Two van der Pol oscillators coupled via a bath

Model description: 

consider a vibration system with nonlinear springs shown in the attached image. The normal form of the system is given by

$$\begin{align*} \dot\xi_{1}^{1} &= \xi_{2}^{1} \\ \dot{\xi}_{2}^{1}&=-\frac{k_1}{m_{1}}(\xi_{1}^{1}-\xi_{1}^{2})-{\bar{k} _{1}\over m_{1}}(\xi_{1}^{1}-\xi_{1}^{2})^{3}-{c_1\over m_{1}}(\xi_{2}^{1}-\xi_{2}^{2})+{u_1\over m_{1}} \\ \dot{\xi}_{1}^{2} &= \xi_{2}^{2} \\ \dot\xi_{2}^{2}&=\frac{k_1}{m_{2}}(\xi_{1}^{1}-\xi_{1}^{2})+{\bar{k} _{1}\over m_{2}}(\xi_{1}^{1}-\xi_{1}^{2})^{3}+{c_1\over m_{2}}(\xi_{2}^{1}-\xi_{2}^{2}) \\ &-{k_2\over m_{2}}(\xi_{1}^{1}-\xi_{1}^{2})-{k_2\over m_{2}}(\xi_{1}^{1}-\xi_{1}^{2})^{3}-{c_2\over m_{2}}(\xi_{2}^{1}-\xi_{2}^{2})+{u_2\over m_{2}} \\ \dot{\eta}_{1} &= \eta_{2} \\ \dot{\eta}_{2} &= {k_2\over m_{3}}(\xi_{1}^{2}-\eta_{1})+{\bar{k} _{2}\over m_{3}}(\xi_{1}^{2}-\eta_{1})^{3}+{c_2\over m_{3}}(\xi_{2}^{2}-\eta_{2}) \\ &-{k_3\over m_{3}}\eta_{1}-{\bar{k} _{3}\over m_{3}}\eta_{1}^{3}-{c_3\over m_{3}}\eta_{2} \\ y_{1} &=\xi_{1}^{1} \\ y_{2} &=\xi_{1}^{2}, \end{align*}$$

where parameters $m_i, k_i$ and $\bar{k} (i=1,2,3)$ are positive constants. The system has the relative degrees $r_1 = 2, r_2 = 2$ and the following zero dynamics.

$\begin{align*} \dot{\eta}_{1} &=\eta_{2} \\ \dot{\eta}_{2} &=-\frac{k_2}{m_3}\eta_{1}-\frac{\bar{k}_2}{m_3}\eta_1^3+\frac{c_2}{m_3}(\xi_2^2-\eta_2) \\ &-\frac{k_3}{m_3}\eta_1-\frac{\bar{k}_3}{m_3}\eta_1^3-\frac{c_3}{m_3}\eta_2 \end{align*}$$

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Publication details: 

TitleSampled-Data Models for Decouplable Nonlinear Multivariable Systems
Publication TypeConference Paper
Year of Publication2010
AuthorsNishi, M., Ishitobi M., Liang Shan, and Kunimatsu S.
Conference NameProceedings of SICE Annual Conference 2010
Date Published08/2010
PublisherIEEE
Conference LocationTaipei
ISBN Number978-1-4244-7642-8
Accession Number11594970
Keywordscontinuous time systems, control system synthesis, MIMO systems, nonlinear control systems, sampled data systems
AbstractOne 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.
URLhttp://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5602233