# A fourth-order heat exchanger process

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# Continuous flow stirred tank reactor

## Model description:

Coupled nonlinear differential equations describing a process involving a continuous flow stirred tank reactor are given by

\begin{align*} \dot{C}_1 &= -C_1u + C_1(1-C_2)e^{C_2/\Gamma} \\ \dot{C}_2 &= -C_2u + C_1(1-C_2)e^{C_2/\Gamma}\dfrac{1+\beta}{1+\beta-C_2}. \end{align*}

In these equations, the state variables $C_1$ and $C_2$ represent dimensionless forms of cell mass and amount of nutrients in a constant volume tank, bounded between zero and unity. The control $u$ is the flow rate of nutrients into the tank, and is the same rate at which contents are removed from the tank. The constant parameters $\Gamma$ and $\beta$ determine the rates of cell formation and nutrient consumption; these parameters are set to $\Gamma$= 0.48 and $\beta$ = 0.02 for the nominal benchmark specification.

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## Publication details:

 Title Neurocontrol of nonlinear dynamical systems with Kalman filter trained recurrent networks Publication Type Journal Article Authors Puskorius, G.V., and Feldkamp L.A.

# 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.

# Continuously stirred tank reactor system

## Model description:

A schematic of the CSTR plant is shown in the attached image. The process dynamics are described by

\begin{align*} \dot{C}_{a} &=\frac{q}{V}(C_{a0}-c_{a})-a_{0}C_{a}e^{-\frac{E}{RT_{a}}} \\ \dot{T}_{a} &=\frac{q}{V}(T_{f}-T_{a})+a_{1}C_{a}e^{-\frac{E}{RT_{a}}}+a_{3}q_{c}\left(1-e^{\frac{a_{2}}{q_{c}}}\right)(T_{cf}-T_{a}), \end{align*}

where the variables $C_a$ and $T_a$ are the concentration and temperature of the tank, respectively; the coolant flow rate $q_c$ is the control input and the parameters of the plant are defined in the attached table. Within the tank reactor, two chemicals are mixed and react to produce a product compound $A$ at a concentration $C_a(t)$ with the temperature of the mixture being $T(t)$. The reaction is both irreversible and exothermic.

In the paper, authors assumed that plant parameters $q, C_{a0}, T_f$ and $V$ are at the nominal values given in the attached table. The activation energy $E/R = 1 \times 10^4K$ is assumed to be known. The state variables the input and the output are defined as $x=[x_1,x_2]^{\mathrm T}=[C_a,T_a]^{\mathrm T},u=q_c,y=C_a$. Using this notation, the CSTR plant can be re-expressed as

\begin{align*} \dot{x}_{1} &=1-x_{1}-a_{0}x_{1}e^{-\frac{10^4}{{\rm a}_2}} \\ \dot{x}_{2} &=T_{f}-x_{2}+a_{1}x_{1}e^{-\frac{10^4}{{\rm a}_2}}+a_{3}u\left(1-e^{-\frac{a_2}{u}}\right)(T_{cf}-x_{2}) \\ y &= x_{1}, \end{align*}

where the unknown constant parameters are $a_0, a_1, a_2$ and $a_3$.

## Publication details:

 Title Adaptive nonlinear control of continuously stirred tank reactor systems Publication Type Conference Paper Authors Zhang, T., and Guay M.

# A fourth-order heat exchanger process

## Model description:

The system under study consists of two sets of single shell heat exchangers filled with water, placed in parallel and cooled by a liquid saturated refrigerant flowing through a coil system, as it is illustrated in the attached image. The saturated vapour generated in the coil system is separated from the liquid phase in the stages $S1$ and $S2$, both of neglected volumes. This vapour, withdrawn in $S1$ and $S2$, reduces the refrigerant mass flow rate along the cooling system, and only the saturated liquid portion is used for cooling purposes. Table below provides the fluid properties and equipment dimensions. The temperature of the refrigerant remains constant at $T_C$ as the liquid is saturated, and the energy exchanged with water is used to vapourise a small portion of the refrigerant fluid. The idividual heat exchanger energy balances can be expressed in terms of deviation variables to define the following LTI system:

\begin{align*} A &= \begin{bmatrix} -\dfrac{1+\nu_A}{\tau_1} & \dfrac{1}{\tau_1} & 0 & 0\\ 0 & -\dfrac{1+\nu_A}{\tau_2} & 0 & 0 \\ 0 & 0 & -\dfrac{1+\nu_B}{\tau_3} & \dfrac{1}{\tau_3} \\ 0 & 0 & 0 & -\dfrac{1+\nu_B}{\tau_4} \\ \end{bmatrix} \\ B &= \begin{bmatrix} 0 & 0 & 0 & 0\\ \dfrac{k_1}{\tau_1} & 0 & \dfrac{k_3}{\tau_2} & 0\\ 0 & 0 & 0 & 0\\ 0 & \dfrac{k_2}{\tau_t} & 0 & \dfrac{k_4}{\tau_4}\\ \end{bmatrix}, C = \begin{bmatrix} \mu & 0 & \mu & 0\\ 0 & \mu & 0 & \mu\\ \mu & 0 & \mu & 0\\ 0 & \mu & \mu & 0\\ \end{bmatrix}, D = \begin{bmatrix} 0 \end{bmatrix}, \end{align*}

where: $\nu_A \triangleq (hA/C_p(\dot{m}_1 + \dot{m}_3)), \\ \nu_B \triangleq (hA/C_p(\dot{m}_2 + \dot{m}_4)),\\ \tau_1 \triangleq (M_1/(\dot{m}_1 + \dot{m}_3)),\\ \tau_2 \triangleq (M_2/(\dot{m}_1 + \dot{m}_3)),\\ \tau_3 \triangleq (M_3/(\dot{m}_2 + \dot{m}_4)),\\ \tau_4 \triangleq (M_4/(\dot{m}_4 + \dot{m}_3)),\\ k_1 \triangleq(\dot{m}_1/(\dot{m}_1 + \dot{m}_3)),\\ k_2 \triangleq(\dot{m}_2/(\dot{m}_2 + \dot{m}_4)),\\ k_3 \triangleq(\dot{m}_3/(\dot{m}_1 + \dot{m}_3)),\\ k_4 \triangleq(\dot{m}_4/(\dot{m}_1 + \dot{m}_3)),\\ \mu \triangleq(hA/h_{lv}).\\$

This system was discretised with a sampling time of $T = 1 s$ and was discretised while rounding the input and output delays to the closest integer-multiples of $T$.

 $C_p$ 4.217 kJ/kg K water specific heat $h_{lv}$ 850 kJ/kg refrigerator heat vapourisation $T_S(0)$ 40$^{\circ}$C initial temperature in $E_s$ $T_{jin}(0)$ 40$^{\circ}$C initial water inlet temperature $j$ $T_c$ 40$^{\circ}$C refrigerant temperature $\dot{m}_j$ 1 kg/s water mass flow $j$ $M_s$ 50 kg mass of water in $E_s$ $hA$ 8 kJ/kg overall surface heat transfer $V_1$ 0.5 $\times$ 10$^{-3}$m$^4$ inlet water pipe volume 1 $V_2$ 2 $\times$ 10$^{-3}$m$^4$ inlet water pipe volume 2 $V_4$ 1.5 $\times$ 10$^{-3}$m$^4$ inlet water pipe volume 4 $\rho$ 1000 kg/m$^3$ water density

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## Publication details:

 Title Discretisation of continuous-time dynamic multi-input multi-output systems with non-uniform delays Publication Type Journal Article Authors Kassas, Z.M.