Quasi sliding mode control of chaos in fractional order duffing system

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Quasi sliding mode control of chaos in fractional order duffing system

  1. 1. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014) February 27 – 28, 2014 Quasi-sliding mode control of chaos in Fractional- order Duffing system Kishore Bingi Electrical Engineering Department National Institute of Technology Calicut, India kishore9860@gmail.com Susy Thomas Professor and Head of the department Electrical Engineering Department National Institute of Technology Calicut, India susy@nitc.com Abstract—In this paper a quasi-sliding mode controller for control of chaos in Fractional-order duffing system is designed. Here, the designed sliding mode control law is to make the Fractional-order duffing system globally asymptotically stable and it also guarantees the system globally asymptotically in the presence of uncertainties and external disturbances. Finally numerical results demonstrate the effectiveness of the proposed controller. Index Terms—Chaos, Fractional-order duffing system, Quasi- sliding mode I. INTRODUCTION Fractional calculus is three centuries old as conventional calculus, but not very popular among engineering and sciences. However, its applications to physics and engineering have just started in recent decades. The beauty of the subject is the solution of the Fractional derivative (or) integral. After the invention of Grunwald-Letnikov derivative, Riemann-Liouville and Caputo definition the applications are rapidly grown up because it was found that many of the systems can be elegantly modeled with the help of Fractional derivative. Chaotic behavior of dynamic systems can be utilized in many real-world applications such as engineering, finance, microbiology, biology, physics, robotics, mathematics, economics, philosophy, meteorology, computer science, and civil engineering and so on. From the investigation of researchers it was found that Functional-order chaotic systems possess memory and display more sophisticated dynamics compared to its Integer-order systems. Recently, the control of chaos in Fractional-order systems has been one of the most interesting topics, and many researchers have made great contributions. For example, in [1], a state feedback control law was proposed for control of chaos in Fractional-order Chen system. In [2], a control algorithm is proposed for Fractional-order Liu system to improve the projective synchronization in the integer order systems. In [3], an active control methodology for controlling chaotic behavior of a Fractional-order version of Rossler system was presented. The main feature of the designed controller is its simplicity for practical implementation. In [4], the Fractional Routh-Hurwitz conditions are used to control chaos in Fractional-order modified autonomous Van der Pol-duffing system to its equilibrium. In [5], a non-linear state feedback control in ODE system to Fractional-order systems is studied. In [6], a classical PID controller is designed for Fractional-order systems with time delays. In this paper, the Fractional-order duffing system is introduced and to control chaos in this system, a Quasi-sliding mode controller is proposed. The proposed control law makes the states of the system asymptotically stable. Simulation results illustrate that the controller can easily eliminate chaos and stabilize the system on the sliding surface. This paper is organized as follows. Section II contains the basic definitions about Fractional Calculus. Section III describes about Fractional-order duffing system. A Quasi- sliding mode controller is proposed to control chaos in Fractional-order duffing system in Section IV. In Section V the concluding comments are given. II. BASIC DEFINITIONS Definition 1: The continuous integral-differential operator is defined as                 t 0 α α α α t 0,dτ 01, 0, dt d D    (1) Definition 2: The Grunwald-Letnikov derivative definition of order  is describes as    jhtf jh tfD j j h t            0 0 1 1 lim)(    (2) Definition 3: Suppose that the unstable Eigen values of a focus points are 2,12,12,1  j . The necessary condition to exhibit double scroll attractor is the Eigen value 2,1 remaining in the unstable region. The condition for commensurate derivative is
