Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment 9789811622663, 9789811622670

Table of contents :
Foreword
Preface
Summary
Contents
1 Introduction
1.1 Background
1.2 Equipment Life Prediction
1.2.1 Fundamental Concept of Life Prediction
1.2.2 Literature Review on Life Prediction
1.3 Maintenance Decision of Equipment
1.3.1 Fundamental Concept and Classification of Maintenance
1.3.2 Literature Review on Maintenance Decision of Single-Component System
1.3.3 Literature Review on Maintenance Decision of Multi-component System
References
2 Residual Life Prediction Based on Wiener Process with Nonlinear Drift
2.1 Introduction
2.2 Definition of Wiener Process
2.3 Degradation Modeling Based on Wiener Process with Nonlinear Drift
2.4 Probability Density Function of the Residual Life
2.5 Parameter Estimation
2.5.1 Off-Line Estimation of Common Parameters and Hyper-Parameters
2.5.2 Real-Time Updating of Random Parameter
2.6 Case Study
2.6.1 Problem Description
2.6.2 Results and Discussions
2.7 Summary of This Chapter
References
3 Residual Life Prediction Based on Wiener Process with Abrupt Changepoint
3.1 Introduction
3.2 Degradation Model with Abrupt Changepoint Based on Wiener Process
3.2.1 Wiener-Process-Based Degradation Model
3.2.2 Changepoint Detection in Performance Degradation Process
3.3 Conjugate Distribution of Prior Distribution of Exponential Family
3.4 Bayesian Online Changepoint Detection Algorithm
3.5 Empirical Bayesian Method for Determining Prior Distribution
3.5.1 Improved Forward–Backward Algorithm
3.5.2 Joint Distribution of Changepoint Markers at Adjacent Detection Moments
3.5.3 EM Algorithm
3.6 Residual Life Prediction Based on Bayesian Online Changepoint Detection
3.7 Case Study
3.8 Summary of This Chapter
References
4 Gamma Process-Based Degradation Modeling and Residual Life Prediction
4.1 Introduction
4.2 Definition of Gamma Process
4.3 Parameter Estimation for Gamma Process
4.3.1 Method of Moments
4.3.2 Maximum Likelihood Estimate
4.4 Residual Life Prediction Based on Gamma Process
4.4.1 Life Distribution
4.4.2 Residual Life Distribution
4.4.3 Reliability Function
4.4.4 Example Verification
4.5 Degradation Modeling Based on Gamma Process with Environmental Impact
4.5.1 Problem Description
4.5.2 Residual Life Distribution
4.5.3 Maintenance Decision
4.6 Summary of This Chapter
References
5 Inverse Gaussian Process-Based Degradation Modeling and Residual Life Prediction
5.1 Introduction
5.2 Definition of Inverse Gaussian Process
5.3 ER-Based Parameter Estimation
5.3.1 Parameter Estimation Based on Single Specific Equipment’s Degradation Data
5.3.2 Fusion of Fixed Parameters Based on ER
5.4 Derivation of Residual Life Distribution
5.5 Experimental Verification
5.6 Summary of This Chapter
References
6 Degradation Modeling and Residual Life Prediction Based on Support Vector Machine
6.1 Introduction
6.2 SVR Principle
6.2.1 Primal and Dual Problems
6.2.2 Sparsity of SVR
6.2.3 Kernel Function
6.3 Residual Life Prediction Method Based on GA-Optimized SVR
6.3.1 Problem Description
6.3.2 Basic Ideas
6.3.3 Specific Steps of the Method
6.3.4 Case Study
6.4 Residual Life Prediction Method Based on SVR and FCM Clustering
6.4.1 Problem Description
6.4.2 Basic Ideas and Specific Steps
6.4.3 Case Study
6.5 Summary of This Chapter
References
7 Degradation Modeling and Residual Life Prediction Based on Fuzzy Model of Relevance Vector Machine
7.1 Introduction
7.2 Fuzzy Model Based on Relevance Vector Machine
7.2.1 Mathematical Description of Fuzzy Model
7.2.2 Fuzzy Model Based on Relevance Vector Machine
7.2.3 Uniform Approximation of Fuzzy Model Based on Relevance Vector Machine
7.3 Fuzzy Model Identification Based on Relevance Vector Machine
7.3.1 Structure Identification
7.3.2 Parameter Identification
7.3.3 Fuzzy Model Identification Algorithm Based on RVM and Gradient Descent Method
7.4 Degradation Modeling and Residual Life Prediction
7.5 Case Study
7.5.1 Description of Simulation System for Continuous Stirred Tank Reactor
7.5.2 Simulation Experiment and Results
7.5.3 Result Analysis
7.6 Summary of This Chapter
References
8 Degradation Modeling and Reliability Prediction Based on Evidence Reasoning
8.1 Introduction
8.2 Degradation Modeling Based on Evidence Reasoning
8.2.1 Structure and Expression Form of Prediction Model
8.2.2 Degradation Modeling and Prediction Under the ER Framework
8.2.3 Utility Based Construction of the Numerical Outputs
8.3 Recursive Algorithms for Updating the ER-Based Prediction Model
8.3.1 Recursive Parameter Estimation Algorithm Based on Judgment Output
8.3.2 Recursive Parameter Estimation Algorithm Based on Numerical Output
8.4 Case Study
8.4.1 Problem Description
8.4.2 Reference Points of Teliability Data
8.4.3 Degradation Modeling and Prediction Model
8.4.4 Simulation Results Based on Judgment Output
8.4.5 Simulation Results Based on Numerical Output
8.5 Summary of This Chapter
References
9 Weight Optimization-Based Particle Filter Algorithm for Degradation Modeling and Residual Life Prediction
9.1 Introduction
9.2 Particle Filter Algorithm Based on Weight Optimization
9.2.1 Particle Filter Algorithm and Characteristic Analysis
9.2.2 Particle Filter Algorithm Based on Weight Optimization
9.3 Degradation Modeling with Weight Optimization-Based Particle Filter
9.3.1 Description of Degradation Process
9.3.2 Parameter Estimation
9.4 Residual Life Prediction
9.5 Numerical Simulation
9.6 Summary of This Chapter
References
10 Degradation Modeling and Residual Life Prediction Based on Grey Predcition Model
10.1 Introduction
10.2 Grey Predcition Model
10.2.1 Classical Grey GM (1, 1) Model [12]
10.2.2 Improved Grey Predcition Model
10.3 Residual Life Prediction Based on Improved Grey Predcition Model
10.4 Case Study
10.5 Summary of This Chapter
References
11 Optimal Inspection Policy for Deteriorated Equipment Based on Life Prediction Information
11.1 Introduction
11.2 Inspection Strategy and the Optimization Objective Function
11.3 Optimal Inspection Policy Based on Residual Life Prediction
11.3.1 Optimal Inspection Period of Equipment When Unknown G(X)
11.3.2 Optimal Inspection Period of Equipment with G(X) Known
11.4 Optimal Inspection Policy for Inertial Platform
11.5 Summary of This Chapter
References
12 Cooperative Predictive Maintenance of Two-Component System with Limited Resources
12.1 Introduction
12.2 Cooperative Predictive Maintenance Model
12.3 Maintenance Decision Modeling and Optimization
12.3.1 Estimation of Expected Failure Times
12.3.2 Cost Rate Model
12.3.3 Maintenance Optimization
12.4 Numerical Simulation
12.5 Summary of This Chapter
References

Citation preview

Changhua Hu · Hongdong Fan · Zhaoqiang Wang

Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment

Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment

Changhua Hu · Hongdong Fan · Zhaoqiang Wang

Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment

Changhua Hu Xi’an Research Institute of High-Technology Xi’an, Shaanxi, China

Hongdong Fan Xi’an Research Institute of High-Technology Xi’an, Shaanxi, China

Zhaoqiang Wang Xi’an Research Institute of High-Technology Xi’an, Shaanxi, China

ISBN 978-981-16-2266-3 ISBN 978-981-16-2267-0 (eBook) https://doi.org/10.1007/978-981-16-2267-0 Jointly published with National Defense Industry Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: National Defense Industry Press. © National Defense Industry Press 2022 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

Residual life prediction and predictive maintenance of complex engineering equipment exist as a significant engineering problem demanding a prompt solution. For the applications in aviation, aerospace and other related sectors requiring high reliability and long life, this problem is a very challenging one. In consideration of the high cost and long lifecycle of this kind of equipment, it is difficult to acquire massive failure data or life data of the equipment through tests. As a result, the traditional residual life prediction and optimal maintenance methods based on statistical analysis of failure distribution become unavailable or unfeasible. While performing R&D, type test, storage and operation of the equipment, we have accumulated some monitoring and test data indicating the operation status and performance of the equipment. These data contain abundant information about the residual life of equipment. Unfortunately, the existing residual life prediction methods have not applied these data reasonably. Fortunately, as early as 2002, the author and his research team have begun to explore the comprehensive application of life data and degradation data in equipment history monitoring and inspection information to model the performance degradation rule, predict the residual life and make optimal maintenance decisions. A great number of original achievements, such as the first passage time (FPT) distribution and residual life prediction of nonlinear Wiener degradation process, self-detection of abrupt changepoint, multi-stage degradation modeling and residual life prediction, evidence reasoning degradation modeling and residual life prediction based on subjective and objective information, optimal detection strategy of degraded equipment based on life prediction information and cooperative predictive maintenance of two-component system under limited resources, have been made. These research results have been published on IEEE Transactions on Reliability, European Journal of Operational Research, Science China and other top academic journals in the field and attracted wide attention from peers both at home and abroad. It means that the theoretical results of this book have produced a wide range of international academic influence with superiority, systematization and originality. Meanwhile, the author and his team attach great importance to integrating theory with practice and use the methods proposed in the book to handle the residual life prediction and maintenance decision of aerospace products, such as gyroscopes and platforms, and obtain some prediction and decision results more applicable to engineering practices. This v

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Foreword

book has strong specific applicability and high reference value. In this sense, it is a rare monograph on equipment residual life prediction and optimal maintenance decision in recent years. The publication of this book will promote the development of residual life prediction and maintenance decision technologies for complex engineering equipment based on degradation modeling and also provide an important theoretical basis for solving major engineering problems related to the residual life prediction and maintenance decision of complex engineering equipment. Moreover, this book is an excellent monograph in the fields of reliability engineering, maintenance engineering and management engineering and brings a significant reference value to numerous scientific workers engaged in the research in related fields. March 2021

Jiancheng Fang Academician of Chinese Academy of Sciences Beijing, China

Preface

Equipment performance degradation and even failure will inevitably occur during equipment operation. Maintenance has been widely applied as an indispensable approach to ensure normal system operations. After years of development, maintenance has evolved from the breakdown maintenance at the earliest stage to the condition-based maintenance at the current stage. In recent years, predictive maintenance based on condition-based maintenance has attracted wide attention from researchers. Life prediction, which is known as the core technology for realizing predictive maintenance, has become one of top priorities for domestic and overseas researchers. Both the Outline of the National Program for Long- and Medium-Term Scientific and Technological Development (2006–2020) promulgated by the State Council in February 2006 and the field of advanced manufacturing technology in Program 863 have listed the life prediction technology of major products and facilities as one of the cutting-edge technologies for instant development. The traditional life prediction technology takes class-I products as the research object, and performs statistical analysis on the life data by statistical methods, and then obtains the life distribution. However, this method has ignored the influence of environment and other factors during equipment operation. Therefore, the life distribution obtained by statistical methods cannot accurately describe the life change of equipment, resulting in the unreasonable arrangement of maintenance activities. With the development of science and technology, equipment life is getting longer and longer, and the reliability is getting higher and higher. Therefore, it is difficult to collect massive life data. This fact will inevitably degrade the accuracy of statistical results. With the development of sensor technology, it is extremely urgent to evaluate the residual life of equipment online through data monitoring, which has attracted the attention of scholars. As a kind of monitoring data, performance degradation data, which contain a large number of reliable and useful key life-related information, can directly reflect the performance degradation process of equipment. Degradation process modeling and prediction using performance degradation data has become an important research in the field of life prediction. Starting from the practical engineering requirements of residual life prediction and optimal maintenance decision of key missile components, this book systematically discusses the theories and applications of equipment performance degradation vii

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Preface

rule modeling, residual life prediction and optimal maintenance decision based on historical equipment monitoring data. This book is also a systematic summary of the research results of the author and his team in this field for more than one decade. As early as 2002, the author and his team have started researches on the theories and applications of equipment performance degradation rule modeling, residual life prediction and optimal maintenance decision. However, the research on residual life and predictive maintenance of missiles and other aerospace equipment from this perspective can be seldom found in China, and the related monographs are rare. This book consists of 12 chapters. Chapter 1 presents an overview of researches on life prediction and maintenance decision methods. Chapter 2 proposes the real-time residual life prediction method based on Wiener process for the equipment suffering nonlinear degradations. With respect to abrupt changepoints during degradation, Chap. 3 has given the algorithm of abrupt changepoint detection and the real-time prediction method of residual life. In Chap. 4 and Chap. 5, performance degradation modeling and residual life prediction methods based on Gamma process and inverse Gaussian process have been studied, respectively. In Chap. 6, the residual life prediction method based on support vector machine (SVM) is studied for the small sample size of degradation data. Chapter 7 proposes a fuzzy model identification method based on relevance vector machine (RVM) and studies the corresponding performance degradation modeling and prediction methods. The degradation modeling of evidence reasoning and the prediction of residual life based on subjective and objective information are systematically studied in Chap. 8. In Chap. 9, the related algorithms of particle filter are introduced, and an excellent weight selected particle filter algorithm is proposed and applied to the residual life prediction of equipment. Chapter 10 introduces the grey model theory and its application in performance degradation modeling and prediction. In Chap. 11, the optimal detection strategy based on residual life prediction information is studied. In Chap. 12, a cooperative predictive maintenance model is proposed for the equipment with two dependent failure modes. The publication of this book is supported by the National Defense Industry Press, Springer Verlag and partially supported by National Natural Science Fund under grants 61873273, 61833016, and 61973046. Here, I would like to express my sincere thanks. Due to my limited knowledge, it is inevitable that there is something inappropriate with the book. Any comment from the readers will be appreciated. Xi’an, China March 2021

Changhua Hu

Summary

This book is an academic monograph that systematically discusses the residual life prediction and optimal maintenance decision method based on performance degradation modeling. It mainly covers the overview of life prediction and maintenance decision modeling and optimization, real-time residual life prediction based on Wiener process for nonlinearly degraded equipment, performance degradation modeling and remaining life prediction (respectively based on Wiener process with abrupt changepoint, Gamma process, inverse Gaussian process, support vector machine, relevance vector machine fuzzy model, weight selected particle filter and gray prediction model), performance degradation modeling and reliability prediction based on evidence reasoning, optimal inspection strategy of degraded equipment based on life prediction information and cooperative predictive maintenance of two-component system under limited resources. This book can provide reference for the great majority of scientific and technical personnel engaged in theoretical research or applied research on equipment fault diagnosis and fault-tolerant control, life prediction and maintenance decision, etc. For the great majority of engineering and technical personnel, teachers, graduates and senior undergraduates engaged in reliability engineering, maintainability engineering, management engineering, testing and measurement technologies and instruments, inspection technologies and automation devices, this is a systematic, innovative and practical reference book with cutting-edge knowledge and outstanding theories.

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Contents

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Equipment Life Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Fundamental Concept of Life Prediction . . . . . . . . . . . . . 1.2.2 Literature Review on Life Prediction . . . . . . . . . . . . . . . . 1.3 Maintenance Decision of Equipment . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Fundamental Concept and Classification of Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Literature Review on Maintenance Decision of Single-Component System . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Literature Review on Maintenance Decision of Multi-component System . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual Life Prediction Based on Wiener Process with Nonlinear Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Definition of Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Degradation Modeling Based on Wiener Process with Nonlinear Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Probability Density Function of the Residual Life . . . . . . . . . . . . . 2.5 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Off-Line Estimation of Common Parameters and Hyper-Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Real-Time Updating of Random Parameter . . . . . . . . . . . 2.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Residual Life Prediction Based on Wiener Process with Abrupt Changepoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Degradation Model with Abrupt Changepoint Based on Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Wiener-Process-Based Degradation Model . . . . . . . . . . . 3.2.2 Changepoint Detection in Performance Degradation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Conjugate Distribution of Prior Distribution of Exponential Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Bayesian Online Changepoint Detection Algorithm . . . . . . . . . . . 3.5 Empirical Bayesian Method for Determining Prior Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Improved Forward–Backward Algorithm . . . . . . . . . . . . . 3.5.2 Joint Distribution of Changepoint Markers at Adjacent Detection Moments . . . . . . . . . . . . . . . . . . . . . 3.5.3 EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Residual Life Prediction Based on Bayesian Online Changepoint Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gamma Process-Based Degradation Modeling and Residual Life Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Definition of Gamma Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Parameter Estimation for Gamma Process . . . . . . . . . . . . . . . . . . . . 4.3.1 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Maximum Likelihood Estimate . . . . . . . . . . . . . . . . . . . . . 4.4 Residual Life Prediction Based on Gamma Process . . . . . . . . . . . . 4.4.1 Life Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Residual Life Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Reliability Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Example Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Degradation Modeling Based on Gamma Process with Environmental Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Residual Life Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Maintenance Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Inverse Gaussian Process-Based Degradation Modeling and Residual Life Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Definition of Inverse Gaussian Process . . . . . . . . . . . . . . . . . . . . . . 5.3 ER-Based Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Parameter Estimation Based on Single Specific Equipment’s Degradation Data . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Fusion of Fixed Parameters Based on ER . . . . . . . . . . . . . 5.4 Derivation of Residual Life Distribution . . . . . . . . . . . . . . . . . . . . . 5.5 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degradation Modeling and Residual Life Prediction Based on Support Vector Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 SVR Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Primal and Dual Problems . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Sparsity of SVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Kernel Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Residual Life Prediction Method Based on GA-Optimized SVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Specific Steps of the Method . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Residual Life Prediction Method Based on SVR and FCM Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Basic Ideas and Specific Steps . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degradation Modeling and Residual Life Prediction Based on Fuzzy Model of Relevance Vector Machine . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Fuzzy Model Based on Relevance Vector Machine . . . . . . . . . . . . 7.2.1 Mathematical Description of Fuzzy Model . . . . . . . . . . . 7.2.2 Fuzzy Model Based on Relevance Vector Machine . . . . . 7.2.3 Uniform Approximation of Fuzzy Model Based on Relevance Vector Machine . . . . . . . . . . . . . . . . . . . . . . 7.3 Fuzzy Model Identification Based on Relevance Vector Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Structure Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

7.3.3

Fuzzy Model Identification Algorithm Based on RVM and Gradient Descent Method . . . . . . . . . . . . . . 7.4 Degradation Modeling and Residual Life Prediction . . . . . . . . . . . 7.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Description of Simulation System for Continuous Stirred Tank Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Simulation Experiment and Results . . . . . . . . . . . . . . . . . . 7.5.3 Result Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

9

Degradation Modeling and Reliability Prediction Based on Evidence Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Degradation Modeling Based on Evidence Reasoning . . . . . . . . . 8.2.1 Structure and Expression Form of Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Degradation Modeling and Prediction Under the ER Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Utility Based Construction of the Numerical Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Recursive Algorithms for Updating the ER-Based Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Recursive Parameter Estimation Algorithm Based on Judgment Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Recursive Parameter Estimation Algorithm Based on Numerical Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Reference Points of Teliability Data . . . . . . . . . . . . . . . . . 8.4.3 Degradation Modeling and Prediction Model . . . . . . . . . 8.4.4 Simulation Results Based on Judgment Output . . . . . . . . 8.4.5 Simulation Results Based on Numerical Output . . . . . . . 8.5 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weight Optimization-Based Particle Filter Algorithm for Degradation Modeling and Residual Life Prediction . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Particle Filter Algorithm Based on Weight Optimization . . . . . . . 9.2.1 Particle Filter Algorithm and Characteristic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Particle Filter Algorithm Based on Weight Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Degradation Modeling with Weight Optimization-Based Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 162 163 163 164 166 168 169 171 171 172 172 173 174 176 176 181 185 185 186 186 187 189 190 191 193 193 194 194 200 202

Contents

9.3.1 Description of Degradation Process . . . . . . . . . . . . . . . . . 9.3.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Residual Life Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Degradation Modeling and Residual Life Prediction Based on Grey Predcition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Grey Predcition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Classical Grey GM (1, 1) Model . . . . . . . . . . . . . . . . . . . . 10.2.2 Improved Grey Predcition Model . . . . . . . . . . . . . . . . . . . 10.3 Residual Life Prediction Based on Improved Grey Predcition Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Optimal Inspection Policy for Deteriorated Equipment Based on Life Prediction Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Inspection Strategy and the Optimization Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Optimal Inspection Policy Based on Residual Life Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Optimal Inspection Period of Equipment When Unknown G(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Optimal Inspection Period of Equipment with G(X) Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Optimal Inspection Policy for Inertial Platform . . . . . . . . . . . . . . . 11.5 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Cooperative Predictive Maintenance of Two-Component System with Limited Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Cooperative Predictive Maintenance Model . . . . . . . . . . . . . . . . . . 12.3 Maintenance Decision Modeling and Optimization . . . . . . . . . . . . 12.3.1 Estimation of Expected Failure Times . . . . . . . . . . . . . . . 12.3.2 Cost Rate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Maintenance Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Summary of This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

1.1 Background With the high-tech development and increasing global competitions, the modern industrialization process intends to be more and more extensive and complex [1]. However, the related equipment in the fields of industrial production, transportation, communication, aerospace and missile weapon system is becoming more and more advanced. Moreover, most of them are becoming more and more complex. The failure of a tiny component may cause a failure to the whole system and even a great disaster. For example, on September 8, 1994, a Boeing 737 aircraft of American Airlines crashed near Pittsburgh due to the non-command deflection of its rudder, and 131 people died [2]. In 2005, due to the blockage of Tower P-102 in the nitration unit of Jilin Petrochemical Company Aniline Plant, a severe explosion occurred, which caused great economic losses [3]. Therefore, it is of great significance to ensure the reliability and security of such complex systems. This also determines the necessity of repairing these kinds of complex equipment. Maintenance can improve the system reliability, availability and safety, thus reducing losses and ensuring personnel safety. But maintenance needs massive financial supports. According to related statistics, maintenance costs account for 15% of total production costs in the manufacturing industry and 40% in the steel industry. In the USA, the annual maintenance cost of enterprises is more than US$ 200 billion [4]. It is estimated that about 30% of the maintenance cost is caused by low-efficiency maintenance methods [5]. In order to keep maintenance as an effective role in ensuring correct system operations with the lowest cost, researchers have accomplished massive studies on maintenance policies. Meanwhile, maintenance has also evolved from the original breakdown maintenance to scheduled maintenance, and then to the widely used condition-based maintenance at present. With the rapid development of sensor technology and prediction technology, the predictive maintenance, as an important solution, has evolved from condition-based maintenance. At present, the scholars both at home and abroad have paid close attention to predictive maintenance. Some foreign © National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_1

1

2

1 Introduction

countries have paid attention to predictive maintenance as early as the end of last century. In China, both the Outline of the National Program for Long- and MediumTerm Scientific and Technological Development (2006–2020) promulgated by the State Council in February 2006 and the field of advanced manufacturing technology in Program 863 have listed the life prediction technology of major products and facilities as one of the cutting-edge technologies for instant development [6]. Life prediction, which is known as the core technology for realizing predictive maintenance, has become one of top priorities for domestic and overseas researchers [7–11]. The traditional life prediction technology takes class-I products as the research object and performs statistical analysis on the life data by statistical methods and then obtains the life distribution. However, this method has ignored the influence of environment and other factors during equipment operation. Therefore, the life distribution obtained by statistical methods cannot accurately describe the life change of equipment, resulting in the unreasonable arrangement of maintenance activities. With the development of science and technology, equipment life is getting longer and longer, and the reliability is getting higher and higher. Therefore, it is difficult to collect massive life data. This fact will inevitably degrade the accuracy of statistical results. With the development of sensor technology, it is extremely urgent to evaluate the remaining life of equipment online through data monitoring, which has attracted the attention of scholars. Jardine et al. [12] summarized the main research results of life prediction in recent years. They pointed out that the current researches on remaining life prediction mainly focus on predicting the probability distribution and expectation, involving statistical method, artificial intelligence (AI) and mechanism model. Both statistical method and AI are data-driven methods. Xiaosheng Si et al. systematically reviewed the statistical data-driven remaining life prediction methods in the reference [13]. According to the types of status monitoring data, the obtained data are divided into direct monitoring data and indirect monitoring data. Based on such data classification, the existing methods are divided into direct monitoring data-based methods and indirect monitoring data-based methods. They also described and commented all existing prediction methods from the perspective of remaining life modeling. In this book, the author has made a more comprehensive summary and comment on the life prediction methods in the following aspects, including life determination of newly developed equipment, the remaining life prediction of equipment under working conditions and the storage life of equipment [14]. In particular, the performance degradation path modeling and maintenance decision modeling and optimization are described as follows.

1.2 Equipment Life Prediction

3

1.2 Equipment Life Prediction 1.2.1 Fundamental Concept of Life Prediction Residual life (RL), usually called remaining useful life (RUL) and also called remaining service life and remaining life, is the normal operation time of equipment from the current moment. Life prediction refers to the prediction on how much time is left before one failure (or multiple failures) when the current equipment status and historical status data are known. In most cases, remaining life is defined as a conditional random variable [12]: T − t|T > t, Z (t)

(1.1)

where T is the random variable of failure time, t indicates the current operation time and Z (t) represents the historical monitoring condition up to the current moment. Since RUL is a random variable, its distribution is very meaningful to gain a full understanding toward RUL. In relevant references, remaining useful life estimate (RULE) is defined in two different ways, including the calculation of RUL probability distribution and the expectation of RUL in some cases [12]: E(T − t|T > t, Z (t))

(1.2)

The accurate definition of equipment failure is a significant prerequisite for life prediction based on performance degradation data. It is generally believed that a failure occurs when the performance degradation data reach the preset failure threshold. For example, the equipment suffering fatigue failure is defined as an incident that the fatigue crack data reach the prescribed threshold.

1.2.2 Literature Review on Life Prediction 1.2.2.1

Life Determination Technology for Newly Developed Equipment

There are two types of newly developed equipment, including the equipment upgraded from the existing equipment and the equipment obtained by redesign. For the former type, the available information mainly comes from similar equipment, so the similar product reasoning method is usually used for equipment lifetime prediction. For the latter type, the available information mainly includes mechanism information, component and equipment structure information, information obtained through accelerated life test and life information in environmental test. The

4

1 Introduction

corresponding lifetime prediction methods include mechanism analysis, component reliability synthesis, accelerated lifetime test and environmental factor conversion. However, most lifetime prediction results obtained through the above methods are not reliable. Under such circumstance, it is necessary to study other methods that can realize accurate lifetime prediction. At present, most related researches focus on model-based methods and data-driven methods. I.

Lifetime prediction method based on similar equipment information

This method makes comprehensive use of the prior information obtained by similar equipment in the long-term operation and the information in the lifetime test of newly developed equipment for lifetime prediction. The basic model of this method is shown as follows: h(λ) = ρh(λ|H ) + (1 − ρ)h(λ|N ) where h(λ) indicates the posterior information related to the reliability of newly developed equipment, h(λ|H ) refers to the information obtained in the operation process of similar equipment, h(λ|N ) represents the new information obtained during the test for newly developed equipment, ρ is inheritance factor and reflects the similarity of reliability between new equipment and old equipment and can be determined by test information or experts and 1 − ρ represents update factor and reflects the uncertainty caused by the new equipment when improving the old equipment. II.

Lifetime prediction method based on mechanism analysis

When this method is adopted, it needs to analyze the physical and chemical factors that lead to equipment failure. The relationship between equipment failure and physical and chemical factors, such as component wear, is established through physical factor analysis, and physical and chemical factor analysis, thus acquiring the lifetime evolution rule and predicting the equipment lifetime. This method has the advantage of predicting the equipment lifetime more accurately. Tanaka and Mura have proposed a mechanism model that can describe fatigue cracking along slip band [15]. Mu and Lu further established a three-dimensional simulation model to describe fatigue cracking and performed lifetime prediction based on the model [16]. However, most engineering equipment is very complex. It is difficult to obtain the mechanism model, thus restricting the application of this method. III.

Lifetime prediction method based on component reliability synthesis

In this method, the reliability relationship between equipment and its components is established at first. Then the reliability of the whole equipment is evaluated and analyzed according to the reliability of components. Reference [17] proposed a reliability evaluation and lifetime prediction method based on reliability competitions and applied this method to reliability evaluation and lifetime prediction of some

1.2 Equipment Life Prediction

5

circuits. Chen et al. improved the method in Reference [17] and applied it to reliability evaluation and lifetime prediction of aviation power circuits [18]. This method has the disadvantage of establishing the relationship between equipment and all its components. It is difficult to establish such relationship for some very complex equipment. IV.

Lifetime prediction method based on accelerated lifetime test

If the failure mechanism of equipment under different stress levels remains unchanged, the lifetime test can be performed at the stress level higher than the normal, and the lifetime test data can be obtained in a short period of time. The lifetime distribution and environmental factor of equipment under the environment of accelerated lifetime test can be obtained by analyzing the lifetime data and converted into the lifetime distribution under normal environment. This method has been widely used in aerospace and other fields [19]. The main difficulties in using this method are as follows: first, how to ensure the consistency of failure mechanism between accelerated lifetime test and normal status; second, how to extrapolate the result of accelerated lifetime test to normal status. It is also difficult to create the model of relationship between accelerated lifetime test and normal status. V.

Lifetime prediction method based on environmental factor conversion

In this method, it needs to convert the test data in different environments into the test results under the same environment and use the data for reliability prediction. Like the method based on accelerated lifetime test, keeping the failure mechanism under different environments unchanged is a premise of using this method. The determination of environmental impact factors under different environments plays a decisive role in this method. The test data under different environments can be converted, which has expanded the source of available data. However, this method has the disadvantage that the type of lifetime distribution must be known. In practical engineering, it is generally assumed that the lifetime of electronic equipment and mechanical equipment follows exponential distribution and Weibull distribution, respectively. Based on final products and failure products, Wengeng Pan [20] discussed the data processing of sampling test for ammunition storage reliability by using the environmental factor method and performed evaluation before and after conversion by Bayesian method. According to the result of comparison with classical methods, the data after conversion can improve the status of ammunition storage and avoid waste. Dongpao Hong et al. [21] used the proportional risk model to describe the relationship between reliability and environmental factor and gave a method of determining environmental conversion coefficient based on the comprehensive use of variable environmental test data.

6

1 Introduction

1.2.2.2

Research Status of Residual Lifetime Prediction of Working Equipment

The residual lifetime prediction of equipment under working conditions refers to the RUL prediction of equipment based on relevant information after the equipment has been operated for a period of time. The aforementioned relevant information includes historical information during aforementioned operation period, lifetime information of similar equipment and lifetime information obtained during accelerated lifetime test. Furthermore, the aforementioned three types of information are mainly divided into failure time data and performance degradation amount. Accordingly, the remaining lifetime prediction method can be divided into remaining lifetime prediction method based on failure time data, remaining lifetime prediction method based on performance degradation amount and remaining lifetime prediction method based on multi-source information fusion. Detailed division is shown in Fig. 1.1. I.

Residual lifetime prediction method based on failure time data

If the failure time data of the equipment are obtained, the parameters of equipment lifetime distribution can be estimated by using statistical inference method based on the assumed equipment lifetime distribution, and then the remaining lifetime distribution of the equipment after being operated for a period of time can be obtained. The commonly used lifetime distribution form includes exponential distribution, normal distribution and Weibull distribution. The suitability of the equipment lifetime distribution directly affects the accuracy of lifetime prediction results. Marshall and Olkin [22] summarized the commonly used lifetime distribution functions and discussed the method of estimating the parameters for the corresponding distribution functions; Residual life prediction of equipment under working conditions

Residual life prediction method based on degradation

Residual life prediction method based on failure time data

Residual life prediction method based on direct monitoring data

Residual life prediction method based on multi- source information fusion

Residual life prediction method based on indirect monitoring data

Re maining life prediction method ba s ed on proportional hazard model

Re maining life prediction method ba s ed on multi-state reliability model

Re maining life prediction method ba s ed on hidden s e miMarkov chain

Re maining life prediction method ba s ed on hidden Markov chain

Re maining life prediction method ba s ed on random filtering

Re maining life prediction method ba s ed on stocha stic proce s s

Re maining life prediction method ba s ed on time s erie s modeling

Fig. 1.1 Residual lifetime prediction of equipment under working conditions

1.2 Equipment Life Prediction

7

however, this method only applies to the overall lifetime distribution of equipment and fails to consider the equipment degradation information during the operation period. Consequently, it cannot well reflect the remaining lifetime distribution of the equipment after being operated for a period of time. II.

Residual lifetime prediction method based on performance degradation amount

According to the historical operation information of the equipment, the performance degradation path of the equipment can be established. On this basis, the time when the equipment performance degradation amount exceeds the failure threshold can be determined, and then RUL of the equipment can be determined. This method can be divided into the remaining lifetime prediction based on direct monitoring data and the remaining lifetime prediction based on indirect monitoring data. (I)

Residual lifetime prediction method based on direct monitoring data

Direct monitoring data mainly refer to monitoring data that can directly reflect the performance or health status of the equipment. The commonly mentioned performance degradation amount such as wear and fatigue crack data fall into this category. Therefore, the remaining lifetime prediction based on direct monitoring data is to predict the time when the monitoring data reach the failure threshold for the first time. The remaining lifetime prediction method based on direct monitoring data can be divided into the remaining lifetime prediction method based on time series modeling and the remaining life prediction method based on stochastic process. 1.

Residual life prediction method based on time series modelling

The direct monitoring data obtained at the monitoring time constitute time series. Therefore, the equipment performance degradation rule can be established by applying the remaining life prediction method based on time series modeling, and finally, the time when the equipment performance degradation amount reaches the failure threshold for the first time can be determined on this basis, so that the remaining life of equipment can be obtained. The commonly used time series model includes auto-regressive moving model, gray model, artificial neural network, support vector machine (SVM) and their combined prediction model. The remaining life prediction method based on time series modeling has been widely used to predict the remaining life of bearings, gyroscopes and other equipment; however, such method cannot reflect the uncertainty of prediction results well since it can only obtain the amplitude of remaining life instead of the distribution form of remaining life. 2.

Residual life prediction method based on stochastic process

Based on this method, it is considered that the equipment performance degradation rule can be described by applying stochastic process, and then the distribution of the time when the performance degradation amount reaches the failure threshold

8

1 Introduction

for the first time can be determined, so that the remaining life distribution of equipment can be obtained. Different from the remaining life prediction method based on time series modeling, this method provides the equipment life under the probabilistic framework. Therefore, the probability distribution is obtained, which can well reflect the uncertainty of prediction results and provide convenience for subsequent maintenance decision. Specifically, the stochastic process model commonly used in this method includes random coefficient model, Gamma process, inverse Gaussian process, Wiener process and Markov chain. The random coefficient model is one of the models early used in performance degradation amount modeling. In 1993, Lu and Meeker [23] firstly proposed the random coefficient regression model, and then Lu [24] and Tseng [25] developed and applied such model into the modeling of semiconductor industry and LED brightness degradation. Wang [26] and Bae [27] respectively studied the remaining life prediction of the same type of equipment with common characteristics under the modeling and the nonlinear degradation conditions. On this basis, Gebraeel [28– 31] et al. further proposed the remaining life prediction method based on Bayesian update and used such method to describe the variation in brake pad thickness. Park [32] et al. analyzed the remaining life prediction based on the accelerated degradation model. If the failure threshold of the equipment is known, then the random coefficient regression model of the equipment can be easily obtained. The random coefficient regression model and the statistical analysis method, which are relatively simple, have been widely applied in industry and chemical fields. Gamma process is a stochastic process model commonly used for the remaining life prediction of the equipment. This process is usually used to model the degradation path of monotone data, such as metal wear and crack growth. In 1975, Abdel-Hameed firstly proposed and used Gamma process to model the continuously monotonous degradation amount [33]. In 2000, Wang et al. applied Gamma process in the research of remaining life for large pumps [34]. In 2000, Bagdonavicius incorporated the impact of dynamic environment into the degradation model and proposed a remaining life prediction method based on Gamma process which considered dynamic environment [35]. In 2004, Lawless and Crowder considered the parameters in the Gamma process as random variables [36]. In 2009, Noortwijk summarized the relevant research and application of Gamma process in the field of life prediction over recent years [37]. The basic idea of inverse Gaussian process is to describe the degradation process based on the variation in increment by assuming that degradation is strictly monotonous and that the increment of degradation follows an inverse Gaussian distribution. Inverse Gaussian process was firstly proposed by Wasan [38] in 1968; however, it was not until 2010 that the inverse Gaussian process was firstly applied in equipment degradation modeling by Wang [39]. Inverse Gaussian process was used to describe the monotonous degradation process. Owing to the connection between the inverse Gaussian distribution and the Wiener process with linear drift, inverse Gaussian process can be derived and implemented mathematically in an easier way and is more flexible and applicable compared with Gamma process.

1.2 Equipment Life Prediction

9

The remaining life prediction method based on Wiener process is mainly applicable to non-monotonic equipment performance degradation process. By this method, the mathematical models in the forms shown below are mainly used to describe the degradation process.  X (t) = x0 +

t

λ(s)ds + σ B(t)

0

Wherein: x0 refers to the initial performance degradation amount; λ(t) refers to the drift parameter; σ refers to the diffusion coefficient; B(t) refers to the standard Brownian motion. After the equipment performance degradation model is obtained, the remaining life distribution of the equipment can be calculated by using the given failure threshold and the Wiener process-related theory. In order to realize accurate and real-time remaining life prediction of the equipment, generally, the remaining life prediction results can be updated from time to time based on the real-time monitoring information of the equipment. Gebraeel et al. [28] firstly established the equipment degradation model based on Wiener process with linear drift (or linearizability) and realized the online update of random drift coefficient by assuming that the drift coefficient followed normal distribution and applying the degradation amount observed in a real-time way and Bayesian inference method. Gebraeel method has a great influence on the field of life prediction and health management for the equipment; however, the remaining life prediction results obtained by applying Gebraeel method are only applicable to the linearly degraded equipment or the equipment of which the performance degradation amount can be directly linearized. Moreover, the Brownian motion term in the degradation model utilized in this method is only considered as an observation error. Consequently, the remaining life distribution obtained is not an accurate solution in the sense of first passage time. Therefore, with regard to the shortcomings of Gebraeel method, a performance degradation modeling and remaining life prediction method based on Wiener process with nonlinear drift is studied in Chapter 2 herein, so as to carry out the remaining life prediction of the nonlinearly degraded equipment. The remaining life prediction method based on Markov chain is often used for the modeling of degradation processes which are characterized by continuous-time discrete states. It involves two assumptions. In one assumption, the future degradation state is only depending on the current degradation state (namely, memoryless); in the other assumption, the monitoring data of the system can reflect the operating state of the system [40]. In the remaining life prediction method based on Markov chain, the first passage time can be defined as the time when the degradation process reaches failure state for the first time, and the remaining life can be calculated based on the first passage time. From 2003 to 2012, Kharoufeh carried out a series of studies on this method and proposed a degradation model based on Markov chain which considered environmental impact [40–43]. In 2010, Lee et al. incorporated Markov property of degradation process into remaining life prediction based on regression model [44].

10

(II)

1 Introduction

Residual life prediction based on indirect monitoring data

Indirect monitoring data mainly refer to monitoring data that can only indirectly or partially reflect the performance or health status of equipment, including vibration analysis data, oil analysis data, etc. The remaining life prediction methods based on indirect monitoring data include random filtering, proportional hazard model (PHM), hidden Markov model, hidden semi-Markov model, etc. 1.

Residual life prediction based on random filtering

This method has become a hot spot in current research and attracted the attention of many researchers. This method is usually applied to the equipment that has not been maintained or replaced and has been degrading. In addition, the performance degradation data of equipment should show a certain trend. The model usually used in this method is as follows xt = αxt−1 + εt yt = βxt + ηt where xt , yt are the actual performance degradation and performance monitoring data of equipment at t; εt , ηt are corresponding noises; α, β are parameters related to the model. According to expert knowledge and indirect monitoring data, Wang and Zhang predicted the remaining life of bearings by using random filtering method [45, 46]. 2.

Residual life prediction based on proportional hazard model

The proportional hazard model was proposed by Cox in 1972, which was used in the medical field at first. The proportional hazard model was introduced into the field of reliability in 1980s and has attracted the attention of researchers since then and has been widely used in the field of life prediction. Generally, the proportional hazard model can relate the failure rate function of working equipment with the overall failure rate function and performance monitoring data of the equipment and then calculate the remaining life distribution of the equipment according to the failure rate function of the equipment [47]. On the basis of this model, Jardine studied the problem of condition-based maintenance decision and determined the optimal replacement time of equipment [48]. 3.

Residual life prediction based on hidden Markov model

The hidden Markov model (HMM) is developed on the basis of Markov chain, which is often used to predict the life of equipment with hidden performance degradation. Bunks et al. [49] proposed a remaining life prediction method based on HMM and expectation maximization algorithm. In order to model complex systems better, Baruah and Chinnam [50] combined HMM with dynamic Bayesian network and used it to predict the remaining life.

1.2 Equipment Life Prediction

4.

11

Residual life prediction based on hidden semi-Markov model

The hidden semi-Markov model (HSMM) is an improved hidden Markov model, which assumes that the residence time of equipment in a degradation state follows arbitrary distribution, such as normal distribution. Dong and He [51, 52] applied this model to the life prediction of equipment and achieved good results. Liu et al. [53] described the degradation state transition probability of equipment and the residence time of each state by using HSMM and then predicted the remaining life of equipment based on sequential Monte Carlo simulation. (III)

Residual life prediction based on multi-source information fusion

In the life test of equipment, usually only a part of equipment fails within the specified time, while the other part of equipment can still work normally. At this time, the data obtained include not only the failure time data of equipment but also the performance degradation data of non-failed equipment. Although the overall life distribution of this kind of equipment can be obtained only by using the failure time data, if the performance degradation data can also be used, a more accurate remaining life prediction result can be obtained. Therefore, making full use of failure time data and performance degradation data to predict the remaining life of equipment is mainly considered in the remaining life prediction method based on multi-source information fusion. By using the characteristic that the first passage time distribution of Wiener process was inverse Gaussian distribution, Pettit and Young [54] integrated the failure time data and performance degradation data in Bayesian framework and predicted the remaining life distribution of equipment. Lee and Tang [55] further estimated the parameters in the model proposed by Pettit and Young by using EM algorithm and applied them to predict the remaining life of LED.

1.2.2.3

Research Status of Equipment Storage Life Prediction

As early as 1950s, the USA conducted many storage tests for missiles and obtained a large number of failure time data and performance degradation data of missiles [56]. In 1980s the Soviet Union also conducted many accelerated life tests for missiles and improved the missiles so that the missiles could be used normally within ten years without test [57]. Two aspects of information can be used for equipment in storage. One is the failure time data of the equipment; the other is the performance degradation information obtained during the periodic inspection of equipment. With the development of science and technology, there is equipment with high reliability and long life, especially missiles, whose storage life is generally long, which makes it difficult to obtain enough storage life data in a short time. Therefore, it is often necessary to use accelerated storage life test or accelerated degradation test to shorten the test time to obtain storage life data or performance degradation data. According to the different types of data obtained, the current storage life prediction methods can be divided into

12

1 Introduction

two categories: one is the storage life prediction method based on failure time data; the other is the storage life prediction method based on performance degradation data. I.

Equipment storage life prediction based on failure time data

By statistically analyzing the failure time data of equipment during storage, the storage life distribution form of equipment can be obtained. Due to the long life of some of the equipment, it is difficult to obtain enough failure time data in a short time through field storage test. Therefore, the problem that it is difficult to obtain failure time data can be solved through accelerated storage life test. Accordingly, this kind of method can be divided into: the field storage test-based method and the accelerated storage life test-based method. (I)

Field storage test-based method

Store the equipment under normal conditions until it fails. By analyzing these failure time data, the life distribution of equipment during storage can be obtained. The equipment life obtained by this method is very close to the actual life. Therefore, this method was widely used to predict the storage life of military equipment in the twentieth century. However, due to slow performance degradation during storage, it takes a lot of time to obtain enough test data. (II)

Accelerated storage life test-based method

In view of the shortcomings of the field storage test method, people consider using accelerated storage life test to obtain failure time data and then predict the storage life of equipment. This method tests the storage life of equipment under the stress level exceeding the normal storage environment conditions. Since the test environment of equipment becomes harsh, which accelerates the degradation of equipment, shortens the test time and reduces the cost, this method has been widely used. For mechanical equipment, van Dorp [58] studied the statistical properties of equipment when failure data follows exponential distribution and Weibull distribution. Furthermore, Xiufeng Zhou et al. [59] proposed a new method to predict the storage life of electronic communication equipment. It should be pointed out that the research object of accelerated storage life test can be equipment with degradation failure mode or equipment with sudden failure mode, but the test mainly records the failure time data of equipment, not the performance degradation data. II.

Storage life prediction based on performance degradation data

The storage life of equipment with degradation failure mode can be obtained by analyzing the performance degradation data. However, under normal storage conditions, the performance degradation process of equipment is very slow, and the performance degradation data do not change obviously, so it is difficult to use them to predict the storage life. In order to solve this problem, accelerated degradation test

1.2 Equipment Life Prediction

13

came into being. The purpose of accelerated degradation test is to study the performance degradation rule of equipment, determine the performance degradation path of equipment and obtain the storage life information of equipment by extrapolation. This method has been developed rapidly because it does not need a large number of test samples and does not need to test the equipment until it fails. Nelson first studied the accelerated degradation test [60]. Padgett et al. extended this method to equipment such as LED, logic integrated circuits and power supplies [61].

1.3 Maintenance Decision of Equipment 1.3.1 Fundamental Concept and Classification of Maintenance Maintenance refers to “all technical and management activities, including monitoring, carried out to maintain or restore the state in which the product is capable of performing specified functions.” Maintenance is short for service and repair [62]. Service refers to all activities taken to keep the system in good working condition when the system still works normally, including cleaning, wiping, lubricating, oiling, etc. Repair refers to activities taken after system failure, such as fault detection, troubleshooting and repair. Researchers began to pay attention to this problem in 1950s and proposed a large number of maintenance models to solve the maintenance problems of different systems. Until now, there are still a large number of references related to maintenance every year, which shows that maintenance decision modeling and optimization are still hot and difficult points at present. Maintenance can be divided into corrective maintenance (CM) and preventive maintenance according to the timing of maintenance. Corrective maintenance, also known as failure maintenance, was the main maintenance method before 1940s, which mainly involved repairing the system after it has failed. Obviously, this kind of maintenance was driven by failure events, which made people mistakenly think that breakdown maintenance was a cost-saving maintenance method [5]. Later, people gradually realized that the maintenance cost needed would be higher than that of arranging relevant maintenance operations before failure if minor faults were allowed to develop until failure. This was because the system needed to be maintained immediately after failure, which would interrupt the normal production plan and bring losses. Moreover, since there was no prediction method at that time, the management personnel could not know when the failure would occur, which led to the sudden occurrence of failure events and made the enterprises unable to prepare the materials, tools and maintenance personnel needed for maintenance in time, which would increase the losses caused by failure to a certain extent [63]. These shortcomings of breakdown maintenance promoted the emergence and development of preventive maintenance policy. However, due to the uncertainty of the

14

1 Introduction

failure process, failure usually occurred during the operation of the system. Therefore, breakdown maintenance was considered in the policy formulation process in the maintenance theory developed later. Preventive maintenance refers to finding fault symptoms through inspection and detection on the premise that the system can still work normally, and taking appropriate maintenance actions to eliminate possible faults in the future. According to the types of information used in maintenance decision, maintenance methods can be further divided into scheduled maintenance (SM) and condition-based maintenance (CBM). Recently, on the basis of condition-based maintenance, predictive maintenance has gradually attracted the attention of researchers. Scheduled maintenance refers to the arrangement of maintenance activities by management personnel according to the characteristics such as failure rate or life distribution obtained based on the statistics of failure time data. After the World War II, there was a shortage of materials and personnel. In order to improve the material supply capacity, highly automated equipment has been put into use one after another. Meanwhile, the urgency of war required that the production equipment must be shut down as little as possible, so the maintenance of the equipment became important. However, since the traditional failure maintenance was driven by failure events, maintenance operations aimed at restoring the specified functions of the system could be carried out only after the system has failed. Obviously, this maintenance method could no longer meet the needs at that time. In order to prevent the occurrence of failure, the researchers proposed the idea of preventive maintenance in 1950s [64], that is, carry out maintenance operations on the system according to predetermined intervals or specified criteria, so as to reduce the probability of system failure or prevent the function degradation. It should be pointed out that preventive maintenance at this time actually referred to the maintenance arranged according to the time, that is, time-based preventive maintenance (TBPM), also known as scheduled maintenance. At that time, China also introduced the scheduled maintenance system from the Soviet Union and applied it to the power industry [65]. Compared with the failure maintenance, this maintenance method has the advantage of improving the system reliability, reducing the frequency of fault and increasing the productivity through a series of maintenance operations (inspection, repair, replacement, cleaning and lubricating, etc.). However, the introduction of scheduled maintenance not only improved the reliability and availability of equipment but also increased the maintenance cost of enterprises. According to the survey, domestic enterprises in the USA spent nearly USD 600 billion to maintain their key equipment in 1981, and the number doubled over 20 years [64]. Germany spent 13–15% of its GDP on maintenance, while the Netherlands spent 14% [66, 67]. Specifically, 15–70% of enterprises’ total expenditure was spent on the maintenance of production equipment [68]. More notably, one-third of the high maintenance cost was wasted in the maintenance implementation process [5]. Main reasons are described as follows. First, in the implementation of scheduled preventive maintenance, the maintenance interval is mainly determined by statistical analysis of the failure time data of the same type of system, without considering the actual performance of the system during operation, resulting in that the maintenance

1.3 Maintenance Decision of Equipment

15

interval obtained is the optimal interval for the same type of system as a whole, but it may not be the optimal interval for individual equipment. Second, maintenance is carried out blindly according to the established interval, regardless of the actual health status of the system, which is easy to cause a lot of unnecessary maintenance, that is, maintenance is carried out when it is not needed; or cause insufficient maintenance, that is, maintenance is not carried out when it is needed, so that failure cannot be effectively avoided. In addition, the traditional scheduled maintenance is to disassemble and maintain equipment in the system at intervals, and the failure rate of the components just maintained is generally high, which will lead to a very high failure rate of the whole system after accumulation [65]. According to statistics, in 1996, the proportion of unscheduled shutdown and output reduction caused by improper maintenance for 100, 125 and 200 MW thermal power generating units in China accounted for 36, 31 and 41%, respectively [65]. Considering the disadvantages of scheduled preventive maintenance, such as low efficiency, and benefiting from the rapid development of sensor technology, equipment status-based maintenance, also called condition-based maintenance, has gradually attracted the attention of researchers. Condition-based maintenance is also preventive maintenance in nature, but this maintenance method mainly evaluates the health status of current system by monitoring and analyzing some indexes (such as temperature, pressure and metal content in oil) closely related to the health status of equipment, and the optimal maintenance decision is made on this basis [69]. This maintenance method based on the status of the system during operation greatly improves the maintenance efficiency, reduces unnecessary maintenance and saves maintenance costs. At present, condition-based maintenance has been widely used in manufacturing fields such as machinery, power, petrochemicals and military fields. According to reports, in order to monitor the health status of weapons and equipment in real time, the US Army has equipped its Black Hawk Helicopter with a status and usage monitoring system in 2004 [70]. According to the different meanings of condition in condition-based maintenance, condition-based maintenance can be further divided into narrow condition-based maintenance and broad condition-based maintenance. The narrow condition-based maintenance is still called condition-based maintenance herein, which emphasizes the use of only the immediate results obtained from monitoring to determine whether maintenance is needed and what kind of maintenance methods (repair, replacement, etc.) are arranged. The broad condition-based maintenance, that is, predictive maintenance (PdM), refers to “providing information about taking correct measures at the right time and for the right reasons through a prediction and condition management system, which can safely determine the remaining life of degraded components during their use, clearly indicate when they should be maintained and automatically provide the parts list and tools needed to restore any event that is causing performance or safety limit degradation to normal” [71]. The specific classification of maintenance methods is shown in Fig. 1.2. The maintenance decision references of single-component system and multicomponent system will be briefly summarized as follows. Considering that the maintenance theory has been developed for several decades, there are many references

16

1 Introduction

Maintenance

Preventive Maintenance (PM)

Corrective Maintenance (CM)

Scheduled Maintenance (SM)

Condition-based Maintenance (CBM)

Predictive Maintenance (PdM)

Fig. 1.2 Specific classification of maintenance methods

related to maintenance decision, and there are always inextricable links between references. Therefore, the summary herein can only try to basically and comprehensively expound the basic problems in maintenance decision.

1.3.2 Literature Review on Maintenance Decision of Single-Component System In this part, according to the classification methods of scheduled maintenance, condition-based maintenance and predictive maintenance, the references on maintenance decision of single-component system will be summarized, respectively.

1.3.2.1

Scheduled Maintenance

After a large number of researches on the references, it is found that the scheduled maintenance activities are usually started after a certain characteristic of the system reaches the threshold. Based on the types of characteristics, the references on scheduled maintenance will be briefly summarized as follows. I.

Age-dependent maintenance policy

The age-dependent maintenance policy refers to the policy of arranging maintenance activities according to the age of the system after it is put into operation. The research on age-based maintenance policy can be traced back to Morse’s work in 1958 [72]. Later, Barlow [73] proposed a policy which is still widely used today, namely age replacement policy, which means that the preventive replacement will be carried out when the age of a component reaches a preset T, and the failure replacement will be carried out if failure occurs before the age of a component reaches T. In

1.3 Maintenance Decision of Equipment

17

1965, Barlow [74] summarized the previous research results and studied the optimal policy in detail. Later, the introduction of the concept of imperfect maintenance [75] greatly enriched the age replacement policy and produced many extended models [76, 77]. For the maintenance effect, imperfect maintenance means that maintenance has produced certain effects and restored certain functions of the system, but it is not perfect maintenance that the system can be repaired as new. The core idea of these extended models is similar to that of the basic age replacement policy, except that according to the different maintenance effects, different maintenance methods, i.e., minimal repair, imperfect maintenance and perfect maintenance, are used to replace preventive replacement and failure replacement in the basic age replacement policy for maintenance modeling and optimization. Scarf et al. [78] proposed an age-based inspection and replacement policy for a component in inhomogeneous component groups. By minimizing the maintenance loss per unit time, the optimal maintenance age T, the optimal inspection interval and the number of inspecting components K were obtained. II.

Periodic maintenance policy

The periodic maintenance policy refers to the policy that the components will be maintained after a fixed time interval, i.e., T. Here, T is a constant. If breakdown maintenance is carried out for the failure occurred between two maintenances, this maintenance policy becomes block replacement policy [77]. As with the age replacement policy, if the maintenance in the periodic maintenance policy is replaced by minimal repair, imperfect maintenance and perfect maintenance, various extended models will be produced. For example, Berg and Epstein [79] modified the block maintenance policy by introducing age limit. In this policy, the component will be replaced when it fails, and at the scheduled replacement time point kT (k = 1, 2,…), if the age of the component is still less than or equal to t0 (0 ≤ t0 ≤ T ), the preventive replacement will not be carried out, and the component will continue operating until failure or next scheduled replacement time point (k + 1) T ( k = 1, 2,…). Obviously, when t 0 = T, this policy degrades into the block replacement policy. Tango [80] considered that if failure occurred before the scheduled time, the failed components would be replaced with the used components. Brezavscek et al. [81] considered the joint optimization problem of periodic maintenance policy and spare parts supply policy. Huang et al. [82] considered the case when the arrival time of components was random and presented the analytical results. Other extended models can refer to the corresponding references [83–87]. III.

Failure limit policy

The failure limit policy refers to the policy that the preventive maintenance will be carried out for the system only after its failure rate or other reliability indexes reach a specified threshold [88]. In this policy, maintenance can be arranged according to failure rate information, and imperfect maintenance is often characterized by failure rate and effective age [89, 90]. Therefore, it is natural to combine various imperfect maintenance models with failure limit policy to form various maintenance models.

18

1 Introduction

On the basis of introducing adjustment factors for failure rate and effective age, Lie and Chun [91] considered that preventive maintenance would be carried out when the failure rate of components reached a certain threshold, while the failure occurred during the operation of components would be corrected by minimal repair. Lin et al. [92] proposed a mixed failure rate model and studied the failure limit policy based on the model. IV.

Sequential maintenance policy

If preventive maintenance is carried out on a single-component system at unequal time intervals, this policy is called sequential maintenance policy. Barlow and Proschan [93] compared the sequential maintenance policy with the age replacement policy and considered that the flexibility of the sequential maintenance policy made the cost in the maintenance process lower than that of the optimal age replacement policy. Under the framework of sequential maintenance policy, the next maintenance cycle was obtained by minimizing the expected cost in the remaining time. Therefore, the maintenance interval in the future was not determined at the beginning, and the next maintenance interval was determined after preventive maintenance. Nakagawa [94, 95] discussed a policy of implementing preventive maintenance at x k (k = 1, 2,…, N) and replacing the components during the Nth preventive maintenance, and the failures occurred between the two preventive maintenance would be corrected by minimal repair. Here, the decision variables are N and x k (k = 1, 2,…, N), respectively. As the age increases, most components need to be maintained frequently, which makes the sequential maintenance policy more practical. Compared with failure limit policy, the sequential maintenance policy directly controls the variable x k , while the failure limit policy directly controls indexes such as failure rate and reliability. XXII.

Repair limit policy

The repair limit policy refers to the policy that after a component fails, a given index value under the condition of performing maintenance is first evaluated, and when the index value exceeds the preset threshold value, replacement will be carried out. Otherwise, the system will be maintained. Here, the given indexes are usually selected as cost and time, and the corresponding policies are also called repair cost limit policy and repair time limit policy, respectively. The repair cost limit policy has an obvious disadvantage, that is, the maintenance or replacement of components is determined only according to the cost that maintenance is carried out every time. In view of this, Beichelt [96] considered taking the maintenance cost rate as a standard to determine whether replacement or maintenance was needed: when the maintenance cost rate reached or exceeded a threshold, the components would be replaced. Otherwise, they would be maintained. Yun and Bai [97] considered imperfect maintenance based on Beichelt’s work. The repair time limit policy was originally proposed by Nakagawa and Osaju [98], who considered that if the failed components were still not maintained within the time of T, they would be replaced. Otherwise, the maintained components would be put into operation. Here, T was the maintenance time threshold.

1.3 Maintenance Decision of Equipment

1.3.2.2

19

Condition-Based Maintenance

Since the condition-based maintenance is mainly decided according to the performance state of the system, it is necessary to model the degradation rule of the performance variables of the system. Considering the randomness of the degradation process of the system, most of the existing condition-based maintenance references use probability-related degradation models as the basis for decision. According to the types of degradation models used in condition-based maintenance, the references on condition-based maintenance will be briefly summarized as follows. It should be noted that although some references on condition-based maintenance also use the traditional scheduled maintenance policy, we still classify these references into the references on condition-based maintenance. I.

