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Optimal State Estimation for Process Monitoring, Fault Diagnosis and Control

Optimal State Estimation for Process Monitoring, Fault Diagnosis and Control

Ch. Venkateswarlu Chief Scientist (Retd.), Indian Institute of Chemical Technology (CSIR-IICT), Hyderabad, India

Rama Rao Karri Petroleum and Chemical Engineering, Faculty of Engineering, Universiti Teknologi Brunei, Gadong, Brunei Darussalam

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2022 Elsevier B.V. All rights reserved.

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This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein).

Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-323-85878-6 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Susan Dennis Editorial Project Manager: Sam Young Production Project Manager: Sruthi Satheesh Cover Designer: Greg Harris Typeset by MPS Limited, Chennai, India

Contents About the authors Preface

Part I Basic details and state estimation algorithms 1. Optimal state estimation and its importance in process systems engineering 1.1 Introduction 1.2 Significance of state estimation 1.3 Role of state estimation in process systems engineering 1.4 Outline of this book 1.5 Summary

2. Introduction to stochastic processes and state estimation filtering 2.1 Introduction 2.2 Probability and stochastic variables 2.2.1 Probability theorems 2.2.2 Conditional probability 2.3 Probability distributions and distribution functions 2.3.1 Discrete random variables and discrete probability distributions 2.3.2 Continuous random variables and continuous probability distributions 2.4 White Gaussian noise and colored noise 2.5 Stochastic/random processes 2.5.1 Stochastic model scheme 2.5.2 Stochastic representation of real processes 2.5.3 Stochastic representation of general dynamic models 2.6 Filtering, estimation, and prediction problem 2.6.1 Filtering, prediction, and smoothing 2.6.2 Bayes filtering approach for state estimation

xiii xv

1

3 3 3 4 4 4

5 5 5 6 7 7 8 10 13 14 15 16 18 18 18 19

2.6.3 Stochastic filtering approach for state estimation 2.7 Summary References

3. Linear filtering and observation techniques 3.1 Introduction 3.2 Representation of a system and associated variables 3.2.1 Definition of system state 3.2.2 System variables 3.2.3 State space representation of linear systems 3.2.4 Output equations 3.2.5 Discrete time representation of linear systems 3.3 Concepts of observability and controllability 3.3.1 Observability of a continuous system 3.3.2 Observability of a discrete system 3.3.3 Controllability of a continuous system 3.3.4 Controllability of a discrete system 3.4 Recursive weighted least squares estimator 3.4.1 Estimation procedure 3.5 Luenberger observer for state estimation 3.5.1 Discrete form of observer design 3.5.2 Discrete observer based state feedback control 3.5.3 Continuous form of observer 3.5.4 Continuous observer based state feedback control 3.6 Reduced order Luenberger observer for state estimation 3.7 Kalman filter for state estimation 3.7.1 Discrete Kalman filter 3.7.2 Filter covariance matrices 3.7.3 Continuous Kalman filter 3.8 State estimation applications of linear filtering and observation techniques 3.9 Summary References

19 20 20

21 21 21 21 21 22 22 23 24 24 25 25 25 27 30 31 32 32 37 38 41 44 44 46 52 56 56 56 v

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4. Mechanistic model-based nonlinear filtering and observation techniques for optimal state/parameter estimation 59 4.1 Introduction 59 4.2 General nonlinear system and system models 59 4.2.1 Mechanistic/first principle models 59 4.2.2 General representation of dynamic models 61 4.3 Observability of nonlinear systems 61 4.3.1 Local observability 62 4.3.2 Global observability 64 4.4 Extended Kalman filter 65 4.4.1 Process representation for state estimation 66 4.4.2 Process representation for state and parameter estimation 66 4.4.3 Extended Kalman filter for continuous time nonlinear systems 66 4.4.4 Extended Kalman filter for discrete time nonlinear systems 68 4.4.5 Emphasis on covariance matrices of extended Kalman filter 68 4.5 Steady state extended Kalman filter 73 4.6 Two-level extended Kalman filter 74 4.6.1 State estimation filter 74 4.6.2 Parameter identification filter 75 4.7 Adaptive fading extended Kalman filter 75 4.8 Unscented Kalman filter 76 4.9 Square root unscented Kalman filter 78 4.10 Ensemble Kalman filter 80 4.11 Particle filter 81 4.12 Reduced order Luenberger observer 84 4.12.1 Determination of observer coefficient matrices 84 4.13 Reduced order extended Luenberger observer 85 4.14 Nonlinear observer 86 4.15 State estimation applications of nonlinear filtering and observation techniques 88 4.16 Summary 88 References 88

5. Data-driven modeling techniques for state estimation 91 5.1 Introduction 5.2 Principal component analysis 5.2.1 Basic principles 5.2.2 Geometric interpretation of PCA 5.2.3 Eigen structure for data matrix 5.2.4 PCA model establishment

91 91 91 92 92 93

5.3 Projection to latent structures 5.3.1 The regression problem 5.3.2 Scaling and centering the data 5.3.3 PLS model 5.4 Artificial neural networks 5.4.1 ANN structure and its components 5.4.2 Neuron processing functions 5.4.3 Learning paradigms 5.4.4 Learning algorithms and training procedure 5.4.5 Information processing 5.4.6 ANN architectures 5.5 Radial basis function networks 5.5.1 Structure of RBFN 5.5.2 Automatic configuration of RBFN 5.5.3 Implementation procedure 5.6 Nonlinear iterative partial least squares 5.6.1 NIPALS algorithm for PLS 5.6.2 Nonlinear PLS within PLS framework 5.6.3 NIPALS algorithm for RBFN 5.7 State estimation applications of data-driven modeling techniques 5.8 Summary References

6. Optimal sensor configuration methods for state estimation

94 94 94 94 95 95 97 97 97 100 100 102 103 105 106 107 107 108 108 110 110 110

113

6.1 Introduction 113 6.2 Brief review on sensor configuration methods 113 6.3 Optimal sensor configuration: classical methods 114 6.3.1 Sensitivity index 114 6.3.2 Singular value decomposition 114 6.3.3 Principal component analysis 115 6.4 Optimal sensor configuration: gramian-based methods for linear systems 116 6.4.1 Observability gramian for linear systems 116 6.4.2 Observability gramian-based sensor selection procedure 117 6.5 Optimal sensor configuration for nonlinear systems 117 6.5.1 Empirical observability gramian for nonlinear systems 118 6.5.2 Empirical observability gramian metrics 119 6.5.3 Empirical observability gramian-based sensor selection procedure 119 6.6 Summary 119 References 121

