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Abstract：We investigate the stabilization of a memristorbased chaotic system.Based on the system，a TS model is established and a switching controller is designed.Then，by Lyapunov stability theory，one can acquire a criterion to guarantee that the conclusion holds.Finally，numerical results are demonstrated to verify the effectiveness of our method.
Key words：chaotic system； memristor； fuzzy modeling； switching control
CLC number： TM 132 Document code： A Article ID： 10005137（2016）01000106
1 Introduction
During the recent decades，memristor has been focused on by many researchers.It was postulated as the missing fourth passive circuit element by Chua in 1971[1].In the following almost 40 years，scientists had aimed at inventing a practical device.In 2008，a large development that a passive two terminal physical implementation was immediately related to memristor theory was achieved by HewlettPackard Labs[2].Many researchers show great interest in its physical properties and applications on computer[3-17].For instance，Saeed Afshar presented a memristorbased neuromorphic competitive （mNCC） circuit which utilized a single sensor and could control the output of N actuators delivering optimal scalable performance，and immunity from device variation and environmental noise[3].Adam Rak and Gyorgy Cserey introduced a new simulation program with integrated circuit emphasis macromodel of the recently physically implemented memristor[4].Sung Hyun Jo experimentally displayed a nanoscale siliconbased memristor device and showed a hybrid system composed of complementary metaloxide semiconductor neurons[5].
In addition，numerous authors have investigated the dynamical properties and synchronization of memristorbased systems[18-23].Wu analyzed the dynamic behaviors for a class of memristorbased Hopfield networks and supplied several sufficient conditions to guarantee a novel memristive neural network for realizing winnertakeall behavior[18-19].Some others considered the problem of fuzzy modeling and impulsive control chaotic system and memristorbased chaotic system[24-28]and Tanaka K provided approaches to fuzzy modeling and design[29].In Ref.[28]，Huang researched into the problem of intermittent control of a memristorbased Chua′s oscillator and presented the oscillator as the TS fuzzy model system.Nevertheless，in this paper we will design a switching controller which is different from the controller in letter[28]and construct a Lyapunov function for stabilizing a fourthorder memristorbased chaotic system[30].We also utilize fuzzy modeling and analysis for this system. The rest of this paper is presented as follows.In Section 2，a memristorbased system is introduced and a TS fuzzy model is built according to the system.Furthermore，in Section 3，a switching controller is given to stabilize the memristorbased chaotic system and a criterion obtained by Lyapunov function theory is provided.Numerical simulation examples are given to demonstrate the scheme in Section 4. And finally，conclusions are drawn in Section 5.
5 Conclusions
In this paper，the control problem and fuzzy modeling of memristorbased chaotic system are considered.Furthermore，a switching controller is designed to stabilize the original system.Finally，numerical results are acquired to illustrate the effectiveness of our method.
References：
[1]Chua L.Memristorthe missing circuit element[J].IEEE Trans Circuit Theory，1971，18（5）：507-519.
[2]Strukov DB，Snider GS，Stewart GR，et al.The missing memristor found[J].Nature，2008，453（7191）：80-83.
[3]Afshar S，Kavehei O，Van Schaik A，et al.Emergence of competitive control in a memristorbased neuromorphic circuit[J].International Joint Conference on，Neural Networks，2012，9（10-15）：1-8.
[4]Rak A，Cserey G.Macromodeling of the memristor in SPICE[J].Computeraided design of integrated circuits and systems，IEEE Transactions on，2010，29（4）：632-636.
[5]Jo S H，Chang T，Ebong I，et al.Nanoscale memristor device as synapse in neuromorphic systems[J].Nano letters，2010，10（4）：1297-1301.
[6]Indiveri G，LinaresBarranco B，Legenstein R，et al.Integration of nanoscale memristor synapses in neuromorphic computing architectures[J].arXiv preprint，2013，arXiv：1302.7007.
[7]Atkin K.An introduction to the memristor[J].Physics Education，2013，48（3）：317.
[8]Di Ventra M，Pershin Y V，Chua L O.Circuit elements with memory：memristors，memcapacitors，and meminductors[J].Proceedings of the IEEE，2009，97（10）：1717-1724.
[9]FaghihNassiri J，Maddy N，Buckland L，et al.Development system for memristor circuits[C].Nanotechnology （IEEENANO），2013 13th IEEE Conference on，Beijing：IEEE，2013：95-98.