  2. 2. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014) February 27 – 28, 2014          i i aq    tan 2 (3) III. FRACTIONAL-ORDER DUFFING SYSTEM Duffing system was introduced by Georg. Duffing with negative linear stiffness, damping and periodic excitation is often written in the form  tFxxxx  cos3   (4) The equation (4) is rewritten as a system of first order autonomous differential equations in the form:          tFtxtxty dt dy ty dt dx   cos 3  (5) From equation (5), the Fractional Duffing system is obtained by replacing conventional derivatives by fractional derivatives.              tFtxtxtytyD tytxD q t q t   cos 32 1  (6) 1q , 2q are Fractional derivatives and F,,,  are system parameters. Here if 21 qq  , then the Fractional-order duffing system is called commensurate Fractional-order duffing system. Otherwise we call the system as non-commensurate Fractional- order duffing system. The Jacobian matrix of the duffing system (5) is          2 3 10 x J The fixed points (equilibrium) of the Integer-order duffing system with parameters 1,3.0,15.0,1,1  F are A  0,0728.1 , B  0,1667.0 and C  0,9061.0 and their corresponding Eigen values are, For A we get 9278.0,0778.12,1  , For B we get j4122.1075.02,1  , and For C we get j4122.1075.02,1  . Here the Eigen value for corresponding equilibrium point A is saddle points which satisfy the stability condition of chaotic behavior. Figure 1 shows the chaotic attractor of Integer-order duffing system with parameters simulation time stsim 200 and with initial condition )13.0,21.0( . We choose 98.021 qq Figure 2 shows the chaotic attractor of commensurate Fractional-order duffing system and Figure 3 shows the time response of the states of the commensurate Fractional-order duffing system with parameters, simulation time 05.0,200  hstsim and with initial condition )13.0,21.0( . For 98.0,95.0 21  qq Figure 4 shows the chaotic attractor of non-commensurate Fractional-order duffing system and Figure 5 shows the time response of the states of the non- commensurate Fractional-order duffing system with parameters, simulation time 05.0,200  hstsim and with initial condition )13.0,21.0( . -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 X YFigure 1: Chaotic attractor of Integer-order duffing system with parameters, simulation time stsim 200 and with initial condition )13.0,21.0( . -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 X Y Figure 2: Chaotic attractor of commensurate Fractional-order duffing system with parameters, 98.0 21 qq simulation time stsim 200 , 05.0h and with initial condition )13.0,21.0( . IV. QUASI SLIDING-MODE CONTROL OF FRACTIONAL-ORDER DUFFING SYSTEM The sliding mode control scheme involves: 1) selection of sliding surface that represents a desirable system dynamic
  3. 3. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014) February 27 – 28, 2014 behavior, 2) finding a switching control law that a sliding mode exists on every point of the sliding surface. The control input )(tu is added to the last state equation in order to control chaos. 0 20 40 60 80 100 120 140 160 180 200 -2 -1 0 1 2 Time X 0 20 40 60 80 100 120 140 160 180 200 -1 -0.5 0 0.5 1 Time Y Figure 3: Time response of the states of the commensurate Fractional- order duffing system with parameters, 98.0 21 qq simulation time stsim 200 , 05.0h and with initial condition )13.0,21.0( . -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 X Y Figure 4: Chaotic attractor of non-commensurate Fractional-order duffing system with parameters, 98.02,95.01  qq simulation time stsim 200 , 05.0h and with initial condition )13.0,21.0( . Therefore the Fractional-order duffing system can be described as follows:               )(cos32 1 tutFtxtxtytyD tytxD q t q t    (7) The sliding mode control )(tu in equation (7) has following structure: )()()( tututu sweq  (8) Where )(tueq , the equivalent control and )(tusw , the switching control of the system. 0 20 40 60 80 100 120 140 160 180 200 -2 -1 0 1 2 Time X 0 20 40 60 80 100 120 140 160 180 200 -1 -0.5 0 0.5 1 Time YFigure 5: Time response of the states of the non-commensurate Fractional- order duffing system with parameters, 98.02,95.01  qq simulation time stsim 200 , 05.0h and with initial condition )13.0,21.0( . Let us choose the sliding surface )(ts as     t q t dttytxtyDts 0 1 )()()()( 2  (9) For sliding mode control method, the sliding surface and its derivative must be zero.   0)()()()( 0 12    t q t dttytxtyDts  (10) 0)()()()( 2  tytxtyDts q t  (11) Therefore         0)()()(cos )()()()( 3 2   tytxtutFtxtxty tytxtyDts eq q t     )cos()(1)()( 3 tFtxtxtueq   (12) The switching control )(tusw is chosen in order to satisfy the sliding condition )()( ssignKtusw  (13) Where K is the gain of the controller and )(ssign is the Signum function. Therefore, the total control law can be defined as )()cos()()1)(()( 3 ssignKtFtxtxtu   (14)
