been introduced and the generalized Liu chaotic control system has been studied. Based on the time-domain approach al inequalities, a suitable control has   a a ( ) ( ) − − α ≤ + ⋅ − 2 1 bt , Q t Q e  (8) b b ∀ ≥ . 0 generalized stabilization Liu chaotic system can be achieved. Besides, not only the guaranteed exponential convergence rate can be arbitrarily pre-specified but itical time can be correctly estimated. Finally, some numerical simulations have been offered to show the feasibility and effectiveness of the ct ( ) α − 1 1     a a ( ( ) t ) ( ) 0 ( ) − α 2 2 (ii) If (ii) If − α − ≤ + ⋅   − t ≥ 2 1 bt   ; , W x x e ct   b b ( ) t . = 0 x Consquently, we conclude that (i) If , 0 t ≤ ≤ ct ( ) − α 1 2 2   a Ministry of Science and China for supporting this grants MOST 106-2221-E-214-007, E, and MOST 107-2221- ( ) t ( ) 0 − α 2 2 − ≤ + ⋅ bt ; x x e  t ≥  b , 0 ( ) x (ii) If in view of (5) with above condition (i) completes the proof. □ 3.NUMERICAL SIMULATIONS Consider the generalized Liu chaotic system with 10 2 1 = − = a a , 40 a , , 4 a and ( ) [ x 2 4 0 = example, is to design a feedback control such that the system (1) realize the generalized stabilization with the guaranteed exponential convergence rate 5 . 0 = b . From (2), with = a 8 . 0 = α , ( ) 100 10 5 . 10 2 1 1 x t x t x t u − + − = ( ) ( ), 100 2 t x − ( ) 100 4 2 3 1 3 3 t x t x t x t u − − = Consequently, by Theorem 1, we conclude that system (1) achieves generalized stabilization parameters of 10 2 1 = − = a a , a 5 . 2 7 − = a , 4 a , and feedback control law of ( Furthermore, the exponential convergence rate guaranteed critical time are given by 047 . 0 = . The typical state trajectories of uncontrolled systems and controlled systems are depicted in Fig Figure 2, respectively. From the foregoing simulations , respectively. From the foregoing simulations with above condition (i). This = , ct t and C. Gong, “An efficient Liu chaotic system of (1) 6= a simulation of the fractional chaotic system and its Journal of the Franklin Institute, , 2018. 3= = , 1 − . Our objective, in Our objective, in this 5= a = 5 . 2 − , , 0 a a a 4 ]T 7 − 8= 2 example, is to design a feedback control such that the and Y. Zhang, “New stabilization with memritive chaotic system and the application in with the guaranteed exponential convergence rate , vol. 172, pp. 873-878, , 2 we deduce = = 100 , , 3 p q ( ) + ( ) t ( ), t (9a) R. Lan, J. He, S. Wang, T. Gu, and X. Luo, Integrated chaotic Signal Processing, vol. 147, pp. 133- 6 . 0 1 ( ) 2 t ( ) 6 . 0 ( ) ( ) x t systems systems for for image image = − 5 . 0 − 40 u t x t x x 2 1 1 3 (9b) ( ) ( ) ( ), (9c) , we conclude that the 2 6 . 0 Fuzzy synchronization of chaotic systems via intermittent control,” Chaos, 106, pp. 154-160, 2018. achieves generalized stabilization with 3= 4= , 1 − = a = 40 , , 0 a a 5 6 feedback control law of (9). exponential convergence rate and the iven by Q. Lai, A. Akgul, M. Varan, J. Kengne, and A.T. Dynamic analysis and synchronization control of an unusual chaotic system with exponential term and coexisting attractors,” Chinese Journal of Physics, vol. 56, pp. 2837- 8= and = 5 . 0 b ct of uncontrolled systems and controlled systems are depicted in Figure 1 and .T. Huynh, “Self-evolving @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1114

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