[10]Younis A，Chu D，Lin X，et al.HighPerformance nanocomposite based memristor with controlled quantum dots as charge traps[J].ACS applied materials and interfaces，2013，5（6）：2249-2254.
[11]Soltiz M，Kudithipudi D，Merkel C，et al.Memristorbased neural logic blocks for nonlinearly separable functions[J].IEEE Transactions on Computers，2013，62（8）：1597-1606.
[12]Fouda M E，Radwan A G.Memristorbased voltagecontrolled relaxation oscillators[J].International Journal of Circuit Theory and Applications，2014，42（10）：1092-1102. [13]Valov I，Linn E，Tappertzhofen S，et al.Nanobatteries in redoxbased resistive switches require extension of memristor theory[J].Nature communications，2013，4：1771.
[14]Di Ventra M，Pershin Y V.On the physical properties of memristive，memcapacitive and meminductive systems[J].Nanotechnology，2013，24（25）：406-417.
[15]Biolek Z，Biolek D，Biolkova V.SPICE model of memristor with nonlinear dopant drift[J].Radioengineering，2009，18（2）：210-214.
[16]Joglekar Y N，Wolf S J.The elusive memristor：properties of basic electrical circuits[J].European Journal of Physics，2009，30（4）：661-675.
[17]Kavehei O，Iqbal A，Kim Y S，et al.The fourth element：characteristics，modelling and electromagnetic theory of the memristor[J].Proceedings of the Royal Society A：Mathematical，Physical and Engineering Science，2010，466（2120）：2175-2202.
[18]Wu A，Zhang J，Zeng Z.Dynamic behaviors of a class of memristorbased Hopfield networks[J].Physics Letters A，2011，375（15）：1661-1665.
[19]Wu A，Zeng Z，Chen J.Analysis and design of winnertakeall behavior based on a novel memristive neural network[J].Neural Computing and Applications，2014，24（7-8）：1595-1600.
[20]Elsayed A M A，Elsaid A，Nour H M，et al.Dynamical behavior，chaos control and synchronization of a memristorbased ADVP circuit[J].Communications in Nonlinear Science and Numerical Simulation，2013，18（1）：148-170.
[21]Ho Y，Huang G M，Li P.Dynamical properties and design analysis for nonvolatile memristor memories[J].Circuits and Systems I：Regular Papers，IEEE Transactions on，2011，58（4）：724-736.
[22]Riaza R.Dynamical properties of electrical circuits with fully nonlinear memristors[J].Nonlinear Analysis：Real World Applications，2011，12（6）：3674-3686.
[23]Tenreiro Machado J.Fractional generalization of memristor and higher order elements[J].Communications in Nonlinear Science and Numerical Simulation，2013，18（2）：264-275.
[24]Zhong Q，Bao J，Yu Y，et al.Impulsive control for TCS fuzzy modelbased chaotic systems[J].Mathematics and Computers in Simulation，2008，79（3）：409-415.
[25]Zhong Q，Bao J，Yu Y，et al.Exponential stabilization for discrete TakagiCSugeno fuzzy systems via impulsive control[J].Chaos，Solitons and Fractals，2009，41（4）：2123-2127.
[26]Zhong Q S，Yu Y B，Yu J B.Fuzzy modeling and impulsive control of a memristorbased chaotic system[J].Chinese Physics Letters，2010，27（2）：22-24.
[27]Wen S，Zeng Z，Huang T.Fuzzy modeling and synchronization of different memristorbased chaotic circuits[J].Physics Letters A，2013，377（s34-36）：2016-2021.
[28]Huang J，Li C，He X.Stabilization of a memristorbased chaotic system by intermittent control and fuzzy processing[J].International Journal of Control，Automation and Systems，2013，11（3）：643-647.
[29]Tanaka K，Wang HO.Fuzzy control systems design and analysis：a linear matrix inequality approach[M].New York：Wiley，2004.
[30]Itoh M，Chua L O.Memristor oscillators[J].International Journal of Bifurcation and Chaos，2008，18（11）：3183-3206.
摘要：研究了以忆阻器为基础的混沌系统的稳定性.基于该系统建立了一个TS模型并设计了一个开关控制器.通过李雅普诺夫稳定性理论，取得了一个以保证结论成立的准则.最后用数值结果验证了本研究方法的有效性.