Part II Optimal state estimation for process monitoring 7

123

Application of mechanistic model-based nonlinear filtering and observation techniques for optimal state estimation in multicomponent batch distillation 125 7.1 Introduction 125 7.2 Batch distillation process and its dynamic model 125 7.3 Simplified dynamic model of batch distillation 128 7.3.1 Equilibrium relations 128 7.4 The application system 128 7.5 Measurements configuration for state estimation 129 7.6 Performance criteria 132 7.7 Extended Kalman filter for compositions estimation 132 7.7.1 Design of EKF estimator 132 7.7.2 EKF implementation results 134 7.8 Steady state Kalman filter for compositions estimation 136 7.8.1 Design of SSKF estimator 136 7.8.2 SSKF implementation results 136 7.9 Adaptive fading extended Kalman filter for compositions estimation 137 7.9.1 Design of AFEKF estimator 137 7.9.2 AFEKF implementation results 137 7.10 Comparative performance of composition estimators 137 7.11 Summary 139 References 140

8

Application of mechanistic model-based nonlinear filtering and observation techniques for optimal state estimation in multicomponent reactive batch distillation with optimal sensor configuration 141 8.1 Introduction 8.2 Reactive batch distillation process and its dynamic model 8.2.1 Equilibrium relations 8.2.2 Enthalpy relations 8.3 Simplified dynamic model of reactive batch distillation 8.4 The application system

141 142 144 144 144 145

9

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8.5 Sensor configuration for state estimation 8.5.1 Sensor configuration using sensitivity index 8.5.2 Sensor configuration using singular value decomposition 8.5.3 Sensor configuration using principal component analysis 8.6 Performance criteria 8.6.1 Mean integral squared error 8.7 Extended Kalman filter for compositions estimation 8.7.1 Design of extended Kalman filter estimator 8.7.2 Extended Kalman filter implementation results 8.8 Summary References

145 146 147 149 152 152 152 154 155 160 161

Application of mechanistic model-based nonlinear filtering and observation techniques for optimal state estimation in complex nonlinear dynamical systems 163 9.1 Introduction 9.2 Nonlinear dynamical CSTR 9.2.1 Mathematical model 9.2.2 Stability analysis 9.2.3 Bifurcation analysis 9.3 Optimal state estimation in nonlinear dynamical CSTR 9.3.1 State estimator design 9.3.2 State estimation results 9.4 Nonlinear dynamical homopolymerization reactor 9.4.1 Mathematical model 9.4.2 Stability analysis 9.4.3 Steady state solution 9.4.4 Bifurcation analysis 9.5 Optimal state estimation in nonlinear dynamical homopolymerization reactor 9.5.1 State estimator design 9.5.2 State estimation results 9.6 Summary References

163 164 164 165 166 166 166 170 170 171 173 174 177 177 179 181 181 183

10 Application of mechanistic model-based nonlinear filtering and observation techniques for optimal state estimation of a kraft pulping digester 185 10.1 Introduction

185

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10.2 Experimental system and dynamic modeling 10.2.1 Batch pulping process 10.2.2 Development of a dynamic model for kraft pulping digester 10.3 Optimal state estimation of kraft pulping digester 10.3.1 Design of EKF 10.3.2 Design of nonlinear observer 10.4 State estimation results 10.5 Summary References

186 186 188 191 191 193 193 196 198

11 Application of mechanistic model-based nonlinear filtering and observation techniques for optimal state estimation of a continuous reactive distillation column with optimal sensor configuration 201 11.1 Introduction 11.2 The process and its mathematical model 11.2.1 Total mass balance 11.2.2 Component balance 11.2.3 Energy balance 11.2.4 Tray hydraulics 11.3 Optimal sensor configuration using empirical observability grammians 11.3.1 Significance of optimal sensor configuration in a reactive distillation column 11.3.2 Measurement specific empirical observability grammians 11.3.3 Optimal sensor configuration using empirical observability grammians 11.4 State estimator design 11.5 Estimator performance measure for optimality of sensor configuration 11.6 Analysis of results 11.7 Summary References

201 202 202 202 203 203 204

204 204

206 207 210 211 214 215

12 Application of mechanistic model-based nonlinear filtering and observation techniques for optimal state estimation of a catalytic tubular reactor with optimal sensor configuration 217 12.1 Introduction 12.2 The process and its mathematical model

217 218

12.2.1 Process 12.2.2 Mathematical model 12.3 Method of solution 12.4 Results of numerical solution 12.5 Optimal sensor configuration in a catalytic tubular reactor 12.5.1 Significance of sensor configuration in a tubular reactor 12.5.2 Empirical observability Grammians 12.5.3 Optimal sensor configuration using empirical observability Grammians 12.5.4 Sensor configuration results 12.6 Optimal state estimation using unscented Kalman filter 12.6.1 State estimator design 12.6.2 State estimation results 12.7 Summary References

226 227 227 229 229

13 Applications of data-driven model-based methods for process state estimation

231

13.1 Introduction 13.2 Projection to latent structures model-based compositions estimator for multicomponent batch distillation 13.2.1 State estimator development 13.2.2 State estimation results 13.3 Artificial neural network model-based compositions estimator for multicomponent batch distillation 13.3.1 State estimator development 13.3.2 State estimation results 13.4 Radial basis function network model-based compositions estimator for multicomponent batch distillation 13.4.1 State estimator development 13.4.2 State estimation results 13.5 NIPALS RBFN model-based compositions estimator for multicomponent batch distillation 13.5.1 State estimator development 13.5.2 State estimation results 13.6 Summary References

218 218 219 221 222

223 223

223 224

231

231 232 234

236 236 238

238 238 239

243 243 243 246 246

Part III Application of data driven model-based methods for process state estimation

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14 Optimal state and parameter estimation for fault detection and diagnosis in continuous stirred tank reactor 14.1 Introduction 14.2 General structure of model-based fault detection and diagnosis 14.3 General process description for fault detection and diagnosis 14.4 Nonlinear CSTR, its mathematical model and fault cases considered 14.5 Method of extended Kalman filter 14.5.1 Design strategy 14.5.2 Analysis of results 14.6 Method of reduced order extended Luenberger observer and extended Kalman filter 14.6.1 Design strategy 14.6.2 Analysis of results 14.7 Method of two-level extended Kalman filter 14.7.1 Design strategy 14.7.2 Analysis of results 14.8 Method of a discrete version of extended Kalman filter and sequential least squares 14.8.1 Design strategy 14.8.2 Analysis of results 14.9 Method of discrete version of extended Kalman filter and simultaneous least squares 14.9.1 Design strategy 14.9.2 Analysis of results 14.10 Summary References

