# 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*}

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## 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*}

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## 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*}

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## Publication details:

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

# Coupled electric drives

## Model description:

This particular laboratory-scale process simulates the actual industrial problems in tension and speed controls as they occur in magnetic tape drives, textile machines, paper mills, strip metal production plants, etc. To simulate these problems. the coupled electric drives consists of two similar servo-motors which drive a jockey pulley via a continuous flexible belt (see attached image).The jockey pulley assembly constitutes a simulated work station. The basic control problem is to regulate the belt speed and tension by varying the two servo-motor torque.

The structure of the transfer function matrix model (numerator and denominator orders of each transfer function) have been determined from the theoretical modelling of the coupled electric drive system which has given

$$\begin{bmatrix} Y_1(s)\\ Y_2(s) \end{bmatrix} =G(s) \begin{bmatrix} U_1(s)\\ U_2(s) \end{bmatrix}$$

with

$G(s) = \begin{bmatrix} \dfrac{b_{1,1,0}}{s^2+a_{11}s+a_{12}} & \dfrac{b_{1,2,0}}{s^2+a_{11}s+a_{12}}\\ \dfrac{b_{2,1,0}s+b_{2,1,1}}{s^3+a_{21}s^2+a_{22}s+a_{23}} & \dfrac{b_{2,2,0}}{s^3+a_{21}s^2+a_{22}s+a_{23}} \end{bmatrix}$

The inputs $U(s)$ to the system are the drive voltages to the servo-motor power amplifiers. The outputs $Y_1(s)$ and $Y_2(s)$ are the jockey pulley velocity and the belt tension respectively.

## Publication details:

 Title A new bias-compensating least-squares method for continuous-time MIMO system identification applied to a laboratory-scale process Publication Type Conference Paper Authors Garnier, H., Sibille P., and Nguyen H.L.

# 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 $|\upsilon| \leq 1$.

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## 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