This work is developed by Simulink, Matlab R2020b.
A three-tank system, as sketched in Fig. 1, has typical characteristics of tanks, pipelines and pumps used in chemical industry and thus often serves as a benchmark process in laboratories for process control.
The model and the parameters of the three-tank system introduced here are from the laboratory setup DTS200 (see [3]).
| Parameters |
Symbol |
Value |
Unit |
| cross section area of tanks |
$\mathcal A$ |
$154$ |
$cm^2$ |
| cross section area of pipes |
$s_n$ |
$0.5$ |
$cm^2$ |
| max. height of tanks |
$H_{max}$ |
$62$ |
$cm$ |
| max. flow rate of pump 1 |
$Q_{1max}$ |
$150$ |
$cm^3/s$ |
| max. flow rate of pump 2 |
$Q_{2max}$ |
$150$ |
$cm^3/s$ |
| coeff. of flow for pipe 1 |
$a_1$ |
$0.46$ |
|
| coeff. of flow for pipe 2 |
$a_2$ |
$0.60$ |
|
| coeff. of flow for pipe 3 |
$a_3$ |
$0.45$ |
|
$ s_{13} = s_{23} = s_0 = s_n = 0.5\ cm^2$
Input variables:
$ u = [Q1\ Q2]^T$
Measurements:
$ y = [h_1\ h_2\ h_3]^T $
Applying the incoming and outgoing mass flows under consideration of Torricelli’s law, the dynamics of DTS200 is modeled by
$$
\begin{equation}
\begin{split}
&\mathcal A\dot{h_1}=Q_1-Q_{13},\mathcal A\dot{h_2}=Q_2+Q_{32}-Q_{20},\mathcal A\dot{h_3}=Q_{13}-Q_{32}\\
&Q_{13}=a_1s_{13}sign(h_1-h_3)\sqrt{2g|h_1-h_3|} \\
&Q_{32}=a_3s_{23}sign(h_3-h_2)\sqrt{2g|h_3-h_2|},Q_{20}=a_2s_0\sqrt{2gh_2} \\
\end{split}
\end{equation}
$$
The linear form of the above model can be achieved by a linearization at an operating point as follows:
$$
\begin{equation}
\begin{split}
\begin{array}{l}
\dot x = Ax + Bu,y = Cx\\
x = \left[ {\begin{array}{_{20}{c}}
{{h_1} - {h_{1,o}}}\\
{{h_2} - {h_{2,o}}}\\
{{h_3} - {h_{3,o}}}
\end{array}} \right],u = \left[ {\begin{array}{_{20}{c}}
{{Q_1} - {Q_{1,o}}}\\
{{Q_2} - {Q_{2,o}}}
\end{array}} \right],{Q_o} = \left[ {\begin{array}{_{20}{c}}
{{Q_{1,o}}}\\
{{Q_{2,o}}}
\end{array}} \right]\\
A = \frac{{\partial f}}{{\partial g}}\left| {_{h = {h_o}},B = } \right.\left[ {\begin{array}{_{20}{c}}
{\begin{array}{_{20}{c}}
{\frac{1}{{\cal A}}}\\
0\\
0
\end{array}}&{\begin{array}{_{20}{c}}
0\\
{\frac{1}{{\cal A}}}\\
0
\end{array}}
\end{array}} \right],C = \left[ {\begin{array}{*{20}{c}}
1&0&0\\
0&1&0\\
0&0&1
\end{array}} \right]
\end{array}
\end{split}
\end{equation}
$$
$$
\begin{equation}
\begin{split}
f\left( h \right) = \left[ {\begin{array}{_{20}{c}}
{\frac{-a_1 s_{13}sign(h_1-h_3)\sqrt{2g|h_1-h_3|})}{{\cal A}}}\\
{\frac{a_3 s_{23}sign(h_3-h_2)\sqrt{2g|h_3-h_2|}-a_2s_0\sqrt{2gh_2})}{{\cal A}}}\\
{\frac{a_1 s_{13}sign(h_1-h_3)\sqrt{2g|h_1-h_3|}-a_3s_{23}sign(h_3-h_2)\sqrt{2g|h_3-h_2|}}{{\cal A}}}
\end{array}} \right],h = \left[ {\begin{array}{_{20}{c}}
{h_1}\\
{h_2}\\
{h_3}
\end{array}} \right]
\end{split}
\end{equation}
$$
Twelve different faults were injected starting at the 200th sample.
| Fault ID |
Description |
Location |
| 1 |
$f_1(t) = 6 \sin\frac{t-200}{6\pi}$ |
Actuator |
| 2 |
$f_2(t) = 0.02(t-200)$ |
Actuator |
| 3 |
$f_3(t) = 0.005(t-200) + (-1)^{\lfloor \frac{t}{48} - \frac{5}{4}\lfloor \frac{t}{60} \rfloor\rfloor }$ |
Actuator |
| 4 |
$f_4(t) = -0.005(t-200)$ |
Sensor |
| 5 |
$f_5(t) = 0.005(t-200)$ |
Sensor |
| 6 |
$f_6(t) = 0.008(t-200)$ |
Sensor |
| 7,8,9 |
$f_7(t) =f_8(t) =f_9(t) = 0.00005 (t-200)$ |
Process |
| 10 |
$f_{10}(t) =-0.0002(t-200)$ |
Process |
| 11 |
$f_{11}(t) =-0.0003(t-200)$ |
Process |
| 12 |
$f_{12}(t) =-0.0005(t-200)$ |
Process |
The system contianed fault signals can be represented by
$$
\begin{equation}
\begin{split}
&\mathcal{A} \dot{h}_1 = Q_1 + f_1 + f_2 - Q_{13} - f_7 \check Q_{10} + \omega_1\\
&\mathcal{A} \dot{h}_2 = Q_2 + f_1 + f_3 + Q_{32} - f_8 (f_{11}+1) \check Q_{20} + \omega_2 \\
&\mathcal{A} \dot{h}_3 = Q_{13} - Q_{32} - f_9 \check Q_{30} + \omega_3\\
&Q_{13} = a_1 s_{13} (f_{10}+1) sign(h_1-h_3) \sqrt{2g \left| h_1 - h_3 \right| } \\
&Q_{32} = a_3 s_{23} (f_{12}+1) sign(h_3-h_2) \sqrt{2g \left| h_3 - h_2 \right| } \\
&\check Q_{i0} = a_i s_{0} \sqrt{2g h_i }, i = 1,2,3
\end{split}
\end{equation}
$$
The collected training and testing data sets can be describled as
[train].mat -> normal (16008 × 5)
model1[train].mat -> fault01 (2001 × 5), ..., fault12 (2001 × 5)
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