249 249 250 251 252 253 253 255

257 257 260 261 261 263

264 264 266

267 267 269 269 270

15 Optimal state and parameter estimation for fault detection and diagnosis of a nonlinear batch beer fermentation process 271 15.1 Introduction 15.2 General structure and general process description for model-based fault detection and diagnosis 15.3 Batch beer fermentation process, its mathematical model and fault cases

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Contents

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15.3.1 Mathematical model 15.3.2 Fault cases in batch beer fermentation 15.4 Method of extended Kalman filter 15.4.1 Design strategy 15.4.2 Analysis of results 15.5 Method of reduced-order extended Luenberger observer and extended Kalman filter 15.5.1 Design strategy 15.5.2 Analysis of results 15.6 Method of two-level extended Kalman filter 15.6.1 Design strategy 15.6.2 Analysis of results 15.7 Method of discrete version of extended Kalman filter and sequential least squares 15.7.1 Design strategy 15.7.2 Analysis of results 15.8 Method of discrete version of extended Kalman filter and simultaneous least squares 15.8.1 Design strategy 15.8.2 Analysis of results 15.9 Summary References

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16 Optimal state and parameter estimation for fault detection and diagnosis of a high-dimensional fluid catalytic cracking unit 16.1 16.2 16.3 16.4

Introduction Process representation Fluid catalytic cracking unit Mathematical model of fluid catalytic cracking unit 16.4.1 Feed and preheat systems 16.4.2 Reactor 16.4.3 Regenerator 16.4.4 Air blowers 16.4.5 Catalyst circulation 16.4.6 Nomenclature and numerical data 16.5 Fluid catalytic cracking unit system variables 16.5.1 State variables 16.5.2 Algebraic variables 16.5.3 Measured variables 16.6 Fault cases considered in fluid catalytic cracking unit 16.7 Design of discrete version of extended Kalman filter

273 273 273 274

276 276 279 279 279 282

284 284 284

285 286 287 287 287

289 289 290 290 291 291 292 296 300 302 303 305 305 305 305 306 306

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16.8 Design of unscented Kalman filter 16.9 Analysis of results 16.10 Summary References

307 308 310 311

Part IV Optimal state estimation for process control

313

17 Optimal state estimator-based inferential control of continuous reactive distillation column

315

17.1 Introduction 17.2 Process and the dynamic model 17.2.1 Process description 17.2.2 The dynamic model of the process 17.3 The process characteristics 17.3.1 Nonlinearity analysis 17.3.2 Interaction and stability aspects 17.4 Classical proportional-integral/ proportional-integral-derivative controllers for distillation column 17.5 Brief description of genetic algorithms 17.6 Design of genetically tuned proportional-integral controllers 17.6.1 Controller tuning 17.6.2 Formulation of the objective function 17.6.3 Desired response specifications for genetic-algorithm tuning 17.7 Design of composition estimator 17.8 Analysis of results 17.9 Summary References

18 Optimal state estimation for nonlinear control of complex dynamic systems 18.1 Introduction 18.2 Optimal state estimation and estimator-based control of chaotic chemical reactor 18.2.1 Chaotic chemical reactor and its mathematical model 18.2.2 Estimator design and estimation results 18.2.3 Controller algorithm 18.2.4 Controller design

315 315 316 317 317 317 317

319 319 320 320 320 321 322 325 330 330

333 333

333 333 334 334 334

18.2.5 Analysis of results 18.3 Optimal state estimation and estimator-based control of homopolymerization reactor 18.3.1 Homopolymerization reactor and its mathematical model 18.3.2 Estimator design and estimation results 18.3.3 Controller design 18.3.4 Analysis of results 18.4 Summary References

19 Optimal state estimator based control of an exothermic batch chemical reactor 19.1 Introduction 19.2 Experimental system and its mathematical model 19.2.1 Experimental batch reactor 19.2.2 Analysis of reaction components 19.2.3 Esterification reactor model 19.3 State/parameter estimation 19.4 Control algorithms 19.4.1 Extended Kalman filter control 19.4.2 Generic model control 19.4.3 Model predictive control 19.5 Design of estimator based controllers for the esterification reactor 19.5.1 Extended Kalman filter control 19.5.2 Generic model control 19.5.3 Model predictive control 19.6 Analysis of results 19.7 Summary References

335

338 338 338 338 339 342 342

343 343 343 344 345 345 347 348 348 349 349 350 350 350 351 352 356 357

Part V Optimal state estimation for online optimization

359

20 Optimal state and parameter estimation for online optimization of an uncertain biochemical reactor

361

20.1 Introduction 20.2 The process and its mathematical model

361 362

Contents

20.3 State and parameter estimation using extended Kalman filter 20.4 State and parameter estimation using two-level extended Kalman filter 20.5 Online optimization problem 20.6 Functional conjugate gradient method 20.7 Extended Kalman filter-assisted online optimizing control of the biochemical reactor 20.7.1 Designing the strategy 20.7.2 Analysis of results 20.8 Two-level extended Kalman filter-assisted online optimizing control strategy 20.8.1 Designing the strategy

362 362 363 363

364 364 365

366 366

20.8.2 Analysis of results 20.9 Summary References

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367 368 371

21 Overview, opportunities, challenges, and future directions of state estimation 373 21.1 21.2 21.3 21.4 21.5 Index

Overview Opportunities Challenges Future directions Summary

373 373 374 374 375 377

Preface This book is addressed to students, researchers, and industry professionals in multiple domains of science, engineering, and technology. This book covers the fundamentals and advanced topics of state estimation with a number of solved examples and case studies, which are beneficial to the personnel of different disciplines to gain knowledge and apply to the problems encountered in their respective domains. General readers of initial chapters that cover fundamentals, algorithms, basic problems, and solved examples on state estimation are expected to have familiarity with the basics of mathematics. Readers of lateral chapters that deal with the design and real applications of state estimators are expected to have familiarity with the fundamentals of mathematics, mathematical modeling, and control concepts along with their basic domain knowledge in engineering and science.

State estimation or soft sensing is an approach that deals with the methods that are used to estimate the unmeasured process variables through the use of known measurements and process knowledge. The enhanced degree of process automation and the growing demand for higher performance in industrial systems make state estimation an integral part of various operational strategies associated with the process systems engineering field. It occupies a greater role in the design, modeling, monitoring, fault diagnosis, optimization, control and operation of all kinds of physical, chemical, biological, and other engineering and technological processes through the use of systematic computer-aided approaches.

The task of state estimation has become more challenging in the face of growing demand to meet the requirements of process safety, environmental regulations, energy efficiency, better product quality, and optimum utilization of available resources.

