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.

Table of Contents:
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