1. KE-90.5100 Process Monitoring (4cr)

2. Course information The course consists of Lectures: Exercises: Exam:
Tue 14 – 16, Ke3
Thu 12 – 14, Ke3
Exercises:
Fri 10:00 – 12:00, computer class room
Exam:
Oct
Jan

3. Course information The course consists of The grade consists of
5 obligatory homeworks (presented during exercises):
Assistant: M.Sc. Cheng Hui
submit report by within 2 weeks
All exercises have to be OK before exam
Assignments: Group work to be presented at the end of the course
The grade consists of
Assignment (30%)
Exam (70%)

4. Course material Course web pages Slides Handouts Exercises/Homework
Material from assignment

5. Course Staff Hui Cheng, Alexey Zakharov Fernando Dorado Di Zhang
reception Thu 10 –11, F302
Alexey Zakharov
Fernando Dorado
Di Zhang

6. Scope of the course Tools of the process control engineer Classical
-PID
-Bode,
-Nyqvist -…
Modern
Control
- Multivariable
control
- MPC
- IMC -…
Modeling and
simulation
- First principles
modeling
- Identification
Simulation of
dynamic systems
- …
Intelligent
methods
- Neural
Networks
- Fuzzy logic
- GA
Multivariate data analysis
PCA
PLS
SOM KE
Basics of process
automation
KE Control
applications in
process industries KE
Process Modelling
and Simulation KE
Process Monitoring

7. Scope of the course After the course you will know
How and when to use some statistical process monitoring methods
The basics of neural networks and fuzzy systems and how to utilize them in monitoring and control
The basics of genetic algorithms

8. Introduction The idea of process monitoring
Goals of process monitoring
The process monitoring loop
Process monitoring methods
Data selection
Data pretreatment
Univariate vs. multivariate statistics

9. The idea
To monitor or monitoring generally means to be aware of the state of a system

10. The idea
Multivariate data in industrial processes: impossible for human operator to monitor hundreds of measurements for possible faults
Costs and safety issues with equipment malfunctions / process disturbances
Shutdowns expensive
Amount of maintenance breaks
New equipment: delivery time
Safe working environment for plant staff

11. The idea Process equipment malfunctions and process disturbances
E.g. contamination of sensors, faults of analyzers, clogging of filters, degradation of catalyst, changing properties of feed stock, leaks, actuator faults etc…
How to detect early enough?
How to distinguish between?

12. The goals Get indication of As early as possible to
process disturbances and
malfunctions in process equipment
As early as possible to
Increase uniform quality of the products
Improve safety
Minimize maintenance costs

13. The process monitoring loopno
Fault
Detection
yes
Fault
Identification
Fault
Diagnosis
Process
Recovery
Fault Detection = determining whether a fault has occurred
Fault Identification = identifying the variables most relevant for diagnosing the fault
Fault Diagnosis = determining which fault occurred
Process Recovery = Removing the effect of the fault

14. Process monitoring methods
+ Easier to implement
− Don’t include
process knowledge
+ Includes process
knowledge
− Needs process models
Model-Based
Data-Based
Qualitative
Quantitative
Statistical
Neural
Networks

15. Process monitoring methods
Model-Based
Residual based observers
Parity-space based
Causal models
Signed digraphs
Data-Based
Qualitative
Quantitative
Trend analysis
Rule-based
Neural
Networks
introductory
Statistical
PCA
PLS
RPCA
Scope of the course
PCA
PLS
RPCA..
SOM
MLP
RBFN..
+ GA and some
Control

16. Data selection Training data includes: Testing data
If goal is to verify that process is in normal state (monitoring) the process data used for training should represent normal conditions
If the goal is to identify if the process is in a normal or some specified faulty state the process data used for training should include all the possible faulty states as well as the normal state
Testing data
Completely independent from training data!