Condition-based maintenance based on regression model

This kind of references mainly studies the use of regression analysis method to model the performance degradation process of the system and the provision of the optimal condition-based maintenance policy on this basis. Wang [26] described the degradation path of the system by using a regression model whose coefficients were random variables and followed a certain known distribution and assumed that the maintenance operation would be started when the degradation value of the system exceeded its maintenance threshold. Then, according to the needs, the author selected appropriate objective function, such as loss cost, downtime or system reliability, and proposed a condition-based maintenance model to present the optimal maintenance threshold and inspection time interval on this basis. Jardine [99] proposed a proportional hazards model (PHM), which took Weibull distribution as the benchmark failure rate function and used Markov process to describe the law of state change, and made the optimal maintenance decision based on the model. The policy used here is to replace the system immediately after it fails. If the system does not fail, replace it when its failure rate reaches a certain threshold. Ghasemi [69] considered the problem of condition-based maintenance when monitoring information was noisy. In this reference, the author described the law of real state change of the system by using continuous-time discrete state Markov process and considered that the real state was unknown and could only be estimated by observation information. Then, PHM was used to establish the relationship between the real state of the system and the failure rate and to describe the degradation process of the system. Finally, the problem was transformed into a partial observed Markov decision process (POMDP) and solved by dynamic programming method. Other related references are [100, 101]. Although PHM makes use of the operating state data of the system, information obtained through the model is still the statistical information such as reliability. Therefore, the condition-based maintenance based on PHM often uses the traditional scheduled maintenance policy, but the obtaining method of information such as reliability is changed from historical failure time data to PHM. Moreover, the benchmark failure rate function needs to be given in advance in PHM, and the failure rate function is only related to time. In addition, PHM only uses the state

20

1 Introduction

information at the current time rather than all the historical information. Sun et al. [102] pointed out that the change of the system covariates was caused by the change of its failure rate and proposed a proportional covariates model (PCM) to estimate the failure rate of the system on this basis. Compared with PHM, PCM does not need historical failure data. The change of state is considered in the maintenance decision process for condition-based maintenance based on probability statistical model, and the results are dynamically updated with the change of state. However, because only the state information at the current time is used, the uncertainty of the results is relatively large. II.

Condition-based maintenance based on Markov process

This kind of references mainly studies the maintenance decision problem when the degradation process of the system is described by Markov process. At present, the control limit rule (CLR) is mainly used for the condition-based maintenance based on Markov process, that is, preventive maintenance will be carried out when the degradation degree of the system reaches a certain threshold. Otherwise, the system can continue operating. Here, this threshold is called preventive maintenance threshold. Markov process, including Brownian motion, Gamma process and Poisson process, as a special stochastic process, has good mathematical properties, which makes it widely used in maintenance [37]. References [103–106] studied the condition-based maintenance policy when the degradation process of the system was described by Gamma process. In these references, there was not only a preventive maintenance threshold, but also a failure threshold. When the degradation degree of the system exceeded the failure threshold, the system was judged to have failed, and the replacement would be carried out immediately. In addition, the time interval of state inspection was not fixed, and the next inspection time was determined by the degradation state of the current system. Finally, the author determined the optimal preventive maintenance threshold and the next inspection time by minimizing the expected maintenance cost per unit time or maximizing the availability. Liao et al. [107] proposed a condition-based availability limit policy under the condition of imperfect maintenance for a class of system whose degradation process could be described by Gamma process, in which the imperfection of maintenance was reflected in the fact that the degradation state of the system would not degrade to 0 after maintenance, but to a random quantity that followed normal distribution. Finally, the author found the optimal maintenance threshold by searching algorithm. Monplaisir and Arumugadasan [108] considered using a continuous-time Markov process with seven discrete states to describe the degradation process of crankcase of locomotive diesel engine, and then used the process for maintenance support. For the degraded system subject to periodical monitoring, Amari and McLaughlin [109] discretized the degradation process into multiple states, and then used discrete state Markov chain to model maintenance, and finally obtained the optimal preventive maintenance threshold and inspection frequency by maximizing the availability of the system. Chen and Trivedi [110] proposed to use semi-Markov decision process (SMDP) to jointly model the change of inspection rate and maintenance type in condition-based maintenance and

1.3 Maintenance Decision of Equipment

21

presented an optimization method, that is, the inspection rate was taken as the input parameter of SMDP, and then an optimal condition-based maintenance policy was presented for each inspection rate. Other related references are [111, 112, 114, 115]. For the condition-based maintenance based on Markov process, the degradation process is described by Markov process in the modeling process, and the relevant objective function is established on this basis, and then the optimal preventive maintenance threshold is obtained after optimization. However, after these thresholds are determined, they will not be updated with the change of system state. Moreover, when Markov process is used to model degradation process, the real degradation process of the system shall have Markov property, which reduces the application scope of this method. In addition, a lot of historical data are needed to support the modeling of Markov process, so a lot of data need to be collected and analyzed before the Markov process is used as the basis of condition-based maintenance.

1.3.2.3

Predictive Maintenance

The predictive maintenance mentioned in this book refers to activities that focus on the use of predictive information to arrange future maintenance operations. For predictive maintenance, the future change trend is mainly predicted by monitoring and analyzing the performance degradation process of the system in real time and through certain technical means. Then, based on this, an effective maintenance policy is designed according to the specified requirements. Compared with condition-based maintenance, predictive maintenance makes full use of historical performance degradation information in the modeling process, and its decision result can be updated in real time with the change of system state, and the corresponding maintenance operation can be arranged according to the change of health status of equipment. In addition, predictive maintenance can further reduce maintenance losses and prolong the service life of equipment. This is because management personnel can judge when the failure occurs according to the predictive information under the predictive maintenance policy, so that they can arrange personnel to carry out maintenance on the system at a suitable time before the system failure occurs to avoid major accidents, and they can reduce the number of spare parts stored and storage costs. Life prediction technology has been developed rapidly in recent years because it is necessary to have prediction information when predictive maintenance is carried out. Donghua Zhou et al. [113] summarized the real-time reliability evaluation and prediction technology of engineering system in detail. Heng et al. [64] summarized the references on life prediction of rotating machinery. Although there are more and more references on life prediction in recent years, there are few references using the prediction information to arrange maintenance policy, and the development is relatively slow. Christer et al. [116] predicted the corrosion degree of inductor in induction furnace by using Kalman filtering under the framework of linear state space model, and presented the cost loss model when the replacement maintenance policy was used based on this, and finally presented the optimal replacement time by simulation. Lu et al. [117] also considered the work similar to that in reference [116] but used a

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1 Introduction

different objective function. For the nonlinear non-Gaussian system, Cadini [118] estimated the hidden performance variables by using particle filter and then presented the optimal replacement policy of components. Based on the exponential degradation path model proposed by Gebraeel, Kevin et al. [119] proposed a predictive maintenance policy based on the degradation model. Firstly, according to the information obtained from real-time monitoring, the author estimated the random parameters in the exponential degradation model by Bayesian updating method and presented the remaining life distribution of the system. Finally, the maintenance operation was arranged according to this. Elwany et al. [120] also used the exponential degradation path model to estimate the remaining life distribution and updated it in real time by using the sensor data. On this basis, the author also considered the component maintenance and replacement policy and the spare parts ordering policy. Finally, the optimal component replacement time and the optimal spare parts ordering time were obtained by simulation. For the degraded system subject to continuous monitoring, Li et al. [121] considered using time series prediction technology to predict the equipment reliability in real time and based on this designed a predictive maintenance policy which could dynamically update the maintenance threshold. On the basis of reference [121], Sun et al. [122] further considered using the modified two-stage degradation model to describe the degradation process of the system and dynamically determined the maintenance threshold and its confidence interval on this basis. For a class of system subject to continuous monitoring and degradation, Zhou et al. [123] considered combining sequential imperfect maintenance policy with predictive maintenance policy and proposed a reliability-centered predictive maintenance policy. Here, the imperfect maintenance was modeled by introducing age reduction factor and failure rate growth factor into the failure rate model, and the failure limit policy was used, that is, preventive maintenance would be carried out when the failure rate reached a certain threshold. The author presented the optimal preventive maintenance threshold through simulation. On the basis of previous references, You et al. [124] considered the reliability of prediction based on performance degradation data and the application of failure rate obtained based on failure time statistics in maintenance decision and presented the existence and uniqueness of the optimal solution through theoretical derivation. On the basis of failure prediction, Djurdjanovic [125] proposed an intelligent maintenance system, so as to achieve the goal that the system downtime was almost zero. For a class of unrepairable single-component system with exponential growth in performance degradation, Elwany [126] first considered the predictive distribution of performance degradation obtained based on sensor measurement information with Markov decision process and proved that the optimal replacement policy was a threshold policy.

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1.3.3 Literature Review on Maintenance Decision of Multi-component System The actual engineering systems involved in the fields of industrial production, transportation and national defense are usually multi-component systems. While the researchers study the maintenance decision of single-component system, the maintenance decision of multi-component system has been valued and developed gradually. The so-called multi-component system refers to a system composed of multiple identical or different components in a certain relationship (series, parallel and series– parallel). If there is no correlation between these components, the maintenance of multi-component system can be arranged according to the statistical characteristics of each component. But in fact, there are always dependencies between components. Thomas [127] pointed out that there were three kinds of dependencies in multi-component systems, namely economic dependence, stochastic dependence and structural dependence. At present, researches on multi-component systems are mainly carried out around the first two kinds of dependencies, so only the references related to the first two kinds of dependencies are summarized herein.

1.3.3.1

Economic Dependence

Economic dependence is manifested in the fact that the cost of joint maintenance of several components in the system is not equal to the cost of individual maintenance of these components. In general, since simultaneous maintenance of components requires only removal and installation for one time, simultaneous maintenance can save cost. However, in some cases, simultaneous maintenance of components will increase the maintenance cost. For example, group maintenance of components will require more maintenance personnel, which will increase the labor cost. Nicolai [128] called the economic dependence in the former case as positive economic dependence and called that in the latter case as negative economic dependence. In view of the few research reference on the latter situation, this paper only summarizes the related research on positive economic dependence and no longer considers negative economic dependence. For the sake of convenience, the economic dependence mentioned below refers to the positive economic dependence. Because joint maintenance of components saves cost, maintenance management personnel do everything possible to consider how to achieve joint maintenance. Then, there are two kinds of maintenance policies: group maintenance and opportunistic maintenance.

1.3.3.2

Group Maintenance

Group maintenance refers to the joint maintenance of components in the same group to save maintenance cost.

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1 Introduction

Okumoto and Elsayed [129] proposed a T-life maintenance policy for a class of system in which multiple components are connected in parallel: when the system has been put into operation for more than a certain threshold T, all failed components will be replaced. The life of each component in the system is an independently and identically distributed random variable, and if the component fails and it is not replaced, production loss will occur. Okumoto and Elsayed gave sufficient conditions for the existence of definite, unique and limited maintenance time T. Assaf and Shanthikumar [130] studied the system similar to that in Reference [129]. On the basis of further assuming that the life of components is an independently, identically and exponentially distributed random variable, they gave sufficient conditions for the existence of an m-failure maintenance policy. The so-called m-failure maintenance policy means that when the total number of failed components in the system exceeds m, all failed components will be replaced in groups. Ritchken and Wilson [131] comprehensively considered the advantages and disadvantages of T-life maintenance policy and m-failure maintenance policy, and proposed a (m, T ) group maintenance policy, that is, as long as the system running time reaches time T or the number of failed components reaches m, the system will be replaced. For a class of system composed of independent repairable components, Sheu and Jhang [132] proposed a two-stage group maintenance policy (T, W, k). In the first stage (0, T ], the components with minor failure are corrected by minimal repair, while the components with catastrophic failure are replaced. In the second stage (T, T + W ], the components with minor failure are also corrected by minimal repair, but the components with catastrophic failure are temporarily shelved until the time reaches T + W or the number of catastrophic failures in the second stage reaches k. Recently, Heidergott [133] proposed a new group maintenance policy for multicomponent system: once the number of failed components in the system reaches F, the components that have failed and those that have not failed but whose age exceeds the threshold θ will be replaced. The author gave various unbiased estimators by using the measure-valued differentiation approach. In view of the fact that the traditional group policy does not consider the shortterm information changes, Dekker, Wildeman and Smit, et al. proposed the dynamic group maintenance policy [134–137]. The core idea of their research is to consider group maintenance in a limited time period based on rolling time domain method. First, determine the optimal replacement time of each component, then delay or advance the penalty cost function caused by maintenance time and finally find out the maintenance group with the most cost savings on this basis.

1.3.3.3

Opportunistic Maintenance

Because group maintenance is to jointly maintain the components in the same group without considering the actual degradation state of the components, it is easy to cause excessive maintenance. If you take advantage of the opportunity of system shutdown due to breakdown maintenance or preventive maintenance of a certain component

1.3 Maintenance Decision of Equipment

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to carry out preventive maintenance on other components, the maintenance cost will be reduced accordingly, which is called opportunistic maintenance. Jorgenson [138] first studied the opportunistic maintenance of ballistic missile. Berg [139, 140] proposed an opportunistic maintenance policy for a system composed of two components whose life follows the exponential distribution: when one of the two components fails, if the other component’s age exceeds the threshold L, this component is also replaced. Zheng and Fard [141] proposed a (L-u, L) policy based on failure rate for a system composed of k components. When the failure rate of a component reaches the threshold L or the component fails and the failure rate is within the interval (L-u, L), it will be replaced. At the same time, it is necessary to check the failure rate of other normally working components and replace the components whose failure rate is within the interval (L-u, L). Similar to the policy adopted in Reference [141], Tian and Liao [142] considered the opportunistic maintenance of multi-component system based on the model with proportional failure rate under the condition-based maintenance framework. In order to optimize and obtain the two thresholds in the policy, the author gave an approximate method to calculate the cost of implementing the policy. Zhijun Cheng and Bo Chen [143] of National Defense University studied the opportunistic maintenance optimization model of a class of system in which two components are connected in series under the premise of economic dependence between components. In this reference, the author deduced the system reliability model under the (t, T ) opportunistic maintenance policy and gave an analytical method for solving the expected cost and steady-state availability per unit time. Wenrui Hou et al. [144] of Shanghai Jiaotong University proposed an opportunistic maintenance optimization model considering risks in view of the fact that component failure may lead to multiple system failures. Considering the huge economic losses caused by accident shutdown of nuclear power plants and the simultaneous maintenance of equipment outside the nuclear island, Yanqiao Chen et al. [145] of Naval University of Engineering studied the opportunistic maintenance policy of equipment outside the nuclear island and solved the optimal maintenance interval length and the coefficient of opportunistic maintenance by Monte Carlo simulation. Different from previous references, Dekker [146, 147] considered the opportunistic maintenance when the interval of maintenance opportunities follows the exponential distribution and has no relation with components in the system. Dagpunar [148] considered the opportunistic maintenance when the opportunity occurs in the Poisson process which is of a strength function (_). Under the condition-based maintenance framework, it is difficult to obtain the analytical expression of the threshold in the opportunistic maintenance of multi-component system. At this time, heuristic algorithm or Monte Carlo simulation plus stochastic simulation is usually used to optimize the objective function. Camci [149] studied the maintenance arrangement of multi-component system with economic dependence among components by using the predicted reliability. Different from the traditional CBM, there is no control limit, but the maintenance

26

1 Introduction

risk is expressed by the predicted reliability and then optimized by genetic algorithm, and finally, the optimal maintenance time is obtained. However, this method only aims at unrepairable system and only considers the maintenance arrangement in a limited period of time in the future.

1.3.3.4

Stochastic Dependence

Stochastic dependence means that the failure of one component in a multi-component system will affect the performance indicators of other components. Here, the performance indicators are usually the component age, failure rate, state variables, existence of failure, etc. Murthy and Nguyen [150] proposed two basic stochastic dependence description models for a class of two-component system: one is that the failure of one component in the system will cause other components to fail with a certain probability p, but not fail with a probability of 1-p; the other is that the failure of one component in the system will increase the failure rate of other components to a certain extent. For the two-component system with the above-mentioned Type I stochastic dependence and economic dependence, Scarf and Deara studied the age-based replacement policy and group replacement policy, respectively, in References [151] and [152]. Nakagawa and Murthy [153] specifically studied the Type II stochastic dependence. Lai and Chen [154] studied the two-component system with both Type I and Type II stochastic dependence. In this system, when Component 1 fails, the failure rate of Component 2 will increase by a certain amount, while the failure of Component 2 will make Component 1 fail immediately. On the basis of Type II stochastic dependence, Zequeira and Bérenguer studied the periodic imperfect maintenance policy when there were competitive repairable and unrepairable failure modes in the system. Castro [155] proposed a new stochastic dependence model for studying similar situation.

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143. Cheng Z, Guo Bo (2007) The optimal analysis of opportunistic maintenance model of multicomponents system. Ind Eng J 10(5):66–69 144. Hou W, Jiang Z, Jin Y (2008) An opportunistic maintenance model considering risk. J Shanghai Jiaotong Univ (Chin Ed) 42(7):1095–1099 145. Chen Y, Jin J, Huang Z (2009) Study on opportunity maintenance policy of equipment outside nuclear island based on operational availability. Nucl Power Eng 30(6):108–111 146. Dekker R, Smeitink E (1991) Opportunity-based block replacement. Eur J Oper Res 53(1):46– 63 147. Dekker R, Dijkstra M (1992) Opportunity-based age replacement: exponentially distributed times between opportunities. Nav Res Logist 39(2):175–190 148. Dagpunar J (1996) A maintenance model with opportunities and interrupt replacement options. J Oper Res Soc 47(11):1406–1409 149. Camci F (2009) System maintenance scheduling with prognostics information using genetic algorithm. IEEE Trans Reliab 58(3):1–14 150. Murthy DNP, Nguyen DG (1985) Study of two-component system with failure interaction. Nav Res Logist Q 32(2):239–247 151. Scarf PA, Deara M (1998) On the development and application of maintenance policies for a two-component system with failure dependence. IMA J Manag Math 9(2):91–107 152. Scarf PA, Deara M (2002) Block replacement policies for a two-component system with failure dependence. Nav Res Logist 50(1):70–87 153. Nakagawa T, Murthy DNP (1993) Optimal replacement policies for a two unit system with failure interactions. RAIRO. Recherche Opérationnelle 27(4):427–438 154. Lai M T, chen Y C. Optimal periodic replacement policy for a two-unit system with failure rate interaction. The International Journal of Advanced Manufacturing Technology, 2006, 29(3):367–371. 155. Castro I (2009) A model of imperfect preventive maintenance with dependent failure modes. Eur J Oper Res 196(1):217–224

Chapter 2

Residual Life Prediction Based on Wiener Process with Nonlinear Drift

2.1 Introduction Wiener process, also known as Brownian motion, was first proposed by Robert Brown, a biologist of the UK, in 1827 based on observing the physical phenomenon that pollen particles “move irregularly” on the liquid surface. Einstein gave a mathematical description of the physical law of this phenomenon for the first time in 1905, which made a remarkable progress in this subject. With the efforts of Smolucliowski, Fokker, Planck, Burger, Furth Ornstein, Ublenbeck et al., the physical theoretical work in this field developed rapidly, but the mathematical progress was slow due to the difficulty in accurate description. It was not until 1918 that Wiener made an accurate mathematical description of this phenomenon in theory. Furthermore, the properties of the orbit of Wiener process were studied, and the definition of measure and integral in the space of Wiener process was proposed, which made the research on Wiener process and its universal implication get rapid and deep development, and gradually penetrate into the various fields of probability theory and mathematical analysis, making it an important part of modern probability theory. Because of its advantages in mathematics and physics, Wiener process has become a widely used degradation modeling and remaining life prediction method [1–3]. To simplify the model, for the early models requiring remaining life prediction based on Wiener process, it was assumed that the degradation rate of the same kind of equipment was a fixed constant; that is, different degraded individuals of the same kind had the same degradation rate, without considering the uncertainty of degradation process. Obviously, due to the similarity in design, manufacture, materials and functions, the degradation paths of the same kind of equipment will have certain similarity. However, it is often seen from engineering practice that, due to the variability in manufacture, transportation and use and the influence of random environment (such as differences in equipment manufacture, different use intensity and random dynamic environment), although they are the same kind of equipment, there are obvious differences in degradation rates or degradation paths among different © National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_2

33

34

2 Residual Life Prediction Based on Wiener Process …

degraded individuals. In other words, there may be great difference in degradation paths among different degraded individuals of the same kind, which is often called heterogeneity [4]. In order to describe the similarity and individual difference of degraded equipment simultaneously, Wiener process model based on mixed effect (including common effect and random effect) has been widely concerned in the references. The common effect is used to describe the same characteristics among the same kind of equipment, and the random effect is used to describe the heterogeneity among different individuals of the same kind [4, 5]. It shall be pointed out that the introduction of random effect in the degradation model will have a direct impact on the predicted results of the remaining life of equipment, that is, the addition of a certain uncertainty. The existing Wiener process-driven degradation model can be roughly divided into offline model and online model. The former mainly focuses on fitting the overall degradation trend of a class of equipment, so the predicted remaining life is actually the mean time to failure (MTTF) of the remaining life of such equipment. The predicted remaining life distribution is also the overall remaining life distribution of such equipment. These life characteristics have important reference value in the design stage of equipment or the test stage of new equipment. However, for a specific degraded equipment, due to the randomness and dynamics of the actual operating environment, as well as the variability of the use intensity and task, its degradation path may be quite different from the degradation evolution trend of other equipment of the same kind. There may be a large deviation in evaluating the remaining life of a specific equipment by using the remaining life results obtained in the equipment design or test stage. In order to accurately predict the remaining life of a specific degraded equipment in real time, a feasible method is to dynamically update the predicted results of the remaining life by using the information observed in real time for a specific equipment. Gebraeel et al. [5] firstly established the equipment degradation model based on Wiener process with linear drift (or linearizability) and realized the online updating of random drift coefficient by assuming that the drift coefficient followed normal distribution and applying the degradation data observed in real time and Bayesian inference method. Gebraeel method has a great influence on the field of life prediction and health management for the equipment. In recent years, this research has been widely used in actual maintenance arrangement and inventory control [6, 7]. Although the Gebraeel method has been widely used in equipment life prediction and health management practice, there are still several problems in the Gebraeel method: (1) The degradation model is based on a linear (or linearized) model, which causes the remaining life prediction and health management results are only applicable to the linearly degraded equipment or the equipment whose degradation data can be directly linearized; (2) there is no systematic prior parameter estimation method, and the prior information can only be obtained by simple averaging; (3) Brownian motion term in the degradation model is only treated as observation error, which makes the remaining life distribution not an exact solution in the sense of first passage time, but an approximate solution in the Bernstein distribution class. It is well known that the moment of this class of Bernstein distribution does not exist, which limits the application of corresponding results in life prediction and health

2.1 Introduction

35

management. In order to overcome the above problems of Gebraeel method, based on a class of Wiener process model with nonlinear drift, this chapter systematically studies the methods of offline estimation of prior parameters and online updating of random parameters, which includes two stages: (1) offline estimation of common parameters and hyper-parameters in prior distribution based on historical degradation data of other equipment of the same kind; (2) online real-time updating of random parameters based on data observed in real time for a specific degraded equipment. Furthermore, the estimated results of parameters are used to update the remaining life distribution in real time in the sense of first passage time. The accuracy of predicted results of the remaining life is verified by degradation data of bearing. In this chapter, firstly, the definition of Wiener degradation process is presented, then a class of generalized Wiener process model with nonlinear drift is presented, and the analytical expression of remaining life distribution is derived. Furthermore, the methods of offline estimation of prior parameters and online updating of random parameters are presented, and the real-time prediction of remaining life of equipment is realized. Finally, the effectiveness of the proposed degradation modeling and remaining life prediction method is verified by a bearing example.

2.2 Definition of Wiener Process As a stochastic process with continuous time parameters and continuous state space, Wiener process is a class of the most basic, simplest and most important stochastic process. Many other processes can often be regarded as the fonctionnelle or extension of Wiener process in some sense. Wiener process is one of the clearest and most diversified stochastic process. At present, Wiener process and its extension have been widely used in many scientific fields, such as physics, economics, communication theory, biology, management science and mathematical statistics. Simultaneously, because Wiener process is closely related to differential equation (such as heat conduction equation), it also becomes an important channel for the connection between probability and analysis. Wiener process with drift has been widely used in degradation modeling and life prediction. Firstly, the definition of standard Wiener process is presented.   Definition 2.1 If a stochastic process X˜ (t), t ≥ 0 satisfies: (1) X˜ (t) is an inde  pendent incremental process; (2) ∀s, t > 0, X˜ (s + t) − X˜ (s) ∼ N 0, c2 t ; that is, X˜ (s + t) − X˜ (s) is the normal distribution with expectation 0 and variance c2 t; (3)  X˜ (t) is a continuous function with respect to t, then X˜ (t), t ≥ 0 is called Wiener process (or Brownian motion). In fact, (3) in the definition can be deduced from (1) and (2) in the definition, so X˜ (t), t ≥ 0 can be judged as Brownian motion only by satisfying (1) and (2).

36

2 Residual Life Prediction Based on Wiener Process …

  When c = 1, X˜ (t), t ≥ 0 is called the standard Wiener process, which is denoted as B(t). A class of simplest degradation model based on Wiener process is Wiener process with linear drift, and its mathematical description is as follows X (t) = X (0) + vt + σ B(t) where X (t) represents the degradation amount of degraded equipment at t; X (0) represents the degradation amount at 0 (initial time), which is usually known constant; v is drift coefficient; σ is diffusion coefficient.

2.3 Degradation Modeling Based on Wiener Process with Nonlinear Drift Section 2.2 presents the basic definition of Wiener process and the model of Wiener degradation process with linear drift. Because nonlinear degradation is more common in practice, this section further presents a class of model of Wiener degradation process with nonlinear drift. Let X (t) be the degradation of actual equipment at t, which can be characterized by the following Wiener process with nonlinear drift [8] t u(τ ; β)dτ + σ B(t),

X (t) = X (0) + α

(2.1)

0

t where X (0) is the initial degradation at 0, α 0 u(τ ; β)dτ represents the average cumulative effect of equipment degradation process and is called nonlinear drift of degradation amount X (t), σ is diffusion coefficient, and B(t) represents standard Brownian motion. Specifically, α is a random parameter, and β and σ are common parameters. Without losing generality, it is assumed that X (0) = 0, and α ∼ N (μα , σα2 ), and it is assumed that α and B(t) are statistically independent. These assumptions are common in degradation modeling and remaining life prediction. It is worth mentioning that Formula (2.1) has a generalized function form, and the Wiener process with linear drift described in Sect. 2.2 and the common degradation model based on Wiener process in References [5, 6, 9] can be taken as special cases.

2.4 Probability Density Function of the Residual Life

37

2.4 Probability Density Function of the Residual Life Based on the Wiener process model with nonlinear drift presented in Sect. 2.3, the probability density function of remaining life is derived in the sense of first passage time. In order to obtain the probability density function of the remaining life of degraded equipment described by Formula (2.1), it is necessary to deduce the probability density function of the corresponding life first. Based on the concept of first passage time [10], the life T of degraded equipment described by Formula (2.1) can be defined as  T = inf t : X (t) ≥ X f |X (0) < X f ,

(2.2)

where x f is the known failure threshold. Because of the existence of nonlinear part in Formula (2.1), it is usually difficult to get the analytical solution of probability density function of equipment life T defined by Formula (2.2). In order to obtain the probability density function of the analytical form of T , the method in Reference [8] is adopted herein. Considering the randomness of parameters, the conditional probability density function of life T defined by Formula (2.2) is f T |α (t|α) ∼ =

Xf −α

t 0

⎧  2 ⎫ t ⎪ ⎨ ⎬ X f − α 0 u(τ ; β)dγ ⎪ u(τ ; β)dγ + αu(t; β)t . exp − √ ⎪ ⎪ 2σ 2 t σ t 2π t ⎩ ⎭ (2.3)

According to Formula (2.3), the probability density function of life T still depends on random parameter α. Further, the unconditional probability density function f T (t) of life T can be obtained by the total probability formula on the basis of Formula (2.3), namely  f T (t) =

f T |α (t|α) f (α)dα,

(2.4)



where f (α) and  are the probability density function and parameter space of random parameter α, respectively. In practice, f (α) needs to be estimated by parameter estimation method based on observed data. Once the probability density function of life T is obtained; that is, Formula (2.4), the probability density function of the remaining life of a specific degraded equipment can also be  obtained  accordingly. The specific process is as follows. Let tk k ∈ N + represent the current time and L k represent the remaining life of the equipment at tk . If the degraded equipment reaches the failure threshold for the first time at t, lk = t − tk . By using the independent increment of Brownian motion,

38

2 Residual Life Prediction Based on Wiener Process …

the following formula can be obtained from Formula (2.1) t k +lk

Y (lk ) = Y (0) + α

u(τ ; β)dτ + σ B(lk ),

(2.5)

tk

where Y (lk ) = X (tk + lk ) − X (tk ), and Y (0) = 0. According to the above-mentioned, Formulas (2.5) and (2.1) have similar function forms. Therefore, based on the results of Formulas (2.3) and (2.4), the probability density function of the remaining life L k can be obtained. Theorem 2.1 The probability density function of the remaining life of the degraded equipment expressed by the degradation model (2.1) at tk is f L k (lk ) ∼ = ⎧ ⎨ ⎩

1    2 lk +tk 2 2 2 2πlk σα,k tk u(τ ; β)dτ + σ lk

⎡ l +t ⎤ k k u(τ ; β)dτ − u(lk + tk ; β)lk ⎦ X f − xk − ⎣ tk

⎫ ⎪ X f − xk tk u(τ ; β)dτ + μα,k σ lk ⎬  2 ⎪ lk +tk 2 ⎭ σα,k u(τ ; β)dτ + σ 2 lk tk ⎧  2 ⎫  lk +tk ⎪ ⎪ ⎪ ⎨ X f − xk − μα,k t ⎬ u(τ ; β)dτ ⎪ k  , exp −    2  lk +tk ⎪ ⎪ ⎪ 2 ⎩ 2 σα,k ⎭ u(τ ; β)dτ + σ 2 lk ⎪ t 2 σα,k



  lk +tk

2

(2.6)

k

2 where μα,k and σα,k are the mean and variance of random parameter α updated at tk , respectively, and xk is the degradation observation of degraded equipment at tk . Theorem 2.1 can be easily obtained by Formulas (2.2)–(2.5), and the proof process is omitted. According to Formula (2.6), in order to obtain the probability density function of the remaining life of the equipment at tk , it is necessary to know the relevant parameters in the model. Next, the parameter estimation process is presented.

2.5 Parameter Estimation The estimation of unknown parameters in degradation model includes the following two steps. (1) Offline estimation: Estimate the common parameters β and σ , and

2.5 Parameter Estimation

39

2 hyper-parameters μα,0 and σα,0 in the prior distribution of random parameter α based on the historical degradation data of the same kind of equipment; (2) online updating: at any tk , update the parameters in the random parameter α distribution f (α) by using 2 the degradation data xk observed in real time; that is, μα,k and σα,k .

2.5.1 Off-Line Estimation of Common Parameters and Hyper-Parameters Assuming that there are degradation data from M same kind of equipment, record the number of degradation data of the i-th equipment as Ni (1 ≤ i ≤ M). Let X (ti, j ) (abbreviated as X i, j ) represent the degradation data (i.e., the j-th) of the i-th equipment at ti, j , and 1 ≤ j ≤ Ni . Based on Formula (2.1), ti, j X i, j = X (0) + α0

  u(τ ; β)dτ + σ B ti, j ,

(2.7)

0

where the prior of random parameter α, and its distribution is π0 (α) ∼   α0 represents 2 . Further, it is assumed that the degradation data of different degraded N μα,0 , σα,0 equipment are uncorrelated, but the degradation data of the same degraded equipment are correlated, and the correlation is characterized by covariance matrix. The following theorem can be obtained according to Formula (2.1) and the properties of Brownian motion. Theorem 2.2 If the degradation datum of the i-th degraded equipment is X i = (X i,1 , X i,2 , · · · , X i,Ni ) , X i obeys multivariate Gaussian distribution, and its mean and covariance are, respectively. µi =μα,0 Ii 2 i =σα,0 Ii Ii + σ 2 K i ,

where Ii = ⎡ ti,1 ti,1 · · · ti,1 ⎢ ti,1 ti,2 · · · ti,2 ⎢ ⎢ . . . . ⎣ .. .. . . ..

 ti,1 ⎤

0

u(τ ; β)dτ ,

 ti,2 0

u(τ ; β)dτ , · · · ,

(2.8)  ti,Ni 0

 u(τ ; β)dτ , K i

=

⎥ ⎥ ⎥. ⎦

ti,1 ti,2 · · · ti,Ni It is proved that, based on the Formula (2.7) and the properties of standard Brownian motion, the mean of X i can be easily obtained, namely µi = μα,0 Ii . Next, we focus on deriving the covariance matrix i of X i . According to the definition of covariance matrix, i is expressed as

40

2 Residual Life Prediction Based on Wiener Process …

    ⎤ ⎡  D X i,1 cov Xi,1 , X i,2 · · · cov X i,1 , X i,Ni  ⎢ · · · cov X i,2 , X i,Ni ⎥ D X i,2 ⎢ ⎥ i = ⎢ ⎥. .. .. ⎣ ⎦ . .   D X i,Ni

(2.9)

The variance D(X i, j )(1 ≤ j ≤ Ni ) can be easily calculated by the following formula, 

⎛



D X i, j = D ⎝α0 ⎡ 2 ⎣ = σα,0

ti, j

⎞  u(τ ; β)dτ + σ B ti, j ⎠

0

ti, j



⎤2 u(τ ; β)dτ ⎦ + σ 2 ti, j .

(2.10)

0

Let ti,1 ≤ ti, jm < ti, jn ≤ ti,Ni , then the covariance between X i, jm and X i, jn can be calculated by the following formula, cov(X i, jm , X i, jn ) = E(X i, jm X i, jn ) − E(X i, jm )E(X i, jn ).

(2.11)

Based on the correlation properties of standard Brownian motion and conditional expectation, we can get  "  #  E X i, jm X i, jn = E X i, jm X i, jn − X i, jm + X i, jm   "   # 2  = E E X i, jm X i, jn − X i, jm |α0 + E X i, jm   " # " # 2  = E E X i, jm |α0 E X i, jn − X i, jm |α0 + E X i, jm ⎧ ⎡ ti, j ⎛ ti, j ⎞⎤⎫ m n ti, jm ⎨ ⎬ = E α02 ⎣ u(τ ; β)dτ ⎝ u(τ ; β)dτ − u(τ ; β)dτ ⎠⎦ ⎩ ⎭ 0 0 0   2 + E X i, jm ⎡ ti, j ⎛ ti, j ⎞⎤ m n ti, jm = ⎣ u(τ ; β)dτ ⎝ u(τ ; β)dτ − u(τ ; β)dτ ⎠⎦ 

0

α02





0

0

2 

E + E X i, jm ⎡ ti, j ⎛ ti, j ⎞⎤ m n ti, jm = ⎣ u(τ ; β)dτ ⎝ u(τ ; β)dτ − u(τ ; β)dτ ⎠⎦ 

0

μ2α,0

+

2 σα,0



0



+ D X i, jm



0

"  #2 + E X i, jm

2.5 Parameter Estimation

⎡ ⎢ =⎣

41

ti, jm

ti, jn u(τ ; β)dτ

0

⎛ ti, j ⎞2 ⎤ m  ⎥ 2 u(τ ; β)dτ − ⎝ u(τ ; β)dτ ⎠ ⎦ μ2α,0 + σα,0

0

0

⎡ ti, j ⎤2 ⎡ ⎤2 m ti, jm 2 + ⎣ u(τ ; β)dτ ⎦ σα,0 + σ 2 ti, jm + ⎣μα,0 u(τ ; β)dτ ⎦ 0

0

ti, jm =

ti, jn u(τ ; β)dτ

0

  2 u(τ ; β)dτ μ2α,0 + σα,0 + σ 2 ti, jm .

(2.12)

0

and 

   E X i, jm E X i, jn = μ2α,0

ti, jm

ti, jn u(τ ; β)dτ

0

u(τ ; β)dτ .

(2.13)

0

By substituting the results of Formulas (2.12) and (2.13) into Formula (2.11), we can get 



cov X i, jm , X i, jn =

ti, jm 2 σα,0

ti, jn u(τ ; β)dτ

0

u(τ ; β)dτ + σ 2 ti, jm .

(2.14)

0

Further, by substituting Formulas (2.10) and (2.14) into Formula (2.9), the covariance matrix i shown in Formula (2.8) can be obtained. The proving is completed. Based on Theorem 2.2, the log-likelihood function generated by degradation data from M degraded equipment is M M M −1 $ $   1 $ 1$ 1

(|X1:M ) = − ln(2π ) X i − µi X i − µi , Ni − ln| i | − 2 2 i=1 2 i=1 i=1 i

(2.15) where  = (μα,0 , σα,0 , β, σ ) is an unknown parameter, and X 1:M = (X 1 , X 2 , · · · , X M ) represents degradation data of M degraded equipment. The optimal value of unknown parameter  can be obtained by maximizing loglikelihood function expression (2.15). Next, by taking the estimated  value as a priori, at any tk , the online observed datum xk can be used to update the parameters in the random parameter α distribution in real time.

42

2 Residual Life Prediction Based on Wiener Process …

2.5.2 Real-Time Updating of Random Parameter For a specific degraded equipment, at any tk in its life cycle, the random parameter α of the degradation model can be obtained from all observed data x 1:k = {x1 , x2 , · · · , xk } (corresponding to all observed data of the equipment from t1 to tk ) of the equipment before tk . By means of Bayesian rules, the posterior distribution of the degraded equipment at tk can be obtained as follows, p(α|x 1:k ) ∝ p(x 1:k |α)π0 (α),

(2.16)

where p(x 1:k |α) represents the likelihood function under the given random parameter α, and the prior distribution π0 (α) has been estimated by Formula (2.15). Specifically, p(x 1:k |α) can be obtained by using the basic properties of Brownian motion based on Formula (2.1). p(x 1:k |α) = % k

&

1

  2π σ 2 tq − tq−1 ⎧  2 ⎫  tq ⎪ k ⎨ $ ⎬ xq − xq−1 − α tq−1 u(τ ; β)dτ ⎪   exp − , ⎪ ⎪ 2σ 2 tq − tq−1 ⎩ q=1 ⎭ q=1

(2.17)

For the degradation amount x0 = 0 at the initial time (t0 = 0), see the definition of Formula (2.1). It is worth mentioning that, because p(x 1:k |α) and π0 (α) are in normal distribution, p(α|x 1:k ) is also in normal distribution. Based on Formulas (2.16) and (2.17), the mean and variance of p(α|x 1:k ) can be obtained, respectively, as follows μα,k =

C+D , A+B

(2.18)

2 σα,k =

1 , A+B

(2.19)

and

 2 (   ) 2 2 σ tq − tq−1 , B = 1 σα,0 where A = , C = q=1 tq−1 u(τ ; β)dτ   *   ) 2     'k tq xq − xq−1 tq−1 μα,0 σα,0 , D = q=1 u(τ ; β)dτ σ 2 tq − tq−1 . The probability density function of the remaining life of the equipment at tk can be obtained by substituting the parameters estimated from Formulas (2.15), (2.18) and (2.19) into Formula (2.6). 'k

  tq

2.5 Parameter Estimation

43

In order to better understand and grasp the parameter estimation and remaining life process proposed above, the main steps are summarized as follows: Step 1: Offline estimation of parameters. Based on the observed historical degradation data of M same kind of equipment, the common parameters and hyperparameters in the prior distribution of random parameters are estimated by using Formula (2.15). Step 2: Online updating of parameters. For a specific in-service equipment, at any tk in its life cycle, after observing the degradation data xk , the parameters μα,k and 2 are updated in real time by using Formulas (2.18) and (2.19). σα,k Step 3: Online prediction of remaining life. By substituting the estimated results of parameters obtained in Steps 1 and 2 into Formula (2.6), the probability density function of the remaining life of such in-service equipment at any tk can be obtained. Step 4: Once the observed datum xk+1 of the degraded equipment is obtained at tk+1 , return to Step 2 and repeat Steps 2 and 3 to obtain the probability density function of the remaining life of the equipment at tk+1 .

2.6 Case Study The effectiveness and superiority of the real-time prediction method of remaining life proposed above are verified by the actual vibration data of bearings in this section, and this method is compared with the following two typical methods in the existing references: Wiener process with linear drift [5] and Wiener process with nonlinear drift [8]. Here, the two methods are called Gebraeel method and Si method, respectively. The method in this chapter is obviously different from Gebraeel method; that is, the degradation model in this chapter is nonlinear, while Gebraeel method is limited to linear model or linearizable model. Although the Si method also has nonlinear structure, compared with the remaining life prediction method studied in this chapter, the Si method has no real-time updating capability.

2.6.1 Problem Description The bearing is a key equipment in many rotating machinery systems. As shown in Fig. 2.1, the degradation of bearing equipment begins with tiny cracks under the bearing raceway. With the continuous use of bearing and the aggravation of degradation, the cracks gradually spread to the surface of the raceway, resulting in pits or peeling in the raceway. These pits will increase the friction between the raceway and the ball, thus increasing the vibration degree of the bearing. Usually, the vibration degree of the bearing can be reflected by the vibration data. The larger the vibration data are, the more serious the bearing degradation is. In engineering

44

2 Residual Life Prediction Based on Wiener Process …

Fig. 2.1 Normal and failed bearings [11]

practice, once the vibration data of the bearing increase to an established failure threshold, the bearing performance will be considered invalid. The data used in this example come from Reference [12]. The vibration data of five bearings are shown in Fig. 2.2, and the selected feature is root mean square (rms) of vibration data. In this example, the failure threshold of the bearing is set as 20. According to Fig. 2.2, the vibration data increase gradually with the increase of bearing operation time. In addition, although the vibration data of all bearings are gradually increasing as a whole, the specific degradation paths (or degradation rates) of different bearings are different due to individual differences. Next, the vibration data from bearing 1 to bearing 4 are selected as training data, and the vibration data of bearing 5 are selected as test data to verify the effectiveness of degradation modeling and remaining life prediction method proposed in this chapter. The main reason for selecting bearing 5 as the test bearing herein is that the vibration datum of the last (i.e., 280 h) test of the bearing is 19.6412, which is very close to the failure threshold (20) of vibration. Therefore, for the convenience of verifying the accuracy of the predicted results, the actual life of the test bearing 5 can be approximately 285 h. Specifically, in addition to the initial vibration datum 0, 22 sets of vibration data have been measured during the life cycle of the bearing 5, as shown in Fig. 2.2.

2.6 Case Study

45

35 Bearing 1 Bearing 2 Bearing 3 Bearing 4 Bearing 5 Failure threshold

30

libration data(r)

25

20 failure

15

10

5

0 0

Test bearing

150

100

50

200

250

300

Operation time (h)

Fig. 2.2 Vibration data of bearings

2.6.2 Results and Discussions In order  t to apply the results of this chapter, it is necessary to specify the function form of 0 u(τ ; β)dτ in Formula (2.1) for degradation model first. Based on the evolution t trend of bearing degradation data in Fig. 2.2, 0 u(τ ; β)dτ is selected as the power t function herein, namely 0 u(τ ; β)dτ = t β . The common parameters and hyperparameters in the prior distribution of random parameters can be estimated by using the vibration data from bearing 1 to 4 based on the maximum likelihood estimation method. The estimated results are shown in Table 2.1. By taking the estimations of the parameters in Table 2.1 as prior information, the parameters in the random parameter α distribution can be updated dynamically by using the vibration data of the test bearing (bearing 5) observed in real time. The dynamic updating results of mean and variance of random parameter α are shown in Fig. 2.3. The following two points can be observed from Fig. 2.1. The mean of random parameter α can be dynamically updated with the continuous acquisition of observed data of the test bearing. Specifically, the overall evolution trend of the mean of α decreases rapidly at the beginning of the bearing life profile, especially before the operation time of 100 h. The main reason is that the degradation trend or degradation rate of the test bearings is slower or smaller than that of the training bearings (from bearing 1 to Table 2.1 Estimated results of parameters

μˆ α,0

σˆ α,0

βˆ

σˆ

0.1994

0.1003

1.2121

1.1000

46

2 Residual Life Prediction Based on Wiener Process … 0.25

0.2

Updating results of mean of α

0.15

0.1

0.05

0 0

50

100

150

200

250

300

Operation time (h)

(a) Updating results of mean of α 0.01 0.009 0.008 Updating results of variance of α

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0

50

100

150

200

250

Operation time (h)

(b) Updating results of variance of α Fig. 2.3 Dynamic updating results of mean and variance of random parameter α

300

2.6 Case Study Table 2.2 Prediction error of residual life of bearing 5 with the method in this chapter and two comparison methods

47 Gebraeel method

Si method

Method in this chapter

RMSE

127.6651

137.9437

106.2919

MAE

106.3661

116.5225

85.2469

4), which can be seen from the vibration data in Fig. 2.2. The main reason for the slight increase in the mean of random parameter α in the operation time of about 150 h is that, the vibration data of the test bearing increase in the operation time of about 150 h. This phenomenon can also be seen from the vibration data of bearing 5 in Fig. 2.2, which also shows that the parameter estimation method proposed in this chapter can track the immediate health status of the equipment in real time. (2) The updating results of variance of random parameter α decrease gradually with the increase of operation time. The evolution trend shows that the uncertainty of the estimation of random parameter α decreases gradually with the increase of observed data. Based on the above-estimated results of parameters, the remaining life of the test bearing 5 is predicted in real time; that is, the predicted results of the remaining life of bearing 5 can be updated dynamically every observation of a vibration datum. As mentioned above, in addition to the initially observed value (0), a total of 22 sets of observed data are obtained in the life profile of bearing 5. Accordingly, 22 sets of predicted results of remaining life of the bearing can be obtained. The RMSE and MAE indexes are used to quantitatively compare the prediction accuracy of the remaining life of the bearing by the method in this chapter with that by two benchmark methods. Specifically, the number of predicted positions of remaining life is selected as N R = 22. The comparison results of RMSE and MAE with the three methods are shown in Table 2.2. According to Table 2.2, the method in this chapter has the smallest prediction error, followed by Gebraeel method, and the Si method has the largest prediction error. As mentioned earlier, the reason why the prediction accuracy of the method in this chapter is better than that of the other two methods is obvious, and the reason why the prediction accuracy of Si method is lower than that of Gebraeel method mainly includes the following two aspects. (1) As shown in Fig. 2.2, the degradation path of the test bearing 5 is quite different from the degradation path (or degradation rate) of the training bearings 1 to 4. Therefore, although the degradation model in Si method has a nonlinear structure, it cannot adaptively update the model parameters with the real-time degradation data of specific observation equipment. (2) The degradation path of bearing 5 is approximately linear in the early stage (before the operation time of 250 h), so it is easy to understand that the prediction accuracy of Gebraeel method with linear model is better than that of Si method with nonlinear model. Therefore, in the practice of degradation modeling and remaining life prediction, it is necessary to select a degradation model with generalized nonlinear model structure and realize the immediate updating of degradation model parameters with the data observed in real time for the degraded equipment.

48

2 Residual Life Prediction Based on Wiener Process …

The following conclusions can be drawn based on the results of the above two examples. (1) The real-time prediction method of remaining life proposed in this chapter is effective and practical. (2) Compared with the remaining life prediction methods commonly used in the references, the method in this chapter has greatly improved the RMSE and MAE indexes.

2.7 Summary of This Chapter In this chapter, a real-time prediction method of remaining life of the nonlinearly degraded equipment is proposed based on a class of Wiener process model with nonlinear drift. In degradation model, the common parameters and hyper-parameters in the prior distribution of random parameters can be estimated by using maximum likelihood estimation method from degradation data of the same kind of equipment. For a specific in-service equipment, the posterior updating of random parameters can be realized by Bayesian method based on its degradation data observed in real time, which makes the predicted results of remaining life reflect the health status of the equipment in real time. Experimental results show that the remaining life prediction method proposed in this chapter can significantly improve the prediction accuracy of remaining life of the equipment.

References 1. Wang ZQ, Hu CH, Wang W et al (2014) An additive Wiener process-based prognostic model for hybrid deteriorating systems. IEEE Trans Reliab 63(1):208–222 2. Si XS, Wang W, Hu CH et al (2011) Remaining useful life estimation—a review on the statistical data driven approaches. Eur J Oper Res 213(1):1–14 3. Zhang ZX, Si XS, Hu CH (2015) An age-and state-dependent nonlinear prognostic model for degrading systems. IEEE Trans Reliab 64(4):1214–1228 4. Ye ZS, Wang Y, Tsui KL et al (2013) Degradation data analysis using Wiener processes with measurement errors. IEEE Trans Reliab 62(4):772–780 5. Gebraeel NZ, Lawley MA, Li R et al (2005) Residual-life distributions from component degradation signals: a Bayesian approach. IIE Trans 37(6):543–557 6. Elwany AH, Gebraeel NZ (2008) Sensor-driven prognostic models for equipment replacement and spare parts inventory. IIE Trans 40(7):629–639 7. You MY, Liu F, Wang W et al (2010) Statistically planned and individually improved predictive maintenance management for continuously monitored degrading systems. IEEE Trans Reliab 59(4):744–753 8. Si XS, Wang WB, Hu CH et al (2012) Remaining useful life estimation based on a nonlinear diffusion degradation process. IEEE Trans Reliab 61(1):50–67 9. Peng CY, Tseng ST (2009) Mis-specification analysis of linear degradation models. IEEE Trans Reliab 58(3):444–455 10. Lee M-LT, Whitmore GA (2006) Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary. Stat Sci 21(4):501–513

References

49

11. FEMTO-ST, IEEE PHM (2012) Data Challenge, online website. http://www.femto-st.fr/en/ Research-departments/AS2M/Research-groups/PHM/IEEE-PHM-2012-Data-challenge.php. 12. Wang W, Zhang W (2008) Early defect identification: application of statistical process control methods. J Qual Maint Eng 14(3):225–236

Chapter 3

Residual Life Prediction Based on Wiener Process with Abrupt Changepoint

3.1 Introduction Affected by the physical or chemical property or structure of material itself, environmental stress change or other factors during various lifetime, the degeneration process of equipment will also show obvious differences in various life stages [1]. For example, in the two-stage model proposed by Wang in Reference [2], the performance degradation process of equipment is largely different before and after the defect point, and the degradation of equipment accelerates after the defect point. If the point with significant change of degradation process can be detected in real time, viz. the changepoint in degradation process, and be integrated into the degradation modeling process, our model will be more accurate. At present, degradation models which take such stage differences into consideration mainly include hidden (semi-) Markov model (HMM/HSMM) [3–8], two-stage model [2], Wiener process with self-adaption drift coefficient [9–12], etc. However, the number of hidden variables needs to be determined for HMM/HSMM. Hidden variables reflect commonly the state quantity of the equipment itself. At present, the state of equipment is artificially classified into several ones based on its performance before failure. Typical classification is as shown in Reference [8]: the performance of drill bits is classified into good, medium, bad and worst, which is not necessarily consistent with the stage difference presented by degraded performance of equipment in engineering practice. A more reasonable practice is to judge in real time whether the degraded performance of equipment is subject to periodic change based on the detected data and then confirm the number of hidden variables. In the two-stage models, assuming that the equipment degrades by the same rule after the defect point, when the degradation rule of equipment changes after the defect point, such as significant change in degradation speed and variance of degradation increment, the model parameter estimated by all data after the defect point cannot reflect the degradation rule of equipment accurately at the latest moment. Therefore, the consistency of degradation rule shall be taken in account to improve the accuracy of models. Wiener process with self-adaption © National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_3

51

52

3 Residual Life Prediction Based on Wiener Process …

drift coefficient considers that drift coefficient is decided by historical observation data. By taking the drift coefficient as the state quantity and equipment performance detection information as the observation, the optimal estimated drift coefficient is obtained by establishing and solving the state-space equation. Kalman filter (KF) [9], strong tracking filter (STF) [10] and others have been successfully applied in such models, with favorable effect achieved. The duration information of each state is of great significance to understanding the life-cycle degradation rule of equipment, but this is not taken into account in this approach. It can be known from the above analysis that, the key of modeling for the degradation process with possible change in degradation rule is to detect the point of degradation rule change, viz. the changepoint. However, the current modeling approach lacks consideration of changepoint detection, state duration and other important information. Statistical analysis theory of changepoint refers to a nonlinear statistic theory developing in recent years and used to study the abrupt change phenomenon in real world, which develops rapidly both in theory and application in recent years [13– 15]. The existing relatively mature changepoint analysis methods mainly include: likelihood ratio (LR) test, cumulative sum (CUSUM) test, Bayes test, etc. LR test features great hysteresis as it needs the data after changepoint to calculate the LR; CUSUM test requires certain data amount to construct the test statistics, so accurate test is hard to conduct when the interval between changepoints is relatively small; Bayes test can accurately test the changepoint of data sequence when a certain prior information is available. Based on Bayesian Online Changepoint Detection, this chapter introduces the approach of performance degradation modeling and remaining life prediction with abrupt changepoint based on Wiener process. It emphasizes on stating how to determine the online updating of prior distribution and posterior distribution parameters of Bayesian Online Changepoint Detection and calculate the remaining life distribution of equipment with historical data, and presenting the application example of the proposed approach on inertial platform.

3.2 Degradation Model with Abrupt Changepoint Based on Wiener Process 3.2.1 Wiener-Process-Based Degradation Model The degradation process of some equipment shows continuity and independent increment characteristics which is consistent with the nature of Wiener process. Wiener process is a common method of applying Brownian motion in degradation models. It is assumed in the method that Brownian motion has additivity effect on degradation process. The linear form of Wiener process as shown in Formula (3.1) has been extensively researched in such fields as finance and reliability prediction.

3.2 Degradation Model with Abrupt Changepoint Based on Wiener Process

yt = y0 + ut + σ Bt

53

(3.1)

where yt represents the degradation measurement at sampling time point t with y0 as it is the initial value. Bt is standard Brownian motion, viz. Bt ∼ N (0, t); N (0, t) represents the normal distribution with 0 and t as its mean and variance, respectively. u is drift coefficient and is related to the stress borne by the product. σ is the diffusion coefficient decided by the impact on the performance of product and test equipment by such random factors as product inconsistency and instability, measuring error and stability of measuring equipment, and external noise during test. It can be learnt from the normality of Wiener that the increment of equipment performance degradation follows the normal distribution with mean uti and variance σ 2 ti [16], so its probability density function is   f yi |u, σ 2 =

  1 (yi − uti )2 exp − √ 2σ 2 ti σ 2π ti

(3.2)

where ti = ti+1 − ti represents the performance detection interval of system and yi = yti+1 − yti is the increment of performance degradation amount. According to the definition of life given in Formula (2.2), lifetime of equipment is defined as the first time to reach the failure threshold D [17]. Reference [18] proves that the first passage time of Brownian motion with drift follows inverse Gaussian distribution whose probability density function is shown in Formula (3.3).   D − y0 (D − y0 − ut)2 f (t; y0 , D) = √ exp − 2σ 2 t σ 2π t 3

(3.3)

By integral of Formula (3.3), the cumulative distribution function of lifetime can be obtained by  F(t) = 1 − 

D − y0 − ut √ σ t



 − exp

   2u(D − y0 ) D − y0 + ut (3.4)  − √ σ2 σ t

where  refers to the standard normal cumulative probability distribution function. Parameters u and σ 2 can be estimated with Bayesian method according to the increment information of equipment performance detection, which will be detailed later. Furthermore, according to the definition of residual life and Formula (3.3), probability density function of the residual life based on the monitored degradation data yt0 = {y0 , y1 , . . . , yt } is calculated by     D − yt (D − yt − ult )2 f L t lt | yt0 =  × exp − 2lt σ 2 σ 2πlt3

(3.5)

54

3 Residual Life Prediction Based on Wiener Process …

3.2.2 Changepoint Detection in Performance Degradation Process Expectation and variance are two important indices describing random variable. As shown in Figs. 3.1 and 3.2, the changepoint is defined as the point with significant change of expectation or variance in time series in the stochastic process. Of course, there is also probability of simultaneous change of expectation and variance. Lots of engineering practice indicates that, similar changepoint also exists during equipment service. In order to improve the accuracy of degradation model and better describe the rule of degradation process, it is necessary to detect the changepoint of equipment before estimation of model parameter. The change of equipment performance is caused by the combined action of its internal structure and material performance change as well as environmental impact, the internal change of equipment cannot be directly observed generally, and the environmental condition is random, so the position of these changepoints is usually unknown. Assuming that only the parameter θ of degradation model changes before and after the changepoint and the model type remains unchanged, the detection of changepoint is to determine the point for significant change of model parameter θ in degradation process based on the existing performance detection information and 1

Performance value

0.5

0

-0.5

Change point 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1.2

1.4

1.6

1.8

Detection time

Fig. 3.1 Change of mean in random sequence

0.5

Performance value

0

-0.5 Change point

-1

0

0.2

0.4

0.6

0.8

1

Detection time Fig. 3.2 Change of variance in random sequence

3.2 Degradation Model with Abrupt Changepoint Based on Wiener Process

55

then estimate the model parameter before and after the changepoint. Take r τ = [r1τ , r2τ , · · · , rτ τ ]T as the changepoint mark at τ in degradation process. Then rτ k = 1 represents the latest changepoint k at instance τ , and the test of changepoint in degradation process with Bayesian method actually refers to the solution of Formula (3.6).  

tC(τ ) = argmax p rkτ | yt1 , 1 ≤ k ≤ τ , 1 ≤ τ ≤ t

(3.6)

k

where tC(τ ) represents the latest changepoint as of τ . Due to changes in the degradation after the changepoint, it  rule of equipment  t t is more reasonable to use f L t lt | y (t) instead of f L t lt | y0 as the remaining life tC prediction distribution density of equipment.

3.3 Conjugate Distribution of Prior Distribution of Exponential Family For convenient calculation, it is assumed that the increment of performance detection data is generated by the exponential family distribution with conjugate prior distribution. If prior distribution can be written in the form of hyper-parameter η and sufficient statistic φ(θ ) as shown below:   p(θ |η) = exp η T φ(θ ) − log Z (η)

(3.7)



where Z (η) = exp η T φ(θ ) dθ , and when the overall distributed likelihood function is in the following form, p



y1T |θ



=

T  t=1

p(yt |θ ) =

T 



exp u(yt )T φ(θ ) − log  Z(u(yt ))

(3.8)

t=1

  The posterior distribution p θ | y1T of parameters is in the form same as prior distribution p(θ|η), and the hyper-parameter satisfies the following relation [19] ηpost = η +

T 

u(yt )

(3.9)

t=1

  In Wiener process, the increment is yi ∼ N uti , σ 2 ti , and Formula (3.4) can be rewritten into the form in Formula (3.10).

