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China University of Petroleum. Performance Monitoring of MPC Based on Dynamic 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: [email protected]. Outline. - PowerPoint PPT Presentation
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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
China University of Petroleum
College of Information and Control Engineering
Qingdao 266555, China E-mail: [email protected]
Outline
Introduction
Performance assessment using dynamic PCA
Performance diagnosis using unified weighted
dynamic PCA similarity
Performance monitoring procedure
Case study
Conclusions
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.
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 T
1 1 1( ) [ ( ) ( ) ( ) ( ) ( ) ( )]m n nx k u k u k e k e k y k y k
Control variables
Model predictive errors
Controlled variables
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
T
T
0 0
T
( )
( 1)( )
( 1)
x k
x kk
x k n
X X
0 0 0 0[ ( ) ( 1) ( )]sk k k k X X X X
PCA training data
Dynamic PCA training data
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
T
T( )
new
new
x
x
t P
r I PP
2 T 2
TSPE
T
t Σ t
r r
2. Performance assessment using dynamic PCA
The performance indexes for assessing the MPC controller are defined as follows
)()(
2
2
2
kT
Lk
II
TI
T
Performance benchmark, the threshold for T2 calculated by using the data of the benchmark period
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
SPE
SPE ( )SPE ( )
I
II
Lk
k
If performance indexe is smaller than 1, it is considered that the current controller performance has deteriorated.
3. Performance diagnosis using unified weighted dynamic PCA similarity
The main causes for MPC performance deterioration
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 SP
CA is defined by
C1, C2 : the principal component subspaces corresponding to the two
data sets,
a: the number of principal components,
θij : the angle between the ith principal component of C1 and the
jth principal component of C2.
2 T TPCA 1 2 2 1
1 1
1 1cos tr
a a
iji j
Sa a
C C C C
It describes the degree of similarity between the two data sets X1 and X2.
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
T T1 2 2 1
PCA(1) (2)
1
tra
i ii
S
C C C C
( ) ( ) ( )1 2diag , , ,i i i
i a
( ) ,1ij j a T , 1, 2i i i X X
i i i C C
The more consistent the two data sets are in the principal component subspaces, the closer to 1 theWPCA similarity measure is.
1X 2X
T T1 2 2 1
DPCA-PCS(1) (2)
1
tra
i ii
S
C C C C
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.
T T1 2 2 1
DPCA-RS(1) (2)
1
tra
i ii
S
G G G G
i i i G G
iG
3. Performance diagnosis using unified weighted dynamic PCA similarity
We are now introduce the proposed unified-weighted DPCA (UWDPCA) similarity measure
β : the weighting factor, should appropriately be selected according to the specified monitored process.
DPCA DPCA-PCS DPCA-RS(1 )S S S
Therefore, not only the similarity of the principal component subspaces, but also the similarity of the residual subspaces, are considered.
4. Performance monitoring procedure
Establish subspaces of each performance class.
Store them in the database of performance patterns.
Calculate performance benchmark.
Calculate the DPCA based performance indexes.
Online Performance monitoring
If performance indexes are greater or equal to 1, No
Yes
A poor performance is detected.
Find the root cause based on the unified-weighted dynamic PCA similarity.
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.
27 28 27 27
1 118 14 15
2 2
20 223 3
4.05 1.77 5.88 3.60
50 1 60 1 50 1( ) ( )5.39 5.72 6.90
( ) ( )50 1 60 1 40 1
( ) ( )4.38 4.42 7.20
33 1 44 1 19 1
s s s
s s s
s s
e e e e
s s sy s u se e e
y s u ss s s
y s u se e
s s s
27
15 151
25
1.44
45 1 40 1( )1.52 1.83
( )25 1 20 1
1.14 1.26
27 1 32 1
s s
s s
s
e
s sd se e
d ss s
e
s s
Output variables
Input variables
Disturbance variables
5. Case study
Five prior-known causes to the performance deterioration
Table 1. Classes of performance deteriorationand 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
5. Case study
Performance deterioration detection results
Table 2. Comparison of detection time for thePCA and DPCA based performance assessment methods.
ClassPCA DPCA
SPE T2 SPE T2
C1 340 312 322 312
C2 315 316 313 314
C3 338 336 330 333
The DPCA based performance assessment methoddetected the performance deterioration earlier.
5. Case study
Performance diagnosis results
Table 3. Performance diagnosis results for the FP1 period.
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.
It belongs to the C1 class of performance deterioration.
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
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.
China University of Petroleum
College of Information and Control Engineering
Qingdao 266555, China E-mail: [email protected]