Statistical Monitoring of Complex Multivatiate Processes With Applications in Industrial Process Control

Uwe Kruger, The Petroleum Institute, Abu Dhabi, United Arab Emirates

Preface xiii

Acknowledgements xvii

Abbreviations xix

Symbols xxi

Nomenclature xxiii

Introduction xxv

PART I FUNDAMENTALS OF MULTIVARIATE

STATISTICAL PROCESS CONTROL 1

1 Motivation for multivariate statistical process control 3

1.1 Summary of statistical process control 3

1.1.1 Roots and evolution of statistical process control 4

1.1.2 Principles of statistical process control 5

1.1.3 Hypothesis testing, Type I and II errors 12

1.2 Why multivariate statistical process control 15

1.2.1 Statistically uncorrelated variables 16

1.2.2 Perfectly correlated variables 17

1.2.3 Highly correlated variables 19

1.2.4 Type I and II errors and dimension reduction 24

1.3 Tutorial session 26

2 Multivariate data modeling methods 28

2.1 Principal component analysis 30

2.1.1 Assumptions for underlying data structure 30

2.1.2 Geometric analysis of data structure 33

2.1.3 A simulation example 34

2.2 Partial least squares 38

2.2.1 Assumptions for underlying data structure 39

2.2.2 Deflation procedure for estimating data models 41

2.2.3 A simulation example 43

2.3 Maximum redundancy partial least squares 49

2.3.1 Assumptions for underlying data structure 49

2.3.2 Source signal estimation 50

2.3.3 Geometric analysis of data structure 52

2.3.4 A simulation example 58

2.4 Estimating the number of source signals 65

2.4.1 Stopping rules for PCA models 65

2.4.2 Stopping rules for PLS models 76

2.5 Tutorial Session 79

3 Process monitoring charts 81

3.1 Fault detection 83

3.1.1 Scatter diagrams 84

3.1.2 Non-negative quadratic monitoring statistics 85

3.2 Fault isolation and identification 93

3.2.1 Contribution charts 95

3.2.2 Residual-based tests 98

3.2.3 Variable reconstruction 100

3.3 Geometry of variable projections 111

3.3.1 Linear dependency of projection residuals 111

3.3.2 Geometric analysis of variable reconstruction 112

3.4 Tutorial session 119

PART II APPLICATION STUDIES 121

4 Application to a chemical reaction process 123

4.1 Process description 123

4.2 Identification of a monitoring model 124

4.3 Diagnosis of a fault condition 133

5 Application to a distillation process 141

5.1 Process description 141

5.2 Identification of a monitoring model 144

5.3 Diagnosis of a fault condition 153

PART III ADVANCES IN MULTIVARIATE

STATISTICAL PROCESS CONTROL 165

6 Further modeling issues 167

6.1 Accuracy of estimating PCA models 168

6.1.1 Revisiting the eigendecomposition of Sz0z0 168

6.1.2 Two illustrative examples 171

6.1.3 Maximum likelihood PCA for known Sgg 172

6.1.4 Maximum likelihood PCA for unknown Sgg 177

6.1.5 A simulation example 182

6.1.6 A stopping rule for maximum likelihood PCA models 187

6.1.7 Properties of model and residual subspace estimates 189

6.1.8 Application to a chemical reaction process – revisited 194

6.2 Accuracy of estimating PLS models 202

6.2.1 Bias and variance of parameter estimation 203

6.2.2 Comparing accuracy of PLS and OLS regression models 205

6.2.3 Impact of error-in-variables structure upon PLS models 212

6.2.4 Error-in-variable estimate for known See 218

6.2.5 Error-in-variable estimate for unknown See 219

6.2.6 Application to a distillation process – revisited 223

6.3 Robust model estimation 226

6.3.1 Robust parameter estimation 228

6.3.2 Trimming approaches 231

6.4 Small sample sets 232

6.5 Tutorial session 237

7 Monitoring multivariate time-varying processes 240

7.1 Problem analysis 241

7.2 Recursive principal component analysis 242

7.3 Moving window principal component analysis 244

7.3.1 Adapting the data correlation matrix 244

7.3.2 Adapting the eigendecomposition 247

7.3.3 Computational analysis of the adaptation procedure 251

7.3.4 Adaptation of control limits 252

7.3.5 Process monitoring using an application delay 253

7.3.6 Minimum window length 254

7.4 A simulation example 257

7.4.1 Data generation 257

7.4.2 Application of PCA 258

7.4.3 Utilizing MWPCA based on an application delay 261

7.5 Application to a Fluid Catalytic Cracking Unit 265

7.5.1 Process description 266

7.5.2 Data generation 268

7.5.3 Pre-analysis of simulated data 272

7.5.4 Application of PCA 273

7.5.5 Application of MWPCA 275

7.6 Application to a furnace process 278

7.6.1 Process description 278

7.6.2 Description of sensor bias 279

7.6.3 Application of PCA 280

7.6.4 Utilizing MWPCA based on an application delay 282

7.7 Adaptive partial least squares 286

7.7.1 Recursive adaptation of

7.7.2 Moving window adaptation of

7.7.3 Adapting the number of source signals 287

7.7.4 Adaptation of the PLS model 290

7.8 Tutorial Session 292

Sx0x0 and Sx0y0 287Sx0x0 and Sx0y0 287

8 Monitoring changes in covariance structure 293

8.1 Problem analysis 294

8.1.1 First intuitive example 294

8.1.2 Generic statistical analysis 297

8.1.3 Second intuitive example 299

8.2 Preliminary discussion of related techniques 304

8.3 Definition of primary and improved residuals 305

8.3.1 Primary residuals for eigenvectors 306

8.3.2 Primary residuals for eigenvalues 307

8.3.3 Comparing both types of primary residuals 307

8.3.4 Statistical properties of primary residuals 312

8.3.5 Improved residuals for eigenvalues 315

8.4 Revisiting the simulation examples of Section 8.1 317

8.4.1 First simulation example 318

8.4.2 Second simulation example 321

8.5 Fault isolation and identification 324

8.5.1 Diagnosis of step-type fault conditions 325

8.5.2 Diagnosis of general deterministic fault conditions 328

8.5.3 A simulation example 328

8.6 Application study of a gearbox system 331

8.6.1 Process description 332

8.6.2 Fault description 332

8.6.3 Identification of a monitoring model 334

8.6.4 Detecting a fault condition 338

8.7 Analysis of primary and improved residuals 341

8.7.1 Central limit theorem 341

8.7.2 Further statistical properties of primary residuals 344

8.7.3 Sensitivity of statistics based on improved residuals 349

8.8 Tutorial session 353

PART IV DESCRIPTION OF MODELING METHODS 355

9 Principal component analysis 357

9.1 The core algorithm 357

9.2 Summary of the PCA algorithm 362

9.3 Properties of a PCA model 363

10 Partial least squares 375

10.1 Preliminaries 375

10.2 The core algorithm 377

10.3 Summary of the PLS algorithm 380

10.4 Properties of PLS 381

10.5 Properties of maximum redundancy PLS 396

References 410

Index 427

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