56

3 Residual Life Prediction Based on Wiener Process …

   uti2 1 yi2 yi uti 2 f yi |u, σ = exp − + − 2 ti σ 2 ti 2σ 2 ti    1 1 1 − ln(2π ) + log 2 2 σ ti 2

(3.10)

  If the prior distribution of θ = u, σ 2 is normal-inverse Gamma distribution, viz. θ ∼ N /I Ga(m, v, a, B)    vu 2 vmu B vm 2 2 π u, σ |m, v, a, B = exp − 2 + 2 − 2 + 2 2σ σ σ σ    1 1 ln 2 − ln Z N I g (η) + a− 2 σ

(3.11)

where Z N I g (η) = (2π/v) 2 |B|−a (a), and (a) is Gamma

function. p(yi |θ ) p(θ|η)dθ that It can be calculated according to p(yi |η) = the marginal distribution of samples is in T distribution, viz. yi |η ∼  T T (yi |m, (v + 1)B/va, 2a). Take ϕ(θ ) = −u 2 /2σ 2 , u/σ 2 , 1/σ 2 , − log σ 2 and  T η = v, vm, B + vm 2 /2, a − 1/2 as the sufficient statistic and hyper-parameter, T  respectively, and there is u(yi ) = 1, yi , yi2 /2, 1/2 . Formulas (3.10) and (3.11) are in the form of Formulas (3.7) and (3.8), respectively, so the posterior density distribution function of parameters can be calculated according to Formula (3.9). The calculation result is shown below. ⎧ τ ⎪ 1 1 1  2 ⎪ yk ⎨ vτ = v0 + τ, Bτ + 2 vτ m 2τ = B0 + 2 v0 m 20 + 2 k=1 (3.12) τ  ⎪ τ ⎪ yk ⎩ aτ = a0 + 2 , vτ m τ = v0 m 0 + 1

k=1

where ητ refers to the parameter of posterior distribution acquired with data y1τ . In order to improve the calculation efficiency, Formula (3.12) can be rewritten into the recursive form as shown in Formula (3.13), and ητ +1 can be calculated by using the posterior parameter ητ of τ and the observation datayt+1 at τ + 1, viz. 

τ −yτ +1 ) vτ +1 = vτ + 1, Bτ +1 = Bτ + vτ (m2(v τ +1) vτ m τ +yτ +1 1 aτ +1 = aτ + 2 , m τ +1 = vτ +1

2

(3.13)

3.4 Bayesian Online Changepoint Detection Algorithm

57

3.4 Bayesian Online Changepoint Detection Algorithm According to the nature of exponential family distribution introduced in this section, the parameter of posterior distribution can be quickly calculated by Bayesian inference, making it possible to quickly detect the changepoint of data sequence online. The essence of Bayesian Online Changepoint Detection algorithm is to determine   the probability distribution p r τ |y1τ . It can be learnt from Bayes formula and total probability formula that     p r τ |y1τ ∝ p r τ , y1τ    p r τ , r τ −1 , y1τ −1 , yτ ∝ r τ −1

   ∝ p r t , r τ −1 , y1τ −1 , yτ r τ −1

     p r τ , yτ |r τ −1 , y1τ −1 p r t−1 , y1τ −1 ∝ r τ −1

     p r τ −1 , y1τ −1 p yτ |r τ −1 , y1τ −1 p(r τ |r τ −1 ) ∝

(3.14)

r τ −1

For the degradation process with changepoint, η(k) τ is defined as the parameter of posterior distribution calculated based on ykτ , and η(k) τ can be recursed by η k according to Formula (3.13). Since     p yτ |r τ −1 , y1τ −1 = p yτ |rk,τ −1 = 1, y1τ −1     = p yτ |ykτ −1 = p(yτ |θ ) p θ |η(k) τ −1 dθ

(3.15)

  (k) (k) (k) (k) (k) (k) it can be known that yτ |η(k) τ −1 ∼ T yτ |m τ −1 , vτ −1 + 1 Bτ −1 /vτ −1 aτ −1 , 2aτ −1 . It can be known from the definition of rτ k that the degradation rule of equipment changes when rτ τ = 1; otherwise, the equipment will continue the degradation by previous rule and the duration of such degradation rule increases accordingly. Figure 3.3 presents the change structure of the degradation rule. Solid line means the equipment continues the degradation by previous rule, while dashed line means the degradation rule changes. It can be learnt from Fig. 3.3 that ⎧    ⎨ H rτ, j  ,  k = τ + 1  p rτ +1,k |rτ, j = 1 − H rτ, j , k = j + 1 ⎩ 0, else

(3.16)

  In Formula (3.16), H rτ, j means the probability for the degradation rule of equipment changes when  rτ, j = 1. When the duration of degradation rule follows certain distribution, H rτ, j refers to the hazard rate function of such distribution. When

58

3 Residual Life Prediction Based on Wiener Process …

Fig. 3.3 Structure of changepoint

r51

r11

r 41

r52

r31

r 42

r53

r 21

r32

r 43

r54

r 22

r33

r 44

r55

the duration  ofdegradation rule follows geometric distribution, it can be learnt that there is H rτ, j = 1/λ, and λ means the scale parameter of geometric distribution. According to the above analysis and the results of Formulas (3.14), (3.15) and (3.16), Bayesian Online Changepoint Detection algorithm for performance degradation data sequence is presented below. Algorithm 3.1 Online Bayesian Detection algorithm Step 1

Step 2 Step 3

Select the distribution parameters v0 , m 0 , B0 , a0 of prior density function of degradation increment and the parameter λ of hazard rate function, and take r11 = 1;  (r ) Calculate ωτ τ −1 = p yτ |r τ −1 , y1τ −1 according to Formula (3.15) during acquisition of new data yτ ; Calculate the joint probability between the increased degradation rule duration and acquisition of existing detection data, viz.  r       p rkτ = 1, y1τ = p rk,τ −1 = 1, y1τ −1 ωτ k,z−1 1 − H rk,τ −1 , 1 ≤ k ≤ τ − 1

Step 4

Calculate the joint probability between changepoint occurrence of degradation rule and acquisition of existing detection data, viz. τ −1       (rk,τ −1 )  p rτ τ = 1, y1τ = p rk,τ −1 , y1τ −1 H rk,τ −1 ωτ k=1

Step 5

Calculate the joint probability distribution of inspection data, viz. τ      p y1τ = p rkτ = 1, y1τ k=1

3.4 Bayesian Online Changepoint Detection Algorithm

Step 6

59

Determine the marked probability and maximum probability of changepoint.       p rkτ = 1|y1τ = p rkτ = 1, y1τ / p y1τ , 1 ≤ k ≤ τ

Step 7

Determine the position of the latest changepoint.  

tC(τ ) = argmax p rkτ |y1τ , 1 ≤ k ≤ τ k

Step 8

Update the parameter of posterior distribution according to Formula (3.13), and then carry out Step 2.

3.5 Empirical Bayesian Method for Determining Prior Distribution Studies show that the selection of prior distribution will largely affect the accuracy of Bayesian Changepoint Detection algorithm. However, as the duration of degradation rule cannot be observed directly, traditional parameter estimation methods, such as moment estimation and maximum likelihood estimation, cannot effectively finish the estimation of prior parameter. Therefore, the method of determining the prior distribution by Bayesian Changepoint Detection is presented in this section based on empirical Bayesian method and approximate Bayesian inference and EM algorithm. First, it improves the forward–backward algorithm [19] of HMM parameter estimation based on the structural features of Bayesian changepoint for processing the hidden variable r τ ; then it presents the method of determining the prior distribution parameter by Bayesian Online Changepoint Detection algorithm based on historical observation data via approximate Bayesian inference and EM algorithm.

3.5.1 Improved Forward–Backward Algorithm When the data set of performance degradation increment used for training is set as y1t , it can be known from Bayesian theorem that:         p r τ |y1t ∝ p y1t |r τ p(r τ ) ∝ p yτt +1 |y1τ , r τ p r τ , y1τ

(3.17)

    τ (i) (i) = p yτt +1 |y1τ , rτ i = 1 and γτ(i)  When αtτ = p rτ i = 1, y1 , βτ p rτ i = 1|y1 , it can be known according to Formula (3.17) that

=

60

3 Residual Life Prediction Based on Wiener Process …

γτ(i) ∝ ατ(i) βτ(i)

(3.18)

It can be known from conditional probability formula and total probability formula that       p r τ , y1τ = p yτ |r τ , y1τ −1 p r τ , y1τ −1     p(r τ |r τ −1 ) p r τ −1 , y1τ −1 = p yτ |r τ , y1τ −1

(3.19)

rτ −1

Thus, for i = 1, 2, . . . τ − 1, −1  τ      p riτ = 1, y1τ = p yτ |ri,τ = 1, y1τ −1 p riτ = 1|r j,t−1 = 1 p r j,τ −1 , y1τ −1 j=1

(3.20) For i = τ , −1    τ    p rτ τ = 1, y1τ = p yτ |r j,τ −1 = 1, y1τ −1 p rτ τ = 1|r j,τ −1 = 1 p r j,τ −1 , y1τ −1 j=1

(3.21) It can be known by combining Formulas (3.20) and (3.21) above that

ατ(i)

⎧   (ri,τ −1 )  ⎪ 1 − H r j,τ −1 ατ(i)−1 , 1 ≤ i ≤ τ − 1 ⎨ ωτ −1  ( j) = τ (r j,τ −1 )  ⎪ ωτ H r j,τ −1 ατ −1 , i = τ ⎩

(3.22)

j=1

Similarly,     t    p yτt +1 |y1τ , r τ = p yτ +2 |y1τ +1 , r τ +1 p yτ +1 |y1τ , r τ +1 p(r τ +1 |r τ ) rτ +1

(3.23) Then,  (i) βτ = p yτt +1 |y1τ , ri,τ = 1 =

τ +1 

    t |y τ +1 , r τ p yt+2 j,τ +1 = 1 p yτ +1 |y1 , r j,τ +1 = 1 p r j,τ +1 = 1|ri,τ = 1 1

j=1

(3.24) i.e.,

3.5 Empirical Bayesian Method for Determining Prior Distribution

   (ri,τ )  +1) (rτ,τ )  βτ(i) = βτ(i)+1 ωτ +1 1 − H ri,τ + βτ(τ+1 ωτ +1 H ri,τ

61

(3.25)

Based on the above analysis, the improved forward–backward algorithm is given below. Algorithm 3.2 Improved Forward–Backward Algorithm Step 1

Step 2 Step 3 Step 4 Step 5 Step 6 Step 7

Select the parameter value v0 , m 0 , B0 , a0 of prior distribution and the distribution parameter λ of degradation rule duration to determine the prior hyper-parameter η0 , and then take α1(1) = 1, τ = 0; When τ ≤ t − 1, there is τ = τ + 1;   (r ) Calculate ωτ τ −1 = p yτ |r τ −1 , y1τ −1 , and also calculate ατ(i) by Formula (3.22), and then save them, respectively; Update parameter η(k) τ , 1 ≤ k ≤ τ , carry out Step 2; or else carry out Step 5; Take βt(i) = 1, where, 1 ≤ i ≤ t, τ = t; When τ ≥ 2, there is τ = τ − 1; τ) saved in Step 3 and Formula (3.25), and then Calculate βτ(i) with ωτ(r+1 carry out Step 5 after saving.

Solve ατ(i) and βτ(i) with algorithm 3.2. It can be seen from algorithm that, the improved algorithm makes full use of the relation between r τ and r τ −1 to reduce the calculation.

3.5.2 Joint Distribution of Changepoint Markers at Adjacent Detection Moments The joint distribution of changepoint markers at adjacent moments will be needed when EM algorithm is applied in the next chapter, i.e.,   (i j) ξτ,τ +1 = p riτ = 1, r j,τ +1 = 1|y1t   Due to p r τ , r τ +1 |y1t ∝    y1τ p(r τ +1 |r τ ) p yτt +2 |r τ +1 , y1τ +1 , there exists

(3.26)

  p r τ , y1τ p(yτ +1 |r τ +1

  ( j)   (i j) ξτ,τ +1 ∝ ατ(i) p yτ +1 |rτ +1, j = 1, y1τ p rτ +1, j = 1|rτ i = 1 βτ +1

(3.27)

The following can be obtained by using the relation between r τ and r τ −1 and the result calculated by Algorithm 3.2:

62

3 Residual Life Prediction Based on Wiener Process … ( j)

(ii) (i) (riτ ) ξτ,τ +1 ∝ ατ ωτ +1 (1 − H (riτ ))βτ +1

(3.28)

(i,τ +1) (τ +1) (i) (rτ τ ) ξτ,τ +1 ∝ ατ ωτ +1 H (riτ )βτ +1

(3.29)

(i j)

Save ξτ,τ +1 for future application.

3.5.3 EM Algorithm EM algorithm is commonly applied for hidden variable parameter estimation [20]. Generally, we use EM algorithm to solve the parameter of model via maximum likelihood function. In degradation model, y1t represents the set of all observable t represents all hidden variables, and prior distribution hypervariables, R = {r i }i=1 parameter v0 , m 0 , B0 , a0 and degradation rule duration  distribution parameter λ are to be estimated. As there is no analytical form for p y1t , R|η0 , λ , the calculation amount shows an exponential increase along with the increase of observation data. As to this problem, q(R) is decomposed into the product of disjoint subset event probability by applying Bayesian approximate reasoning idea in Reference [21], and the form of EM algorithm is reconsidered—preset all parameters in Step E, solve the decomposition item of q(R) that makes L(q, ) the maximum with variational method item by item, and then acquire the value of q(R); fix q(R) in Step M and calculate the parameter value that makes L(q, ) the maximum by using the property of exponential family conjugate distribution and the concave–convex procedure (CCCP). Steps E and M of the improved EM algorithm are shown here in below. Step E: Fix parameter value and solve the q(R) convenient calculation, decompose q(R) into  t τ t For q(r q = and substitute q(R) into L(q, ) to obtain ) (r ) τ iτ iτ τ =1 τ =1 i=1 L(q, ) =

 t  

=



···



···

 t  

rτ

rt







···



−

rτ t 

τ =1

 rt

···



q(r τ ) ⎡

q(r τ )⎣

τ =1

t 

 q(r τ ) ln p y1t , R|η0 , λ



lnq(r τ )

τ =1 t 

rk,k=τ k=τ



t    q(r τ ) ln p y1t , R|η0 , λ − lnq(r τ )

τ =1

rt t 



τ =1

r1

r1

=

t    t q(r τ ) ln p y1 , R|η0 , λ − lnq(r τ )

τ =1

R

=



⎤   t q(r k ) ln p y1 , R|η0 , λ ⎦

 

τ =1

{}



3.5 Empirical Bayesian Method for Determining Prior Distribution

−



q(r τ ) ln q(r τ ) −

 k=τ

rt

q(rk ) ln q(r k )

63

(3.33)

rk

  $  % Define the distribution of ln p˜ y1t , q(r τ )|η0 , λ = ln p y1t , R|η0 , λ −q(r τ ) + const. · ∼q(r τ ) represents solving the expectation of variables except q(r τ ), t %   $    q(r k ) ln p y1t , R|η0 , λ , and the items irrelln p y1t , R|η0 , λ −q(rτ ) = rk,k=τ k=τ

evant with q(r τ ) are regarded as the constant. Then Formula (3.33) can be expressed as    q(r τ ) ln p˜ y1t , q(r τ )|η0 , λ L(q, ) = rτ

−



q(r τ ) ln q(r τ ) + const

(3.34)

rτ

  Due to q(r τ ) ≥ p˜ y1t , q(rτ )|η0 , λ , it can be learnt that the maximum value of L(q, ) is obtained when the equality holds. Then there is % $  ln q(r τ ) = ln p y1t , R|η0 , λ ∼q(r τ ) + const

(3.35)

When exponent is taken for both sides and it is normalized, it is learnt that q(r τ ) can be solved according to the formula below % $  exp ln p y1t , R|η0 , λ −q(rτ ) $  t % q(r τ ) =  rτ exp ln p y1 , R|η 0 , λ −q(rτ )

(3.36)

In the changepoint model presented in this paper, as r τ is only related to r τ −1 and r τ −1 in changepoint mark, there is         t , y1t = p r τ |r τ +1 , r τ −1 , y1t p {r i }i=τ |y1t p y1t p {r i }i=1

(3.37)

% $  Substitute Formula (3.37) into ln p y1t , R ∼q(rτ ) .As we only care about q(r τ ) and add those excluding q(r τ ) to the fixed value at the end of the formula, there is $

    % % $   ln p y1t , R ∼q(rτ ) = ln p r τ |r τ +1 , r τ −1 , y1t p {r i }i=τ |y1t p y1t ∼q(rτ )    $  = ln p r τ |r τ +1 , r τ −1 , y1t + ln p {r i }i=τ |y1t  % + ln p y1t ∼q(rτ ) % $  = ln p r τ |r τ +1 , r τ −1 , y1t ∼q(rτ ) + const (3.38)

  In Formula (3.38), p r τ |r τ +1 , r τ −1 ,y1t can be rewritten in the following form

64

3 Residual Life Prediction Based on Wiener Process …

p



r τ |r τ +1 , r τ −1 , y1t



  p r τ , r τ +1 , r τ −1 , y1t   = p r τ +1 , r τ −1 , y1t       p r τ +1 , r τ −1 |r τ , y1t p r τ |y1t p y1t   = p r τ +1 , r τ −1 , y1t         p r τ −1 |r τ , y1t p r τ +1 |r τ , y1t p r τ |y1t p y1t   = p r τ +1 , r τ −1 , y1t (3.39)

Substitute Formula (3.39) into Formula (3.38). Similarly, as we only care about the variables related to r τ and consider placing the variables excluding r τ into the constant at the end of the formula for normalization, there is  %   $ ln p r τ −1 |r τ , y1t q(r τ −1 ) + ln p r τ |y1t $  t % % $  ln p y1 , R ∼q(r τ ) = + ln p r τ +1 |r τ , y1t q(r τ +1 ) + const ⎧   ⎫  ⎪ q(r τ −1 ) ln p r τ −1 |r τ , y1t + ln p r τ |y1t ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨r τ −1 =    ⎪ ⎪ ⎪ ⎪ + q(r τ +1 ) ln p r τ +1 |r τ , y1t + const ⎪ ⎪ ⎭ ⎩ r τ +1

(3.40) Substitute Formula (3.40) into Formula (3.36), we can learnt that the value of q(r τ ) is ⎧  ⎫  t ⎪ ⎪ |r q(r , y log p r ) ⎪ ⎪ τ +1 τ +1 τ 1 ⎪ ⎪ ⎨r ⎬   τ +1 t q(r τ ) ∝ p r τ |y1 exp    ⎪ ⎪ ⎪ + q(r τ −1 ) log p r τ −1 |r τ , y1t ⎪ ⎪ ⎪ ⎩ ⎭

(3.41)

r τ −1

(i j)

Previous calculation results ξτ,τ +1 and γτ(i) are required in calculation of Formula (i j) ( ji) (3.41). The intermediate variables πτ,τ +1 and χτ −1,τ are defined as follows: (i j) πτ,τ +1



= p rk,τ +1

 ( ji) χτ −1,τ = p ri,τ −1

)   ( jk) )  ξτ,τ +1 p r jτ = 1, rk,τ +1 = 1)y1t t ) t  = 1)r jτ = 1, y1 = = ( j) p r jτ = 1)y1 γτ ) t  (i j) )  ξτ −1,τ p r jτ = 1, ri,τ −1 = 1)y1 ) t  = 1)r jτ = 1, y1t = = ( j) p r jτ = 1)y1 γτ

It can be known from Formula (3.41) that:

3.5 Empirical Bayesian Method for Determining Prior Distribution   q r jτ = 1 =

65

+ *   τ −1  ( jk) ( ji) ( j) τ +1  k=1 q rk,τ +1 = 1 log πτ,τ +1 + i=1 q ri,τ −1 = 1 log χτ −1,τ γτ   τ +1    τ −1  τ ( j k) ( j i ) ( j ) q rk,τ +1 = 1 log πτ,τ +1 + i=1 q ri,τ −1 = 1 log χτ −1,τ γτ j =1 exp exp

k=1

(3.42) q(R) can be calculated successively by using Formula (3.42). Assume that the result of calculation by circulating Step E for the kth time is q (k) (R). Step M: Fix q (k) (R)to solve the parameter value that makes L(q, )the maximum. The parameter $   that )makes% L(q, ) the maximum also refers to the parameter that makes ln p y1t , R)η0 , λ q (k) (R) the maximum. Because: ) )     ln p y1t , R)η0 , λ = ln p( R|λ) + ln p y1t )η0 , R

(3.43)

)   ,   In Formula (3.43), P y1t )η0 , R and p R|1 λ , respectively, satisfy P



)

y1t )η0 ,



R =

t 

 exp −

i=1

-

+ ln Z

t 

  rti ln Z˜ (u(yt )) − ln Z η0

τ =i t 

.

rti u(yt ) + η0

(3.44)

τ =i

  t −1   , riτ ,    , rτ τ τ ri,τ −1 1 − 1 λ p R|1 λ = 1 λ 

τ =2

(3.45)

i=1

Further decompose Formulas (3.44) and (3.45), respectively, into Formulas (3.46) and (3.47)   t τ −1   ,    ,    ,  riτ ln 1 − 1 λ + riτ ln ri,τ −1 ln p R|1 λ = rτ τ ln 1 λ + 

=

τ =2

i=1

t 

t  τ −1 

 ,  rτ τ ln 1 λ +

τ =2

t  τ −1  ,   riτ ln 1 − 1 λ + riτ ln ri,τ −1

τ =2 i=1

τ =2 i=1

(3.46) ln p



)

yt1 )η,



R =− +

t  t 

t     ˜ rτ i ln Z u yτ − ln Z (η)

i=1 τ =1 - t t  

ln Z

i=1

  rτ i u yτ + η

.

i=1

(3.47)

τ =i

substitute Formulas (3.43), ) % $ Then  ln p y1t , R)η0 , λ q (k) (R) . There exists

(3.46)

and

(3.47)

into

66

3 Residual Life Prediction Based on Wiener Process … t t  τ −1  )  %  ,    ,  ln p y1t , R)η0 , λ q (k) (R) = qτ(k) ln 1 λ + qiτ(k) ln 1 − 1 λ τ

$

τ =2

+

τ =2 i=1

t  τ −1  τ =2 i=1

−

t t   i=1 τ =1

+

/ t 

t  $ %   riτ ln ri,τ −1 q (k) (R) − ln Z η0 i=1

˜ qτ(k) i ln Z (u(yτ )) -

ln Z

t 

.0 rτ i u(yτ ) + η0

τ =i

i=1

(3.48) q (k) (R)

It can be known from Jensen Inequality that / t 

ln Z

t 

.0 ≥

rτ i u(yτ ) + η

τ =i

i=1

t 

q (k) (R)

ln Z

t 

. qτ(k) i u(yτ )

+η

(3.49)

τ =i

i=1

The following can be obtained by substituting Formula (3.49) into Formula (3.48) ) %    ln p y1t , R)η0 , λ q (k) (R) ≥  η0 , λ

$

(3.50)

where t t  t  τ −1 τ −1  $ %    ,   ,   (k)  (k)  η0 , λ = riτ ln ri,τ −1 q (k) (R) qτ τ ln 1 λ + qiτ ln 1 − 1 λ + τ =2 t t  

−

i=1 τ =1

+

/ t  i=1

τ =2 i=1

τ =2 i=1

˜ qτ(k) i ln Z (u(yτ )) −

ln Z

- t 

t 

  ln Z η0

i=1

.0

qτ(k) i u(yτ )

+η

τ =i

(3.51) q (k) (R)

 ,  ,  , It can be known by taking ∂ η0 , λ ∂ 1 λ = 0 that the estimation of 1 λ is t t  τ   , (k+1)  (k) 1 λ = qτ τ / qiτ(k) τ =2

τ =2 i=1

When exponential family conjugate distribution is selected, it is convenient to       solve η(k) 0 with CCCP. First, take  η 0 =  η 0 +  η 0 as the variable relating to       η0 in  η0 , λ , where  η0 is the convex function of η0 , and  η0 is the concave function of η0 , and

3.5 Empirical Bayesian Method for Determining Prior Distribution

67

     η0 = −t ln Z η0

(3.53)

- t . t  (k)     η0 = ln Z qτ i u(yτ ) + η0

(3.54)





i=1

τ =i

   ,    η0 and  η0 , λ share the same stagnation point that satisfies ∂ η0 ∂η0 = 0, viz.     ∂  η0 /∂η0 = −∂  η0 /∂η0

(3.55)

Update the parameter η0 according to the rules presented in the formula below, till convergence.     ← −∇  η(k) ∇  η(k+1) 0 0

(3.56)

In exponential group distribution, make full use of the relation between Z (η) and sufficient statistic φ(θ), to make the calculation simpler and more convenient. Such relation will be given by deduction I. Collary In exponential family distribution, Z (η) bears the following relations with sufficient statistic φ(θ ): ∂ ln Z (η)/∂η = φ(θ ) p( θ |η)

(3.57)



Proof By substituting Z(η) = exp ηT φ(θ ) dθ into the left side of Formula (3.57), we can obtain



,

, , ∂ exp ηT φ(θ ) ∂ηdθ ∂ exp ηT φ(θ ) dθ ∂η





∂ ln Z (η) ∂η = = exp ηT φ(θ ) dθ exp ηT φ(θ) dθ





φ(θ ) exp ηT φ(θ ) dθ

= φ(θ ) exp ηT φ(θ) − ln Z (η) dθ = T exp ln exp η φ(θ ) dθ  = φ(θ ) p( θ |η)dθ  φ(θ ) p( θ |η)dθ = φ(θ ) p( θ|η) By substituting Formulas (3.53), (3.54) and (3.57) into (3.56), we can know the recursion calculation criterion of parameter η0 under the exponential family conjugate distribution is

68

3 Residual Life Prediction Based on Wiener Process …

1 φ(θ ) p θ|η(k) +i t i=1 t

φ(θ) p θ |η(k+1) ← 0

In Formula (3.58), there is η(k) +i = (k) v+i

=

v0(k)

+

t 

qτ i , m (k) +i

=

t i=1

v0(k) m (k) 0

τ =i

+

(3.58)

(k) qτ(k) i u(yτ )+η 0 , viz.

t 

(k) qτ i yτ /v0(k) , a+i

=

a0(k)

τ =i

1 + qτ i 2 τ =i

1 1 1 (k) (k)2 (k) B+i = B0(k) + v0(k) m (k)2 + qτ i yτ2 − v+i m +i 0 2 2 τ =i 2

t

t

(3.59)

In the exponential family conjugate distribution, φ(θ ) p( θ |η) also has a simple analytical form. For the degradation model selected in the paper, the calculation result of φ(θ ) p( θ |η) is 1

2

3

4

$

%

2 u2 u = 21 ν1 + mσ 2 , 2σ 2 p( θ|η) σ 2 p( θ|η) $ ) 1 )% $ % log) σ 2 ) p( θ|η) = (a) − ln|B|, σ12 p( θ|η)

=m

$

%

1 σ 2 p( θ|η) −1

= aB

(3.60)

where (a) refers to the derivative of Gamma function (a). It can be learnt by substituting Formulas (3.60) and (3.59) into Formula (3.58) that 1

2

u2 2σ 2 p( θ|η ) 0

t 1 

2

$

%

u2 u ,  2σ 2 p θ|η(k) σ 2 p( θ|η0 ) +i i=1 t $ ) )% $ ) 1 )% $ %  log) σ 2 ) p( θ|η0 ) = 1t log) σ12 ) p θ|η(k) , σ12 p( θ|η0 ) +i i=1

=

1 t

=

1 t

=

1 t

t $  i=1 t $  i=1

%

u  (k) σ 2 p θ|η+i

%

p( θ|η)

1  (k) σ 2 p θ|η+i

(3.61) The estimation of v0 ,m0 ,B0 ,a0 can be calculated according to Formulas (3.60) and (3.61). 5 6 1u 2 1 / 2 = 2 σ p(θ|η0 ) σ p(θ |η0 )  5 −1 6 5 6 u2 (k+1) (k+1) 1 (k+1) mˆ vˆ0 = 2 − mˆ 0 2σ 2 p(θ |η) σ 2 p(θ |η0 ) 0 5 6 5 ) )6   ) 1 ) 1 (k+1) (k+1) − ln aˆ 0 + ln 2  aˆ 0 − log)) 2 )) σ p(θ|η0 ) σ p(θ|η0 ) 5 6 1 B0(k+1) = aˆ 0(k+1) / 2 σ p(θ|η0 )

mˆ (k+1) 0

(3.62)

3.5 Empirical Bayesian Method for Determining Prior Distribution

69

In combination with the above analysis and reasoning, the EM algorithm for determining the prior distribution parameter of performance increment in degradation model based on historical information is given below. Algorithm 3.3 EM algorithm for determining the prior distribution parameter of performance increment in degradation model based on historical information Step 1 Step 2

Step 3 Step 4

Step 5

(0) (0) (0) Select v0(0) , m (0) and take k = 0; 0 , B0 , a0 , λ Calculate the variables α and β with algorithm 3.2 according to (k) (k) (k) v0(k) , m (k) 0 , B0 , a0 , λ , calculate the variables γ , ξ , π and χ with α and β, and then initialize q (k) (R) = γ ; Carry out Step E, calculate the value of q (k+1) (R) item by item by using ξ , π , χ and q (k) (R) according to Formula (3.42); Carry out Step M, keep q (k+1) (R) unchanged and then calculate , B0(k+1) , a0(k+1) , λ(k+1) with Formulas (3.59), (3.60), v0(k+1) , m (k+1) 0 (3.61) and (3.62); The algorithm will end if the iteration condition is satisfied; or else take k = k + 1 and carry out Step 2.

3.6 Residual Life Prediction Based on Bayesian Online Changepoint Detection Attributing to the good property of exponential family conjugate distribution, the Bayesian Online Changepoint Detection algorithm and the EM algorithm for determining the prior distribution parameter of performance increment in degradation models based on historical information are given in above chapters. This section will introduce the remaining life prediction method based on Bayesian Online Changepoint Detection. The basic frame of the remaining life prediction method based on Bayesian Online Changepoint Detection is first shown below. Basic frame of remaining life prediction method based on Bayesian Online Changepoint Detection Step 1 Solve the prior distribution of models with algorithm 3.3 based on historical data; Step 2 Acquire the latest equipment performance degradation data; Step 3 Detect whether the degraded performance of equipment at the latest observation point changes with the Bayesian Online Changepoint Detection algorithm and prior distribution parameters in Step 1 and

70

3 Residual Life Prediction Based on Wiener Process …

determine the latest degradation rule changepoint as of the detection ending time; Estimate parameters in Wiener process with the prior distribution obtained in Step 1 and equipment performance degradation data after the latest changepoint; Predict the remaining life distribution of equipment with the first passage time distribution data in Wiener process.

Step 4

Step 5

Refer to corresponding chapters above for Step 1–Step 3. Parameters in Wiener process in Step 1 can be estimated with the method below. After calculating and detecting the latest changepoint tC(t) with theBayesian Online

Changepoint Detection algorithm, the latest posterior distribution π u, σ 2 |y t(t) = tc  2 2 π u, σ |ηt (t) + of u, σ can be calculated based on the performance detection C

information y t(t) after tC(t) . So the posterior marginal distribution of u, σ 2 is tc

   π u|ηt (t) + = π u, σ 2 |ηt (t) + dσ 2 C C    π σ 2 |ηt (t) + = π u, σ 2 |ηtc(t) + du C

(3.63)

The expected posterior value of u, σ 2 can be calculated according to Formula (3.63), and by using it as the estimation of u, σ 2 , viz. 2 1  uˆ = π u|ηt(t) + C u 2 1  ) 2 2) (t) σˆ = π σ ηt + C

σ2

(3.64)

Substitute uˆ and σˆ 2 into Formula (3.64) into Formula (3.5) in Step 5 to calculate the remaining life distribution of equipment.

3.7 Case Study Gyroscope is a key component of inertial navigation system and also the core component deciding the performance of inertial platform. It is widely applied in missile weapon system and features long storage period, non-reuse, failure upon degradation and periodic detection. It fails always due to degradation in the actual application. To ensure its service performance, the gyroscope is always inspected regularly (monthly) after delivery. Drift coefficient is an important parameter reflecting the performance of gyroscope, so the 72 groups of monthly performance inspection data about linear

3.7 Case Study

71

0.37

Linear term drift coefficient /°/(hg0)

0.4 0.29 0.25 0.21 0.17 0.13 0.09 0.05

1

10

19

28

37

46

55

64

72

Detection time / month

Fig. 3.4 Monthly detection data for linear term drift coefficient of X-axis gyroscope on certain inertial platform

term drift coefficient of X-axis gyroscope on certain inertial platform as shown in Fig. 3.4 are selected as the object for modeling for the degradation process of gyroscope with the method proposed in this paper. First, determine the prior distribution of parameters u and σ 2 in performance incremental distribution with the first 64 groups of data by EM algorithm, next, ascertain the latest changepoint tC(t) as of the sampling time of the rest eight groups of data with Bayesian Online Changepoint Detection algorithm, determine the posterior distribution and Bayesian estimations uˆ and σˆ 2 of parameters u and σ 2 with the data after the latest changepoint tC(t) and calculate the first passage time distribution of failure threshold in Wiener process with uˆ and σˆ 2 and take it as the remaining life distribution. Figure 3.5 shows the relationship between the estimation of main parameters and iteration steps in EM algorithm. According to the figure, the estimation of prior parameters shows convergence in less iteration steps, proving that prior distribution parameters of u and σ 2 and parameters of hazard rate function can be effectively obtained by EM algorithm. With EM algorithm, the prior distribution parameters of u and σ 2 are estimated as v0 = 0.0523, m 0 = 0.0044, B0 = 0.0015 and a0 = 17.7865, and the parameter of hazard rate function is λ = 7.1981. Substitute parameters calculated by EM algorithm into Bayesian Online Changepoint Detection algorithm to detect the changepoint for performance degradation increment of gyroscope, viz. the changepoint of degradation rule as shown in Fig. 3.6; the grayscale at lower section of Fig. 3.6 indicates that the duration of performance degradation rule is related to the probability at the corresponding ordinates, and the deeper the color is, the larger the probability will be. According to Fig. 3.6, the performance degradation rule of

72

3 Residual Life Prediction Based on Wiener Process … -3

m

0

5

x 10

4.5 4

20

0

60

40

80

100

120

140

160

180

200

-3

B

0

2 1 0

α0

x 10

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

20 10 0

γ0

0.15

0.1

Number of iterations

Fig. 3.5 Main estimated results of parameters by EM algorithm

0.02

K

sx1

0.04

0

-0.02

10

30

20

40

50

60

70

50

60

70

Performance detection time /month

Duration / month

60 50 40 30 20 10 10

20

30

40

Performance detection time /month

Fig. 3.6 Changepoint during performance degradation of gyroscope

3.7 Case Study

73

Table 3.1 Estimated parameters with changepoint considered Detection time   uˆ ×10−3   σˆ 2 ×10−5

65

66

67

68

69

70

71

72

3.336

3.332

3.616

3.248

2.954

3.162

3.570

3.403

7.5003

7.4566

7.4831

7.4632

7.3972

7.5011

7.4229

7.5003

equipment changes significantly at the 26th, the 36th and the 40th detection points, which is consistent visually with the performance degradation curve in Fig. 3.4. Table 3.1 shows the Bayesian estimation of all parameters at different detection times with changepoint considered by using the method in this paper. When no changepoint is considered, the estimations uˆ = 4.326×10−3 and σˆ 2 = 1.7152×10−5 are obtained with the maximum likelihood estimation (MLE) in Reference [22]. It is obvious by comparing all parameters in the two conditions that when the changepoint during performance degradation is considered, parameters in Wiener process can reflect more accurately the present degradation rule of equipment, and the residual life of equipment predicted with the parameter is more rational. Calculate the residual life distribution of gyroscope with the model parameters with and without changepoint considered, respectively, and the results are shown in Fig. 3.7. According to the figure, the mean residual life of gyroscope with changepoint considered is significantly greater than that without changepoint considered. It can be seen from Fig. 3.4 that the slope of performance degradation curve reduces after the 38th performance detection point, and the estimation uˆ of performance degradation parameters with changepoint considered is less than that without changepoint considered in Table 4.1; it shows that the performance degradation of gyroscope decelerates and the mean residual life time of equipment increases. Moreover, it is 0.05 Remaining life distribution with changepoint considered Expected remaining life with changepoint considered

0.04

Residual life distribution without changepoint considered Expected remaining life without changepoint considered

0.03 Probability density 0.02 0.01 0 72

200 71

150 70 69

100 68

Performance inspection time /month

67

50 66

Fig. 3.7 Residual life distribution of gyroscope

65

0

Residual life / month

74

3 Residual Life Prediction Based on Wiener Process …

obvious that, after considering the changepoint, the variance σˆ 2 of degradation increment gets larger which is because that the performance stability of gyroscope reduces along with the gyroscope’s service time. Also, it can be learnt from Fig. 3.7 that, the variance for residual life prediction is greater slightly than that without changepoint considered.

3.8 Summary of This Chapter This chapter studies the equipment degradation modeling method based on Bayesian Online Changepoint Detection algorithm. It first estimates the prior distribution parameter of characteristics in degradation process with the improved forward–backward algorithm and EM algorithm based on the equipment performance degradation increment, and after obtaining the latest observed value, acquires the latest changepoint of performance degradation characteristics with Bayesian Online Changepoint Detection algorithm and calculates the posterior distribution and Bayesian estimation of the characteristics with the performance increment detection data after changepoint; finally, it substitutes the estimated characteristics into degradation models, viz. Wiener process, to predict the residual life of equipment based on the first passage time distribution of failure threshold. It is proved after its application in performance detection data during service of gyroscope that, the method can not only detect effectively the changepoint of degradation rule in equipment degradation process, but also overcome the impact of historical data on predicted results, and it is of certain actual application value.

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10. Hu C, Si X, Maoyin C, Wang W (2011) An Adaptive Wiener-maximum-process-based model for remaining useful life estimation. In: Prognostics and system health management conference, Shenzhen, China 11. Hu C, Si X, Chen M et al (2011) An adaptive and nonlinear drift-based wiener process for remaining useful life estimation. In: Prognostics and system health management conference, Shenzhen, China 12. Wang Z, Changhua Hu, Wang W et al (2014) Wiener process-based online prediction method of remaining useful life for draught fans in steel mills. J Univ Sci Technol Beijing 36(10):1361– 1368 13. Robbins M (2011) Change-point analysis: asymptotic theory and applications. ProQuest, UMI Dissertation Publishing 14. Chen J, Wang Y (2009) A statistical change point model approach for the detection of DNA copy number variations in array CGH data. IEEE/ACM Trans Comput Biol Bio-Inform 6(4):529–541 15. Dong H, Gee T (2005) Comparison of the CUSCORE, GLRT and CUSUM control charts for detecting a dynamic mean change. Inst Stat Math 7(3):531–552 16. Doksum KA, Hoyland A (1992) Models for variable-stress accelerated life testing experiments based on wiener processes and the inverse Gaussian distribution. Technimetrics 34:74–82 17. Guo L, Chen J, Zhao F (2008) Application of SVM based geometric distance method in equipment performance degradation assessment. J Shanghai Jiaotong Univ (Chin Ed) 12(7):1077–1080 18. Chen L, Changhua Hu (2010) Degradation reliability analysis based on phase-type approximation. J Chin Inertial Technol 18(2):255–260 19. Adams RP, MacKay DJC (2007) Bayesian online change point detection. Technical report, University of Cambridge, http://www.inference.phy.cam.ac.uk/rpa23/papers/rpachangepoint. pdf 20. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via EM algorithm. J Roy Stat Soc 39(1):1–38 21. Scarf PA (2007) A framework for condition monitoring and condition based maintenance. Qual Technol Quant Manage 4(2):301–312 22. Feng J, Zhou J (2008) Life prediction for circulation Pump of manned spacecraft based on wiener process. Chin Space Sci Technol 4:53–58

Chapter 4

Gamma Process-Based Degradation Modeling and Residual Life Prediction

4.1 Introduction Degradation is a stochastic physical or chemical process in which the internal material of equipment changes asymptotically under external stress. The performance degradation modeling method based on Brownian motion is discussed in sections above, but the performance degradation process shown with the model is not a strict and monotonous process. In engineering practice, the performance degradation amount of equipment presents always a monotonically increasing trend. Gamma process can describe well the change rule of data on monotonic characteristic degradation process, such as metal wear and crack growth [1–3]. In 1975, Abdel-Hameed put forward first and developed modeling for continuous and monotonic degradation data with Gamma process [4], and in 2009, Noortwijk summarized the study and application of Gamma process in life prediction field in recent years [5]. Zhang et al. developed detailed study on the reliability of fluid floated gyroscope with Gamma process [6]. In this chapter, we will study the equipment performance degradation path modeling and residual life prediction method based on Gamma process as well as the optimal equipment maintenance decision method under external environmental influence.

4.2 Definition of Gamma Process Gamma process is a stochastic process with independent and non-negative increment that follows the Gamma distribution with identical scale parameters. Assuming that X is a random variable following Gamma distribution, its probability density function is [7].

© National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_4

77

78

4 Gamma Process-Based Degradation … 1 Shape parameter is 1

0.9

Shape parameter is 2 Shape parameter is 5

0.8

Shape parameter is 0.2

0.7

f(x)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

Gamma Distribution

Fig. 4.1 Schematic diagram for Gamma distribution of different shape parameters of unit scale parameters

Ga(x|α, β) =

β α α−1 x exp{−βx}I(0,∞) (x) (α)

(4.1)

where α > 0 and is a shape parameter, β > 0 and is a scale parameter; when x ∈ A, ∞ I A (x) = 1; if x ∈ / A, I A (x) = 0; (a) = 0 t a−1 e−t dt is the Gamma function when a > 0. Moreover, when α(t) is a non-decreasing and right continuous real-valued function, t ≥ 0, α(0) ≡ 0. Figure 4.1 shows the Gamma distribution of different shape parameters of unit scale parameters. The Gamma process with the shape parameter α (t) > 0 and the scale parameter β > 0 is a continuous-time stochastic process {X(t), t ≥ 0}. It features as follows [5, 7]: (1) (2) (3)

X(0) = 0; X (t) = X (t + t) − X (t) ∼ Ga(α(t + t) − α(t), β) for t ≥ 0 and t > 0; When n ≥ 1 and 0 ≤ t 0 < t 1 < … < t n < ∞ in any cases, the random variables X (0), X (t1 ) − X (t0 ), …, X (tn ) − X (tn−1 ) are independent mutually.

In short, Gamma process is a continuous-time stochastic process with nonnegative and independent increment that is identical to Gamma distribution. When the shape parameter α(t) is the linear function of t, the stochastic process is a steady Gamma process. According to the definition of Gamma process, the probability density function of X(t) is f X (t) (x) = Ga(x|α(t), β )

(4.2)

4.2 Definition of Gamma Process

79

The mean and variance are E(X (t)) =

α(t) β

Var(X (t)) =

α(t) β2

(4.3) (4.4)

It can be learnt from dtd E(X (t)) = α  (t)/β that, the derivative of shape parameter α(t) reflects the mean degradation rate of equipment at t and reveals the relationship between equipment degradation degree and system age [8]. Empirical studies show that the expected degradation level of equipment at t can always be described with the expression below [5]. E(X (t)) =

ct b α(t) = ∝ tb β β

(4.5)

Moreover, the parameter b in Formula (4.5) can be determined based on engineering experience for specific study objects; for example, in the event of concrete degradation due to reinforcement corrosion, the parameter b is always 1 [9], in the study of the depth of scour pit, the parameter b is always set as 0.4 [10, 11].

4.3 Parameter Estimation for Gamma Process When neglecting the impact of covariate, the n records (x ij , t ij ), i = 0,1,…,n; j = 0,1,…, mi of sample track can be combined directly for steady Gamma n process and ti,m i . At this are thought from the same sample track, and the total time is i=1 moment, the shape parameter α and the scale parameter β can be estimated based on the data given with method of moments and MLE. When estimating parameters of non-steady Gamma process, it is necessary to know in advance the change rule of shape parameters along with time. In performance degradation modeling field, the expected degradation level of non-steady Gamma can always be described with Formula (4.5). With such a description method, the nonsteady Gamma process can be transferred into steady Gamma process via appropriate conversion. In detail, the Formulas (4.3) and (4.4) can be transferred into below after selecting z(t) = t b E(X (t)) = Var(X (t)) =

cz β

(4.6)

c β2

(4.7)

80

4 Gamma Process-Based Degradation …

After the conversion, the non-steady Gamma process about t is transferred into the steady process about transfer time z(t). At this moment, the unknown parameters c and β can be estimated directly with the parameter estimation method for steady Gamma process when the parameter b is known [5]. Therefore, the method of moments and MLE frequently used in parameter estimation method for steady Gamma process are highlighted in this section.

4.3.1 Method of Moments The basic idea of method of moments lies in that matching the corresponding statistics with the moment of a random variable. For example, when the independent observation of n Gaussian random variables (with the mean of μ and the variance of σ 2 ) is u 1 , · · · , u n , the mean and variance of samples are n n 2 1  1 u i , SU2 = u i − U¯ U¯ = n i=1 n − 1 i=1

(4.8)

Since E[U¯ ] = μ, E[SU2 ] = σ 2 , the estimation of μ and σ 2 is U¯ and SU2 , respectively. For steady Gamma process, assuming xi = xi − xi−1 , ti = ti − ti−1 , i = 1, 2, · · · . Define the degradation rate Ri with xi /ti which are mutually independent and follow the Gamma distribution, so, the same as the Gaussian process above, we can calculate the sample mean and sample variance of degradation rate as 2 n n  1  xi 1  xi 2 ¯ ¯ ,S = −R R= n i=1 ti R n − 1 i=1 ti  

Clearly, E R¯ = α/β, Var R¯ =

1 α n2 β 2

n i=1



1 ti

(4.9)

. Therefore,

 2 n X i 1  E − R¯ n − 1 i=1 ti   

 n 1  X i α α 2 ¯ − R− = E − n − 1 i=1 ti β β

  

n 

X i 2 X i 1  Var − Var + Var R¯ = n − 1 i=1 ti n ti   n 1 α  1 = n β 2 i=1 ti



E S R2 =

(4.10)

4.3 Parameter Estimation for Gamma Process

81

According to the sample moment and expected value, there is α = R¯ β

(4.11)

α nS R2 = n 2 β i=1 (1/ti )

(4.12)

Moreover, we can get the estimation of parameters below after solving Formulas (4.11) and (4.12) R¯ 2

αˆ =

R¯

βˆ =

n

(1/ti ) nS R2

(4.13)

(1/ti ) nS R2

(4.14)

i=1

n

i=1

Cinlar et al. put forward another strategy of moment method [12], and its moment for calculating increment is n xi xn ¯ = Y = i=1 n tn i=1 ti SY2 =

(4.15)

n   2 xi − Y¯ ti

(4.16)

i=1

Because 

n

xi E[X (tn )] α = (4.17) = E Y¯ = E i=1 n tn β i=1 ti  n   n     2    2 α α E X i − ti − Y¯ − ti X i − Y¯ ti =E β β i=1 i=1 =

n  i=1

Var(X i ) − 

α = 2 tn 1 − β

n  i=1

  ti2 Var Y¯

 n   ti 2 i=1

tn

 (4.18)

The results below can be obtained with the expected value by matching corresponding sample moments xn α = Y¯ = β tn

(4.19)

82

4 Gamma Process-Based Degradation …

α SY2  = n   2  β2 tn 1 − i=1 ti tn

(4.20)

Thus, we can get the estimation of parameters α and β as αˆ =

βˆ =

 n   2  xn2 1 − i=1 ti tn

(4.21)

tn SY2

 n   2  xn 1 − i=1 ti tn

(4.22)

SY2

It is clear that when ti , i = 1, · · · , n, and n is equal, the same estimation is got with the two strategies. If the parameter b is unknown, it is also necessary to estimate it based on data for non-steady Gamma process. In detail, take logarithm on both ends of Formula (4.5) to get log E[X (t)] = b log t + log(c/β)

(4.23)

If log t is the x-axis and log E[X (t)] is the y-axis, a straight line with the slope of b is got with Formula (4.23). So according to the least square method, there is: n bˆ =

n 

(log ti )(log xi ) − (

i=1

n

n 

n 

log ti )(

i=1 n 

(log ti )2 − (

i=1

n 

i=1

log xi ) (4.24)

log ti )2

i=1

After getting the estimation bˆ of parameter b, take z = bˆ t to get the steady Gamma process after transfer, and then estimate the parameter with the method of moments above.

4.3.2 Maximum Likelihood Estimate For steady Gamma process {X(t), t ≥ 0} with the shape parameter of α and the scale parameter of β, the likelihood function of α and β can be established easily based on the increment (x i , t i ), i = 1,…,n got. It is a steady Gamma process, so the shape parameter α(t) can be expressed as αt, viz. α(t) = αt. We can get according to definitions of Gamma process that

4.3 Parameter Estimation for Gamma Process

X i ∼ Ga(αti , β) =

83

β αti (xi )αti −1 −xi β e (αti )

(4.25)

The log-likelihood function is l(α, β) =

n 

(αti − 1) log xi + αtn log β −

i=1

n 

log (αti ) − xn β

(4.26)

i=1

Take partial derivatives for α and β, respectively, of l(α, β), and the MLE Formula is below:  ∂l ti (log xi − ψ(αti ) + log β) = 0 = ∂α i=1

(4.27)

∂l αtn = − xn = 0 ∂β β

(4.28)

n

where ψ(u) is the derivative about u of logarithmic Gamma function log (u), viz. ψ(u) =

∂ log (u)   (u) = ∂u (u)

(4.29)

When u > 0, the Formula (4.29) is called digamma function. According to Formula (4.28), there is β=

αtn xn

(4.30)

Then, substitute Formula (4.30) into Formula (4.27) to get n 

αtn  − ti ψ(αti ) = 0 xn i=1 n

ti log xi +tn log

i=1

(4.31)

It is easy to know that Formula (4.31) is a transcendental equation and is always required to be solved with numerical method. For non-steady Gamma process {X(t), t ≥ 0} with the shape parameter of α = ct b and the scale parameter of β, the likelihood function of b, c and β can also be easily established with the same method above: (x1 , · · · , xn |c, β, b ) =

n  i=1

b b β c[ti −ti−1 ] c t b −t b −1 b

xi [ i i−1 ] exp{−βxi } b  c(ti − ti−1 ) (4.32)

84

4 Gamma Process-Based Degradation …

Take logarithm on both ends of Formula (4.32), then take partial derivative for variables b, c and β, respectively, and make respective results of partial derivatives as 0, viz. ∂ log (x1 , · · · , xn |c, β, b ) = 0 ∂β

(4.33)

∂ log (x1 , · · · , xn |c, β, b ) ∂c n   b





 b b = ti − ti−1 log(β) − ψ c tib − ti−1 + log xi i=1

= tnb log(β) +

n  b



  b b ti − ti−1 log xi − ψ c tib − ti−1 =0

(4.34)

i=1

∂ log (x1 , · · · , xn |c, β, b ) ∂b n   b−1





 b−1 b = bc ti − ti−1 log(β) − ψ c tib − ti−1 + log xi i=1

= tnb−1 log(β) +

n   b−1



 b−1 b ti − ti−1 log xi − ψ c tib − ti−1 =0

(4.35)

i=1

It can be known after further solution that the estimation of all parameters satisfies the Formula below βˆ =

ˆ

ct ˆ nb xn

n         ˆ ˆ ˆ bˆ bˆ tnb log βˆ = tib − ti−1 ψ cˆ tib − ti−1 − log xi

(4.36)

(4.37)

i=1 n         ˆ ˆ ˆ ˆ b−1 b tnb−1 log βˆ = tib−1 − ti−1 ψ c tib − ti−1 − log xi

(4.38)

i=1

Formulas (4.36–4.38) can be solved with numerical method [13] to obtain the estimation of three unknown parameters, and we will not go into detail here.

4.4 Residual Life Prediction Based on Gamma Process

85

4.4 Residual Life Prediction Based on Gamma Process 4.4.1 Life Distribution Performance failure (substandard equipment performance) of lots of equipment is first passage failure. In other words, once the failure threshold ξ is achieved in equipment degradation process X(t), the equipment can be thought failed. The failure time is defined as the time when the sample degradation path of X(t) exceeds ξ for the first time, while the life is the length of time from the commissioning of equipment to reaching failure threshold for the first time [14]. Therefore, we can get the formula below T ≡ inf{X (t) ≥ ζ } = {t|X (t) ≥ ζ, X (s) < ζ, 0 ≤ s < t }

(4.39)

According to the monotonic property of the sample path, if X(t) is a Gamma process, then the life distribution is calculated as FT (t) = Pr(T ≤ t) = Pr(X (t) ≥ ζ ) =

γ [α(t), ζβ] [α(t)]

(4.40)

where γ (a, x) means the upper incomplete Gamma function and is defined as ∞ γ (a, x) =

t a−1 e−t dt

(4.41)

x

The first and the second moment of life is expressed as ∞ E[T ] =

(1 − FT (t))dt

(4.42)

0



E T

2



∞ =2

t(1 − FT (t))dt

(4.43)

0

Formulas (4.42) and solved with numerical method [15]. In   (4.43) are always detail, define qn = Pr tn < Tξ ≤ tn+1 , take tn = n/α, n = 0, 1, 2, · · · and perform discrete-time approximation for Formula (4.40), and there is qn =

γ [n + 1, ζβ] γ [n, ζβ] (ζβ)n −ζβ − = e [n + 1] [n] n!

(4.44)

86

4 Gamma Process-Based Degradation …

Gamma process Poisson approximation

Probability density function First passage time

Fig. 4.2 Probability density function for life distribution based on Gamma process and its Poisson approximation

It is a Poisson distribution. In other words, the probability of equipment failure within time interval (n/α, (n + 1)/(n + 1)] is the Poisson distribution with parameter of ζ /β [15]. According to characteristics of Poisson distribution, the mean and variance of life are equal to ζβ/α and ζβ/α 2 . Figure 4.2 shows the probability density function of the Gamma process with the shape parameter of 1, and the scale parameter and failure threshold of 10 at the first passage time as well as its Poisson approximation. The true value of mean and variance is 10.495 and 10.022, respectively, while the both are 10 based on Poisson approximation.

4.4.2 Residual Life Distribution When equipment has operated till to current time tc , its residual life distribution can generally be calculated based on its life distribution function, viz. FT (t|tc ) = Pr(T ≤ t|T > tc ) =

FT (t) − FT (tc ) 1 − FT (tc )

(4.45)

However, for the performance degradation process descried with Gamma process, the distribution function at the time reaching failure threshold ξ for the first time in degradation process can also be established based on the performance degradation value x c = X(t c ) < ξ at t c and independent increment characteristics of Gamma process.

4.4 Residual Life Prediction Based on Gamma Process

87

FT (t|tc ) = Pr(T ≤ t|X (tc ) = xc ) = Pr(X (t) − X (tc ) ≥ ζ − xc ) γ [α(t − tc ), (ζ − xc )β] = [α(t − tc )]

(4.46)

It shares the same type with Formula (4.40), but it substitutes ξ and t with ξ -x c and t- t c respectively. The residual life distribution calculated with Formula (4.45) uses only the equipment’s hours worked, while the equipment residual life distribution calculated with Formula (4.46) uses comprehensively the equipment degradation value at the current moment and hours worked, and thus can better reflect the change of equipment health condition. Figure 4.3 shows the residual life distribution of equipment in three different circumstances after working for 10 unit time, where the shape parameter and scale parameter of Gamma process are 1, and the failure threshold is 10. According to the figure, in case of smaller current performance degradation value, the failure probability of equipment will be less than that of equipment with larger performance degradation value in the same period in the future. The equipment residual life distribution calculated only with hours worked is not affected by current performance degradation level.

Cumulative distribution function FT(t/tc) Residual life (t)

Fig. 4.3 Equipment residual life distribution function based on Gamma process in different circumstances

88

4 Gamma Process-Based Degradation …

4.4.3 Reliability Function It is assumed that T is the life of equipment, the reliability function at t is below for monotonic increasing degradation process: R(t) = P(T > t) = P{X (t) < ζ }

(4.47)

For the Gamma process with the shape parameter of α and the scale parameter of β, the reliability function can be expressed as R(t) = P(T ≥ t) = P{X (t) ≤ ζ } ζ = 0

= where γl (a, x) = function.

x 0

β α α−1 −βx x e dx (α)

γl (α, ζβ) (α)

(4.48)

t a−1 e−t dt, x ≥ 0, a ≥ 0 is called the lower incomplete Gamma

4.4.4 Example Verification In this section, the theoretical result is verified with the fatigue crack length data of A2017-T4 type aluminum alloy. A2017-T4 type aluminum alloy is widely applied in aerospace and military field. Its quality is generally assessed with its fatigue length. When the crack length equals to or is 6 mm more than a preset threshold, the aluminum alloy can be considered failure. Reference [16] shows the fatigue crack data of four groups of A2017-T4 type aluminum alloy samples under the pressure of 200 MPa. Each of the samples is tested for 10 times at a time interval of 105 period of rotation, with 10 crack length data obtained. The first group of data is hereby used only among the four groups of data for verification. It is shown in Fig. 4.4. The non-steady Gamma process with the shape parameter of α = ct b and the scale parameter of β is used for modeling for the degradation path shown in Fig. 4.4. In consideration of the less samples, it is inadvisable to estimate unknown parameters with method of moments. Therefore, the parameters b, c and β are estimated with MLE in this section, with the results shown in Table 4.1. Then, the residual life probability density function and mean residual life of this group of samples are estimated with the parameter estimation obtained, as shown in Fig. 4.5. The figure shows the residual life probability density function at each detection time and also the actual residual life and the predicted mean residual life at the corresponding detection time. We may find after comparison that the predicted

4.4 Residual Life Prediction Based on Gamma Process

89

Fatigue crack length /mm Operation time/ (105 cycle)

Fig. 4.4 Fatigue crack growth track of A2017-T4 type aluminum alloy Table. 4.1 Estimated parameters in gamma process

Parameters

bˆ

Cˆ

βˆ

Estimation

5.85

0.071

1.9831

Residual life probability density function

Estimated mean remaining life

Detection time/ (105 cycle)

Actual remaining life

Residual life/ (105 cycle)

Fig. 4.5 Curve of residual life distribution probability density function at different detection times

90

4 Gamma Process-Based Degradation …

residual life mean is very close to the actual residual life, verifying the effectiveness of the performance degradation modeling and residual life prediction method based on Gamma process.

4.5 Degradation Modeling Based on Gamma Process with Environmental Impact In the earlier part of this chapter, the residual life prediction method for describing the equipment performance degradation process with Gamma process and the corresponding parameter estimation method are mainly discussed, with no impact from external environment on the performance degradation process taken into account. Actually, the health state of equipment is usually closely associated with its operation environment, and the environment usually changes randomly and dynamically. For example, troops will always transport missiles to launch sites thousands miles away by vehicles during actual training. In the process, bumping of vehicles and changes in ambient temperature will exert adverse impact on the health state of inertial platform in missiles to further affect the combat effectiveness of weaponry. All of those factors will increase to different extents the difficulty in rationally making maintenance plan for such kind of complex equipment. Therefore, this chapter focuses on the study of equipment performance degradation modeling based on Gamma process under external environmental impact and also the optimal maintenance policy.