17. Data pretreatment Main procedures in pretreatment are:
Removing variables
The data set may contain variables that have no relevant information for monitoring
Removing outliers
Outliers are isolated measurement values that are erroneous and will cause biased parameter estimates for the method used. Methods for removing outliers include visual inspection and statistical tests.
Autoscaling
Process data often needs to be scaled to avoid particular variables dominating the process monitoring method (autoscaling = subtract mean and divide by standard deviation)

18. Univariate vs. multivariate statistics
The most simple type of monitoring is based on univariate statistics
Individual thresholds are determined for each variable (Shewart charts)
upper control limit
lower control limit
target
out-of-control
in-control

19. Univariate vs. multivariate statistics
Tight thresholds result in a high false alarm rate but low missed alarm rate
Limits too spread apart result in low false alarm rates but high missed alarm rates
A trade off between false alarms and missed alarms
normal
faulty
threshold
missed alarms
false alarms

20. Univariate vs. multivariate statistics
Univariate methods determine threshold for each variable individually without considering other variables
The fact that there are correlations between the variables is ignored
The multivariate T2 statistic takes into account these correlations
The T2 statistic is based on an eigenvector decompostion of the covariance matrix

21. Univariate vs. multivariate statistics
Comparison of univariate statistics and T2 statistic x2
T2 statistical confidence
region
Abnormal data
can be classified
as ok! x1
univariate statistical
confidence region
More on T2 later in the PCA part

22. Principal Component Analysis
Process monitoring
Model-Based
Data-Based
Qualitative
Quantitative
Statistical
Neural
Networks
PCA

23. Principal component analysis
Linear method
Greatly reduces the number of variables to be monitored – data compression
Based on eigenvalue and eigenvector decomposition of the covariance matrix
Simple indexes for monitoring

24. Principal component analysis
The idea of PCA is to form a minimum number of new variables to describe the variation of the data by using linear combinations of the original variables x2
PCA model
PC1: Direction of
largest variation
PC2: Direction of
2. largest variation x1
PC1
PC2

25. PCA- principal components
The new axes = principal components are selected according to the direction of highest variation in the original data set
The new axes are orthogonal x2 x1
PC2
PC1
PCA

26. PCA- principal components
Principal components will rotate the data set so that the different groups might be separable
Separation of faulty data
on pc2-axis
No separation of
faulty data (red)
with original axis
pc2 x2
pc1 x1

27. PCA- scores
The PCA scores are the values of the original data points on the principal component axes
PC2
Sample 1
PC1

28. PCA model calculation
The direction of the principal components = eigenvectors
The variation of the data along a eigenvector is given by the corresponding eigenvalue x2
PC1=e1=w11x1+w12x2
PCA model x1

29. Principal component analysis
Principal Component Analysis is based on a eigenvalue/eigenvector decomposition of the covariance matrix of the data set (X) with n observations (rows) and m variables (columns).
X is centered to have zero mean and scaled so that each variable has the same variance (especially if the variables have different units
Covariance
matrix C
The covariance matrix
can be decomposed:

30. Principal component analysis
The decomposition can be done by solving
In Matlab the eigenvalues and eigenvectors are obtained by:
[eig_vec,eig_val] = eig(C)
Each eigenvector is a column in eig_vec and the eigenvalues are on the diagonal of eig_val
A score-matrix T can be calculated: T = XVk
Vk is a transformation matrix containing the eigenvectors corresponding to the k largest eigenvalues. The user chooses k (more later)
T is a ”compressed” version of X
and

31. Principal component analysis
The compressed data can be decompressed:
There is a residual matrix E between the original data and the decompressed data
Combining the equations and rewriting the TVkT term:
pi are the principal components
The scores ti are the distances along the principal component pi

32. Principal component analysis
Matlab demo: Compressing and decompressing data via eigenvalue decomposition

33. Principal Component Analysis
Example: Calculate (by hand) the eigenvalues and eigenvectors for:
Specify that
also verify
Recall that

34. Monitoring indexes
SPE – variation outside the model, distance off the model plane
Hotelling T2 – variation inside the model, distance from the origin along the model plane
Points with SPE
violation
SPE
Points with
T2 violation
PC2
PCA model
plane
PC1

35. Monitoring indexes
T2: Measures systematic variations of the process. For individual observation:
SPE: Measures the random variations of the process