The contents of this book are organized such that the readers in a step-by-step approach can learn the basics of state estimation, acquaint themselves with the relevant algorithms, and understand the problem-solving approaches. As they go further, they will be getting familiar with the design concepts and real applications of state estimation towards process monitoring, fault diagnosis, online optimization, and control. Based on the knowledge gained from the real applications, the audience can create or apply the concepts and strategies of state estimation to solve the problems in their respective domains of science and engineering. Therefore this book covers the pedagogical aspects such as knowledge, understanding, application, analysis, evaluation, and creation.

This book is presented in five parts, wherein the details regarding the description, algorithms, examples, procedures, real system applications, and case studies concerning different approaches and methods of state estimation are elaborated in detail. The first part of this book consists of six chapters covering the basic details, a variety of state estimation algorithms in different approaches, and their implementation procedures, along with solved examples. Moreover, various methods of sensor configuration for state estimation and their implementation procedures are covered in this part.

The second part comprises seven chapters covering mechanistic model-based filtering and observation techniques as well as data-driven model-based state estimation methods. The design and implementation of various state estimation strategies for monitoring applications of several real processes are presented and discussed thoroughly in this part. The third part of the book contains three chapters emphasizing the design and application of various quantitative modelbased state estimation methods for process fault detection and diagnosis in various nonlinear processes. The fourth part consists of three chapters dealing with the methods and applications of state estimation toward process control. The algorithmic implementation, design, and evaluation of different state estimation methods with several real applications are presented in detail in this part. The fifth part consists of a chapter explaining the importance of state estimation in online optimization. Different state estimation methods with design and implementation procedures for online optimization of real processes are presented in this part. The last chapter at the end of this book briefs an overview, opportunities, challenges, and future directions in state estimation.

Formulation, design, and implementation of various state estimation strategies to solve a wide variety of base cases as well as real engineering problems make this book more beneficial to researchers working in multiple domains. Many references with a variety of state estimation and related theme titles are included at the end of each chapter of this

xv

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Preface

book. These references will be immensely useful to the readers to advance their general knowledge and domain knowledge in the field of state estimation.

Finally, the first author takes this opportunity to thank his colleague Dr. K. Yamuna Rani, beloved PhD students Dr. Rama Rao Karri, Dr. C. Sumana, Dr. N. Anil, Dr. P. Anand, Dr. J.S. Eswari, and several graduate and postgraduate students at the Indian Institute of Chemical Technology (CSIR-IICT) for their contributory research support to this book. Also, thanks to CSIR-IICT management for providing this opportunity to pursue the research and publish its major outcomes.

Ch. Venkateswarlu and Rama Rao Karri

About the authors Dr. Ch. Venkateswarlu has formerly worked as a scientist, senior principal scientist, and chief scientist at Indian Institute of Chemical Technology (IICT), Hyderabad, a premier research and development (R&D) institute of Council of Scientific and Industrial Research (CSIR), India.

Recently, he retired as the Director R&D at BV Raju Institute of Technology (BVRIT), Narsapur, Greater Hyderabad. Prior to this, he has also worked as a professor, principal, and head of Chemical Engineering Department of the same institute. He completed his graduation from Andhra University, India, and from Indian Institute of Chemical Engineers, and postgraduation and PhD in Chemical Engineering from Osmania University, Hyderabad, India. He holds 35 years of R&D and industry experience along with 20 years of teaching experience.

His research interests lie in the areas of dynamic process modeling and simulation; process identification and dynamic optimization; process monitoring and fault diagnosis; state estimation and soft sensing; statistical process control and advanced process control; applied engineering mathematics and evolutionary computing; artificial intelligence and expert systems; and bioprocess engineering and bioinformatics. He has published more than 120 research papers in peer journals of repute along with some more international and national proceeding publications. He is also credited with 150 technical paper presentations and invited lectures. He has authored a book published by Elsevier along with a few book chapters. He is also in editorial boards of a few international journals. He has executed several R&D projects sponsored by DST and Industry. He is a reviewer of several international research journals and many national and international research project proposals. He has guided several postgraduate and PhD students. He has served as a long-term guest faculty for premier institutes such as Bhaba Atomic Research Centre Scientific Officers Training, BITS Pilani MS (off-campus), and IICT-CDAC Bioinformatics Programs. He is a fellow of Andhra Pradesh Akademi of Sciences and Telangana State Academy of Sciences. He received various awards in recognition to his R&D and academic contributions.

Dr. Rama Rao Karri working as a professor (Sr. Asst.) in Petroleum and Chemical Engineering, Faculty of Engineering, Universiti Teknologi Brunei (UTB), Brunei Darussalam.

He has obtained his PhD in chemical engineering from Indian Institute of Technology (IIT) Delhi, Masters from IIT Kanpur, and Bachelors from Andhra University College of Engineering, Visakhapatnam, India. He has worked as a postdoctoral research fellow at NUS, Singapore, for about 6 years and has over 18 years of working experience in academics, industry, and research. He has led the water resources cluster team at UTB (2015 17), which has focused on water energy nexus, smart water, and environmental management. His research activities are mainly focused in the development of modeling methods for systems with inherent uncertainties in multidisciplinary fields. He has experience of working in multidisciplinary fields and has expertise in various evolutionary optimization techniques and process modeling.

He has published around 100+ research articles in reputed journals, book chapters, and conference proceedings with a combined impact factor of 312.2 and has an h-index of 23 (Scopus and Google Scholar). He is an editorial board member in 6 renowned journals and a peer-review member for more than 81 reputed journals from Elsevier (22), Springer (18), Taylor & Francis (9), Scientific Reports, and other reputed journals (32), and has peer reviewed more than 300 manuscripts. He is a recipient of Publons Peer Reviewer Award as top 1% of global peer reviewers for Environment & Ecology and Crossfield categories for the year 2019. He is also delegating as an advisory board member for many international conferences. He held a position as Editor-in-Chief in International Journal of Chemoinformatics and Chemical Engineering, IGI Global, USA (April 2019 August 2020). He is also a Managing Guest Editor for special issues (1) “Magnetic nano composites and emerging applications,” in Journal of Environmental Chemical Engineering xiii

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About the authors

(IF: 5.909), and (2) “Novel CoronaVirus (COVID-19) in Environmental Engineering Perspective,” in Journal of Environmental Science and Pollution Research (IF: 4.223), Springer. He has also been a guest editor for the special issue “Nanocomposites for the Sustainable Environment” in Applied Sciences Journal (IF: 3.056), MDPI. He is the associate editor in Scientific Reports (IF: 4.379), Springer Nature, and International Journal of Energy and Water Resources (IJEWR), Springer Inc. He is also the coeditor and managing editor for seven Elsevier, one Springer, and one CRC edited books. Recently he has been listed in the 2021 world 2% scientists report compiled by Stanford University Researchers.

