Exponential State Observer Design for a Class of Uncertain Chaotic and Non Chaotic Systems

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 Exponential State Observer Design Uncertain Chaotic Uncertain Chaotic and Non-Chaotic Systems Exponential State Observer Design for a Class Class of Chaotic Systems Yeong-Jeu Sun Professor, Department of Electrical Engineering, f Electrical Engineering, I-Shou University, Kaohsiung, Taiwan Shou University, Kaohsiung, Taiwan ABSTRACT In this paper, a class of uncertain chaotic and non chaotic systems is firstly introduced and the state observation problem of such systems is explored. Based on the time-domain approach with integra differential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and effectiveness of the obtained results. Key Words: Chaotic system, state observer design, uncertain systems, exponential convergence rate 1.INTRODUCTION In recent years, various chaotic systems widely explored by scholars; see, for example, [1 and the references therein. Frequently, chaotic signals are often the main cause of system instability and violent oscillations. Moreover, chaos often occurs in various engineering systems and applied physics instance, ecological systems, secure communicati and system identification. Based on practical considerations, not all state variables of most systems can be measured. Furthermore, the design of the state estimator is an important work when the sensor fails. Undoubtedly, the state observer design of with both chaos and uncertainties tends to be more difficult than those without chaos and uncertainties. For the foregoing reasons, the observer design of uncertain chaotic systems is really significant essential. In this paper, the observability problem for uncertain nonlinear chaotic or non-chaotic investigated. By using the time-domain approach with integral and differential equalities observer for a class of uncertain will be provided to ensure the global exponential stability of the resulting error system. In guaranteed exponential convergence rate can be precisely calculated. Finally, simulations will be given to demonstrate the effectiveness of the main result. This paper is organized as follows. The problem formulation and main results 2. Several numerical simulations are 3 to illustrate the main result. Finally, conclusion remarks are drawn in Section paper, ℜ denotes the n-dimensional = : denotes the Euclidean norm of the vector x, and a denotes the number a. 2.PROBLEM FORMULATION AND MAI RESULTS In this paper, we explore the following nonlinear systems: ( ) , , 3 2 1 1 1 t x t x t x f t x ∆ = & ( ) , 3 1 2 2 2 t x t x f t ax t x + = & ( ) 1 4 3 3 3 t x f t x f t x + = & ( ) , 0 , 3 ≥ ∀ = t t bx t y ( ) x x x x x 20 10 3 2 1 0 0 0 = where ( ) 3 2 1 : = t x t x t x t x ( ) ℜ ∈ t y is the system output, initial value, indicate the parameters of the system 0 ≠ b . Besides, in order to ensure the existence and uniqueness of the solution, we assume that uniqueness of the solution, we assume that all the In this paper, a class of uncertain chaotic and non- chaotic systems is firstly introduced and the state observation problem of such systems is explored. domain approach with integral and differential equalities, an exponential state observer for a class of uncertain nonlinear systems is established to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential convergence rate can be calculated correctly. Finally, numerical simulations are presented to exhibit the feasibility and equalities, a new state uncertain nonlinear systems ensure the global exponential stability of the resulting error system. In addition, the guaranteed exponential convergence rate can be precisely calculated. Finally, simulations will be given to demonstrate the some some numerical numerical ult. This paper is organized as follows. The problem are presented in Section simulations are given in Section to illustrate the main result. Finally, conclusion remarks are drawn in Section 4. Throughout this Chaotic system, state observer design, uncertain systems, exponential