关键词：混沌系统； 忆阻器； 模糊建模； 切换控制
（责任编辑：冯珍珍，包震宇）
Key words：chaotic system； memristor； fuzzy modeling； switching control
CLC number： TM 132 Document code： A Article ID： 10005137（2016）01000106
1 Introduction
During the recent decades，memristor has been focused on by many researchers.It was postulated as the missing fourth passive circuit element by Chua in 1971[1].In the following almost 40 years，scientists had aimed at inventing a practical device.In 2008，a large development that a passive two terminal physical implementation was immediately related to memristor theory was achieved by HewlettPackard Labs[2].Many researchers show great interest in its physical properties and applications on computer[3-17].For instance，Saeed Afshar presented a memristorbased neuromorphic competitive （mNCC） circuit which utilized a single sensor and could control the output of N actuators delivering optimal scalable performance，and immunity from device variation and environmental noise[3].Adam Rak and Gyorgy Cserey introduced a new simulation program with integrated circuit emphasis macromodel of the recently physically implemented memristor[4].Sung Hyun Jo experimentally displayed a nanoscale siliconbased memristor device and showed a hybrid system composed of complementary metaloxide semiconductor neurons[5].
In addition，numerous authors have investigated the dynamical properties and synchronization of memristorbased systems[18-23].Wu analyzed the dynamic behaviors for a class of memristorbased Hopfield networks and supplied several sufficient conditions to guarantee a novel memristive neural network for realizing winnertakeall behavior[18-19].Some others considered the problem of fuzzy modeling and impulsive control chaotic system and memristorbased chaotic system[24-28]and Tanaka K provided approaches to fuzzy modeling and design[29].In Ref.[28]，Huang researched into the problem of intermittent control of a memristorbased Chua′s oscillator and presented the oscillator as the TS fuzzy model system.Nevertheless，in this paper we will design a switching controller which is different from the controller in letter[28]and construct a Lyapunov function for stabilizing a fourthorder memristorbased chaotic system[30].We also utilize fuzzy modeling and analysis for this system. The rest of this paper is presented as follows.In Section 2，a memristorbased system is introduced and a TS fuzzy model is built according to the system.Furthermore，in Section 3，a switching controller is given to stabilize the memristorbased chaotic system and a criterion obtained by Lyapunov function theory is provided.Numerical simulation examples are given to demonstrate the scheme in Section 4. And finally，conclusions are drawn in Section 5.
5 Conclusions
In this paper，the control problem and fuzzy modeling of memristorbased chaotic system are considered.Furthermore，a switching controller is designed to stabilize the original system.Finally，numerical results are acquired to illustrate the effectiveness of our method.
References：
[1]Chua L.Memristorthe missing circuit element[J].IEEE Trans Circuit Theory，1971，18（5）：507-519.
[2]Strukov DB，Snider GS，Stewart GR，et al.The missing memristor found[J].Nature，2008，453（7191）：80-83.
[3]Afshar S，Kavehei O，Van Schaik A，et al.Emergence of competitive control in a memristorbased neuromorphic circuit[J].International Joint Conference on，Neural Networks，2012，9（10-15）：1-8.
[4]Rak A，Cserey G.Macromodeling of the memristor in SPICE[J].Computeraided design of integrated circuits and systems，IEEE Transactions on，2010，29（4）：632-636.
[5]Jo S H，Chang T，Ebong I，et al.Nanoscale memristor device as synapse in neuromorphic systems[J].Nano letters，2010，10（4）：1297-1301.
[6]Indiveri G，LinaresBarranco B，Legenstein R，et al.Integration of nanoscale memristor synapses in neuromorphic computing architectures[J].arXiv preprint，2013，arXiv：1302.7007.
[7]Atkin K.An introduction to the memristor[J].Physics Education，2013，48（3）：317.
[8]Di Ventra M，Pershin Y V，Chua L O.Circuit elements with memory：memristors，memcapacitors，and meminductors[J].Proceedings of the IEEE，2009，97（10）：1717-1724.
[9]FaghihNassiri J，Maddy N，Buckland L，et al.Development system for memristor circuits[C].Nanotechnology （IEEENANO），2013 13th IEEE Conference on，Beijing：IEEE，2013：95-98.
