Chaos Suppression and Practical Stabilization of Uncertain Generalized Duffing Holmes Control Systems with Unknown Actua

1. International Research Research and Development (IJTSRD) International Open Access Journal International Open Access Journal International Journal of Trend in Scientific Scientific (IJTSRD) ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume Chaos Suppression and Practical Stabilization of Uncertain Generalized Duffing-Holmes Control Systems Holmes Control Systems ISSN No: 2456 | www.ijtsrd.com | Volume - 2 | Issue – 1 Chaos Suppression and Practical Stabilization of Uncertain Generalized Duffing with Unknown Actuator Nonlinearity with Unknown Actuator Nonlinearity Chaos Suppression and Practical Stabilization of Uncertain Yeong-Jeu Sun Professor, I-Shou University Shou University, Kaohsiung, Taiwan Professor, Department of Electrical Engineering, ABSTRACT In this paper, the concept of practical stabilization for nonlinear systems is introduced and the practical stabilization of uncertain generalized Duffing control systems with unknown actuator nonlinearity is explored. Based on the time-domain approach differential inequalities, a single control is presented such that the practical stabilization for a class of uncertain generalized Duffing-Holmes systems with unknown actuator nonlinearity can be achieved. Moreover, both of the guaranteed exponential convergence rate and convergence radius can be correctly calculated Finally, simulations are given to demonstrate the feasibility and effectiveness of the obtained results. his paper, the concept of practical stabilization for nonlinear systems is introduced and the practical stabilization of uncertain generalized Duffing-Holmes control systems with unknown actuator nonlinearity is control approach, backstepping control approach, adaptive sliding mode control appro adaptive sliding mode control approach, and others. control approach, backstepping control approach, In this paper, the concept of practical stabilizability for uncertain dynamic systems is introduced and the practical stabilizability of uncertain generalized Holmes control systems with unknown actuator nonlinearity will be studied. Using the time- domain approach with differential inequality, a single control will be designed such that the practical stabilization can be achieved for a class of uncertain Holmes systems with unknown In this paper, the concept of practical stabilizability for uncertain dynamic systems is introduced and the practical stabilizability of uncertain generalized Duffing-Holmes control systems with unknown actuator nonlinearity will be stud domain approach with differential inequality, a single control will be designed such that the practical stabilization can be achieved for a class of uncertain generalized Duffing-Holmes systems with unknown actuator nonlinearity. Not only the convergence radius and guaranteed exponential convergence rate can be arbitrarily pre-specified, but also the unknown actuator nonlinearity and mixed simultaneously overcome by the proposed single control. Several numerical simu provided to illustrate the use of the main results. provided to illustrate the use of the main results. domain approach with differential inequalities, a single control is presented such that the practical stabilization for a class of Holmes systems with unknown actuator nonlinearity can be achieved. Moreover, both of the guaranteed exponential convergence rate and convergence radius can be correctly calculated Finally, simulations are given to demonstrate the feasibility and some some numerical numerical ly the convergence radius and guaranteed exponential convergence rate can be specified, but also the unknown actuator nonlinearity and mixed simultaneously overcome by the proposed single control. Several numerical simulations will also be Keywords: Practical synchronization, chaotic system, uncertain generalized Duffing-Holmes unknown actuator nonlinearity, chaos suppression haotic system, Holmes haos suppression uncertainties uncertainties can can be be systems, systems, 1. INTRODUCTION The rest of paper is organized formulation and main results are presented in Section 2. Numerical simulations are illustrate the effectiveness of the developed results. Finally, some conclusions are drawn in Section Finally, some conclusions are drawn in Section 4. organized as follows. The problem are presented in Section 2. simulations are given in Section 3 to veness of the developed results. Chaotic dynamic systems have been extensively investigated in past decades; see, for instance, [1 and the references