convergence rate dimensional real space, the Euclidean norm of the column absolute value of a real n T⋅ x x x chaotic systems have been ; see, for example, [1-8] Frequently, chaotic signals ROBLEM FORMULATION AND MAIN are often the main cause of system instability and Moreover, chaos often occurs in various engineering systems and applied physics; for instance, ecological systems, secure communication, the following uncertain ( ( ) ( ) ( ( ) ), ( ) ), (1a) (1b) (1c) (1d) Based on practical ( ) ( ) ( ), considerations, not all state variables of most systems Furthermore, the design of the state estimator is an important work when the sensor fails. ( ( ) ( ) ( ) ) ( ) design of systems ]T [ ] [ ( ) x x30 , (1e) T with both chaos and uncertainties tends to be more without chaos and uncertainties. the foregoing reasons, the observer design of [ ] ( ) ( ) ( ) is the state vector, T ∈ ∈ ℜ 3 is the system output, [ ]T significant and is the ∈ b a, 0 < and x x x 10 20 30 1f ∆ is uncertain function, is uncertain function, and parameters of the systems, with n order to ensure the existence and ℜ a lity problem for a class of chaotic systems is domain approach with @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1158

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 () 4 , 3 , 2 , ∈ ∀ ⋅ i fi are smooth the inverse function of system is a special case of systems (1) with f = , 1 f , 1 4 4x f − = , and 1 − = a . It is a well fact that since states are not always available for direct measurement, particularly in the event of sensor failures, states must be estimated. The paper is to search a novel state observer uncertain nonlinear systems (1) such that exponential stability of the resulting error systems can be guaranteed. Before presenting the main result, the state reconstructibility is provided as follows. Definition 1 The uncertain nonlinear systems (1) are state reconstructible if there exist a state observer ( 0 , , = y z z f & and positive numbers κ and ( ) 0 , exp : ≥ ∀ − ≤ − = t t t z t x t e α κ , where ( ) t z represents the reconstructed state of systems (1). In this case, the positive number called the exponential convergence rate. Now, we are in a position to present the main results for the exponential state observer of uncertain (1). Theorem 1. The uncertain systems (1) are exponentially state reconstructible. Moreover, a suitable state observer is given by ( ) 3 4 1      b b ( ) , 3 1 2 2 2 t z t z f t az t z + = & (2b) ( ) 0 , 3 ≥ ∀ = t t y b In this case, the guaranteed exponentia rate is given by a − = : α . Proof. For brevity, let us define the observer error ( ) , : ∈ ∀ − = i t z t x t e i i i and 0 ≥ t . (3) From (1d) and (2c), one has ( ) ( ) y b b . 0 , 0 ≥ ∀ = t (4) International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 { } From (1c), it can be readily obtained that From (1c), it can be readily obtained that ( ) t x f t x t x f 3 3 3 1 4 − = & It results that ( )    b b Thus, one has ( ) ( )    b b 1 3 4 & functions of 1f ∆ and are smooth and ( ) ( ) ( ( ) ) . 4f exists. The chaotic sprott E chaotic sprott E (1) with . It is a well-known , ∆ f = x x 1 2 3 3= 2 1 x ( ( ) t ( ( ) t 1 ) ) 2 − = − 1 x t f x & f x fact that since states are not always available for direct 1 4 3 3 3 (5)    1 ( ) t ( ) vent of sensor he aim of this state observer for the (1) such that the global the resulting error systems can   − = − 1   f y & f y t 4 3  ( ) t ( ) t = − e t x z 1 1 1     1 1 ( ) t −   = − 1    f y & t f y 4 3 nting the main result, the state reconstructibility is provided as follows.     1 ( ) t ( ) t −   , 0 − − 1     f y f y b b  (6) = in view of (5) and (2a). In addition (4), and (6), it is easy to see that ( ) ( ) ( ) ( ) ( ) ( ) t z t x a 2 2 − = ( ) . 