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### Summary

were published in the series as the contributed volume, Process Control Performance Assessment: From Theory to Implementation with Andrzej Ordys, Damian Uduehi, and Michael Johnson as Editors (ISBN 978-1-84628-623-0, 2007). Along with this good progress in process controller assessment methods, researchers have also been investigating techniques to diagnose what is causing the process or control loop degradation. This requires the use of on-line data to identify faults via new diagnostic indicators of typical process problems. A significant focus of some of this research has been the issue of valve problems; a research direction that has been motivated by some industrial statistics that show up to 40% of control loops having performance degradation attributable to valve problems. Shoukat Choudhury, Sirish Shah, and Nina Thornhill have been very active in this research field for a number of years and have written a coherent and consistent presentation of their many research results as this monograph, Diagnosis of Process Nonlinearities and Valve Stiction. The Advances in Industrial Control series is pleased to welcome this new and substantial contribution to the process diagnostic literature. The reader will find the exploitation of the extensive process data archives created by today's process computer systems one theme in the monograph. From another viewpoint, the use of higher-order statistics could be considered to provide a continuing link to the earlier methods of the statistical process control paradigm.

### Author Notes

M. A. A. Shoukat Choudhury received his B. Sc. Engineering (Chemical) from Bangladesh University of Engineering and Technology (BUET), Dhaka, Bangladesh in 1996. He was awarded a gold medal for his outstanding results in B. Sc. Engineering. He obtained an M. Sc. Engineering (Chemical) in 1998 from the same university. He has completed his PhD degree in process control (Chemical Engineering) at the University of Alberta, Canada. For his outstanding research performance during the course of PhD program he has been awarded several awards such as University of Alberta PhD Dissertation Fellowship, Andrew Stewart Memorial Prize and ISA Educational Foundation Scholarship. He is the principal inventor of an internation patent (applied, 2005) on "Methods for Detection and Quantification of Control Valve Stiction". The methodologies and algorithms described in this patent are implemented and available in the commercial software ProcessDoctor from Matrikon Inc. His main research interests include diagnosis of poor control performance, stiction in control valves, data compression, control loop performance assessment and monitoring, and diagnosis of plant wide oscillations.

Sirish Shah received his B.Sc. degree in control engineering from Leeds University in 1971, a M.Sc. degree in automatic control from UMIST, Manchester in 1972, and a Ph.D. degree in process control (chemical engineering) from the University of Alberta in 1976. During 1977 he worked as a computer applications engineer at Esso Chemicals in Sarnia, Ontario. Since 1978 he has been with the University of Alberta, where currently holds the NSERC-Matrikon-ASRA Senior Industrial Research Chair in Computer Process Control. In 1989, he was the recipient of the Albright & Wilson Americas Award of the Canadian Society for Chemical Engineering in recognition of distinguished contributions to chemical engineering. He has held visiting appointments at Oxford University and Balliol College as a SERC fellow in 1985-86 and at Kumamoto University, Japan as a senior research fellow of the Japan Society for the Promotion of Science (JSPS) in 1994. The main area of his current research is process and performance monitoring, system identification and design and implementation of softsensors. He has recently co-authored a book titled, Performance Assessment of Control Loops: Theory and Applications. He has held consulting appointments with a wide variety of process Industries and has also taught many industrial courses.