4.5.1 Problem Description In this subsection, the performance degradation modeling for equipment with single component under external random environmental impact and also the optimal maintenance problem are mainly discussed. Equipment performance will degrade usually during operation due to its own reason when no impact from external factors is taken into account, and when the performance degradation value is beyond the given threshold ξ, the equipment is considered failure. Similarly, the Gamma process with the shape parameter of α and the scale parameter of β is used to describe the equipment performance degradation process {X(t), t ≥ 0}; where α > 0, β > 0, and X(t) is the performance degradation amount of equipment at t. It is assumed furthermore that the equipment will not degrade during storage, and its performance degradation amount is X(0) = 0 when the equipment is just put into operation. Besides being affected by the internal degradation process, the equipment is affected by external impact process. Marking the times of external impact for equipment within [0, t] as N(t) and assuming that {N(t), t ≥ 0} is a non-homogeneous Poisson process, the intensity function is λ(t). Each impact externally will speed

4.5 Degradation Modeling Based on Gamma Process with Environmental …

91

up the performance degradation process of components, to be specific, increasing the equipment performance degradation amount to certain extent. For easy study, it is assumed that the damage amplitude caused by impact is unrelated to the impact times and is a random variable following Gamma distribution. Mark the increase amplitude of degradation amount caused by impact as Y, and there is Y ~ Ga(α Y , β Y ).

4.5.2 Residual Life Distribution This subsection introduces first the accurate formulate for calculating the residual life distribution by mathematical derivation and then performs simulation calculation for the residual life distribution via numerical simulation. In the event of influence from external impact, the performance degradation amount of equipment at t is Z (t) = X (t) + Y (t)

(4.49)

where Y (t) means the increment of components for performance degradation process within the period [0, t] due to external impact, with the specific expression of Y (t) = N(t)Y. As described above, N(t) refers to the times of external impact for components within [0, t], and {N(t), t ≥ 0} is a non-homogeneous Poisson process, with the intensity function of λ(t). Y refers to the increase amplitude of degradation amount caused by impact, which is denoted as Y ~ Ga(α Y , β Y ). If the degradation value za at age a is observed for the degradation process with a monotonically increasing degradation performance, the reliability at time a + t is obtained with the Formula (4.47), as shown below: R(t|a, z a ) = Pr{Tr ≥ t|a, z a } = Pr{Z (a + t) ≤ D|a, z a } = Pr{X (a, a + t) + Y (a + t) − Y (a) + z a ≤ D} = Pr{X (a, a + t) + (N (a + t) − N (a))Y ≤ D − z a } =

∞ 

Pr{X (a, a + t) + kY ≤ D − z a } × Pr{N (a + t) − N (a) = k}

k=0

=

∞  k=0

 a+t R(a, t, k, z th )

a

λ(s)ds

k

  a+t exp − a λ(s)ds k!

(4.50)

where Tr means the residual useful life of components; k ∈ N. According to Formula (4.48), there is

92

4 Gamma Process-Based Degradation …

R(a, t, k, z a ) = Pr{X (a, a + t) + kY ≤ D − z a }   = Pr X (a, a + t) + Y (k) ≤ D − z a ∞     = Pr X (a, a + t) + Y (k) ≤ D − z a |Y (k) = y (k) f Y (k) y (k) dy (k) 0

∞ =

    Pr X (a, a + t) ≤ D − z a − y (k) |Y (k) = y (k) f Y (k) y (k) dy (k)

0

∞

(k) D−z a −y

= 0

0

  β αt αt−1 x exp(−βx)dx f Y (k) y (k) dy (k) (αt)

The probability density function of Y (k) is hereby given first. Since Y (k) = kY, and Y follows the Gamma distribution with parameters of α Y and β Y , so Y (k) ~ Ga (kα Y , β Y ), namely the probability density function of Y (k) is [17] f Y (k) (y (k) ) =

  βYkα kαY −1 y (k) exp −βY y (k) (kαY )

(4.51)

Finally, substitute Formula (4.51) into Formula (4.43) and substitute the results obtained into Formula (4.50) to get the reliability of equipment at a +t under external impact. In consideration of the complex calculation with Formula (4.50), Monte Carlo simulation method is used hereby for solution. Generate a sample track of nonhomogeneous Poisson process according to Reference [18] to get the internal and external impact times k of [a, a + t], then generate in simulation M 1 samples following Ga(kα Y , β Y ) distribution and substitute them into Formula (4.51) to calculate the probability of no failure for equipment within [a, a + t] under the sample track, repeat the above process for M times to get the reliability of equipment at a + t. Figure 4.6a shows the performance degradation sample track of equipment, respectively, with and without external impact when it is just put into operation (viz. a = 0, z a = 0), where the increment of performance degradation process caused by external impact is shown in Fig. 4.6b. Parameters of degradation process {X (t), t ≥ 0} are α = 1, β = 1, and the failure threshold ξ is 30. The intensity function of external impact process is λ(t) = αs βs (αs t)βs −1 , where αs = 0.2, βs = 1.2. Parameters about the increment of performance degradation level caused by external impact are αY = 0.5, βY = 1.2. The intensity function of homogeneous Poisson process used to generate the non-homogeneous Poisson process is λ = 1, T = 50. Figure 4.7 shows the life distribution curve of equipment with/without external environmental impact calculated by Monte Carlo simulation; the simulation times N = 1000 and other parameters are the same as above. According to the figure, the failure probability with external environmental impact is greater than that without external environmental impact at the same time.

4.5 Degradation Modeling Based on Gamma Process with Environmental …

93

No impact With impact

erform ance degra dation level

Time t (a) Equipment performance degradation curve with/without impact

ncrem ent of degra dation level

Time t (b) Degradation increment caused by impact

Fig. 4.6 Influence of external impact on equipment performance degradation process 1 Failure probability withl impact Failure probability without impact

0.9

0.8

0.7

Failure probability

0.6

0.5

0.4

0.3

0.2

0.1

0

0

10

20

30

40

50

Time t

Fig. 4.7 Life distribution of components with/without external impact

60

70

94

4 Gamma Process-Based Degradation …

4.5.3 Maintenance Decision In order to understand the degradation state of equipment accurately and help the management personnel to make reasonable maintenance decisions, monitor equipment performance variables at the interval of h(h > 0) and judge whether to keep the equipment running until time tk+1 , or to immediately implement the preventive replacement for the equipment at a cost of c p based on the monitoring results of the sensor at time tk , k = 0, 1, 2, · · · . In the event that the equipment fails from time tk to time tk+1 , the failure replacement will be implemented at a cost of c f . Extra losses may be caused due to equipment failure; therefore, c f > c p > 0. In addition, it is assumed that the time and cost required for equipment monitoring and the time spent in equipment replacement are negligible, and the performance degradation process of components will not be affected during equipment monitoring. To determine the optimal replacement time of equipment, it is required to obtain the expected average cost (also called the expected cost rate) model of equipment after long-time operation. Since the equipment replacement time is negligible, the replacement of the whole equipment means an equipment renewal. According to the renewal process theory [19], the expected cost rate is calculated as follows: expected cost rate =

Expected maintence cost of a renewal cycle Expected length of a renewal cycle

(4.52)

Of which, the interval between consecutive replacement is called a renewal cycle. The numerator and denominator of Formula (4.52) are calculated, respectively, as follows. It is assumed that the preventive replacement will be implemented for equipment after tr unit time from the current time a, and equipment will be replaced immediately when it fails before the preventive replacement, the expected length of the equipment renewal cycle is tr Tr = a +

R(t|a, z a )dt

(4.53)

0

Expected cost in the renewal cycle is Cr = c p R(tr |a, z a ) + cr [1 − R(tr |a, z a )]

(4.54)

Then, the expected cost rate of equipment can be obtained as follows by substituting Formulas (4.53) and (4.54) into Formula (4.52):

4.5 Degradation Modeling Based on Gamma Process with Environmental …

Cr Tr c p R(tr |a, z a ) + cr [1 − R(tr |a, z a )] = t a + 0r R(t|a, z a )dt

95

C(tr |a, z a ) =

(4.55)

Therefore, the optimal replacement time of equipment can be obtained by minimizing Formula (4.55), namely tr∗ = arg min C(tr |a, z a ) tr ≥0

(4.56)

The optimization objective in Formula (4.56) is a one-variable function, and the existence and uniqueness of its solution are easy to be proved, so it is not described in this paper. However, it is difficult to get the analytical form of the optimal solution because the functional form involved in Formula (4.56) is complex, and the approximate optimal solution can be obtained by the numerical method. A numerical example is used to verify the theoretical results concluded in this section. It is assumed that parameters of degradation process {X (t), t ≥ 0} are α = 1, β = 1, the failure threshold ξ is 30. The intensity function of external impact process is λ(t) = αs βs (αs t)βs −1 with αs = 0.2, βs = 1.2. Parameters about the increment of performance degradation level caused by external impact are: αY = 0.5, βY = 1.2. The intensity function of homogeneous Poisson process used to generate the non-homogeneous Poisson process is λ = 1, T = 50. a and z a are set as 0, the preventive maintenance cost is c p = 50 and the failure maintenance cost is cr = 500. Figure 4.8 shows the curve of the expected cost rate with time. According to the figure, it is easy to find the point (as shown by the circle in the figure) on the curve corresponding to the minimum cost rate, which represents the approximate optimal replacement time of components. Here, the optimal replacement time is tr∗ = 15, and the minimum cost per unit time is C ∗ = 3.4242.

4.6 Summary of This Chapter This chapter introduces firstly the related knowledge of Gamma process and the common methods for estimating unknown parameters, then discusses the performance degradation modeling based on Gamma process and residual life distribution computation method and also verifies the feasibility and effectiveness of the method by fatigue crack length data of A2017-T4 aluminum alloy. In addition, this chapter further studies the optimal maintenance decision method of equipnment under the influence of external environment, derives the life distribution of Gamma performance degradation process under the influence of external environment by modeling the environmental influence and calculates the expected

96

4 Gamma Process-Based Degradation … 50

45

40

35

Point corresponding to the minimum cost rate

30

Cost rate

25

20

15

10

5

0

0

10

20

30

40

50

60

Time t

Fig. 4.8 Curve of cost rate with time

cost rate model under the given maintenance policy, and finally, verifies the theoretical results with a numerical example.

References 1. Chen L (2010) Research on gyroscope reliability prediction method based on degradation modeling. Rocket Force University of Engineering, Xi’an 2. Chen L, Changhua Hu (2010) Simulation study on the effect of measurement error on estimation of gamma process degradation model. Syst Simul Technol 6(1):1–5 3. Zhang J (2011) Research on equipment degradation path and reliability modeling method fusing multi-source data. Rocket Force University of Engineering, Xi’an 4. Abdel-Hameed M (1975) A Gamma wear process. IEEE Trans Reliab 24(2):152–153 5. Van Noortwijk JM (2009) A survey of the application of gamma processes in maintenance. Reliab Eng Syst Saf 94(1):2–21 6. Zhang J, Zhang W (2011) Research on reliability of liquid floated gyro based on gamma process degradation model. J Rocket Force Univer Eng 25(2):38–41 7. Singpurwalla N (1997) Gamma processes and their generalizations: an overview. In: Cooke R, Mendel M, Vrijling H (eds) Engineering probabilistic design and maintenance for flood protection. Dordrecht, Kluwer Academic Publishers 8. Liu B, Yu M, Hu Y (2005) Analysis on reliability of system degradation process in storage environment. In: The 7th global conference on reliability, Beijing, pp 247–254 9. Ellingwood BR, Mori Y (1993) Probabilistic methods for condition assessment and life prediction of concrete structures in nuclear power plants. Nuclear Eng Des 142(2–3):155–66 10. Van Noortwijk JM, Klatter HE (1999) Optimal inspection decisions for the block mats of the Easter-Scheldt barrier. Reliab Eng Syst Saf 65:203–211

References

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11. Hoffmans GJCM, Pilarczyk KW (1995) Local scour downstream of hydraulic structures. J Hydraul Eng 121(4):326–340 12. Çinlar E, Bažant ZP, Osman E (1977) Stochastic process for extrapolating concrete creep. J Eng Mech Div ASCE 103(EM6):1069–1088 13. Nicolai RP, Dekker R, van Noortwijk JM (2007) A comparison of models for measurable deterioration: an application to coatings on steel structures. Reliab Eng Syst Saf 92:1635–1650 14. Ying Gu (2004) Mathematical methods in reliability engineering. Publishing House of Electronics Industry, Beijing 15. Yuan X (2007) Stochastic modeling of deterioration in nuclear power plant components. University of Waterloo 16. Si X, Changhua Hu (2016) Data-driven remaining useful life prediction theory and applications for equipment. National Defense Industry Press, Beijing 17. Moschopoulos PG (1985) The distribution of the sum of independent Gamma random variables. Annals Instit Stat Math 37(Part A):541–544 18. Ning R (2012) Simulation method of nonhomogeneous poisson process. Stud College Math 01:465–470 19. Lin Y (2001) Applied random process. Tsinghua University Press, Beijing

Chapter 5

Inverse Gaussian Process-Based Degradation Modeling and Residual Life Prediction

5.1 Introduction Wiener process and Gamma process are widely used in the fields of degradation modeling and residual life prediction. However, in some specific applications (such as GaAs laser), Wiener process and Gamma process cannot accurately describe the performance degradation process of equipment. In view of this situation, Wang and Xu considered using the inverse Gaussian process to model the performance degradation process of equipment [1]. Reference [2] describes an adaptive inverse Gaussian degradation model established by Bayesian rule. In existing references, the maximum likelihood estimation method is usually used to comprehensively utilize multiple groups of degradation data in the same batch to estimate the unknown parameters of the inverse Gaussian process. In contrast, this chapter proposes the evidential reasoning (ER) method to fuse these data and introduces the concept of attribute weight to estimate parameters of inverse Gaussian model more accurately for further prediction of the residual life [3, 4]. In this chapter, the parameters of degradation model are estimated first by using the Bayesian method and expectation maximization algorithm based on the performance degradation data of single equipment. And then, the estimated results for parameters of each equipment are fused by the ER algorithm to obtain the fused parameters which are used as the initial value of parameters in the degradation model of new equipment. On this basis, the random parameters and fixed parameters are updated in real time with the Bayesian method and expectation maximization algorithm for new equipment based on the real-time observed data and then the residual life of new equipment is predicted.

© National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_5

99

100

5 Inverse Gaussian Process-Based Degradation …

5.2 Definition of Inverse Gaussian Process The inverse Gaussian process {Y (t), t ≥ 0} is mathematically defined as follows: (1) (2)

When t 2 > t 1 ≥ s2 > s1 , increments Y (t 2 ) −Y (t 1 ) and Y (s2 ) −Y (s1 ) are independent of each other, which means Y (t) has an independent increment; When t > s ≥ 0, the increment Y (t)−Y (s) obeys the inverse Gaussian distribution of IG((t) −(s) and η[(t)−(s)]2 ).

Where (t) is a monotonically increasing function about t; IG(a,b) represents the inverse Gaussian distribution, and the probability density function is [1]  f I G (y; a, b) =

  b b(y − a)2 ,y >0 exp − 2π y 3 2a 2 y

(5.1)

The cumulative distribution function is:         by b y 2b − 1 + exp  − + 1 , y > 0 (5.2) FI G (y; a, b) =  y a a y a where (•) represents the cumulative distribution function of normal distribution. Without loss of generality, let (0) = 0 and Y (0) = 0. Therefore, Y (t) obeys the inverse Gaussian distribution IG((t) and η(t)2 ) with the mean value of (t) and the variance of (t)/η. The application of inverse Gaussian degradation process originates from the analytical investigation of Wiener degradation process in the fields of residual life prediction and health management. For the Wiener process for the following expressions: W (x) = vx + σ B(x)

(5.3)

If a failure threshold  is given, the first passage time distribution of Wiener process described by Formula (5.3) is the inverse Gaussian distribution IG(/v and 2 /σ 2 ). If the failure threshold is defined as a variable with time, which is denoted as (t), then the first passage time will be an inverse Gaussian process with the mean function of (t)/v and the scale function of (t)/σ 2 . The inverse Gaussian process can be used to fit the degradation process of equipment. Ye et al. discussed it in detail in reference [5] and defined the relation between Wiener process and inverse Gaussian process as reciprocal relationship, so it will not be repeated here. In practical engineering applications, the equipment failure time T D can generally be defined as the time reaching the failure threshold D for the first time. If the monotonicity of the inverse Gaussian degradation process is considered, the cumulative distribution function of T D can be expressed as

5.2 Definition of Inverse Gaussian Process

P(TD < t) = P(Y (t) > D) = 1 − FI G (D; (t), η2 (t))

101

(5.4)

In Formula (5.4), η(t) will be greater when t is greater, and Y (t) can be approximately regarded as a normal distribution with the mean value of (t) and the variance of (t)/η. Therefore, the cumulative distribution function of T D can be approximately expressed as follows:   √   D η D − (t) P(TD < t) ≈ 1 −  √ =  η(t) − √ (t)/η (t) 

(5.5)

This distribution is called Birnbaum–Saunders distribution, a degradation model generally used to study crack life [5]. Degradation phenomena such as corrosion and fatigue crack growth develop gradually under the irregular and random external impact, which continuously consume the life of equipment. The laws of impact can be described in detail by Poisson process. For example, every charge and discharge behavior of a lithium battery is a random small impact behavior. With the increasing charge and discharge times, the life of the lithium battery will be shortened. This process can be approximately described as a Poisson process. The track wear process of hard disk can also be described as a Poisson process. In fact, the compound Poisson process and its variation forms can fit many degradation processes effectively. N (t) The compound Poisson process can be described as C(t) = i=1 Di . In the Formula, N (t) represents a uniform Poisson process with its arrival rate of ν; while Di represents the strength of each impact process. If the arrival rate ν increases and the impact strength Di decreases continuously, the compound Poisson process can be evolved into various expressions which are convenient for computation. For example, the Gamma process, which is generally used to describe the monotonic degradation process, is a special case when the arrival rate ν approaches to infinite while the impact strength Di converges to zero at a certain rate. Similarly, the inverse Gaussian process is a special case of the compound Poisson process. Based on the contents of this chapter, the Wiener process described by the formula below w(x) = μ−1 x + η−1/2 B(x)

(5.6)

and the inverse Gaussian process Y (t) ~ IG(μ(t), 2 (t)) derived thereof are considered. Where  (t) is the threshold function that increases monotonically with time. The inverse Gaussian process is also used for modeling for the performance degradation law and prediction of the residual life of equipment. To solve this problem, in this chapter, the relevant parameters of the degradation model are estimated, and then the residual life of equipment is predicted.

102

5 Inverse Gaussian Process-Based Degradation …

5.3 ER-Based Parameter Estimation The ER method can fuse uncertain information from multi-sources effectively, so it has certain advantages in dealing with probabilistic uncertainty and fuzzy uncertainty. Yang et al. put forward the ER method in 1990s, which can efficiently solve the multiattribute decision problem [6]. Yang improved the original ER method partially in 2002, and then further analyzed the nonlinear characteristics of this reasoning algorithm [7]. Changhua Hu et al. applied the belief rule base (BRB) to the health status test of inertial platform and achieved good application results [8]. Since the degradation process of multiple equipment in the same batch can be described by the inverse Gaussian process, in principle, the parameter results obtained from multiple single-group of data can be fused. In this section, the information of multiple groups of data in the same batch is fused by the ER method to obtain the parameters of degradation model. Specifically, for the degradation data of single equipment, the unknown parameters of the inverse Gaussian degradation model are estimated by the Bayesian method and expectation maximization method. In this way, the parameters of multiple degradation models can be obtained by processing multiple groups of degradation data, respectively. Then, the ER algorithm is used to carry out data fusion processing on parameters of multiple groups of degraded model, and the final estimated results of parameters are obtained. The general idea is shown in Fig. 5.1.

Degradation data of multiple pieces of equipment in the same batch Model parameter determination algorithm based on inverse Gaussian process

Test to obtain the 1" group of degradation data

Test to obtain the nth group of degradation data

Determine model parameters of the equipment

Determine model parameters of the equipment

ER data fusion

Model parameter fusion algorithm based on ER

Parameters of equipment degradation model in the same batch

Fig. 5.1 General idea for parameter estimation

5.3 ER-Based Parameter Estimation

103

5.3.1 Parameter Estimation Based on Single Specific Equipment’s Degradation Data In consideration that the performance parameters of the equipment will surely change with the operation time due to the inevitable change of degradation model parameters during equipment degradation, (these changes are usually reflected by the testing data obtained in real time), these data can be used to update the model parameters in real time [9]. Similar to reference [5], the model parameter μ−1 in the Wiener process w(x) = μ−1 x + η−1/2 B(x) in the previous section is regarded as random variables; thus, the inverse Gaussian process model induced by the Wiener process can be applied for the study of degradation modeling of a batch of similar products. In this section, the prior knowledge and real-time monitoring data are fused by the Bayesian parameter updating method to obtain the posterior distribution of parameters. The parameters of prior distribution and other fixed parameters such as η−1/2 are solved by the EM algorithm on the basis of construction of likelihood function. EM algorithm is essentially a maximum likelihood estimation method. Every step of the EM algorithm consists of an expectation step (Step E) and a maximization step (Step M). The missing of some data or hidden of parameters often occur due to practical problems or limited observation conditions in the process of parameter estimation. EM algorithm can solve this kind of problem effectively. Firstly, the initial operation value of one (group) missing data is set, and then the model parameters are estimated through expectation step and maximum step, respectively. After that, the approximate solution of the missing data is obtained by the estimated values of the parameters, and the iteration is repeated until the beginning of converge of model parameters, indicating the iterative process ends. The final parameter solution is now obtained. However, although EM algorithm can ensure that every step of iterative operation is the optimal solution, a local optimal solution may be obtained generally, even an analytical solution cannot be obtained [10]. Therefore, the Bayesian parameter updating method is used to ensure that the solution of analytical form can be obtained by finally estimated parameter results [11]. (1)

Bayesian updating of random parameters According to the theory in Sect. 5.1, for the Wiener process X (t) = μt +σ B(t) with drift, the life distribution of equipment can be expressed as below when the failure threshold is l [12]: f (t) = √

  (l − μt)2 exp − 2σ 2 t 2π σ 2 t 3 l

(5.7)

Similarly, for the Wiener model w(x) = μ−1 x +η−1/2 B(x) used  in this chapter and the inverse Gaussian process Y (t) ∼ I G μ(t), η2 (t) induced thereof, it is assumed that the performance degradation data obtained at the monitoring time t i (i = 0, 1, …) is Y (t i ), let

104

5 Inverse Gaussian Process-Based Degradation …

λi = (ti ) − (ti−1 ) yi = Y (ti ) − Y (ti−1 )

(5.8)

where λi represents the increment of converted time at the i th monitoring time, which is a monitoring data related to the monitoring time ti ; Y (ti ) is the observed data of performance parameters at the ith monitoring time; yi is the increment of degradation amount at the ith monitoring time. The Formula below can be obtained according to related properties of the inverse Gaussian process:  2  n  −1   λi − μ−1 yi λi   = exp − p Y1:n μ 2η−1 yi 2π η−1 yi3 i=1

(5.9)

where Y1:n = {Y (t1 ), Y (t2 ), · · · , Y (tn )} represents all monitoring data collected by t n . It is assumed that the prior distribution of random parameter μ−1 is normal distribution on the account of the Bayesian conjugation and simple calculation, namely  2  −1

−1  − ω μ 1 g μ ; w, κ −2 = √ exp − 2κ −2 2π κ −1

(5.10)

where ω, κ −1 represent the mean and variance of normal distribution. And then, the posterior distribution of parameter μ−1 is obtained by using the Bayes formula based on its prior distribution and real-time monitoring data:   



  

−1 p Y1:n μ−1 g μ−1 ∝ p Y1:n μ−1 g μ−1 p μ |Y1:n = p(Y1:n )

(5.11)

Substituting Formulas (5.9) and (5.10) into Formula (5.11), we can get the formula below   2  2  n 

−1  μ−1 − ω λi − μ−1 yi p μ |Y1:n ∝ exp − exp − (5.12) 2κ −2 2η−1 yi i=1 The formula below can be obtained after simplified calculation   

−1 μ−1 − ω˜ n2 p μ |Y1:n ∝ exp − 2k˜n−2 The mean and variance parameters are, respectively, calculated by

(5.13)

5.3 ER-Based Parameter Estimation

105

 ηY (tn ) + κ 2 

ω˜ n = η(tn ) + ωκ 2 /κ˜ n2 κ˜ n =

(5.14)



According to the properties of normal we can know that μ−1 |Y1:n 

distribution, obeys a normal distribution of N ω˜ n , κ˜ n−2

−1



1

p μ |Y1:n = √ 2π κ˜ n−1

  μ−1 − ω˜ n2 exp − 2κ˜ n−2

(5.15)

The expression for posterior distribution of parameters including prior information and real-time observed data can be obtained from Formula (5.15). Formula (5.15) shows that as the operation time goes on, we can get the monitoring data Yi continuously, and the random parameter μ−1 will be updated correspondingly. In this way, the posterior distribution of parameters will be updated whenever a monitoring data is obtained, filling the gap of low accuracy of prior knowledge and improving the adaptability of degradation model. (2) Expectation maximization estimation of fixed parameter 

Take the fixed variable as  = ω, κ −2 , η−1 , q . Where ω and κ −2 are the mean and variance of parameter μ−1 under prior normal distribution, η−1 is the volatilization coefficient of Wiener process and q is the hidden parameter in the threshold function (t). According to the prior parameter distribution and performance degradation data, the joint likelihood function is     n  1 η(yi − μλi )2 κ 2 (1/μ − ω)2 + ln η + ln λi − l = ln κ − 2 2 2μ2 yi i=1

(5.16)

The solving process can be divided into Step E and Step M by the EM algorithm. Step E: in order to simplify the expression form of all steps to facilitate further solution, calculate the following formula firstly. 

a = E μ−1 = ω˜ n  2  = ω˜ n + κ˜ n−2 b = E μ−1     ˆ (i) = E According to l  n E is obtained as

 (i)  ˆ μ−1 Y1:n , n

(5.17)

 

log p Y1:n , μ−1 | , the result of Step

106

5 Inverse Gaussian Process-Based Degradation …

      κ2  ˆ (i) 2 b − 2aω + ω l n = ln κ − 2  n    1 η ln η + ln λ j − by j − 2aλ j + λ2j /y j + 2 2 j=1

(5.18)

ˆ (i) where  n represents the result of the i th step calculated with the monitoring data Y1:n . Step M: it is required to calculate the value of the formula below:     ˆ (i) ˆ (i+1)  = arg max l  n n

(5.19)

Take partial derivatives, respectively, for parameters ω, κ −2 , η−1 and q according to Formula (5.19), and the following recursive Formula of each fixed parameter can be obtained ⎧ i+1 ω =a ⎪ ⎪ ⎪ −2(i+1) ⎪ ⎪ = b − ω2 κ ⎪ ⎨ n  (5.20) byi − 2aλi + λi2 /yi η−1(i+1) = n1 ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ ⎪ ⎩ (λi /λi + aηλi  + ηλi λi /yi ) = 0 i=1

ˆ (i) ˆ (i+1) and  The iterative relation of  n n is established by Formula (5.20) in this way, and the iterative process ends when a convergence condition is satisfied. And then the estimated values of four fixed parameters are obtained.

5.3.2 Fusion of Fixed Parameters Based on ER Theoretically speaking, equipment in the same batch shares the same or slightly different parameters. It is hereby assumed that the degradation models of equipment in the same batch are the same, and it is necessary to fuse the degradation model parameters obtained above from each group of data. It is now assumed that the given reference set is D = {D1 , D2 , · · · , D N }(usually given in advance by experts), the parameter data of the kth model can be expressed with the confidence level below:   βk = βk,1 , βk,2 , · · · , βk,N , k = 1, 2, · · · , n

(5.21)

It is assumed that the attribute weights of the k th group of data can be expressed as ωk (k = 1, 2, · · · , n), firstly, convert the confidence level

5.3 ER-Based Parameter Estimation

107

β j,k ( j = 1, · · · , N , k = 1, · · · , n) in the output part of the ER model into basic probability mass of models, with the detailed expressions as below: m j,k = ωk β j,k m D,k = 1 − ωk

N 

(5.22)

β j,k

(5.23)

j=1

m¯ D,k = 1 − ωk ⎛ m˜ D,k = ωk ⎝1 −

N 

(5.24) ⎞ β j,k ⎠

(5.25)

j=1

where m j,k represents the basic probability mass of model parameter data relative to the evaluation result D j ; m D,k represents the basic probability mass of model parameter data relative to set D = {D1 , · · · , D N }. And there is m D,k = m¯ D,k + m˜ D,k [13], the specific meaning represented by each item is explained as follows: (1)

(2)

m¯ D,k is caused by the importance of the k th evidence (the so-called activation weight). If the kth evidence is absolutely important, that is ωk = 1, now make m¯ D,k = 0; result of the k m˜ D,k represents the imperfection expression for the evaluation th evidence; if the k th evidence is complete, that is 1 − Nj=1 β j,k = 0, now make the m˜ D,k = 0.

And then, combine n evidences to obtain the confidence level relative to the evaluation result D j ( j = 1, ..., N ). The specific process is as follows: It is assumed that the m j,I (k) represents the basic probability of the corresponding model parameter data relative to D j obtained after corresponding combination for previous k evidences by Dempster criterion, and m D,I (k) = 1 − Nj=1 m j,I (k) . Here, let m j,I (1) = m j,1 and m D,I (1) = m D,1 , respectively. The following expressions can be obtained after combination of previous k evidences by the iterative operation with Dempster criterion:   m j,I (k+1) = K I (k+1) m j,I (k) m j,k+1 + m j,I (k) m D,k+1 + m D,I (k) m j,k+1 m D,I (k) = m¯ D,I (k) + m˜ D,I (k)   m˜ D,I (k+1) = K I (k+1) m˜ D,I (k) m˜ D,k+1 + m˜ D,I (k) m¯ D,k+1 + m¯ D,I (k) m˜ D,k+1   m¯ D,I (k+1) = K I (k+1) m¯ D,I (k) m¯ D,k+1

(5.26) (5.27) (5.28) (5.29)

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5 Inverse Gaussian Process-Based Degradation …

⎤−1

⎡ ⎢ K I (k+1) = ⎣1 −

N  N 

⎥ m j,I (k) m t,k+1 ⎦ , k = 1, · · · , L − 1

(5.30)

j=1 t=1 t= j

βj =

m j,I (L) , 1 − m¯ D,I (L) βD =

j = 1, · · · , N

(5.31)

m˜ D,I (L) 1 − m¯ D,I (L)

(5.32)

where β j represents the confidence level of model parameters relative to the evaluation result D j ; β D represents the confidence level of model parameters not set with any evaluation result D j . Reference [6] has proved the Formula β D + Nj=1 β j = 1, so the confidence level of model parameters here can be regarded as a generalized confidence probability. The above iterative algorithm is the ER algorithm, and based on that, reference [7] further proposes ER analytical algorithm.  α× βj =





 N N n ' ωk β j,k + 1 − ωk 1 − ωk βi,k − βi,k k=1 i=1 k=1 i=1   n ' 1−α× (1 − ωk ) n '

(5.33)

k=1

⎡ α=⎣

n N  

 ωk β j,k + 1 − ωk

j=1 k=1

N 

 βi,k

− (N − 1)

i=1

n  k=1

 1 − ωk

N 

⎤−1 βi,k ⎦

i=1

(5.34) In Formula (5.33), β j represents the confidence level of model parameters relative to the evaluation result D j , which is a function of the evidence weight ωk (k = 1, · · · , n) and corresponding confidence level β j,k ( j = 1, · · · , N , k = 1, · · · , n). All evidence in the belief rule base (BRB) are combined by the ER analytical algorithm to obtain the final output expression of the BRB which is S=



 Dj, βj ,

j = 1, · · · , N



(5.35)

In this way, the estimated results of the fused model parameters can be obtained as follows. T =

N  i=1

βi u(Di )

(5.36)

5.3 ER-Based Parameter Estimation

109

where u(Di ) represents the effectiveness of the i th evaluation grade. Based on all of the discussion above, taking the fixed parameter η as an example, the specific steps for the fusion of fixed parameters of inverse Gaussian model based on ER are given below. Step 1: Obtain the model parameter (η1 , η2 , · · · , ηn ) based on degradation data of each group by using the algorithm as described in Sect. 5.3.1. Step 2: Obtain the confidence level of model parameters relative to reference value according to Formula (5.21) on the premise that the reference value set D = {D1 , D2 , · · · , D N } is given. Step 3: It is assumed that the weight of each group of degradation model parameter is expressed as ω1 , · · · , ωn , integrate the confidence level of n groups of parameters by using the ER analytical algorithm shown in Formulas (5.33) and (5.34) to obtain the comprehensive evaluation result of parameters shown in Formula (5.35). Step 4: Obtain the evaluation results of fused parameters by Formula (5.36). With the above model parameter fusion algorithm based on ER, the degradation model parameter of multiple pieces of equipment in the same batch can be effectively fused to fully utilize existing data information. Compared with the degradation data of single equipment, the fused results possess more universal significance and are more suitable for analyzing all equipment in the same batch.

5.4 Derivation of Residual Life Distribution After the posterior distribution of random parameters and the estimated value of fixed parameters are obtained, the specific cumulative distribution function for the first passage time of the inverse Gaussian degradation process can be obtained by Formula (5.2): ( F(t|μ, Y1:n ; η, (t)) p(μ|Y1:n ; ω, k)d μ (5.37) F(t|Y1:n ) = where F(t|μ, Y1:n ; η, (t)) = P(Y (t + tn ) > D|Y1:n , μ; η, (t)) = P(Y (t) > D − yn |μ; η, (t))

 ) D − yn η (t) − =1− D − yn μ



 ) D − yn η 2η(t)  − + (t) − exp μ D − yn μ (5.38) In this case, if (t) is differentiable, the probability density function of the first passage time can be further expressed as:

110

5 Inverse Gaussian Process-Based Degradation …

( f (t|Y1:n ) =

f (t|μ, Y1:n ; η, (t)) p(μ|Y1:n ; ω, k)dμ

(5.39)

where

 )

) D − yn η η f (t|μ, Y1:n ; η, (t)) = φ (t) −  (t) D − yn μ D − yn



 ) D − yn 2η(t) 2η  η  (t) − + (t) − exp μ μ D − yn μ

 )

 ) 2η(t) D − yn η η + exp  (t) (5.40) φ − + (t) μ D − yn μ D − yn The probability density function when the equipment reaches the failure thread for the first time can be obtained by substituting the estimated results of parameters obtained in Sect. 5.3 into Formula (5.39). The detailed steps are as follows: Step 1: Determine the prior distribution for random parameter μ−1 j of inverse Gaussian degradation model of the jth (j = 1, 2, …, M) equipment, and assign initial values to the parameters w j and k −1 j in the distribution.   ˆ j = ωˆ j , κˆ −2 ˆ −1 ˆ j of fixed parameStep 2: Obtain the estimated value  j ,η j , p j

ters by substituting the performance degradation data Y1:n of the jth equipment into Formula (5.20). 

ˆ = ω, Step 3: Obtain the estimated value  ˆ κˆ −2 , ηˆ −1 , pˆ of all fused parameters by the fixed parameter fusion algorithm of inverse Gaussian model based on ER algorithm. Step 4: Record the key performance parameter value Y (ti ) of new equipment at each monitoring time ti (i = 0, 1, · · · , n), and calculate the corresponding threshold function increment λi and degradation amount increment yi , i = 1, · · · , n according to Formula (5.2). Step 5: Substitute λi and yi , i = 1, · · · , n into Formulas (5.14) and (5.15), and carry out the Bayesian updating for the distribution of random parameter μ−1 of the model. Then, take the estimated result of parameters obtained in Step 3 as the initial value of the fixed parameter in its inverse Gaussian degradation model and calculate ˆ n of the fixed parameter at t n according to Formula (5.20). the estimated value  Step 6: Substitute the estimated result of parameters into Formulas (5.37) and (5.39) to obtain the probability distribution function and probability density function of equipment failure time based on degradation data Y 1:n .

5.5 Experimental Verification In this section, the simulation verification is carried out for theoretical results introduced above. Firstly, it is assumed that the function (t) of the inverse Gaussian degradation process is in the form of power exponent, that is (t) = t q , and further

5.5 Experimental Verification

111

let q = 1, and other parameters are ω = 2, κ −2 = 0.9 and η = 3. Then, obtain the realized value of n random parameters μ based on the given parameters and then obtain n groups of performance degradation data with the realized values. Finally, randomly select three samples from n groups of data for parameter fusion estimation and take another group of data as the performance monitoring data of new equipment. Here, only parameters ω and η are fused by ER. The incremental curve of these three groups of performance degradation data is respectively shown in Figs. 5.2, 5.3 and 5.4. In the process of parameter estimation, estimate parameters ω and η by using related methods in Sect. 5.3.1 and then carry out fusion processing based on the ER algorithm using the method in Sect. 5.3.2. The reference values are given as [1.7, 1.8, 1.9, 2, and 2.1] and [2, 2.5, 3, 3.5, and 4], respectively, for parameters ω and η. The estimated parameter results of each data are shown in Table 5.1. In addition, when taking the absolute value of the deviation between the estimated value and the true value of parameters as the standard to measure the estimation accuracy, the results shown in Table 5.2 can be obtained. The curve for confidence level of parameters ω and η is shown in Figs. 5.5 and 5.6, respectively. The results show that the parameters fused by ER have smaller deviation and are closer to the true value than those not being fused, which demonstrates the effectiveness of the proposed algorithm. Fig. 5.2 Incremental curve of the first group of degradation data

10 8 Increment

6 4 2 0

Fig. 5.3 Incremental curve of the second group of degradation data

0

20

40

60

80

100

60

80

100

7 6 5 Increment

4 3 2 1 0 0

20

40

112

5 Inverse Gaussian Process-Based Degradation …

Fig. 5.4 Incremental curve of the third group of degradation data

10 8 Increment

6 4 2 0 0

20

40

60

80

100

Table 5.1 Estimated results of parameters Group I:

Group II:

Group III:

ω

2.0986

1.8860

1.7967

ER fusion 1.8917

η

3.8052

3.0051

2.3905

2.9677

Table 5.2 Comparison of estimated results of parameters Group I:

Group II:

Group III:

Average of the first three groups

ER fusion

Absolute value of ω deviation

0.0986

0.1140

0.2033

0.1386

0.1083

Absolute value of η deviation

0.8052

0.0051

0.0323

0.2809

0.0323

Fig. 5.5 Confidence level of parameter ω

0.5

Confidence level

0.4 0.3 0.2 0.1 0

1.7

1.8 1.9 2 Reference value

2.1

Next, predict the residual life of new equipment by using the estimated results of parameters above. The incremental curve for performance degradation data of new equipment is shown in Fig. 5.7.

5.5 Experimental Verification

113

0.8

Confidence level

0.6

0.4

0.2

0

4

3.5 3 2.5 Reference value

2

Fig. 5.6 Confidence level of parameter η 7.5 7 6.5 6

Data

5.5 5 4.5 4 3.5

0

10

20

30

40

50

60

70

Fig. 5.7 Incremental curves of performance degradation data

Estimate the posterior distribution of random parameters and the fixed parameters by taking the estimated parameter result fused by ER as the initial value and using the parameter estimation method introduced in Sect. 5.3.1, and then predict the performance degradation path in one step with the estimated results; the results are shown in Fig. 5.8. Multi-step prediction is not used here because that although the dynamic performance of performance degradation parameters of equipment can be obtained with the multi-step prediction, inaccurate results may be predicted due to the existence of accumulative errors [14, 15]. For comprehensive comparison, Fig. 5.8 also gives a one-step predicted path of performance degradation based on Wiener process. The parameters of Wiener process are estimated by the method in Ref. [9]. It can be seen from Fig. 5.8 that the method discussed in this chapter can be used to track and predict in one step the inverse Gaussian degradation data effectively. The Wiener process can also be used to track the real degradation path, but the fluctuation is great, and the fitting effect of monotonically increasing data is not as good as the

114

5 Inverse Gaussian Process-Based Degradation … 200

Real degradation path Predicted path of inverse Gaussian Wiener predicted path 150

Degradation path

100

50

0 0

20

40

80 60 Testing time (h)

100

120

140

Fig. 5.8 Real degradation path and predicted path of equipment performance parameters

fitting effect of monotonically increasing data completed by the inverse Gaussian process. Especially, it takes a long time to reach stability in the early stage. After obtaining the updated model parameters and their prior distribution parameters in each step, the failure probability function curve and residual life probability density function curve of equipment can be obtained, respectively, according to the residual life calculation method given in Sect. 5.4, as shown in Figs. 5.9 and 5.10. It can be seen from the experimental simulation results that the algorithm proposed in this chapter can fit the inverse Gaussian degradation data effectively to obtain more reasonable predicted results of residual life. According to Fig. 5.9, the equipment fails completely at about the 200th hour. The above prediction results can also provide reference for equipment maintenance.

Probability of failure Testing time (h)

Fig. 5.9 Failure probability curve of equipment

5.6 Summary of This Chapter

115

Probability density function Testing time (h)

Residual life (h)

Fig. 5.10 Curve of residual life probability density function of equipment

5.6 Summary of This Chapter For equipment with monotonic degradation process, the inverse Gaussian model can effectively simulate the change path of key performance parameters, which has attracted the attention of some scholars at home and abroad. In this chapter, an inverse Gaussian degradation model that can be updated adaptively is proposed in combination with some modeling ideas of foreign scholars to further investigate the problem of inverse Gaussian degradation modeling theoretically. In this chapter, the inverse Gaussian process is used to model the degradation process of equipment, with multi-source data, including prior information and performance degradation data of multiple pieces of equipment in the same batch, adopted. Firstly, the random parameters in degradation model of single equipment are updated by the Bayesian method based on the corresponding performance degradation data, and then other fixed parameters of the degradation model are estimated by expectation maximization algorithm. After parameters of each equipment degradation model are obtained, the estimated parameter results of each equipment are fused by the ER algorithm to obtain the fused parameters which are used as the initial value of the parameters in the degradation model of new equipment. On this basis, the random parameters and fixed parameters are updated in real time for new equipment with the Bayesian method and expectation maximization algorithm based on the real-time observed data, and then the residual life of new equipment is predicted. Finally, the feasibility of the method proposed in this chapter is verified by the numerical simulation.

116

5 Inverse Gaussian Process-Based Degradation …

References 1. Wang X, Xu D (2010) An inverse Gaussian process model for degradation data. Technometrics 52(2):188–197 2. Xu W, Wang W (2013) RUL estimation using an adaptive inverse Gaussian model. Chem Eng Trans 33:331–336 3. Li M, Changhua Hu, Zhou Z et al (2015) A degradation modeling method based on inverse gaussian process and evidential reasoning. Electron Opt Control 22(1):92–96 4. Li M (2014) Research on remaining life prediction and maintenance decision method based on inverse gaussian process. Rocket Force University of Engineering, Xi’an 5. Ye ZS, Chen N (2014) The inverse Gaussian process as a degradation model. Technometrics 56(3):302–311 6. Yang JB, Sing MG (1994) An evidential reasoning approach for multiple attribute decision making with uncertainty. IEEE Trans Syst Man Cybern A 24:1–18 7. Yang JB, Xu DL (2002) Nonlinear information aggregation via evidential reasoning in multiattribute decision analysis under uncertainty. IEEE Trans Syst Man Cybern A 32:376–393 8. Changhua Hu, Si X (2010) Real-time parameters estimation of inertial platform’s health condition based on reliability rule base. Acta Aeronautica ET Astronautica Sinica 7:1454–1465 9. Si XS, Wang WB, Chen MY et al (2012) A degradation path-dependent approach for remaining useful life estimation with an exact and closed-form solution. Eur J Oper Res 226(1):53–66 10. Lian J (2006) Application of EM algorithm and its improvement in estimation of hybrid model parameter. Xi’an, Chang’an University 11. Wen J, Luo S, Zhao J et al (2001) The BEM algorithm: an EM method based on bayesian. Comput Res Develop 7:821–825 12. Peng B (2010) Research on reliability modeling method based on wiener process. National University of Defense Technology, Changsha 13. Zhou Z, Yang J (2011) Modeling of expert system and complex system of belief rule base. Science Press, Beijing 14. Zhang S, Li W, Ding K (2009) A novel approach to evidence combination based on the evidence credibility. Control Theory Appl 26(7):812–814 15. Gertsbackh IB, Kordonskiy KB (1969) Models of failure. Springer, New York, USA

Chapter 6

Degradation Modeling and Residual Life Prediction Based on Support Vector Machine

6.1 Introduction In general, the performance degradation data of high reliability and long life products are usually small samples due to the high price and measurement damage, while the performance degradation rule of complex products is usually nonlinear. How to establish the performance degradation path model of individual products under small sample and nonlinear data is a basic problem to be solved in this chapter. Based on the principle of Vapik–Chervonemkis (VC) dimension of statistical learning theory and structural risk minimization, support vector machine (SVM) has the advantages of simple structure, strong generalization ability and strong small sample learning ability. By introducing kernel function, it transforms the feature space from linear inseparability into linear separability and finally turns it into a convex quadratic programming solution problem with linear constraints. SVM algorithm was originally proposed for classification problem. Vapnik extended it to nonlinear regression estimation by introducing ε -insensitive loss function and got support vector regression (SVR). At present, SVR has been widely used in regression analysis, function estimation, probability density estimation, subspace analysis, time series prediction and other fields. For this, we can find numerous references, but only a few is on degradation modeling [1–4]. Jun Wu [5, 6] studied a degradation path modeling method based on least square support vector regression (LS-SVR), and proposed a two-step parameter optimization and search method of grid search and cross-validation. Guoping Xu [7– 9] proposed a gray SVR model based on the combination of gray theory and SVR, modeled and predicted the gyro vibration energy data and studied the life prediction method of dynamic-tuned gyroscope based on SVM. It can be seen that there are very few studies on the application of SVR in degradation modeling, so it is necessary to carry out in-depth research on the application of SVR in the field of degradation modeling.

© National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_6

117

118

6 Degradation Modeling and Residual Life Prediction …

Therefore, taking SVR algorithm as the basic theoretical tool, this chapter mainly considers two modeling problems under small sample performance degradation data: (1) There are only small sample data of particular individuals; (2) there are rich data of similar products at the same time [10–14]. For degradation modeling with small sample data of particular individuals, in order to improve the performance of SVR, the genetic algorithm (GA) is used to optimize the selection of model parameters, and the kernel functions are compared and selected. A degradation path modeling and life prediction method based on GAoptimized SVR is then proposed [11]. When there are rich performance degradation data of similar products, in order to make full use of these data to further improve the model accuracy, a modeling idea based on weighted similarity of degradation path is proposed from the perspective of considering the similarity of degradation path [12]. The degradation data of similar products are normalized, and the fuzzy c-means (FCM) clustering algorithm is used to summarize the normalized data. The model of particular individuals is then obtained by weighting the model of similar products based on the similarity of degradation path between particular individuals and similar products. Finally, the weights are updated with the real-time measurement data of particular individuals to realize real-time life prediction. According to the condition whether the measurement time of particular individuals is normalized, two real-time degradation path modeling and life prediction methods based on SVR and FCM clustering are proposed—degree-of-membership-based weighted method and error-based-weighted method.

6.2 SVR Principle 6.2.1 Primal and Dual Problems l training data {(x i , yi ), i = 1, 2, . . . , l} are given, where the input is x i ∈ Rh , the output is yi ∈ R. Firstly, SVR maps the input into a high-dimensional Hilbert space H through nonlinear mapping ϕ and then constructs a linear function in H that can fit the training sample set: f (x) = ω T ϕ(x i ) + b, ω ∈ H, ϕ : Rh → H where ω is the weight vector in H; b is the bias term. The ε-insensitive loss function is selected as L ε (x, y, f (x)) = max{0, |y − f (x)| − ε}. According to the structural risk minimization criterion, the primal optimization problem of SVR algorithm is

6.2 SVR Principle

119 l  min 21 ω T ω + cl (ξi + ξi∗ ) ω,b,ξ (∗) i=1  ⎧ T ⎨ ω ϕ(x  Ti ) + b − yi ≤ ε + ξi∗ s.t. yi − ω ϕ(x i ) + b ≤ ε + ξi ⎩ ξi , ξi∗ ≥ 0

(6.1)

where ξ (∗) = [ξ1 , ξ1∗ , · · · , ξl , ξl∗ ] ∈ R2l is the relaxation factor, (*) means that the cases without * and with * are both included; ε(ε > 0) is the insensitive parameter; c(c > 0) is the penalty parameter, which controls the penalty degree of the sample with error exceeding ε. Since the primal problem (6.1) is convex and its feasible region is not empty, the solution about (ω, b) exists. For linear mapping ϕ(x) = x, i.e., linear SVR algorithm, Reference [15] has proved that the solution about ω of primal problem (6.1) exists and is unique. And for the nonlinear mapping ϕ to be studied here, it can be similarly proved that the solution about ω of primal problem (6.1) exists and is unique, which will not be repeated here. After introducing Lagrange function, it becomes L(ω, b, ξ (∗) , α (∗) , η(∗) ) =

l l  1 T c ω ω+ (ξi + ξi∗ ) − (ηi ξi + ηi∗ ξi∗ ) 2 l i=1 i=1

−

l 

αi (ε + ξi + yi − ω T ϕ(x i ) − b)−

i=1

l 

αi∗ (ε + ξi∗ + yi − ω T ϕ(x i ) − b)

i=1

(6.2) where α (∗) , η(∗) ∈ R2l is Lagrange multiplier and satisfies the condition of ≥ 0. According to Wolf’s duality theory, partial derivatives of ω, b and ξ (∗) are calculated, respectively, and made be equal to 0 αi∗ , ηi∗

⎧ l  ⎪ ∂L ⎪ =ω− (αi∗ − αi )ϕ(x i ) = 0 ⎪ ∂ω ⎪ ⎪ i=1 ⎨ l  ∂L = (αi − αi∗ ) = 0 ⎪ ∂b ⎪ ⎪ i=1 ⎪ ⎪ ⎩ ∂ L(∗) = c − η(∗) − α (∗) = 0 i i l ∂ξ i

From Formula (6.3)

(6.3)

120

6 Degradation Modeling and Residual Life Prediction …

⎧ l  ⎪ ⎪ ω = (αi∗ − αi )ϕ(x i ) ⎪ ⎪ ⎨ i=1 l  (αi − αi∗ ) = 0 ⎪ ⎪ ⎪ i=1 ⎪ ⎩ (∗) ηi = cl − αi(∗)

(6.4)

Formula (6.4) is substituted into Formula (6.2), and α (∗) ‘s maximum is obtained from Formula (6.2) l l l    (αi∗ − αi )(α ∗j − α j )(ϕ(x i ) · ϕ(x j )) − ε (αi∗ + αi ) + yi (αi∗ − αi ) max − 21 α (∗) i, j=1 i=1 i=1 ⎧ l ⎨ ∗ (α − αi ) = 0 s.t. i=1 i ⎩ 0 ≤ αi , αi∗ ≤ cl

where (·) is the inner product of a vector. The kernel function K (·, ·) is used to express the inner product of nonlinear mapping ϕ in Hilbert space H, i.e., K (x i , x j ) = (ϕ(x i ) · ϕ(x j )) = ϕ(x i )T ϕ(x j ), the dual problem of the primal problem (6.1) is then obtained l l l    min 21 (αi∗ − αi )(α ∗j − α j )K (x i , x j ) + ε (αi∗ + αi ) − yi (αi∗ − αi ) α (∗) i, j=1 i=1 i=1 ⎧ l ⎨ ∗ (α − αi ) = 0 s.t. i=1 i ⎩ 0 ≤ αi , αi∗ ≤ cl (6.5)

It can be seen that SVR transforms the linear inseparability of the original input space into the linear separability of Hilbert space through the kernel function and finally turns it into a convex quadratic programming solution problem with linear constraints. Since the dual problem (6.5) is convex and its feasible region is not empty, its solution exists. Since the dual problem (6.5) is not strictly always convex, its solution is not necessarily unique. However, starting from any solution of dual problem (6.5), the solution about (ω, b) of the primal problem (6.1) can be obtained. As for the relationship between the solutions of dual problem (6.5) and primal problem (6.1), there are the following theorems: Theorem 6.1 Let α¯ (∗) = [α¯ 1 , α¯ 1∗ , · · · , α¯ l , α¯ l∗ ]T be the solution of dual problem (6.5), then the unique solution about ω of primal problem (6.1) can be expressed as follows:

6.2 SVR Principle

121

ω¯ =

l 

∗  αi − αi ϕ(x i )

(6.6)

i=1



Prove: Let K = K (x i , x j ) l×l , α = [α1 , · · · , αl ]T , α ∗ = [α1∗ , · · · , αl∗ ]T , e = [1, · · · , 1]T ∈ Rl , y = [y1 , · · · , yl ]T , then the dual problem (6.5) can be expressed as min 21 (α ∗ − α)T K (α ∗ − α) + εeT (α ∗ + α) − y T (α ∗ − α) α (∗) eT (α ∗ − α) = 0 s.t. 0 ≤ αi , αi∗ ≤ cl If α¯ (∗) is an arbitrary solution of dual problem (6.5), then according to KKT ¯ s¯ (∗) and ξ¯ (∗) condition, it can be known that there are Lagrange multipliers b, ⎧ ¯ − s¯ ∗ + ξ¯ ∗ = 0 ⎪ ¯ + εe − y + be K (α¯ ∗ − α) ⎪ ⎪ ⎨ −K (α¯ ∗ − α) ¯ − s¯ + ξ¯ = 0 ¯ + εe + y − be (∗) ¯ s ≥ 0 ⎪ ⎪ ⎪ ⎩ ¯ (∗) ξ ≥0

(6.7)

c ξ¯i∗ (α¯ i∗ − ) = 0, s¯i∗ α¯ i∗ = 0 l

(6.8)

And

Formulas (6.7) and (6.8) mean that

Let ω¯ =

l

 i=1

∗

¯ ≥ −εe − ξ¯ ¯ − y + be K (α¯ ∗ − α) ∗ ¯ ≥ −εe − ξ¯ ¯ + y − be −K (α¯ − α)

 αi∗ − αi ϕ(x i ), then the above results are equivalent to

¯ ≤ ε + ξ¯i∗ yi − [ω¯ T ϕ(x i ) + b] T ¯ − yi ≤ ε + ξ¯i [ω¯ ϕ(x i ) + b]

(∗) (∗) ¯ b, ξ¯ ) satisfies the It can be seen from ξ¯ ≥ 0 in Formula (6.7) that (ω, constraints of the primal problem (6.1), and it is the feasible solution. Further, according to Formula (6.8), l 1 T c ∗ ω¯ ω¯ + (ξ¯ + ξ¯i ) 2 l i=1 i

122

6 Degradation Modeling and Residual Life Prediction …

=

l c ∗ 1 T ¯ − s¯ ∗ + ξ¯ ∗ ] ¯ + εe − y + be ω¯ ω¯ + (ξ¯ + ξ¯i ) − α¯ ∗T [K (α¯ ∗ − α) 2 l i=1 i

¯ − s¯ + ξ¯ ] ¯ + εe + y − be − α¯ ∗T [−K (α¯ ∗ − α) 1 ¯ T (α − α ∗ ) ¯ T K (α¯ ∗ − α) ¯ − εeT (α¯ ∗ + α) ¯ + y T (α¯ ∗ − α) ¯ − be = − (α¯ ∗ − α) 2 c T T − ξ (∗) (α¯ (∗) − e) + s¯ (∗) α¯ (∗) l 1 ∗ T ¯ K (α¯ ∗ − α) ¯ − εeT (α¯ ∗ +α) ¯ + y T (α¯ ∗ − α) ¯ = − (α¯ − α) 2 which means the objective function value of the primal problem (6.1) at point (∗) ¯ b, ξ¯ ) is equal to the optimal value of the dual problem (6.5). Therefore, it can (ω, (∗) ¯ b, ξ¯ ) is the solution of the be seen according to the strong duality theorem that (ω, primal problem (6.1). In the proof of Theorem 6.1, it has been clarified that the Lagrange multiplier b¯ is the solution about b of the primal problem (6.1), and next the expression of b¯ will be discussed. Theorem 6.2 Let α¯ (∗) = [α¯ 1 , α¯ 1∗ , · · · , α¯ l , α¯ l∗ ]T be the solution of dual problem (6.5). Let

S10 = {i|α¯ i = 0}, S1 = {i|α¯ i ∈ (0, c/l)}, S1

c/l

= {i|α¯ i = c/l},

S20 = {i|α¯ i∗ = 0}, S2 = {i|α¯ i∗ ∈ (0, c/l)},

c/l S2

= {i|α¯ i∗ = c/l}

For j ∈ {1, 2} and k ∈ {0, c/l}, define dmax (S kj ) = max{yi − ω¯ T ϕ(x i )}, dmin (S kj ) = min{yi − ω¯ T ϕ(x i )} i∈S kj

i∈S kj

Then the set of solutions about (ω, b) of the primal problem (6.1) can be expressed as 

Among them, ω¯ =

 ¯ b ∈ [bd , bu ] (ω, b)|ω = ω,

(6.9)

l

  αi∗ − αi ϕ(x i ), and bd , bu is calculated according to the

i=1

following conditions: (i)

If S1 = ∅ or S2 = ∅, then take j ∈ S1 or k ∈ S2 and calculate accordingly

6.2 SVR Principle

123

⎧ l

  ⎪ ⎪ α¯ i∗ − α¯ i K (x i , x j ) + ε, S1 = ∅ ⎨ yj −

bd = bu =

i=1

l

  ⎪ ⎪ ⎩ yk − α¯ i∗ − α¯ i K (x i , x k ) − ε, S2 = ∅

(6.10)

i=1

(ii)

If S1 = ∅ or S2 = ∅, then

c/l

bu = min{dmin (S2 ) − ε, dmin (S10 ) + ε} c/l bd = max{dmax (S1 ) + ε, dmax (S20 ) − ε}

(6.11)

Proof It can be seen according to Theorem 6.1 that if α¯ (∗) is the solution of dual (∗) (∗) ¯ b, ξ¯ ) the solution of primal problem (6.5), then b¯ and ξ¯ exist and make (ω, problem (6.1). The necessary and sufficient condition for b¯ as a solution is to meet KKT conditions (6.7) and (6.8). The following two cases will be discussed, respectively: (1)

S1 = ∅ or S2 = ∅

If there exists j ∈ S1 or k ∈ S2 , then the KKT condition means that ξ¯ j = s¯ j = ¯ = ε. Then, conclusion 0, ω¯ T ϕ(x j ) + b¯ − y j = ε or ξ¯k∗ = s¯k∗ = 0, yk −[ω¯ T ϕ(x k ) + b] (i) can be drawn. (2)

S1 = ∅ and S2 = ∅ The KKT condition is equivalent to ⎧ c/l s¯i = 0, ξ¯i = −ε + ω¯ T ϕ(x i ) + b¯ − yi ≥ 0, i ∈ S1 ⎪ ⎪ ⎨ ∗ c/l T s¯i = 0, ξ¯i∗ = −ε − ω¯ ϕ(x i ) − b¯ + yi ≥ 0, i ∈ S2 ⎪ ξ¯ = 0, s¯i = ε − ω¯ T ϕ(x i ) − b¯ + yi ≥ 0, i ∈ S10 ⎪ ⎩ i∗ ξ¯i = 0, s¯i∗ = ε + ω¯ T ϕ(x i ) + b¯ − yi ≥ 0, i ∈ S20

which can be solved together to obtain the conclusion (ii) of the theorem. Thus, after solving the optimal solution α (∗) of dual problem (6.5), and determining the value of b according to Theorem 6.2, the nonlinear regression function constructed by SVR algorithm can be obtained f (x) =

l 

∗  αi − αi K (x i , x) + b i=1

(6.12)

124

6 Degradation Modeling and Residual Life Prediction …

6.2.2 Sparsity of SVR ¯ of the ¯ b) It can be seen from the conclusion of Theorem 6.2 that the solution (ω, , y corresponding to primal problem (6.1) only depends on the sample points ) (x i i 

nonzero αi∗ − αi in the training set, while other sample points have no effect on the results. Thus, the calculation of nonlinear regression function can be simplified, which reflects the sparsity of SVR. Reference [15] discusses the values of component α (∗) and their relationship with corresponding sample points, which are directly cited below. Definition 6.1 (support vector) Call the input x i in training set {(x i , yi ), i = 1, 2, · · · , l} a support vector if αi = 0 or αi∗ = 0 is obtained in the solution of dual problem (6.5). Definition 6.2 (ε-band) Give the positive integer ε, ε−-band of a hyperplane y = f (x) is the region that the hyperplane moves up and down ε along y-axis. Theorem 6.3 Deng and Tian [15] Let α (∗) ∈ R 2l be the solution of optimization problem (4.5), then αi ∈ [0, c/l] and αi∗ ∈ [0, c/l] when i = 1, · · · , l, and in αi and αi∗ , at most one of them is not zero. Theorem 6.4 Deng and Tian [15] Let α (∗) ∈ R 2l be the solution of optimization problem (6.5), (i) if αi = αi∗ = 0, the corresponding sample point (xi , yi ) must be inside or on the boundary of the ε-band; (ii) if αi ∈ (0, c/l), αi∗ = 0 or αi = 0, αi∗ ∈ (0, c/l), the corresponding sample point (xi , yi ) must be on the boundary of the ε-band; (iii) if αi = c/l, αi∗ = 0 αi = 0, αi∗ = c/l, the corresponding sample point (x i , yi ) must be outside or on the boundary of the ε-band. Corollary 6.1 Deng and Tian [15] (i) If the sample point (x i , yi ) is inside the εband, αi = αi∗ = 0; (ii) if the sample point (x i , yi ) is on the boundary of the ε-band, αi ∈ [0, c/l], αi∗ = 0 or αi = 0, αi∗ ∈ [0, c/l]; (iii) if the sample point (x i , yi ) is outside the ε-band, αi = c/l, αi∗ = 0 or αi = 0, αi∗ = c/l. From the conclusion (i) of Corollary 6.1, it can be seen that all the sample points (x i , yi ) inside the ε-band satisfy the condition of αi = αi∗ = 0, that is to say, they are not support vectors and have no contribution to the constructed nonlinear regression function. Then, removing these sample points which are not support vectors will not affect the final nonlinear regression function. Conversely, only those sample points (x i , yi ) corresponding to αi = 0 or αi∗ = 0 will affect the results of nonlinear regression function. The sparsity shown in the above cases may simplify the calculation of nonlinear regression function. This advantage is obviously inseparable from the ε-insensitive loss function. It can also be found from Corollary 6.1 that the value of ε determines the width of the ε-band and affects the number of support vectors, which is manifested in the complexity of the nonlinear regression function (6.12) constructed by SVR. If the value of ε is too small, all the sample points will be support vectors, then SVR

6.2 SVR Principle

125

algorithm does not have sparsity, that is, the structure of Formula (6.12) is the most complex; if the value of ε is too large, there will be no support vector. According to Theorem 6.2, it can be seen that the form of regression function constructed by SVR algorithm is a line y = b, and the number of such line is infinite, but the fitting effect of these functions on training samples is obviously not ideal. In short, the smaller the value of ε is, the more the number of support vectors is, the more complex the Formula (6.12) is, the smaller the training error is and the higher the fitting accuracy is; on the contrary, the greater the value of ε is, the less the number of support vectors is, the simpler the Formula (6.12) is, the greater the training error is and the lower the fitting accuracy is.

