# Pendulum system with Coulomb friction

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# Cart plus crane plus hammer

## Model description:

The Euler-Lagrange equations of motion of the system are given as follows

$$\begin{pmatrix} (M+m) & mL\cos{q_1} & 0\\ mL\cos{q_1} & mL^2+\Theta & \dfrac{\Theta}{2}\\ 0 & \dfrac{\Theta}{2} & \Theta \end{pmatrix} \begin{pmatrix} \ddot{x}\\ \ddot{q_1}\\ \ddot{q_2} \end{pmatrix} + \begin{pmatrix} -mL\sin{q_1\dot{q_1}^2}\\ -mLg_g\sin{q_1}\\ 0 \end{pmatrix} = \begin{pmatrix} Q_x\\ Q_1\\ Q_2 \end{pmatrix},$$

where $Q_x (N)$ is the generalized force pushing the cart in the horizontal “$x$” direction, $Q_1$ and $Q_2$ are torques in $(N · m)$ rotating the beam of the crane around a horizontal axis orthogonal to “$x$” and counter-rotating the hamper at the free end of the beam to avoid turning out the worker from the hamper. $L (m)$ denotes the lenght of the crane’s beam, $g_g$ ($m/s^2$) is the gravitational acceleration, $m$ ($kg$) and $\Theta$ $(kg · m^2)$ denote the momentum (with respect to its own center of mass that was supposed to be on the rotational axle) and the mass of the hamper.

## Publication details:

 Title Analysis of the Fixed Point Transformation Based Adapive Robot Control Publication Type Conference Paper Authors Tar, J.K., and Rudas I.J.

# Third-order nonlinear discrete-time system #2

## Model description:

Image below shows the block diagram of a discrete-time system. \begin{align*} H_1(z) &=\dfrac{0.2z^{-1}}{z^{-1}-0.21z^{-2}} \\ H_2(z) &=\dfrac{0.1z^{-1}}{1-1.1z^{-1}+0.3z^{-2}} \\ H_3(z) &=\dfrac{0.3z^{-1}}{1-0.4z^{-1}} \end{align*}

3

## Publication details:

 Title Nonlinear system identification using genetic algorithms with application to feedforward control design Publication Type Conference Paper Authors Luh, Guan-Chun, and Rizzoni G.

# Third-order nonlinear discrete-time system #1

## Model description:

The block diagram of a third-order nonlinear discrete time system adopted by Fakhouri for identification evaluation is shown below. \begin{align*} H_1(z) &=\dfrac{0.1z^{-1}}{1-0.5z^{-1}} \\ H_2(z) &=\dfrac{0.1z^{-1}}{1-1.3z^{-1}+0.42z^{-2}} \\ H_3(z) &=\dfrac{1.0z^{-1}}{1-0.7z^{-1}} \end{align*}

3

## Publication details:

 Title Nonlinear system identification using genetic algorithms with application to feedforward control design Publication Type Conference Paper Authors Luh, Guan-Chun, and Rizzoni G.

# Continuous stirred-tank reactor system

## Model description:

The following CSTR system developed by Liu(1967). The reaction is exothermic first-order, $A \rightarrow B$, and is given by the following mass and energy balances. One should notice that the energy balance includes cooling water jacket dynamics. The following model was identified using regression techniques on the energy balance equations:

\begin{align*} y(k) &= 1.3187y(k-1) - 0.2214y(k-2) - 0.1474y(k-3) \\ &- 8.6337u(k-1) + 2.9234u(k-2) + 1.2493u(k-3) \\ &- 0.0858y(k-1)u(k-1) + 0.0050y(k-2)u(k-1) \\ &+ 0.0602y(k-2)u(k-2) + 0.0035y(k-3)u(k-1) \\ &- 0.0281y(k-3)u(k-2) + 0.0107y(k-3)u(k-3). \end{align*}

3

## Publication details:

 Title Identification and Control of Bilinear Systems Publication Type Conference Paper Authors Bartee, James F., and Georgakis Christos

# Pendulum system with Coulomb friction

## Model description:

Consider a pendulum system with Coulomb friction and external perturbation

$$\ddot {\theta} = \frac{1}{J}u - \frac{g}{L}\sin \theta - \frac{V_s}{J}\dot{\theta } - \frac{P_s}{J}\mathrm{sgn}(\dot{\theta}) + \upsilon,$$

where parameters have the following values $M=1.1$, $L=0.9$, $J=ML^2=0.891$, $V_s=0.18$, $P_s=0.18$, $P_s=0.45$, $g=9.815$, and $v$ is an uncertain external perturbation $|\upsilon| \leq 1$.

2

## Publication details:

 Title A Simple Nonlinear Observer for a Class of Uncertain Mechanical Systems Publication Type Journal Article Authors Su, Yuxin, Müller P.C., and Zheng Chunhong