### Table of Contents

1 Introduction | p. 1 |

1.1 Concepts in Data-Driven Analysis of Chemical Processes | p. 2 |

1.1.1 Linear and Nonlinear Time Series | p. 3 |

1.1.2 Statistics and Randomness | p. 3 |

1.1.3 Frequency Content and Spectral Methods | p. 6 |

1.2 Nonlinearity in Control Valves | p. 8 |

1.3 The Layout of the Book | p. 10 |

1.3.1 Part I Higher-Order Statistics | p. 10 |

1.3.2 Part II Data Quality - Compression and Quantization | p. 10 |

1.3.3 Part III Nonlinearity and Control Performance | p. 11 |

1.3.4 Part IV Control Valve Stiction - Definition, Modelling, Detection and Quantification | p. 12 |

1.3.5 Part V Plant-wide Oscillations - Detection and Diagnosis | p. 13 |

1.3.6 References | p. 14 |

1.4 Summary | p. 14 |

Part I Higher-Order Statistics | |

2 Higher-Order Statistics: Preliminaries | p. 17 |

2.1 Introduction | p. 17 |

2.2 Time Domain Analysis | p. 18 |

2.2.1 Moments | p. 18 |

2.2.2 Cumulants | p. 20 |

2.2.3 The Relationship Between Moments and Cumulants | p. 22 |

2.2.4 Properties of Moments and Cumulants | p. 22 |

2.2.5 Moments and Cumulants of Stationary Signals | p. 25 |

2.3 Spectral Analysis | p. 25 |

2.3.1 Power Spectrum, n=2 | p. 26 |

2.3.2 Bispectrum, n=3 | p. 27 |

2.4 Summary | p. 28 |

3 Bispectrum and Biocherence | p. 29 |

3.1 Bispectrum | p. 29 |

3.1.1 Estimation of the Bispectrum | p. 30 |

3.1.2 Properties of Estimators and Asymptotic Behaviour | p. 32 |

3.1.3 Bicoherence or Normalized Bispectrum | p. 34 |

3.1.4 Properties of Bispectrum and Bicoherence | p. 35 |

3.2 Bispectrum or Bicoherence Estimation Issues | p. 37 |

3.2.1 Choice of Window Function | p. 38 |

3.2.2 Choice of Data Length, Segment Length and Fourier Transform Length | p. 40 |

3.3 Summary | p. 41 |

Part II Data Quality - Compression and Quantization | |

4 Impact of Data Compression and Quantization on Data-Driven Process Analyses | p. 45 |

4.1 Introduction | p. 45 |

4.2 Data Compression Methods | p. 47 |

4.2.1 Overview of Data Compression | p. 47 |

4.2.2 Box-Car (BC) Algorithm | p. 47 |

4.2.3 Backward-Slope (BS) Algorithm | p. 47 |

4.2.4 Combined Box-Car and Backward-Slope (BCBS) Method | p. 49 |

4.2.5 Swinging Door Compression Algorithm | p. 49 |

4.2.6 The Compression Factor | p. 49 |

4.3 Measures of Data Quality | p. 50 |

4.3.1 Statistical Measures | p. 50 |

4.3.2 Nonlinearity Measures | p. 51 |

4.3.3 Performance Index (Harris) Measures | p. 51 |

4.4 Process Data for Compression Comparison | p. 52 |

4.4.1 Industrial Example 1 | p. 52 |

4.4.2 Industrial Example 2 | p. 55 |

4.5 Results and Discussions for Industrial Example 2 | p. 56 |

4.5.1 Visual Observations | p. 56 |

4.5.2 Statistical Properties | p. 57 |

4.5.3 Nonlinearity Assessment | p. 58 |

4.5.4 Performance (Harris) Index | p. 58 |

4.6 Summary of Data Quality Measures | p. 59 |

4.7 Automated Detection of Compression | p. 59 |

4.7.1 Motivation | p. 59 |

4.7.2 Compression Detection Procedure | p. 60 |

4.7.3 Implementation Considerations | p. 61 |

4.8 A Recommendation for Harmless Storing of Data | p. 63 |

4.9 Quantization | p. 63 |

4.10 Summary | p. 65 |

Part III Nonlinearity and Control Performance | |

5 Measures of Nonlinearity - A Review | p. 69 |

5.1 Definition of Nonlinear Systems | p. 69 |

5.2 Nonlinearity in Process Time Trends | p. 70 |

5.3 Various Measures of Nonlinearity | p. 70 |

5.3.1 Model-Based Measures of Nonlinearity | p. 71 |

5.3.2 Time Series-Based Measures of Nonlinearity | p. 71 |