6.2.3 Kernel Function In Formula (6.5), the selection of mapping is very important. After the mapping is selected, the kernel function K (·, ·) can be constructed from its inner product to calculate SVR. A mapping can only correspond to one Hilbert space and get a kernel function. However, the inner product of different Hilbert spaces can get the same kernel function. Therefore, a kernel function can correspond to many mappings. We only need to select the kernel function K (·, ·) and do not need to know what the specific mapping is. K (·, ·) has a wide range of choices, but it must meet the following definition. Definition 6.3 (kernel function) Let χ be a subset of Rh . Call the function K(x, x ) defined on χ × χ is a kernel function, if there is a mapping from χ to a Hilbert space H.

ϕ:

χ →H x → ϕ(x)

Then

 K (x, x ) = ϕ(x) · ϕ(x ) = ϕ(x)T ϕ(x ) The kernel function defined in Definition 6.3 is also called kernel or positive definite kernel. Reference [15] has proved its necessary and sufficient conditions and studied the properties of positive definite kernel, which will not be repeated here. Common kernel functions include (1)

Gaussian radial basis kernel, also known as Gaussian kernel (GK) or radial basis kernel (RBK):  2 K (x, x ) = exp(− x − x  /σ 2 )

(6.13)

126

(2)

6 Degradation Modeling and Residual Life Prediction …

Wavlet kernel (WK) [16]:     xik − x jk 2 xik − x jk K (x, x ) = cos(1.75 ) exp − σk 2σk2 k=1

h 

(6.14)

where σk is the magnification factor. (3)

Polynomial kernel (PK) K (x, x ) = [(x · x ) + σ1 ]σ2 , σ1 ≥ 0, σ2 ∈ N ≥ 2

(6.15)

where it is called non-homogeneous polynomial kernel when σ1 > 0, and it is called homogeneous polynomial kernel when σ1 = 0. For the convenience of expression, the following content uniformly uses σ to represent the parameter of the kernel function, and the number of components it contains is determined according to the type of kernel function selected or the dimension of the input.

6.3 Residual Life Prediction Method Based on GA-Optimized SVR 6.3.1 Problem Description Assume that the product works at a constant stress level, and the performance parameter y of the product will increase or decrease with the running time t (or the travel distance, the number of stress cycles, etc.). It is defined that when y reaches the failure threshold η, the product will fail. During the running of the product, its performance parameter y is measured for m times, and the historical measurement degradation data set is {(ti , yi ), i = 1, 2, . . . , m}. The mathematical problem to be solved in this section is as follows: m data {(ti , yi ), i = 1, 2, . . . , m} are known, in which ti , yi ∈ R, it is required to use the SVR algorithm in Sect. 6.2 construct the function of y for t. The specific function form is y = f (t) =

m 

∗  αi − αi K (ti , t) + b i=1

(6.16)

6.3 Residual Life Prediction Method Based on GA-Optimized SVR

127

6.3.2 Basic Ideas The basic idea of SVR algorithm for degradation path modeling and life prediction is as follows: Firstly, select the appropriate kernel function and determine the model parameters in SVR algorithm; secondly, substitute the training data into QP (6.5) and obtain Lagrange multiplier, and calculate the bias b according to Theorem 6.2, so as to build the degradation path model shown in Formula (6.16); finally, predict the degradation in the future time by using Formula (6.16), estimate the failure time of the product in combination with the failure threshold η and obtain the remaining life of the product by subtracting the current time. Since this section studies the degradation path modeling problem under small sample data, there is no large-scale training sample set problem in SVR training and learning. The optimization solution of QP can be well completed by using Newton method, interior point method and other mature classical optimization algorithms. Therefore, this section directly calls the QUADPROG function in MATLAB to solve the QP (6.5). Considering the input dimension h = 1 of training data, the kernel function parameters have only one component when RBK or WK is selected, and the kernel function parameters have two components when PK is selected. According to the SVR principle in Sect. 6.2.1, it can be seen that there are three parameters in QP (6.5): kernel function parameter σ , insensitive parameter ε and penalty parameter c. Different kernel functions and kernel function parameters σ correspond to different feature spaces and feature mappings. Penalty parameter c is a balance between empirical risk and generalization ability. An ideal value of c is to minimize both the empirical risk and generalization error. If c is too small, the penalty for support vector will be small, which will increase the empirical risk and the generalization error of the model, resulting in the phenomenon of “under-learning”; if c is too large, the inner product of weight ω will be small, and the number of support vectors will be large, which will decrease the empirical risk and the generalization ability of the model, resulting in the phenomenon of “overlearning." It can be seen that the value of parameter (σ, ε, c) is related to the solution of QP (6.5) and ultimately affects the model established by SVR algorithm. In other words, the determination of parameter (σ, ε, c) is an important factor affecting the prediction accuracy of model (6.16). Therefore, it is necessary to optimize the parameter (σ, ε, c) to ensure that the degradation path model has good generalization ability. In view of the excellent global searching ability of GA, GA is used to optimize the parameter (σ, ε, c) here. The basic idea of parameter optimization based on GA is as follows: Firstly, select l data from m data for training and learning and take the residual (m − 1) data as test set to test the generalization ability of nonlinear regression function constructed by SVR algorithm according to fitness function; secondly, set the value range of each parameter variable, search population individuals in parameter change space with GA and take each population individual as the parameter of SVR algorithm; finally, establish the model with 1 data, select the reserved population individuals according to the fitness function value with GA and cross and mutate to

128

6 Degradation Modeling and Residual Life Prediction …

generate new population individuals. The iteration is repeated until the iteration stop condition is met. The population individual with the minimum fitness function value is the optimal parameter. After GA finds the optimal parameter, the optimal parameter is substituted into the QP (6.5) and is solved with all m data, and the nonlinear regression function constructed is the degradation path model based on GA-optimized SVR algorithm. The most important thing of using GA is to design the fitness function, that is, the absolute mean of the relative estimation error of the test set by the model established by SVR algorithm:  r  1   g j − gˆ i j  × 100% e(g, gˆ i ) = r j=1  g j 

(6.17)

where g is the real output value of the test set, gˆ i is the predicted value of the test set after the i-th iteration, and r is the number of samples in the test set.

6.3.3 Specific Steps of the Method The degradation path modeling method based on GA-optimized SVR is listed as follows: Step 1: Determine the training set and test set: select the first l data of the data set {(ti , yi ), i = 1, 2, . . . , m} as the training set, and the last r(r = m − l) data as the test set. Step 2: Let the algebraic variable i = 0, and set imax and emax , determine the crossover probability value, mutation probability value and value range of optimization parameters; randomly generate M groups of initial values (initial populations) of optimization parameters, take each individual of the initial population as the parameters of SVR algorithm, construct and solve QP (6.5), calculate b according to Theorem 6.2, establish the degradation path model shown in Formula (6.12), predict the test set and calculate the fitness value of each individual according to Formula (6.17). Step 3: Let i = i + 1; select the better individuals according to the fitness value, generate new individuals by uniform crossover according to the crossover probability and form a new population by mutation operation on the individuals according to the mutation probability; take each individual of the new population as the parameters of SVR algorithm, establish the degradation path model and calculate the fitness value of each individual. Step 4: If both imax = i and max e(g, gˆ i ) > emax are satisfied, turn to Step 3; otherwise, the individual corresponding to the minimum fitness value is selected as the optimal parameter (σ¯ , ε¯ , c). ¯ Step 5: Take all m data as training samples, solve QP (6.5) under (σ¯ , ε¯ , c) ¯ and establish the final degradation path model (6.16) according to Theorem 6.2.

6.3 Residual Life Prediction Method Based on GA-Optimized SVR

129

According to the degradation path model (6.16), the degradation in the future time can be predicted and the life can be predicted accordingly. When the predicted degradation value reaches η for the first time, it is considered that the product has failed. If tT is defined as the life (failure time) of the product, then tT = min{t| f (t) = η}. Therefore, the residual life of the product at time t is tT − t.

6.3.4 Case Study Fatigue crack growth data [17] have been studied and analyzed by many scholars as performance degradation data, as shown in Fig. 6.1. There are 21 crack growth data, the initial crack length y0 is 2.286 cm, and the failure threshold η is 4.064 cm. The measurement is conducted every 104 stress cycles, and the test deadline is 12 × 104 stress cycles, so each crack can be measured up to 12 times (i.e., measurement time m = 12). If a crack length y ≥ η is found in a certain measurement, the crack will exit the test immediately. It can be found from Fig. 6.1 that 1# ~ 12# cracks have failed before the test deadline, during which 1# crack is measured for 9 times, 2# crack is measured for 10 times, 3# ~ 8# cracks are measured for 11 times, and 9# ~ 21# cracks are measured for 12 times. For all the 21 cracks, the last three measurements are taken as the test set, i.e., the number of samples in the test set is r = 3. According to Steps 1–4 of degradation path modeling method based on GA-optimized SVR in Sect. 6.3.3, the SVR algorithm, respectively, uses RBK-SVR, WK-SVR and PK-SVR for training modeling and prediction and searches for the optimal parameters of each crack under each kernel function. The value ranges of parameters of SVR algorithm defined in GA are as follows: ε ∈ [10−4 , 1], c ∈ [1, 100], σ ∈ [0.01,20] in RBK-SVR and WK-SVR, and σ1 ∈ [0,20], σ2 ∈ [2,10] in PK-SVR. In order to compare the prediction results, RBF Fig. 6.1 Fatigue crack growth data

4.5

3 9 1

η

2

8

12

4

y/cm

13

3.5

18 21

3

2.5

m

130

6 Degradation Modeling and Residual Life Prediction …

Table. 6.1 Relative estimation error results of four models on crack test set Model name

Time

Relative estimation error/% el

RBK-SVR

WK-SVR

PK-SVR

RBFNN

eh

em

m−2

−1.10

2.60

0.59

m−1

−1.29

5.15

2.17

m

0.01

12.96

6.14

m−2

−1.01

1.65

0.52

m−1

−0.07

3.89

1.77

m

0.83

9.49

5.38

m−2

−1.30

2.82

0.89

m−1

−1.61

6.53

3.12

m

−0.23

14.08

6.96

m−2

−2.32

3.01

1.31

m−1

−2.51

7.23

3.84

m

−0.86

15.12

7.56

neural network (RBFNN) is used to train and predict all the 21 cracks. Table 6.1 shows the relative estimation error results of four models on the test set, in which m−2, m−1 and m, respectively, represent the three measurement times of the test set. The relative estimation errors of the test samples at the same position of 21 cracks are statistically analyzed. el and eh are the minimum and maximum, respectively, and em is the absolute mean. The following results can be obtained from Table 6.1. (1)

(2)

(3)

The relative estimation error range (i.e., eh − el ) and the value of em of the four models on the test set increase with the increase of the measurement time, indicating that the generalization ability of the models decreases with the increase of the extrapolation distance. Therefore, it is required to minimize the extrapolation distance in order to obtain higher prediction accuracy. At each measurement time of the test set, the relative estimation error range of RBFNN model is larger than that of the other three SVR models, and the corresponding em value is much larger, indicating that the learning and generalization abilities of SVR algorithm are better than those of RBFNN algorithm in the case of small samples. Among the three SVR models, the relative estimation error range of WKSVR model is smaller than that of RBK-SVR and PK-SVR models, and the corresponding em value is also smaller, indicating that the mapping ability of WK is better than that of RBK and PK for fatigue crack growth data. Therefore, in this chapter, WK function is used in the analysis of fatigue crack growth data in order to obtain higher modeling and prediction accuracy.

Table 6.2 shows the optimal parameters determined by GA when WK-SVR algorithm is used to model the degradation path of 21 cracks. According to Step 5 of the

6.3 Residual Life Prediction Method Based on GA-Optimized SVR

131

Table 6.2 Parameters of GA-optimized WK-SVR algorithm Crack

ε (×10–4 )

σ

Crack

1

26.3672

c

σ

91.1719

0.2357

12

5.8828

91.2500

0.1116

2 3

16.6403

90.0000

0.1016

13

1.9531

81.2500

0.1058

9.8501

100.0000

0.0863

14

1.0000

94.9375

4

0.1387

1.0000

99.2500

0.0725

15

1.0000

78.9922

0.1038

5

1.0000

97.5508

0.0725

16

15.0757

96.5000

0.1272

6

28.8086

99.6719

0.0803

17

1.3428

92.0313

0.3282

7

30.8595

100.0000

0.0719

18

76.4160

96.0000

0.0935

8

27.5879

92.5938

0.3030

19

12.9394

94.9844

0.1038

9

1.9531

99.1133

0.1272

20

19.0430

97.1875

0.1139

10

1.0000

83.5000

0.0920

21

20.2637

86.0000

0.0869

11

1.0000

99.2500

0.0982

c

ε (×10–4 )

degradation path modeling method based on GA-optimized SVR in Sect. 6.3.3, a degradation path model (6.16) is established for 21 cracks under optimal parameters, and then the life prediction is carried out according to model (6.16). The results are shown in Table 6.3. Table 6.3 also shows the life prediction results of RBK-SVR, PK-SVR and RBFNN models, and the life estimated by Paris law (PL) model shown in reference [17]. The form of PL model is y = 0.9(1 − 0.9θ2 θ1 θ2 t)−1/θ2 Table 6.3 Life prediction results of five models Model name

Crack no 1

2

3

4

5

WK-SVR

8.8

RBK-SVR

8.8

PK-SVR

6

7

8

9

10

11

10.0

10.2

10.4

10.0

10.3

10.4

10.4

10.5

10.6

10.8

11.3

11.5

11.8

10.5

10.6

10.7

10.9

11.4

11.5

8.8

10.0

10.0

11.7

10.4

10.4

10.5

10.7

10.9

11.4

11.5

12.0

RBFNN

8.8

10.0

PL

8.8

10.0

10.1

10.3

10.2

10.5

10.5

10.8

11.2

11.4

11.8

10.1

10.3

10.3

10.6

10.6

10.9

11.3

11.5

Model name

Crack no

11.8

12

13

14

15

16

17

18

19

20

21

WK-SVR

11.8

12.7

13.7

13.8

14.6

14.9

15.6

16.2

16.8

17.0

RBK-SVR

11.8

12.5

14.0

13.5

14.8

15.1

16.0

16.6

17.2

17.4

PK-SVR

11.9

12.6

13.6

13.1

14.0

14.2

14.8

15.4

16.3

17.5

RBFNN

11.7

12.9

13.5

12.8

13.8

14.0

14.4

14.8

15.2

15.8

PL

11.8

12.9

13.3

13.8

14.4

14.6

15.1

16.0

16.7

17.0

132

6 Degradation Modeling and Residual Life Prediction … 17.5

WK-SVR RBK-SVR

17

PK-SVR 16.5

RBFNN PL

16 15.5 15 14.5 14 13.5 13 12.5 13

14

15

16

17

18

19

20

21

Fig. 6.2 Life prediction results of 13# ~ 21# cracks by five models

where θ1 and θ2 are fixed parameters, θ1 and θ2 among individuals are independently and identically normally distributed. It can be found from Table 6.3 as follows: (1) Due to the failure of 1# ~ 12# cracks, all the five models estimate life by interpolation, and the difference of their life prediction results is very small, indicating that the fitting ability of all the five models is very good. (2) For 13# ~ 21# cracks do not fail during the test, all the five models estimate the life by extrapolation, and the prediction results are quite different. For a more intuitive comparison, the life prediction results are shown in Fig. 4.2. From Fig. 6.2, the following results can be obtained. (1)

The predicted life values of 16# ~ 20# cracks are sorted from small to large in the order of RBFNN, PK-SVR, PL, WK-SVR and RBK-SVR models. The values of 15# and 21# cracks are out of order only because of one model. The extrapolation distance of 13# and 14# cracks is very small, which does not reflect the prediction rules of the five models. This indicates that the estimation increases in the order of RBFNN, PK-SVR, PL, WK-SVR and RBK-SVR models when the five models predict crack life by extrapolation. That is to say, the life prediction value of RBFNN model is the most conservative, which reflects that RBFNN model has the highest predicted degradation in the future time, which is consistent with the maximum relative estimation error of RBFNN model on the test set in Table 6.1. Again, this reflects that the generalization ability of RBFNN algorithm in the case of small samples is not as good as that of SVR algorithm.

6.3 Residual Life Prediction Method Based on GA-Optimized SVR

(2)

(3)

133

Based on the life prediction results of PL model, both WK-SVR and RBKSVR models overestimate the crack life, and WK-SVR model is closer to the prediction result of PL model than RBK-SVR model; both RBFNN and PKSVR models underestimate the crack life, and the underestimation of RBFNN model is more serious, which is still related to the poor generalization ability of RBFNN model in the case of small samples. By comparing the life prediction results of the three SVR models, most of the life prediction results of WK-SVR model are between RBK-SVR and PK-SVR models and are the closest to PL model. This indicates that WK-SVR model has the highest accuracy in life prediction, that is, the WK function is the most suitable.

6.4 Residual Life Prediction Method Based on SVR and FCM Clustering 6.4.1 Problem Description In the last section, we have obtained preferable prediction results in studying the degradation path model and life prediction method based on GA-optimized SVR aiming at the performance degradation data only for small samples of individual products. However, the method only takes advantage of the historical performance degradation data for small samples of the product itself and does not comprehensively consider the degradation data information of similar products. Nevertheless, we could also collect performance degradation data of products of the same kind (or the same batch) in engineering practice. Generally speaking, these data are abundant. Although there are differences among the degradation paths for individuals of similar products, the overall degradation trend is consistent. For example, (1) (2)

Fatigue crack growth data [17] (refer to Sect. 6.3.4); Degradation data of CG36A transistors [18]. Degradation life tests were carried out on 100 CG36A transistor samples. Each sample had been measured for 9 times separately at [0, 1, 3, 10, 30, 100, 250, 500, 1000] measurement time. While the products are determined to be failed if the performance parameter z increases by 30% relative to the initial value z 0 , the degradation amount is defined as y = (z − z 0 )/z 0 . Then, η is 0.3, and the degradation data are shown in Fig. 6.3.

According to Figs. 6.1 and 6.3, it can be found that the performance degradation data of similar products contain abundant information, especially much the same degradation paths and largely varied degradation paths exist side by side. By studying the similarity of degradation paths between specific individuals and similar products, we can find the degradation paths which are the most similar to specific individuals. If the degradation paths of similar products are weighted based on the similarity and taken as that of specific individuals for life prediction, a much more credible

134

6 Degradation Modeling and Residual Life Prediction …

100

80

y/%

60

40

20

0 0

100

200

300

400

500 t/h

600

700

800

900

1000

Fig. 6.3 Degradation data of CG36A transistors

result may be obtained. Moreover, degradation path models for specific individuals could be updated in real time on the basis of the similarity updated as per the realtime measurements of specific individuals. Therefore, based on the performance degradation data of similar products, we will study the real-time degradation path modeling and life prediction method for specific individuals in terms of the similarity among degradation paths. Considering the situation that the degradation paths of some similar products may be almost the same, merging similar degradation paths at offline stage is beneficial to reduce the weighted calculation at real-time life prediction stage when the situation is quite obvious. Fuzzy c-means (FCM) clustering algorithm is a clustering method based on an objective function, which realizes dynamic iterative clustering by repeatedly modifying the clustering center set and membership matrix, so as to achieve the maximum similarity among objects divided into the same cluster and the minimum similarity among those in different clusters [19]. In this section, it is proposed to take the normalized measurement time as the benchmark and the corresponding degradation measurement vector as the eigenvector while determining the similarity among degradation paths. Therefore, the Euclid distance among eigenvectors can reflect the similarity, that is, the shorter the Euclid distance, the greater the similarity; the longer the Euclid distance, the smaller the similarity. In conclusion, effective merging and classification could be achieved by applying FCM clustering analysis to the eigenvectors of similar products, regardless of whether or not its degradation path structure is the same. As a result, taking the degradation path modeling method based on GA-optimized SVR and FCM clustering algorithm as theoretical bases, a real-time degradation path modeling and a life prediction method based on

6.4 Residual Life Prediction Method Based on SVR and FCM Clustering

135

SVR and FCM clustering are proposed in this section in order to make full use of performance degradation data of similar products. Definition 6.4 Assuming [t1 , · · · , tm ]T is the normalized measurement time, there are n similar products in total, ti j is the jth measurement time for the ith product, yi j is the degradation measurement, mi is the total measurement times of the ith product, Di = {(ti j , yi j ), j = 1, · · · , m i } is the performance degradation data of the ith product, and Ds = {(t s j , ys j ), j = 1, · · · , m s } is the historical measurements of a specific individual. If mj=1 P(ti j = t j ) = 1 is true, Di is named as normalized  s P(ts j = t j ) = 1 is true, Ds is named as performance degradation data. If mj=1 normalized historical measurements. Problems to be solved in this section include (1) build a function model y = f s (t) for y in Ds of t by making full use of Di (i = 1, · · · , n) and applying SVR algorithm while both data sets Di (i = 1, · · · , n) and Ds are known; (2) predict the failure time tT for a specific individual combining the failure threshold η, that is, solve η = f s (tT ) so as to predict the residual life (tT − t) of a specific individual; (3) update the model y = f s (t) according to the real-time measurements (t, ys ) of a specific individual and make real-time life prediction. In the following sections, real-time degradation modeling and residual life prediction method based on SVR and FCM clustering are introduced in detail.

6.4.2 Basic Ideas and Specific Steps Degradation path curves in similar products are abundant. Degradation paths of specific individuals are very likely to be exactly similar or even identical to that of similar products or between that of several similar products. Therefore, a degradation path model for a specific individual could be established by weighting the model of similar products based on the similarity as long as the similarity of degradation paths between specific individuals and similar products can be determined. And the calculation on similarity can be updated in combination with real-time measurements of specific individuals, thus realizing real-time modeling of specific individuals. Among them, an accurate degradation path model of similar products is a prerequisite, and model accuracy can be ensured by the degradation path modeling method based on GA-optimized SVR at offline stage. The quantification of similarity is the key point. When Ds is a kind of normalized historical measurements, the similarity among degradation paths can be reflected by the Euclid distance between the data and the truncated eigenvectors of similar products; however, if Ds is not a kind of normalized historical measurements, it is not suitable for applying normalized treatment to Ds because of fewer measurement times considering that the more similar the two degenerate paths are, the closer the performance degradation data would be at any same time. Therefore, the differences between the performance degradation value of specific individuals at the last measurement time and that of similar products can reflect the similarity among degradation paths. As a result, we propose

136

6 Degradation Modeling and Residual Life Prediction … Historical measurements of specific individuals

Degradation data of similar products

Online stage Normalization treatment based on SVR

No

Is data normalized?

Yes

Is data normalized?

No

Real-time measurement

Error-based weighted method

Yes Clustering

Clustering center set

Offline stage

Degree-ofmembershipbased weighted method

SVR modeling

Real-time modeling and life prediction

Fig. 6.4 Flowchart of real-time degradation modeling and life prediction method based on SVR and FCM clustering

two real-time degradation path modeling and life prediction methods—degreeof-membership-based weighted method and error-based-weighted method—in this section considering whether Ds is a kind of normalized historical measurements. Complete flowchart of the method proposed in this section is shown in Fig. 6.4, and the method is divided into offline and real-time stages. (1)

(2)

Offline stage. First, determine whether the performance degradation data of similar products are normalized performance degradation data. If yes, optimal FCM clustering is directly performed; if not, normalized treatment based on SVR is applied before performing optimal FCM clustering; and finally, degradation path model for each clustering center is established based on SVR. Real-time stage. First, determine whether the historical measurements of specific individuals are normalized historical measurements. If yes, degree-ofmembership-based weighted method is adopted to get degradation path models of specific individuals; otherwise, error-based-weighted method is adopted; and then, carry out model update and life prediction in combination with real-time measurements of specific individuals.

6.4 Residual Life Prediction Method Based on SVR and FCM Clustering

1.

137

Offline stage

First, determine appropriate normalized measurement time as per the measurement time and times of n products. Determination principles of normalized measurement time are as follows: (1) try to make measurement times as many as possible and the interval of measurement time as equal as possible; (2) try to minimize the number of products requiring normalized treatment. In engineering practice, it is possible that not all collected data are normalized performance degradation data owning to the differences in measurement time and times for similar products of each user. Normalization determination should be made first before applying FCM clustering to degradation paths of similar products. If they are not normalized data, the model would be established by a degradation path modeling method based on GA-optimized SVR in Sect. 6.3, and then normalized treatment is conducted. Assuming Di is not a kind of normalized data, steps for normalized treatment are as follows: Step 1: The degradation path model f i (t) for Di is established by a degradation path modeling method based on GA-optimized SVR in Sect. 6.3.3; Step 2: Taking normalized measurement time as the benchmark, the corresponding missing time t j of Di is substituted in f i (t) for estimating the performance degradation data. Di is denoted as Dˆ i = {(ti j , yˆi j ), j = 1, · · · , m i } after normalized treatment and the eigenvector of Dˆ i is denoted as yi = [ yˆi1 , · · · , yˆim ]T . Eigenvectors yi of n samples form a matrix Y ∈ Rm×n and are divided into the c(2 ≤ c < n) Class. Assuming V ∈ Rm×c is a clustering center set, v k is the k th clustering center, dki is the Euclid distance between yi and v k , U ∈ Rc×n is a membership matrix, and u ki is the membership of yi to v k . Establish a clustering validity function F(U, c) based on possibility distribution [19]:  n  n c c n  1  2 1   −1 2 F(U, c) = u − u ki ) u ki ( n k=1 i=1 ki c k=1 i=1 i=1 c, denoted as c, ¯ is selected as the optimal number of clusters to minimize the value of F(U, c). And using this value, the optimal clustering center set and membership ¯ respectively. matrix are obtained and denoted as V¯ and U, Algorithm 6.1 (optimal FCM clustering algorithm) Step 1: Set a weighted index q and an iterative cease threshold δ to let c = 2. Step 2: Let the number of iterations p = 0, and initialize V (p) = [v1,…, vc]. Step 3: Obtain the membership matrix U according to Formula (6.19) and the n n   q q new clustering center Vˆ ( p) according to vˆ k = ( u ki )−1 ( u ki yi ). i=1

i=1

138

6 Degradation Modeling and Residual Life Prediction …

u ki =

⎧ 0, ⎪ ⎪ ⎨ c ⎪ ⎪ ⎩ j=1 1,

dk j = 0, i = j 2/(q−1) (dki d−1 , kj )

dk j = 0, dki = 0

(6.19)

otherwise

    Step 4: If  Vˆ ( p) − Vˆ ( p − 1) > δ, let p = p + 1 and forward to Step 3; otherwise, obtain the value of the clustering validity function according to Formula (6.18), let c = c + 1 and forward to Step 5. Step 5: If c < n, forward to Step 2; otherwise, denote c as c¯ which minimizes the ¯ value of F(U, c), and obtain corresponding V¯ and U. After FCM clustering, degradation path models { f¯1 (t), · · · , f¯c¯ (t)} for c¯ clustering centers are established accordingly by a degradation path modeling method based on GA-optimized SVR in Sect. 6.3 with the normalized measurement time [t1 , · · · , tm ]T as input and the clustering center v¯ i (i = 1, · · · , c) ¯ as output. What is worth mentioning is that: FCM clustering can neither well summarize information, nor effectively reduce the amount of calculation at the real-time stage if similar products are not in large quantities or the resemblance of degradation paths is not obvious; and if normalized treatment is required by most similar products, that is, SVR modeling is required by most similar products, better modeling accuracy can be obtained through directly applying aforementioned models to the calculation at real-time modeling stage. Therefore, FCM clustering is not necessary under above cases, and SVR models and normalized data of similar products are directly used in the calculation at real-time modeling stage. 2. (1)

Real-time stage Degree-of-membership-based weighted method

Degree-of-membership-based weighted method (DWM) is suitable for the case in which Ds is a kind of normalized historical measurements. The basic idea of DWM is to calculate the Euclid distance between the historical performance degradation vector of a specific individual and the vector composed of the first ms elements in each clustering center, so as to determine the membership value of a specific individual to each clustering center, thus weighting the degradation path model of each clustering center to obtain the degradation path model of a specific individual. Finally, Euclid distance, membership value and its degradation path model are updated in turn according to the real-time measurements of a specific individual, and the failure threshold is substituted into the model to obtain the failure time, thus realizing the real-time residual life prediction. The procedures are as follows. Step 1: Let p = m s , t0 = t p , ys = [ys1 , · · · , ysp ]T . Step 2: Take the first p rows of V¯ as a matrix V¯ p , calculate the Euclid distance of each column vector between ys and V¯ p , and obtain the membership zvector w = [w1 , · · · , wc¯ ] of a specific individual to the clustering center by substituting the distance into Formula (6.19).

6.4 Residual Life Prediction Method Based on SVR and FCM Clustering

Step 3: Establish a degradation path model f s (t) =

c¯ 

139

wi f¯i (t) of a specific

i=1

individual. Step 4: Obtain the predicted failure time tT by solving η = f s (tT ) for calculating the residual life tT − t0 . Step 5: Collect real-time measurements (t, ys ), let t0 = t, p = p + 1, ys = [ ys ; ys ] and forward to Step 2. (2)

Error-based-weighted method

Error-based-weighted method (EWM) [20] has no restrictions on Ds and is applicable to any situation. The basic idea of EWM is to substitute the last measurement time of a specific individual into the degradation path model of each clustering center, estimate the performance degradation data, obtain a degradation path model of a specific individual by weighting that of each clustering center (the weight is determined as per the square of each estimation error) and update the performance degradation estimation, weight and its degradation path model accordingly in combination with real-time measurements of a specific individual at least. The failure threshold is substituted into the model to obtain the failure time and realize real-time life prediction. The procedures are as follows: Step 1: Let p = m s , t0 = tsp , y0 = ysp . Step 2: Substitute t0 into { f¯1 (t), · · · , f¯c¯ (t)} to obtain { f¯1 (t0 ), · · · , f¯c¯ (t0 )}, calculate the square of the estimation error ei = [ f¯i (t0 ) − y0 ]2 and obtain the weight according to Formula (4.20) ⎧ 0, e = 0, j = i ⎪ ⎪ −1 j ⎪ ⎨  c¯ ei wi = , ei = 0, e j = 0 ; ej ⎪ ⎪ j=1 ⎪ ⎩ 1, otherwise Step 3: Establish a degradation path model f s (t) =

c¯  i=1

wi f¯i (t) of a specific

individual. Step 4: Obtain the predicted failure time tT by solving η = f s (tT ) for calculating the residual life tT − t0 . Step 5: Collect real-time measurements (t, ys ), let t0 = t, y0 = ys and forward to Step 2.

140

6 Degradation Modeling and Residual Life Prediction …

6.4.3 Case Study 6.4.3.1

Application in Fatigue Crack Growth Data

We will further study on the fatigue crack growth data which has been analyzed in Sect. 6.3.4. Select 5#, 10# and 15# cracks as specific individuals for life prediction while the residual 18 cracks are used as the performance degradation data of similar products. 1.

Offline stage

It is determined that the normalized measurement time is the first 11 measurement times (i.e., m = 11), and only 1# and 2# cracks need normalized treatment (supplemented). As the optimal parameters for each crack based on WK-SVR model have been obtained in Sect. 6.3.4, 1# and 2# cracks will conduct normalized treatment under their optimal parameters, respectively. Then, FCM clustering is carried out on the normalized eigenvectors of 18 cracks. Let the weighted index be q = 2 and the iterative cease threshold is δ = 10−5 in FCM clustering algorithm, the optimal number of clusters is determined to be 12 for the 18 cracks. Therefore, clustering will reduce the weighted calculation amount at the real-time stage by 1/3. Figure 6.5 shows the optimal clustering center obtained by FCM clustering. Compared with Fig. 6.1, the optimal clustering center keeps the characteristics of the original data, such as fewer cracks on the outer side and dense cracks in the middle; besides, the cracks with great differences have been classified separately. Finally, the modeling of degradation path for the 12 clustering centers is made by the degradation path modeling method based on GA-optimized SVR in Sect. 6.3.3. Fig. 6.5 Optimal FCM clustering center of fatigue crack growth data

5 4.5

y/cm

4

η

3.5 3 2.5

m

6.4 Residual Life Prediction Method Based on SVR and FCM Clustering

2.

141

Online stage

First, let t0 = 6, which means the first six measurements of a specific individual are used for modeling, and the measurements after the sixth measurement are used for testing the model accuracy; then, a measurement is added successively to realize realtime modeling, and the subsequent measurements are used to test the model accuracy. Use DWM and EWM for modeling experiments, respectively. Figure 6.6a and c shows the prediction results of 5# crack after 5 times of real-time modeling by the two methods, respectively. In order to compare the prediction results, real-time modeling and prediction are carried out based on the PL model with the same data. Figure 6.6b shows the prediction results of 5# crack after 5 times of real-time modeling. Table 6.4 gives the prediction errors among three specific individuals at the future normalized measurement time by the three methods, in which, el and eh represent the minimum and maximum values of the relative estimation errors, respectively, and em represents the absolute mean of the relative estimation errors. Table 6.5 shows the failure time, mean value and standard deviation of the six predictions by the three methods. In view of the wide applicability of EWM, simulation and interpolation are applied to the historical measurements of 5# crack based on the WK-SVR model, and realtime modeling is carried out by EWM at the interpolation time. The added nonnormalized measurement time is [5.5, 6.5, 7.5, 8.5, 9.5, 10.5], and the degradation amounts for corresponding interpolation are [2.9345, 3.0777, 3.2812, 3.5034, 3.7770, 4.1272]. Figure 6.6d shows the prediction results of 5# crack for 6 times of real-time modeling. Comparing Fig. 6.6a ~ c, it is found that the modeling accuracy of DWM, EWM and PL decreases in turn within the range of normalized historical measurements for 5# crack. And comparing Fig. 6.6b and d, it is found that the modeling accuracy of EWM is higher than that of PL within the range of non-normalized historical measurements for 5# crack. As we can see, both DWM and EWM consider the similarity of degradation path between specific individuals and similar products, and the modeling accuracy has been improved by weighting the degradation path model of similar products as per the similarity. Predictions are made for specific individuals at the 15 future normalized measurement time by the three methods by analyzing Table 6.4, and statistical analyses are applied to the 15 relative estimation errors. By comparing the relative estimation error range (i.e., eh − el ) and em , it is found that the prediction accuracy for the three specific individuals of DWM is the highest; the prediction accuracy for 10# crack of EWM is the worst, while the accuracy is higher for 5# and 15# cracks; and the prediction accuracy for 10# crack of PL is relatively higher, while the accuracy is the worst for 5# and 15# cracks. In conclusion, the modeling accuracy of DWM is higher than that of EWM. This might be caused by some relatively low measurement accuracy at the current time considering DWM which determines the weight based on all normalized historical data of specific individuals, and EWM determines the weight based on the current real-time data of specific individuals. Therefore, DWM is obviously more robust.

142

6 Degradation Modeling and Residual Life Prediction … Real track

4.4

Real track

4.4

t0 = 6

t0 = 6 4.2

t0 = 7

4.2

t0 = 7

t0 = 8

t0 = 8 4.0

t0 = 9

y/cm

y/cm

4.0

t0 = 10 3.8

t0 = 10 3.8

3.6

3.6

3.4

3.4

3.2

3.2

7

8

9

10

t0 = 9

7

11

8

9

m (a) Degradation Path Prediction of DWM 4.4

Real track

t0 = 5.5

t0 = 6

t0 = 6.5

4.2

t0 = 7

4.2

t0 = 7.5

t0 = 8

4.0

t0 = 9

y/cm

y/cm

4.0

t0 = 10 3.8

3.8

3.6

3.4

3.4

3.2

3.2 8

9

10

11

t0 = 8.5 t0 = 9.5

3.6

7

11

(b) Degradation Path Prediction of PL

Real track

4.4

10

m

t0 = 10.5

7

8

9

10

m

m (c) Degradation Path Prediction of EWM

(d) Degradation Path Prediction of

at the Normalized Time

EWM at the Non-normalized Time

Fig. 6.6 Degradation path prediction of DWM, PL and EWM for 5# crack

11

6.4 Residual Life Prediction Method Based on SVR and FCM Clustering

143

Table 6.4 Prediction accuracy of DWM, EWM and PL Crack No

Method

5#

10#

15#

Relative estimation error/% el

eh

em

DWM

−1.778

1.407

0.953

EWM

−2.859

1.580

1.113

PL

−9.658

0.924

3.019

DWM

−2.209

0.864

1.080

EWM

−3.071

1.025

1.466

PL

−0.754

2.859

1.157

DWM

−0.990

1.436

0.895

EWM

−2.848

0.504

1.174

PL

−3.888

2.841

1.682

Table 6.5 Prediction results of crack failure time μ

Crack no Methods t 0 /time 5#

10#

5#

8

8

9

10

σ

6

7

11

DWM

10.24

10.35 10.32 10.28 10.25 10.30 10.24 10.2900 0.0420

EWM

10.28

10.52 10.32 10.23 10.35 10.36 10.28 10.3433 0.0989

PL

10.98

11.60 10.63 10.51 10.41 10.32 10.98 10.7417 0.4791

DWM

12.11

12.02 11.79 11.67 11.78 11.61 12.11 11.8300 0.1963

EWM

12.33

12.17 12.00 11.86 11.84 11.73 12.33 11.9883 0.2259

PL

11.51

11.34 11.22 11.25 11.50 11.62 11.51 11.4067 0.1605

DWM

13.34

13.54 13.80 14.07 13.98 13.61 13.34 13.7233 0.2776

EWM

13.97

15.14 15.09 14.54 14.48 14.00 13.97 14.5367 0.5065

PL

14.64

12.43 12.55 12.68 12.98 13.15 14.64 13.0717 0.8138

Table 6.5 shows: (1)

(2)

The maximum standard deviations among the several prediction results of DWM and EWM are 0.2776 and 0.5065, respectively, while that of PL is 0.8138. It shows that DWM has the best robustness, followed by EWM, and PL has the worst robustness. The failure time is overestimated due to the underestimation of degradation path. Combined with Fig. 4.6, we can see that PL always underestimates the degradation path for 5# crack, so the prediction result of failure time for 5# crack by PL is higher than the true value; the real-time estimation of degradation path for 5# crack by DWM and EWM varies from high to low, so it is more credible to take the mean value of multiple prediction results of failure time as the life estimation value.

144

6 Degradation Modeling and Residual Life Prediction …

6.4.3.2

Application of Degradation Data in CG36A Transistors

The characteristics of degradation data for CG36A transistors have been introduced in the first paragraph of Sect. 6.3.1. Chen et al. [21] fit the degradation path by y = θ1 t θ2 model (Ch&Zh). Considering the symbolic constancy of the model, 11 samples containing negative growth stages have been eliminated, and a sample with excessive long life found in the individual life prediction has also been eliminated. Finally, 88 samples were used to estimate overall life distribution. Reference [21] pointed out that it may be inappropriate to eliminate 12 samples, the validity of the data shall be verified, and the reasons leading to individual differences shall be found. These individuals shall be treated specially. In view of the method proposed in this section, all kinds of nonlinear curves can be better fitted by the degradation path modeling method based on GA-optimized SVR. FCM clustering algorithm can separate special individuals into a class. Therefore, the method proposed in this section does not need to eliminate samples. 1.

Offline stage

Define nine measurement times as the normalized measurement time, and the data for the hundred samples are all normalized performance degradation data. Select three samples as specific individuals, and number them in 1#, 2# and 3#, respectively. Carry out optimal FCM clustering on the residual 97 samples. The optimal number of clusters is determined to be 29, and clustering will reduce the weighted calculation at the real-time stage by 70%. Figure 6.7 shows the optimal clustering center obtained by FCM clustering. Compared with Fig. 6.3, the optimal clustering center keeps the

100

80

y/%

60

40

20

0 0

200

400

600 t/h

Fig. 6.7 Optimal FCM clustering center for transistor data

800

1000

6.4 Residual Life Prediction Method Based on SVR and FCM Clustering

145

Table 6.6 Prediction accuracy of DWM, EWM and Ch&Zh Crack no 2#

3#

3#

Method

Relative estimation error/% el

eh

em

DWM

−6.710

−2.262

4.317

EWM

−15.840

1.464

6.394

Ch&Zh

22.266

60.169

35.725

DWM

1.162

6.652

4.762

EWM

−0.760

33.548

11.210

Ch&Zh

11.269

34.639

21.811

DWM

−3.525

1.131

1.398

EWM

−6.777

1.890

2.973

Ch&Zh

12.928

31.311

18.976

characteristics of the original data, such as fewer curves on the outer side and dense curves in the middle; besides, the curves with great differences have been classified separately. During degradation path modeling based on GA-optimized SVR, zero hour and its corresponding zero initial degradation amount are removed to reduce the influence caused by unequal time intervals. And take logarithm based on 10 of the rest measurement time, that is, lg t is taken as the input. 2.

Real-time stage

First, the first five measurements of a specific individual (which means, t0 = 100, lg t0 = 2 is adopted for the starting point) are used for modeling, and the next three measurements are used for testing the model accuracy; then, a measurement is added successively to realize real-time modeling, and the subsequent measurements are used to test the model accuracy. Use DWM and EWM in modeling experiments, respectively. And use Ch&Zh mode for real-time modeling and prediction with the same measurements. Table 6.6 gives the prediction errors among three specific individuals at the future normalized measurement time by the three methods. Table 4.7 shows the failure time, mean value and standard deviation of the four predictions by the three methods. Predictions are made for specific individuals at the six future normalized measurement time by the three methods by analyzing Table 6.6, and statistical analyses are applied to the six relative estimation errors. By comparing the relative estimation error range (i.e., eh − el ) and em , it is found that: (1)

For the three specific individuals, the prediction accuracy decreases in the order of DWM, EWM and Ch&Zh. It shows that both DWM and EWM consider the similarity of degradation path between specific individuals and similar products, and the modeling accuracy has been improved through weighting the degradation path model of similar products based on the similarity; moreover, the inaccuracy of real-time data has a greater impact on the prediction accuracy

146

6 Degradation Modeling and Residual Life Prediction …

Table. 6.7 Prediction results of transistor failure time Transistor no 1#

2#

3#

(2)

μ

σ

53.333

53.713

1.185

53.456

45.375

15.430

58.108

63.649

53.745

9.311

176.198

177.828

172.584

174.207

3.447

203.704

186.209

177.419

185.579

13.009

126.613

150.915

186.029

216.535

170.023

39.451

DWM

27.670

27.353

27.040

27.227

27.323

0.265

EWM

28.576

26.792

27.102

29.235

27.926

1.169

Ch&Zh

33.498

38.477

42.472

46.020

40.117

5.382

Method

t 0 /h 100

250

500

1000

DWM

55.463

53.211

52.845

EWM

22.233

52.845

52.966

Ch&Zh

42.073

51.151

DWM

170.216

EWM

174.985

Ch&Zh

of EWM than that of DWM considering DWM determines the weight based on all historical data of specific individuals, and EWM determines the weight based on the current real-time data of specific individuals. The prediction errors of CH&ZH method among the three specific individuals are very large, all relative estimation errors are higher than 11%, and the maximum error is up to 60%. It shows that the generalization ability of Ch&Zh method is very poor. Consequently, the method is not suitable for extrapolation prediction of the three transistors selected in this section even it can accurately describe the degradation rule of most transistors. Table 6.7 shows the following prediction results:

(1)

(2)

(3)

(4)

(5)

All three specific individuals failed in the experiments. Both 1# and 3# transistors have failed for no more than 100 h, which represents as t0 at the real-time stage, and the failure time of 2# transistor did not exceed the next measurement time, that is to say, they are all failure time estimated by interpolation. The differences among lives estimated by all previous interpolation shall be very small if Ch&Zh model is applicable as the estimation is based on the interpolation. However, the differences among the thre life predictions for 2# transistor are large. It shows that Ch&Zh model is not suitable due to the greatly varied parameter estimations with the increase in test data. It can be seen from the original data that the each prediction result of Ch&Zh model is greater than 30 while the failure time of 3# transistor is less than 30 h. Therefore, Ch&Zh model cannot accurately describe the degradation rule of 3# transistor. In terms of 1# transistor, most prediction results of the three methods are about 50 h. After eliminating 22.233, the mean value of the residual 11 prediction results is 53.555, which is the closest to the mean value of previous predictions of DWM, indicating that the prediction results of DWM are the most credible. The maximum standard deviations among the several prediction results of DWM and EWM are 3.447 and 15.430, respectively, while that of PL is 39.451.

6.4 Residual Life Prediction Method Based on SVR and FCM Clustering

147

It shows that DWM has the best robustness, followed by EWM, and PL has the worst robustness.

6.5 Summary of This Chapter In this chapter, we studied the problem of nonlinear degradation path modeling under small sample data and considered two kinds of data sources, respectively: (1) small sample data for specific individuals only and (2) abundant data for similar products at the same time. In view of degradation modeling under small sample data, a degradation path modeling and life prediction method based on GA-optimized SVR is proposed in order to improve the performance of SVR algorithm by applying GA optimization to model parameters selection and comparing and selecting kernel functions. The analysis of fatigue crack growth data shows that the prediction accuracy of SVR model is higher than that of RBF neural network model, and the mapping ability of WK function to this kind of data is determined to be the best. When there are rich degradation data of similar products, in order to make full use of these data to further improve the model accuracy, a modeling idea based on weighted similarity is proposed from the perspective of considering the similarity of degradation path. On account of the degradation data for normalized similar products of SVR model, FCM clustering algorithm is adopted to summarize the normalized data. To determine whether the historical measurement time of a specific individual is normalized, two methods for real-time degradation path modeling and life prediction based on SVR and FCM clustering are proposed: (1) degree-of-membership-based weighted method (DWM) applicable to specific individuals with normalized historical measurement time, the membership weight of which is determined in accordance with the Euclid distance between the degradation paths of special individuals and similar products; (2) error-based-weighted method (EWM) with universality and no restriction on historical measurement time, the weight of which is determined as per the prediction error of special individuals based on the degradation path model of similar products at the last measurement time. Finally, the proposed method is applied to fatigue crack growth data and degradation data of CG36A transistors to verify its validity. And the result shows that the prediction accuracy of DWM is higher than that of EWM and existing models while the robustness of DWM is the best.

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Chapter 7

Degradation Modeling and Residual Life Prediction Based on Fuzzy Model of Relevance Vector Machine

7.1 Introduction Generally, the core of realizing the prediction, control and decision-making of the actual system is to establish the mathematical model of the system. However, most systems are characterized by complexity, morbidity and nonlinearity, so it is difficult to establish accurate mathematical model by traditional modeling methods. In contrast, fuzzy system has obvious advantages in dealing with the problems of complexity, nonlinearity and fuzzy uncertainty. Fuzzy system is composed of a series of “If-Then” rules, which is widely used in the modeling of complex nonlinear systems and brings good effect because it can use language information, data information and fuzzy uncertainty information in the real world simultaneously and can easily select initial parameters to accelerate the convergence of identification algorithm. The key to degradation modeling and predictive problems based on fuzzy system is to describe the mathematical model of system degradation rule with historical information of equipment (qualitative or quantitative) and optimize and update the model with real-time measured information, so as to achieve degradation modeling and prediction. Obviously, the core of the problem is to establish a fuzzy system model reflecting the degradation of equipment. Generally, there are two methods for fuzzy modeling [1]: (1) based on prior knowledge, that is, “empirical method.” The fuzzy system established based on this method is usually called fuzzy expert system: (2) based on data, that is, “data-driven method.” The process of establishing fuzzy model based on this method is called fuzzy model identification, which is the most commonly used fuzzy modeling method. The method includes two stages: structure identification and parameter identification. Among them, structure identification is the difficulty and core of fuzzy model identification [2]. In addition, some scholars have also studied the combination of empirical method and data-driven method [3]. When the prior knowledge of the identified object is insufficient and there is only a large number of input and output data, the clustering method is considered to be the most appropriate method for structure identification [4]. Especially, the © National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_7

149

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7 Degradation Modeling and Residual Life Prediction Based …

fuzzy model identification based on fuzzy clustering has always been the research focus in the field of fuzzy modeling. However, the data we obtained are limited in most cases. Especially in the early stage of equipment operation, only a small amount of test data can be obtained. In this case, it is difficult to guarantee the effect of the fuzzy model based on fuzzy clustering. Therefore, we need to further consider a fuzzy modeling method with good performance in the case of small sample data. In recent years, the current machine learning ability is greatly improved, and the generalization ability and modeling quality of fuzzy model are improved by integrating support vector machine (SVM) and kernel function into the research of fuzzy system theory. Therefore, the fuzzy modeling method based on SVM and kernel function becomes a new research hotspot in the field of fuzzy modeling [1, 5–8]. However, SVM algorithm also has such disadvantages as weak sparsity, large amount of calculation and kernel function must meet Mercer condition [9]. Relevance vector machine (RVM) is a nonlinear machine learning algorithm based on positive definite kernel proposed by Tipping [10, 11]. It is based on sparse Bayesian learning theory. It not only has the advantages of SVM such as avoiding overlearning and small sample modeling, but also uses fewer kernel functions than SVM while achieving the same performance as SVM, and the kernel function does not need to meet Mercer conditions, so it is provided with better generalization ability [12]. It is a new idea to apply RVM method to fuzzy model identification. Kim et al. first constructed T-S fuzzy inference system (FIS) with RVM method [13]. However, they did not explain the internal relationship between RVM and FIS, did not theoretically prove the uniform approximation of constructed FIS and did not study how to construct Mamdani fuzzy inference system with RVM method. At present, there are few studies on fuzzy model identification with RVM method. In view of the above problems, this chapter mainly studies the following contents: firstly, analyze the similarity between RVM and FIS based on the functional form, give the function form of fuzzy model based on RVM and prove the uniform approximation with Stone–Weierstrass theorem; secondly, propose a fuzzy model identification algorithm based on RVM and gradient descent. That is, extract fuzzy rules and obtain the initial values of the model structure and parameters with RVM method, and then optimize and update the model parameters with gradient algorithm to realize the parameter identification of the fuzzy model; thirdly, give the degradation modeling and residual life prediction algorithm based on the established fuzzy model and its parameter identification method; finally, apply the proposed fuzzy model identification and prediction method into the simulation data of continuous stirred tank reactor [14–16].

7.2 Fuzzy Model Based on Relevance Vector Machine

151

7.2 Fuzzy Model Based on Relevance Vector Machine 7.2.1 Mathematical Description of Fuzzy Model Consider a multi-input single-output Mamdani fuzzy model. The general form of fuzzy rules is j

j

R j : If x1 is A1 and x2 is A2 and · · · and xr is Arj , Then z is B j

(7.1)

where R j ( j = 1, 2, · · · , M) represents the fuzzy rules, M is the number of rules; xi (i = 1, 2, · · · , r ) represents the input, r is the input dimension; z represents the j output of the fuzzy model; Ai and B j represent language items represented by fuzzy degree-of-membership-based function u A j (xi ) and u B j (z), respectively. i If product inference engine, single-value fuzzy generator and central average fuzzy canceller are used, the whole fuzzy inference function can be expressed as   r j j (x i ) z ¯ u j=1 i=1 Ai  f (x) =   M r j (x i ) u j=1 i=1 A M

(7.2)

i

where f : R r → R and u A j (xi ) are selected as Gaussian membership functions and i z¯ j is the point where u B j (z) obtains the maximum value in the output space. As mentioned above, the key to the prediction of degradation amount and residual life is to construct the mathematical model of the system. Therefore, the main task of this chapter is to complete the identification of the fuzzy model as shown in Formula (7.2), which can be divided into the following two parts: (1) structure identification: mainly completes the division of fuzzy space and determines the number of rules M in Formula (7.2); (2) parameter identification: mainly completes the optimal estimation of degree-of-membership-based function parameters (including center and width) and conclusion parameter z¯ j in Formula (7.2). A method for identifying fuzzy models with RVM is given below. RVM is mainly used to extract fuzzy rules from training data, determine the initial value of fuzzy model structure and parameters and build the initial fuzzy model.

7.2.2 Fuzzy Model Based on Relevance Vector Machine RVM algorithm combines Markov property, Bayes principle, automatic relevance determination (ARD) theory and maximum likelihood theory, which is commonly used in classification and regression problems. This section first studies the principle

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7 Degradation Modeling and Residual Life Prediction Based …

of RV regression [13–17], then proposes a method to construct fuzzy model with RVM and proves the uniform approximation of the constructed model. (1)

Regression principle of relevance vector machine

  Given a multi-input single-output training sample set (x k1 , tk1 ), k1 = 1, · · · , n , x k1 ∈ Rr and tk1 ∈ R, n is the number of training samples. Assuming that the target value tk1 is independent and identically distributed and contains Gaussian noise of εk1 ∼(0, σ 2 ), then tk1 = f (x k1 ; w) + εk1

(7.3)

The model output of RVM is defined as f (x; w) =

n 

wk1 K (x, x k1 ) + w0 = w

(7.4)

k1 =1

where w = (w0 , w1 , · · · , wn )T is the weight vector;  is the n × (n + 1) order design matrix, and  = [φ(x 0 ), φ(x 1 ), · · · , φ(x n )], φ(x k1 ) = [K (x 1 , x k1 ), K (x 2 , x k1 ), · · · , K (x n , x k1 )]T , φ(x 0 ) = [1, · · · , 1]Tn×1 and K (x, x k1 ) are the kernel functions, and x represents the input vector. Since the target value tk1 is independent, the likelihood function of the whole training sample can be expressed as

1 p(t|w, σ 2 ) = (2π σ 2 )−n/2 exp − 2 t − w2 2σ

(7.5)

where t = [t1 , · · · , tn ]T is the output vector. If the maximum likelihood method is directly used to solve w, it will lead to serious overfitting. Generally, in order to improve the generalization ability of the model, an additional constraint is added to the weight coefficient w. For example, add complex penalty function or error function to likelihood function. RVM defines the Gaussian prior probability distribution for each weight α = [α0 , α1 , · · · , αn ]T directly through the hyper-parameter wk2 p(w|α) =

n  k2 =0

N (wk2 |0, αk−1 )= 2

n  αk2 /2π exp(−αk2 wk22 /2)

(7.6)

k2 =0

where each hyper-parameter αk2 corresponds to a weight wk2 and k2 = 0, 1, · · · , n. Given the prior probability distribution and likelihood distribution, the posterior probability distribution of weight is calculated according to Bayesian criterion

7.2 Fuzzy Model Based on Relevance Vector Machine

153

p(t|w, σ 2 ) p(w|α) p(t|α, σ 2 )

(7.7)

p(t|w, σ 2 ) p(w|α)dw = N (0, C)

(7.8)

p(w|t, α, σ 2 ) = The denominator of Eq. (7.7) is

p(t|α, σ ) = 2

where C = σ 2 I + A−1 T , A = diag(α0 , α1 , · · · , αn ). It can be seen that the denominator has nothing to do with w. Thus, we know that the posterior distribution of the weight also follows the multivariate Gaussian distribution p(w|t, α, σ 2 ) = N (μ, )

(7.9)

with the expressions of mean and covariance as follows: μ = σ −2 T t

(7.10)

 = (σ −2 T  + A)−1

(7.11)

where A = diag(α0 , α1 , · · · , αn ) is the hyper-parametric matrix. In the continuous calculation of RVM, most of wk2 tends to be zero, and the learning samples corresponding to nonzero wk2 are called relevance vectors (RVs). Assuming that  x k3 (k3 = 1, · · · , N ) represents the relevance vectors, N is the number of relevance vectors, K (x, x k3 ) represents the kernel function, (x) is the x 2 ), · · · , K (x, x N )] 1×(N +1) order design matrix and (x) = [1, K (x, x 1 ), K (x, and  w = (w˜ 0 , w˜ 1 , · · · , w˜ N )T are vectors composed of non-zero weight, then Formula (7.10) can be written as f (x;  w) =

N 

wk3 K (x, x k3 ) + w˜ 0 = (x) w

(7.12)

k3 =1

(2)

Fuzzy model based on relevance vector machine

   r j (x) u is defined as a fuzzy basis function, If p j (x) = ri=1 u A j (x)/ M j=1 i=1 Ai i then Formula (7.2) can be written as f (x) =

M  j=1

p j (x)¯z j = P(x)Z

(7.13)

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7 Degradation Modeling and Residual Life Prediction Based …

where P(x) = ( p1 (x), · · · , p M (x)) represents the fuzzy basis function matrix and Z = (¯z 1 , · · · , z¯ M )T represents the conclusion parameter matrix. Comparing Formulas (7.12) and (7.13), it can be seen that they have essential similarity except for the differences in notation, that is, they can be regarded as the product form of a set of basis functions and their corresponding weights. Therefore, some RVM models derived from sparse Bayesian theory can be interpreted as FIS obtained by adding and multiplying fuzzy operators, and vice versa. In this case, each sample data as the relevance vectors corresponds to a fuzzy inference rule, and the value of the relevance vectors in the kernel function corresponds to the membership function value of the antecedent of the fuzzy inference rule, so that the two mechanisms are integrated. This provides a feasible method for identifying fuzzy model based on sample data. Based on the comparative analysis of the relationship between RVM and FIS, a method for constructing fuzzy models with RVM is given below. Firstly, it describes the functional form of fuzzy model based on RVM, then gives the definition of Stone algebra [18] and Stone–weierstrass lemma [18, 19] and finally proves that the fuzzy model has uniform approximation in the case of Gaussian membership function. The fuzzy model based on RVM can be expressed as the following function form M j=1

f (x) =  M

z¯ j K (x, x j)

j=1

K (x, x j)

+ w˜ 0

(7.14)

where K (x, x j ) = ri=1 μ A j (xi )( j = 1, · · · , M) represents the kernel function, r is i the dimension of the input vector, M is thenumber of fuzzy  rules (equal to the number  j 2 1 xi −x˜i is a Gaussian membership of relevance vectors N), μ A j (xi ) = exp − 2 j i

j x˜i

σi

j

function, and σi represent the center and width of the membership function, j j j respectively, x represents the input vector,  x j = (x˜1 , · · · , x˜i , · · · , x˜r ) represents the corresponding conclusion parameters of rule j, z¯ j represents the corresponding conclusion parameters of rule j, and w˜ 0 represents adjustable parameters.

