# Two van der Pol oscillators coupled via a bath

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# 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*}

## 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

# Controlled Van der Pol system (2)

## Model description:

Using

$z_{i+1,k+1}=y_{k+1}^{(i)}\approx y_{k}^{(i)}+Ty_{k}^{(i+1)}+\frac{T^2}{2}y_{k}^{(i+2)}+\cdots+\frac{T^{r-i}}{(r-i)!}y_{k}^{(r)}+\frac{T^{r-i+1}}{(r-i+1)!}y_{k}^{(r +1)}$

for $i=0, \cdots,r-1$ the controlled Van der Pol system from Controlled Van der Pol system (1) can be rewritten as:

\begin{align*} x_{1,k+1} &=x_{1,k}+Tx_{2,k}+\frac{T^2}{2}[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}] \\ &+\frac{T^{2}}{3!}[-c(-cx_{2,k}-d\sin x_{1,k}+u_{1,k})-dx_{2,k}\cos x_{1,k} \\ &\times\{-x_{1,k}+\epsilon(1-x_{1,k}^2)x_{2,k}+u_k] \\ x_{2,k+1} &=x_{2,k}+T[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}] \\ &+\frac{T^2}{2}[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}-dx_{2,k}\cos x_{1,k} \\ &\times\{-x_{1,k}+\epsilon(1-x_{1,k}^2)x_{2,k}+u_k] \\ y_{k} &=x_{1,k} \end{align*}

## Publication details:

 Title Nonlinear sampled-data models and zero dynamics Publication Type Conference Paper Year of Publication 2009 Authors Nishi, M., Ishitobi M., and Kunimatsu S. Conference Name International Conference on Networking, Sensing and Control, 2009. ICNSC '09. Date Published 03/2009 Publisher IEEE Conference Location Okayama ISBN Number 978-1-4244-3491-6 Accession Number 10646009 Keywords closed loop systems, continuous time systems, control system synthesis, nonlinear control systems, poles and zeros, sampled data systems, stability 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. Hence, 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. Recently, Yuz and Goodwin 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 degree of a continuous-time nonlinear plant is two. The reason is that the closed-loop system becomes unstable when the more accurate sampled-data model has unstable sampling zero dynamics and a controller design method based on cancellation of the zero dynamics is applied. This paper derives the sampling zero dynamics of the more accurate sampled-data model and shows a condition which assures the stability of the sampling zero dynamics of the model. DOI 10.1109/ICNSC.2009.4919304

# Pendulum System with Relative Degree Two

## Model description:

Consider the pendulum system with the relative degree two

\begin{align*} \dot{x}_{1} &= x_{2}\\ \dot{x}_{2} &= -cx_{2}-d\sin x_{1}+u \\ y &= x_{1} \end{align*}

with $c > 0$.

2

## Publication details:

 Title Nonlinear sampled-data models and zero dynamics Publication Type Conference Paper Year of Publication 2009 Authors Nishi, M., Ishitobi M., and Kunimatsu S. Conference Name International Conference on Networking, Sensing and Control, 2009. ICNSC '09. Date Published 03/2009 Publisher IEEE Conference Location Okayama ISBN Number 978-1-4244-3491-6 Accession Number 10646009 Keywords closed loop systems, continuous time systems, control system synthesis, nonlinear control systems, poles and zeros, sampled data systems, stability 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. Hence, 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. Recently, Yuz and Goodwin 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 degree of a continuous-time nonlinear plant is two. The reason is that the closed-loop system becomes unstable when the more accurate sampled-data model has unstable sampling zero dynamics and a controller design method based on cancellation of the zero dynamics is applied. This paper derives the sampling zero dynamics of the more accurate sampled-data model and shows a condition which assures the stability of the sampling zero dynamics of the model. DOI 10.1109/ICNSC.2009.4919304

# Controlled Van der Pol system (1)

## Model description:

A controlled Van der Pol system can be described by the following equation

\begin{align*} \dot{x}_{1} &= x_{2} \\ \dot{x}_{2} &= -x_{1}+\epsilon (1-x_{1}^{2})x_{2}+u, \epsilon > 0 \\ y &= x_{1} \end{align*}

2

## Publication details:

 Title Nonlinear sampled-data models and zero dynamics Publication Type Conference Paper Year of Publication 2009 Authors Nishi, M., Ishitobi M., and Kunimatsu S. Conference Name International Conference on Networking, Sensing and Control, 2009. ICNSC '09. Date Published 03/2009 Publisher IEEE Conference Location Okayama ISBN Number 978-1-4244-3491-6 Accession Number 10646009 Keywords closed loop systems, continuous time systems, control system synthesis, nonlinear control systems, poles and zeros, sampled data systems, stability 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. Hence, 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. Recently, Yuz and Goodwin 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 degree of a continuous-time nonlinear plant is two. The reason is that the closed-loop system becomes unstable when the more accurate sampled-data model has unstable sampling zero dynamics and a controller design method based on cancellation of the zero dynamics is applied. This paper derives the sampling zero dynamics of the more accurate sampled-data model and shows a condition which assures the stability of the sampling zero dynamics of the model. DOI 10.1109/ICNSC.2009.4919304

# 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*}\$

## 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