36. Different view
SPE

37. Monitoring indexes
The process can be also monitored by tracking the score values for each principal component

38. PCA - calculate the model
Zero-mean the original data set.
Compute the covariance matrix C
Compute the eigenvalues and eigenvectors. Modify the matrices so, that the eigenvalues are in decreasing order (remember to do the same operations to the eigenvectors)
Choose how many principal components to use. (Plot eigenvalues, captured variance)
Form the transformation matrix Vk (principal components) and eigenvalue matrix Λk.
Compute confidence limits for the scores of every PC
Compute the Hotelling T2 & SPE limits

39. PCA - calculate the model
Scale the original data set X (zero-mean, unit variance if necessary).
Calculate the covariance matrix

40. PCA - calculate the model
Compute the eigenvalues and eigenvectors. The eigenvalues are computed according to
The eigenvectors can be solved from the equation:
Remember to keep the eigenvectors in the same order as the eigenvalues

41. PCA - calculate the model
4. Choose how many principal components to use. (Plot eigenvalues, captured variance) λ
var capt. %
tot. var capt. % 1
3.18
28.91 2
2.34
21.31
50.22 3
1.74
15.81
66.03 4
1.08
9.78
75.80 5
0.91
8.24
84.04 6
0.79
7.15
91.20 7
0.49
4.48
95.67 8
0.22
2.04
97.71 9
0.14
1.30
99.01 10
0.11
0.97
99.98 11
0.00
0.02
100.00

42. PCA - calculate the model
With 7 PCs 96% of the variance is captured

43. PCA - calculate the model
Form the transformation matrix Vk (eigenvectors) and eigenvalue matrix ΛK .

44. PCA - calculate the model
6. Compute confidence limits for the scores of every PC
In Matlab:
sqrt(lamda)*tinv(alfa+(1-alfa)/2,N-1) α
α = confidence level

45. PCA - calculate the model
7. Compute Hotelling T2 limit & SPE limit
where the F(K,N-K,) corresponds to the probability point on the F-distribution with (K,N-K) degrees of freedom and confidence level . N = # of data samples, K = # of PCs
“Unused” eigenvalues
where m= number of original variables, k=number of principal components in the model, cα= upper limit from normal distribution with conf. level α

46. PCA – the new data
Scale the new data set with training data scaling values .
Compute PCA transformation (i.e. scores for all the chosen principal components) using Vk.
Compare the scores to confidence limits. If inside the limits, then OK.
Compute the Hotelling T2 & SPE values for the new data set
Compare the Hotelling T2 & SPE values to the limits. If under the limit, then OK

47. PCA – the new data
Scale the new data set with training data scaling values
Compute PCA transformation (i.e. scores for all the chosen principal components) using Vk.

48. PCA – the new data
3. Compare the scores to confidence limits. If inside the limits, then OK
4. Compute the Hotelling T2 & SPE values for the new data set
5. Compare the Hotelling T2 & SPE values to the limits. If under the limit, then OK

49. Fault identification
Once a fault has been detected, the next step is determine the cause of the out-of-control status
One way to handle this is by using so called contribution plots

50. Contribution plots
In response to T2 violations we can obtain contribution plots according to:
For observation xi, find the r cases when the normalized scores ti2/λi > T2/a
Calculate the contribution of each variable xj to the out-of-control scores ti
When conti,j is negative set it equal to zero
Calculate the total contribution of the jth process variable
Plot CONTj for all process variables in a single plot

51. Contribution plots

52. Contribution plots
Can also be made from the SPE index

53. PCA derivation
We want a linear combination of the elements of x that has maximal variance
Next we look for a linear function eT2x that is uncorrelated with eT1x but has maximum variance and so on
We can write
We want to have unit length for the scaling vector e (constraint)

54. PCA derivation The maximization can be done using Lagrange multipliers
We have
Differentiation gives
This is an eigenvalue problem. It is the largest eigenvalue that corresponds to the solution since
Unit lenght
Maximizing the variance = maximizing λ

55. PCA derivation For the second component we can write
Differentiation gives
Multiplication with eT1 gives
uncorrelated
(A)
introduce in (A)
Again an eigenvalue problem

56. PCA derivation Since e2 must be different from e1 and hence λ ≠ λ1
We still want to maximize variation  λ is the second largest eigenvalue
A similar analysis can be done for the third, fourth etc. PCs.