Design of State Estimator for a Class of Generalized Chaotic Systems

1. International Journal of Trend in Scientific Research and Development (IJTSRD) Volume 3 Issue 6, October 2019 2019 Available Online: www.ijtsrd.com e International Journal of Trend in Scientific Research and Development (IJTSRD) International Journal of Trend in Scientific Research and Development (IJTSRD) e-ISSN: 2456 – 6470 Design of State Estimator Generalized Chaotic Systems Generalized Chaotic Systems Yeong-Jeu Sun f State Estimator for a Class Class of Professor, Department of Electrical Engineering, I f Electrical Engineering, I-Shou University, Kaohsiung, Taiwan University, Kaohsiung, Taiwan ABSTRACT In this paper, a class of generalized chaotic systems is considered and the state observation problem of such a system is investigated. Based on the time-domain approach with differential inequality, a simple state estimator for such generalized chaotic systems is developed to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be correctly estimated. Finally, several numerical simulations are given to show the effectiveness result. KEYWORDS: Chaotic system, state estimator, generalized chaotic systems, exponential decay rate How to cite this paper: Yeong-Jeu Sun "Design of State Estimator for a Class of Generalized Chaotic Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456- 6470, Volume-3 | 6, October 2019, pp.1029- 1032, www.ijtsrd.com/papers/ijtsrd2 How to cite this paper "Design of State Estimator for a Class of Generalized Chaotic Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456 6470, Volume Issue-6, October 2019, pp.1029 1032, https://www.ijtsrd.com/papers/ijtsrd2 9270.pdf Copyright © 2019 International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0) In this paper, a class of generalized chaotic systems is considered and the state observation problem of such a system is investigated. Based on the domain approach with differential inequality, a simple state estimator chaotic systems is developed to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be correctly estimated. Finally, several numerical simulations are given to show the effectiveness of the obtained IJTSRD29270 Chaotic system, state estimator, generalized chaotic systems, URL: URL: Copyright © 2019 by author(s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ (CC (CC BY BY 4.0) 4.0) will be given to illustrate the result. 2.PROBLEM FORMULATION AND MAIN RESULTS In this paper, we consider the following systems ( ) bx t ax t x 2 1 1 + = & ( ) t x t x t x f t x 3 2 1 1 2 , , = & ( ) 3 1 2 3 3 x f t x f t cx t x ⋅ + = & ( ) 2 1 t x t x t y β α + = where ( ) 2 1 := t x t x t x ( ) ℜ ∈ t y is the system output, with ( ) , 1 deg ∈ ∀ ≥ i fi , and state equation, with 0 < c and α and β , the parameters of output equation, satisfying b − > β Remark 1: In fact, certain well as generalized Lorenz chaotic system [12], generalized Chen chaotic system system [14], and Pan chaotic system of the system (1). 1.INTRODUCTION In recent years, various forms of chaotic systems have been extensively and intensively studied; see, for example, [1 and the references therein. The study of chaotic systems not only allows us to understand the chaotic characteristics, but also we can use the research results in various applications; such as chaotic synchronization design and controller design of chaotic systems. As we know, signals are highly unpredictable and the initial values are highly sensitive to the output signal. Due to the defects of the measuring instrument or the uncertainties of the system, not all state variables