# Nonlinear benchmark system

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# 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:

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

## 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

# Inverted Pendulum on a Cart

## Model description:

The inverted pendulum on cart to be controlled is shown in the attached image. Its structure consists of a cart and pendulum where the pendulum is hinged to the cart via a pivot and only the cart is actuated, where:

 $\theta$ is the pendulum angle $rad$ $x$ is the cart position $m$ $M$ is the mass of the cart $kg$ $m$ is the mass of the pendulum $kg$ $l$ is distance from turning center to center of mass of the pendulum $m$ $f$ is the cart's friction coefficient $kg/s$ $F$ is force applied to the cart $N$

Lagrange's equations are applied with respect to $\theta$ and $x$ coordinates. The nonlinear state space model of inverted pendulum on cart system can then be obtained as:

\begin{align*} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \dfrac{mg \sin{x_1}+ \dfrac{ m\cos{x_1}(u-f\dot{x})-m^2lx_2^2\cos{x_1}\sin{x_1}}{(M + m)}}{\dfrac{4}{3}ml - \dfrac{m^2l\cos^2{x_1}}{(M+m)}} \\ \dot{x}_3 &= x_4 \\ \dot{x}_4 &= \dfrac{u - f x_1 + \dfrac{3}{4}mg\cos{x_1}\sin{x_1} - mlx_2^2\sin{x_1}}{(M+m) - \dfrac{3}{4}m\cos^2{x_1}}, \end{align*}

where the state variables are consequently assigned as $x_1 = \theta$, $x_2 = \dot{\theta}$, $x_3 = x$ an $x_4=\dot{x}$, and where the input $u$ is the applied force $F$.

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## Publication details:

 Title Hybrid Controller for Swinging up Inverted Pendulum System Publication Type Conference Paper Year of Publication 2005 Authors Nundrakwang, S, Benjanarasuth T, Ngamwiwit J, and Kominet N Conference Name 2005 Fifth International Conference on Information, Communications and Signal Processing Publisher IEEE Conference Location Bangkok ISBN Number 0-7803-9283-3 Accession Number 9097400 Keywords control system synthesis, linear quadratic control, PD control, pendulums, position control, servomechanisms, state feedback Abstract A hybrid controller for swinging up inverted pendulum system is proposed in this paper. The controller composes of two parts. The first part is the PD position control for swinging up the pendulum from the natural pendent position by moving the cart back and forth until the pendulum swings up around the upright position. The second part is a servo state feedback control designed by LQR which will be switched to stabilize the inverted pendulum in its upright position. The effectiveness and reliability of the proposed hybrid controller for swinging up inverted pendulum on cart are also shown by the experimental results DOI 10.1109/ICICS.2005.1689094

# Simplified Schmid pendulum

## Model description:

Simplified Schmid pendulum:

\begin{align*} \ddot{\psi} + a_{21}\omega - a_{11}\sin{\psi} &= -b_1u, \\ \dot{\omega} + a_{22}\omega + a_{12}\sin{\psi} &= b_2u, \end{align*}

where $\psi$ is the pendulum angle; $\omega$ is the wheel angular rate; $u$ is the controlling voltage, applied to the motor; $a_{11},a_{21},a_{12},b_1,b_2$ are positive constants, depending on the design parameter of the pendulum. It is assumed that the upper (unstable) equilibrium point corresponds to $\psi=0$.

## Publication details:

 Title Hybrid quantised observer for multi-input-multi-output nonlinear systems Publication Type Conference Paper Year of Publication 2008 Authors Fradkov, Alexander L., Andrievsky Boris, and Evans Robin J. Conference Name IEEE International Conference on Control Applications, 2008. CCA 2008. Date Published 09/2008 Publisher IEEE Conference Location San Antonio, Texas, USA ISBN Number 978-1-4244-2222-7 Accession Number 10235153 Keywords IMO systems, nonlinear control systems, observers, oscillators, pendulums Abstract Limit possibilities of state estimation under information constraints (limited information capacity of the coupling channel) for multi-input-multi-output (MIMO) nonlinear systems are evaluated. We give theoretical analysis for state estimation of nonlinear systems represented in Lurie form (linear part plus nonlinearity depending only on measurable outputs) with a first-order coder-decoder. It is shown that the upper bound of the limit estimation error is proportional to the upper bound of the transmission error. As a consequence, the upper bound of limit estimation error is proportional to the maximum rate of the coupling signal and inversely proportional to the information transmission rate (channel capacity). The results are applied to state estimation of a nonlinear self-excited mechanical oscillator and a reaction-wheel pendulum. DOI 10.1109/CCA.2008.4629572