Chapter 1

Optimal state estimation and its importance in process systems engineering 1.1

Introduction

Monitoring, diagnosis, and control of process plants heavily rely on accurate and quickly accessible real-time information of the states that describe the operating behavior of the process. The variables that provide a complete representation of the internal status of the system at a given instant of time are referred to the states of the system. In several processes, the full state of the plant is not directly measurable, and the process measurements are usually corrupted with noise. Therefore, reducing the noise in the measurements and reconstructing the state of the plant is a crucial issue from the process monitoring, fault diagnosis and control point of view. Filtering and prediction are elementary concepts to state estimation. The aim of the filtering and estimation problem is to establish the best estimate for the true variable of a system from an incomplete and noisy set of observations of the system. Optimal estimation of unmeasured process states is attractive, when conventional hardware sensors are not available, or the cost and technical limitations prohibit their usage. The methods that are employed to estimate the unmeasured process variables are called state estimators/soft sensors. These soft sensors use known measurements and process knowledge to provide the estimates of the unmeasured process variables, which are incorporated in process operation schemes to achieve economic optimum. The advancement in computer technology and cheap computing power leads to new developments in soft sensor technology.

1.2

Significance of state estimation

State estimators or soft sensors can be built by using currently available online, offline, and historical information of the process. The behavior of any process is indicated by the states of the output variables, which are dependent on the inputs to the process. A subset of these process outputs represents the primary variables that quantify the productivity in terms of the purity, quality, or properties of the product. These so-called primary variables are often the ones that are difficult to measure online. The other outputs (e.g., temperature, flow, and pressure) are called secondary variables, which can be easily measured online. Because of the intrinsic feature of the process, the states of the secondary variables have a certain dependency on the states of the primary variables. For example, pressures and temperatures can define the liquid composition in a system. It is not easy to access the measurements in time for states like concentrations and the parameters like reaction rates and heat transfer coefficients in chemical and biochemical processes. If such states and parameters are estimated using the known measurements of temperatures and pressures, these estimated variables can be used in process operation schemes to improve plant performance. For instance, separation systems like multicomponent distillation columns are to be operated as precisely as possible to meet the desired product purity specifications. If the current compositions of the columns are available in time, they can form a base for improving the process performance through operator decision-making or in developing an automatic closed-loop control scheme. An automatic sensor system for composition measurement is not economical since component analyzers like gas chromatograph involve large measurement delays and high investment and maintenance costs. As an alternative, component concentrations can be estimated from the temperature measurement using the state estimators. Thus, it should be possible to use the readily available secondary variables to infer the primary variable through the use of a soft sensor. State estimation has become an integral part of various schemes such a process design, modeling, monitoring, fault diagnosis, optimization, control, and operation of all kinds of processes.

Optimal State Estimation for Process Monitoring, Fault Diagnosis and Control. DOI: https://doi.org/10.1016/B978-0-323-85878-6.00013-0 © 2022 Elsevier B.V. All rights reserved.

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PART | I Basic details and state estimation algorithms

1.3

Role of state estimation in process systems engineering

The technological advancements have made the process industry undergo significant changes emphasizing optimally running the plants with due consideration to safety, monitoring, and control with minimal manual intervention. The increased degree of automation and the growing demand for higher performance, efficiency, and reliability in industrial systems makes the task of state estimation an integral part of various operational strategies such as monitoring, diagnosis, and control concerning the field of process systems engineering. State estimation is a subject of immense interest with applications in conventional and advanced process control, supervisory control, process monitoring, online optimization, product quality monitoring, data reconciliation, fault detection, and diagnosis, etc. The state estimation-based applications have become more crucial in the face of growing demand to meet process safety requirements, environmental regulations, energy efficiency, better product quality, and optimum utilization of available resources. To meet various goals concerning plant operations, detailed information about the number of process variables and parameters is required. Installation of additional instrumentation and sensors to monitor the plant’s operation can be expensive both in terms of the capital cost as well as maintenance cost. With the use of state estimator/soft sensor, it is possible to estimate many other variables and process parameters by strategically measuring few key variables of the process.

However, to gain the maximum benefit from the state estimation techniques, the sensors that provide vital information about the measured variables are to be placed at optimal locations in the process plant. Thus, optimal selection of measurements has become an important prerequisite for the successful estimation of process states. Various sensor configuration methods have been introduced for the purpose of process state estimation. Thus, state estimation with optimal sensors selection has attained greater significance in monitoring, diagnosis, and control of high dimensional systems.

1.4

Outline of this book

Approaches for state estimation can be broadly classified into mechanistic/first principle model-based approach and datadriven model-based approach. The mechanistic model-based approach includes various stochastic model-based filtering and observation techniques such as Kalman filter, extended Kalman filter, unscented Kalman filter, square root unscented Kalman filter, particle filter, and ensemble Kalman filter. The data-driven model-based approach includes the state estimators derived based on principal component analysis, partial least squares, artificial neural networks, and radial basis function networks. Further various linear and nonlinear observers are also applied for optimal state estimation. The details regarding the description, algorithms, examples, applications, and case studies concerning different approaches and methods of state estimation are elaborated in this book to benefit a wider audience of different disciplines.

The book is presented in five sections highlighting several applications of state estimation in process systems engineering field. The first section covers the fundamental details about state estimation, where in a wide variety of state estimation algorithms in different approaches and their implementation procedures, along with solved examples are presented in-detail. Also, various methods of sensor configuration for state estimation and their implementation procedures are covered. The second section covers process monitoring applications of state estimation, where in design and implementation of various state estimation strategies for monitoring applications of several real processes are discussed. The third section of the book, emphasizes the methods and approaches of state estimation towards process fault detection and diagnosis. The significance of state estimation towards this direction is dealt with different real applications of fault detection and diagnosis in design, execution and performance evaluation. The fourth section of this book presents the methods and approaches of state estimation towards process control with algorithmic implementation, design, and evaluation procedures. The importance of state estimation towards process control is described with several real applications. The fifth section explains the importance of state estimation in online optimization of engineering systems. In this section, different state estimation methods are presented with design and implementation procedures for online optimization of real processes.

1.5

Summary

A state estimator or soft sensor is a mathematical technique used to infer important process variables that are not physically measured. Optimal estimation of the states of a plant from its model in conjunction with its measured outputs constitutes a dominant role in the process systems engineering domain. State estimation is highly useful and supportive to various process operation schemes such as advanced process control, fault detection, process monitoring, and product quality control that aim to achieve higher process performance. This introductory chapter explains the significance and importance of state estimation and briefs about the theme of this book that deals with the details of fundamentals, algorithms, examples, applications, and case studies concerning different approaches and several methods of state estimation.

