# Monitoring and Control of Information-Poor Systems: An Approach based on Fuzzy Relational Models

# Monitoring and Control of Information-Poor Systems: An Approach based on Fuzzy Relational Models

ISBN: 978-1-119-94028-9 February 2012 352 Pages

**E-Book**

$124.99

## Description

The monitoring and control of a system whose behaviour is highly uncertain is an important and challenging practical problem. Methods of solution based on fuzzy techniques have generated considerable interest, but very little of the existing literature considers explicit ways of taking uncertainties into account. This book describes an approach to the monitoring and control of information-poor systems that is based on fuzzy relational models which generate fuzzy outputs.The first part of *Monitoring and Control of Information-Poor Systems* aims to clarify why design decisions must take account of the uncertainty associated with optimal choices, and to explain how a fuzzy relational model can be used to generate a fuzzy output, which reflects the uncertainties associated with its predictions. Part two gives a brief introduction to fuzzy decision-making and shows how it can be used to design a predictive control scheme that is suitable for controlling information-poor systems using inaccurate measurements. Part three describes different ways in which fuzzy relational models can be generated online and explains the practical issues associated with their identification and application. The final part of the book provides examples of the use of the previously described techniques in real applications.

Key features:

- Describes techniques applicable to a wide range of engineering, environmental, medical, financial and economic applications
- Uses simple examples to help explain the basic techniques for dealing with uncertainty
- Describes a novel design approach based on the use of fuzzy relational models
- Considers practical issues associated with applying the techniques to real systems

*Monitoring and Control of Information-Poor Systems* forms an invaluable resource for a wide range of graduate students, and is also a comprehensive reference for researchers and practitioners working on problems involving mathematical modelling and control.

**Preface xi**

**About the Author xv**

**Acknowledgements xvii**

**I ANALYSING THE BEHAVIOUR OF INFORMATION-POOR SYSTEMS**

**1 Characteristics of Information-Poor Systems 3**

1.1 Introduction to Information-Poor Systems 3

*1.1.1 Blast Furnaces* 3

*1.1.2 Container Cranes* 3

*1.1.3 Cooperative Control Systems* 4

*1.1.4 Distillation Columns* 4

*1.1.5 Drug Administration* 4

*1.1.6 Electrical Power Generation and Distribution* 4

*1.1.7 Environmental Risk Assessment Systems* 4

*1.1.8 Financial Investment and Portfolio Selection* 5

*1.1.9 Health Care Systems* 5

*1.1.10 Indoor Climate Control* 5

*1.1.11 NOx Emissions from Gas Turbines and Internal Combustion Engines* 6

*1.1.12 Penicillin Production Plant* 6

*1.1.13 Polymerization Reactors* 6

*1.1.14 Rotary Kilns* 6

*1.1.15 Solar Power Plant* 7

*1.1.16 Wastewater Treatment Plant* 7

*1.1.17 Wood Pulp Production Plant* 7

1.2 Main Causes of Uncertainty 7

*1.2.1 Sources of Modelling Errors* 8

*1.2.2 Sources of Measurement Errors* 8

*1.2.3 Reasons for Poorly Defined Objectives and Constraints* 9

1.3 Design in the Face of Uncertainty 9

References 9

**2 Describing and Propagating Uncertainty 13**

2.1 Methods of Describing Uncertainty 13

*2.1.1 Uncertainty Intervals and Probability Distributions* 13

*2.1.2 Fuzzy Sets and Fuzzy Numbers* 14

2.2 Methods of Propagating Uncertainty 15

*2.2.1 Interval Arithmetic* 15

*2.2.2 Statistical Methods* 16

*2.2.3 Monte Carlo Methods* 16

*2.2.4 Fuzzy Arithmetic* 17

2.3 Fuzzy Arithmetic Using *α*-Cut Sets and Interval Arithmetic 18

2.4 Fuzzy Arithmetic Based on the Extension Principle 21

2.5 Representing and Propagating Uncertainty Using Pseudo-Triangular Membership Functions 24

2.6 Summary 27

References 27

**3 Accounting for Measurement Uncertainty 29**

3.1 Measurement Errors 29

3.2 Introduction to Fuzzy Random Variables 29

*3.2.1 Definition of a Fuzzy Random Variable* 30

*3.2.2 Generating Fuzzy Random Variables from a Knowledge of the Random and Systematic Errors* 30