5.4 Summary | p. 75 |

6 Linear or Nonlinear? A Bicoherence-Based Measure of Nonlinearity | p. 77 |

6.1 Introduction | p. 77 |

6.2 Bispectrum and Biocherence | p. 78 |

6.2.1 Spurious Peaks in the Estimated Bicoherence | p. 78 |

6.2.2 Illustrative Example 1 | p. 79 |

6.2.3 How to Choose [epsilon]? | p. 80 |

6.3 Test of Gaussianity and Linearity of a Signal | p. 81 |

6.3.1 Total Nonlinearity Index (TNLI) | p. 85 |

6.4 Illustrative Example 2: Bicoherence of a Linear and a Nonlinear Signal | p. 85 |

6.5 Illustrative Example 3: Bicoherence of a Nonlinear Sinusoid Signal with Noise | p. 87 |

6.5.1 Mild Nonlinearity (n[subscript l] = 0.05) | p. 88 |

6.5.2 Strong Nonlinearity (n[subscript l] = 0.25) | p. 89 |

6.5.3 Extent of Nonlinearity and Effect of Noise | p. 90 |

6.6 Summary | p. 91 |

7 A Nonlinearity Measure Based on Surrogate Data Analysis | p. 93 |

7.1 Surrogate Time Series | p. 93 |

7.1.1 Nonlinearity Detection Using Surrogates | p. 93 |

7.1.2 Predictability in Nonlinear Time Series | p. 93 |

7.2 Algorithm for Nonlinearity Diagnosis | p. 95 |

7.2.1 Construction of the Data Matrix for Nonlinear Prediction | p. 95 |

7.2.2 Calculation of Prediction Error | p. 96 |

7.2.3 Calculation of Surrogate Data | p. 96 |

7.2.4 Statistical Testing | p. 98 |

7.2.5 Algorithm Summary | p. 98 |

7.3 Selection of the Parameter Values | p. 99 |

7.3.1 Recommended Default Parameter Values | p. 99 |

7.3.2 Choice of Embedding Parameters E and H | p. 99 |

7.3.3 Choice of Parameters C and k | p. 100 |

7.3.4 Default Data Ensemble Size, Q and Number of Samples Per Feature, S | p. 101 |

7.3.5 Choice of the Number of Surrogates, M | p. 101 |

7.4 Data-Preprocessing and End-Matching | p. 102 |

7.4.1 False-Positive Results with Cyclic Data | p. 102 |

7.4.2 End-Matching | p. 103 |

7.4.3 Summary of the Data-Preprocessing Steps | p. 104 |

7.4.4 Application to Oscillating Time Grends | p. 104 |

7.5 Worked Examples | p. 106 |

7.5.1 Identification of Nonlinear Root Causes | p. 106 |

7.5.2 Application to the SE Asia Data Set | p. 106 |

7.5.3 The Mechanisms of Propagation in the SE Asia Process | p. 106 |

7.5.4 An Example with No Nonlinearity | p. 108 |

7.6 Summary | p. 110 |

8 Nonlinearities in Control Loops | p. 111 |

8.1 Process Nonlinearity | p. 111 |

8.1.1 Nonlinearity of a Spherical Tank | p. 111 |

8.1.2 Nonlinearities of a Continuous Stirred Tank Reactor (CSTR) | p. 115 |

8.2 Nonlinear Valve Characteristic | p. 117 |

8.2.1 Linear Valves | p. 118 |

8.2.2 Equal Percentage Valves | p. 118 |

8.2.3 Square-Root Valve | p. 119 |

8.2.4 Remarks on Nonlinear Valve Characteristic | p. 120 |

8.3 Nonlinear Disturbances | p. 121 |

8.4 Summary | p. 121 |

9 Diagnosis of Poor Control Performance | p. 123 |

9.1 Introduction | p. 123 |

9.2 Problem Description | p. 124 |

9.3 Usual Causes of Poor Performance | p. 125 |

9.4 Diagnosis of Poor Control Performance | p. 126 |

9.4.1 Well Tuned Controller | p. 127 |

9.4.2 Tightly Tuned Controller or Excessive Integral Action | p. 128 |

9.4.3 Presence of an External Oscillatory Disturbance | p. 129 |

9.4.4 Presence of Stiction | p. 129 |

9.5 Industrial Case Studies | p. 129 |

9.5.1 Stiction in a Furnace Dryer Temperature Control Valve | p. 130 |

9.5.2 Valve Saturation | p. 131 |

9.5.3 Valve Problems in Some Flow Control Loops | p. 132 |

9.6 Summary | p. 134 |

Part IV Control Valve Stiction - Definition, Modelling, Detection and Quantification | |