Chapter 2

Introduction to stochastic processes and state estimation filtering 2.1

Introduction

In many autonomous systems, the knowledge of the system state is essentially required to monitor the status of the process. In a realistic situation, the state of a process is often not directly obtainable but it is usually inferred or estimated based on the system outputs measured by instruments (sensors) along with the support of a dynamic model representing the system. In most cases, building a perfect model to capture all the dynamic phenomenon is not possible. To compensate for the unmodelled dynamics, process noise is often added to the dynamic model. Moreover, to account the measurement errors in realistic situation, proper measurement noise is added to the measurement model. Processes that are associated with such a random noise phenomenon are defined as stochastic or random processes. For most engineering applications, the process noise and measurement noise are assumed to follow zero-mean Gaussian or normal distribution. The general idea of filtering and estimation problem is to establish the best estimate for the true variable of a system from an incomplete and noisy set of observations of the system.

This chapter describes and elaborates the basic concepts relating to stochastic variables, noise, probability, probability distributions, random processes, stochastic representation of general dynamic models, filtering, prediction, and estimation.

The concepts and definitions presented in this chapter provide a basic framework for other chapters of this book.

2.2

Probability and stochastic variables

A probability variable is a random variable, whereas a stochastic variable is a chance variable. The probability variable is usually denoted by a capital letter X or Y. The name suggests that the variable has something to do with the concept of probabilities. Suppose, a die is rolled and X is the outcome, then the outcome varies for every turn. Thus X represents a random variable. The possible values of X are 1, 2, 3, 4, 5, and 6. Occurrence of each of these values has a probability of 1/6. In a statistical sense, probability represents the relative frequency of an event, when it is observed for more number of times. Assume that a discrete random variable X takes on the values x1, x2, . . ., xk as a result of an experiment. If X takes the value xi for m observations and the total number of trials is n, then the ratio m/n is called the relative frequency of the event X 5 xi. The relative frequency m/n itself is a random number and changes according to the number of trials performed. As an example of relative frequency, consider the tossing of a coin. Here the events are heads and tails. Considering both the events occur equally likely, that implies the coin is unbiased and it is tossed a large number of times such that the event heads appear exactly half the number of tails. The relative frequency of occurrence of heads is thus 0.5. Similarly, the probability of tails is 0.5.

In general, the probability is a nonnegative number and its values lie between 0 and 1. A value of “zero” indicates that the event will not occur and “one” signifies that the event certainly occurs. Suppose we have a case of n possible events, x1, x2, . . ., xn, which are mutually exclusive in the sense that the occurrence of one event excludes the other.

Consider the probabilities of occurrence of these events as p1, p2, . . ., pn, respectively, such that the combined probability of all the events is unity, that is, p1 1 p2 1 ? 1 pn 5 1.

Probability measure: The probability of an event is measured by assigning a numerical value for each event. Consider a sample space S with n(S) outcomes in which the event A occurs n(A) times. The probability measure is expressed as Probability of anevent 5

Number of favorable outcomes Number of possible outcomes

pðAÞ 5

nðAÞ nðSÞ

Optimal State Estimation for Process Monitoring, Fault Diagnosis and Control. DOI: https://doi.org/10.1016/B978-0-323-85878-6.00020-8 © 2022 Elsevier B.V. All rights reserved.

(2.1)

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PART | I Basic details and state estimation algorithms

If the sample space S has n equally liked outcomes, then p1 5 p2 5 . . . 5 pn 5 1=n

(2.2)

This is called a uniform distribution on S. Example 1: Consider rolling a fair die 3 times, find the probability when all the numbers are the same in each trail.

Solution When the die is rolled three times, the sample space has 216 ordered triplets as (i, j, k). Since the die is fair, the event of interest is the equal probability of the same numbers with six outcomes:

A 5 ð1; 1; 1Þ; ð2; 2; 2Þ; . . .; ð6; 6; 6Þ Thus we have a uniform probability distribution. Here n(A) 5 6 and n(S) 5 216.

Thus, P(A) 5 6/216 5 1/36.

2.2.1 Probability theorems The addition and multiplication theorems of probability are briefed as follows: Addition theorem: Let A and B are two independent events in the sample space of an experiment.

PðA , BÞ 5 PðAÞ 1 PðBÞ 2 PðA - BÞ

(2.3)

If A and B are mutually exclusive events, A - B 5 null set. Thus PðA , BÞ 5 PðAÞ 1 PðBÞ

(2.4)

If the sample space consists of n events such that S 5 A1 , A 2 , . . . , A n

(2.5)

Thus, according to the addition theorem, PðSÞ 5 PðA1 Þ 1 PðA2 Þ 1 ? 1 PðAn Þ 5

n X

PðAi Þ

(2.6)

i51

Multiplication theorem: Let A and B are two independent events in the sample space of an experiment. According to the multiplication theorem, PðA - BÞ 5 PðAÞ:PðBÞ

(2.7)

If A1, A2,. . ., An are independent events in the sample space, then PðA1 - A2 - ? - An Þ 5 PðA1 Þ 3 PðA2 Þ? 3 PðAn Þ

(2.8)

Example 2: Consider throwing of a single die with sample space S 5 {1, 2, 3, 4, 5, 6}. What is the probability of getting an odd number ,3 and even number .3?

Solution The sample space S is represented by S 5 f1; 2; 3; 4; 5; 6g Let A and B are mutually exclusive events in the sample space.

A 5 fodd number , 3g 5 f1g B 5 feven number . 3g 5 f4; 6g PðA , BÞ 5 PðAÞ 1 PðBÞ 1 2 1 5 1 5 6 6 2

Introduction to stochastic processes and state estimation filtering Chapter | 2

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2.2.2 Conditional probability It is a probability measure that deals with dependent events. Suppose A and B are two dependent events; the P(A/B) defines the probability of event A after the occurrence of B. Then P(A/B) can be interpreted as a probability of A given B. Similarly, P(B/A) defines the vice versa. This concept is known as conditional probability since a condition on the occurrence of B or A is specified.

