Fractional-Order Control for a Novel Chaotic System Without Equilibrium
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摘要
The control problem is discussed for a chaotic system without equilibrium in this paper. On the basis of the linear mathematical model of the two-wheeled self-balancing robot, a novel chaotic system which has no equilibrium is proposed. The basic dynamical properties of this new system are studied via Lyapunov exponents and Poincar′e map. To further demonstrate the physical realizability of the presented novel chaotic system, a chaotic circuit is designed. By using fractional-order operators,a controller is designed based on the state-feedback method.According to the Gronwall inequality, Laplace transform and Mittag-Leffler function, a new control scheme is explored for the whole closed-loop system. Under the developed control scheme,the state variables of the closed-loop system are controlled to stabilize them to zero. Finally, the numerical simulation results of the chaotic system with equilibrium and without equilibrium illustrate the effectiveness of the proposed control scheme.
The control problem is discussed for a chaotic system without equilibrium in this paper. On the basis of the linear mathematical model of the two-wheeled self-balancing robot, a novel chaotic system which has no equilibrium is proposed. The basic dynamical properties of this new system are studied via Lyapunov exponents and Poincar′e map. To further demonstrate the physical realizability of the presented novel chaotic system, a chaotic circuit is designed. By using fractional-order operators,a controller is designed based on the state-feedback method.According to the Gronwall inequality, Laplace transform and Mittag-Leffler function, a new control scheme is explored for the whole closed-loop system. Under the developed control scheme,the state variables of the closed-loop system are controlled to stabilize them to zero. Finally, the numerical simulation results of the chaotic system with equilibrium and without equilibrium illustrate the effectiveness of the proposed control scheme.
The control problem is discussed for a chaotic system without equilibrium in this paper. On the basis of the linear mathematical model of the two-wheeled self-balancing robot, a novel chaotic system which has no equilibrium is proposed. The basic dynamical properties of this new system are studied via Lyapunov exponents and Poincar′e map. To further demonstrate the physical realizability of the presented novel chaotic system, a chaotic circuit is designed. By using fractional-order operators,a controller is designed based on the state-feedback method.According to the Gronwall inequality, Laplace transform and Mittag-Leffler function, a new control scheme is explored for the whole closed-loop system. Under the developed control scheme,the state variables of the closed-loop system are controlled to stabilize them to zero. Finally, the numerical simulation results of the chaotic system with equilibrium and without equilibrium illustrate the effectiveness of the proposed control scheme.
引文
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[2] O.E.R6ssler,"An equation for continuous chaos,"Physics Letters A vol.57,no.5,pp.397-398,1976.
[3]C.X.Liu,T.Liu,L.Liu,and K.Liu,"A new chaotic attractor,"Chaos Solitons&Fractals,vol.22,no.5,pp.1031-1038,2004.
[4]Y.X.Li,W.K.S.Tang,and G.R.Chen,"Generating hyperchaos via state feedback control,"Int.J.Bifurcation and Chaos,vol.15,no.10,pp.3367-3375,2005.
[5]A.M.Chen,J.N.Lu,J.H.Lv,and S.M.Yu,"Generating hyperchaotic Lu attractor via state feedback control,"Physica A:Statistical Mechanics and Its Applications,vol.364,pp.103-1 10,2006.
[6]W.C.Chen,"Nonlinear dynamics and chaos in a fractional-order financial system,"Chaos,Solitons&Fractals,vol. 36,no.5,pp.1305-13 14,2008.
[7]S.Das and. K.Gupta,"A mathematical model on fractional LotkaVolterra equations,"J.Theoretical Biology,vol.277,no.1,pp.1-6,2011.
[8]X.Y.Wang and J.M.Song,"Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control,"Communications in Nonlinear Science and Numerical Simulation,vol.14,no.8,pp.3351-3357,2009.
[9]L.Liu,D.L.Liang,and C.X.Liu,"Nonlinear state-observer control for projective synchronization of a fractional-order hyperchaotic system,"Nonlinear Dynamics,vol.69,no.4,pp.1929-1939,2012.
[10]S.Jafari,J.C.Sprott,and S.M.R.H.Golpayegani,"Elementary quadratic chaotic flows with no equilibria,"Physics Letters A,vol.377,no.9,pp.699-702,2013.
[11]V.T.Pham,C.Volos,S.Jafari,Z.C.Wei,and X.Wang,"Constructing a novel no-equilibrium chaotic system,",Int.J.Bifurcation and Chaos,vol.24,no.5,pp.1450073,2014.
[12]X.Wang and G.R.Chen,"Constructing a chaotic system with any number of equilibria,"Nonlinear Dynamics,vol.71,no.3,pp.429-436,2013.
