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:
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 |