The measure for conditional probability is given as follows: PðA=BÞ 5

PðA - BÞ ; PðBÞ

provided PðBÞ 6¼ 0

(2.9)

PðA=BÞ 5

PðB - AÞ ; PðAÞ

provided PðAÞ 6¼ 0

(2.10)

or

If A and B are independent events, then PðA=BÞ 5

PðAÞ 3 PðBÞ 5 PðAÞ PðBÞ

(2.11)

It is often required to find the probability of event B after the occurrence of event A. This probability is called the conditional probability of B given A and denoted as P(B/A). The probability measure for this case is given by PðB=AÞ 5

PðA - BÞ PðAÞ

(2.12)

In case if the events A and B occur in a sample space S, and P(A) 6¼ 0, P(B) 6¼ 0, then PðA - BÞ 5 PðAÞ 3 PðB=AÞ 5 PðBÞ 3 PðA=BÞ

(2.13)

If A and B are independent events, then PðA - B 5 PðAÞ 3 PðBÞ

(2.14)

PðA=BÞ 5 PðAÞ; PðB=AÞ 5 PðBÞ

(2.15)

If P(A) 6¼ 0, P(B) 6¼ 0, then

This means that the probability of A does not depend on the occurrence or nonoccurrence of the probability of B, and vice versa.

Example 3: A problem in an engineering subject is given to three students A, B, and C. The chances of solving the problem by the students are 1/3, 1/4, and 1/5, respectively. What is the probability that the problem is solved?

Solution The probability that A can solve the problem, P(A) 5 1/3.

The probability that A cannot solve the problem 5 1 1/3 5 2/3 The probability that B can solve the problem, P(B) 5 1/4.

The probability that B cannot solve the problem 5 1 1/4 5 3/4 The probability that C can solve the problem, P(C) 5 1/5.

The probability that C cannot solve the problem 5 1 1/5 5 4/5 The probability that A, B, and C cannot solve the problem 5 2/3 3 3/4 3 4/5 5 2/5 The probability that the problem is solved by at least one student 5 1 2/5 5 3/5

2.3

Probability distributions and distribution functions

The probability functions and the probability distributions are described as follows:

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PART | I Basic details and state estimation algorithms

2.3.1 Discrete random variables and discrete probability distributions Random variables can be discrete or continuous. If a random variable takes a finite set of countable values, it is called a discrete random variable. Let X be a discrete random variable. Our interest is to compute the probabilities of the form P(X 5 xk) for various values of xk in the range of X. As xk varies as x1, x2,. . ., etc., in the range of X, the probability P (X 5 xk) also varies. Thus P(X 5 xk) is a function of xk. The probability function is represented by PðX 5 xk Þ 5 f ðxk Þ; k 5 1; 2; . . .

(2.16)

The probability function is also called a probability distribution which is given by PðX 5 xÞ 5 f ðxÞ

(2.17)

For x 5 xk, this function follows Eq. (2.16); while for other values of x, f(x) 5 0.

In general, f (x) is a probability function if (i) fP(x) $ 0 (ii) f ðxÞ 5 1, where the sum is taken as the overall possible values of x.

Example 4: Suppose that a coin of the head (H) and tail (T) is tossed twice, the sample space becomes S 5 {HH, HT, TH, TT}. Let X be the random variable representing the number of heads that can come up in the sample space. Find the probability function corresponding to the random variable X. Solution We have X is a random variable representing each of the sample point HH, HT, TH, and TT in the sample space.

Thus, we have PðHH Þ 5 1=4; PðHT Þ 5 1=4; PðTH Þ 5 1=4 ; and PðTT Þ 5 1=4: Then PðX 5 0Þ 5 PðTT Þ 5 1=4 PðX 5 1Þ 5 PðHT , THÞ 5 PðHT Þ 1 PðTH Þ 5 1=4 1 1=4 5 1=2 PðX 5 2Þ 5 PðHH Þ 5 1=4 The probability function is given in Table 2.1.

Example 5: Find whether the following functions f1(x) and f2(x) represent the probability distribution functions for discrete random variables.

8 > < 3=4; x 5 2 2 f1 ðxÞ 5 1=4; x 5 3 ; > : 0; elsewhere

8 < 2=3; f2 ðxÞ 5 2 1=3; : 0;

x521 x54 elsewhere

Solution The function f1(x) takes nonzero values 3/4 and 1/4 at the points x 5 22 and 3, respectively, and x takes all other values with zero probability, that is, f1(22) 5 3/4, f1(3) 5 1/4, and f1(x) 5 0, elsewhere. Condition (i) is satisfied since f1(x) $ 0 for all values of x. Condition (ii) is also satisfied because 3/4 1 1/4 1 0 5 1. Hence f1(x) represents a probability function for a discrete random variable x.

TABLE 2.1 Probability function for example 4. x f(x)

0

1

2

1/4

1

/2

1/4

Introduction to stochastic processes and state estimation filtering Chapter | 2

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P Similarly, in the case of f2(x), x f2 ðxÞ 5 1, thus condition (ii) is satisfied. However, f2(x) at x 5 4, that is, f2(4) 5 21/3, which is negative. Hence condition (i) is violated. Therefore f2(x) cannot be the probability function of any random variable.

Cumulative distribution function: The distribution function or cumulative distribution function for a discrete random variable X can be obtained from its probability function by noting that, for all x in (2 ~ , ~ ), X FðxÞ 5 PðX # xÞ 5 f ðuÞ (2.18) u#x

where the sum is the overall values of u taken on by X for which u # x.

If X takes on only a finite number of values x1, x2, . . ., xn, then the distribution function is given by 8 0 2 ~ , x , x1 > > > > x1 # x , x2 < f ðx1 Þ FðxÞ 5 f ðx1 Þ 1 f ðx2 Þ x2 # x , x3 > > ^ ^ > > : f ðx1 Þ 1 f ðx2 Þ 1 ? 1 f ðxn Þ xn # x , ~

(2.19)

Example 6: Find the distribution function for the random variable x in Example 4.

Solution The distribution function is

8 0 > > < 1=4 FðxÞ 5 3=4 > > : 1

; 2 ~ #x#0 ; 0#x,1 ; 1#x,2 ; 2#x, ~

The distribution function [1] is sketched, as shown in Fig. 2.1.

From the distribution function (Fig. 2.1), it can be observed that the magnitude of jumps at 0, 1, and 2 are the probabilities given in Table 2.1. This shows that the probability function can be obtained from the distribution function.

Accordingly, the probability function of a discrete random variable obtained from the distribution function is given by: f ðxÞ 5 FðxÞ 2 Lim2 FðuÞ u-x

FIGURE 2.1 Plot of the distribution function.

(2.20)

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PART | I Basic details and state estimation algorithms

2.3.2 Continuous random variables and continuous probability distributions A random variable that assumes an infinite number of values is called a continuous random variable. The distribution function of a continuous random variable X is defined as FðxÞ 5 PðX # xÞ.