[13]V.T.Pham,F.Rahma,M.Frasca,and L.Fortuna,"Dynamics and synchronization of a novel hyperchaotic system without equilibrium,"Int.J.Bifurcation and Chaos,vol.24,no.6,pp.1450087,2014.
[14]L.M.Pecora and T-L.Carroll,"Synchronization in chaotic systems,"Physical Review Letters,vol.64,no.8,pp.821-824,1990.
[15]T.L.Carroll and L.M.Pecora,"Synchronizing chaotic circuits,"IEEE Trans.Circuits and Systems,vol.38,no.4,pp.453-456,1991.
[16]G.R.Chen and X.H.Yu,Chaos Control:Theory and Applications,Berlin Heidelberg:Springer,2003.
[17]J.M.Gonzalez-Miranda,Synchronization and Control of Chaos:an Introduction for Scientists and Engineers.London:Imperial College Press,2004.
[18]S.K.Yang,C.L.Chen,and H.T.Yau,"Control of chaos in Lorenz system,"Chaos,Solitons&Fractals,vol.13,no.4,pp.767-780,2002.
[19]J.M.Nazzal and A.N.Natsheh,"Chaos control using sliding-mode theory,"Chaos,Solitons&Fractals,vol.33,no.2,pp.695-702,2007.
[20]N.Cai,Y.W.Jing,and S.Y.Zhang,"Modified projective synchronization of chaotic systems with disturbances via active sliding mode control,"Communications in Nonlinear Science and Numerical Simulation,vol.15,no.6,pp.1613-1620,2010.
[21]M.Sun,L.X.Tian,S.M.Jiang,and J.Xu,"Feedback control and adaptive control of the energy resource chaotic system,"Chaos,Solitons&Fractals,vol.32,no.5,pp.1725-1734,2007.
[22]M.T.Yassen,"Chaos control of chaotic dynamical systems using backstepping design,"Chaos,Solitons&Fractals,vol.27,no.2,pp.537-548,2006.
[23]C.C.Hua,X.P.Guan,and P.Shi,"Adaptive feedback control for a class of chaotic systems,"Chaos,Solitons&Fractals,vol.23,no.3,pp.757-765,2005.
[24]R.Z.Luo and Y.H.Zeng,"The adaptive control of unknown chaoticsystems with external disturbance via a single input,"Nonlinear Dynamics,vol.80,no.1-2,pp.989-998,2015.
[25]M.P.Aghababa and B.Hashtarkhani,"Synchronization of unknown uncertain chaotic systems via adaptive control method,"J.Computational and Nonlinear Dynamics,vol.10,no.5,pp.051004,2015.
[26]S.G.Gao,H.R.Dong,and B.Ning,"Neural adaptive control of uncertain chaotic systems with input and output saturation,"Nonlinear Dynamics,vol.80,no.1-2,pp.375-385,2015.
[27]W.X.Xie,C.Y.Wen,and Z.G.Li,"Impulsive control for the stabilization and synchronization of Lorenz systems,"Physics Letters A,vol.275,no.1-2,pp.67-72,2000.
[28]I.NDoye,H.Voos,and M.Darouach,"Observer-based approach for fractional-order chaotic synchronization and secure communication,"IEEE J.Emerging and Selected Topics in Circuits and Systems,vol.3,no.3,pp.442-450,2013.
[29]L.W.Deng and S.M.Song,"Synchronization of fractional order hyperchaotic systems based on output feedback sliding mode control,"Acta Automatica Sinica,vol.40,no.11,pp.2420-2427,2014.
[30]Y.Xu,H.Wang,Y.G.Li,and B.Pei,"Image encryption based on synchronization of fractional chaotic systems,"Communications in Nonlinear Secience and Numerical Simulation,vol.19,no.10,pp.3735-3744,2014.
[31]A.Oustaloup,J.Sabatier,and P.Lanusse,"From fractal robustness to the CRONE control,"Fractional Calculus and Applied Analysis,vol.2,no.1,pp.1-30,1999.
[32]I.Podlubny,"Fractional-order systems and PIλDμ-controller,"IEEE Trans.Automatic Control,vol.44,no.1,pp.208-214,1999.
[33]T.Z.Li,Y.Wang,and M.K.Luo,"Control of fractional chaotic and hyperchaotic systems based on a fractional order controller,"Chinese Physics B,vol.23,no.8,pp.80501,2014.
[34]N.N.Son and H.P.H.Anh,"Adaptive backstepping self-balancing control of a two-wheel electric scooter,"Int.J.Advanced Robotic Systems,DOI:10.5772/59100,2014.