7.2.3 Uniform Approximation of Fuzzy Model Based on Relevance Vector Machine Define 7.1 [18] Stone algebra Let Z be a set of real continuous functions defined on compact universe U. If Z satisfies the following conditions, we consider Z to be a Stone algebra on compact universe U. (a)

Z is the algebra, that is, Z is closed to addition, multiplication and scalar multiplication;

7.2 Fuzzy Model Based on Relevance Vector Machine

(b) (c)

155

Z can divide the points on U, that is, for every x, y ∈ U , if x = y, there must be f ∈ Z , so that f (x) = f ( y); Z does not disappear at any point on U, that is, for every x ∈ U , there is f ∈ Z , so that f (x) = 0.

Lemma 7.1 [18, 19] Stone-Weierstrass theorem If the set U of real continuous functions defined on compact universe Z is a Stone algebra, then Z is compact everywhere in the set U of all continuous real functions on C(U), that is, any continuous function on Z can be approximated arbitrarily by the elements in Ug(x). Based on the above definition of Stone algebra and Stone–Weierstrass theorem, the uniform approximation theorem and proof of RVM based fuzzy model are given below. Theorem 7.1 Uniform approximation theorem of fuzzy model based on RVM If the set of fuzzy model Y based on relevance vector machine contains all functions in the form of function described in Formula (7.14), then Y has uniform approximation to any continuous function on compact universe U. Proof According to Definition 7.1 and Lemma 7.1, we only need to prove Y to be a Stone algebra on compact universe U. (1)

Firstly, prove that Y is the algebra. Assuming f 1 , f 2 ∈ Y , it can be written as  M1

j =1

f 1 (x) = 1 M1

z¯ j1 K (x, x j1 )

j1 =1

 M2

j =1

f 2 (x) = 2 M2

z¯ j2 K (x, x j2 )

j2 =1

(a)

K (x, x j1 )

K (x, x j2 )

+ w˜ 01

(7.15)

+ w˜ 02

(7.16)

Addition closure  M1  M2 f 1 (x) + f 2 (x) =

(b)

z j1 + z¯ j2 )K (x, x j1 )K (x, x j2 ) j2 =1 (¯ + (w˜ 01 + w˜ 02 )  M1  M2 x j1 )K (x, x j2 ) j1 =1 j2 =1 K (x, (7.17)

j1 =1

Multiplication closure  M1  M2 f 1 (x) f 2 (x) =

j1 =1

+ (w˜ 01 w˜ 02 ) (c)

(w˜ 02 z¯ j1 + w˜ 01 z¯ j2 + z¯ j1 z¯ j2 )K (x, x˜ j1 )K (x, x˜ j2 )  M1  M2 ˜ j1 )K (x, x˜ j2 ) j1 =1 j2 =1 K (x, x

j2 =1

Scalar multiplication closure

(7.18)

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7 Degradation Modeling and Residual Life Prediction Based …

In terms of any c0 ∈ R, it is possible to obtain  M1

j1 x j1 ) j1 =1 (c0 z¯ )K (x,  M1 x j1 ) j1 =1 K (x,

c0 f 1 (x) =

+ (c0 w˜ 01 )

(7.19)

Since membership functions are Gaussian membership functions and their product is still Gaussian type, Formulas (7.17), (7.18), (7.19) and (7.14) have the same form, that is, f 1 (x) + f 2 (x) ∈ Y, f 1 (x) f 2 (x) ∈ Y, c0 f 1 (x) ∈ Y , so Y is an algebra. (2)

Secondly, prove that Y can divide the points on U. Here, we prove that Y can divide the points on U by constructing a function satisfying the requirements.

Let any x 0 , y0 ∈ U , if x 0 = y0 . To construct f ∈ Y , so that f (x 0 ) = f ( y0 ). There are only two rules in current design f : R1 : If x1 is A11 and x2 is A12 and · · · and xr is Ar1 , Then z is z¯ 1 ; R2 : If x1 is A21 and x2 is A22 and · · · and xr is Ar2 , Then z is z¯ 2 . Assuming x 0 = (x10 , x20 , · · · , xr0 ) and y0 = (y10 , y20 , · · · , yr0 ), define two fuzzy sets Ai1 , Ai2 (i = 1, · · · , r ), the corresponding membership function is     (xi − xi0 )2 (xi − yi0 )2 , μ Ai2 (xi ) = exp − μ Ai1 (xi ) = exp − 2 2

(7.20)

In this way, the design parameters of part f are defined, only z¯ 1 , z¯ 2 and wx0 , w y0 are to be determined. Therefore,   z¯ 1 + z¯ 2 ri=1 exp −(xi0 − yi0 )2 /2   f (x 0 ) = (7.21) + wx0 1 + ri=1 exp −(xi0 − yi0 )2 /2   z¯ 2 + z¯ 1 ri=1 exp −(xi0 − yi0 )2 /2   + w y0 f ( y0 ) = (7.22) 1 + ri=1 exp −(xi0 − yi0 )2 /2 0 0 0 Because of x 0 = there is  y , there is always a i, which makes xi 1= yi , so  0 0 2 exp −(xi − yi ) /2 = 1. In this case, we only need to select z¯ = 0, z¯ 2 = 1 and select the appropriate wx0 , w y0 , and then we can get f (x) = f ( y).

(3)

Finally, prove that Y does not disappear at any point on U.

In Formula (7.14), we only need to select z¯ j ≥ 0( j = 1, · · · , M) and w˜ 0 > 0, and then we can get M j=1

f (x) =  M

z¯ j K (x, x j)

j=1

K (x, x j)

+ w˜ 0 ≥ 0 + w˜ 0 > 0

(7.23)

7.2 Fuzzy Model Based on Relevance Vector Machine

157

According to the above proof process, Y is a Stone algebra on compact universe U. Therefore, it can be concluded as follows: The fuzzy model Y based on RVM can approximate any real continuous function on compact universe U with arbitrary accuracy, that is, for any real continuous function U on compact universe g(x) a RVM-based fuzzy model f (x) based on RVM can always be found to make f (x) uniformly approximate to g(x). Remark 7.1 The function form shown in Formula (7.14) increases adjustable parameters w˜ 0 compared with the fuzzy inference model shown in Formula (7.2), which improves the schedulability of the model, but has no effect on the function type. That is, Formula (7.14) is still a fuzzy model. From Theorem 7.1, it can be seen that the fuzzy model constructed by RVM method as shown in Formula (7.14) can approximate to any real continuous function with arbitrary accuracy, and it is necessary to optimize the model parameters reasonably. Based on the function form of Formula (7.14), a fuzzy model identification method based on relevance vector machine and gradient descent algorithm is given below.

7.3 Fuzzy Model Identification Based on Relevance Vector Machine In terms of the identification of the fuzzy model shown in Formula (7.14), it can be realized from the following two aspects: (1) structure identification, that is, determines the number of fuzzy rules M in Formula (7.14); (2) parameter identification, j j that is, estimates the center x˜i and width σi of membership function, conclusion j parameters z¯ and adjustable parameters w˜ 0 in Formula (7.14) through learning algorithm. For the fuzzy model based on RVM constructed in the previous section, RVM is mainly used to realize the structure identification and initial parameter identification of the fuzzy model. The learning algorithm is needed to further optimize and adjust the model parameters. The identification process (as shown in Fig. 7.1) and the identification algorithm are given below.

7.3.1 Structure Identification In the process of identifying fuzzy models with RVM method, RVM extracts relevance vectors (RVs) from training samples, and each relevance vector corresponds to a rule. The structure identification algorithm—Algorithm 7.1, which based on RVM fuzzy model can be obtained. The algorithm flowchart is shown in Fig. 7.2, and the algorithm steps are as follows:

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7 Degradation Modeling and Residual Life Prediction Based … Sample data

Extract relevance vectors in RVM, RV1, RV2, ···, RVN

Parameter learning

If x is RV1, then z is B1 Input x

If x is RV2, then z is B2

�

Defuzzifi cation

If x is RVN, then z is BN

Output z

Fig. 7.1 Identification process of fuzzy model based on RVM

End

Input and output data preprocessing

The number of relevance vectors is the number of rules M, and the initial values of the corresponding parameters are obtained

Initialization of model parameters α, σ2, kmax, αmax

Delete the weight and kernel function of ≥αmax, and set the remaining samples as relevance vectors

k =1

Yes Calculate A and Φ

Calculate μ and Σ

Calculate γk

Update αk and σ2

k = k +1

k = kmax No

Fig. 7.2 Flowchart of fuzzy model structure identification based on RVM

Algorithm 7.1: Structure identification algorithm of fuzzy model based on RVM Step 1. Input and output data preprocessing: determine the dimension of input and output variables and generate training samples and test samples. Step 2. System initialization: including hyper-parameter α = [α0 , α1 , · · · , αn ]T , kernel function width σ 2 , maximum number of cycles kmax and upper limit of hyper-parameter αmax . The membership function shall be Gaussian membership function and let k = 1.

7.3 Fuzzy Model Identification Based on Relevance Vector Machine

159

Step 3. Calculate the hyper-parametric matrix A = diag(α0 , α1 , · · · , αn ) and design matrix . Step 4. Calculate the posterior statistics μ and  of the weight, respectively, with Formula (7.10) and Formula (7.11), so that γ k2 = 1 − αk2 (k2 +1)(k2 +1) , and (k2 +1)(k2 +1) is the diagonal elements of the weight posterior covariance matrix  k2 + 1, k2 = 0, 1, · · · , n. = Step 5. Update the hyper-parameter and kernel function width with αknew 2   n 2 2 2 new γk2 /μ(k2 +1) and (σ ) = t − μ / n − k2 =0 γ k2 , μ(k2 +1) is the k2 + 1 term of posterior mean. Step 6. k = k + 1, if k = kmax , then transfer to Step 7. Otherwise, return to Step 3. Step 7. Delete the weight and its corresponding kernel function corresponding to αk2 ≥ αmax of αk2 wk2 in the hyper-parameter, and set the residual samples as relevance vectors. Each relevance vectors corresponds to a fuzzy rule, and the sample data corresponding to the relevance vector are ( x j , z¯ j )( j = 1, · · · , M), where M is the number of fuzzy rules (equal to the number of relevance vectors),  x j is the initial value of the center vector of the membership function of rule j, and z¯ j is the initial value of the corresponding conclusion parameter. Meanwhile, the updated kernel function width σ 2 and weight w0 are used as the initial values of membership j function width σi and adjustable parameter w˜ 0 , respectively.

7.3.2 Parameter Identification After determining the structure of the fuzzy model and the initial values of the parameters with RVM method, how to learn and optimize the parameters is also an important aspect affecting the identification accuracy. Gaussian membership function is used here. If θ is used to represent all parameters to be estimated, then θ = (z 1 , · · · , z M , x˜11 , · · · , x˜rM , σ11 , · · · , σrM , w˜ 0 ). The gradient descent algorithm is used to optimize and update all parameters, and the parameter learning process is shown in Fig. 7.3.   Assuming that any input–output data pair obtained is x k1 , z k1 , f (x k1 ; θ ) represents the output of the fuzzy model based on RVM, z k1 represents the actual output, e(θ ) represents the difference between the model output and the actual output, k1 = 1, · · · , n, and n is the number of training data. Take the criterion function as J=

n n 2 1  1 2 f (x k1 ; θ ) − z k1 e (θ ) = 2 k =1 2 k =1 1

1

(7.24)

160

7 Degradation Modeling and Residual Life Prediction Based … Actual output z Actual system

e(θ )

Input x

Fuzzy model based on RVM

Model output

Parameter learning j σ i j w0 z j xi

Fig. 7.3 Parameter learning diagram of fuzzy model based on RVM

If gradient descent algorithm is used, for conclusion parameter z¯ j ( j = 1, 2, · · · , M), then  ∂ J  j j z¯ (η + 1) = z¯ (η) − λ1 j  (7.25) ∂ z¯ θ=θ (η) If K (x, x j ) is abbreviated as K j , let a = z = b/a + w˜ 0 . So

M j=1

M

K j, b =

j=1

n  ∂J ∂ J ∂z ∂b 1 = = ( f (x k1 ; θ ) − z k1 ) K j j j ∂ z¯ ∂z ∂b ∂ z¯ a k =1

z¯ j K j , then

(7.26)

1

Substitute Formula (7.26) into Formula (7.25), the iterative formula of z¯ j is  n   1 j z¯ (η + 1) = z¯ (η) − λ1 ( f (x k1 ; θ (η)) − z k1 ) K (η) a(η) k =1 j

j

(7.27)

1

j

j

In the same way, the iterative formulas of x˜i , σi (i = 1, 2, · · · , r ) and w˜ 0 are as follows: j x˜i (η

+ 1) =

j x˜i (η)

n   ( f (x k1 ; θ (η)) − z k1 ) − λ2 k1 =1 j

xi (k1 ) − x˜i (η) z¯ j (η) − f (x k1 ; θ (η)) j K (η) j a(η) (σi (η))2

 (7.28)

7.3 Fuzzy Model Identification Based on Relevance Vector Machine

j

j

σi (η + 1) = σi (η) − λ3

n  

161

( f (x k1 ; θ (η)) − z k1 )

k1 =1 j

xi (k1 ) − x˜i (η) z¯ j (η) − f (x k1 ; θ (η)) j K (η) j a(η) (σi (η))3 w˜ 0 (η + 1) = w˜ 0 (η) − λ4

n 



[( f (x k1 ; θ (η)) − z k1 )]

(7.29)

(7.30)

k1 =1

where, xi (k1 ) is the x k1 dimension of input i, λ1 ∼λ4 represents the learning rate, and η represents the number of iteration steps. Because the optimal initial parameters can be obtained in the process of structure identification with RVM method, the convergence speed of parameter optimization with gradient descent algorithm is accelerated, and the parameters are further optimized.

7.3.3 Fuzzy Model Identification Algorithm Based on RVM and Gradient Descent Method After summarizing the above steps of fuzzy model structure identification and parameter identification, the fuzzy model identification algorithm—Algorithm 7.2, which based on RVM and gradient descent algorithm is obtained. The algorithm steps are as follows. Algorithm 7.2: Fuzzy model identification algorithm based on RVM and gradient descent Step 1. Experimental data collection and preprocessing: collect and preprocess the experimental data to generate sample data. Step 2. Model structure identification: use Algorithm 7.1 to determine the number of fuzzy model rules and initial parameters. Step 3. Model parameter identification: train and optimize the parameters with Formulas (7.27) ~ (7.30) to obtain the optimized parameter model. Step 4. Model validation: verify the performance of the model with test samples. If it meets the requirements, the final model is determined; otherwise, return to Step 1 to learn and train again.

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7 Degradation Modeling and Residual Life Prediction Based …

7.4 Degradation Modeling and Residual Life Prediction The essence of using the fuzzy model based on relevance vector machine for degradation modeling and prediction is to identify the system model according to the existing observation data and predict the characteristic parameters of the system degradation, so as to predict the degradation data and residual life. It is assumed that the input–output observation data of the system can be expressed in the form (x(t), z(t)) of input–output data pairs. Here, x(t) is the input vector of the system at t, which is composed of variables related to the system output, and z(t) is the output variable of the system at t, which is the characteristic parameter describing the degradation of the system. The predicted function can be expressed in the following form [20]: zˆ (t) = f (xt−m , xt−m−2 , . . . , xt−m−r )

(7.31)

where the predictive function f (·) is obtained by the fuzzy model identified in the previous section, zˆ (t) is the predicted output, m is the number of predicted steps, and r is the number of vectors related to the output. It should be noted that, in case of m = 1, it shall be one-step prediction. The flowchart of degradation modeling and residual life prediction based on adaptive fuzzy system is shown in Fig. 7.4. Its basic principle is based on the effective information (xt−m−1 , xt−m−2 , . . . , xt−m−r ) obtained before t, predict the characteristic parameter value t with Formula (7.31) at zˆ (t). Let the predicted characteristic parameter zˆ (t) be equal to the preset failure threshold ω, and calculate the time at tω , then the residual life RULt−m of degraded equipment at tt−m can be calculated by the following formula: RULt = tω − tt−m

(7.32)

where tt−m is the current time. Based on the analysis of the above degradation modeling and prediction principle, the degradation modeling and prediction algorithm of fuzzy system—Algorithm 7.3, which based on RVM and gradient descent algorithm is obtained. The algorithm steps are as follows.

Measured data

Adaptive fuzzy system

Predicted value

+

×

Remaining life

− Threshold

Fig. 7.4 Flowchart of degradation modeling and prediction

7.4 Degradation Modeling and Residual Life Prediction

163

Algorithm 7.3: Degradation modeling and prediction algorithm of fuzzy system based on RVM Step 1. Data collection and preprocessing: select the characteristic parameters representing the degradation of the system as the output z(t), and the related variables constitute the input vector x(t). Step 2 Use Step 1 ~ Step 3 in Algorithm 7.3 to identify the system model as shown in Formula (7.31), and let tk = 1. Step 3. Select the input variables x(tk ) and use Formula (7.31) of fuzzy model to predict the future degradation amount zˆ (tk + m) of the system. Step 4. Select the appropriate threshold ω, and calculate the residual life of the equipment with Formula (7.32). Step 5. Let tk = tk + 1, return to Step 3.

7.5 Case Study In order to verify the effectiveness of the fuzzy model identification and prediction method proposed in this paper, the nonlinear system CSTR is identified by the established fuzzy model (RVM-FIS). The identification accuracy is measured by mean absolute percentage error (MAPE) and root mean square error (RMSE), and the prediction results are compared with those of fuzzy model (RVM-FIS) based on SVM. SVM-FIS draws on the modeling method of reference [5]: firstly, adopt SVM method for structure identification and then adopt the gradient descent algorithm for parameter identification, where C = 60 and the insensitive coefficient ε = 0.5. The function shall be Gaussian kernel function, and the kernel function width σsvm = 0.6.

7.5.1 Description of Simulation System for Continuous Stirred Tank Reactor Continuous stirred tank reactor (CSTR) is a multivariable nonlinear system, in which the reactants have irreversible exothermic reaction. The concentration of reactants q (temperature Tc ) can meet the production requirements by controlling the coolant flow rate C A (temperature T ) flowing through the reactor. The reaction equation can be described by the following equations:   q E dC A = (C A f − C A ) − k0 exp − CA dt V RT   q dT A −H E UA CA + = (T f − T A ) + k0 exp − (Tc − T A ) dt V ρC p RT A VρC p

(7.33)

164

7 Degradation Modeling and Residual Life Prediction Based …

The above model is discretized by Euler, and the following discrete time models are obtained by considering the system noise x(k + 1) = x(k) + dt · g(x(k), u(k)) + wk g(x(k), u(k))   ⎤ ⎡q  E C A f − x1 (k) − k0 exp − x1 (k) ⎥ ⎢V Rx2 (k) ⎥   =⎢ ⎦ ⎣q  −H E UA T f − x2 (k) + k0 exp − x1 (k) + (u(k) − x2 (k)) V ρC p Rx2 (k) VρC p y(k + 1) = [x1 (k + 1), x2 (k + 1)]T + vk+1

(7.34)

The significance and values of the parameters are given in reference [21]. By selecting appropriate input and output variables, the effectiveness of the proposed fuzzy model identification method in the paper can be verified with the model.

7.5.2 Simulation Experiment and Results The initial state of the system is set as follows [21]: When the sampling time is dt = 0.2 min, the initial state values of the system are x1 (0) = 0.22 mol/L and x2 (0) = 447K; the process  noise wk is Gaussian white noise with zero mean covariance matrix 0.0052 0 ; vk is Gaussian white noise with zero mean covariance matrix of of 0 0.52   0.0052 0 . The control law in this experiment is to keep the concentration of 0 0.52 reactants at x1d (k) = 0.2 mol/L. PID control law is adopted as follows: u(k) = u(k − 1) + A0 ε(k) − A1 ε(k − 1) + A2 ε(k − 2) ε(k)  xˆ1 (k|k) − x1d (k)     dt Td Td Td , A1 = K p 1 + 2 , A2 = K p A0 = K p 1 + + Ti dt dt dt

(7.35)

where K p = 100, Ti = 0.4, Td = 0.1, and u(0) = 419. When the system runs to Step 100, set a change to the system to make the coolant flow rate change according to this trend: q(s) = q(100) − (s − 100) × 0.2, where, s ≥ 100, representing the step number of operation of the system. It is assumed that the system contains two inputs (input cold water temperature Tc and coolant flow rate q) and one output (reactor temperature T A ). 300 data are taken as sample data. The first 200 data are used for training, and the last 100 d5ata are tested to verify the identification accuracy and effectiveness of the model. In the

7.5 Case Study

165

training stage, the number of fuzzy rules generated by RVM-FIS and SVM-FIS is recorded; in the test stage, the reactor temperature T A is predicted by the two models, and the prediction error and running time are counted. The simulation is carried out on a computer with main frequency of 1.91 GHZ and memory of 1.00 G, and the simulation results are shown in Figs. 7.5 , 7.6, 7.7 and 7.8. The quantitative indexes of the prediction results of the two fuzzy models are shown in Table 7.1. In order to clearly explain the accuracy of the prediction, considering the randomness of noise, 100 times of Monte Carlo simulation experiments are carried out. When the prediction residuals are 0.03 and 0.05, respectively, the degradation modeling and predicted accuracy rate are counted and the predicted accuracy rate ζ is calculated by the following formula num_MC ζ =

m=1

num_a(m) num_t

num_MC

× 100%

(7.36)

where num_MC represents the number of times of Monte Carlo simulation experiments, num_t represents the number of test points in each experiment, and num_a(m) represents the number of accurately predicted points in the mth experiment. The statistical results of this experiment are shown in Table 7.2.

Tc (K)

460

440

420

400

0

50

100

150 Step

200

250

300

0

50

100

150 Step

200

250

300

100

q (L/min)

90 80 70 60

Fig. 7.5 Curves of input cold water temperature and coolant flow rate

166

7 Degradation Modeling and Residual Life Prediction Based … Taining data Testing data RVM-FIS output SVM-FIS output

500

Reactor temperature (K)

480 460 440 420

Training 400 380

Testing 360 340 320 0

50

100

150 Step

200

250

300

Fig. 7.6 Training and testing curves of two fuzzy models

500 Actual data RVM-FIS prediction SVM-FIS prediction

480

Reactor temperature (K)

460 440 420 400 380 360 340 320 300

0

10

20

30

40

50 Step

60

70

80

90

100

Fig. 7.7 Prediction curves of two fuzzy models

7.5.3 Result Analysis It can be seen from the figure that CSTR has strong nonlinearity, and the two fuzzy modeling methods have good approximation ability, so the prediction results are

7.5 Case Study

167

0.5 Errors of RVM-FIS Errors of SVM-FIS

0.4 0.3

Error = 0.1

0.2

Errors

0.1 0 -0.1

Error = -0.1

-0.2 -0.3 -0.4 -0.5

0

10

20

30

40

50 Step

60

70

80

90

100

Fig. 7.8 Prediction error curves of two fuzzy models

Table 7.1 Comparison of prediction results of two fuzzy models Algorithm

Final rule number (pcs)

Prediction error MAPE

RMSE

SVM-FIS

163

0.0134

0.1160

80.53

RVM-FIS

30

0.0122

0.1059

1.15

Table 7.2 Statistical results of predicted accuracy rate

Prediction time (s)

Prediction residuals

Average prediction time (s)

Accuracy rate (%)

0.03

1.24

92

0.05

1.01

96

close to the real data, the prediction accuracy is high, and the error is small. From the quantitative comparison in Table 7.1, it can be seen that the prediction error of RVM-FIS is smaller than that of MAPE or RMSE. RVM-FIS only uses 30 rules, the model structure is simpler, the prediction time is only 1.15 s, and the speed is greatly improved compared with SVM-FIS. The reason is that RVM training is based on sparse Bayesian theory, and the hyper-parameter prior distribution of weight is introduced to train RVM weight during the training. During the training process, most of the weights will tend to zero quickly, and fewer fuzzy rules can be obtained, so as to ensure the sparsity of the solution [11]. That is to say, the structure of fuzzy model is more concise and the prediction time is shorter when RVM-FIS maintains the same accuracy as SVM-FIS. From the statistics of predicted accuracy rate in Table 7.2,

168

7 Degradation Modeling and Residual Life Prediction Based …

it can be seen that the established fuzzy model and its identification algorithm have high prediction accuracy, and the following conclusions are drawn: (1)

(2)

The fuzzy model identification method based on RVM and gradient descent algorithm is feasible and has high identification accuracy and generalization ability; Compared with SVM-FIS, RVM-FIS has the following advantages: The fuzzy model structure is simpler and the prediction time is faster, because hyperparameter α is introduced for RVM method. As a result, the number of fuzzy rules is less, and the sparsity of the model is stronger with high accuracy; There is no need to calculate the normalization parameters C and insensitive coefficients ε, so as to reduce the amount of calculation; in addition, the RVM kernel function does not need to meet Mercer conditions.

Obviously, the prediction accuracy of fuzzy model for residual life is directly proportional to the prediction accuracy of degradation value, that is to say, the more accurate the predicted degradation value by degradation model is, the higher the prediction accuracy of residual life of equipment will be. The above experiments mainly verify the prediction ability of fuzzy model based on relevance vector machine for equipment degradation value (characteristic parameters). Based on the above prediction results, it is only necessary to give a certain failure threshold ω and use Formula (7.32) to accurately predict the residual life of equipment at any time.

7.6 Summary of This Chapter In this chapter, a method of fuzzy model identification based on RVM is proposed, and its function form is given. The uniform approximation of fuzzy model based on RVM is proved by Stone-Weierstrass theorem, and then a fuzzy model identification method based on RVM and gradient descent algorithm is proposed and applied to degradation modeling and prediction. Through theoretical analysis and simulation experiments, the following conclusions can be drawn. (1)

(2) (3)

The fuzzy model based on RVM has uniform approximation in theory, and it can use both qualitative knowledge and quantitative data, which provides a basis for applying expert experience knowledge to model identification and control; The fuzzy model identification method based on RVM and gradient descent algorithm can construct concise fuzzy model with high identification accuracy; The degradation modeling and prediction algorithm based on RVM fuzzy model can accurately predict the degradation of the system.

References

169

References 1. Chen Y (2005) Support vector machine method and fuzzy system. Fuzzy Syst Math 19(1):1–11 2. Dragan K (2002) Design of adaptive Takagi–Sugeno–Kang fuzzy models. Appl Soft Comput 2(2):89–103 3. Chen SW, Wang J, Wang DS (2008) Extraction of fuzzy rules by using support vector machines. In: Proceedings of the 2008 fifth international conference on fuzzy systems and knowledge discovery, IEEE Computer Society Washington, DC, USA. IEEE, pp 438–442 4. Wong CC, Chen CC (1999) A hybrid clustering and gradient descent approach for fuzzy modeling. IEEE Trans Syst Man Cybern Part B 29(6):686–693 5. Huang XX, Shi FH, Gu W et al (2009) SVM-based fuzzy rules acquisition system for pulsed GTAW process. Eng Appl Artif Intell 22(8):1245–1255 6. Cai Q, Hao Z, Liu W (2009) TSK fuzzy system based on fuzzy partition and support vector machine. Pattern Recogn Artif Intell 22(3):411–416 7. Liu H, Zhou D, Qian F (2008) Control of double inverted pendulum based on fuzzy inference of support vector machine. Chin J Sci Instrum 29(2):330–335 8. Wei L (2009) Study on fuzzy model identification based on kernel method. Doctoral Dissertation of Shanghai Jiaotong University, Shanghai 9. Xu XM, Mao YF, Xiong JN, et al (2007) Classification performance comparison between RVM and SVM. In: IEEE international workshop on anti-counterfeiting, security, identification, Fujian, China. IEEE, pp 208–211 10. Tipping ME (2000) The relevance vector machine. In: Solla SA, Leen TK, Müller K-R (eds) Advances in neural information processing systems, vol 12. MIT Press, Cambridge, pp 652–658 11. Tipping ME (2001) Sparse Bayesian learning and the relevance vector machine. J Mach Learn Res 1(3):211–244 12. Zhang X, Chen F, Gao J et al (2006) Prediction of sparse Bayesian time series. Control Decis Making 21(5):585–588 13. Kim J, Suga Y, Won S (2006) A new approach to fuzzy modeling of nonlinear dynamic systems with noise: relevance vector learning mechanism. IEEE Trans Fuzzy Syst 14(2):222–231 14. Wang ZQ, Hu CH et al (2013) A new online fuzzy modelling method considering prior information with its application in PHM. Int J Adv Comput Technol 5(6):694–703 15. Changhua Hu, Wang Z et al (2011) A RVM fuzzy model identification method and its application in fault prediction. Acta Automatica Sinica 37(4):503–512 16. Wang Z (2010) Adaptive fuzzy system and its application in fault prediction of inertial devices. Rocket Force University of Engineering, Xi’an 17. Yang G, Zhou X, Xuchu Yu (2010) Study of sparse Bayesian model and relevance vector machine. Comput Sci 37(7):225–228 18. Translated by Shen X, Fang Q, Lou Y et al (1989) (author, Dzdnk BK) Introduction to uniform approximation of polynomials. Peking University Press, Beijing 19. Wang LX, Mendel JM (1992) Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Trans Neural Netw 3(5):807–814 20. Si X (2009) Research on fault prediction method of nonlinear system based on evidence reasoning and its application. Master’s Thesis of Rocket Force University of Engineering, Xi’an 21. Zhou ZJ, Hu CH, Xu DL et al (2010) A model for real-time failure prognosis based on hidden Markov model and belief rule base. Eur J Oper Res 207(1):269–283

Chapter 8

Degradation Modeling and Reliability Prediction Based on Evidence Reasoning

8.1 Introduction In the previous chapter, we discussed the performance degradation modeling and residual life prediction methods based on stochastic process, support vector machine, relevance vector machine and other data driving. These methods show good results in their respective application conditions. However, human beings play an irreplaceable role in the final decision-making process. It is very important to combine numerical information with subjective information when making decision, although this subjective information is likely to be incomplete and inaccurate [1–3]. In order to deal with the uncertain information in the modeling and prediction process, Hu Changhua et al. carried out a lot of related research [7–10] based on evidence reasoning method [4–6] and proposed a degradation modeling and prediction model [11] based on evidence reasoning. Compared with the traditional model, the method based on evidence reasoning provides a more practical knowledge representation scheme and can show nonlinear causality. The method has been applied to reliability prediction in Ref. [11], and satisfactory results have been obtained. Aiming at the parameters including attribute weight and grade utility in the model, several optimization methods are adopted in Ref. [11] to select and adjust them, but these methods are all offline. This leads to the running time of the algorithm in the case of a large number of data is too long, thus affecting its actual use. In order to solve this problem, it is necessary to design an online algorithm to continually update the parameters of ER model. Once the model is established and its parameters are estimated, the ER model can be used for modeling and prediction of the degradation data. Inspired by Ref. [11], this chapter will propose two recursive updating algorithms for ER model parameters in data output and judgment output, respectively. In order to apply expectation maximization (EM) [12–14] algorithm, it is assumed that when the inputs of ER model are independent of each other, the output results are also independent of each other. Based on the assumption that the actual output obeys © National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_8

171

172

8 Degradation Modeling and Reliability Prediction Based …

normal distribution, the updating algorithm of ER model parameters is obtained based on recursive EM algorithm, and the effectiveness of the proposed algorithm is verified by an example. In the following content, this chapter first introduces the basis of evidence theory. Second, the degradation modeling method based on ER model is briefly reviewed. Third, corresponding recursive update algorithms are proposed for the case of judgment output and data output, respectively. Finally, a case study is carried out and the research contents of this chapter are summarized.

8.2 Degradation Modeling Based on Evidence Reasoning The purpose of degradation modeling and prediction is to establish a model that can accurately describe the change law of performance degradation data and use the established model to predict the trend of performance degradation, and then accurately predict the failure and estimate the residual useful life. Firstly, this paper introduces the general prediction model and then gives the problem description of performance degradation modeling.

8.2.1 Structure and Expression Form of Prediction Model There is a set of time series {x(t)|x(t) ∈ R }, t = 1, · · · , L composed of input variable values. L ∈ Z + represents the length of the entire time series. Then, the general prediction model is yˆ (t + k − 1) = f (xt−1 , xt−2 , . . . , xt− p )

(8.1)

Here, yˆ (t + k − 1) represents the output value at the t + k + 1 time; (xt−1 , xt−2 , . . . , xt− p ) is an input vector composed of p input values, p ∈ Z + is the dimensions of the embedded space, and its value can be determined by using the method proposed in Ref. [15]. For convenience, let X(t) = (xt−1 , xt−2 , . . . , xt− p ) represent the input vector of the model. Similar to Ref. [11], only one-step prediction is considered here. Then, the model (8.1) can be simplified to the following formula yˆ (t) = f (xt−1 , xt−2 , . . . , xt− p )

(8.2)

In order to apply the evidence reasoning method, xt−1 , xt−2 , . . . , xt− p is assumed to be p basic attributes related to the performance degradation prediction value y(t). Then, the evidence reasoning method aims to identify the internal relationship between X(t) and yˆ (t), and its essence is how to obtain function f (·).

8.2 Degradation Modeling Based on Evidence Reasoning

173

For specific performance degradation modeling and prediction, yˆ (t + k − 1) and yˆ (t) in models (8.1) and (8.2) can be replaced by x(t ˆ + k − 1) and x(t), ˆ while the input vector X(t) is composed of p performance degradation values at the previous time. After the function f (·) is identified, it can be used to describe the change trend of performance degradation for modeling and prediction. In this sense, the essence of performance degradation modeling and prediction is to use the time series composed of performance degradation values to obtain the function f (·) that can describe the change law of performance degradation value, then the function f (·) can be called performance degradation model. Therefore, the rest of this chapter will no longer distinguish between yˆ (t) and x(t). ˆ

8.2.2 Degradation Modeling and Prediction Under the ER Framework This section describes how to use evidence theory to model and predict performance degradation. First, assume that there are p basic attributes xt−1 , xt−2 , . . . , xt− p related to the output value y(t) at the next time. Here, the set composed of these p basic attributes is called evidence set. Then, set the weight of the ith basic attribute of xt−i to wi , i = 1, . . . , p, and make w = {w1 , · · · wi , · · · , w p }. It should be noted that the weights of these basic attributes need to be normalized to meet the following conditions: 0 ≤ wi ≤ 1 and

p 

wi = 1

(8.3)

i=1

If the degradation degree of equipment is divided into N, F1 , . . . , FN , then the degradation status identification framework is F = {F1 , · · · Fn , · · · , FN }

(8.4)

It should be noted that the determination of degradation state depends on specific problems, so the identification framework of degradation state also needs to be determined according to the actual situation. In engineering practice, data may be in the form of numerical value or subjective distribution. In order to apply evidence theory, engineering data need to be transformed into the form of mass function distribution. Under the identification framework F, using the evidence transformation technology provided by Ref. [5], each basic attribute xt−i in the input vector X(t) = (xt−1 , xt−2 , . . . , xt− p ) can be expressed in the following form S(xt−i ) =



  Fn , βn,i (xt−i ) , n = 1, · · · , N , i = 1, · · · , p

(8.5)

174

8 Degradation Modeling and Reliability Prediction Based …

N where βn,i ≥ 0, n=1 βn,i ≤ 1, and βn,i represents the confidence level that the the attribute attribute xt−i is evaluated as Grade Fn . Formula (8.5) indicates that N βn,i = 1, the xt−i is evaluated as Grade Fn based on confidence level βn,i . If n=1 evaluation of the attribute xt−i is complete; otherwise, it is incomplete. How to obtain βn,i depends on the characteristics of the attribute xt−i (i = 1, . . . , p). For example, the data of quantitative attributes are expressed in the form of numerical value, while the data of qualitative attributes are expressed in the semantic form [5, 6, 11]. In order to deal with qualitative and quantitative attributes under a unified confidence level framework, Yang [6] and other scholars have proposed equivalent information transformation technology, through which numerical data, random data or qualitative information can be transformed into mass function form. The technology is practical, and the specific algorithm is shown in Ref. [6]. After each attribute is represented according to Formula (8.5), all attributes can be combined directly by evidence reasoning method to get the final conclusion. Using the ER analytic algorithm given by Wang et al. [16], the final prediction result x(t) can be expressed as 

 

 O y(t) = Fn , βˆn (t) , n = 1, . . . , N 

(8.6)

where ER analytic algorithm can be used to obtain βˆn (t), n = 1, . . . , N : βˆn (t) = D(t) =



p

p ˆ β w − i=1 (1 − wi ) (x ) + 1 − w i n,i t−i i i=1 D(t) p p N      wi βn,i (xt−i ) + 1 − wi − N (1 − wi ) n=1 i=1

(8.7)

(8.8)

i=1 

The comprehensive evaluation value y(t) describes the state of the system at time t. According to y(t), it can be distinguished which evaluation level is used to evaluate the degradation status of the system and the confidence level of the evaluation level. 

8.2.3 Utility Based Construction of the Numerical Outputs 

In engineering practice, the output y(t) of the system is often accurate. In this case, we hope to get the data output equivalent to O(y(t)) in Formula (8.6). The output of all functions of evidence theory is in the form of confidence level, so it needs to be transformed into numerical form equivalently. By introducing utility expectation, we can realize the transformation from confidence level distribution output to data output [5, 11]. u(Fn ) is assumed to be the utility of the identification framework Fn , n = 1, . . . , N . If F j is compared with 

8.2 Degradation Modeling Based on Evidence Reasoning

175

Fi , the decision-maker prefers to evaluate Grade F j , then u(Fi ) < u(F j ), u(Fn ) is generally obtained through expert prior knowledge or objective knowledge. In order to realize the transformation between mass function and data output, the maximum, minimum and average utility are introduced in Ref. [6]. Assuming that the utility of the evaluation grade Fn is u(Fn ), the expected utility of output evaluation O(y(t)) can be defined as follows: 

N     βˆn (t)u(Fn ) u O y(t) = 

n=1

where βˆn (t) represents the lower bound of the confidence level that Fn is evaluated as y(t), and (βˆn (t) + βˆF (t)) is its upper bound. Without losing generality, assuming that F1 obtains the minimum preference of the decision-maker and FN obtains the maximum preference of the decision-maker, the specific calculation of the maximum, minimum and average utility of O(y(t)) is as follows: 





u max (O(y(t)) =

N −1 

βˆn (t)u(Fn ) + (βˆN (t) + βˆF (t))u(FN )

n=1

u min (O(y(t)) = (βˆ1 (t) + βˆF (t))u(F1 ) + 

N 

βˆn (t)u(Fn )

n=2

  u avg (O(y(t)) = u max (O(x(t)) ˆ + u min (O(x(t)) ˆ /2 



Therefore, the defined predicted data output y(t) can be calculated as follows: 

y(t) = u avg (O( yˆ (t)) =

n=N 

βˆn (t)u(Fn ) +

n=1

u(F1 ) + u(FN ) βˆF (t) 2



where y(t) is the prediction result. If the overall evaluation results are complete and accurate, then βˆF (t) = 0; the calculation of the prediction results y(t) can be simplified as 



y(t) =

N 

βˆ j (t)u(F j ).

(8.9)

j=1

Obviously, the more accurate the parameters (including sum) obtained by the evidence reasoning method, the closer the prediction result is to the true value.

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8 Degradation Modeling and Reliability Prediction Based …

8.3 Recursive Algorithms for Updating the ER-Based Prediction Model At present, Ref. [11] has studied the optimal learning algorithm for the model parameters of evidence theory. However, the algorithm is offline in nature and needs a lot of time for training and retraining after obtaining new input and output data. Therefore, it is necessary to design an online parameter updating algorithm, and expectation maximization (EM) algorithm [13] proposed by Dempster et al. is usually used to achieve this purpose. Zhijie Zhou et al. studied the use of EM algorithm to update the parameters of the belief rule base [8] proposed by Professor Jianbo Yang. Based on the work of Refs. [8, 11], this section will study the recursive updating algorithm of evidence reasoning model parameters from the perspective of probability. In the recursive algorithm, the observed values of input and output are needed. Therefore, it is assumed that the data set composed of observation data pairs (X(n), y(n)) can be obtained, where X(n) is the given input vector, y(n) is the output vector obtained by instrument observation or expert evaluation and y(n) is the simulation output generated by the prediction model based on evidence reasoning. In performance degradation modeling and prediction, the y(n) in the data pair (X(n), y(n)) is x(n) and the corresponding simulation output is x(n). A later section will no longer distinguish between x(n) and y(n). Accordingly, we will study the parameter updating algorithm when the output is numerical and subjective judgment, respectively. In order to use recursive EM algorithm to design recursive updating algorithm of ER model parameters, it is assumed that when the input vectors of ER model are independent of each other, the actual output of ER model is also independent of each other. On the basis of this assumption, this chapter designs two recursive algorithms to update ER model in the following content. 



8.3.1 Recursive Parameter Estimation Algorithm Based on Judgment Output In this case, the judgment output y(n) can be expressed as y(n) =



 F j , β j (n) ,

j = 1, . . . , N



(8.10)

where β j (n) indicates the matching degree between the time n and evaluation grade F j . In fact, y(n) obtained from Formula (8.10) is the default output format of ER model. Let B(n) = [β1 (n), . . . , β N (n)]T represent the real output corresponding to the input X(n). Here, β j ( j = 1, . . . , N ) is regarded as a random variable. So, under the given X(n) and Q, the conditional probability density of B(n) is f ( B(n)|X(n), Q), where Q is the unknown parameter vector. According to the independence assumption given above, the following formula can be obtained directly

8.3 Recursive Algorithms for Updating the ER-Based Prediction Model

f ( B(1), . . . , B(n)|X(1), . . . , X(n), Q) =

n 

f ( B(τ )|X(τ ), Q)

177

(8.11)

τ =1

From Formula (8.11), we can get the expectation of the log-likelihood function at the time n    n    (8.12) Ln+1 (Q) = E log f ( B(τ )|X(τ ), Q)X(1), . . . , X(n), Q(n)  τ =1

where E(·|·) is conditional expectation. According to the above, Formula (8.12) can be rewritten into recursive form Ln+1 (Q) = Ln (Q) + E{ log f ( B(n)|X(n), Q)|X(n), Q(n)}

(8.13)

In order to make a reasonable approximation for Ln+1 (Q), this chapter considers Taylor expansion of the first term on the right side of Formula (8.13). So,   Ln (Q) ≈ Ln (Q(n)) + ∇Q Ln (Q(n)) (Q − Q(n))   1 + (Q − Q(n))T ∇Q ∇QT Ln (Q(n)) (Q − Q(n)) 2

(8.14)

Here, ∇Q is the operator for the gradient of each column element of the vector Q. According to the definition of Ln (Q), ∇Q ∇QT Ln (Q(n)) can be approximated as [8, 12, 14] ∇Q ∇QT Ln (Q(n)) ≈ −(n − 1)1 (Q(n))

(8.15)

where 1 (Q(n)) can be calculated by the following formula    1 (Q(n)) = E −∇Q ∇QT log f( B(n)|X(n), Q)X(n), Q(n)

(8.16)

When Q = Q(n), Ln (Q) represented by Formula (8.14) is the maximum, therefore, ∇Q Ln (Q(n)) = 0

(8.17)

Substituting Eqs. (8.16) and (8.17) into Eq. (8.14), we can get the equation below. Ln (Q) ≈ Ln (Q(n)) 1 − (Q − Q(n))T [(n − 1)1 (Q(n))](Q − Q(n)) 2

(8.18)

178

8 Degradation Modeling and Reliability Prediction Based …

Take 1 (Q(n)) = ∇Q log f( B(n)|X(n), Q(n))

(8.19)

where ∇Q log f ( B(n)|X(n), Q(n)) represents the gradient vector at Q(n). Therefore, according to Taylor expansion and expectation of Q(n) and X(n) in the log f ( B(n)|X(n), Q) Ln+1 (Q) = Ln (Q(n)) + E{ log f ( B(n)|X(n), Q(n))|X(n), Q(n)} + 1 (Q(n))(Q − Q(n)) n − (Q − Q(n))T [1 (Q(n))](Q − Q(n)). 2

(8.20)

Since Q = Q(n + 1) is the point obtaining the maximum value of Ln+1 (Q), and the first two terms on the right side of Formula (8.20) are constants, ∇Q Ln+1 (Q(n + 1)) = 0

(8.21)

Therefore, the recursive estimation form of the parameter Q(n + 1) is Q(n + 1) = Q(n) +

1 [1 (Q(n))]−1 1 (Q(n)) n

(8.22)

Since the attribute weight must satisfy the normalization condition, it must be between 0 and 1, and the sum of all weights is 1. Then, Formula (8.22) can be modified as   1 (8.23) Q(n) + [1 (Q(n))]−1 1 (Q(n)) Q(n + 1) = n H 1

where  H1 is projected onto the constraint set H1 represented by Formula (8.3). Ideally, for a given input vector X(n), the predicted output y(n) represented by Formula (8.6) can be as close to its true value y(n) as possible. That is to say, for the data pair (X(n), y(n)) at time n, the ER model can be updated by minimizing the gap between β j (n) and βˆ j (n) generated by ER model. Here, β j (n) can be regarded as  T a random variable with an expected value βˆ j (n). Let B(n) = βˆ1 (n), . . . , βˆN (n) is the predicted output at time n, and it is assumed that B(n) obeys the following complex normal distribution 



f ( B(n)|X(n), Q)  −1/2  T −1

1

 −N /2  (8.24) = (2π) B(n) − B(n) exp − B(n) − B(n)   2 



8.3 Recursive Algorithms for Updating the ER-Based Prediction Model

179

 T where Q = VT , σ1 , σ2 is the parameter vector, and V = [wi ]T and σ1 , σ2 are the  elements of the covariance matrix . Since each term and term of the vector V are independent of each other, 1 (Q(n)) and 1 (Q(n)) in the Formula (8.23) can be rewritten as T  1 (Q(n)) = 1 (Q(n))T , 1 (Q(n))T  1 (Q(n)) =

0 1 (Q(n)) 0 1 (Q(n))

(8.25)

 (8.26)

 where 1 (Q(n)) and 1 (Q(n)) are the derivative with respect  to V, and 1 (Q(n)) element. Clearly, and 1 (Q(n)) are the derivative with respect to the matrix

[1 (Q(n))]−1

  −1 1 (Q(n)) 0   −1 = 0 1 (Q(n))

(8.27)

When only the parameter vector V is considered, Formula (8.23) can be rewritten as follows according to Formulas (8.24) and (8.27) V(n + 1) = V(n) +

−1 1  1 (Q(n)) 1 (Q(n)) n

(8.28)

where V(n) is the known quantity. According to Formulas (8.16) and (8.19), the a term of the gradient vector 1 (Q(n)) and the elements of the matrix 1 (Q(n)) can be calculated by the following formula      ∂B(n)T  −1

 1 (Q(n)) a = (n) B(n) − B(n)   ∂ Va 



    ∂B(n)T  −1 ∂B(n)  1 (Q(n)) a,b = (n)  ∂ Va ∂ Vb  

(8.29) V=V(n)



(8.30) V=V(n)

where a = 1, . . . , p, b = 1, . . . , p.  Both Formulas (8.29) and (8.30) need (n).  N to know the covariance matrix Because β1 (n), . . . , β N (n) must meet j=1 β j (n) = 1, they are not independent of each other. In order  simplify the calculation without loss of generality, it is  to assumed that = ai, j N ×N meets 

ai, j = σ1 , i = j ai, j = σ2 , i = j

(8.31)

180

8 Degradation Modeling and Reliability Prediction Based …

 T Based on this assumption, there is Q = VT , σ1 , σ2 . When the parameter V(n) is known, σi (n) can be calculated by the following formula   σi (n) = arg max log f ( B(n)|X(n), Q) σi

(8.32)

V=V(n)

where i = 1, 2. The equation can be solved by FSOLVE function in MATLAB. Since V must meet the conditions given by Formula (8.3), the recursive formula given by Formula (8.23) must be modified appropriately. First of all, let V = T  V1 , . . . , V p . Then, the constraints given by Formula (8.3) can be expressed as 



h(V) = h V1 , . . . , V p =

p 

Vj − 1 = 0

(8.33)

j=1

0 ≤ Vi ≤ 1, i = 1, . . . , p

(8.34)

Let h(V) = [h(V)]T , and H(V) is Jacobian matrix of h(V). Then, by modifying Formula (8.23), the recursive algorithm satisfying Formula (8.33) can be obtained [8, 17–19] V(n + 1) = V(n) +

 −1 1 π1 {V(n)} 1 (Q(n)) 1 (Q(n)) n

(8.35)

 −1 where V(n + 1) = V(n) + n1 π1 {V(n)} 1 (Q(n)) 1 (Q(n)). Let’s say I p is the identity matrix of dimension p. So,  −1 H(V(n)) π1 {V(n)} = I p − H(V(n))T H(V(n))H(V(n))T p−1 = Ip p

(8.36)

According to the constraints given in Formula (8.34), the parameters V j ( j = 1, . . . , p) estimated by Formula (8.35) are between the upper and lower bounds. Therefore, the projection algorithm in Ref. [8] can be used. The projection  operator π2 V(n + 1) is defined as follows    V j (n + 1)e j π2 V(n + 1) = p



(8.37)

j=1



where e j is column vector of the j item equal to 1 and other items equal to 0. V j (n + 1) can be calculated by the following formula

8.3 Recursive Algorithms for Updating the ER-Based Prediction Model

181

 j (n + 1) V V j (n + 1) =  N  i=1 V j (n + 1)

(8.38)

⎧ V j (n + 1) < 0 ⎨ 0,  V j (n + 1) = 1, V j (n + 1) > 1 ⎩ V j (n + 1), 0 ≤ V j (n + 1) ≤ 1

(8.39)



where

In addition, it is possible that only some parameters will be updated, which may lead to non-singularity of the matrix 1 (Q(n)). Therefore, it must be modified accordingly. The final recursive algorithm is as follows:  −1 α V(n + 1) = π2 V(n) + π1 {V(n)} 1 (Q(n)) + γ Ip 1 (Q(n)) n

(8.40)

where α > 0 is to change the convergence speed, γ I p can modify the matrix 1 (Q(n)) to make it a positive definite matrix, γ ≥ 0 is correction factor. To sum up, the recursive updating Falgorithm of ER model parameters based on judgment output is Initialization: V(0), (0), α,γ Update: If X(n), y(n) and V(n) are known, then (n) is calculated by Formula (8.32) V(n + 1) is calculated by Formula (8.40) 

Prediction: Calculate y(n + 1) according to Formulas (8.6)–(8.8) and return to the update step

8.3.2 Recursive Parameter Estimation Algorithm Based on Numerical Output In this case, y(n) is a data output. When the input of ER model is independent of each other, the real output y(1), . . . , y(n) of ER model is also independent of each other, f ( y(1), . . . , y(n)|X(1), . . . , X(n), Q) =

n 

f ( y(τ )|X(τ ), Q)

(8.41)

τ =1

where f ( y(τ )|X(τ ), Q) is the probability density function of random variable y(τ ). In addition, the effect of evaluation grade is non-negative, for example

182

8 Degradation Modeling and Reliability Prediction Based …

u i ≥ 0, i = 1, . . . , N

(8.42)

Moreover, if F j is compared with Fi , the decision-maker prefers the evaluation grade F j , then u j > ui

(8.43)

The procedure is similar to that in the previous section, and the following recursive formula can be obtained    1 −1 (8.44) Q(n) + [2 (Q(n))] 2 (Q(n)) Q(n + 1) = H2 n where Q is composed of attribute weight and other related parameters, which H2 represents the constraint set composed of Formulas (8.3), (8.42) and (8.43). And then 2 (Q(n)) = ∇Q log f ( y(n)|X(n), Q(n))

(8.45)

   2 (Q(n)) = E −∇Q ∇QT log f( y(n)|X(n), Q)X(n), Q(n)

(8.46)

The output result of Formula (8.6) is distributed. The construction method of utility-based data output is given in Sect. 8.2.3. Therefore, the average score of data output can be calculated by Formula (8.9). Similarly, we also hope that the output y(n) of ER model corresponding to the given input X(n) can be as close as possible to its true value y(n). Here, y(n) will be regarded as a random variable, and its expectation is y(n). Therefore, assuming that y(n) obeys normal distribution, its probability density function is 



  2  y(n) − y(n) f ( y(n)|X(n), Q) = √ exp − 2σ 2π σ 

1

(8.47)

T T   where the parameter vector Q = WT , σ and W = VT , u 1 , . . . , u N represent the utility of attribute weight parameter and evaluation grade, σ is variance. Similar to Formulas (8.25)–(8.28), when only W is considered, since the items in W and σ are independent of each other, the recursive formula (8.44) can be rewritten as follows   −1  1  (8.48) W(n) + 2 (Q(n)) 2 (Q(n)) W(n + 1) = n H 2

where 2 (Q(n)) and 2 (Q(n)) are derivatives of W.

8.3 Recursive Algorithms for Updating the ER-Based Prediction Model

183

  Take W = W1 , . . . , W p+N , according to Formulas (8.45) and (8.46), the 2 (Q(n)) and 2 (Q(n)) in Formula (8.48) have the following form: (1)

If a, b = 1, . . . , p, then the a term of the gradient vector 2 (Q(n)) and the elements of the matrix 2 (Q(n)) are   N    ˆ    y(n) − y(n) ∂ β j (n)  u j (n) 2 (Q(n)) a = σ (n) ∂ Wa  j=1 

   2 (Q(n)) a,b = (2)

(8.49) W=W(n)

⎤  N N ∂ βˆ j (n) ⎦⎣ ∂ βˆ j (n) ⎦ 1 ⎣ u j (n) u j (n) σ (n) j=1 ∂ Wa ∂ Wb  j=1 ⎡

⎤⎡

(8.50) W=W(n)

If a, b = p + 1, . . . , p + N , then



  ˆa− p (n) y(n) − y(n)  β 2 (Q(n)) a =   σ (n) W=W(n)     βˆa−L (n)βˆb− p (n)  2 (Q(n)) a,b =   σ (n) 



(8.51)

(8.52)

W=W(n)

Similarly, it is required σ (n) in the calculation of Formulas (8.49)–(8.52). In the case that X(n), y(n) and W(n) are known, the estimation can be made for σ (n) by the following equation   σ (n) = arg max log f ( y(n)|X(n), Q) σ W=W(n)  2  = y(n) − y(n)  

W=W(n)

(8.53)

In Formula (8.48), the constraint set H2 consists of Formulas (8.3), (8.42) and (8.43). Similarly, the constraint represented by Formula (8.3) can be written as 



h W1 , . . . , W p =

p 

Wi − 1 = 0

(8.54)

i=1

0 ≤ Wi ≤ 1, i = 1, . . . , p

(8.55)

The inequality constraints given by Formula (8.43) can be expressed as   pg W p+i , W p+ j = u i − u j < 0, i = 1, . . . , N − 1, j = i + 1, . . . , N

(8.56)

184

8 Degradation Modeling and Reliability Prediction Based …

 where g = (i − 1)(N − 1) − i−2 k=1 (i − k − 1) + j − i. = (N − 1)(N − 2) − . . . , pG (W)]T , G Take p(W) = p [ (W), 1  N −3 − 2 − k) + 1. If the Jacobian matrix of P(W) is p(W), then the inequality (N k=1 constraint can be treated by the projection operator expressed in the following formula [12, 14].  −1

+ (W(n)) = IN − P(W(n))T P(W(n))P(W(n))T P(W(n))

(8.57)

Therefore, the recursive formula can be obtained  −1 α W(n + 1) = π2+ W(n) + π1+ {W(n)} 2 (Q(n)) + γ Ip+N 2 (Q(n)) (8.58) n where α ≥ 1 can change the convergence speed, and γ ≥ 0 can modify the matrix 2 (Q(n)). So, π1+ {W(n)}



0 π1 {W(n)} = + 0

(W(n))



p+N    W j (n + 1)e j π2+ W(n + 1) =

(8.59)



(8.60)

j=1

where π1 {W(n)} adopts a similar definition and is used to deal with the equality constraint expressed by Formula (8.54). If j = 1, . . . , L, W j (n + 1) is defined in a similar way to Formulas (8.38) and (8.39), and it is used to deal with inequality constraints represented by Formula (8.55). To sum up, the recursive updating algorithm of ER model parameters based on data output is 

Initialization: W(0), σ (0), α, γ Update: if X(n), y(n) and W(n) are known, then σ (n) can be obtained by Formula (8.53), W(n + 1) can be obtained by Formula (8.58) 

Prediction: y(n + 1) is calculated by Formula (8.9), and return to the update step

Since the proposed algorithm is based on the random approximation algorithm, and the convergence of the random approximation has been proved by Kushner et al., the convergence proof of the algorithm proposed in this chapter will not be given here. In addition, it should be noted that since the EM algorithm is used in the proposed algorithm, the proposed algorithm in this chapter can only achieve local optimization and cannot guarantee global optimization.

8.4 Case Study

185

8.4 Case Study In this section, the reliability modeling and prediction of missile weapon is taken as an example to verify the effectiveness of the proposed recursive algorithm.