FðxÞ 5 PðX # xÞ

(2.21)

The F(x) is called the Probability Distribution Function of the random variable X. The following are the properties of the probability distribution function: F ðxÞA½0; 1 Fð2 NÞ 5 0 FðNÞ 5 1 F ðaÞ # F ðbÞ if a # b Pða , X # bÞ 5 F ðbÞ 2 F ðaÞ

(2.22)

The conditional probability distribution function of the random variable X for event A is defined as Fðx=AÞ 5 PðX # x=AÞ PðX # x=AÞ 5 PðAÞ

(2.23)

The distribution function F(x) 5 P(X # x) can be considered as a monotonically increasing function which increases from 0 to 1 and is represented as shown in Fig. 2.2.

On the basis of integral calculus, the probability distribution function can be defined as the derivative of F(x) as dFðxÞ 5 f ðxÞ dx

(2.24)

Ðf ~ðxÞ $ 0 2 ~ f ðxÞdx 5 1 Ð b Pða , X # bÞ 5 a f ðxÞdx

(2.25)

The function f (x) has the properties

FIGURE 2.2 Monotonically increasing function from 0 to 1.

Introduction to stochastic processes and state estimation filtering Chapter | 2

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The function f(x) is also called a probability density function(PDF) or simply density function of a continuous random variable. If f(x) is the density function for a random variable X, then we can represent y 5 f(x) graphically by a curve as in Fig. 2.3. Since f(x) $ 0, the curve cannot fall below the x-axis. The entire area bounded by the curve and the x-axis must be 1. Geometrically the probability that X is between a and b, that is, P(a , X , b), is then represented by the area shown shaded in Fig. 2.3.

The conditional PDF of the random variable X for event A is defined as f ðx=AÞ 5

dFðx=AÞ dx

(2.26)

Suppose we have two random variables X1 and X2, the conditional probability density of the first given that the second takes on the variable x2 as f ðx1 ; x2 Þ 5

f ðx1 ; x2 Þ f ðx2 Þ

(2.27)

where f(x1, x2) and f(x2) are the joint probability density of the two random variables and the marginal density of the second variable.

Example 7: The probability density is represented by an exponential decay function of the form 8 < 1 e2t=3 ; 0 # t , α f ðtÞ 5 3 : 0; elsewhere Compute the probabilities (i) P(0 # t # 2), and (ii) P(3 # t # 8).

Solution Probabilities are calculated as the areas under the density curve between the corresponding ordinates or the integral of the density over the given interval. (i) The probability P(0 # t # 2) is given by ð2 2 1 2t=3

e dt 5 2e2t=3 5 1 2 e22=3 3 0 0

FIGURE 2.3 Probability that X is between a and b.

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PART | I Basic details and state estimation algorithms

(ii) The probability P(3 # t # 8) is given by ð8 8 1 2t=3

e dt 5 2e2t=3 5 e21 2 e28=3 3 3 3 The probabilities are shown in the shaded areas covered by dotted lines as shown in Fig. 2.4.

Example 8: Consider the following density function

f ðxÞ 5

kx3 ; 0 , x , 4 0; otherwise

(a) Find the constant k (b) Compute P(2 , X , 3) Solution Ð~ The function f (x) has the properties such as f (x) $ 0, and 2 ~ f ðxÞdx 5 1. (a) f (x) $ 0 if k . 0

Ð4 3 2 ~ f ðxÞdx 5 0 kx dx kx4 4 5 64k 5 4 0

Ð~

Since this must be equal to 1, which means 64k 5 1 . k 5 1/64. Ð3 1 3 4 3 x dx 5 x 5 81 2 16 5 65 (b) Pð 2 , X , 3Þ 5 2 64

256 2

256

256

256

If f(x) is continuous, we know the probability of X is equal to any particular value is zero. In such a case, we can replace either or both of the signs , in Eq. (2.25) by # . Thus, for this example, Pð 2 # X # 3 Þ 5

FIGURE 2.4 Probabilities of the exponential function.

65 256

Introduction to stochastic processes and state estimation filtering Chapter | 2

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Example 9: If two random variables have the joint probability density 8 < 3 ðx 1 3x Þ; 1 2 f ðx1 ; x2 Þ 5 2 : 0;

for 0 , x1 , 1 & 0 , x2 , 1 otherwise

Find the conditional probability density of the first variable, given that the second variable takes on the value x2.

Solution First, we find the marginal density of the second random variable by integrating the function with respect to x1. Ð1 3 0 2 ðx1 1 3x2 Þdx1 3 5 ð1 1 6x2 Þ; for ð0 , x2 , 1Þ 4

f ðx2 Þ 5

and f(x2) 5 0, otherwise.

Hence, by definition, the conditional probability of the first random variable given that the second takes on the value x2 is 3 ðx1 1 3x2 Þ 2x1 1 6x2 5 f ðx1 ; x2 Þ 5 23 1 1 6x2 4 ð1 1 6x2 Þ

for 0 , x1 , 1; 0 , x2 , 1Þ

and f(x1, x2) 5 0, otherwise.

2.4

White Gaussian noise and colored noise

In filtering and estimation problems, most of the physical systems under consideration are noisy. The noise may arise in a number of ways. For example, the inputs to the system may associate with noise which is unknown/unpredictable.

The outputs from the system may be noisy due to sensor signal measurement inaccuracies. White noise and colored noise are important signals in stochastic systems.

White noise: In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. White noise draws its name from white light, although light that appears white, generally it does not have a flat power spectral density over the visible band. In discrete-time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with zero mean and finite variance. If each sample has a normal distribution with zero mean, the signal is said to be additive white Gaussian noise.

Consider the discrete model of a typical system is represented by the following equations: xk11 5 f ðxk ; θÞ 1 Gw yk 5 Hxk 1 v

(2.28)

where xARn denote the state variables, θ denote system parameters, and yARm denotes the outputs. H is m 3 n measurement matrix, G is state noise coefficient matrix, and the symbols w and v refer to Gaussian white noises. White noises are some uncertainties that are independent time series and distributed identically, which means no autocorrelation between them. In a particular case, the “Gaussian white noise” has a normal distribution with zero mean and standard variation σ.

For the system in Eq. (2.28), the Gaussian white noise {w} on states and Gaussian white noise {v} on output has the following statistical expectations with respect to their mean and autocorrelations. The mean vector and autocorrelation matrix of w and v are:

μw 5 Efwg 5 0; Rww 5 E ww T 5 σ2 I; (2.29) μv 5 Efvg 5 0; Rvv 5 E vvT 5 σ2 I where E is the expected value operator and I is an identity matrix.

Colored noise: Colored noises are some uncertainties that are dependent on their past states and have an autocorrelation. In particular, passing a “Gaussian white noise” from first-order filter results colored noise.