  4. 4. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014) February 27 – 28, 2014 Selecting a Lyapunov function 2 )( 2 1 tsV  Here, the time derivative of the Lyapunov function is given by                 0 ))()()cos()(1)( cos()( )()(cos)( )()( 3 3 3      sK txssignKtFtxtx tFtxtxts txtutFtxtxts tstsV     Therefore, it confirms the existence of sliding mode dynamics and the closed loop system is globally asymptotically stable. Consider the system (7) being perturbed by uncertainties and external disturbance which can be modeled as               )()(),( cos 21 32 1 tutdyxd tFtxtxtytyD tytxD q t q t     (15) Where ),(1 yxd , the uncertainties in the states and )(2 td , the external disturbance are assumed to be bounded i.e. 11 ),( dyxd  and 22 )( dtd  . For the Lyapunov function 2 2 1 sV                 0)( )()( )(),()cos()( 1)(cos)( )()(cos)( 21 21 3 3 3       sddK txssignK tdyxdtFtx txtFtxtxts txtutFtxtxts ssV     Therefore, it confirms the closed loop system in the presence of uncertainties and external disturbance with the sliding mode controller is globally asymptotically stable when 21 ddK  . In order to avoid the chattering effect in the control input )(tu , one of the obvious solutions to make the control function continuous/smooth is to approximate the discontinuous signum function by continuous/smooth sigmoid function.   )( )( ))(( ts ts tssign (16) Here  is a small positive scalar. Therefore, the modified control input )(tu can be defined as     )( )( )cos()()1)(()( 3 ts ts KtFtxtxtu (17) For commensurate Fractional-order Duffing system, the states of the system (7) under the controller (17) and the sliding surface (10) are illustrated in Figure 6 and with uncertainties in the states )sin()cos(45.0),(1 yxyxd  and with external disturbance )sin(5.0)(2 ttd  is illustrated in Figure 7 when gain of the controller K=1.0 and with initial condition )13.0,21.0( . 0 50 100 -0.1 0 0.1 0.2 0.3 State X(t) X Time 0 50 100 -0.1 0 0.1 0.2 0.3 State Y(t) Y Time 0 50 100 -0.5 0 0.5 1 Sliding surface S Time 0 50 100 -2 -1 0 1 Controller U Time Figure 6: Time response of the states of the controlled commensurate Fractional-order duffing system with simulation time s sim t 100 . 0 50 100 -0.1 0 0.1 0.2 0.3 State X(t) X Time 0 50 100 -0.2 -0.1 0 0.1 0.2 State Y(t) Y Time 0 50 100 -0.5 0 0.5 1 Sliding surface S Time 0 50 100 -2 -1 0 1 Controller U Time Figure 7: Time response of the states of the controlled commensurate Fractional-order duffing system in the presence of uncertainties and external disturbance with simulation time s sim t 100 . For non-commensurate Fractional-order Duffing system, the states of the system (7) under the controller (17) and the sliding surface (10) are illustrated in Figure 8 and with uncertainties in the states )sin()cos(45.0),(1 yxyxd  and with external disturbance )sin(5.0)(2 ttd  is illustrated in
  5. 5. Second International Conference on Power, Control and Embedded Systems (ICPCES – 2014) February 27 – 28, 2014 Figure 9 when gain of the controller K=1.0 and with initial condition )13.0,21.0( . From the obtained results, it is clear that the proposed controller is good at controlling the chaos in Fractional order duffing system. 0 50 100 -0.1 0 0.1 0.2 0.3 State X(t) X Time 0 50 100 -0.1 0 0.1 0.2 0.3 State Y(t) Y Time 0 50 100 -0.5 0 0.5 1 Sliding surface S Time 0 50 100 -2 -1 0 1 Controller U Time Figure 8: Time response of the states of the controlled non-commensurate Fractional-order duffing system with simulation time s sim t 100 . 0 50 100 -0.1 0 0.1 0.2 0.3 State X(t) X Time 0 50 100 -0.2 -0.1 0 0.1 0.2 State Y(t) Y Time 0 50 100 -0.5 0 0.5 1 Sliding surface S Time 0 50 100 -2 -1 0 1 Controller U Time Figure 9: Time response of the states of the controlled non-commensurate Fractional-order duffing system in the presence of uncertainties and external disturbance with simulation time s sim t 100 . V. CONCLUSION In this paper, According to Lyapunov stability theorem, the quasi-sliding mode controller is designed to control chaos in Fractional-order duffing system. Based on the sliding mode control method the states of the Fractional-order duffing system have been stabilized. Finally, the numerical results will demonstrate the effectiveness of the proposed controller. REFERENCES [1] Chunguang Li a, Guanrong Chen, “Chaos in the fractional order Chen system and its control”, Chaos, Solutions and Fractals, vol.22, pp. 549–554, 2004. [2] YS Deng, “Fractional order Liu-system synchronization and its application in multimedia security”, Communications, Circuits and Systems (ICCCAS), 2010 International Conference, pp.769- 772, 2010. [3] Alireza K. Golmankhaneh, Roohiyeh Arefi, Dumitru Baleanu, “The Proposed Modified Liu System with Fractional Order”, Advances in Mathematical Physics, Article ID 186037, 2013. [4] Matouk, A.E. “Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol–Duffing circuit” Commun. Nonlinear Science and Numerical Simulation, Vol. 16, pp. 975–986. 2011. [5] Yamin Wang, Xiaozhou Yin, Yong Liu, “Control Chaos in System with Fractional Order, “Journal of Modern Physics, Vol. 3, pp. 496-501, 2012. [6] Hitay Ozbay, Catherine Bonnet, Andre Ricardo Fioravanti, “PID controller design for fractional-order systems with time delays, “Systems & Control Letters, Vol. 61 pp. 18–23, 2012.

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