3.3 A Hybrid Approach to the Propagation of Uncertainty 32

3.4 Fuzzy Sensor Fusion Based on the Extension Principle 34

3.5 Fuzzy Sensors 38

3.6 Summary 39

References 39

**4 Accounting for Modelling Errors in Fuzzy Models 41**

4.1 An Introduction to Rule-Based Models 41

4.2 Linguistic Fuzzy Models 41

*4.2.1 Fuzzy Rules* 41

*4.2.2 Fuzzy Inferencing* 42

*4.2.3 Compositional Rules of Inference* 43

4.3 Functional Fuzzy Models 47

4.4 Fuzzy Neural Networks 48

4.5 Methods of Generating Fuzzy Models 50

*4.5.1 Modifying Expert Rules to Take Account of Uncertainty* 50

*4.5.2 Identifying Fuzzy Rules from Data* 56

4.6 Defuzzification 58

4.7 Summary 60

References 60

**5 Fuzzy Relational Models 63**

5.1 Introduction to Fuzzy Relations and Fuzzy Relational Models 63

5.2 Fuzzy FRMs 65

5.3 Methods of Estimating Rule Confidences from Data 67

5.4 Estimating Probability Density Functions from Data 70

*5.4.1 Probabilistic Interpretation of RSK Fuzzy Identification* 71

*5.4.2 Effect of Structural Errors on the Output of a Fuzzy FRM* 78

*5.4.3 Estimation Based on Limited Amounts of Training Data* 83

5.5 Generic Fuzzy Models 86

*5.5.1 Identification of Generic Fuzzy Models* 87

*5.5.2 Reducing the Time Required to Generate the Training Data* 91

5.6 Summary 92

References 92

**II CONTROL OF INFORMATION-POOR SYSTEMS**

**6 Fuzzy Decision-Making 97**

6.1 Risk Assessment in Information-Poor Systems 97

6.2 Fuzzy Optimization in Information-Poor Systems 99

*6.2.1 Fuzzy Goals and Fuzzy Constraints* 99

*6.2.2 Fuzzy Aggregation Operators* 99

*6.2.3 Fuzzy Ranking* 100

6.3 Multi-Stage Decision-Making 101

*6.3.1 Fuzzy Dynamic Programming* 102

*6.3.2 Branch and Bound* 103

*6.3.3 Genetic Algorithms* 106

6.4 Fuzzy Decision-Making Based on Intuitionistic Fuzzy Sets 106

*6.4.1 Definition of an Intuitionistic Fuzzy Set* 106

*6.4.2 Multi-Attribute Decision-Making Using Intuitionistic Fuzzy Numbers* 107

6.5 Summary 108

References 108

**7 Predictive Control in Uncertain Systems 111**

7.1 Model-Based Predictive Control 111

7.2 Fuzzy Approaches to Model-Based Control of Uncertain Systems 112

*7.2.1 Inverse Control of Fuzzy Interval Systems* 112

*7.2.2 Fuzzy Model-Based Predictive Control* 114

7.3 Practical Issues Associated with Multi-Step Fuzzy Decision-Making 115

*7.3.1 Limiting the Accumulation of Uncertainty* 115

*7.3.2 Avoiding Excessive Computational Demands When Using Enumerative Search Optimization* 115

*7.3.3 Avoiding Excessive Computational Demands When Using Evolutionary Algorithms* 116