10 Different Types of Faults in Control Valves | p. 137 |

10.1 What Is a Control Valve | p. 137 |

10.2 Faults in Control Valve | p. 138 |

10.2.1 Oversized Valve | p. 139 |

10.2.2 Undersized Valve | p. 139 |

10.2.3 Corroded Valve Seat | p. 139 |

10.2.4 Faulty Diaphragm | p. 139 |

10.2.5 Packing Leakage | p. 139 |

10.2.6 Valve Hysteresis | p. 140 |

10.2.7 Valve Stiction | p. 140 |

10.2.8 Large Deadband | p. 140 |

10.2.9 Valve Saturation | p. 141 |

10.3 Summary | p. 141 |

11 Stiction: Definition and Discussions | p. 143 |

11.1 Introduction | p. 143 |

11.2 What Is Stiction? | p. 143 |

11.2.1 Definition of Terms Relating to Valve Nonlinearity | p. 144 |

11.2.2 Discussion of the Term 'Stiction' | p. 145 |

11.2.3 A Formal Definition of Stiction | p. 146 |

11.3 Practical Examples of Valve Stiction | p. 148 |

11.4 Summary | p. 151 |

12 Physics-Based Model of Control Valve Stiction | p. 153 |

12.1 Introduction | p. 153 |

12.2 Physical Modelling of Valve Friction | p. 153 |

12.2.1 Physics of a Control Valve | p. 153 |

12.2.2 Friction Model | p. 154 |

12.2.3 Model Parameters | p. 155 |

12.2.4 Detection of Zero Velocity | p. 156 |

12.2.5 Model of the Pressure Chamber | p. 156 |

12.3 Valve Simulation | p. 157 |

12.3.1 Open-Loop Response | p. 157 |

12.3.2 Closed-Loop Response | p. 158 |

12.4 Summary | p. 160 |

13 Data-Driven Model of Valve Stiction | p. 161 |

13.1 One-Parameter Data-Driven Stiction Model | p. 161 |

13.2 Two-Parameter Data-Driven Model of Valve Stiction | p. 163 |

13.2.1 Model Formulation | p. 163 |

13.2.2 Dealing with Stochastic or Noisy Control Signals | p. 166 |

13.2.3 Open-Loop Response of the Model Under a Sinusoidal Input | p. 166 |

13.2.4 Stiction in Reality | p. 167 |

13.2.5 Closed-Loop Behaviour of the Model | p. 167 |

13.3 Comparison of Physics-Based Model and Data-Driven Model | p. 171 |

13.4 Summary | p. 171 |

14 Describing Function Analysis | p. 173 |

14.1 Introduction | p. 173 |

14.2 Describing Function Analysis for Two-Parameter Stiction Model | p. 174 |

14.2.1 Derivation of the Describing Function | p. 174 |

14.3 Asymptotes of the Describing Function | p. 177 |

14.4 Insights Gained from the Describing Function | p. 178 |

14.4.1 The Impact of the Controller on the Limit Cycle | p. 179 |

14.5 Summary | p. 180 |

15 Automatic Detection and Quantification of Valve Stiction | p. 181 |

15.1 Introduction | p. 181 |

15.2 Stiction Detection - A Literature Review | p. 182 |

15.3 Detection of Stiction Using Nonlinearity Information and the pv-op Mapping | p. 183 |

15.3.1 Detection of Loop Nonlinearity | p. 184 |

15.3.2 Use of pv-op Plot | p. 185 |

15.4 Stiction Quantification | p. 187 |

15.4.1 Clustering Techniques of Quantifying Stiction | p. 187 |

15.4.2 Fitted Ellipse Technique for Quantifying Stiction | p. 190 |

15.5 An Illustrative Example | p. 192 |

15.5.1 Validation of the Results | p. 193 |

15.6 Automation of the Method | p. 193 |

15.7 Simulation Results | p. 195 |

15.7.1 A Worked Example | p. 195 |

15.7.2 Distinguishing Limit Cycles Caused by Stiction and Those Caused by a Sinusoidal Disturbance | p. 196 |

15.7.3 Detecting Stiction When Its Impact Propagates as Disturbance | p. 198 |

15.8 Practical Implementation Issues | p. 201 |

15.8.1 Bicoherence Estimation | p. 201 |

15.8.2 Nonstationarity of the Data | p. 201 |

15.8.3 Problems of Outliers and Abrupt Changes | p. 201 |

15.8.4 Dealing with Short Length Data | p. 202 |

15.8.5 Dealing with Longer Oscillations | p. 202 |