8.4.1 Problem Description As a strategic tool, the reliability and availability of missile weapon are closely related to national security. The composition of missile weapon is very complex, which includes many electronic devices, among which the most important and vulnerable is inertial navigation system. Under the influence of external environment, the performance of the system often degrades. According to statistics, 70% of the fault distribution of missile weapon is closely related to inertial navigation system. Therefore, it is very important to model and predict the performance degradation of the system. Considering that the reliability of missile weapon often decreases with the increase of gyro drift, its reliability can be regarded as a key index to measure the performance of missile weapon. In view of this, this chapter will choose reliability as the performance degradation, and model and predict its change rule. Figure 8.1 shows the reliability variation law of certain missile during the test. The law is obtained by statistical analysis of the data of 75 missiles in recent years. The ordinate of Fig. 8.1 shows the reliability of the missile, and the abscissa is time. Because it takes a lot of time and cost to carry out such test, it is necessary to model and predict the reliability change process of the missile. In the following content, we will use the obtained data to verify the algorithm proposed in this chapter. Fig. 8.1 Reliability variation curve obtained by test

1

Reliability of Missile

0.95

0.9

0.85

0.8

0.75

0

10

20

30 40 50 Times(x30days)

60

70

186

8 Degradation Modeling and Reliability Prediction Based …

Table 8.1 Reference points of reliability data

Semantic

High (F1 )

Average (F2 )

Low (F3 )

Value

1.0

0.8

0.65

8.4.2 Reference Points of Teliability Data For the equipment closely related to safety, researchers not only care about the specific value of its reliability, but also care about the reliability of the equipment reflected behind the reliability value. Usually, semantic variables such as “very high,” “high,” “average,” “low,” “very low” can be used to evaluate reliability level. For example, an expert often says that he has 20% confidence that the reliability of the system is low, while 80% of him thinks that the reliability of the system is high. Therefore, it is a reasonable way to model and predict degradation with the help of semantic variables. In this chapter, to keep things simple, reliability levels are divided into three categories: “high” (F 1 ), “average” (F 2 ) and “low” (F 3 ). Therefore, a state recognition framework is defined as follows: F = {F j , j = 1, 2, 3} = {High, Average, Low} All factors related to reliability are evaluated by rule-based information conversion technology under this framework. By analyzing the characteristics of reliability data, the equivalent rules of data conversion are determined by the method similar to that in Ref. [5], as shown in Table 8.1. For example, the 35th reliability value on the curve shown in Fig. 8.1 is 0.9412, which can be converted into S(y(35)) = {(High, 0.706), (Average, 0.294), (Low, 0)} according to the reference points given in Table 8.1 by using rule-based conversion technology [5]. It should be noted that the reference points given in Table 8.1 need to be selected according to specific problems. In the research of this chapter, because of the high security requirement of missile weapon, its reliability is usually required to be no less than 0.8. Through the above rules, all reliability related factors can be transformed into the identification framework F. This process is similar to the fuzzification step in fuzzy set theory [20].

8.4.3 Degradation Modeling and Prediction Model It can be seen from Fig. 8.1 that the reliability data of missile weapon is a time series, and the reliability value of the next moment depends on the reliability value of the current time to a certain extent. Therefore, it can be assumed that the current drift yt is related to the nearest drift yt−1 , and further extended to be related to the historical value yt−2 , . . . , yt− p . Therefore, the prediction model based on evidence reasoning proposed in this chapter can be used to model the reliability variation law and predict the reliability of missile weapon. Firstly, the embedding dimension p is

8.4 Case Study

187

determined by the method proposed in Ref. [15]. In this study, p = 5 is determined. Thus, yt−2 , . . . , yt− p is as the input vector of the model, so 75 observed values can be transformed into 70 sets of input and output data. Then, the prediction model is yˆ (t) = yˆt = f (yt−1 , yt−2 , yt−3 , yt−4 , yt−5 )

(8.61)

In Formula (8.61), all input and output data are numerical data. Therefore, it is necessary to use rule-based conversion technology to transform numerical data into distribution structure [5].

8.4.4 Simulation Results Based on Judgment Output In this case, according to the algorithm summarized in Sect. 8.3.1 of this chapter, the initial attribute weight value is set to w = [0.6, 0.1, 0.1, 0.1, 0.1], and the stepping factor α and correction factor γ are set to 1500 and 0.015 respectively. Then, the ER model with the above initial values is used for modeling and prediction, and the results are shown in Fig. 8.2a–c. The results show that, in the case of the above initial values, the predicted results by ER model do not match the measured values very well. This shows that the setting of the above initial value is not reasonable. Therefore, it is necessary to update the initial parameters online to make them reasonable gradually. At present, the above initial values are still selected as the initial parameters of the ER model. Then, the input and output data are used to update the parameters according to the update steps and prediction steps in the algorithm proposed in Sect. 8.3.2, and the reliability is modeled and predicted. The results are also shown in Fig. 8.2a–c. By comparing the actual reliability value with the reliability prediction value obtained after updating and prediction and the reliability value obtained only by using the initial value, it can be found that the results obtained by the parameter recursive updating algorithm can well describe the variation law of system reliability. It can also be seen from Fig. 8.2a–c that the matching degree with level “High” monotonically decreases from 0.9803 to 0. Therefore, it can be said that the probability of evaluating the system reliability level as “High” decreases monotonically from 0.9803 to 0. At the same time, the probability of evaluating the system reliability level as “Average” increases monotonically from 0.0197 to 0.997, and then decreases to 0.6025. The probability of evaluating the system reliability level as “Low” increases monotonically from 0 to 0.3975. Absolute percentage error (MAPE) and root mean square error (RMSE) are used to further measure the accuracy of the proposed algorithm. After calculation, the MAPE and RMSE between the observed value corresponding to the semantic “High” and the estimated value generated by the initial ER model are 0.1327 and 0.0731, respectively, while the MAPE and RMSE between the observed value corresponding to the semantic “high” and the estimated value generated by the updated ER model are 0.0198 and 0.0209, respectively. Obviously, the updated ER model can better describe the relationship between yt and yt−1 , yt−2 , . . . , yt−5 . In addition, the results

188

8 Degradation Modeling and Reliability Prediction Based …

1.2

1.4

Reliability of Missile

0.8

0.6

0.4

0.2

Online updating Off-line traing Actual data Initial data Bayesian forecasting Fuzzy method

1.2

Reliability of Missile

Online updating Off-line traing Actual data Initial data Bayesian forecasting Fuzzy method

1

0

1

0.8

0.6

0.4

0.2

-0.2 0

10

20

30

40

50

60

0

70

0

10

20

Times(x30days)

30

40

50

60

70

Times(x30days)

(a) Prediction results corresponding to "High”

(b) Prediction results corresponding to "Average”

0.7 Online updating Off-line traing Actual data Initial data Bayesian forecasting Fuzzy method

Reliability of Missile

0.6

0.5

0.4

0.3

0.2

0.1

0 0

10

20

30

40

50

60

70

Times(x30days)

c) Prediction results corresponding to "Low”

Fig. 8.2 Prediction results corresponding to “High,” “Average” and “Low,” respectively

are compared with the offline model proposed in Ref. [11], classic Bayesian prediction method [21, 22] and T-S fuzzy rule base [20, 23]. The structure of the model used in Ref. [11] is the same as that in this chapter, and the assumption of Gaussian distribution is adopted in the latter two aspects. In the simulation process, the parameters of Bayesian prediction method [21] are updated by Kalman filtering, while the parameters of T-S fuzzy rule base are estimated by recursive least square algorithm [20]. The simulation results are shown in Fig. 8.2, and the comparison results are shown in Table 8.2. Table 8.2 shows that the performance of online learning algorithm is better than Bayesian prediction method and fuzzy method in terms of accuracy, but the latter two methods have certain advantages in time. In addition, online learning algorithm, Bayesian prediction and fuzzy method can get more satisfactory results in accuracy and speed than offline learning method. This is because the offline learning method needs to solve the multi-objective optimization problem [11].

8.4 Case Study

189

Table 8.2 Comparison of simulation results under judgment output

MAPE

RMSE

Preliminary model

Offline learning [11]

Online updating

Bayes [21]

Model system [20]

High

0.1327

0.0408

0.0198

0.1096

0.0363

Average

0.2096

0.0756

0.0032

0.1549

0.0516

Low

0.1489

0.1296

0.0659

0.0685

0.0309

High

0.0731

0.0174

0.0209

0.0447

0.0237

Average

0.0964

0.0745

0.0269

0.0651

0.0443

Low

0.0637

0.0725

0.0169

0.0233

0.0090

0.3946

2.4965

0.8041

0.8041

0.3274

Training/learning time (s)

8.4.5 Simulation Results Based on Numerical Output In order to verify the effectiveness of the proposed recursive algorithm in the case of data output, the following simulations are given in this subsection. According to the algorithm summarized in Sect. 8.3.2, all the input initial attribute weights are equal and the initial utility is set to [u 1 , u 2 , u 3 ] = [1, 0.9, 0.65]. Meanwhile, the stepping factor α and correction factor γ are assumed to 0.5 and 0.02, respectively. As shown in Fig. 8.3, the reliability estimated by the initial ER model does not match the actual value. Therefore, it is necessary to carry out online updating for the ER model parameters. 1

Reliability of Missile

0.95

0.9

0.85 Online updating Off-line traing Actual data Initial data Bayesian forecasting Fuzzy method

0.8

0.75

0

10

20

30 40 Times(x30days)

Fig. 8.3 Prediction results in the case of data output

50

60

70

190

8 Degradation Modeling and Reliability Prediction Based …

Table 8.3 Comparison of simulation results under data output Preliminary model

Offline learning [11]

Online updating

Bayes [21]

Model system [20]

MAPE

0.0141

0.0101

0.0023

0.0063

0.0097

RMSE

0.0167

0.0149

0.0027

0.0092

0.0040

Training/learning time (s)

0.0175

2.0151

0.0890

0.0335

0.0425

After converting the input value (yt−1 , yt−2 , . . . , yt−5 ) according to the reference points shown in Table 8.1, the algorithm summarized in Sect. 8.3.2 will be used to update the parameters of ER model, and the results are shown in Fig. 8.3. After calculation, the MAPE and RMSE between the observed value and the estimated value generated by the initial ER model are 1.41% and 0.0167, respectively, while the MAPE and RMSE between the estimated values generated by the updated ER model are 0.23% and 0.0027, respectively. Obviously, compared to the initial ER model, the updated ER model can better describe the relationship between yt and yt−1 , yt−2 , . . . , yt−5 . Similar to Sect. 8.4.4, the results are compared with offline learning method, Bayesian prediction method and model method. The results are shown in Table 8.3. Similar to Sect. 8.4.4, Table 8.3 clearly indicates the differences between the various methods. Obviously, by using the online updating algorithm can obtain more satisfactory results than other methods. On the other hand, the offline learning method needs more time than other methods, that is to say, the time efficiency of other methods is better, which is very important for the reliability modeling and prediction with high real-time requirements. From the above simulation study, it can be seen that the accuracy of the initial model given by experts is not high, but it can be updated by the algorithm proposed in this chapter, regardless of whether the output result of ER model is judgmental or numeric type. Compared with other methods, the result shows that the method proposed in this chapter is better than Bayesian method and fuzzy method in terms of accuracy. Therefore, the method proposed in this chapter can well model and accurately predict the reliability of missile weapon, which will lay a solid foundation for the subsequent predictive maintenance.

8.5 Summary of This Chapter This chapter studies the recursive updating algorithm of ER model parameters for performance degradation modeling from the perspective of probability. The algorithm provides a novel idea for increasing the predictive ability of ER model. Different from the existing optimized methods for ER model, the recursive algorithm proposed in this chapter can well adjust the parameters of ER model when new information

8.5 Summary of This Chapter

191

is obtained. Through the case study, this chapter also explains how to realize the proposed algorithm. The test results show that the ER model through online updating has broad application prospects in degradation modeling and prediction methods. Although the simulation results show that the algorithm proposed in this chapter has good effect, it has not been proved theoretically. Therefore, in the future work, the performance of the algorithm will be further analyzed in terms of algorithm convergence and parameter sensitivity.

References 1. Pearl J (1988) Probabilistic reasoning in intelligence systems. Morgan Kaufmann, San Mateo, CA 2. Shafer G (1976) A Mathematical theory of evidence. Princeton Univ. Press, Princeton, NJ 3. Walley P (1996) Measures of uncertainty in expert system. Artif Intell 83(1):1–58 4. Yang JB, Singh MG (1994) An evidential reasoning approach for multiple attribute decision making with uncertainty. IEEE Trans Syst Man Cybern Part A Syst Humans 24(1):1–18 5. Yang JB (2001) Rule and utility based evidential reasoning approach for multi-attribute decision analysis under uncertainties. Eur J Oper Res 131(1):31–61 6. Yang JB, Xu DL (2002) On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty. IEEE Trans Syst Man Cybern Part A Syst Humans 32(3):289–304 7. Hu CH, Si XS, Yang JB et al (2011) Online updating with a probability-based prediction model using expectation maximization algorithm for reliability forecasting. IEEE Trans Syst Man Cybern Part A Syst Humans 41(6):1268–1277 8. Zhou ZJ, Hu CH, Yang JB et al (2009) Online updating belief-rule-based systems for pipeline leak detection under expert intervention. Exp Syst Appl 36(4):7700–7709 9. Si XS, Hu CH, Zhou ZJ (2010) Fault prediction model based on evidential reasoning approach. Sci China Inf Sci 53(10):2032–2046 10. Si X, Changhua Hu, Zhang Qi et al (2012) Fault prognosis based on evolving belief-rule-base system. Control Theor Appl 29(12):1589–1586 11. Hu CH, Si XS, Yang JB (2010) Systems reliability forecasting based on evidential reasoning algorithm with nonlinear optimization. Exp Syst Appl 37(3):2550–2562 12. Chung PJ, Bohme JF (2005) Recursive EM and SAGE-Inspired algorithms with application to DOA estimation. IEEE Trans Sig Process 53(8):2664–2677 13. Dempster AP, Laird N, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J Roy Stat Soc B 39(1):1–38 14. Titterington DM (1984) Recursive parameter estimation using incomplete data. J Roy Stat Soc B 46(2):257–267 15. Cao L (1997) Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110:43–50 16. Ying-Ming, Wang Jian-Bo, Yang Dong-Ling, Xu (2006) Environmental impact assessment using the evidential reasoning approach. Eur J Oper Res 174(3):1885–1913 https://doi.org/10. 1016/j.ejor.2004.09.059 17. Kushner HJ, Kelmanson MZ (1976) Stochastic approximation algorithms of the multiplier type for the sequential Monte Carlo optimization of stochastic systems. SIAM J Control Optim 14(5):827–842 18. Kushner HJ, Lakshmivarahan S (1977) Numerical studies of stochastic approximation procedures for constrained problems. IEEE Trans Autom Control 22(3):428–439 19. Kushner HJ, Yin GG (1997) Stochastic approximation algorithms and applications. Springer, New York 20. Passino KM, Yurkovich S (1998) Fuzzy control. Addison-Wesley, Menlo Park, Calif

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21. Harrison PJ, Stevens CF (1976) Bayesian forecasting. J Roy Stat Soc B 38(3):205–247 22. West M (1981) Robust sequential approximate Bayesian estimation. J Roy Stat Soc B 43(2):157–166 23. Ordonez R, Spooner JT, Passino KM (2006) Experimental studies in nonlinear discrete-time adaptive prediction and control. IEEE Trans Fuzzy Syst 14(2):275–286

Chapter 9

Weight Optimization-Based Particle Filter Algorithm for Degradation Modeling and Residual Life Prediction

9.1 Introduction Particle filter (PF) [1–5] is often used to estimate and predict the system state when the system model is known. This algorithm is a nonlinear filtering algorithm based on Bayesian estimation, which has become the mainstream method to solve the parameter estimation and state filtering problems of nonlinear non-Gaussian system in recent years. The main problem of particle filter algorithm in performance degradation modeling and application is how to improve the tracking ability of the algorithm and reduce the amount of calculation. When the system is in normal operation, its state remains unchanged for a long time. Degradation phenomenon and sample impoverishment are particularly prominent in estimating the quantities that remain unchanged for a long time. How to solve the problems of degradation and sample impoverishment? How to improve the tracking ability of the algorithm for the states with large changes and maintaining invariants for a long time? How to reduce the amount of calculation of particle filter algorithm? These problems need to be solved for the fault prediction application of particle filter algorithm. Professor Changhua Hu and his team have carried out in-depth studies on particle filter algorithm and achieved a series of results [6–12]. A new algorithm [7] is formed by introducing the weight optimization into particle filter algorithm, which is applied to performance degradation modeling in order to accurately estimate the residual life of the system. Firstly, the particle filter algorithm and the particle filter algorithm based on weight optimization are introduced. Then, on this basis, the performance degradation modeling and life prediction method according to particle filter algorithm based on weight optimization are given, and the simulation study is carried out.

© National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_9

193

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9 Weight Optimization-Based Particle Filter Algorithm …

9.2 Particle Filter Algorithm Based on Weight Optimization 9.2.1 Particle Filter Algorithm and Characteristic Analysis 9.2.1.1

Sequential Importance Sampling Algorithm

Sequential importance sampling algorithm is a Monte Carlo method. It is a recursive importance sampling algorithm and the basis of particle filter. So far, basically various particle filter algorithms are the different improved forms of sequential importance sampling. Considering the nonlinear dynamic system [13]: 

xk = f (xk−1 , νk−1 ) yk = h(xk , n k )

(9.1)

where xk ∈ R n x is the state vector of the system at k, yk ∈ R n y is the observation output, νk ∈ R n ν is the system noise, n k ∈ R n n is the observation noise, and the mapping f (·) and h(·) are as follows: 

f : R n x × R n y → R n x h : R n x × R nn → R n y

(9.2)

They constitute the state equation and observation equation of the system, respectively. Posterior density p( x0:k |y1:k ) is the complete solution of sequential estimation problem. According to the principle of Monte Carlo simulation, the posterior density p( x0:k |y1:k ) can be approximately expressed as p( x0:k |y1:k ) ≈

N 

  i wki δ x0:k − x0:k

(9.3)

i=1 i The importance density q( x0:k |y1:k ) is introduced, and the sample x0:k is assumed to be obtained from the importance density sampling, then

xki ∼ q( x0:k |y1:k )

(9.4)

Importance weight wki

 i    y1:k p x0:k    ∝ q x i  y1:k 0:k

(9.5)

9.2 Particle Filter Algorithm Based on Weight Optimization

195

The importance density is assumed to be decomposed as q( x0:k |y1:k ) = q( xk |x0:k−1 , y1:k )q( x0:k−1 |y1:k−1 )

(9.6)

i ∼ q( x0:k |y1:k ) can be obtained by adding new That is, the sample set x0:k i i ∼ q( x0:k−1 |y1:k−1 ). And p( x0:k |y1:k ) can particle xk ∼ q( xk |x0:k−1 , y1:k ) to x0:k−1 be expressed in the following recursive form. According to Bayes formula and total probability formula

p( y1:k |x0:k ) p(x0:k ) p(y1:k ) p( yk | y1:k−1 |x0:k ) p( y1:k−1 |x0:k ) p(x0:k ) = p( yk |y1:k−1 ) p(y1:k−1 ) p( y1:k−1 |x0:k ) p(x0:k ) p( yk | y1:k−1 |x0:k ) × = p( yk |y1:k−1 ) p(y1:k−1 )

p( x0:k |y1:k ) =

(9.7)

Use Bayes formula p( yk | y1:k−1 |x0:k ) × p( x0:k |y1:k−1 ) p( yk |y1:k−1 ) p( yk | y1:k−1 |x0:k ) × p( xk | x0:k−1 |y1:k−1 ) p( x0:k−1 |y1:k−1 ) = p( yk |y1:k−1 )

p( x0:k |y1:k ) =

(9.8)

The system state is subject to the first-order Markov process, and the system observation is independent, so p( yk |xk ) × p( xk |xk−1 ) p( x0:k−1 |y1:k−1 ) p(yk ) ∝ p( yk |xk ) p( xk |xk−1 ) p( x0:k−1 |y1:k−1 )

p( x0:k |y1:k ) =

(9.9)

If the importance density satisfies the following condition q( xk |x0:k−1 , y1:k ) = q( xk |xk−1 , yk )

(9.10)

Combine Formula (9.5)–(9.10) wki

∝

i wk−1

   i   p yk |xki p xki xk−1    q x i x i , yk k

(9.11)

k−1

Then   i , yk xki ∼ q xk |xk−1 Weight normalization

(9.12)

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9 Weight Optimization-Based Particle Filter Algorithm …

wki

=

wki

 N 

wki

(9.13)

i=1

Generally    i    i , yk = p xki xk−1 q xki xk−1

(9.14)

  i xki ∼ p xk |xk−1

(9.15)

  i p yk |xki wki ∝ wk−1

(9.16)

i.e.,

So,

The above is the basis of SIS particle filter algorithm. According to each measurement value of the system, samples and weights are calculated recursively according to SIS method, thus forming SIS particle filter algorithm. Algorithm 9.1 Process of SIS particle filter algorithm Step 1: sampling according to the importance density at k = 0, take k = 1; Step 2: prediction   i , νk−1 xki = f xk−1

(9.17)

Step 3: weighting wki

=

i wk−1

   i   p yk |xki p xki xk−1    i × q xki xk−1 , yk

(9.18)

Step 4: weight normalization wki

=

wki

N 

wki

(9.19)

xki × wki

(9.20)

i=1

Step 5: state estimation xk* =

N  i=1

9.2 Particle Filter Algorithm Based on Weight Optimization

197

Step 6: return to Step 2.

9.2.1.2

Degradation Phenomenon of the Particle Filter Algorithm

Doucet proved the inevitability of degradation phenomenon of SIS algorithm theoretically [14]. Degradation hinders the development of particle filter algorithm. The degradation degree of particle is usually measured by effective particle number Neff [15] N   1 + Var wk∗i    p xki  y1:k ∗i  wk =  i  i q x x , yk Neff =

k

(9.21)

(9.22)

k−1

where wk∗i is called “real weight.” However, in many cases, it is difficult to calculate the value accurately, and the estimate can be obtained from the following formula 

Neff

1 ≈ round N  2 i i=1 wk

(9.23)

where round(•) represents the rounding operation to the nearest integer, wki is the normalized weight, and the above formula shall meet 1 ≤ Neff ≤ N . In the following two extreme cases, Neff is taken as 1 and N, respectively: (1) (2)

When the weight is uniformly distributed, wki = N1 , then Neff = N ; j If ∃ j ∈ {1, 2, · · · , N }, so that wk = 1, and wki = 0, ∀i = j, then Nef f = 1.

The smaller Neff , the more serious the degradation phenomenon. Obviously, the degradation is an undesired result for the particle filter algorithm. The common method to reduce the effect of this degradation is to select appropriate importance density and adopt resampling algorithm. (1)

Importance density selection

The criterion to select the importance density is to minimize the variance of importance weight, and the optimal importance density is q( xk |x0:k−1 , y1:k )= p( xk |x0:k−1 , y1:k ). However, the optimal importance density needs to be sampled directly from p( xk |x0:k−1 , y1:k ) for posterior density, and the integral is calculated in the whole state space. From the perspective of application, most of the importance density adopts q( xk |x0:k−1 , y1:k )= p( xk |xk−1 ), which is suboptimal but obtained easily by the algorithm.

198

(2)

9 Weight Optimization-Based Particle Filter Algorithm …

Resampling

Resampling algorithm is another method to solve the degradation problem. When a serious degradation phenomenon is observed; that is, when the effective sample number is less than a critical value, resampling is used to realize the redistribution of computing resources. The basic idea of resampling is to eliminate the particles with small weights and concentrate the computing resources on the important particles with large weights, so as to avoid the situation that the weight of most particles is almost zero. The idea of resampling is clearly described in detail in Refs. [16, 17]. The most commonly used resampling method is the system resampling method. When the total number of samples remains n, the samples with larger weight are copied many times, so as to realize the resampling process. Obviously, the resampling process reduces degradation at the expense of amount of calculation and robustness.

9.2.1.3

SIR Particle Filter Algorithm

Algorithm 9.2 Process of resampling particle filter algorithm (SIR) [18] Step 1: sampling N particles according to the importance density at k = 0, i , 1/N > , take assuming that each particle sampled is represented by < xk−1 k = 1; Step 2: prediction   i , νk−1 xki = f xk−1

(9.24)

Step 3: weighting wki

=

i wk−1

   i   p yk |xki p xki xk−1    × q x i x i , yk k

(9.25)

k−1

Step 4: weight normalization wki

=

wki

N 

wki

(9.26)

xki × wki

(9.27)

i=1

Step 5: state estimation xk* =

N  i=1

Step 6: if Neff < N /3, resampling; Step 7: return to Step 2.

9.2 Particle Filter Algorithm Based on Weight Optimization

199

SIR particle filter algorithm usually adopts system resampling to conduct resampling for particle set. The main idea of system resampling is to conduct sampling for particles in the particle set according to the weight, which can be selected repeatedly. Algorithm 9.3 System resampling algorithm Initialize cumulative distribution function:c1 = 0 for i = 2 : N ci = ci−1 + wki end for Cycle: start from i=1 Sampling in the uniformly distributed [0, 1/N ] : μl ∼ U [0, 1/N ] for j = 1 : N u j = u l + ( j − 1)/N ; while u j > ci i =i +1 end while j∗ Sample: xk = xki j∗ Weight: wk = 1/N end for As shown above, because the resampling algorithm can repeatedly sample when sampling, the samples with larger weight are selected many times, which will inevitably lead to the sample impoverishment.

9.2.1.4

Convergence

Berzuini et al. [19] established the central limit theorem for SIR algorithm, and Crisan et al. [20] established more general convergence results from two aspects, one is the empirical distribution generated by particle filter algorithm pˆ =

N 

  i w˜ ki δ x0:k − x0:k

(9.28)

i=1

which almost certainly converges (weak convergence) to state posterior distribution p( x0:k |y1:k ). The other is the convergence of mean square error of state estimation;

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9 Weight Optimization-Based Particle Filter Algorithm …

that is ∀k ≥ 0, there is a constant ck independent of N, for any bounded measurable function f k 

N 1   i  f k (x0:k ) p( x0:k |y1:k )d x0:k E f k x0:k − N i=1

2 ≤ ck

 f k 2 N

(9.29)

This conclusion shows that the convergence rate of the particle filter algorithm is 1/N , and independent of the dimension of the state space under relatively weak assumptions. However, ck usually increases exponentially with time. Under some stronger assumptions, uniform convergence results can be obtained [20] 

n 1  i  E f k (x0:k ) p( x0:k |y1:k )d x0:k f k x0:k − n i=1

2 ≤c

 f k 2 N

(9.30)

The existing results show that [21], under some conditions, with the increase of time k, if N increases by k 2 , the approximation error remains stable. However, for a fixed N, whether the error is stable or not has not been general conclusion [22].

9.2.2 Particle Filter Algorithm Based on Weight Optimization Although SIR algorithm can solve the degradation problem, it brings sample impoverishment problem. To solve this problem, the scholars both at home and abroad have made some research achievements, such as resampling moving algorithm, simulated annealing particle filter algorithm and auxiliary particle filter algorithm. Although they can solve the sample impoverishment problem to a certain extent, they cannot solve the estimation problem of long-term invariants [23]. In order to solve the above problems, this chapter proposes a particle filter algorithm based on weight optimization. The algorithm selects the particle with relatively large weight from a large number of candidate particles for state estimation, so as to improve the diversity of the sample set and solve the degradation problem to a certain extent, thus improve the estimation and tracking ability of the particle filter algorithm.

9.2.2.1

Basic Idea of the Algorithm

The basic idea of particle filter algorithm based on weight optimization: If the required number of particles is N, when the system is initialized, N s (N s > N) samples are extracted and given a weight of 1/N . After N s samples are predicted, the corresponding weights are calculated and sorted separately according to the weights. The weights of the N samples with larger weights are normalized. The N samples are

9.2 Particle Filter Algorithm Based on Weight Optimization

201

used to estimate the state of the system. Then the weights of N samples are restored to the values before normalization, and then the weights of all samples are normalized. Finally, the N s samples enter the next iteration. In this way, the diversity of the samples involved in the state estimation can be guaranteed to the maximum extent, so as to solve the problem of sample impoverishment and degradation problem to a certain extent.

9.2.2.2

Particle Filter Algorithm Based on Weight Optimization

Definition 9.1 Weights are assigned to the sample set at k = 0 and the sample in the sample set after resampling wki

=

i wk−1

   i   p yk |xki p xki xk−1 1   i i × =  N q xk xk−1 , yk

(9.31)

where i ∈ [1, Ns ], and Ns > N .

Algorithm 9.4 Particle filter algorithm process based on weight optimization Step 1: sampling N s particles according to the importance density at k = 0, i each particle sampled shall represented < xk−1  , 1/N >, and take k = 1;  by i i Step 2: according to Formula xk = f xk−1 , νk−1 , calculate the state of N s particles at k; p y |x i p x i |x i i × ( qk xki )|x i( k,y k−1 ) , calculate the Step 3: according to Formula wki = wk−1 ( k k−1 k ) weight of N s particles; Step 4: sort the N s particles according to the weight, and select previous N particles; Step 5: normalize the weight of N particles taken out according to xk* = N i ˜ ki ; i=1 x k × w Step 6: use the N particles to estimate the state according to Formula selected N w˜ ki ; wki = w˜ ki × i=1 Step 7: restore the weight N of Ni particles selected to those according to w˜ k before normalization, then conduct the Formula wki = w˜ ki × i=1 N s i normalization for the weights of all N s particles: wki = wki / i=1 wk ; Step 8: return to Step 2 for the next iteration.

202

9.2.2.3

9 Weight Optimization-Based Particle Filter Algorithm …

Analysis of Computation Complexity

Lemma 9.1 [24] In the particle renewal stage, the computation complexity of SIR particle filter algorithm is O(N ). Theorem 9.1 [24] In the particle renewal stage, the computation complexity of particle filter algorithm based on weight optimization is O(Ns ). Proof When the system is initializing, for the particle filter algorithm based on weight optimization, the extracted particle number is Ns (Ns > N ), and Ns particles are involved in the subsequent particle renewal process. In the particle renewal stage, the computation complexity of SIR particle filter algorithm is O(N ); therefore, the computation complexity of particle filter algorithm based on weight optimization is O(Ns ), Q.E.D. The common resampling algorithm resamples the particles in the particle set according to the weight, which will inevitably lead to the sample impoverishment. However, all Ns particles in the particle filter algorithm based on weight optimization are involved in the particle update at any time, and each particle is statistically independent from each other, which makes the particle set contain more different particle paths and improves the diversity of the particle set. The particle filter algorithm based on weight optimization will inevitably, slightly increase the amount of calculation. However, if the conditions permit, the algorithm improves the diversity of the sample set, reduces the influence of degradation and makes the algorithm have better tracking ability.

9.3 Degradation Modeling with Weight Optimization-Based Particle Filter 9.3.1 Description of Degradation Process This chapter mainly aimed at the dynamic system with hidden performance degradation process, as shown in the following Formula [25]: 

x n = f (x n−1 , ϕ) + ωn−1 yn = h(x n ) + vn

(9.32)

where x n ∈ R p is the state vector of the system, yn ∈ Rq is the observation output, {ωn ∈ R p , n ∈ N} and {vn ∈ Rq , n ∈ N} are independent of each other. ϕ ∈ Rr is the parameter vector of the system related to the variable of performance degradation φ ∈ R. When the performance degradation occurs on a component of the system, the parameter vector of the system ϕ will change directly, which will indirectly affect

9.3 Degradation Modeling with Weight Optimization-Based …

203

the state and observation vector of the system. Since the performance degradation amount of the system cannot be directly measured, the system is called a dynamic system with hidden degradation process. Considering that Brownian motion with drift is often used in performance degradation modeling and reliability analysis. Therefore, this chapter further assumes that the performance degradation amount φ is Brownian motion with drift; that is φ(τ ) = η0 + ητ + σ B(τ )

(9.33)

where η0 = φ(0) is the initial value of performance degradation, η ∈ R, σ ∈ R + , B(0) = 0. Before the system is put into use, the initial value of performance degradation process can be measured in some ways, Therefore, η0 can be considered as a known quantity, while parameters η, σ are unknown quantity to be estimated. The Brownian motion with drift has the following properties: φ(τ ) ∼ N (η0 + ητ, σ 2 τ ) φ(τ + τ ) − φ(τ ) ∼ N (ητ, σ 2 τ ) Therefore, the continuous performance degradation process can be rewritten as follows φn+1 = φn + ηT + εn φ0 = η0

(9.34)

where φn = φ(nT ) is the performance degradation value at nT, with error of εn ∼ N (0, σ 2 T ). In conclusion, the system Eq. (9.32) can be rewritten as follows: ⎧ ⎨ x n+1 = f (x n , ϕ(φn )) + ωn−1 φ = φn + ηT + εn ⎩ n+1 yn+1 = h(x n+1 ) + vn+1

(9.35)

9.3.2 Parameter Estimation The system shown in Formula (9.34) is a nonlinear non-Gaussian system; therefore, particle filter algorithm can be used to estimate its state and parameters. Doucet and Tadi´c [26] considered the application of recursive maximum likelihood estimation to parameter estimation on the basis of minimizing the mean log-likelihood function by using random approximation method. In order to calculate the derivative of the log-likelihood function, they use the sampled particles to calculate the derivative of

204

9 Weight Optimization-Based Particle Filter Algorithm …

the state prediction distribution and the posterior distribution. This chapter proposes the parameter estimation method similar to that in this reference. Denote θ = [η, σ ] as the unknown parameter vector of the system, and x˜ = [x  , φ] as the expanded state vector of the system. So, the likelihood function expression of the system is



L(θ ) =

···

n         p yk  x˜ k pθ x˜ k  x˜ k−1 d x˜ 0 · · · d x˜ n . p0 x˜ 0 k=1

Since x k is related to φk−1 , pθ ( x k | x˜ k−1 ) has nothing to do with parameter vector θ , and φk only depends on φk−1 . So, the above formula can be written as



L(θ) =

···

p0 ( x˜ 0 )

n 

 p( yk  x˜ k ) p( x k | x˜ k−1 ) pθ ( φk |φk−1 )d x˜ 0 · · · d x˜ n .

k=1

where the form of pθ ( φk |φk−1 ) can be determined by Formula (9.34); that is   (φk − φk−1 − ηT )2 . exp − pθ ( φk |φk−1 ) = √ 2σ 2 T 2π T σ 1

(9.36)

Therefore, the log-likelihood function can be written as l(θ ) =

n 

       log p yk  x˜ k p x k | x˜ k−1 pθ ( φk |φk−1 )d x˜ k

k=1

The purpose of the recursive maximum likelihood estimation method is to maximize lim (1/n)l(θ ). The specific parameter updating formula is n→∞



 p( yn  x˜ n ) p( x n | x˜ n−1 ) pθ

 ( φn |φn−1 )d x˜ n /∂θ n−1 n−1

θ n =θ n−1 + γn ∂ log   p( yn  x˜ n ) p( x n | x˜ n−1 )∇θ n−1 pθ n−1 ( φn |φn−1 )d x˜ n  =θ n−1 + γn  . p( yn  x˜ n ) p( x n | x˜ n−1 ) pθ ( φn |φn−1 )d x˜ n

(9.37)

n−1

a positive iteration size sequence of non-decreasing sequence, where {γn }n≥0 is step γn2 = ∞. According to Formula (9.36), the which satisfies γn = ∞ and expression of gradient ∇θ n−1 pθ n−1 ( φn |φn−1 ) in Formula (9.37) can be obtained. Next, the weight particle filter algorithm is used to estimate the parameters of the performance degradation process.

9.3 Degradation Modeling with Weight Optimization-Based …

205

Algorithm 9.5 Performance degradation parameter estimation algorithm according to particle filter algorithm based on weight optimization (i) Step 1: extract particle xˆ 0|0 ∼ p(x 0 ), i = 1, · · · , Ns , at n = 0 and let φˆ 0|0 = η0 , then give the appropriate parameter vector for initial estimation of θˆ 0 ; (i) Step 2: assume the current time nT, extract particle xˆ n|n−1 ∼     (i) (i) (i) (i) , φˆ n|n−1 φn |φˆ n−1|n−1 , i = 1, · · · , Ns , ∼ pˆ p x n | xˆ n−1|n−1 , φˆ n−1|n−1 θ n−1

respectively;    (i) Step 3: according to ωn(i) ∝ p yn  xˆ n|n−1 , calculate the weight of each particle; Step 4: sort Ns particles according to their weight, and select previous (i) (i) N particles, denote as xˆ˜ 1n|n−1 , i = 1, . . . , N , where xˆ˜ 1n|n−1 =   (i) xˆ 1n|n−1 , φ (i) 1n|n−1 , and denote the corresponding weight of each particle (i) as ω1n ; (i) = Step 5: normalize the weight of N particles taken out according to ω˜ 1n (i) N (i) ω1n / i=1 ω1n . N (i) (i) Step 6: estimate the performance degradation value φˆ n = i=1 ω1n φ1n|n−1 ; Step 7: use the selected N particles to update the parameter according to Formulas (9.38) and (9.39) given below, where the step size is γn = γ0 n −β , γ0 > 0, 0.5 < β ≤ 1;

      ˆ (i) (i) ˆ | ˆ φ x ∂ p φ p y  n ηˆ n−1 1n|n−1 i=1 1n|n−1 n−1 θˆ n−1      ηˆ n = ηˆ n−1 + γn (9.38) N  (i) (i) ˆ 1n|n−1 pθˆn−1 φˆ 1n|n−1 φˆ n−1 i=1 p yn | x       N ˆ (i) (i) ˆ | ˆ φ x ∂ p φ p y  n n−1 σ ˆ ˆ n−1 1n|n−1 i=1 1n|n−1 θ n−1      σˆ n = σˆ n−1 + γn (9.39) N ˆ (i) (i) ˆ | ˆ x p φ p y n ˆ 1n|n−1 i=1 1n|n−1 φn−1 N

θ n−1

Step 8: use the relative values in Step 3 to resample all Ns particles, then return to Step 2 for the next iteration.

9.4 Residual Life Prediction Since the performance degradation process is assumed to be the Brownian motion with drift represented by Formula (9.33), the residual life distribution at the current time, that is, the distribution of the first passage time, can be calculated after the parameters at the current time are estimated and given the failure threshold

206

9 Weight Optimization-Based Particle Filter Algorithm …

φth . According to the content in Chapter 3, the distribution is inverse Gaussian distribution, and its probability density function is shown in the following formula: ⎛  2 ⎞ ˆ   φ − φ − η ˆ t th n n φth − φˆ n ⎟ ⎜ exp⎝− f n t; φˆ n , φth = √ ⎠ 2 3 2σˆ n t σˆ n 2π t

(9.40)

Algorithm 9.6 Residual life estimation algorithm according to particle filter algorithm based on weight optimization ˆ Step N 1:(i)at(i)nT, estimate the performance degradation value φn = ˆ i=1 ω1n φ1n|n−1 . If φn < φth (monotone decreasing degradation process) or φˆ n > φth (monotone increasing degradation process), the system fails and the algorithm stops; Step 2: the residual life distribution of the current time can be obtained by substituting the parameters of the current time estimated by algorithm 9.5 and φˆ n into Formula (9.40), then return to Step 1 for the next iteration.

9.5 Numerical Simulation This simulation adopts a three-tank system (DTS200) manufactured by German Amira Company in Reference [27]. As shown in Fig. 9.1, the main body of the

Fig. 9.1 Three-tank system

9.5 Numerical Simulation

207

device is 3 vertically placed organic glass cylinders T 1 , T 2 , T 3 , the cross-sectional area of each cylinder is A. The three cylinders are connected by circular pipes with a cross section of S n , and a water outlet valve is arranged below the cylinder T 2 . The outflow water is collected into the organic glass water tank below and can be recycled. There is a leakage valve with a cross-sectional area of S 1 below T 1 , T 2 and T 3 . Under normal circumstances, these leakage valves are closed. The liquid levels h1 and h2 in T 1 and T 2 are controlled by flow rates Q1 and Q2 of circulating water pumped into T 1 and T 2, respectively, by two pumps. The state variables of the system are liquid levels of h1 , h2 and h3 in the three water tanks. The system model is as follows: ⎧ dh 1 ⎨ A dt = Q 1 − Q 13 A dh 2 = Q 13 − Q 32 ⎩ dhdt3 A dt = Q 2 + Q 32 − Q 20

(9.41)

where   Q 13 = az 1 Sn sgn(h 1 − h 3 ) 2g|h 1 − h 3 |1/2   Q 32 = az 3 Sn sgn(h 3 − h 2 ) 2g|h 3 − h 2 |1/2 Q 20 = az 2 Sn (2gh 2 )1/2

(9.42)

The relevant parameter values of the three-tank system are set as follows: A = 0.0154 m2 , Sn = 5 × 10−5 m2 , g = 9.81 m/s2 , az1 = 0.490471, az2 = 0.611429, az3 = 0.450223. The liquid level selected is the state variable, i.e.,xi = h i , i = 1, 2, 3, and it is assumed that the liquid level h 1 , h 3 can be directly measured, and their noise is Gaussian distribution with mean value of 0 and variance of 1 × 10−4 , while the measurement noise of liquid level h 2 is non-Gaussian noise. ⎧ ⎨ y1,n = x1,n + v1,n y = x2,n + v2,n ⎩ 2,n y3,n = x3,n + v3,n

(9.43)

where v2,n = 1 × 10−2 v∗ v∗ is the random variable which obeys the T distribution of 20 degrees of freedom. It is further assumed that the system noise vector is Gaussian noise with mean value of 0 and variance of 1 × 10−8 I3 . In addition, it is assumed that the performance degradation process of the system is related to the flow system az1 , with the degradation path of az1 (τ ) = η0 + ητ + σ B(τ )

(9.44)

208

9 Weight Optimization-Based Particle Filter Algorithm … 0.5 real value estimated value

performance degradation value

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0

200

400

600

800

1000

1200

1400

1600

1800

2000

Fig. 9.2 Actual performance degradation path and its estimate

where η0 = 0.490471, the true values of the parameter are η = −2 × 10−4 , σ = 4 × 10−5 , respectively, the failure threshold is set to az th = 0.1. The simulation time is 2000s, and the sampling interval is 1 s. The particle number in the relative value particle filter algorithm is N s = 1000, N = 500, the initial distribution of system state vector is Gaussian distribution, with the mean vector of [0.45555, 0.15902, 0.31995] , the variance matrix of 1 × 10−8 I3 , and the initial estimate of degradation process parameters is set to ηˆ 0 = −0.0001, σˆ 0 = 0.0001. The simulation results are shown in Figs. 9.2, 9.3 and9.4. It can be seen from Fig. 9.2 that the estimated performance degradation path can fit the actual performance degradation process well after a period of fluctuation. Figure 9.3 shows the estimated results of performance degradation process parameters. Similar to Fig. 9.2, the estimate of the parameters also fluctuates for a period of time and then gradually converges to their true values, which shows the effectiveness and convergence of the parameter estimation algorithm. Finally, Fig. 9.4 shows the distribution of the residual life when the time is 1920, 1921, 1922, 1923 and 1924, respectively. It can be seen from the figure that the residual life of the system is constantly updated and the mean is getting smaller and smaller with the time goes by, which is consistent with the intuitive sense and the actual situation. Therefore, from the simulation results of the three-tank, it can be seen that the residual life prediction algorithm according to particle filter algorithm based on weight optimization can accurately identify the degradation process of the system and estimate the residual life timely and accurately. To sum up, the residual life prediction method according to particle filter algorithm based on weight optimization is feasible and effective.

9.6 Summary of This Chapter

(a)

209

(b)

Fig. 9.3 Estimated results of degradation process parameters

Fig. 9.4 Residual life distribution at different monitoring times

9.6 Summary of This Chapter In practical application, the conventional particle filter algorithm is easily affected by the sample impoverishment, the influence is particularly serious when estimating the quantities that remain unchanged for a long time. To solve this problem, a particle

210

9 Weight Optimization-Based Particle Filter Algorithm …

filter algorithm based on weight optimization is proposed. The particle with large weight is selected from a large number of particles for estimation; thus, the diversity of the sample set is improved, the degradation problem is solved to a certain extent, and the tracking and estimation ability of the algorithm is improved. The simulation results show that the particle filter algorithm based on weight optimization is proposed for the sample impoverishment, and the particle diversity is relatively good. Therefore, the residual life prediction algorithm according to particle filter algorithm based on weight optimization is more sensitive to the change of the system state and can monitor the small changes. However, because the number of particles in the sampling particle set is limited, with the increase of iteration steps, the number of effective samples in the particle set gradually decreases, which will inevitably affect the tracking ability of the algorithm. If the number of particles in the particle set is increased greatly, this problem will be alleviated, but the amount of calculation will increase exponentially. Therefore, the residual life prediction algorithm according to particle filter algorithm based on weight optimization is suitable for the system which needs to test small changes, has less iteration steps and requires less real-time requirements.

References 1. Chen M, Zhou D (2003) Fault prediction techniques for dynamic systems. Control Theory Appl 20(6):819–820 2. Cappe O, Godsill SJ, Moulines E (2007) An overview of existing methods and recent advances in sequential Monte Carlo. IEEE Proc 95(5):899–924 3. Wu JW, Trivedi MM (2007) Simultaneous eye tracking and blink detection with interactive particle filters. EURASIP J Adv Signal Proc 2008(1):1–17 4. Guo D, Wang X, Chen R (2005) New sequential Monte Carlo methods for nonlinear dynamic systems. Stat Comput 15(2):135–147 5. Kashiwaya S (2007) Chemical reaction rate parameter estimation by MAP particle filter algorithm. In: 2007 IEEE congress on evolutionary computation. pp 4489–4496 6. Changhua Hu, Zhang Qi, Qiao Y (2008) Strong tracking particle filter with application to fault prediction. Acta Automatica Sinica 34(12):1522–1528 7. Zhang Qi, Changhua Hu, Qiao Y (2008) Particle filter algorithm based on weight selected. Control Decis Mak 23(1):117–120 8. Zhang Qi, Xin W, Changhua H, et al. (2008) Research on artificial immune particle filter. Control Decis Mak 23(3):293–296+301 9. Zhang Qi, Changhua Hu, Qiao Y et al (2009) Fault prediction algorithm based on stochastic perturbed particle filter. Control Decis Mak 24(2):284–288 10. Zhang Qi, Changhua Hu, Qiao Y (2009) Fault prediction method based on clustering particle filter. Inf Control 38(1):115–120 11. Zhang Qi, Changhua Hu, Qiao Y et al (2009) Fault prediction algorithm according to particle filter algorithm based on weight optimization. Syst Eng Electron 31(1):221–224 12. Zhang Qi, Changhua Hu (2007) Study of particle filter with dynamic particle number. Control Eng 14(7):32–34 13. Li T (2003) Application of nonlinear filtering method in navigation system. National University of Defense Technology, Changsha 14. Doucet A, Godsill S (1998) On sequential Monte Carlo sampling methods for Bayesian filtering. University of Cambridge, Cambridge

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15. Liu JS, Chen R (1998) Sequential Monte Carlo methods for dynamic systems. J Am Stat 83:1032–1044 16. Carpenter J, Clifford P, Fearnhead P (1999) Improved particle filter for nonlinear problems. IEE Proc-Radar, Sonar Navig 146(1):2–7 17. Arulampalam S, Maskell S, Gordon N (2002) A tutorial on particle filters for online nonGaussian Bayesian tracking. IEEE Trans Signal Proc 50(2):174–188 18. Gordon NJ, Salmond DJ, Smith AFM (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc-F 140(2):107–113 19. Berzuini C, Best NG, Gilks WR et al (1997) Dynamic conditional independence models and Markov chain Monte Carlo methods. J Am Stat Assoc 92:1403–1411 20. Crisan D, Doucet A (2002) A survey of convergence results on particle filtering methods for practitioners. IEEE Trans Signal Proc 50(3):736–746 21. Kunsch HR (2001) State space and hidden Markov models. In: Barndorff-Nielsen OE, Cox DR, Kluppelberg C (eds) Complex stochastic systems. CRC press, London, pp 109–173 22. Yang X (2006) A study of hybrid estimation theory and applications based on particle filtering. Northwestern Polytechnical University, Xi’an 23. Mo Y, Xiao D (2005) Evolutionary particle filter and its application. Control Theory Appl 22(2):269–270 24. Doucet A, Godsill S, Andrieu C (2000) On sequential Monte Carlo sampling methods for Bayesian filtering. Stat Comput 10(1):197–208 25. Xu ZG, Ji YD, Zhou DH (2008) Real-time reliability prediction for a dynamic system based on the hidden degradation process identification. IEEE Trans Reliab 57(2):230–242 26. Doucet A, Tadi´c VB (2003) Parameter estimation in general state-space models using particle methods. Ann Inst Stat Math 55:409–422 27. Chen MZ, Zhou DH (2001) Particle filtering based fault prediction of nonlinear systems. In: IFAC symposium proceedings of safe process. Elsevier Science, Washington, pp 2971–2977

Chapter 10

Degradation Modeling and Residual Life Prediction Based on Grey Predcition Model

10.1 Introduction With the development of system science and control science, the object that under study is more and more complex, and there are more and more factors that affect the state of the research object, which includes not only the complexity of the internal structure of the object, but also the diversity, variability and uncertainty of the external environmental impact. This makes it impossible for us to get the complete information of the system under limited technical means; that is, the system is in the “poor” data information state where part of the information is known and part of the information is unknown, whether it is reliability analysis, life prediction or fault prediction. This makes the modeling analysis of the system must be a kind of modeling under the state of “poor” data information [1–3]. In 1980s, the grey system theory established by Professor Julong Deng is a new method to study the uncertainty problem of insufficient data and poor data information. As for the gray system theory, the uncertain systems of “small sample” and “poor data information” with “part of information known and part of information unknown” are taken as the research objects, and the correct description and effective monitoring of the system operation behavior and evolution law are realized mainly through the generation and development of part of given information and extraction of valuable information. As the important content of grey prediction theory, the grey predcition model plays an important role in solving the prediction of systems with insufficient data and uncertainty and provides an effective and practical prediction process for solving this kind of prediction problems [4–11]. In this chapter, the classical grey GM (1,1) model is firstly introduced, which is followed by the analysis of its existing problems. In order to overcome these problems, a modified grey prediction model is then developed. Finally, a case study is provided to evaluate the effectiveness of the improved model.

© National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_10

213

214

10 Degradation Modeling and Residual Life Prediction …

10.2 Grey Predcition Model 10.2.1 Classical Grey GM (1, 1) Model [12] Assuming that the non-negative original data column is X (0) =   (0) (0) (0) x (1), x (2), . . . , x (n) , its (1-AGO) accumulated generating sequence is   k X (1) = x (1) (1), x (1) (2), . . . , x (1) (n) , where x (1) (k) = i=1 x (0) (i), k = 1, . . . , n. While the generated mean sequence of consecutive neighbors of original data   column can be denoted as Z (1) = z (1) (2), z (1) (3), . . . , z (1) (n) , where z (1) (k) =

 1  (1) x (k) + x (1) (k − 1) , k = 2, 3, . . . , n 2

(10.1)

For X (1) , the whitening differential equation can be established as follows dx (1) + ax (1) (t) = b dt

(10.2)

That is, GM (1, 1) model, where parameter a is development coefficient and b is grey action quantity. The difference form of Formula (10.2) is x (0) (k) + az (1) (k) = b

(10.3)

Denote ⎡

−z (1) (2) ⎢ −z (1) (3) ⎢ B=⎢ .. ⎣ .

⎡ (0) ⎤ ⎤ x (2) 1 ⎢ x (0) (3) ⎥ 1⎥ ⎢ ⎥ ⎥ .. ⎥, Y = ⎢ .. ⎥, ⎣ . ⎦ .⎦

−z (1) (n) 1

x (0) (n)

Take  = (a, b)T as the parameter vector to be identified, then it can be derived according to Eq. (10.2) Y = B where the parameter vector can be obtained by the least square method ˆ T = (B T B)−1 B T Y ˆ = (a,  ˆ b)

(10.4)

10.2 Grey Predcition Model

215

The obtained parameters are substituted into Eq. (10.2), and the discrete solution is obtained as follows

 bˆ −ak bˆ (1) (0) (10.5) xˆ (k + 1) = x (1) − e ˆ + , k = 1, 2, . . . , n aˆ aˆ Restore to original data and obtain

xˆ

(0)

(k + 1) = xˆ

(1)

(k + 1) − xˆ

(1)

(k) = x

(0)

 bˆ ˆ (1) − (e−aˆ − 1)e−a(k−1) aˆ

(10.6)

Definition 10.1 Parameter −a in GM (1, 1) model is development coefficient and b is grey action quantity. −a reflects the development trend of xˆ (1) and xˆ (0) . Under normal circumstances, the system action quantity should be exogenous or predetermined. The grey action quantity b in GM (1, 1) model is the data mined from the background value. It reflects the relationship of data changes, and its exact connotation is grey. Grey action quantity is the concrete embodiment of connotation extension. Its existence is the watershed to distinguish grey modeling from general input and output modeling, and it is also an important symbol to distinguish grey system viewpoint from grey box viewpoint.

10.2.2 Improved Grey Predcition Model 10.2.2.1

Analysis of Problems Existing in Grey GM (1, 1) Model

The grey GM (1, 1) model has the advantages of less sample data, simple principle, convenient operation, high short-term prediction accuracy and verifiability. Therefore, it has been widely used and achieved satisfactory results. However, like other prediction methods, it also has some limitations, mainly in the following aspects [13–16]: (1)

(2)

(3)

GM (1, 1) model is mainly applicable for single exponential growth sequence, but the model cannot be simply applied when the actual sequence data are abnormal; GM (1, 1) model is used to predict a data sequence. In order to predict multiple sequences at the same time, we can only establish GM (1, 1) model for each sequence separately and cannot make full use of the relationship between sequences; GM (1, 1) model requires that the original modeling data must be non-negative sequence. When the original data are negative or positive and negative, the model cannot be used directly;

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10 Degradation Modeling and Residual Life Prediction …

(4)

The simulation and prediction accuracy of GM (1, 1) model depends on the constants a and b, and the values of a and b depend on the original sequence and the structure of background values. Therefore, it is necessary to improve the structure of background value to improve the prediction accuracy and adaptability of the model.

In view of the shortcomings and limitations of the grey GM (1, 1) model, it is necessary to improve the traditional GM (1, 1) model, so as to improve the prediction accuracy and application scope of the model.

10.2.2.2

Residual Error GM (1, 1) Model

In the actual prediction, even when the modeling and prediction data sequence meet the modeling requirements of GM (1, 1) model, it does not mean that it will obtain good prediction effect. Sometimes the prediction accuracy does not meet our requirements. In this case, it can be considered that a residual error GM (1, 1) model can be established for the sequence. Due to the GM (1, 1) model using in the process of prediction, the residual error also contains some useful system information. Therefore, it is necessary to further mine the information contained in the residual error to improve the prediction accuracy [17]. Assuming that the original sequence is X (0) = (x (0) (1), x (0) (2),…, x (0) (n)), (x (0) (n)0, k = 1,2,…,n), the corresponding prediction simulation sequence is Xˆ (0) = (xˆ (0) (1), xˆ (0) (2), . . . , xˆ (0) (n)), denote the difference between the original sequence and its simulated value as ε(0)   ε(0) = ε(0) (1), ε(0) (2), . . . , ε(0) (n) where ε(0) (k) = x (0) (k) − xˆ (0) (k) is the residual error sequence of X (0) . If there exists k 0 which satisfies (1) (2)

∀k ≥ k0 , the symbol of ε(0) (k) = x (0) (k) − xˆ (0) (k) is consistent; n − k0 ≥ 4,

      then call ε(0) (k0 ), ε(0) (k0 + 1), . . . , ε(0) (n) as the modelable residual tail end, and it is still denoted as   ε(0) = ε(0) (k0 ), ε(0) (k0 + 1), . . . , ε(0) (n) , (1) = The GM (1, 1) time response  representation of 1-AGO sequence ε  (1) (1) (1) ε (k0 ), ε (k0 + 1), . . . , ε (n) is



bε εˆ (k + 1) = ε (k0 ) − aε (1)

(0)

 exp[−aε (k − k0 )] +

bε aε

(10.7)

10.2 Grey Predcition Model

217

Thus, the simulated sequence of residual tail end ε(0) is   εˆ (0) = εˆ (0) (k0 ), εˆ (0) (k0 + 1), . . . , εˆ (0) (n) . Then, the time response formula after the correction of Xˆ (1) by εˆ (0) is xˆ (1) (k + 1) =

   (0) k < k0 x − b e−ak + b ,  (0) (1) ab  −ak ab x (1) − a e + a + εˆ (0) (k + 1), k ≥ k0

 where εˆ (0) (k + 1) = ±aε × ε(0) (k0 ) −

bε aε



(10.8)

exp[−aε (k − k0 )], and the sign of

residual correction value should be consistent with the residual tail end ε(0) . Therefore, the final residual GM (1, 1) model obtained through cumulative reduction is as follows:    k < k0 (1 − ea )x (0) (1) − ab e−ak (0)   xˆ (k + 1) = (1 − ea ) x (0) (1) − ab e−ak ± aε ε(0) (k0 ) − ab e−aε (k−k0 ) k ≥ k0 (10.9) The residual GM (1, 1) model can modify the prediction result of GM (1, 1) through the further development of residual sequence, so as to effectively improve the prediction accuracy of GM (1, 1) model. The specific steps are summarized as follows: (1) (2) (3) (4)

The GM (1, 1) model is established based on the original prediction data, and the corresponding simulation prediction value of each data is obtained; The residual sequence of the model is obtained from the original data and the simulated prediction value; The corresponding GM (1, 1) model is established for the residual sequence, and the simulated prediction value of the residual sequence is obtained; The result obtained in (3) can be used to correct the corresponding prediction value in (1), thus the final modified prediction result can be obtained.

10.2.2.3

Grey GM (1, N) Model

In actual system analysis, there is often more than one factor that affects system performance. The performance of a system is often the result of a combination of factors. However, only one single factor can be considered in the grey GM (1, 1) model, which obviously does not conform to the actual situation, so it is difficult to get a good prediction effect. To solve this problem, the grey GM (1, N) model can be adopted, where N represents the number of factors affecting the system performance [18].