*7.3.4 Handling Infeasibility* 117

*7.3.5 Choosing the Weighting in Multi-Criteria Cost Functions* 117

*7.3.6 Dealing with Hard Constraints* 118

7.4 A Simplified Approach to Fuzzy FRM-Based Predictive Control 118

*7.4.1 The Fuzzy Decision-Maker* 119

*7.4.2 Conditional Defuzzification* 120

7.5 FMPC of an Uncertain Dynamic System Based on a Generic Fuzzy FRM 122

7.6 Summary 127

References 128

**8 Incorporating Fuzzy Inputs 129**

8.1 Fuzzy Setpoints and Fuzzy Measurements 129

*8.1.1 Fuzzy Setpoints* 129

*8.1.2 Fuzzy Measurements* 129

8.2 Fuzzy Measures of the Tracking Error and its Derivative 131

8.3 Inference with Fuzzy Inputs 136

8.4 Fuzzy Output Neural Networks 138

8.5 Modelling Input Uncertainty Using a Fuzzy FRM 140

8.6 Summary 151

References 151

**9 Disturbance Rejection in Information-Poor Systems 153**

9.1 Rejecting Unmeasured Disturbances in Uncertain Systems 154

*9.1.1 Robust Fuzzy Control* 154

*9.1.2 Feedback Linearization Using a Fuzzy Disturbance Observer* 155

*9.1.3 Fuzzy Model-Based Internal Model Control* 155

9.2 Fuzzy IMC Based on a Fuzzy Output FRM 157

9.3 Rejecting Measured Disturbances in Non-Linear Uncertain Systems 161

9.4 Fuzzy MPC with Feedforward 162

9.5 Summary 166

References 166

**III ONLINE LEARNING IN INFORMATION-POOR SYSTEMS**

**10 Online Model Identification in Information-Poor Environments 171**

10.1 Online Fuzzy Identification Schemes 171

*10.1.1 Recursive Fuzzy Least-Squares* 171

*10.1.2 Recursive Forms of the RSK Algorithm* 172

10.2 Effect of Poor-Quality and Incomplete Training Data 176

10.3 Ways of Reducing the Computational Demands 177

*10.3.1 Evolving Fuzzy Models* 177

*10.3.2 Hierarchical Fuzzy Models* 181

10.4 Summary 185

References 185

**11 Adaptive Model-Based Control of Information-Poor Systems 187**

11.1 Robust Adaptive Fuzzy Control 187

11.2 Adaptive Fuzzy FRM-Based Predictive Control 188

11.3 Commissioning the Controller 189

*11.3.1 Methods of Incorporating Prior Knowledge* 189

*11.3.2 Initialization Using a Generic Fuzzy FRM* 189

11.4 Generating an Optimal Control Signal Using a Partially Trained Model 192

*11.4.1 Taking the Amount of Training into Account* 192

*11.4.2 Incorporating a Secondary Controller* 194

*11.4.3 Combining the Fuzzy Predictions Generated by More than One Model* 201

11.5 Dealing with the Effects of Disturbances 202

*11.5.1 Adaptive Feedforward Control Based on an Inaccurate Disturbance Measurement* 203

11.6 Summary 209

References 209

**12 Adaptive Model-Free Control of Information-Poor Systems 211**

12.1 Introduction to Model-Free Adaptive Control of Non-Linear Systems 211

12.2 Fuzzy FRM-Based Direct Adaptive Control 211

12.3 Behaviour in the Presence of a Noisy Measurement of the Plant Output 213

12.4 Behaviour in the Presence of an Unmeasured Disturbance 218

12.5 Accounting for Uncertainty Arising from a Measured Disturbance 222

12.6 Summary 227

References 227

**13 Fault Diagnosis in Information-Poor Systems 229**

13.1 Introduction to Fault Detection and Isolation in Non-Linear Uncertain Systems 229

*13.1.1 Model-Based Methods for Non-Linear Systems* 230

*13.1.2 Ways of Accounting for Uncertainty* 232

13.2 A Fuzzy FRM-Based Fault Diagnosis Scheme 233

*13.2.1 Measuring the Similarity of FRMs* 234

*13.2.2 Accumulating Evidence of Fault-Free or Faulty Operation* 236

*13.2.3 Generating Robust Generic Models of Faulty Operation* 239

*13.2.4 Multi-Step Fault Diagnosis* 239

13.3 Summary 242

References 243

**IV SOME EXAMPLE APPLICATIONS**

**14 Control of Thermal Comfort 247**

14.1 Main Sources of Uncertainty and Practical Considerations 248

14.2 Review of Approaches Suggested for Dealing with the Uncertainty 249

14.3 Design of the Fuzzy FRM-Based Control System 249

*14.3.1 The Fuzzy FRM* 250

*14.3.2 The Fuzzy Cost Functions* 252

*14.3.3 The Fuzzy Goals* 252

*14.3.4 The Fuzzy Decision-Maker* 254

*14.3.5 The Conditional Defuzzifier* 254

14.4 Performance of the Thermal Comfort Controller 254

14.5 Concluding Remarks 258

References 259

**15 Identification of Faults in Air-Conditioning Systems 261**

15.1 Main Sources of Uncertainty and Practical Considerations 261

15.2 Design of a Fuzzy FRM-Based Monitoring System for a Cooling Coil Subsystem 263

15.3 Diagnosis of Known Faults in a Simulated Cooling Coil Subsystem 264

*15.3.1 Fault-Free Operation* 264

*15.3.2 Leaky Valve* 264

*15.3.3 Fouled Coil* 265

*15.3.4 Valve Stuck in the Fully Closed Position* 266

*15.3.5 Valve Stuck in the Midway Position* 267

*15.3.6 Valve Stuck in the Fully Open Position* 268

15.4 Commissioning of Air-Handling Units 269

15.5 Concluding Remarks 272

References 272

**16 Control of Heat Exchangers 275**

16.1 Main Sources of Uncertainty and Practical Considerations 275

16.2 Design of a Fuzzy FRM-Based Predictive Controller 276

16.3 Design of a Fuzzy FRM-Based Internal Model Control Scheme 283

16.4 Concluding Remarks 290

References 290

**17 Measurement of Spatially Distributed Quantities 293**

17.1 Review of Approaches Suggested for Dealing with Sensor Bias 293

17.2 An Example Application 294

*17.2.1 Air Temperature Estimation Using a Single-Point Sensor with Bias Correction* 294

*17.2.2 Air Temperature Estimation Based on Mass and Energy Balances* 299

17.3 Using Bias Estimation and Fuzzy Data Fusion to Improve Automated Commissioning in Air-Handling Units 302

*17.3.1 Diagnosis When the Measurement Bias is Estimated Accurately* 303

*17.3.2 Diagnosis When the Estimate of the Measurement Bias is Inaccurate* 303

17.4 Concluding Remarks 305

References 306

**Index 309**