are just measurable for a nonlinear system. At this time, the design of the state estimator is very important and needs to be seriously faced. The state estimator has come to take its pride of place in system identification and filter theory. Besides, the state estimator design for the state reconstruction of dynamic systems with chaos is in general not as easy as that without chaos. Based on the above-mentioned reasons, estimator design of chaotic systems is quite crucial. In this paper, the state reconstructor for a cla generalized chaotic systems is considered. Using the time domain approach with differential inequality for such generalized chaotic systems will be provided guarantee the global exponential stability of the resulting error system. In addition, the guaranteed exponential be accurately estimated. Finally, some numerical example to illustrate the effectiveness of the obtained systems have been extensively and intensively studied; see, for example, [1-15] The study of chaotic systems not only allows us to understand the chaotic characteristics, but also we can use the research results in various chaos applications; such as chaotic synchronization design and ROBLEM FORMULATION AND MAIN RESULTS , we consider the following generalized chaotic As we know, chaotic ( ) ( ) t ( ) ( ( ), (1a) signals are highly unpredictable and the initial values are ( ( ) ( ) ) ) (1b) )), ( ) ( ) ( ) (2 ( ue to the defects of the measuring instrument or the uncertainties of the system, not all state variables are just (1c) t t (1d) At this time, the design of the state estimator is very important and needs to be seriously The state estimator has come to take its pride of place in system identification and filter theory. Besides, the state [ ( )] t ( ) system output, ( ) T ∈ ℜ 3 is the state vector, x 3 3 2f and b a , , 0 ≠ b 3f are polynomials are the parameters of . In addition, we select nstruction of dynamic { } 3 , 2 c is in general not as easy as that without mentioned reasons, the state is quite meaningful and , and and , the parameters of output equation, satisfying α . a c the state reconstructor for a class of the . Using the time- with differential inequality, a state estimator for such generalized chaotic systems will be provided to In fact, certain well-known chaotic systems, such generalized Lorenz chaotic system [11], Lü chaotic system generalized Chen chaotic system [13], unified chaotic chaotic system [15], are the special cases the resulting error exponential decay rate can numerical examples @ IJTSRD | Unique Paper ID – IJTSRD29270 29270 | Volume – 3 | Issue – 6 | September 6 | September - October 2019 Page 1029

2. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 A.System (1) is called the generalized Lorenz chaotic system [11] in case of , 25 10 k a ∆ − − = 25 10 k b ∆ + = 8 k c = with , 8 . 0 0 < ∆ ≤ k ( 1 1 1 29 35 28 x k x k f − ∆ + ∆ − = x f = , 2 3 x f = . B.System (1) is called the Lü chaotic system [12] in case of , 25 10 k a ∆ − − = 25 10 k b ∆ + = 8 k c = with , 8 . 0 = ∆k ( 1 1 1 29 35 28 x k x k f − ∆ + ∆ − = x f = , 2 3 x f = . C.System (1) is called the generalized Chen chaotic system [13] in case of , 25 10 k a ∆ − − = 25 10 k b ∆ + = 8 k c = with , 1 8 . 