# Two van der Pol oscillators coupled via a bath (1)

## Model description:

Consider nonlinear van der Pol oscillators coupled via a bath. The normal form of the system is expressed by

$$\begin{align*} \dot{\xi}_{1}^{1}&=\xi_{2}^{1} \\ \dot{\xi}_{2}^{1}&=-\xi_{1}^{1}+\epsilon\{1-(\xi_{1}^{1})^{2}\}\xi_{2}^{1}+k(\eta_{1}-\xi_{1}^{1})+u_{1} \\ \dot{\xi}_{1}^{2}&=\xi_{2}^{2} \\ \dot{\xi}_{2}^{2}&=-\xi_{1}^{2}+\epsilon\{1-(\xi_{1}^{2})^{2}\}\xi_{2}^{2}+k(\eta_{1}-\xi_{1}^{2})+u_{2} \\ \dot{\eta}_{1}&=k(\xi_{1}^{1}-\eta_{1})+k(\xi_{1}^{2}-\eta_{1}) \\ y_{1}&=\xi_{1}^{1} \\ y_{2}&=\xi_{1}^{2}, \end{align*}$$

where parameters $\epsilon$ and $k$ are positive constants. The system has the relative degrees $r_1 = 2, r_2 = 2$ and the following zero dynamics:

$ \dot{\eta}_{1}=-2k\eta_{1}.$

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## Publication details:

Title | Sampled-Data Models for Decouplable Nonlinear Multivariable Systems |

Publication Type | Conference Paper |

Year of Publication | 2010 |

Authors | Nishi, M., Ishitobi M., Liang Shan, and Kunimatsu S. |

Conference Name | Proceedings of SICE Annual Conference 2010 |

Date Published | 08/2010 |

Publisher | IEEE |

Conference Location | Taipei |

ISBN Number | 978-1-4244-7642-8 |

Accession Number | 11594970 |

Keywords | continuous time systems, control system synthesis, MIMO systems, nonlinear control systems, sampled data systems |

Abstract | One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Few studies have been investigated for obtaining sampled-data models of nonlinear multi-input multi-output systems (MIMO system) though we can find studies which consider nonlinear single-input single-output systems. This paper shows a more accurate sampled-data model than the Euler model for nonlinear multi-input multi-output systems and derives sampling zero dynamics of the model. |

URL | http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5602233 |