• > Brief Introduction
• > System dynamics
• > Linearization
• > Fault description
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  • Test:
• > My blog
• > References

> Brief Introduction

​ 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$

> System dynamics

​ 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

$$ \begin{equation}     \begin{split}         &\mathcal A\dot{h_1}=Q_1-Q_{13},\mathcal A\dot{h_2}=Q_2+Q_{32}-Q_{20},\mathcal A\dot{h_3}=Q_{13}-Q_{32}\\ &Q_{13}=a_1s_{13}sign(h_1-h_3)\sqrt{2g|h_1-h_3|} \\ &Q_{32}=a_3s_{23}sign(h_3-h_2)\sqrt{2g|h_3-h_2|},Q_{20}=a_2s_0\sqrt{2gh_2} \\     \end{split} \end{equation} $$

> Linearization

​ The linear form of the above model can be achieved by a linearization at an operating point as follows:

$$ \begin{equation} \begin{split} \begin{array}{l} \dot x = Ax + Bu,y = Cx\ x = \left[ {\begin{array}{{20}{c}} {{h_1} - {h_{1,o}}}\ {{h_2} - {h_{2,o}}}\ {{h_3} - {h_{3,o}}} \end{array}} \right],u = \left[ {\begin{array}{{20}{c}} {{Q_1} - {Q_{1,o}}}\ {{Q_2} - {Q_{2,o}}} \end{array}} \right],{Q_o} = \left[ {\begin{array}{{20}{c}} {{Q_{1,o}}}\ {{Q_{2,o}}} \end{array}} \right]\ A = \frac{{\partial f}}{{\partial g}}\left| {_{h = {h_o}},B = } \right.\left[ {\begin{array}{{20}{c}} {\begin{array}{{20}{c}} {\frac{1}{{\cal A}}}\ 0\ 0 \end{array}}&{\begin{array}{{20}{c}} 0\ {\frac{1}{{\cal A}}}\ 0 \end{array}} \end{array}} \right],C = \left[ {\begin{array}{*{20}{c}} 1&0&0\ 0&1&0\ 0&0&1 \end{array}} \right] \end{array} \end{split} \end{equation} $$

$$ \begin{equation} \begin{split} f\left( h \right) = \left[ {\begin{array}{{20}{c}} {\frac{-a_1 s_{13}sign(h_1-h_3)\sqrt{2g|h_1-h_3|})}{{\cal A}}}\ {\frac{a_3 s_{23}sign(h_3-h_2)\sqrt{2g|h_3-h_2|}-a_2s_0\sqrt{2gh_2})}{{\cal A}}}\ {\frac{a_1 s_{13}sign(h_1-h_3)\sqrt{2g|h_1-h_3|}-a_3s_{23}sign(h_3-h_2)\sqrt{2g|h_3-h_2|}}{{\cal A}}} \end{array}} \right],h = \left[ {\begin{array}{{20}{c}} {h_1}\ {h_2}\ {h_3} \end{array}} \right] \end{split} \end{equation} $$

> Fault description

​ 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:

[train].mat -> normal (16008 × 5)

Test:

model1[train].mat -> fault01 (2001 × 5), ..., fault12 (2001 × 5)

> My blog

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> References

[1] Z. Pan, H. Chen, Y. Wang, B. Huang, and W. Gui, "A new perspective on ae-and vae-based process monitoring," TechRxiv, Apr. 2022, doi.10.36227/techrxiv.19617534.
[2] S. X. Ding, "Data-driven design of fault diagnosis and fault-tolerant control systems," London, U.K.: Springer, 2014.
[3] Z. Pan, Y. Wang, k. Wang, G. Ran, H. Chen, and W. Gui, "Layer-Wise Contribution-Filtered Propagation for Deep Learning-Based Fault Isolation," Int. J. Robust Nonlinear Control, Jul. 2022, doi.10.1002/rnc.6328
[4] Z. Pan, Y. Wang, K. Wang, H. Chen, C. Yang, and W. Gui, "Imputation of Missing Values in Time Series Using an Adaptive-Learned Median-Filled Deep Autoencoder," IEEE Trans. Cybern., 2022, doi.10.1109/TCYB.2022.3167995
[5] Y. Wang, Z. Pan, X. Yuan, C. Yang, and W. Gui, "A novel deep learning based fault diagnosis approach for chemical process with extended deep belief network," ISA Trans., vol. 96, pp. 457–467, 2020.
[6] Z. Pan, Y. Wang, X. Yuan, C. Yang, and W. Gui, "A classification-driven neuron-grouped sae for feature representation and its application to fault classification in chemical processes," Knowl.-Based Syst., vol. 230, p. 107350, 2021.