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KE-90.5100 Process Monitoring (4cr)

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KE-90.5100Process Monitoring (4cr)

- The course consists of
- Lectures:
- Tue 14 – 16, Ke3
- Thu 12 – 14, Ke3

- Exercises:
- Fri 10:00 – 12:00, computer class room

- Exam:
- Oct 31 2008
- Jan 8 2009

- Lectures:

- The course consists of
- 5 obligatory homeworks (presented during exercises):
- Assistant: M.Sc. Cheng Hui
- submit report by email within 2 weeks ([email protected])
- All exercises have to be OK before exam

- Assignments: Group work to be presented at the end of the course

- 5 obligatory homeworks (presented during exercises):
- The grade consists of
- Assignment (30%)
- Exam (70%)

- Course web pages
- Slides
- Handouts
- Exercises/Homework
- Material from assignment

- Hui Cheng,
[email protected], reception Thu 10 –11, F302

- Alexey Zakharov
- Fernando Dorado
- Di Zhang

Tools of the process control engineer

Classical

Control

-PID

-Bode,

-Nyqvist

-…

Modern

Control

- Multivariable

control

- MPC

- IMC

-…

Intelligent

methods

- Neural

Networks

- Fuzzy logic

- GA

- Multivariate data analysis
- PCA
- PLS
- SOM

- Modeling and
- simulation
- - First principles
- modeling
- - Identification
- Simulation of
- dynamic systems
- - …

KE-90.2100

Basics of process

automation

KE-90.5100

Process Monitoring

KE-90.4510 Control

applications in

process industries

KE-90.3100

Process Modelling

and Simulation

- 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

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

The idea

To monitor or monitoring generally means to be aware of the state of a system

- 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

- 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?

Get indication of

- process disturbances and
- malfunctions in process equipment
As early as possible to

- Increase uniform quality of the products
- Improve safety
- Minimize maintenance costs

no

Fault

Identification

Process

Recovery

Fault

Diagnosis

Fault

Detection

yes

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

Process monitoring

+ Easier to implement

− Don’t include

process knowledge

+ Includes process

knowledge

− Needs process models

Model-Based

Data-Based

Quantitative

Qualitative

Statistical

Neural

Networks

Model-Based

Residual based observers

Parity-space based

Causal models

Signed digraphs

Statistical

PCA

PLS

RPCA

Process monitoring

Data-Based

Quantitative

Qualitative

Trend analysis

Rule-based

Neural

Networks

introductory

Scope of the course

PCA

PLS

RPCA..

SOM

MLP

RBFN..

+ GA and some

Control

- Training data includes:
- 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!

- 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)

- Removing variables

out-of-control

in-control

upper control limit

target

lower control limit

- The most simple type of monitoring is based on univariate statistics
- Individual thresholds are determined for each variable (Shewart charts)

- 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

false alarms

missed alarms

- 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

- 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

Process monitoring

Model-Based

Data-Based

Quantitative

Qualitative

Statistical

Neural

Networks

PCA

- 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

PCA model

PC2

x2

PC1

PC1: Direction of

largest variation

PC2: Direction of

2. largest variation

x1

- 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

PC2

x2

PCA

PC1

x1

- The new axes = principal components are selected according to the direction of highest variation in the original data set
- The new axes are orthogonal

- 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

- The PCA scores are the values of the original data points on the principal component axes

PC2

Sample 1

PC1

- 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

- 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:

- 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

- 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

- Matlab demo: Compressing and decompressing data via eigenvalue decomposition

- Example: Calculate (by hand) the eigenvalues and eigenvectors for:
Specify that

also verify

Recall that

- 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

- T2: Measures systematic variations of the process. For individual observation:
- SPE: Measures the random variations of the process

Different view

SPE

- The process can be also monitored by tracking the score values for each principal component

- 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

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

- 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

4. Choose how many principal components to use. (Plot eigenvalues, captured variance)

With 7 PCs 96% of the variance is captured

- Form the transformation matrix Vk (eigenvectors) and eigenvalue matrix ΛK .

6. Compute confidence limits for the scores of every PC

sqrt(lamda)*tinv(alfa+(1-alfa)/2,N-1)

In Matlab:

α

α = confidence level

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

- 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

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

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

- 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

- 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

- Can also be made from the SPE index

- 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)

- 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

- For the second component we can write
- Differentiation gives
- Multiplication with eT1 gives

uncorrelated

(A)

introduce in (A)

Again an eigenvalue problem

- 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.