15.8.6 Valve Nonlinearity | p. 202 |

15.8.7 Filtering of the Data | p. 203 |

15.8.8 Segmenting Data for pv-op Plot | p. 204 |

15.9 Summary | p. 204 |

16 Industrial Applications of the Stiction Quantification Algorithm | p. 205 |

16.1 Industrial Case Studies | p. 205 |

16.1.1 Loop 1: A Level Loop | p. 205 |

16.1.2 Loop 2: A Linear-Level Control Loop | p. 207 |

16.1.3 Loop 3: A Flow Control Loop | p. 208 |

16.1.4 Loop 4: Flow Control Loop Cascaded with Level Control | p. 209 |

16.1.5 Loop 5: A Pressure Control Loop | p. 210 |

16.1.6 Loop 6: A Composition Control Loop | p. 210 |

16.1.7 Loop 7: A Cascaded Flow Control Loop | p. 211 |

16.1.8 Loop 8: A Temperature Control Loop | p. 212 |

16.1.9 Loops 9 and 10 | p. 212 |

16.2 Online Compensation for Stiction | p. 213 |

16.3 Summary | p. 215 |

17 Confirming Valve Stiction | p. 217 |

17.1 Methods to Confirm Valve Stiction | p. 217 |

17.2 Gain Change Method for Confirming Valve Stiction | p. 218 |

17.2.1 Distinguishing Stiction from External Oscillatory Disturbance | p. 218 |

17.3 Describing Function Analysis | p. 222 |

17.3.1 Comparison of Describing Function Analysis (DFA) Results with Simulation Results | p. 225 |

17.4 Industrial Example | p. 225 |

17.5 Summary | p. 226 |

Part V Plant-wide Oscillations - Detection and Diagnosis | |

18 Detection of Plantwide Oscillations | p. 229 |

18.1 Introduction | p. 229 |

18.2 What is an Oscillation? | p. 230 |

18.2.1 Units of Frequency | p. 230 |

18.2.2 Examples of Oscillatory Signals | p. 230 |

18.3 Detection of Oscillation(s) in a Single Time Series | p. 231 |

18.3.1 The Power Spectrum | p. 231 |

18.3.2 Hagglund's IAE Method | p. 231 |

18.3.3 Autocovariance (ACF) Based Method | p. 232 |

18.3.4 Other Methods | p. 237 |

18.4 What are Plant-wide Oscillations? | p. 237 |

18.5 Classification of Plant-wide Oscillations or Disturbances | p. 237 |

18.5.1 Time scales | p. 237 |

18.5.2 Oscillating and Non-oscillating Disturbances | p. 238 |

18.6 Detection of Plant-wide Oscillations | p. 238 |

18.6.1 High-Density Plots | p. 238 |

18.6.2 ACF-Based Method | p. 239 |

18.6.3 Power Spectral Correlation Map (PSCMAP) | p. 239 |

18.6.4 Spectral Envelope Method | p. 240 |

18.6.5 Spectral Decomposition Methods | p. 241 |

18.7 Summary | p. 250 |

19 Diagnosis of Plant-wide Oscillations | p. 253 |

19.1 Root Cause Diagnosis of Plant-wide Oscillation | p. 253 |

19.1.1 Finding a Nonlinear Root Cause of a Plant-Wide Disturbance | p. 253 |

19.1.2 Finding a Linear Root Cause of a Plant-wide Disturbance | p. 256 |

19.2 Industrial Case Study 1 - Eastman Chemical Plant | p. 257 |

19.2.1 Data Description | p. 258 |

19.2.2 Reduction of the Problem Size | p. 258 |

19.2.3 Detection of Plant-wide Oscillation by PSCMAP | p. 259 |

19.2.4 Nonlinearity Analysis Using Biocherence-Based Indices | p. 260 |

19.2.5 Diagnosis of the Problem in Loop LC2 | p. 262 |

19.3 Industrial Case Study 2 - SE Asia Refinery Data Analysis | p. 263 |

19.3.1 Oscillation Detection by PSCMAP | p. 264 |

19.3.2 Oscillation Detection by Spectral Envelope | p. 265 |

19.3.3 Oscillation Diagnosis | p. 266 |

19.4 Industrial Case Study 3 - Mitshubishi Chemical Corporation | p. 266 |

19.4.1 Scope of the Analysis and Data Set | p. 268 |

19.4.2 Oscillation-Detection Results | p. 268 |

19.4.3 Oscillation Diagnosis | p. 268 |

19.4.4 The Results of Maintenance on the PC1 and LI1 Loops | p. 271 |

19.5 Summary | p. 272 |

References | p. 273 |

Copyright Acknowledgements | p. 281 |

Index | p. 283 |