1. 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 University of Petroleum College of Information and Control Engineering Qingdao 266555, China E-mail: tianxm@upc.edu.cn

2. Outline Introduction Performance assessment using dynamic PCA Performance diagnosis using unified weighted dynamic PCA similarity Performance monitoring procedure Case study Conclusions

3. 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.

4. 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

5. 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 k s steps leads to the augmented data set PCA training data Dynamic PCA training data

6. 2. Performance assessment using dynamic PCA The principal components t and the residual variables r can be obtained as follows The two statistics, T 2 and SPE, are defined by

7. 2. Performance assessment using dynamic PCA The performance indexes for assessing the MPC controller are defined as follows Performance benchmark, the threshold for T 2 calculated by using the data of the benchmark period The T 2 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 If performance indexe is smaller than 1, it is considered that the current controller performance has deteriorated.

8. 3. Performance diagnosis using unified weighted dynamic PCA similarity The main causes for MPC performance deterioration

9. 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 X 1 and X 2, t he PCA similarity measure S PCA is defined by C 1, C 2 : the principal component subspaces corresponding to the two data sets, a: the number of principal components, θ ij : the angle between the ith principal component of C 1 and the jth principal component of C 2. It describes the degree of similarity between the two data sets X 1 and X 2.

10. 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.

11. 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.

12. 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. Therefore, not only the similarity of the principal component subspaces, but also the similarity of the residual subspaces, are considered.

13. 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.

14. 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. Output variables Input variables Disturbance variables

15. 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 ClassOperation conditionRelative parameterValue/ range C1C1 Disturbancemean+0.2 C2C2 Model mismatchGains of first column × 2.0 C3C3 Model mismatchTime constant of first column × 2.0 C4C4 Constraint/SaturationConstraint of outputs(-0.7,0.7) C5C5 DisturbanceStandard variance0.02

16. 5. Case study Performance deterioration detection results Table 2. Comparison of detection time for the PCA and DPCA based performance assessment methods. Class PCADPCA SPET2T2 T2T2 C1C1 340312322312 C2C2 315316313314 C3C3 338336330333 The DPCA based performance assessment method detected the performance deterioration earlier.

17. 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 C 1 class was the root cause of poor performance. It belongs to the C 1 class of performance deterioration. FP1C1C1 C2C2 C3C3 C4C4 C5C5 WPCA0.96211.00000.44900.31080.4876 WDPCA0.96211.00000.44880.31070.4874 Unified- WDPCA 1.00000.88510.45170.49220.6411

18. 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.

19. China University of Petroleum College of Information and Control Engineering Qingdao 266555, China E-mail: tianxm@upc.edu.cn