[35]I.Podlubny,Fractional Differential Equations.London:Academic Press,1999.
[36]Y.Li,Y.Q.Chen,and I.Podlubny,"Mittag-Leffler stability of fractional order nonlinear dynamic systems,"Automatica,vol.45,no.8,pp.1965-1969,2009.
[37]S.J.Sadati,D.Baleanu,A.Ranjbar,R.Ghaderi,and T.Abdeljawad,"Mittag-Leffler stability theorem for fractional nonlinear systems with delay"Abstract and Applied Analysis,vol.2010,pp.Article ID 10865 1,2010.
[38]S.Liu,X.Y.Li,W.Jiang,and X.F.Zhou,"Mittag-Leffter stability of nonlinear fractional neutral singular systems,"Communications in Nonlinear Science and Numerical Simulation,vol.17,no.10,pp.3961-3966,2012.
[39]D.L.Qian,C.P.Li,R.P.Agarwal,and P.J.Y.Wong,"Stability analysis of fractional differential system with Riemann-Liouville derivative,"Mathematical and Computer Modelling,vol.52,no.5-6,pp.862-874,2010.
[40]F.R.Zhang and C.P.Li,"Stability analysis of fractional differential systems with order lying in(1,2),"Advances in Difference Equations,vol.2011,pp.Article ID 213485,2011.
[41]Z.Q.Qin,R.C.Wu,and Y.F.Lu,"Stability analysis of fractionalorder systems with the Riemann-Liouville derivative,"Systems Science&Control Engineering,vol.2,no.1,pp.727-731,2014.
[42]A.A.Kilbas,H.M.Srivastava,and J.J.Trujillo,Theory and Applications of Fractional Differential Equations.New York:Elsevier,2006.
[43]L.P. Chen,Y.G.He,Y.Chai,and R.C.Wu,"New results on stability and stabilization of a class of nonlinear fractional-order systems,"Nonlinear Dynamics,vol.75,no.4,pp.633-641,2014.
[44]H.P. Ye,J.M.Gao,and Y.S.Ding,"A generalized Gronwall inequality and its application to a fractional differential equation,"J.Mathematical Analysis and Applications,vol.328,no.2,pp.1075-1081,2007.
[45]Z.B.Wu and Y.Z.Zou,"Global fractional-order projective dynamical systems,"Communications in Nonlinear Science and Numerical Simulation,vol.19,no.8,pp.2811-2819,2014.
[46]S.M.Yu,Chaotic System.s and Chaotic Circuits:Principle,Design and its Application in Communications.Xi'an:Xidian University Press,2011.
[47]X.J.Wen,Z.M.Wu,and J.G.Lu,"Stability analysis of a class of nonlinear fractional-order systems,"IEEE Trans.Circuits and SystemsⅡ:Express Briefs,vol.55,no.11,pp.1178-1182,2008.
[48]C.C.Hua and X.. Guan,"Adaptive control for chaotic systems,"Chaos,Solitons&Fractals,vol.22,no.1,pp.55-60,2004.
[49]L.P.Liu,Z.Z.Han,and W.L.Li,"Global sliding mode control and application in chaotic systems,"Nonlinear Dynamics,vol.56,no.1-2,pp.193-198,2009.
[50]Q.Jia,"Hyperchaos generated from the Lorenz chaotic system and its control,"Physics Letters A,vol.366,no.3,pp.217-222,2007.
[51]I.Petras,Fractional-Order Nonlinear Systems:Modeling,Analysis and Simulation.Berlin Heidelberg:Springer-Verlag,2011.
[2] O.E.R6ssler,"An equation for continuous chaos,"Physics Letters A vol.57,no.5,pp.397-398,1976.
[3]C.X.Liu,T.Liu,L.Liu,and K.Liu,"A new chaotic attractor,"Chaos Solitons&Fractals,vol.22,no.5,pp.1031-1038,2004.
[4]Y.X.Li,W.K.S.Tang,and G.R.Chen,"Generating hyperchaos via state feedback control,"Int.J.Bifurcation and Chaos,vol.15,no.10,pp.3367-3375,2005.
[5]A.M.Chen,J.N.Lu,J.H.Lv,and S.M.Yu,"Generating hyperchaotic Lu attractor via state feedback control,"Physica A:Statistical Mechanics and Its Applications,vol.364,pp.103-1 10,2006.
[6]W.C.Chen,"Nonlinear dynamics and chaos in a fractional-order financial system,"Chaos,Solitons&Fractals,vol. 36,no.5,pp.1305-13 14,2008.
[7]S.Das and. K.Gupta,"A mathematical model on fractional LotkaVolterra equations,"J.Theoretical Biology,vol.277,no.1,pp.1-6,2011.