0 < ∆ < k ( 1 1 1 29 35 28 x k x k f − ∆ + ∆ − = x f = , 2 3 x f = . D.System (1) is called the unified chaotic system [14] in case of , 25 10 k a ∆ − − = 25 10 k b ∆ + = 8 k c = with , 1 0 ≤ ∆ ≤ k ( 1 1 1 29 35 28 x k x k f − ∆ + ∆ − = x f = , 2 3 x f = . E.System (1) is called the Pan chaotic system [15] in case of 8 − = c , 1 f = Theorem 1: The system (1) is exponentially state reconstructible. Besides, a suitable state estimator is given by ( ) 1 1 t by t z a t z +    β ( ) 1 2 t z t y t z β β ( ) 2 3 1 2 3 3 t z f t z f t cz t z + = & with the guaranteed exponential decay rate b− = β ( ) 3 , 2 , 1 , : ∈ ∀ − = i t z t x t e i i i it can be readily obtained that ( ) ( ) a t bx t ax    β ( ) t x t y b t ax 1  α ,   α b ( ) ( ),  = − (2a) & − − ∆ , 3 α 1 ( ) ( ) ( ), ( ) ) ( ) = − (2b) − x x , 2 1 3 ( ) ( ( ) ), (2c) 2 1 η = : − . c , α α : 1 ( ) Proof. Define , from (1), (2) with a − − ∆ , 3 ( ) { }, (3) ) ( ) − x x , 2 1 3 2 1 ( ) t ( ) t = − e & t x & z & 1 1 1   α b ( ) ( ) t ( ) t  = + − − − z by 1 2 1 , ( ) − − ∆   − β α ( ) = + , 1  3 ) ( ) − x x ,   b 2 1 3 ( ) t ( ) t     − − − a z by 1 β 2 1   α b [ ] ( ) t ( ) t   α   = − − − a x z 1 1 β e ( ) t , = − ∀ ≥ , . 0 t 1 1 − − ∆ It results that d , 3 [ ] ( ) t ( α ) ) ( ) exp dt e t − x x , = 1 1 0 2 1 3 [ e ] ( ) ( ) t ( ) 1 t e ( α ) α − ( ) 0 ( ) 0 2 1 = ⇒ exp = e t t e 1 1 1 ( ) ( ) 0 1 e ( exp ⇒ ⇒ Moreover, form (1)-(4), we have ( ) ( ) − = β ( ) t x 1 − = β ( ) t e1 β ( exp − = β Thus, one has ( ) 0 1 2 = e t e β exp = t 1 1 ) α ⋅ − ∀ ≥ , . 0 (4) e t t 1 1 = − − = 10 , a 16 x x x 10 , , b 1 1 3 3 ( ) t ( ) ( ) t 1 f = f = x x , . = − e t x t z t 2 1 3 2 2 2 2 It is a well-known fact that since states are not always available for direct measurement, states must be estimated. The objective of this paper is to search a state estimator for the system (1) such that the global exponential stability of the resulting error systems can be guaranteed. In what follows, x denotes the Euclidean norm of the column vector x and   α α 1 β ] y x ( ) t ( ) − − y z t  1 β α − [ ( ) t x 2 α − = a denotes the absolute value of a real number a. α − ) ( ) 1 e α ∀ ≥ 0 , . 0 t t Before presenting the main result, let us introduce a definition which will be used in the main theorem. Definition 1: The system (1) is exponentially state reconstructible if there ( ) t y t z h t z E , = & and positive numbers k and η such that ( ) : ≤ − = k t z t x t e where ( ) t z expresses the reconstructed state of the system (1). In this case, the positive number η is called the exponential decay rate. Now we present the main result. 1 − α ( ) ( ) exist a state estimator ⋅ − α ∀ ≥ exp , . 0 t t (5) ( ( ) ( ) ) 1 ( ) x ( ) z ( ) ( ) ( ) − f f ( ) { , 2 }, − η ∀ ≥ exp , 0 , t t = ∀ ∈ i i , : , 3 Define the polynomials h x z i i − x z and form (1)-(5), it yields ( ) t x t e 3 3 & & − = ( ) ( z f 1 2 − ( ) x 3 ( ) t z 3 z & − t 3 [ ] ( ) ( 3 ( ( ) t ) ( ( ) t ) = + ⋅ c t f x f x 2 1 3 2 ( ) t ) ( ) t ) f z 2 @ IJTSRD | Unique Paper ID – IJTSRD29270 | Volume – 3 | Issue – 6 | September - October 2019 Page 1030

3. International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470 ( ) ( ) t z f t x f t z f 2 3 2 3 1 2 − ⋅ + ( ) ( ) ( ) t z t x t z t x h t z f 2 2 2 2 3 1 2 , − ⋅ + ( ) ( ) . 0 , , 2 2 2 3 1 2 ≥ ∀ ⋅ + t t e t z t x h t z f [ ] ( ) = ( ) x ( ( ) ( ( ) t ) ( ( ) ( ( ) t ) ( ( ) t ) = + − = = + ⋅ − , f , x 5 . 3 x 19 3 . , f x x x x x x x f x , ce t f x t f x t f z 1 1 2 ( ) 3 1 1 1 3 2 1 1 3 3 2 2 1 2 1 [ x ] ( t ) ( ) h ⋅ ) ( ) ( ) t 1 ) ) and (7) is exponentially state reconstructible by the state estimator ( ) 5 . 27 541 1 t y t z t z + − = & ( ) 1 2 t z t y t z − = ( ) 9 . 