therein. Very often, chaos in man dynamic systems is an origin of instability and an origin of the generation of oscillation. Generally speaking, the robust stabilization of uncertain dynamic systems with a single controller is in general not as simple as that with multiple controllers. decades, various methodologies in robust control of chaotic system have been offered, such as control approach, adaptive control approach, variable structure approach, adaptive control approach, variable structure Chaotic dynamic systems have been extensively investigated in past decades; see, for instance, [1-13] and the references therein. Very often, chaos in many dynamic systems is an origin of instability and an origin of the generation of oscillation. Generally speaking, the robust stabilization of uncertain dynamic systems with a single controller is in general not as simple as that with multiple controllers. In the past decades, various methodologies in robust control of chaotic system have been offered, such as control @ IJTSRD | Available Online @ www.ijtsrd.com @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Dec 2017 Page: 1285

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 2. PROBLEM FORMULATION AND MAIN RESULTS   t d   t   t   0        2 t 2 t e s dt e s s dt 0 t     2 t e a dt Nomenclature 0   a   the n-dimensional real space n   , 1    2 t e t . 0 2 a the modulus of a complex number a Consequently, one has I the unit matrix   a  a    t   0 .         2 t s s e , t 0  2 2 A the transport of the matrix A T This completes the proof. □ x the Euclidean norm of the vector   n x In this paper, we consider the following uncertain generalized Duffing-Holmes control systems with uncertain actuator nonlinearity described as the minimum eigenvalue of the matrix P with real eigenvalues minP  ( ) the spectrum of the matrix A  (A ) , (1a) x   x 1 2 the matrix P is a symmetric positive definite matrix  P 0     x 3 1 5 1 x  q x q x q x q x 2 1 q 1 2 w 2 t 3 4  ,    (1b)    cos  f , x 5 1 1 2   u     t . 0 Before presenting the problem formulation, let us introduce a lemma which will be used in the proof of the main theorem.    where input, (unmodeled external excitations, and disturbance), and represents the uncertain actuator nonlinearity. For the existence of the solutions of (1), we assume that the unknown terms   2 x f  and functions. It is well known that the system (1) without any uncertainties (i.e.,  f chaotic behavior for certain values of the parameters [1]. In this paper, the concept of practical stabilization will be introduced. Motivated by time-domain approach with differential inequality, a suitable control strategy will be established. Our goal is to design a single control such that the practical stabilization for a class of uncertain nonlinear systems of (1) can be achieved. is the state vector, represents the mixed uncertainties dynamics, parameter is the T       2 1 x f  x x u 1 2  1,x x 2 mismatchings, Lemma 1   u     t If a continuously differentiable real function satisfies the inequality s   u are all continuous   1,x   t   t ,      s  a 2 s , t 0     u ) displays     x , x 0 1 2 with and , then    a 0 0   a  a    t   0 .         2 t s s e , t 0   2 2 Proof. It is easy to see that   t   t       2 t 2 t e s  e 2 s   Throughout this paper, the following assumption is made: d   t         2 t 2 t e s e a , t . 0 dt It results that   0  (A1) There exist continuous function positive number 1r such that, for all arguments, and f x , x 1 2     ,   f x , x f x , x 1 2 1 2   u .     2 u 1u r @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1286

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Remark 1   t     3 1 5 1 h : q x q x q x q x 1 1 2 2 3 4  1     (4) 2     q cos  w t 1 x Generally nonlinearity satisfies speaking, if the uncertain actuator 5 1 1     2 x f , 2   u   p p ,           2 2 r u u r u , u where is the unique solution to the 1 3   P : 0 1 2   p p 3 2 we often ascribe reduction endurance. Thus, we know see Figure 1. 2r as the gain border and 1r as the gain following Lyapunov equation from (A1);   2r T    1 P      2      1 2 (5) A precise definition of the practical stabilization is given as follows, which will be used in subsequent main results.    