[8]X.Y.Wang and J.M.Song,"Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control,"Communications in Nonlinear Science and Numerical Simulation,vol.14,no.8,pp.3351-3357,2009.
[9]L.Liu,D.L.Liang,and C.X.Liu,"Nonlinear state-observer control for projective synchronization of a fractional-order hyperchaotic system,"Nonlinear Dynamics,vol.69,no.4,pp.1929-1939,2012.
[10]S.Jafari,J.C.Sprott,and S.M.R.H.Golpayegani,"Elementary quadratic chaotic flows with no equilibria,"Physics Letters A,vol.377,no.9,pp.699-702,2013.
[11]V.T.Pham,C.Volos,S.Jafari,Z.C.Wei,and X.Wang,"Constructing a novel no-equilibrium chaotic system,",Int.J.Bifurcation and Chaos,vol.24,no.5,pp.1450073,2014.
[12]X.Wang and G.R.Chen,"Constructing a chaotic system with any number of equilibria,"Nonlinear Dynamics,vol.71,no.3,pp.429-436,2013.
[13]V.T.Pham,F.Rahma,M.Frasca,and L.Fortuna,"Dynamics and synchronization of a novel hyperchaotic system without equilibrium,"Int.J.Bifurcation and Chaos,vol.24,no.6,pp.1450087,2014.
[14]L.M.Pecora and T-L.Carroll,"Synchronization in chaotic systems,"Physical Review Letters,vol.64,no.8,pp.821-824,1990.
[15]T.L.Carroll and L.M.Pecora,"Synchronizing chaotic circuits,"IEEE Trans.Circuits and Systems,vol.38,no.4,pp.453-456,1991.
[16]G.R.Chen and X.H.Yu,Chaos Control:Theory and Applications,Berlin Heidelberg:Springer,2003.
[17]J.M.Gonzalez-Miranda,Synchronization and Control of Chaos:an Introduction for Scientists and Engineers.London:Imperial College Press,2004.
[18]S.K.Yang,C.L.Chen,and H.T.Yau,"Control of chaos in Lorenz system,"Chaos,Solitons&Fractals,vol.13,no.4,pp.767-780,2002.
[19]J.M.Nazzal and A.N.Natsheh,"Chaos control using sliding-mode theory,"Chaos,Solitons&Fractals,vol.33,no.2,pp.695-702,2007.
[20]N.Cai,Y.W.Jing,and S.Y.Zhang,"Modified projective synchronization of chaotic systems with disturbances via active sliding mode control,"Communications in Nonlinear Science and Numerical Simulation,vol.15,no.6,pp.1613-1620,2010.
[21]M.Sun,L.X.Tian,S.M.Jiang,and J.Xu,"Feedback control and adaptive control of the energy resource chaotic system,"Chaos,Solitons&Fractals,vol.32,no.5,pp.1725-1734,2007.
[22]M.T.Yassen,"Chaos control of chaotic dynamical systems using backstepping design,"Chaos,Solitons&Fractals,vol.27,no.2,pp.537-548,2006.
[23]C.C.Hua,X.P.Guan,and P.Shi,"Adaptive feedback control for a class of chaotic systems,"Chaos,Solitons&Fractals,vol.23,no.3,pp.757-765,2005.
[24]R.Z.Luo and Y.H.Zeng,"The adaptive control of unknown chaoticsystems with external disturbance via a single input,"Nonlinear Dynamics,vol.80,no.1-2,pp.989-998,2015.
[25]M.P.Aghababa and B.Hashtarkhani,"Synchronization of unknown uncertain chaotic systems via adaptive control method,"J.Computational and Nonlinear Dynamics,vol.10,no.5,pp.051004,2015.
[26]S.G.Gao,H.R.Dong,and B.Ning,"Neural adaptive control of uncertain chaotic systems with input and output saturation,"Nonlinear Dynamics,vol.80,no.1-2,pp.375-385,2015.
[27]W.X.Xie,C.Y.Wen,and Z.G.Li,"Impulsive control for the stabilization and synchronization of Lorenz systems,"Physics Letters A,vol.275,no.1-2,pp.67-72,2000.
[28]I.NDoye,H.Voos,and M.Darouach,"Observer-based approach for fractional-order chaotic synchronization and secure communication,"IEEE J.Emerging and Selected Topics in Circuits and Systems,vol.3,no.3,pp.442-450,2013.
[29]L.W.Deng and S.M.Song,"Synchronization of fractional order hyperchaotic systems based on output feedback sliding mode control,"Acta Automatica Sinica,vol.40,no.11,pp.2420-2427,2014.
