# Three Compartment Model Describing the Dynamics of a Drug in a Tissue

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# 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.

# 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 This vapour, withdrawn in $S1$ and $S2$, reduces the refrigerant mass flow rate along the cooling system, and only the saturated liquid portion is used for cooling purposes.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

4

## 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.

# T-S fuzzy model

## Model description:

Consider a T-S fuzzy model

Plant Rule $i$: If $x_1(t)$ is $F_1^1(x_1(t))$

Then $x(t+1) = A_ix(t)+B_iu(t),$

where

\begin{align*} A_1 &=\left[\matrix{-a & 2\cr -0.1 & b}\right], A_2=\left[\matrix{-a & 2\cr-0.1 & b }\right], A_3=\left[\matrix{-0.9 & 0.5\cr -0.1 & -1.7}\right] \\ B_1 &=\left[\matrix{b\cr 4}\right], B_2=\left[\matrix{b\cr 4.8}\right], B_3=\left[\matrix{3\cr 0.1}\right]. \end{align*}

The parameters $a$ and $b$ are adjusted to compare the relaxation of stabilization conditions.

## Publication details:

 Title Stabilization of discrete-time nonlinear control systems - Multiple fuzzy Lyapunov function approach Publication Type Conference Paper Authors Kau, Shih-Wei, Huang Xin-Yuan, Shiu Sheng-Yu, and Fang Chun-Hsiung

# A linear system

## Model description:

Consider a linear system represented by the transfer function

$$G(s)=\dfrac{c}{s(s+a)}$$

where $a$ and $c>0$ are unknowns constants, and the reference model

$$G_m(s)=\dfrac{\omega^2}{s^2 + 2\zeta\omega s + \omega^2}.$$

2

## Publication details:

 Title Adaptive output feedback control of nonlinear systems represented by input-output models Publication Type Journal Article Authors Khalil, H.K.

# Three Compartment Model Describing the Dynamics of a Drug in a Tissue

## Model description:

Consider the system

\begin{align*} \dot{x}_1 &= p_{13}x_3 + p_{12}x_2 - p_{21}x_1+u \\ \dot{x}_2 &= -p_{12}x_2 + p_{21}x_1 \\ \dot{x}_3 &= -p_{13}x_3 \\ y &= x_2, \end{align*}

where $x=[x_1, x_2, x_3]$ is the state vector, e.g. $x_1,x_2,x_3$ are drug masses in compartment 1, 2 and 3, respectively; initial conditions are $x_1(0)=0$, $x_2(0)=0$, $x_3(0)=0$; $u$ is the drug input; $y$ is the measured drug output; $p=[p_{12},p_{21},p_{13}]$ is the rate parameter vector (assumed constant).

3

## Publication details:

 Title A new differential algebra algorithm to test identifiability of nonlinear systems with given initial conditions Publication Type Conference Paper Authors Saccomani, M.P., Audoly S., Bellu G., and D'Angio L.