1    P 2 I ,      2      1 2 with radius and exponential convergence rate are  and  , respectively. . In this case, the guaranteed convergence   0 Definition 1 The uncertain system (1) is said to realize the practical stabilization, provided that, for any there exist a control : u trajectory satisfies and ,     0 0 Proof. From (1), the state equation can be represented as  ,  such that the state    0 1  x  x   t               t x e , t , 0 2      1 2 2    0           3 1 5 1 f q x q x q x q x for some called the convergence radius and the positive number  is called the exponential convergence rate. Namely, the practical stabilization means that the statesof system (1) can converge to the equilibrium point at 0  x , with any pre-specified convergence radius and exponential convergence rate. Obviously, a control system, having small convergence radius and large exponential convergence rate, has better steady-state response and transient response. . In this case, the positive number  is  q   0 1 1 2 2 3 4 1   x q        q  q 2        cos  w t 1 x 2 2 x 5 1 1 2  q       3 1 Ax B f x x 1 1 2 2 3    , 0   2      5 1 q x cos w t 1 x 4  5 1 1   x     2 2 , t 2     0 1 0 where and . Clearly, one  :  : A B        . This implies 2    1     1 2 2 1 is Hurwitz and     A has the Lyapunov equation of (5) has the unique posive definite solution P. Let    A   I Now we present the main result for the practical stabilization of uncertain systems (1) via time-domain approach with differential inequalities.    t    t   t . (6)  T V x x Px    t  The time derivative of the system (1) with (2)-(6) is given by along the trajectories of V x Theorem 1 The uncertain systems (1) with (A1) realize the practical stabilization under the following control   x    t      T T T V x x  A P  PA  x 2 x PB        3 1  5 1 f q q x q x q x   t    t  1 1  2 1 2 3 4   x , (2)     u r p x p x       3 1 2 2 2     q cos w t x 2 2 5 1 1 2   t  r :   t  ,(3) 2 h   t   P          2 r h p x p x r 1 3 1 2 2 1 min @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1287

4. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470           f  Consequently, we conclude that         T T x 2 P 2 I x 2 x PB     T 3 1 5 1 2 x PB q x q x q x q x 1 1 2 2 3 4    0    P      2 V x   t      2 t 2 min x e 2            q cos w t 1 x 2 2 x   P  5 1 1 2 min    0    P      2 V x      2 t 2 min e       P    t          T T T min 2 x Px 2 x PB 2 h x PB    0    P      2 V x   2     t   min e                   P T T   2 x Px u 2 h x PB  r min     , t . 0     2          2 T   2 V r u 2 h x PB This completes the proof. □ 1 r 3. NUMERICAL SIMULATIONS 2         T T 2 V 2 r r x PB 2 h x PB 1 Consider the following uncertain generalized Duffing- Holmes control systems with unknown actuator nonlinearity described as 2        2 V 2 r r p x p x 1  3 1 2 2   2 h p x p x 3 1 2 2 , (8a) x   x     2 V 1 2 2     2 2 x r p x p  x  h 1 3 1 2 2 r     x 3 1 5 1 x  q x q x q x q x   P        2 r h p p x 2 1 q 1 2 2 3  4  ,    1 3 1 2 2 1 min (8b)   cos  w t f , x 5 1 1 2    2 h p x p x   u 3 1 2 2     t , 0    2 V where     P min              2 2  h p x p x r  P 3 1 2 2 1 min      2 r h p x p x r , , , , , , 1 q  . 0    4 5 q 1 w 1 q 25 q 1 q 0 3 . 0 1 1 3 1 2 2 1 2 3   2 r         2 V     u ,             2 1 3 f x , x a x , b u c u 1 2 1     P             2 r h p x p x r P By the          1 3 1 2 2 1 r min  1 a , 1 1 b , 3 c . 0   ,          2 r h p x p x     1 t 3 1 2 2 1 min   . 0 Our objective, in this example, is to design a feedback control such that the uncertain systems (8) realize the practical stabilization with the exponential convergence rate 3   and the convergence radius condition (A1) is clearly satisfied if we let  . From (5) with 3   , we have inequality   yz . The  2    and . 1 . 0          x xz , x , 0 z 0 y , 0  y z , 2 1 f x 1, x x 1 r 1 It can be deduced that    t          141 26 5 .     V x 2 V x t   P    P , , 5 . 0 (7)   min   P 26 5 . 5 . 5       2 2 , t . 0 min   p , 5 . 5 p 26 . 5 . Thus, from Lemma 1, (6), and (7), it can be readily obtained that 2 3 From (4), it can be readily obtained that     P     V   2   x t V x t   t min    3 . 0    3 1 h : x . 0 25 x x cos t 16 x 8 x       P    P 1 2 1 2       2 t 2 e x 0 min   2 1 x .    2 , t . 0 min @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1288

5. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 (3) with 1 . 0   , one has     015 . 0 5 . 5 5 . 26 2 1    x x t control, given by (2), can now be calculated as From REFERENCES 2 h t   t . Finally, the desired  r : 1)X. Liu, L. Bo, Y. Liu, Y. Zhao, Y. Zhao, J. Zhang, N. Hu, S. Fu, M. Deng, “Detection of micro-cracks using nonlinear lamb waves based on the Generalized Generalized Duffing-Holmes system,” Journal of Sound and Vibration, vol. 405, pp. 175- 186, 2017. h   t    t  . (9)    5 . 5  u r 26 5 . x x 1 2 Consequently, by Theorem 1, we conclude that system (8) with the control (9) is practically stable, with the exponential convergence rate convergence radius 1 . 