0 , 2 ≥ ∀ = t t ae It follows that ( ) , 0 2 2 ≥ ∀ = − t t ae t e & Multiplying with exp ( ) exp 2 2 − − ⋅ t ae at t e & Hence, it can be readily obtained tha ( ) , 0 = dt Integrating the bounds from 0 ( ) exp 2 − ⋅ ∫ dt This implies that ( ) , exp 0 2 2 ⋅ = at e t e Thus, from (4), (6), and (7), we have ( ) ( ) , exp 0 2 ⋅ = at e Consquently, we conclude that t suitable state observer with the guaranted exponential convergence rate a − = α . This completes the proof. . This completes the proof. □ ∀ ≥ , 0 t are exponentially In addition, from (1b), (1c), is easy to see that state reconstructible if there exist a state observer ) and α such that ( ) t ( ) f 2 = − e & t x & z + t & ( ) ( ) ( ) 2 2 2 [ ] ) ] ) ( ( ) ( ) ( ) ( ) ] ( ) ( ) ( ) ( ) t x 3 ) = , ax t x t x , t 2 1 3 [ ] ( x − + az t f z t z t the reconstructed state of 2 2 1 3 [ ( ) = + , ax t f t x , t (1). In this case, the positive number α is called the exponential convergence rate. 2 2 1 3 [ [ ] ( − + az t f x t 2 2 1 ( ) a position to present the main results uncertain systems ( ) . 0 ( ) exponentially state , a suitable state observer is yields ) = at yields − at ( ( ) ( ) ⋅ − ∀ t ≥ exp 0 , 0 . 0     1 1 Hence, it can be readily obtained that ( ) t ( ) ,  t (2a)  − = − 1  z t f y & f y [ ] ( ) ⋅ − exp d e t at ∀ ≥ 2 . 0 t ( ) ( ) ( ( ) ( ) ), 1 . (2c) z t and t , it results [ ] ( ) t t d e t at the guaranteed exponential convergence ∫ . = = ∀ t ≥ 0 , 0 0 dt dt 0 0 ( ) ( ) . (7) observer error ∀ ≥ 0 t ( ) ( ) { } 3 , 2 , 1 , we have ( ) t ( ) ( ) ( ) ∀ = + + 2 1 2 2 2 3 e t e e t e t ( ) t ( ) t 1 = − e t x 1 z ≥ 3 3 3 . 0 t ( ) t = − conclude that the system (2) is a the guaranted exponential y t @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1159

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 3.NUMERICAL SIMULATIONS Consider the uncertain nonlinear system x x c f ⋅ ∆ = ∆ , 1 2 x f = , 4x f − = , 1 − = a , = b By Theorem 1, we conclude that the systems (1) with (8) is exponentially state reconstructible by the state observer ( ) , 4 8 ( ) 1 2 2 t z t z t z + − = & ( ) . 0 , 2 The typical state trajectory of the uncertain (1) with (8) is depicted in Figure 1. Furthermore time response of error states is depicted in Fig From the foregoing simulations results, it is seen that the uncertain systems (1) with (8) are state reconstructible by the state observer o the guaranted exponential convergence rate 4. CONCLUSION In this paper, a class of uncertain chaotic and non chaotic systems has been introduced observation problem of such system studied. Based on the time-domain approach with integral and differential equalities, a observer for a class of uncertain nonlinear been constructed to ensure the global exponential stability of the resulting error system. guaranteed exponential convergence rate precisely calculated. Finally, numerical simulations have been presented to exhibit the effectiveness feasibility of the obtained results. ACKNOWLEDGEMENT The author thanks the Ministry of Science and Technology of Republic of China for supporting this work under grants MOST 106-2221 MOST 106-2813-C-214-025-E, and MOST E-214-030. Besides, the author is grateful to Professor Jer-Guang Hsieh for the useful REFERENCES 1.S. Xiao and Y. Zhao, “A large class of chaotic sensing matrices for compressed sensing Processing, vol. 149, pp. 193-203, 201 2.R. Zhang, D. Zeng, S. Zhong, K. 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4. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 14.A. Sassi, H.S. Ali, M. Zasadzinski, and K. Abderrahim, “Adaptive observer design for a class of descriptor nonlinear systems,” European Journal of Control, vol. 44, pp. 90 Journal of Control, vol. 44, pp. 90-102, 2018. disturbance estimation using Takagi European Journal of Control, vol. 44 2018. ing Takagi-Sugeno,” 44, pp. 114-122, A. Sassi, H.S. Ali, M. Zasadzinski, and K. Abderrahim, “Adaptive observer design for a class of descriptor nonlinear systems,” European 8 6 4 2 x1(t); x2(t); x3(t) 0 -2 -4 x1: the Blue Curve x2: the Green Curve -6 x3: the Red Curve -8 0 50 100 150 t (sec) Figure 1: Typical state trajectory trajectory es of the uncertain nonlinear systems (1) (1) with (8). 2 1.5 e1(t); e2(t); e3(t) 1 e3 0.5 e1=e2=0 0 -0.5 0 5 10 15 t (sec) Figure ure 2: The time response of error states. @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 3 | Issue – 1 | Nov-Dec 2018 Dec 2018 Page: 1161