[10]Younis A，Chu D，Lin X，et al.HighPerformance nanocomposite based memristor with controlled quantum dots as charge traps[J].ACS applied materials and interfaces，2013，5（6）：2249-2254.
[11]Soltiz M，Kudithipudi D，Merkel C，et al.Memristorbased neural logic blocks for nonlinearly separable functions[J].IEEE Transactions on Computers，2013，62（8）：1597-1606.
[12]Fouda M E，Radwan A G.Memristorbased voltagecontrolled relaxation oscillators[J].International Journal of Circuit Theory and Applications，2014，42（10）：1092-1102. [13]Valov I，Linn E，Tappertzhofen S，et al.Nanobatteries in redoxbased resistive switches require extension of memristor theory[J].Nature communications，2013，4：1771.
[14]Di Ventra M，Pershin Y V.On the physical properties of memristive，memcapacitive and meminductive systems[J].Nanotechnology，2013，24（25）：406-417.
[15]Biolek Z，Biolek D，Biolkova V.SPICE model of memristor with nonlinear dopant drift[J].Radioengineering，2009，18（2）：210-214.
[16]Joglekar Y N，Wolf S J.The elusive memristor：properties of basic electrical circuits[J].European Journal of Physics，2009，30（4）：661-675.
[17]Kavehei O，Iqbal A，Kim Y S，et al.The fourth element：characteristics，modelling and electromagnetic theory of the memristor[J].Proceedings of the Royal Society A：Mathematical，Physical and Engineering Science，2010，466（2120）：2175-2202.
[18]Wu A，Zhang J，Zeng Z.Dynamic behaviors of a class of memristorbased Hopfield networks[J].Physics Letters A，2011，375（15）：1661-1665.
[19]Wu A，Zeng Z，Chen J.Analysis and design of winnertakeall behavior based on a novel memristive neural network[J].Neural Computing and Applications，2014，24（7-8）：1595-1600.
[20]Elsayed A M A，Elsaid A，Nour H M，et al.Dynamical behavior，chaos control and synchronization of a memristorbased ADVP circuit[J].Communications in Nonlinear Science and Numerical Simulation，2013，18（1）：148-170.
[21]Ho Y，Huang G M，Li P.Dynamical properties and design analysis for nonvolatile memristor memories[J].Circuits and Systems I：Regular Papers，IEEE Transactions on，2011，58（4）：724-736.
[22]Riaza R.Dynamical properties of electrical circuits with fully nonlinear memristors[J].Nonlinear Analysis：Real World Applications，2011，12（6）：3674-3686.
[23]Tenreiro Machado J.Fractional generalization of memristor and higher order elements[J].Communications in Nonlinear Science and Numerical Simulation，2013，18（2）：264-275.
[24]Zhong Q，Bao J，Yu Y，et al.Impulsive control for TCS fuzzy modelbased chaotic systems[J].Mathematics and Computers in Simulation，2008，79（3）：409-415.
[25]Zhong Q，Bao J，Yu Y，et al.Exponential stabilization for discrete TakagiCSugeno fuzzy systems via impulsive control[J].Chaos，Solitons and Fractals，2009，41（4）：2123-2127.
[26]Zhong Q S，Yu Y B，Yu J B.Fuzzy modeling and impulsive control of a memristorbased chaotic system[J].Chinese Physics Letters，2010，27（2）：22-24.
[27]Wen S，Zeng Z，Huang T.Fuzzy modeling and synchronization of different memristorbased chaotic circuits[J].Physics Letters A，2013，377（s34-36）：2016-2021.
[28]Huang J，Li C，He X.Stabilization of a memristorbased chaotic system by intermittent control and fuzzy processing[J].International Journal of Control，Automation and Systems，2013，11（3）：643-647.
[29]Tanaka K，Wang HO.Fuzzy control systems design and analysis：a linear matrix inequality approach[M].New York：Wiley，2004.
[30]Itoh M，Chua L O.Memristor oscillators[J].International Journal of Bifurcation and Chaos，2008，18（11）：3183-3206.
摘要：研究了以忆阻器为基础的混沌系统的稳定性.基于该系统建立了一个TS模型并设计了一个开关控制器.通过李雅普诺夫稳定性理论，取得了一个以保证结论成立的准则.最后用数值结果验证了本研究方法的有效性.
关键词：混沌系统； 忆阻器； 模糊建模； 切换控制
（责任编辑：冯珍珍，包震宇）