# Self-excited nonlinear oscillator

## Model description:

Self-excited nonlinear oscillator:

\begin{align*} \dot{x}_1 &=x_2,\\ \dot{x}_2 &=-\omega_1^2\sin{y_1}-\varrho x_2 + k_p \arctan(k_c y_2),\\ \dot{x}_3 &=\omega_2 \cdot (x_1 - x_3), \\ y_1 &=x_1, \\ y_2 &=x_1-x_3, \end{align*}

where $y(t)\in \mathbb{R}^2$ is the sensor output vector (to be transmitted over the communication channel), $\omega_1, \omega_2, \varrho, k_p, k_c$ are system parameters, $x=[x_1,x_2,x_3]^{\mathrm T}\in\mathbb{R}^3$ of the system state vector $x(t)$ based on the signals $y_1(t),y_2(t)$, transmitted over the communication channel.

The system has the form

$\dot{x}(t)=Ax(t)+\varphi(y(t)),$ $y(t)=Cx(t),$

where

$A=\begin{bmatrix}0 & 1 & 0 \\ 0 & -\varrho & 0 \\ \omega_2 & 0 & -\omega_2\end{bmatrix},$

$C = \begin{bmatrix}1, & 0, & 0\\ 1, & 0, & -1\end{bmatrix},$

$\varphi(y) = \begin{bmatrix} 0\\ \omega_1^2 \sin{y_1} + k_p \arctan{(k_c y_2)} \\ 0 \end{bmatrix}.$

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## Publication details:

 Title Hybrid quantised observer for multi-input-multi-output nonlinear systems Publication Type Conference Paper Year of Publication 2008 Authors Fradkov, Alexander L., Andrievsky Boris, and Evans Robin J. Conference Name IEEE International Conference on Control Applications, 2008. CCA 2008. Date Published 09/2008 Publisher IEEE Conference Location San Antonio, Texas, USA ISBN Number 978-1-4244-2222-7 Accession Number 10235153 Keywords IMO systems, nonlinear control systems, observers, oscillators, pendulums Abstract Limit possibilities of state estimation under information constraints (limited information capacity of the coupling channel) for multi-input-multi-output (MIMO) nonlinear systems are evaluated. We give theoretical analysis for state estimation of nonlinear systems represented in Lurie form (linear part plus nonlinearity depending only on measurable outputs) with a first-order coder-decoder. It is shown that the upper bound of the limit estimation error is proportional to the upper bound of the transmission error. As a consequence, the upper bound of limit estimation error is proportional to the maximum rate of the coupling signal and inversely proportional to the information transmission rate (channel capacity). The results are applied to state estimation of a nonlinear self-excited mechanical oscillator and a reaction-wheel pendulum. DOI 10.1109/CCA.2008.4629572

# Nonlinear benchmark system

## Model description:

\begin{align*} x_1(t+1) &=\left(\dfrac{x_1(t)}{1+x_1^2(t)}+1\right)\sin{x_2(t)} \\ x_2(t+1) &=x_2(t)\cos{x_2(t)}+x_1(t)e^{-((x_1^2(t)+x_2^2(t))/8} + \dfrac{u^3(t)}{1+u^2(t)+0.5\cos{x_1(t)+x_2(t)}} \\ y(t) &=\dfrac{x_1(t)}{1+0.5\sin{x_2(t)}}+\dfrac{x_2(t)}{1+0.5\sin{x_1(t)}}+e(t), \end{align*}

where $e(t)$ is the noise term, has a variance of 0.1.

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## Publication details:

 Title Nonlinear system identification via direct weight optimization Publication Type Journal Article Year of Publication 2005 Authors Roll, Jacob, Nazin Alexander, and Ljung Lennart Journal Automatica Volume 41 Pagination 475 - 490 Date Published 01/2005 ISSN 0005-1098 URL http://dx.doi.org/10.1016/j.automatica.2004.11.010