KE-90.5100 Process Monitoring (4cr)

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 31 2008 Jan 8 2009

Course information The course consists of The grade consists of 5 obligatory homeworks (presented during exercises): Assistant: M.Sc. Cheng Hui submit report by email within 2 weeks (di.zhang@tkk.fi) 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%)

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

Course Staff Hui Cheng, Alexey Zakharov Fernando Dorado Di Zhang hui.cheng@hut.fi, reception Thu 10 –11, F302 Alexey Zakharov Fernando Dorado Di Zhang

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-90.2100 Basics of process automation KE-90.4510 Control applications in process industries KE-90.3100 Process Modelling and Simulation KE-90.5100 Process Monitoring

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

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

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

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

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?

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

The process monitoring loop no 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

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

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

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!

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)

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

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

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

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

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

Different view SPE

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

Contribution plots

Contribution plots Can also be made from the SPE index

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)

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

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

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.