# Ball and Plate System

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# A nonlinear ARX (NARX) system

## Model description:

\begin{align*} y(t)&=\begin{bmatrix}0.1 & -0.1 & 0.25 & 0.5 \end{bmatrix} \varphi (t) +\\ &+ \frac{L}{2}(\|\varphi (t)\|^2 - 2(max\{\|\varphi(t)\|^2, 1\} -1) +\\ &+ 2(max\{\|\varphi(t)\|^2, 2\} - 2) - (max\{\|\varphi(t)\|^2, 3\} -3)) + e(t), \end{align*}

where

$\varphi(t)=\begin{bmatrix} y(t-1) & y(t-2) & u(t-1) & u(t-2)\end{bmatrix}^{\mathrm T}$

$L = 0.1$

$e(t) \in N(0, 0.01)$, i.e. $\sigma = 0.1$.

## 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

# Bilinear system of non-minimum phase

## Model description:

\begin{align*} y(t) &= y(t-1) + u(t-1) + 1.3u(t-2) + 0.3u(t-1)y(t-1) \\ &+0.5u(t-2)y(t-2)+e(t)/\Delta, \end{align*}

where $e(t)$ is normal school white noise signal with covariance 0.1.

2

## Publication details:

 Title Generalized Predictive Control for a Class Of Bilinear Systems Publication Type Conference Paper Year of Publication 1970 Authors Liu, Guizhi, and Li and Ping Conference Name Control, Automation, Robotics and Vision Date Published 2006 Abstract A new generalized predictive control algorithm for a kind of input-output bilinear system is proposed in the paper (BGPC). The algorithm combines bilinear and linear terms of I/O bilinear system, and constitutes an ARIMA model analogous to linear systems. Using optimization predictive information fully, the algorithm carries out multi-step predictions by recursive approximation. The heavy computation of generic nonlinear optimization is avoided with control law of analytical form being used to the non-minimum phase bilinear systems. Simulation results show the effectiveness of the algorithm and the performance of the algorithm is better than linear generalized predictive control (LGPC). Key words: bilinear systems; bilinear generalized predictive control (BGPC); recursive approaches; non-minimum phase systems; analytical control laws DOI 10.1109/ICARCV.2006.345181

# Ball and Plate System

## Model description:

The ball and plate system is a system, where a metal ball stays on a rigid square plate with each side length of 1m. The slope of the plate can be manipulated by two perpendicularly installed step motors, so that the tilting of the plate will make the ball moving.

\begin{align*} \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2\\ \dot{x}_3\\ \dot{x}_4\\ \dot{x}_5\\ \dot{x}_6\\ \dot{x}_7\\ \dot{x}_8\\ \end{bmatrix} &= \begin{bmatrix} x_2 \\ B(x_1x_4^2 + x_4x_5x_8 - g \sin x_3)\\ x_4\\ 0\\ x_6\\ B(x_5x_8^2 + x_1x_4x_8 - g \sin x_7)\\ x_8\\ 0\\ \end{bmatrix} + \begin{bmatrix} 0 & 0\\ 0 & 0\\ 0 & 0\\ 1 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 0\\ 0 & 1\\ \end{bmatrix} \begin{bmatrix} u_x\\ u_y\\ \end{bmatrix}, \\ Y &= h(X) = (x_1,x_5)^{\mathrm T}, \end{align*}

where $B= m/(m + J/R^2)$ and $X = (x_1; x_2; x_3; x_4; x_5; x_6; x_7; x_8)^{\mathrm T} = (x; \dot{x}; \theta_x; \dot{\theta}_x; y; \dot{y}; \theta_y; \dot{\theta}_y)^{\mathrm T}$

Parameters are presented in the table below.

 Symbol Description Parameter value and unit $m$ Mass of the ball $0.11$ $Kg$ $R$ Radius of the ball $0.02$ $m$ $S$ Dimension of the ball $1.0 \times 1.0$ $m^2$ $x$ Position of the ball in the $x$-axis $m$ $y$ Position of the ball in the $y$-axis $m$ $\dot{x}$ Velocity of the ball in the $x$-axis $m/s$ $\dot{y}$ Velocity of the ball in the $y$-axis $m/s$ $w$ Rolling angular velocity of the ball $Arc/s$ $\dot{r}$ Velocity of the ball, $\dot{r}^2 = x^2 + y^2$ $m/s$ $v_{max}$ Maximum velocity of the ball $4$ $mm/s$ $\tau_x$ Torque applied to the plate in the $x$-axis $Kg$ $m^2/s^2$ $\tau_y$ Torque applied to the plate in the $y$-axis $Kg$ $m^2/s^2$ $\theta_x$ Angle of the plate in the $x$-axis $Arc$ $\theta_y$ Angle of the plate in the $y$-axis $Arc$ $\dot{\theta}_x$ Angle velocity of the plate in the $x$-axis $Arc/s$ $\dot{\theta}_y$ Angle velocity of the plate in the $y$-axis $Arc/s$ $u_x$ Angle acceleration velocity of the plate from $x$-axis $Arc/s^2$ $u_y$ Angle acceleration velocity of the plate from $y$-axis $Arc/s^2$ $J_P$ Mass moment of inertia of the plate $0.5$ $Kg$ $m^2$ $J$ Mass moment of inertia of the ball $1.76e$ $-$ $5$ $Kg$ $m^2$ $g$ Acceleration due to gravity $9.8$ $m/s^2$

8

## Publication details:

 Title Trajectory planning and tracking of ball and plate system using hierarchical fuzzy control scheme Publication Type Journal Article Year of Publication 2004 Authors Fan, Xingzhe, Zhang Naiyao, and Teng Shujie Journal Fuzzy Sets and Systems Volume 144 Pagination 297-312 Date Published Jun ISSN 0165-0114 DOI 10.1016/S0165-0114(03)00135-0