# Two van der Pol oscillators coupled via a bath (2)

## Model description:

Consider the model described in Two van der Pol oscillators coupled via a bath (1).

The current model is using a slightly different notation:

$$\begin{align*} \dot{\xi}_{1}^{1} &= \dot{x}_1 \\ \dot{\xi}_{2}^{1} &= \dot{x}_2 \\ \dot{\xi}_{1}^{2} &= \dot{x}_3 \\ \dot{\xi}_{2}^{2} &= \dot{x}_4 \\ \dot{\eta}_{1} &= \dot{x}_5 \end{align*}$$

Note that this system is decouplable by static state feedback because the decoupling matrix of this system is

$D_{1}(\xi,\eta)=\left[\matrix{1 &0 \cr 0 &1}\right]$

The authors have proposed the following Yuz and Goodwin type approximate model which is more accurate than the Euler model.

$$\begin{align*} x_{1,k+1}&=x_{1,k}+T_{x_{2,k}}+\frac{T^2}{2}\{u_{1,k}-x_{1,k}+\epsilon\{1-x_{1,k}^{2}\}x_{2,k}+k(x_{5,k}-x_{1,k})\} \\ x_{2,k+1}&=x_{2,k}+T\{u_{1,k}-x_{1,k}+\epsilon\{1-x_{1,k}^{2}\}x_{2,k}+k(x_{5,k}-x_{1,k})\} \\ x_{3,k+1}&=x_{3,k}+T_{x_{4,k}}+\frac{T^2}{2}\{u_{2,k}-x_{1,k}+\epsilon\{1-x_{3,k}^{2}\}x_{4,k}+k(x_{5,k}-x_{1,k})\} \\ x_{4,k+1}&=x_{4,k}+T_{x_{4,k}}+T\{u_{2,k}-x_{2,k}+\epsilon\{1-x_{3,k}^{2}\}x_{4,k}+k(x_{5,k}-x_{3,k})\} \\ x_{5,k+1}&=x_{5,k}+T\{k(x_{1,k}-x_{5,k})+k(x_{3,k}-x_{5,k})\} \\ y_{1,k}&=x_{1,k} \\ y_{2,k}&=x_{3,k} \end{align*}$$

## Type:

## Form:

## Time domain:

## Linearity:

## Publication details:

Title | Sampled-data model for nonlinear coupled Van der Pol oscillators |

Publication Type | Conference Paper |

Year of Publication | 2011 |

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

Conference Name | 2011 Proceedings of SICE Annual Conference (SICE) |

Date Published | 09/2011 |

Publisher | IEEE |

ISBN Number | 978-1-4577-0714-8 |

Accession Number | 12354559 |

Keywords | closed loop systems, continuous time systems, control system synthesis, MIMO systems, nonlinear control systems, relaxation oscillators, sampled data systems, stability |

Abstract | For sampled-data controller design of nonlinear continuous-time systems, it is important to derive a good approximate sampled-data model because the exact sampled-data model for nonlinear systems is often unavailable to the controller designers. In the multi-input multi-output (MIMO) case, the authors have proposed an accurate approximate model which includes extra zero dynamics corresponding to the relative degree of the continuous-time nonlinear system. Such extra zero dynamics are called sampling zero dynamics. A more accurate sampled-data model is, however, required when the relative degrees of a continuous-time nonlinear plant are two. The reason is that the closed-loop system becomes unstable when the more accurate sampled-data model has unstable sampling zero dynamics. This paper derives the sampling zero dynamics of the more accurate sampled-data model for coupled Van der Pol oscillators and analyzes the relationship between the stability of the closed-loop system and the stability of the sampling zero dynamics of a proposed model. |

URL | http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6060620&queryText%3DSampled-data+model+for+nonlinear+coupled+Van+der+Pol+oscillators |