2 2 1 3 3 t z t z t z t z + − = & with the guaranteed exponential decay rate typical state trajectories of (7) and the time response of error states for the systems (7) and (8) are depicted in Figure 1 and Figure 2, respectively. Example 2: Consider the following Pan chaotic system: ( ) 10 10 2 1 1 t x t x t x + − = & ( ) 16 3 1 1 2 t x t x t x t x − = & ( ) 3 ( ) 2 1 t x t x t y + = . By Theorem 1, we conclude that the system 3 2 2 ( t ) ( = + , ce f x z ( ) ( ), 3 3 2 2 1 [ ] ( ) ) ( ) (8a) (8b) (8c) ⋅ − x t z ( ) ( ), ( ) 1 1 [ , ] ( t ( ( ) ( ( ) ) ( ) ( ) ( ) ( ) ( ) ) ( ) t ( ) ( ) ( ), ( ) ( ) 1 e = + ⋅ ce f x t h x t z t 3 3 2 2 1 1 ( ) ( ) η := . The 9 . 2 This implies that ( ) dt [ ] ( ) t c − exp d e t ( ( ) f 2 ( ) t ) ( ) ( ) ) ( ) ) t c − ( ) = ⋅ 3 , f x h x t z t 3 2 2 1 1 ( ⋅ ⋅ ⋅ t c − exp e t 1 ( ) ( ), ( ( ) t exp ) ( ( ) t ) (9a) (9b) + , z ⋅ 0 h x t z 1 3 2 2 ( ) ( ) 3 x ( ) ( ), ( ) ( ), 1 x t x ( ). ( ) t ( ) e 3 ( ⋅ − e 2 ( ) t t ∫ 0 ( ) − 8 − ⇒ exp e t c = + x & t t t (9c) 3 3 2 ( ( ) t ) ( ( ) t ( ) t ) ( ) e 1 ( ) = ⋅ , f x h x z t (9d) 3 2 2 1 1 ( ) ⋅ − exp t c dt − 8 = − = = 10 , 10 , a b c Comparison of (9) with (1), one has , t ( ( ) t ) ( ( ) t ( ) t ) ( ) e 2 3 ∫ 0 + ⋅ , f z h x z t 2 1 3 2 2 α f and (9) is exponentially state reconstructible by the state estimator ( ) 10 201 1 t y t z t z + − = & ( ) 1 2 t z t y t z − = ( ) 3 = β ( 1 x = , 1 , x ) = ( ) x ( exp )dt t c − = − = , f 16 x , ⋅ e − x x x x x f x , exp ( ) 3 t c 1 2 ( ) 3 1 1 3 2 1 1 ( ) ( ) 0 − ⇒ . By Theorem 1, we conclude that the system t e 3 2 2 3 t ( ) ( ) ( ), ∫ 0 ≤ − α t c − exp M t dt (10a) (10b) 1 ( ) 8 − ( ), M M [ ] ( ) = − + − α t c − ≤ 1 exp t ( ) t ( ) ( ), 2 z t 1 − α −  α + = + c c z & t z z t (10c) 1 1 3 3 1  M ( ) t ( ) 0 ( ) t c ≤ + ⋅  c ⇒ 8 exp , e e  η := with the guaranteed exponential decay rate . The 3 3 α + (6) 3 1 ∀ ≥ , 0 t typical state trajectories of (9) and the time response of error states for the systems (9) and (10) are depicted in Figure 3 and Figure 4, respectively. 4.CONCLUSION In this paper, a class of generalized chaotic systems has been considered and the state observation problem of such systems has been investigated. Based on the time-domain approach with differential inequality, a simple state estimator for such generalized chaotic systems has been developed to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be correctly estimated. Finally, several numerical simulations have been proposed to show the effectiveness of the obtained result. ACKNOWLEDGEMENT The author thanks the Ministry of Science and Technology of Republic of China for supporting this work under grant MOST 107-2221-E-214-030. Furthermore, the author is grateful to Chair Professor Jer-Guang Hsieh for the useful comments. REFERENCES [1]Y. Zhang, Z. Liu, H. Wu, S. Chen, and B. Bao, “Two- memristor-based chaotic system and its extreme multistability reconstitution via dimensionality reduction analysis,” Chaos, Solitons & Fractals, vol. 127, pp. 354- 363, 2019. where M ( ( ) t ) ( ( ) t ( ) t ) − ( ) 0 ≥ ⋅ ⋅ , f x h x z e 3 2 2 1 1 1 . α ( ( ) t ) ( ( ) t ( ) t ) ( ) 0 + ⋅ ⋅ , f z h x z e 2 1 3 2 2 1 β Consequently, by (4)-(6), we conclude that ( ) , 0 ≥ ∀ t  = e k 1 max ( ) t ( ) t ( ) t ( ) t c = + + ≤ ⋅ 2 1 2 2 2 3 exp , e t e e e k  − β □ α   M ( ) 0 ( ) 0 ( ) 0 + , , with . This e e 1 3 α +   c 1 completes the proof. 3.NUMERICAL SIMULATIONS Example 1: Consider the following generalized Lorenz chaotic system: ( ) 5 . 27 5 . 27 2 1 1 t x t x t x + − = & ( ) 3 . 19 5 . 3 1 1 1 2 t x t x t x t x − + = & ( ) 9 . 2 2 1 3 3 t x t x t x t x + − = & ( ) 2 1 t x t x t y + = Comparison of (7) with (1), one has 9 . 2 , 5 . 27 , 5 . 27 − = = − = c b a , α ( ) ( ), (7a) (7b) (7c) (7d) ( ) ( ) ( ) ( ), 3 x t ( ) ( ) ( ), ( ) ( ). = β = , 1 @ IJTSRD | Unique Paper ID – IJTSRD29270 | Volume – 3 | Issue – 6 | September - October 2019 Page 1031

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