Project Assignment: Tank System Simulation

Objective

Simulate the dynamics of an interconnected tank system using Simulink and

MATLAB ODE code. This project aims to provide hands$-$on experience in

modeling and simulating a system with two interconnected tanks.

Equations

The dynamics of the tank levels and $\{$:$h_2)$ can be described by the following

differential equations:

$A_1\frac{d^2{h}_1}{d{t}^2}+B_1(\frac{d{h}_1}{dt}{)}^2+C_1\frac{d{h}_1}{dt}=F_{i{n}_1}(h_1,\frac{d{h}_1}{dt},h_2,\frac{d{h}_2}{dt})-F_{au{t}_1}(h_1,\frac{d{h}_1}{dt},h_2,\frac{d{h}_2}{dt})+D_{1}sin(2f_{1}t)$

$A_2\frac{d^2{h}_2}{d{t}^2}+B_2(\frac{d{h}_2}{dt}{)}^2+C_2\frac{d{h}_2}{dt}=F_{i{n}_2}(h_1,\frac{d{h}_1}{dt},h_2,\frac{d{h}_2}{dt})-F_{au{t}_2}(h_1,\frac{d{h}_1}{dt},h_2,\frac{d{h}_2}{dt})+D_{2}sin(2f_{2}t)$

where:

$A_i$: Coefficient for the second$-$order term in Tank $i$

$B_i$ : Coefficient for the non$-$linear term in Tank $i$

$C_i$: Coefficient for the finst$-$order term in Tank $i$

$D_i$ : Amplitude of the disturbance term in Tank i

$f_i$ : Frequency of the disturbance term in Tank $i$

$h_i=$ Tank level in Tank $i$

The inflow and outflow functions are modified as follows:

$F_{in}(h_1,\frac{d{h}_1}{dt},h_2,\frac{d{h}_2}{dt})=0\mathrm{.}1h_1+0\mathrm{.}2(\frac{d{h}_1}{dt})+0\mathrm{.}1h_1^2+0\mathrm{.}05h_2+0\mathrm{.}1(\frac{d{h}_2}{dt})+0\mathrm{.}2h_2^2$

$F_{\text{o}\text{u}\text{t}}(h_1,\frac{d{h}_1}{dt},h_2,\frac{d{h}_2}{dt})=0\mathrm{.}05h_1+0\mathrm{.}1(\frac{d{h}_1}{dt})+0\mathrm{.}2h_1^2+0\mathrm{.}1h_2+0\mathrm{.}2(\frac{d{h}_2}{dt})+0\mathrm{.}15h_2^2$

$F_{in}(h_1,\frac{d{h}_1}{dt},h_2,\frac{d{h}_2}{dt})=0\mathrm{.}15h_2+0\mathrm{.}3(\frac{d{h}_2}{dt})+0\mathrm{.}05h_2^2+0\mathrm{.}1h_1+0\mathrm{.}2(\frac{d{h}_1}{dt})+0\mathrm{.}1h_1^2$

$F_{\text{e}\text{u}\text{t}}(h_1,\frac{d{h}_1}{dt},h_2,\frac{d{h}_2}{dt})=0\mathrm{.}2h_2+0\mathrm{.}1(\frac{d{h}_2}{dt})+0\mathrm{.}15h_2^2+0\mathrm{.}05h_1+0\mathrm{.}1(\frac{d{h}_1}{dt})+0\mathrm{.}2h_1^2$

Initial Conditions

Initial tank levels: $h_1(0)=1m,h_2(0)=2m$

Initial velocities: $\frac{d{h}_1}{dt}(0)=0\mathrm{.}2\frac{m}{s},\frac{d{h}_2}{dt}(0)=0\mathrm{.}1\frac{m}{s}$

Constants

Croes$-$sectional area of Tank $1(A_1)$:$2m^2$

Cross$-$sectional area of Tank $2(A_2)$:$3m^2$

Coefficient for the second$-$order term in Tank $1(B_1)$:$1\mathrm{.}5$

Coefficient for the second$-$order term in Tank $2(\{$:$B_2)$:$2\mathrm{.}0$

Coefficient for the non$-$linear term in Tank $1(C_1)$:$0\mathrm{.}2$

Coefficient for the non$-$linear term in Tank $2(C_2)$:$0\mathrm{.}1$

Amplitude of the disturbance term in Tank $1(D_1)$:$0\mathrm{.}05$

Amplitude of the disturbance term in Tank $2(D_2)$:$0\mathrm{.}03$

Frequency of the disturbance term in Tank $1(f_1)$:$0\mathrm{.}2Hz$

Frequency of the disturbance term in Tank $2(f_2)$:$0\mathrm{.}15Hz$