0   . The typical state trajectories of uncontrolled systems and controlled systems are depicted in Figure 2 and Figure 3, respectively. Besides, the time response of the control signal is depicted in Figure 4. From the foregoing simulations results, it is seen that the uncertain dynamic systems of (8) achieves the practical stabilization under the control law of (9). 2)V. Wiggers and P.C. Rech, “Multistability and organization of periodicity in a Van der Pol-Duffing oscillator,” Chaos, Solitons & Fractals, vol. 103, pp. 632-637, 2017. and the guaranteed   3 3)H. Shi and W. Li, “Research on weak resonance signal detection method based on Duffing oscillator,” Procedia Computer Science, vol. 107, pp. 460-465, 2017. 4)J. Niu, Y. Shen, S. Yang, and S. Li, “Analysis of Duffing oscillator with time-delayed fractional- order PID controller,” International Journal of Non- Linear Mechanics, vol. 92, pp. 66-75, 2017. 4. CONCLUSION 5)J. Kengne, Z.N. Tabekoueng, and H.B. Fotsin, “Coexistence of multiple attractors and crisis route to chaos in autonomous third order Generalized Generalized Duffing-Holmes oscillators,” Communications in Nonlinear Science and Numerical Simulation, vol. 36, pp. 29-44, 2016. In this paper, the concept of practical stabilization for nonlinear systems has been introduced and the practical stabilization of uncertain generalized Duffing-Holmes control systems with unknown actuator nonlinearity has been studied. Based on the time-domain approach with differential inequalities, a single control has been presented such that the practical stabilization for a class of uncertain generalized Duffing-Holmes systems with unknown actuator nonlinearity can be achieved. Not only the unknown actuator nonlinearity and mixed uncertainties can be simultaneously overcome by the proposed control, but also the convergence radius and guaranteed exponential convergence rate can be arbitrarily pre-specified. Finally, some numerical simulations have been offered to show the feasibility and effectiveness of the obtained results. type chaotic 6)J. Li, M. Mao, and Y. Zhang, “Simpler ZD- achieving controller synchronization with parameter perturbation, model uncertainty and external disturbance as compared with other controllers,” Optik-International Journal for Light and Electron Optics, vol. 131, pp. 364- 373, 2017. for chaotic systems 7)H. Tirandaz and A.K. Mollaee, “Modified function projective feedback control for time-delay chaotic Liu system synchronization and its application to secure image transmission,” Optik-International Journal for Light and Electron Optics, vol. 147, pp. 187-196, 2017. 8)H.P. Ren, C. Bai, Q. Kong, M.S. and Baptista, C. Grebogi, “A chaotic spread spectrum system for underwater acoustic communication,” Physica A: Statistical Mechanics and its Applications, vol. 478, pp. 77-92, 2017. ACKNOWLEDGEMENT The author thanks the Ministry of Science and Technology of Republic of China for supporting this work under grants MOST 105-2221-E-214-025, MOST 106-2221-E-214-007, and MOST 106-2813-C-214- 025-E. 9)K.B. Deng, R.X. Wang, C.L. Li, and Y.Q. Fan, “Tracking control for a ten-ring chaotic system with an exponential nonlinear term,” Optik-International Journal for Light and Electron Optics, vol. 130, pp. 576-583, 2017. @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1289

6. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 10)E.M. Bollt, “Regularized forecasting of chaotic dynamical systems,” Chaos, Solitons & Fractals, vol. 94, pp. 8-15, 2017. 1 x1 0.5 11)M.J. Mahmoodabadi, R.A. Maafi, and M. Taherkhorsandi, “An optimal adaptive robust PID controller subject to fuzzy rules and sliding modes for MIMO uncertain chaotic systems,” Applied Soft Computing, vol. 52, pp. 1191-1199, 2017. 0 x1(t); x2(t) -0.5 -1 x2 -1.5 12)W. Jiang, H. Wang, J. Lu, G. Cai, and W. Qin, “Synchronization for chaotic systems via mixed- objective dynamic output feedback robust model predictive control,” Journal of the Franklin Institute, vol. 354, pp. 4838-4860, 2017. -2 0 0.5 1 1.5 2 2.5 3 t (sec) Figure 3: Typical state trajectories of the feedback- controlled system of (8) with (9). 13)G. Xu, J. Xu, C. Xiu, F. Liu, and Y. Zang, “Secure communication based on the synchronous control of hysteretic chaotic neuron,” Neurocomputing, vol. 227, pp. 108-112, 2017.   u   u r1 u Figure 1: Uncertain actuator nonlinearity. 2 x1: the Blue Curve x2: the Green Curve 1.5 1 0.5 x1(t); x2(t) 0 -0.5 -1 -1.5 -2 0 50 100 150 200 250 t (sec) 300 350 400 450 500 Figure 2: Typical state trajectories of the uncontrolled system of (8). @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 1 | Nov-Dec 2017 Page: 1290