Performance Monitoring of MPC Based on Dynamic Principal Component Analysis Professor Xue-Min Tian Co-author: Gong-Quan Chen, Yu-Ping Cao, Sheng Chen China.

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]