In this paper we report the control and synchronisation of chaos in a memristive Murali–Lakshmanan–Chua (MLC) circuit. This circuit, introduced by the present authors in 2013, is basically a non-smooth system having two discontinuity boundaries by virtue of it having a flux-controlled active memristor as its nonlinear element. While the control of chaos has been effected using state feedback techniques, the concept of adaptive synchronisation and observer-based approaches have been used to effect synchronisation of chaos. Both these techniques are based on state space representation theory which is well known in the field of control engineering. As in our earlier works on this circuit, we have derived the Poincaré discontinuity mapping (PDM) and zero time discontinuity mapping (ZDM) corrections, both of which are essential for realising the true dynamics of non-smooth systems. Further, we have constructed the observer- and controller-based canonical forms of the state-space representations, have set up the Luenberger observer, derived the controller gain vector to implement state feedback control and calculated the gain matrices for switch feed back and finally performed parameter estimation for effecting observer-based adaptive synchronisation. Our results obtained by numerical simulation include time plots, phase portraits, estimation of the parameters and convergence of error graphs and phase plots showing complete synchronisation.

中文翻译：

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忆阻 Murali-Lakshmanan-Chua 电路中混沌的状态反馈控制和基于观测器的自适应同步

在本文中，我们报告了忆阻 Murali-Lakshmanan-Chua (MLC) 电路中混沌的控制和同步。该电路由本作者于 2013 年引入，基本上是一个具有两个不连续边界的非光滑系统，因为它具有磁通控制的有源忆阻器作为其非线性元件。虽然已经使用状态反馈技术实现了对混沌的控制，但是自适应同步的概念和基于观察者的方法已经被用于实现混沌的同步。这两种技术都基于控制工程领域众所周知的状态空间表示理论。正如我们之前在该电路上的工作一样，我们已经导出了庞加莱不连续映射 (PDM) 和零时间不连续映射 (ZDM) 校正，这两者对于实现非光滑系统的真实动力学都是必不可少的。此外，我们构建了状态空间表示的基于观测器和控制器的规范形式，设置了 Luenberger 观测器，导出控制器增益向量以实现状态反馈控制并计算开关反馈的增益矩阵，最后执行用于实现基于观测器的自适应同步的参数估计。我们通过数值模拟获得的结果包括时间图、相图、参数估计和误差图的收敛以及显示完全同步的相位图。导出控制器增益向量以实现状态反馈控制并计算开关反馈的增益矩阵，最后执行参数估计以实现基于观测器的自适应同步。我们通过数值模拟获得的结果包括时间图、相图、参数估计和误差图的收敛以及显示完全同步的相位图。导出控制器增益向量以实现状态反馈控制并计算开关反馈的增益矩阵，最后执行参数估计以实现基于观测器的自适应同步。我们通过数值模拟获得的结果包括时间图、相图、参数估计和误差图的收敛以及显示完全同步的相位图。