Performance Monitoring of MPC Based on Dynamic Principal Component Analysis

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Performance Monitoring of MPC Based onDynamic
Principal Component Analysis

• Professor Xue-Min Tian
• Co-author Gong-Quan Chen, Yu-Ping Cao, Sheng
  Chen

College of Information and Control Engineering
Qingdao 266555, China E-mail tianxm_at_upc.edu.cn
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Outline

• Introduction
• Performance assessment using dynamic PCA
• Performance diagnosis using unified weighted
  dynamic PCA similarity
• Performance monitoring procedure
• Case study
• Conclusions

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1. Introduction

• The increasing popularity of model predictive
  control (MPC) in industrial applications has led
  to a high demand for performance monitoring.
• The research for the performance monitoring of
  MPC controllers is not studied as comprehensive
  as that for conventional feedback controllers. It
  mainly focus on performance assessment.
• A unified framework based on the dynamic
  principal component analysis (PCA) is proposed
  for performance assessment and diagnosis of
  constrained multi-variable model predictive
  control systems.

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2. Performance assessment using dynamic PCA

• For MPC, The model predictive error vector is
  affected by the control action and the level of
  process-model mismatch as well as the plant
  disturbances.
• The monitoring variable set can be

Control variables
Model predictive errors
Controlled variables
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2. Performance assessment using dynamic PCA

• For dynamic systems, not only the correlation of
  the process variables but also the correlation of
  the dynamic time series should be taken into
  account.
• The traditional PCA is based on analyzing
• Extending the training data to the previous ks
  steps leads to the augmented data set

Dynamic PCA training data
PCA training data
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2. Performance assessment using dynamic PCA

• The principal components t and the residual
  variables r can be obtained as follows
• The two statistics, T2 and SPE, are defined by

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2. Performance assessment using dynamic PCA

• The performance indexes for assessing the MPC
  controller are defined as follows

Performance benchmark, the threshold for T2
calculated by using the data of the benchmark
period
If performance indexe is smaller than 1, it is
considered that the current controller
performance has deteriorated.
The T2 statistic of the monitored data
Performance benchmark, the threshold for SPE
calculated by using the data of the benchmark
period
The SPE statistic of the monitored data
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3. Performance diagnosis using unified weighted
dynamic PCA similarity

• The main causes for MPC performance deterioration

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3. Performance diagnosis using unified weighted
dynamic PCA similarity

• We propose a similarity measure based
  classification method to realize the performance
  diagnosis.
• For two data sets X1 and X2, the PCA similarity
  measure SPCA is defined by
• C1, C2 the principal component subspaces
  corresponding to the two
• data sets,
• a the number of principal
  components,
• ?ij the angle between the ith
  principal component of C1 and the
• jth principal component of C2.

It describes the degree of similarity between the
two data sets X1 and X2.
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3. Performance diagnosis using unified weighted
dynamic PCA similarity

• Let being the first a
  eigenvalues of
  
• The weighted PCA (WPCA) similarity measure is
  defined as
• If the DPCA is applied to the two augmented data
  sets and ,
• we obtain the weighted DPCA (WDPCA) similarity
  measure

The more consistent the two data sets are in the
principal component subspaces, the closer to 1
the WPCA similarity measure is.
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3. Performance diagnosis using unified weighted
dynamic PCA similarity

• In the traditional process fault detection, the
  principal component subspace is used to reflect
  the main changes of process status or system.
• Noises and unmeasured disturbances are included
  in the residual subspace.
• The similarity measure of the residual subspaces
  should be considered.
• the two weighted residual
  subspaces,
• the two residual
  subspaces.

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3. Performance diagnosis using unified weighted
dynamic PCA similarity

• We are now introduce the proposed
  unified-weighted DPCA (UWDPCA) similarity measure
• ß the weighting factor, should
  appropriately be selected according to the
  specified monitored process.

Therefore, not only the similarity of the
principal component subspaces, but also the
similarity of the residual subspaces, are
considered.
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4. Performance monitoring procedure
Establish subspaces of each performance class.
Store them in the database of performance
patterns.
Calculate performance benchmark.
Online Performance monitoring
Calculate the DPCA based performance indexes.
Yes
If performance indexes are greater or equal to 1,
No
A poor performance is detected.
Find the root cause based on the unified-weighted
dynamic PCA similarity.
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5. Case study

• The Shell tower is a typical multi-variable
  constrained process.
• A constrained MPC strategy was simulated. High
  and low constraints as well as saturation limits
  were imposed on the inputs, outputs and input
  increment velocities.

Disturbance variables
Output variables
Input variables
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5. Case study

• Five prior-known causes to the performance
  deterioration

Table 1. Classes of performance deterioration and
related parameter values in generating the
training data
Class Operation condition Relative parameter Value/ range
C1 Disturbance mean 0.2
C2 Model mismatch Gains of first column 2.0
C3 Model mismatch Time constant of first column 2.0
C4 Constraint/Saturation Constraint of outputs (-0.7,0.7)
C5 Disturbance Standard variance 0.02
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5. Case study

• Performance deterioration detection results

Table 2. Comparison of detection time for the PCA
and DPCA based performance assessment methods.
Class PCA PCA DPCA DPCA
Class SPE T2 SPE T2
C1 340 312 322 312
C2 315 316 313 314
C3 338 336 330 333
The DPCA based performance assessment
method detected the performance deterioration
earlier.
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5. Case study

• Performance diagnosis results

It belongs to the C1 class of performance
deterioration.
Table 3. Performance diagnosis results for the
FP1 period.
FP1 C1 C2 C3 C4 C5
WPCA 0.9621 1.0000 0.4490 0.3108 0.4876
WDPCA 0.9621 1.0000 0.4488 0.3107 0.4874
Unified-WDPCA 1.0000 0.8851 0.4517 0.4922 0.6411
The WPCA and WDPCA similarity measures could not
locate the root cause of performance
deterioration, while the UWDPCA similarity
measure correctly identified that the C1 class
was the root cause of poor performance.
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6. Conclusions

• We have proposed a unified framework based on the
  dynamic PCA for the performance monitoring of
  constrained multi-variable MPC systems.
• The dynamic PCA based performance benchmark is
  adopted to assess the performance of a MPC
  controller.
• The root cause of performance deterioration can
  be located by pattern classification according to
  the maximum unified weighted similarity.
• A case study involving the Shell process has
  demonstrated the effectiveness of the proposed
  MPC performance assessment and diagnosis
  framework.

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Thank you.
College of Information and Control Engineering
Qingdao 266555, China E-mail tianxm_at_upc.edu.cn