[30]Y.Xu,H.Wang,Y.G.Li,and B.Pei,"Image encryption based on synchronization of fractional chaotic systems,"Communications in Nonlinear Secience and Numerical Simulation,vol.19,no.10,pp.3735-3744,2014.
[31]A.Oustaloup,J.Sabatier,and P.Lanusse,"From fractal robustness to the CRONE control,"Fractional Calculus and Applied Analysis,vol.2,no.1,pp.1-30,1999.
[32]I.Podlubny,"Fractional-order systems and PIλDμ-controller,"IEEE Trans.Automatic Control,vol.44,no.1,pp.208-214,1999.
[33]T.Z.Li,Y.Wang,and M.K.Luo,"Control of fractional chaotic and hyperchaotic systems based on a fractional order controller,"Chinese Physics B,vol.23,no.8,pp.80501,2014.
[34]N.N.Son and H.P.H.Anh,"Adaptive backstepping self-balancing control of a two-wheel electric scooter,"Int.J.Advanced Robotic Systems,DOI:10.5772/59100,2014.
[35]I.Podlubny,Fractional Differential Equations.London:Academic Press,1999.
[36]Y.Li,Y.Q.Chen,and I.Podlubny,"Mittag-Leffler stability of fractional order nonlinear dynamic systems,"Automatica,vol.45,no.8,pp.1965-1969,2009.
[37]S.J.Sadati,D.Baleanu,A.Ranjbar,R.Ghaderi,and T.Abdeljawad,"Mittag-Leffler stability theorem for fractional nonlinear systems with delay"Abstract and Applied Analysis,vol.2010,pp.Article ID 10865 1,2010.
[38]S.Liu,X.Y.Li,W.Jiang,and X.F.Zhou,"Mittag-Leffter stability of nonlinear fractional neutral singular systems,"Communications in Nonlinear Science and Numerical Simulation,vol.17,no.10,pp.3961-3966,2012.
[39]D.L.Qian,C.P.Li,R.P.Agarwal,and P.J.Y.Wong,"Stability analysis of fractional differential system with Riemann-Liouville derivative,"Mathematical and Computer Modelling,vol.52,no.5-6,pp.862-874,2010.
[40]F.R.Zhang and C.P.Li,"Stability analysis of fractional differential systems with order lying in(1,2),"Advances in Difference Equations,vol.2011,pp.Article ID 213485,2011.
[41]Z.Q.Qin,R.C.Wu,and Y.F.Lu,"Stability analysis of fractionalorder systems with the Riemann-Liouville derivative,"Systems Science&Control Engineering,vol.2,no.1,pp.727-731,2014.
[42]A.A.Kilbas,H.M.Srivastava,and J.J.Trujillo,Theory and Applications of Fractional Differential Equations.New York:Elsevier,2006.
[43]L.P. Chen,Y.G.He,Y.Chai,and R.C.Wu,"New results on stability and stabilization of a class of nonlinear fractional-order systems,"Nonlinear Dynamics,vol.75,no.4,pp.633-641,2014.
[44]H.P. Ye,J.M.Gao,and Y.S.Ding,"A generalized Gronwall inequality and its application to a fractional differential equation,"J.Mathematical Analysis and Applications,vol.328,no.2,pp.1075-1081,2007.
[45]Z.B.Wu and Y.Z.Zou,"Global fractional-order projective dynamical systems,"Communications in Nonlinear Science and Numerical Simulation,vol.19,no.8,pp.2811-2819,2014.
[46]S.M.Yu,Chaotic System.s and Chaotic Circuits:Principle,Design and its Application in Communications.Xi'an:Xidian University Press,2011.
[47]X.J.Wen,Z.M.Wu,and J.G.Lu,"Stability analysis of a class of nonlinear fractional-order systems,"IEEE Trans.Circuits and SystemsⅡ:Express Briefs,vol.55,no.11,pp.1178-1182,2008.
[48]C.C.Hua and X.. Guan,"Adaptive control for chaotic systems,"Chaos,Solitons&Fractals,vol.22,no.1,pp.55-60,2004.
[49]L.P.Liu,Z.Z.Han,and W.L.Li,"Global sliding mode control and application in chaotic systems,"Nonlinear Dynamics,vol.56,no.1-2,pp.193-198,2009.
[50]Q.Jia,"Hyperchaos generated from the Lorenz chaotic system and its control,"Physics Letters A,vol.366,no.3,pp.217-222,2007.
[51]I.Petras,Fractional-Order Nonlinear Systems:Modeling,Analysis and Simulation.Berlin Heidelberg:Springer-Verlag,2011.