# Isothermal Continuous Stirred Tank Reactor

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# Chemical Reactor Model

## Model description:

The following two-dimensional single-input single-output system represents a chemical reactor model

\begin{align*} \dot{x}_1 &= u(Ce -x_1) - rx_1 \\ \dot{x}_2 &= rx_1 - ux_2 \\ y &= x_1 - x_2, \end{align*}

where coefficients are in $\mathbb{R}$, $x_1$ and $x_2$ denote the reactant and product concentrations, respectively. The input $u$ corresponds to the input flow of reactant, $r$ and $C_e$ denote kinetic and reactor parameters.

## Publication details:

 Title Observer Synthesis for a Class of Bilinear Systems: a Differential Algebraic Approach Publication Type Conference Paper Authors Martinez-Guerra, R., and De Leon-Morales J.

# Laboratory Scale Liquid Level System

## Model description:

The system consists of a DC water pump feeding a conical flask which in turn feeds a square tank, giving the system second-order dynamics. The controllable input is the voltage to the pump motor and the system output is the height of the water in the conical flask. The aim, under simulation conditions, is for the water height to follow some demand signal. The plant model was identified as

\begin{align*}z(t) &=0.9722z(t-1)+0.3578u(t-1)-0.1295u(t-2)-\\ &-0.3103z(t-1)u(t-1)-0.04228z^6(t-2)+0.1663z(t-2)u(t-2)+\\ &+0.2573z(t-2)e(t-1)-0.03259z^2(t-1)z(t-2) - 0.3513z^2(t-1)u(t-2)+\\ &+0.3084z(t-1)z(t-2)u(t-2)+0.2939z^2(t-2)e(t-1)+\\ &+0.1087z(t-2)u(t-1)u(t-2)+0.4770z(t-2)u(t-1)e(t-1)+\\ &+0.6389u^2(t-2)e(t-1)+e(t), \end{align*}

where $e(t)$ is a noise.

2

## Publication details:

 Title Self-tuning control of non-linear ARMAX models Publication Type Journal Article Authors Sales, K. R., and Billings S. A.

# Three Heated Rooms

## Model description:

Consider three heated rooms depicted in the attached image. The first room can be heated directly by the input $u^1$, the heat transfer into the room. The other two rooms are heated via two boilers with the inputs $u^2$ and $u^3$ respectively. The temperature of each room is described by $x^1$, $x^2$, and $x^3$ respectively, the temperature of each boiler by $x^4$ and $x^5$. The heat emission of each room is considered as a nonlinear function of the room temperature. With $x = (x^1,\ldots, x^5)^{\mathrm T}$ the the nonlinear system has the form

$$\dot{x}=\begin{pmatrix} -c_0(x^2-T_0)-c_1(x^1-T_0)^2\\ -c_0(x^2-T_0)-c1(x^2-T_0)^2 + c_2(x^4-x^2)\\ -c_0(x^3-T_0)-c1(x^3-T_0)^2 + c_2(x^5-x^3)\\ -c_2(x^4-x^2)\\ -c_2(x^5-x^3) \end{pmatrix} + \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} u^1\\ u^2\\ u^3 \end{pmatrix},$$

where $c_0$, $c_1$, and $c_2$ are parameters for the heat transmission, and $T_0$ is the temperature outside of the rooms. We will choose our parameters so that the time is measured in hours and $x^i$ is measured in Kelvin. The state manifold is $\mathcal{M}= \mathbb{R}^5$, and the coupling conditions are

$y=h(x)=\begin{pmatrix}x^1-x^2\\x^1-x^3\end{pmatrix}=0,$

i.e. the temperature of the three rooms should be equal.

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## Attachment: threeheatedrooms.PNG

## Publication details:

 Title Attractive Invariant Submanifold-based Coupling Controller Design Publication Type Conference Paper Authors Labisch, Daniel, and Konigorski Ulrich

# Isothermal Continuous Stirred Tank Reactor

## Model description:

\begin{align*} \dot x_1 &= -k_1x_1 - k_3x_1^2 + u(c - x_1) \\ \dot x_2 &= k_1x_1 - k_2x_2 - ux_2 \\ y &= x_2, \end{align*}

where $c$ and $x_1$ are the concentrations of the input and reactant substance, $x_2$ is the concentration of the desired output product, $u$ is the normalized input flow rate of the reactant substance. The output $y$ here reflects the grade of the final product. The parameters $k_1, k_2, k_3$ and $c$ are positive constants under the isothermal considered conditions.

2

## Publication details:

 Title Flatness-based optimal noncausal output transitions for constrained nonlinear systems: case study on an isothermal continuously stirred tank reactor Publication Type Journal Article Authors Wang, G.L., and Allgower F.