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10 Degradation Modeling and Residual Life Prediction …

Definition 10.2 Assuming that X 1(0) =



system characteristic sequence, and X i(0)

 x1(0) (1), x1(0) (2), · · · · · · , x1(0) (n) is the   = xi(0) (1), xi(0) (2), · · · · · · , xi(0) (n) ,

i = 2, · · · , N are the sequence of related factors, X i(1) is the 1-AGO sequence of X i(0) , and Z 1(1) is the generated mean sequence of consecutive neighbors of X 1(1) , then x1(0) (k) + az 1(1) (k) =

N 

bi xi(1) (k)

(10.10)

i=2

is called GM (1, N) model. In the above model, a is the system development coefficient, bi xi(1) (k) the driving term, bi the driving coefficient and aˆ = [a, b2 , · · · b N ]T the parameter sequence, and its least square estimate satisfies −1  aˆ = B T B B T Y where ⎡

−z 1(1) (2) x2(1) (2) ⎢ −z (1) (3) x (1) (3) 1 2 B=⎢ ⎣ ··· ··· −z 1(1) (n) x2(1) (n)

⎡ (0) ⎤ ⎤ x1 (2) · · · x N(1) (2) ⎢ (1) x1(0) (3) ⎥ · · · x N (3) ⎥ ⎥ ⎥, Y = ⎢ ⎢ . ⎥ ⎦ . ⎣ ··· ··· . ⎦ (1) (0) · · · x N (n) x1 (n)

(10.11)

Definition 10.3 Assuming aˆ = [a, b2 , · · · b N ]T , then dx1(1) + ax1(1) = b2 x2(1) + b3 x3(1) + · · · + b N x N(1) dt is called GM (1, N) model x1(0) (k) + az 1(1) (k) = b2 x2(1) (k) + b3 x3(1) (k) + · · · + b N x N(1) (k)

(10.12)

is also known as shadow equation. Definition 10.4 Assuming that X i(0) , X i(1) (i = 1, 2, · · · , N ), Z 1(1) , B and Y are as shown in Eq. (10.11), then:

10.2 Grey Predcition Model

(1)

219

The solution of whitenization equation  x

(1)

(t) = e

−at

 =e

−at

N  

bi xi(1) (t)eat dt

d x1(1) dt

+x

+ ax1(1) =

(1)

i=2

x1(1) (0)

−t

N 

bi xi(1) (0)

+

(0) − N  

i=2

(2)

N  i=2

N  

bi xi(1) is as follows  bi xi(1) (0)dt

i=2



bi xi(1) (t)eat dt

(10.13)

i=2

N When the change range of X i(1) (i = 1, 2, · · · , N ) is very small, i=2 bi xi(1) (k) can be regarded as a grey constant, so when GM (1, N) model is as follows x1(0) (k) + az 1(1) (k) =

N 

bi xi(1) (k).

i=2

its approximate time response formula is as follows

xˆ1(1) (k

+ 1) =

x1(1) (0)

 N N 1 1 (1) −at bi xi (k + 1) e + bi xi(1) (k + 1) − a i=2 a i=2 (10.14)

where x1(1) (0) is taken as x1(0) (1). (3)

The cumulative reduction formula is as follows xˆ1(0) (k + 1) = α (1) xˆ1(1) (k + 1) = xˆ1(1) (k + 1) − xˆ1(1) (k)

(4)

(10.15)

The differential simulation formula of GM (1, N) is as follows xˆ1(0) (k) = −az 1(1) (k) +

N 

bi xˆi(1) (k)

(10.16)

i=2

10.2.2.4

Equal Dimension and New Information Prediction Model

It is successful to use the grey predcition model for short-term prediction, but in the development process of any grey predcition model, there will be some random disturbance or driving factors over time, which will affect the development of the system successively. Therefore, in practical application, we must constantly consider the newly increasing disturbance or driving factors in the system over time. In order to predict the system at a longer time, we can develop and utilize the predicted data obtained by GM (1, 1) model, supplement them to the original sequence and remove

220

10 Degradation Modeling and Residual Life Prediction …

the oldest data at the same time, thus forming a new sequence with equal dimensions. Adopting the new sequence to build GM (1, 1) model for further prediction and repeating the process are the principles of the equal dimension and new information prediction model. The modeling steps are as follows: Assuming that the original sequence is X (0) = (x (0) (1), x (0) (2), · · · x (0) (n)), (0) (x (k ≥ 0), k = 1, 2, · · · , n), for which the GM (1, 1) model is built for prediction, the prediction value at n +1 is denoted as x (0) (n + 1). The data x (0) (1) in the original sequence are removed, and x (0) (n + 1) is added to the residual sequence. The new equal dimension sequence is as follows   X 1(0) = x (0) (2), x (0) (3), · · · , x (0) (n), x (0) (n + 1) . This new sequence is taken as the original sequence of modeling, and the GM (1, 1) model is built for the next prediction. Then, the above steps are repeated continuously for equal dimension metabolism prediction.

10.2.2.5

Background Value Reconstruction Optimized GM (1, 1) Model

Firstly, the influence of model parameters on prediction accuracy is analyzed. Assuming that the once accumulated sequence of the original sequence X (1) = (x (1) (1), x (1) (2), · · · x (1) (n)) is a discrete sequence on the exponential curve y = theorem, there is a point ξk in (k − 1, k) x (1) (t), according to Lagrange mean value  dx (1)  (1) (1) such that x (k) − x (k − 1) = dt  , k − 1 < ξk < k. As the sequence x (1) (k) t=ξk

is monotonic, x (1) (ξk ) = λk x (1) (k − 1) + (1 − λk )x (1) (k), λk ∈ (0, 1). Therefore, when building GM (1, 1) model, the grey derivative x (0) (k) = x (1) (k) − x (1) (k − 1) is the derivative of the point ξk , and its corresponding background value should be x (1) (ξk ) = λk x (1) (k − 1) + (1 − λk )x (1) (k). In the process of the classical GM (1, 1) modeling, Eq. (10.1) is simply used to replace x (1) (ξk ), which is only a special case (λk = 0.5), obviously not optimal, and may bring unacceptable errors to the prediction in many cases. In fact, we have the following theorems. Theorem 10.1 The sufficient and necessary condition for that the sequence X (1) = (x (1) (1), x (1) (2), · · · x (1) (n)) has the white exponential law   b −a(k−1) b e x (1) (k) = x (0) (1) − + a a is that there exists λ = 

1 a

−

1 ea −1

(10.17)

satisfying

   x (1) (k) − x (1) (k − 1) = a λx (1) (k − 1) + (1 − λ)x (1) (k) = b, k = 2, 3, · · · , n (10.18)

10.2 Grey Predcition Model

221

The above theorem shows that for sequences with white exponential law satisfying the Eq. (10.17), the GM (1, 1) model built by taking x (1) (k) − x (1) (k − 1) as the grey derivative and   1 1 (10.19) Z (1) (k) = λx (1) (k − 1) + (1 − λ)x (1) (k), λ = − a a e −1 as the background value of corresponding grey derivative must have white exponential law coincidence property. When the data sequence changes gently and |a| is small, with Eq. (10.1) as the background value, the GM (1, 1) model built has higher accuracy with slight error; however, when the data changes rapidly and |a| is large, GM (1, 1) model based on the background value selection in the classical GM (1, 1) modeling method has a large error. Based on the above analysis, we build a background value reconstruction optimized GM (1, 1) model. The background value reconstruction optimized GM (1, 1) model is discussed in detail below. Assuming that the original sequence is X (0) = (x (0) (1), x (0) (2), · · · , x (0) (n)) and (0) x (k) ≥ 0, k = 1, 2 · · · , n is the discrete point sequence, its once accumulated generating sequence is X (1) = (x (1) (1), x (1) (2), · · · x (1) (n)). The optimized GM (1, 1) model built for the generating sequence X (1) is as follows 

   x (1) (k) − x (1) (k − 1) + a λx (1) (k − 1) + (1 − λ)x (1) (k) = b, k = 2, 3, · · · , n (10.20)

where λ=

1 1 − a , k = 2, 3, · · · a e −1

(10.21)

According to the classical GM prediction model, the original data optimized GM (1, 1) model is as follows xˆ (0) (1) = x (0) (1),    (0)  b −a(k−1) (0) a e x (1) − , k = 2, 3, · · · n. xˆ (k) = 1 − e a However, tn the practical problem, we only know X (0) and a are the parameters to be identified, so the optimal background value corresponding to grey derivative (1) (1) (1) (1) x (k)1 − x 1 (k − 1) cannot be determined by λx (k − 1) + (1 − λ)x (k) and λ = a − ea −1 . However, we can first build GM (1, 1) model according to the traditional modeling method, obtaining the first approximation value of a, namely a1 ; then, substitute a1 into the formula λ = a1 − ea1−1 to get λ1 and the optimal background value of grey derivative; next, build the optimized GM (1, 1) model

222

10 Degradation Modeling and Residual Life Prediction …

again, obtaining the quadratic approximation value of a, namely a2 . In this way, the optimal value of a can be gradually approached. The calculation method of corresponding model parameters a and b is as follows: From Eq. (10.17), the equation x (1) (k) = e−a x (1) (k − 1) + ab (1 − e−a ) can be obtained. Denote b e−a = β1 , (1 − e−a ) = β2 , a

(10.22)

and take k = 2, 3, · · · , n, then there exists x (1) (2) = β1 x (1) (1) + β2 x (1) (3) = β1 x (1) (2) + β2 ··· x

(1)

(n) = β 1 x (1) (n − 1) + β2

(10.23)

The matrix expression of the above equation is C = Aβ, where ⎤ ⎡ x (1) (1) x (1) (2) (1) (1) ⎢ ⎢ x (3) ⎥ ⎥, A = ⎢ x (2) C =⎢ ⎣ ⎣ ······ ⎦ ··· (1) x (n) x (1) (n − 1) ⎡

⎤ 1   1 ⎥ ⎥, β = β1 . β2 ···⎦ 1

Hence, according to the least square method, βˆ = (A T A)−1 A T C. Furthermore, β1 =

(n − 1)

n−1 i=1

x (1) (i)x (1) (i + 1) −

(n − 1)

n−1  i=1

 n−1

x (1) (i)

i=1

2

−

x (1) (i)

 n−1 i=1

 n−1 i=1

x (1) (i)

2

x (1) (i + 1)



(10.24) n−1 β2 =

i=1

x (1) (i + 1) − β1 n−1

n−1 i=1

x (1) (i)

(10.25)

From Eq. (10.22), the following results can be obtained a = − ln β1 , b =

β2 a. 1 − β1

(10.26)

10.3 Residual Life Prediction Based on Improved Grey Predcition Model

223

10.3 Residual Life Prediction Based on Improved Grey Predcition Model Take the current time as k, obtain the one-step prediction value of the performance degradation amount based on the improved grey predcition model from historical data x1:k , which is denoted as xˆk+1 , and then obtain the p-step prediction value of the performance degradation amount based on data x1:k , xk+1 , · · · , xk+ p . For the monotonously increasing degradation process, the residual life can be calculated according to the following formula   RLk = min p : xk+ p ≥ xth , p ∈ N

(10.27)

where xth is the failure threshold given in advance and N represents the natural number set.

10.4 Case Study Inertial device is the key device of missile control system. As for the inertial device, the performance of its gyroscope is directly related to the working quality of missile control system. As an important index for evaluating gyroscope performance, gyroscope error coefficient is widely used in engineering and practice, and it is of great significance to accurately predict it. However, the test times and service time of gyroscopes are strictly limited, so the error coefficient of gyroscopes obtained in practice is a kind of small sample data, and in many cases, it is also a kind of zerofailure data. In order to evaluate the reliability, predict the life and fault of gyroscope, we consider this kind of data as the performance degradation amount, and then use several grey predcition models mentioned above for modeling analysis, so as to study the degradation path of gyroscope. Take 14 groups of historical data (Table 10.1) of an error coefficient of a certain gyroscope as an example, with the first 10 groups of data as modeling training samples of the model and the last 4 groups of data as prediction test samples. Table 10.1 Fourteen groups of historical data of an error coefficient of a certain gyroscope SN

1

2

3

4

5

6

7

Value

2.2917

2.2424

2.2619

2.2522

2.2915

2.2621

2.2311

SN

8

9

10

11

12

13

14

Value

2.2349

2.2518

2.2927

2.2899

2.2496

2.2590

2.2797

224

10 Degradation Modeling and Residual Life Prediction …

Before modeling prediction, smoothness test and quasi-exponential law test should be carried out on the modeling data first. The smoothness test is generally carried out by the following formula: x(k) ρ(k) = k−1 , k = 2, 3, · · · , n, i=1 x(i)

(10.28)

where x(k) represents the original modeling data sequence. If the original sequence < 1, k = 2, 3, · · · , n − 1; (2) ρ(k) ∈ [0, ε], meets the conditions: (1) ρ(k+1) ρ(k) k = 3, 4, · · · , n; (3) ε < 0.5, it can be considered as passing the smoothness test. The quasi-exponential law test is generally carried out by the following formula, σ (1) (k) =

x (1) (k) = 1 + ρ(k) < 1.5, k = 3, 4, · · · , n − 1)

x (1) (k

(10.29)

The results of smoothness test and quasi-exponential law test of the original modeling sequence are shown in Table 10.2. It can be seen from Table 10.2 that the original data completely meets the requirements of modeling, and it can be predicted by building a grey predcition model. Simulation software MATLAB7.0 is used for prediction simulation, and the results are shown in Fig. 10.1. Next, we introduce the average relative error as the evaluation index and compare the prediction performance of these four methods to a certain extent. The results are shown in Table 10.3. From Fig. 10.1 and Table 10.3, we can see that compared with the classical GM (1, 1) prediction model, the three improved prediction models proposed in this paper have better prediction effect and more accurate accuracy. Especially, the equal dimension and new information prediction model has better prediction effect, which can be seen from the trend of the predicted path in Fig. 10.1. It shows that the equal dimension and new information prediction model has a strong tracking ability to the sequence, which also conforms to the objective law. However, through the analysis of Fig. 10.1, we can find that both the classical GM (1, 1) prediction model and the improved prediction model have not strong ability to track the original prediction sequence, and the prediction effect is not ideal. The basic reason for this situation is that the grey predcition model is still a linear prediction model in essence, which has great limitations and inadaptability in the prediction and tracking of nonlinear data. Table 10.2 Smoothness and quasi-exponential law tests of modeling data 2

3

4

5

6

7

8

9

10

ρ(k)

0.9785

0.4989

0.3314

0.2533

0.1995

0.1640

0.1412

0.1246

0.1128

σ (k)

1.9785

1.4989

1.3314

1.2533

1.1995

1.1640

1.1412

1.1246

1.1128

10.4 Case Study

225

2.29 Measured value GM (1, 1) prediction value Equal dimension metabolism prediction value Residual prediction value Background value reconstruction prediction value

2.285 2.28 2.275 2.27 2.265 2.26 2.255 2.25 2.245

2

1.5

1

3

2.5

3.5

4

Fig. 10.1 Prediction results of simulation modeling

Table 10.3 Comparison of prediction results SN

1

2

3

4

Measured value

2.2899

2.2496

2.2590

2.2797

Average relative error

Method GM (1, 1)

2.2655

2.2667

2.2680

2.2693

0.67%

Equal dimension innovation

2.2642

2.2623

2.2637

2.2627

0.65%

Residual

2.2642

2.2651

2.2663

2.2678

0.66%

Background value reconstruction

2.2653

2.2663

2.2678

2.2691

0.66%

By analyzing the original modeling and predicting sequence, it can be found that the original sequence has certain nonlinearity. Therefore, in order to further improve the predicting and modeling ability of the grey predcition model for nonlinear data, it is necessary to combine the grey prediction theory with the existing nonlinear prediction theory to form a new prediction model, which is the next work to be completed.

226

10 Degradation Modeling and Residual Life Prediction …

10.5 Summary of This Chapter This chapter introduces the research situation of grey theory and presents some basic concepts, principles, grey model equations, sequence generation operators, etc. It studies the basic principles of grey modeling and at the same time gives the modeling and prediction process of the classical GM (1, 1) prediction model; on the basis of analyzing the defects of the classical GM (1, 1) prediction model, offers and studies four improved grey predcition models, and provides the modeling and prediction principles and prediction steps of these four prediction models, respectively; takes an error coefficient of a certain missile gyroscope as performance degradation amount, adopts the classical GM (1, 1) prediction model and other three improved prediction models to build the corresponding performance degradation path model, and simulates and predicts the error coefficient, so as to detect the usability of the original model and various improved models.

References 1. Deng JL (1982) Control problems of grey systems. Syst Control Lett 82(5):288–294 2. Deng J (1986) The foundation of grey theory. The Publishing House of Huazhong University of Science and Technology, Wuhan 3. Liu S, Dang Y, Fang Z et al (2004) Grey system theory with application. Science Press, Beijing 4. Huang Y, Changhua Hu (2005) Optimized adaptive grey predcition model and its application in missile fault prediction. J Projectiles, Rockets, Missiles and Guidance 24(3):699–701 5. Kai Lu, Changhua Hu (2010) Application of a new GM (1, 1)-AR model in gyro drift tendency prediction. Electron Opt Control 17(3):88–92 6. Kai Lu, Changhua Hu (2009) Application of a new grey predcition model algorithm to the gyro drift tendency. Syst Simul Technol 5(3):176–181 7. Kai Lu, Liu G, Tian P (2009) Application of grey neural network in power short-term load prediction. North China Electr Power Technol 5:1–5 8. Kai Lu (2010) Performance degradation path modeling of inertial devices based on grey theory. Rocket Force University of Engineering, Xi’ an 9. Zhang D, Bi Y, Bi Y et al (2004) Power load prediction method base on serial grey neural network. Syst Eng Theory Pract 12:128–132 10. Huang Y, Changhua Hu (2006) Third-order grey neural network modeling and the application in missile fault prediction. Electron Opt Control 13(5):39–42 11. Zhou Z, Guo Ke, Chen L (2007) Grey neural network technology for time series data prediction. Knowl Jungle 1:128–129 12. Xie N, Liu S (2005) Discrete GM (1, 1) model and mechanism of grey predcition model. Syst Eng Theory Pract 1:93–98 13. Liu S, Deng J (2000) Application range of GM (1, 1) model. Syst Eng Theory Pract 5:121–124 14. Cheng M (2004) Approach exploration on improving the prediction accuracy of GM (1, 1) model. New Explor Theory 2:16–17 15. Ji P, Huang W, Xiangyong Hu (2001) A study on the properties of grey predcition model. Syst Eng Theory Pract 9:105–108 16. Zhang D, Jiang S, Shi K (2002) Theoretical defect of grey prediction formula and its improvement. Syst Eng Theory Pract 8:140–142

References

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17. Cao J, Liu Y, Dai R (2007) Network traffic prediction based on error advanced grey model. Comput Eng Des 28(21):5144–5146 18. Liu W, Wang F, Liu Y (2007) Multivariate linear regression model based on multivariate grey predcition model. Sci Technol Eng 7(24):6603–6606

Chapter 11

Optimal Inspection Policy for Deteriorated Equipment Based on Life Prediction Information

11.1 Introduction Performance inspection is the most direct and effective method to ensure the reliability of equipment, especially for equipment stored for a long time—before storing the used equipment and after enabling the equipment in store, the equipment performance should be fully tested to ensure its usability [1]. Equipment inspection needs to consider technical level and decision-making level. The technical level mainly studies inspection content and methods, including the selection of equipment performance characteristics, acquisition and processing of test data, etc., to ensure that the test data objectively reflect the health state of equipment; the decision-making level mainly solves the problem of choosing the inspection opportunity and minimizes the impact of inspection on the equipment itself, tasks and costs while obtaining the equipment state information. This chapter focuses on the decision-making level of equipment inspection. At present, the research emphasis of equipment inspection at decision-making level is on determining the inspection interval of equipment. Due to the influence of multiple factors such as production plan and personnel arrangement, the inspection interval of equipment is often arbitrary. On the one hand, in order to ensure the performance of equipment to meet the usage requirements, reduce the probability of equipment failure between two times of inspection and find and deal with equipment failure in time, it is necessary to shorten the inspection interval and increase the inspection times of equipment. However, frequent performance inspection will greatly increase the inspection cost of equipment, consume the service life of key components of equipment and shorten the effective service time of equipment. On the other hand, although the longer inspection interval can reduce the inspection cost and improve the utilization rate of equipment, it increases the possibility of equipment failure between two times of inspection. For degraded systems with high requirements on reliability and safety, such as aerospace crafts, submarines and missiles, the excessive inspection interval will inevitably bring greater risks. In addition, in recent © National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_11

229

230

11 Optimal Inspection Policy for Deteriorated Equipment Based …

years, condition-based maintenance (CBM) and prognostics and health management (PHM) of equipment have developed rapidly [2], and equipment performance state inspection is the foundation and important part of condition-based maintenance and prognostics and health management. Therefore, it is of great significance and value to study the inspection strategy which can make the equipment reach the optimal indexes such as the longest effective operation time, the lowest inspection cost and the longest residual life on the premise of ensuring that the equipment meets the use requirements. Many scholars have studied the optimal inspection of degraded equipment and achieved a series of results. Scraf has summarized the state inspection and conditionbased maintenance and established the model framework of equipment inspection and maintenance [3]. Wang and Pham et al. have summarized the maintenance strategies of degraded equipment [4] and studied the maintenance strategies of single system and equipment under incomplete maintenance conditions [5]. Dekker et al. have studied the maintenance policy of multi-device equipment [6]. As for the inspection strategy of the equipment stored for a long time, K. Ito, T. Nakagawa, et al. have published a series of academic papers, which studied the inspection strategy of equipment in store [7–10] with high reliability requirements and equipment in store with performance degradation tested periodically [11] and gave the determination method of the optimal inspection period. Lam Yeh [12], Kaio and Osaki [13], Barlow [14], Yang and Klutke [15], Ye [16], et al. have also studied different optimal inspection strategies for equipment. For details, please refer to relevant references. However, for some degraded equipment stored for a long time, besides periodic inspection according to the maintenance plan, the performance of the equipment will also be tested after each operation so as to ensure the availability of the equipment. This situation should be considered when solving the optimal inspection period of equipment. At present, Nakagawa et al. have studied this issue [10, 17], but the premise of most of the results is that the operation time of the equipment follows the exponential distribution and the life distribution of the equipment is a simple Weibull distribution or exponential distribution. For degraded systems, especially small batches of degraded systems, on the one hand, the operation time distribution of equipment is not necessarily exponential distribution; on the other hand, due to the lack of failure data, the life distribution of equipment shall be determined according to historical degradation data. Therefore, on the basis of predicting the residual life and estimating the operation time distribution of equipment by using degradation data, this chapter studies the optimal inspection strategy of equipment and applies the research results to inertial platform [18].

11.2 Inspection Strategy and the Optimization Objective Function

231

11.2 Inspection Strategy and the Optimization Objective Function In order to ensure that the performance of the equipment meets the actual use requirements, the maintenance plan of the equipment is generally made according to the operation instructions, and the time for inspecting the equipment is determined according to the plan. Usually, this kind of inspection is carried out on a regular basis. For example, some large-scale equipment shall be tested monthly, quarterly and yearly, and the performance of the missile weapon system in storage needs to be tested periodically. In addition, after each use, the performance of the equipment shall also be tested to determine its state, so as to ensure timely maintenance measures and normal use of the equipment next time. Therefore, a reasonable inspection strategy shall be the combination of inspection based on maintenance plan and postwork inspection. First, based on engineering practice, the following assumptions are made: (1)

(2) (3) (4) (5)

(6)

Whether the equipment fails can be determined only by performance inspection, and whether the equipment fails can be determined only by performance inspection after failure; The performance inspection result of the equipment at the current time only depends on the state of the equipment at the current time; Ignore the influence of inspection on equipment performance and life distribution and the duration of inspection; Take the time of equipment to complete tasks as a random variable; Equipment performance inspection can be divided into two categories: One is based on equipment maintenance plan, the other is the inspection after each equipment use, and the time of the two types of inspection is independent of each other; The residual life distribution of the equipment does not change until it is tested that the degradation process changes significantly again.

According to assumptions (1), (4) and (5), as shown in Fig. 11.1, equipment failure can only be found by performance inspection after work or by performance inspection based on maintenance plan. In Fig. 11.1, the upper diagram shows the process of inspection equipment failure according to the maintenance plan, and the lower diagram shows the process of inspecting equipment failure after work. It can be seen from Fig. 11.1 that when the equipment fails between the k-th and the k + 1-th performance inspection, with the failure time t ∈ Ak+1 k , and after j times of operation is completed before the equipment failure time t, take ≡ (kT, (k + 1)T ), where Ak+1 ≡ (t, (k + 1)T ) and T is the inspection period Ak+1 t k of the equipment. If the accumulated operation time S j+1 for the equipment to finish / Ak+1 , the failure of the equipthe ( j + 1)-th operation meets the condition of S j+1 ∈ t ment will be tested by the performance inspection according to the maintenance plan,

232

11 Optimal Inspection Policy for Deteriorated Equipment Based … Zj

Z j −1

T

T

T

T

T

Zj

Z j −1

T

Z j +1

T

Z j +1

T

T

T

Inspection point based on the maintenance plan

Inspection point after work

Failure

Inspection point where failure is tested

Fig. 11.1 Schematic diagram of equipment performance inspection process

and when S j+1 ∈ Ak+1 , the failure of the equipment will be tested by the perfort mance inspection after the end of the ( j + 1)-th operation. Their probabilities can be calculated according to Eqs. (11.1) and (11.2), respectively.  / P S j+1 ∈

 ∈ Ak+1 , k W (t) = j

t Ak+1 t

 P S j+1 ∈



 ∈ Ak+1 , k W (t) = j

t Ak+1 t

(k+1)T  

= kT



t

0

(k+1)T  

= kT

G((k + 1)T − x)dG j (x)dF(t) (11.1)

0

t



 G((k + 1)T − x) dG j (x)dF(t) −G(t − x) (11.2)

j where S j represents the cumulative operation time S j = k=1 Z k of the equipment after the j-th operation and Z j represents the duration of the equipment after the j-th operation; G(x) represents the probability of the equipment’s duration Z j ≤ x in the j-th operation, namely G(x) ≡ P Z j ≤ x , and g(x) represents its probability density function; G j (x) represents the probability of the j times of operation at least in a time period (0, x ], and the value of G j (x) is the j order convolution of G(x) itself; W (x) is the number of times the equipment operates in time period (0, x ]. For the convenience of discussion, take (x) = 1 − (x), (x) is an arbitrary distribution function. According to the conditional probability formula, the probability Pp of equipment failure tested by the performance inspection according to the maintenance plan and

11.2 Inspection Strategy and the Optimization Objective Function

233

the probability PR of failure tested by the performance inspection after the end of the equipment operation are shown in Formulas (11.3) and (11.4), respectively [11]. Pp =

∞  

PR =

⎣

⎧ ∞  ⎨ ⎩

⎤

∞  

t

0

j=0

t

G((k + 1)T − x)dG j (x)⎦dF(t)

(11.3)

0

j=0

(k+1)T

kT

k=0

⎡

kT

k=0 ∞  

(k+1)T

⎫ ⎬ [G((k + 1)T − x) − G(t − x)]dG j (x) dF(t) ⎭ (11.4)

where F(t) is the distribution function of the residual life of the equipment and its probability density function is f (t). The equipment is tested in accordance with the maintenance plan after each task until the time when failure is tested, let C1 (T ) represent the expected inspection cost of the equipment with the inspection period of T. Then, C1 (T ) is selected as the optimization objective function, and the optimal inspection period T ∗ of the equipment can be obtained by minimizing C1 (T ). Denote C1P = (k + 1)C p + jC R + C D [(k + 1)T − t] and C1R = kC p + ( j + 1)C R + C D (x + y − t), where C p represents the cost of performance inspection according to the equipment maintenance plan, with the unit of RMB 10,000/time, and C R represents the cost of performance inspection after a single operation, with the unit of RMB 10,000/time; C D represents the loss per unit time caused by failure of equipment not tested, with the unit of RMB 10,000/time. Then the optimization objective function C1 (T ) is as follows C1 (T ) =

∞  

⎡ ⎣

kT

k=0

+

(k+1)T

∞  

C1P G((k + 1)T − x)dG j (x)⎦dF(t)

⎧ ∞  t  ⎨ ⎩

kT

k=0

t 0

j=0

(k+1)T

⎤

∞  

j=0

(k+1)T −x t−x

0

⎫  ⎬ C1R dG(y) dG j (x) dF(t) (11.5) ⎭

After simplifying Eq. (11.5), the objective function for optimal inspection of equipment can be obtained as follows: C1 (T ) = C p

∞ 

 F(kT ) + C R

k=0 ∞  (k+1)T 

M(t) dF(t) 0



+ CD

kT

k=0

+

 t  0

(k+1)T −x

t−x

∞

(k+1)T

G(y)dy t

 

G(y)dy dM(x) dF(t) − C p − C R

234

11 Optimal Inspection Policy for Deteriorated Equipment Based … ∞   k=0

(k+1)T

kT



 G((k + 1)T ) − G(t) dF(t) t + 0 [G((k + 1)T − x) − G(t − x)]dM(x) (11.6)

where M(x) represents the expected operation times M(x) =

∞

( j)

j=1

G j (x) of the

equipment in a time period (0, x ]; PW (x) is the probability of equipment operating ( j) for j times in a time period (0, x ], that is, PW (x) = G j (x) − G j+1 (x). The optimal inspection period of equipment is as follows: T ∗ = arg min C1 (T )

(11.7)

T

11.3 Optimal Inspection Policy Based on Residual Life Prediction When the life distribution function F(t) and operation duration distribution function G(x) of equipment are known, the optimal inspection period of equipment can be calculated according to Eqs. (11.6) and (11.7). For a small batch of equipment with high reliability requirements and large differences in individual performance, such as inertial platform, when its performance degradation meets certain requirements, the life distribution can be solved by the method introduced in Chap. 3. If the first passage time distribution of Wiener process is selected as the distribution of equipment residual life, the distribution of equipment residual life is shown in Eq. (11.8).   D − y0 (D − y0 − ut)2 exp − f (t; y0 , D) = √ 2σ 2 t σ 2π t 3

(11.8)

According to the exposition in the Chap. 3, when there is a changepoint in the degradation process, the parameters of the life distribution function of the equipment will change significantly before and after the changepoint, that is, u and σ 2 in Eq. (11.8) have a big difference before and after the changepoint. Then the probability distribution before the change of degradation rule is used to solve the period of equipment inspection, and it is no longer possible to guarantee the minimum C1 (T ). It is necessary to recalculate the optimal inspection period according to the life distribution function after the changepoint. Thus, the method for determining the dynamic optimal inspection period of equipment is shown in Fig. 11.2.

11.3 Optimal Inspection Policy Based on Residual Life Prediction

235

Determine the operation time distribution G(x)

Determine the remaining life distribution F(t)

Calculate the expected cost C(T) of the equipment when it fails

Determine the optimal inspection period T*

Test the equipment at the next optimal inspection point

Yes

No Check if the degradation rule of equipment changes

Fig. 11.2 Schematic diagram of the method for determining the dynamic optimal inspection period of equipment

11.3.1 Optimal Inspection Period of Equipment When Unknown G(X) If the distribution form of each operation time of the equipment is known and the distribution has a single form, for example, if it follows the exponential distribution G(x) = 1 − e−θ x , we can know that M(x) = θ x. Equation (11.6) can be expressed as Eq. (11.9). The parameters of G(x) can be solved by historical data and maximum likelihood estimation (MLE) method. When the life distribution is exponential distribution or Weibull distribution, Eq. (11.9) can be further expressed as an analytical form [19, 20] which is convenient for analysis and calculation. When the life distribution is in the form of Eq. (11.8), it is difficult for Eq. (11.9) to be further converted into an analytical form, so it is necessary to use numerical methods for calculation. C1 (T ) = C p

∞  k=0



∞

F(kT ) + C R θ

t dF(t) 0

 ∞    C D  (k+1)T  1 − e−θ[(k+1)T −t] dF(t) + CR − Cp + θ k=0 kT

(11.9)

Leibniz’s formula [21] is adopted to differentiate Eq. (11.9); the following formula can be obtained

236

11 Optimal Inspection Policy for Deteriorated Equipment Based …

  ∞ ∞ 

 ∂C1 (T ) CD = Cp 1 − e−θ T k f (kT ) + C R − C p + k f (kT ) ∂T θ k=0 k=0  ∞   (k+1)T CD  θ e−θ[(k+1)T −t] dF(t) − CR − Cp + (k + 1) θ kT k=0 (11.10) Take ∂C1 (T )/∂ T = 0 ∞ k=0

(k + 1)

 (k+1)T

θ e−θ[(k+1)T −t] dF(t) Cp − 1 − e−θ T − = C R − C p + C D /θ k=0 k f (kT ) (11.11)

kT∞

Thus, the optimal inspection interval T ∗ of equipment can be obtained by solving Eq. (11.11).

11.3.2 Optimal Inspection Period of Equipment with G(X) Known When the distribution of equipment operation time is unknown or it is difficult to fit the historical inspection data well with a single distribution, the mixture distribution model can be used to describe the distribution of equipment operation time. The existing mixture distribution models include Gaussian mixture model and exponential mixture model. The domain of Gamma distribution model is non-negative real number space, which has been concerned and studied by many experts and has been applied in the aspects of economical efficiency and dependability. In this paper, the Gamma mixture distribution is chosen as the operation time distribution of equipment. The probability density function of Gamma distribution is shown in Eq. (11.12), where α and β are the scale parameter and shape parameter of the Gamma distribution respectively, (·) is the Gamma function, and its expectation and variance are α/β and α/β 2 , respectively. ga( z|α, β) =

β α α−1 −βt z e (α)

(11.12)

The probability density function of the Gamma mixture distribution is shown in Eq. (11.13), where wk represents the component of the k-th Gamma distribution. g(x) =

K  k=1

wk ga( z|αk , βk )

(11.13)

11.3 Optimal Inspection Policy Based on Residual Life Prediction

237

The parameters of g(x) in Eq. (11.13) include the coefficient wk , shape parameter αk and scaleparameter βk of the k-th Gamma distribution, where, for wk , 0 ≤ K wk = 1. The standard EM algorithm can be used for estimation wk ≤ 1 and k=1 based on historical operation data, and the references [21, 22] can be consulted for the detailed and variance of Gamma mixture distribution are  K process. The mean  K wk αk /βk and σ 2 = k=1 wk αk /βk2 , respectively. μ0 = k=1 x The distribution function of equipment operating hours is G(x) = 0 g(t) dt. Assuming that the operating hours of the equipment are independent of each other, M(x) is a renewal process with time interval following Gaussian mixture distribution. The solution methods of M(x) mainly include analytical method, numerical method and moment method [23]. The analytical method needs to use the Laplace transform and its inverse transform to solve the multiple convolution G j (x). When g(x) is the Gamma mixture distribution, it is difficult to obtain an analytical solution for convenient calculation. Numerical method discretizes the probability density function of mixture distribution and then calculates multiple convolution G j (x), which requires a lot of operations. Moment method is also an approximate method. This method first calculates the moments of mixture distribution and its multiple convolution G j (x) and then uses a specific distribution with the same moments (such as normal distribution, Beta distribution) to approximate G j (x). This method, suitable for the case that is difficult to be solved by analytical method, can be adopted when the calculation is relatively simple and the accuracy requirement is not very high. When g(x) is a Gaussian mixture model, the moment method is selected to solve M(x) and m(x). The normal distribution is used to approximate G j (x), and the calculation of M(x) and m(x) can be performed as follows. 2 Step 1: Determine expectation μ j (first moment), variance σ j (second moment) of G j (x). μ j = jμ0 = j

K 

wk αk

(11.14)

k=1 K  j 2 σ = jσ 2 = j wk αk /βk2

(11.15)

k=1

Step 2: Determine the approximate normal distribution function and the surface density distribution function of G j (x). 2 1 j 2 g j (x) ≈ √ e−(x−μ j ) /2(σ ) σ j 2π  x 2 1 j 2 j e−(t−μ j ) /2(σ ) dt G (x) ≈ √ σ j 2π −∞

Step 3: Solve M(x) and m(x).

(11.16) (11.17)

238

11 Optimal Inspection Policy for Deteriorated Equipment Based …

m(x) =

∞ 

g j (x)

(11.18)

G j (x)

(11.19)

k=0

M(x) =

∞  k=0

The calculation result of M(x) is substituted into Eq. (11.6), different inspection interval T is selected to obtain the relationship between C1 (T ) and T, and then the period of the optimal performance inspection T ∗ could be obtained.

11.4 Optimal Inspection Policy for Inertial Platform Inertial platform is in storage state at ordinary times, and its performance shall be tested according to the maintenance plan to ensure that it is in available state at any time. At the same time, due to the implementation of relocation, operation training and other tasks, its performance needs to be tested before storage after each task. In this section, the time between two tasks is regarded as the operation time of inertial platform, and the calculation method of residual life distribution in the previous chapter and the model proposed in the previous section are used to solve the optimal inspection interval of inertial platform. Here, the drift coefficient of gyroscope is chosen to characterize the performance of inertial platform. Information such as inspection cost, task cycle distribution and its parameters of inertial platform is shown in Table 11.1. It is assumed that the time interval between two tasks is independent of each other and follows the exponential distribution of parameter θ , where 1/θ is the average task time interval of the inertial platform, and M(t) = θ t. According to Eqs. (11.9) and (11.11), the relationship between the average inspection cost C1 (T ) and T at different inspection time and the optimal inspection interval T ∗ of inertial platform can be obtained until the time when failure is tested from inertial platform. When the value of 1/θ is 5, 10, 20 and 40, respectively, the inspection cost C1 (T ) and the optimal inspection period T ∗ of the inertial platform are as follows until the time when failure is tested. Figures 11.3, 11.4, 11.5 and 11.6 show the relationship between C1 (T ) and T of the inertial platform and the optimal inspection period T ∗ when the value of 1/θ is 5, 10, 20 and 40, respectively. Figure 11.7 shows the optimal inspection period T ∗ of inertial platform at different inspection times. Table 11.1 Values of parameters related to inertial platform inspection strategy

Inspection cost items

Cost (unit: RMB 10,000)

CP

2

CR

3

CD

40

11.4 Optimal Inspection Policy for Inertial Platform

239

1000 C(t) of the 7th inspection point T* of the 7th inspection point

C(t) / RMB 10, 000

800

C(t) of the 17th inspection point T* of the 27th inspection point C(t) of the 32th inspection point

600

T* of the 32th inspection point C(t) of the 42th inspection point T* of the 42th inspection point

400

C(t) of the 72th inspection point T* of the 72th inspection point

200

0

0

5

10 检测周期 Inspection period / month T/ T ᭶

15

20

Fig. 11.3 C 1 (T ) and T * of inertial platform with 1/θ = 0.2 1000

C(t) of the 7th inspection point

C(t) / RMB 10, 000

T* of the 7th inspection point

800

C(t) of the 17th inspection point T* of the 27th inspection point C(t) of the 32th inspection point

600

T* of the 32th inspection point C(t) of the 42th inspection point T* of the 42th inspection point

400

C(t) of the 72th inspection point T* of the 72th inspection poin

200

0

0

5

10

15

Inspection period T / month

20

Fig. 11.4 C 1 (T ) and T * of inertial platform with 1/θ = 0.1

After analyzing the calculation results, the following conclusions can be drawn: (1)

The optimal inspection period of inertial platform is shortened with the increase of operation time. At the initial stage, the platform has excellent performance and high reliability, so increasing the inspection period properly can reduce the inspection cost. When it is tested that the degradation rule of the platform changes, the residual life distribution of the platform changes accordingly, so does its optimal inspection period. In the middle and later stages, the reliability of the platform decreases and the possibility of failure between two times of inspection increases, so it is necessary to shorten the inspection interval to reduce the risk of failure.

240

11 Optimal Inspection Policy for Deteriorated Equipment Based … 1000

C(t) of the 7th inspection point T* of the 7th inspection point

C(t) / RMB 10, 000

800

C(t) of the 17th inspection point T* of the 27th inspection point C(t) of the 32th inspection point

600

T* of the 32th inspection point C(t) of the 42th inspection point T* of the 42th inspection point

400

C(t) of the 72th inspection point T* of the 72th inspection poin

200

0

0

5

10 Inspection period T / month

15

20

Fig. 11.5 C 1 (T ) and T * of inertial platform with 1/θ = 0.05 1000 C(t) of the 7th inspection point T* of the 7th inspection point

C(t) / RMB 10, 000

800

C(t) of the 17th inspection point T* of the 27th inspection point C(t) of the 32th inspection point

600

T* of the 32th inspection point C(t) of the 42th inspection point

400

T* of the 42th inspection point C(t) of the 72th inspection point T* of the 72th inspection poin

200

0

0

5

10 Inspection period T / month

15

20

Fig. 11.6 C 1 (T ) and T * of inertial platform with 1/θ = 0.025

(2)

(3)

When the time interval between two tasks of inertial platform becomes shorter, the optimal inspection interval of equipment increases; on the contrary, if the time interval between tasks becomes longer, the optimal inspection period will be shortened accordingly. This is because the performance of the platform needs to be tested before it is stored after each task, which reduces the risk of platform failure between two inspection points when inspecting according to the maintenance plan, which is consistent with the actual situation. The performance of the platform studied is tested in a fixed period on a monthly basis, and the calculation results show that the inspection period is well-advised.

11.5 Summary of This Chapter

241

7 Optimal inspection period for θ = 0.05

Optimal inspection period / month

6

Optimal inspection period for θ = 0.05 Optimal testing period for θ = 0.05

5

Optimal testing period for θ = 0.025

4 3 2 1

6

11

16

21

26

31 36 41 46 Inspection time /month

51

56

61

66

71

Fig. 11.7 T ∗ of inertial platform at different inspection time

11.5 Summary of This Chapter In view of the unreasonable inspection interval in equipment performance inspection, this chapter studies the optimal inspection strategy of equipment on the basis of predicting the residual life of equipment. The inspection strategy of combining periodic inspection with postoperation inspection is selected, aiming at minimizing the expected inspection cost of equipment until the time when failure is tested, and the method of determining the optimal inspection period of equipment is given under the conditions of known and unknown operation time distribution, which provides reference and basis for equipment inspection and maintenance. This method is applied to the performance inspection of a certain type of inertial platform, and the optimal inspection period of this type of inertial platform in the current state is obtained. The results show that the strategy of monthly inspection on the platform is too conservative, and it is suggested to increase the inspection interval of the equipment appropriately so as to reduce the inspection cost and the impact of inspection on the equipment life.

References 1. Li M, Changhua Hu, Zhou Z et al (2015) Optimal inspection strategy of equipment in store based on degradation data. Syst Eng Electron 37(5):1219–1223 2. Zhang ZX, Si XS, Hu CH An age-dependent and state-dependent nonlinear prognostic model for degrading systems. IEEE Trans Reliab. https://doi.org/10.1109/TR.2015.2419220

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3. Scarf PA (2007) A framework for condition monitoring and condition based maintenance. Qual Technol Quant Manag 4(2):301–312 4. Wang H (2002) A survey of maintenance policies of deteriorating systems. Eur J Oper Res 139:469–489 5. Pham H, Wang H (1996) Imperfect maintenance. Eur J Oper Res 94:425–438 6. Dekker R, Wildeman RE, Der V et al (1997) A review of multi-component maintenance models with economic dependence. Math Methods Oper Res 45(3):411–435 7. Ito K, Nakagawa T, Nishi K (1995) Extended optimal inspection policies for a system in storage. Math Comput Model 22(10–12):83–87 8. Ito K, Nakagawa T (1995) An optimal inspection policy for a storage system with high reliability. Microelectron Reliab 36(6):875–882 9. Ito K, Nakagawa T (2000) Optimal inspection policies for a storage system with degradation at periodic tests. Math Comput Model 31:191–195 10. Nakagawa T, Yasui K (1980) Approximate calculation of optimal inspection times. J Oper Res Soc 31:851–853 11. Nakagawa T, Mizutani S, Chen M (2010) A summary of periodic and random inspection polices. Reliab Eng Syst Saf 95:906–911 12. Yeh L (1995) An optimal inspection-repair-replacement policy for standby systems. J Appl Probab 32(1):212–223 13. Kaio N, Osaki S (1984) Some remarks on optimum inspection policies. IEEE Trans Reliab R(33):277–279 14. Barlow RE, Hunter LC, Proschan F (1962) Optimum checking procedures. J Soc Ind Apple Math 11:1078–1095 15. Yang Y, Klutke GA (2000) Improved inspection schemes for deteriorating equipment. Probab Eng Inf Sci 14:445–460 16. Yeh RH (1997) Optimal Inspection and replacement policies for multi-state deteriorating systems. Eur J Oper Res 96:248–259 17. Sugiura T, Mizutani S, Nakawaga T (2006) Optimal random and periodic inspection policies. In: Reliability modeling, analysis and optimization. World Scientific, Singapore, pp 393–403 18. Zhang Z (2013) Performance degradation mechanism, rule and optimal inspection and decision method of inertial platform. Rocket Force University of Engineering, Xi ‘an 19. Seal H (1969) Stochastic theory of a risk business. Wiley, New York 20. Nakagawa T (2005) Maintenance theory of reliability. Springer, London 21. Department of Mathematics, Tongji University (1996) Advanced mathematics, vol 2. Higher Education Press, Beijing 22. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via EM algorithm. J Roy Stat Soc 39(1):1–38 23. Saralees N (2008) A review of results on sums of random variables. Acta Appl Math 103:131– 140

Chapter 12

Cooperative Predictive Maintenance of Two-Component System with Limited Resources

12.1 Introduction Preventive maintenance is an effective means to keep the system reliability above a satisfactory level, prolong the service life of equipment and reduce the failure rate [1–6]. However, preventive maintenance actions cannot reduce or eliminate the failure risk caused by system design defects. In view of this situation, Lin et al. [7] think that there are two main failure modes in the system: maintainable failure mode and non-maintainable failure mode. Among them, the degradation corresponding to maintainable failure mode can be eliminated or alleviated by preventive maintenance actions such as cleaning and wiping, lubricating and oiling, and fastening screws, while the degradation or failure corresponding to non-maintainable failure mode can be eliminated only by replacement operation. In view of the independent failure modes, Lin et al. [7] proposed an imperfect sequential preventive maintenance policy. However, in some cases, these failure modes are not statistically independent, but have an interactive relationship. These interactions should be taken into account when modeling maintenance decisions for such systems. In recent years, many references have begun to study the optimal maintenance decision modeling under the condition of interactions of failure modes [7–18]. For example, Murthy and Nguyen [8] proposed two models describing the interdependence of failure modes for a system composed of two components. One is that no matter which component in the system fails, it will cause another component to fail. The other is that the failure of any one component will have a certain impact on the failure rate of another component. Zequeira and Berenguer [19] focused on the periodic preventive maintenance problem when the two failure modes are competitive failure modes and have statistical dependence on each other, and gave the conditions for the existence and uniqueness of the optimal strategy. Castro [16] also studied similar problems, but the difference from Reference [19] mainly lies in the model used to describe the interdependence of failure modes.

© National Defense Industry Press 2022 C. Hu et al., Residual Life Prediction and Optimal Maintenance Decision for a Piece of Equipment, https://doi.org/10.1007/978-981-16-2267-0_12

243

244

12 Cooperative Predictive Maintenance of Two-Component …

It can be seen that most of the existing references focus on how to model and optimize the interaction between failure modes based on failure rate information under the traditional maintenance framework, without considering the maintenance decision modeling based on performance degradation amount. The result of maintenance decision cannot reflect the real-time change of system health state. Considering the rapid development of sensor technology and its wide application in industrial production, it is worth studying how to use the performance data of equipment in operation to make maintenance decision. Moreover, there are usually two-way influences between failure modes. For example, resistors will emit heat during use, which will increase the ambient temperature, while the failure rate of carbon resistors is greatly affected by the ambient temperature [20]. Therefore, when these resistors are assembled together, the temperature emitted by each resistor will affect the failure rate of other resistors. It can be seen from this that it is necessary to study the maintenance policy for the failure mode system with two-way influence. In addition, most references always think that resources are sufficient, but in fact, the resources used for maintenance (maintenance costs, spare parts, etc.) are not inexhaustible. For example, when multiple components fail and these components are distributed in different places, the allocation of maintenance personnel is a problem that needs careful consideration, and the number of personnel will directly affect the quality of maintenance. Therefore, it is necessary to study the influence of resource restriction on maintenance effect in the process of maintenance decision-making, so as to provide theoretical support for rational allocation of maintenance resources. At present, many references have studied the modeling of maintenance effect. Barlow and Hunter [21] first proposed the concept of minimal repair. Lin [22] proposed a hybrid imperfect maintenance model. Wu and Zuo [23] summarized the imperfect maintenance models in the existing reference. Yu et al. [24] considered how to use real data to select a reasonable imperfect maintenance model. In this chapter, considering imperfect maintenance and limited resources, a cooperative predictive maintenance model (CPdM) is proposed for a two-component system with interactions of failure modes [25]. Generally speaking, the information obtained from real-time monitoring is used to predict and estimate the reliability and future failure times, and then the prediction results are taken into account in the maintenance decision-making process, and the reasonable allocation of maintenance resources is considered at the same time, so that the maintenance model is more in line with the actual situation and the maintenance decision-making results are more reasonable.

12.2 Cooperative Predictive Maintenance Model This chapter considers how to use real-time monitoring information to rationally allocate limited maintenance resources and minimize the expected loss cost per unit time when the failure modes of two components of a complex system interact with

12.2 Cooperative Predictive Maintenance Model

245

each other. Before the cooperative predictive maintenance model is given, several precondition assumptions are introduced. Since this chapter studies that a component corresponds to a failure mode, it is no longer necessary to distinguish a single failure mode from a single component in the following contents of this chapter. For convenience of discussion, i is used to mark failure mode, i = 1 is used to mark failure mode 1 and i = 2 is used to mark failure mode 2. Assumption 12.1 The maintainable system is a two-component system, and there is two-way influence between failure modes corresponding to its components. The influence mode is that every failure of one component will have a certain influence on the failure rate of the other component. In addition, each failure mode has a performance variable φi (t), corresponding to i = 1, 2. References [8–12, 16–18] all consider the situation that failure modes have oneway influence, and this chapter considers the situation that failure modes have twoway interaction. Therefore, it can be said that Assumption 12.1 is an extension of the research content in the previous reference. In practical industrial systems, the system is generally composed of multiple components, and each component has at least one failure mode. Therefore, this assumption is a reasonable one and has practical significance. Assumption 12.2 Limited resources for maintenance. Usually, maintenance resources include maintenance personnel, spare parts and corresponding maintenance costs. However, this chapter only considers the limited maintenance cost. Therefore, the subsequent contents no longer strictly distinguish maintenance resources from maintenance costs. Obviously, due to the limitation of equipment running environment and enterprise capability, it is impossible to have all the resources for maintenance. For example, an enterprise’s annual budget for arranging maintenance is definitely limited, and it is impossible for an enterprise to allocate enough resources to maintenance activities regardless of its actual ability. Therefore, when maintaining each component, it is necessary to allocate the insufficient maintenance resources to each component reasonably, so as to achieve better maintenance results with limited resources and then embody the idea of cooperation in the process of resource allocation. Assumption 12.3 The arrangement of future maintenance activities is based on the prediction value of expected failure times in the future and the minimum average cost of long-term operation of the system. As mentioned in the preface, most of the current references cannot satisfy Assumption 12.1. These references mainly study the maintenance problems when failure modes are independent of each other or only have one-way influence. In this case, a possible way to deal with it is to implement competitive maintenance, that is, to use all the limited resources for the maintenance of a certain failure mode. This will cause another component to stop working, thus resulting in losses. In order to overcome the shortcomings of the existing methods and make the maintenance policy better reflect

246

12 Cooperative Predictive Maintenance of Two-Component …

the change of system health state, this chapter focuses on the predictive maintenance problem when assumptions 12.1, 12.2 and 12.3 are met at the same time. Other necessary assumptions are as follows: Assumption 12.4 Two failure modes are maintained at the same time, and this simultaneous maintenance is called a preventive maintenance of the system. The resources for preventive maintenance of the system are limited each time and fixed in number. Only after the previous preventive maintenance is completed will it be replenished. In addition, as a result of limited resources allocated to each failure mode for maintenance, it is impossible for the maintenance operation to restore the failure mode as new. Use η(t) ∈ η to represent the maintenance resource allocation ratio corresponding to failure mode 1 determined according to the performance degradation amount at the current time t. Where η is all possible sets for allocation ratios η(t), i.e., η = {η1 , η2 , . . . , ηq }, η = {η1 , η2 , · · · , ηq }, 0 < η1 < η2 < · · · < ηq < 1 and q ∈ N. An assumption about η is given below. Assumption 12.5 The set η does not change with time. Moreover, all preventive maintenance resources have the same allocation ratio η(t). Assumption 12.6 No matter which failure mode occurs, it will be discovered by management personnel immediately. Furthermore, the failure that has occurred can be corrected by minimal maintenance, so that the components can resume operation again. Furthermore, the minimal repair will not change the changing rule of performance variable φi (t). In addition, the loss cost caused by inspection and the possible additional damage to the system are not considered here. Assumption 12.7 The unknown actual hazard rate h¯ i (s) is piecewise continuous and strictly convex increasing in [tk , tk+1 ). The hazard rate h 0,i (s) is also continuous and strictly convex increasing. Further, it satisfies h 0,i (0) = 0. Assumption 12.8 The time consumed for implementing replacement, minimal repair and preventive maintenance is negligible. On the premise of meeting the above assumptions, this chapter will give a detailed description of the cooperative predictive maintenance model in the following contents combined with Fig. 12.1. (1)

It is assumed that the system has undergone preventive maintenance for k time(s) and has run to the current time t = tk + t L , where t L is the length of time that the system has been running up to now after completing maintenance for the k-th time just now. In order to keep the system above a satisfactory reliability level, it is planned to arrange periodic preventive maintenance for the N (t) − 1 time(s) in the future.

12.2 Cooperative Predictive Maintenance Model

h1 (tk +1 + τ )

h1 (t1 + τ ) z N 2,1 1 0,1

=a

247

Nz

(t )

= a1 2,k +1 h0,1 ( y1,+k +1 + τ )

+ 1,1

h ( y +τ )

h1 (τ ) = h0,1 (τ )

time

tk

t2

t1

Nz

+ = a2 1,1 h0,2 ( y2,1 +τ )

tk + 2

tk +1

h2 (t1 + τ ) current time t = tk + t L

h2 (tk +1 + τ ) Nz

(t )

= a2 1,k +1 h0,2 ( y2,+ k +1 + τ )

h2 (τ ) = h0,2 (τ )

T1|0

tk + N ( t )

time

t1 T2|1

t2

tk t L

Planned PM

T (t ) − t L

tk +1

T (t )

remaining time till the replacement

tk + 2

tk + N ( t )

Planned replacement

Fig. 12.1 Schematic diagram of cooperative predictive maintenance

(2)

Denote the time interval between two times of preventive maintenance determined at t, the maintenance period, as T (t), which shall meet condition of T (t) ≥ t L . After the system has been maintained for the N(t)-1 time(s), it will be replaced with a new system after T (t). In the following, t will be used to represent tk + t L without declaration.

(3)

Periodically sample the performance degradation amount corresponding to each failure mode, with the sampling time interval of t. In time interval (t, tk+1 ], the failure rate h i (s) of failure mode can be estimated by performance degradation amount. However, as the preventive maintenance has an impact on performance degradation amount, the failure rate function h 0,i (s) obtained from life data statistics is still used to represent the degradation process of failure mode after the first maintenance. There is bio-directional interaction between failure modes; that is, the failure of failure mode i will have cumulative influence on the failure rate function of another failure mode i¯ (please refer to h i (tl + τ ) in Fig. 12.1). Here, i and i¯ are indices of failure modes, and i = 1, 2. If i = 1, then i¯ = 2; otherwise, z i¯ = 1. Before the current time t, use N i,l to represent all the failure times that occur before the completed maintenance of the failure mode i for the l-th time (l ≤ k). If the current time is t, the total failure times of the failure mode i within the time period [t0 , tk+ j ] (including the failure times that have occurred within [t0 , t] and the expected failure times that are estimated within (t0 , tk+ j ]) are related to the failure rate function of the failure mode i¯ within (tk+ j , tk+N (t) ]. Due to the interaction of failure modes and limited resources, the effect of preventive maintenance is not perfect. The effect of incomplete maintenance can be described by the change of failure rate function and effective service

(4)

(5)

248

12 Cooperative Predictive Maintenance of Two-Component …

life [7]. The failure rate of the failure mode i after the maintenance for the l-th time is denoted as h i (tl + s) and calculated by the following equation h i (tl + τ ) =

⎧ ⎨

z

N i,l ¯

+ h 0,i (yi,l + τ )ai

if l ≤ k z ⎩ h (y + (t) + τ )a Ni,l¯ (t) if l > k 0,i i,l i

(12.1)

z

N i,l ¯

where 0 < τ < tl+1 − tl . In Formula (12.1), adjustment factors ai

z

(l ≤ k) and

N ¯ (t) ai i,l

(l > k) are introduced to quantify the influence of failure times of failure + + and yi,l (t) of failure mode i¯ on failure mode i [11]. The effective service life yi,l mode i after each maintenance can be calculated and predicted, respectively, by the following Formula.

+ yi,l (t) =

+ + = yi,l−1 + ρi,l Tl|l−1 , 1 ≤ l ≤ k yi,l + + yi,l (t) = yi,l−1 + ρi (t)T (t), l = k + 1 + yi,l−1 (t) + ρi (t)T (t), k + 2 ≤ l ≤ k +

. N (t) − 1

Therefore, the effective service life of failure mode i in any short time before the preventive maintenance for the j-th time not implemented is + yi,k+ j (t) = yi,k + [1 + ( j − 1)ρi (t)]T (t), j = 1, · · · , N (t).

(12.2)

The adjustment factors ρi,l and ρi (t) have the following deterministic functional relationship with the resource allocation ratio [34]: l c p,i

−bi = = ln(1 − ρi,l )



l ≤ k, i = 1; c p · ηl c p · (1 − ηl ) l ≤ k, i = 2.

and c p,i (t) =

−bi = ln(1 − ρi (t))



l > k, i = 1; c p · η(t) c p · (1 − η(t)) l > k, i = 2.

where bi > 0 is constant and can be estimated by historical data, ηl is the resource allocation ratio of completed maintenance and η(t) the allocation ratio of maintenance to be implemented. It should be noted that the formula given above is only used to describe the relationship between resource allocation ratio and adjustment factor. Of course, we can choose other relational expressions for expression according to the actual situation, but this does not affect the results of this chapter. (6)

The maintenance time can be neglected.

According to the assumptions given above and the detailed description of the cooperative predictive maintenance model, the maintenance decision model and its optimization are given below.

12.3 Maintenance Decision Modeling and Optimization

249

12.3 Maintenance Decision Modeling and Optimization 12.3.1 Estimation of Expected Failure Times As the performance degradation related to each failure mode is sampled periodically, all the performance degradation amount gotten within [tk , t] can be used to obtain the predicted reliability Ri (tk+1 |t) of the corresponding failure mode. Further, the predicted reliability is used to estimate the expected number of times that the failure mode i will fail within [t, tk+1 ], which is denoted as Nˆ i,k (t). In this way, the decision of preventive maintenance can change with the actual health state of the system [26]. A core technology involved here is real-time reliability prediction based on performance degradation amount. So far, many references have studied this issue, such as references [27–31]. Like other references, the performance reliability prediction of each failure mode is still based on the concept of soft failure; that is, when the performance degradation amount φi (t) of the failure mode i exceeds the preset threshold φi,th , it is considered that soft failure has occurred. Therefore, the predicted reliability of failure mode i at tk+1 can be calculated by the following Formula [30] φi,th Ri (tk+1 |t) =

f (φi (tk+1 )|φi (τ ), tk ≤ τ ≤ t)dφi (tk+1 ).

(12.3)

−∞

At the same time, Ri (tk+1 |t) can also be calculated according to the following formula ⎛ T (t)−t ⎞  L Ri (tk+1 |t) = exp⎝− h i (t + τ |t)dτ ⎠. (12.4) 0

Furthermore, if every failure of the failure mode is to restore its operation through minimal repair, Nˆ i,k (t) can be expressed by the following Formula [26] Nˆ i,k (t) =

T (t)−t L

h i (t + τ |t)dτ

(12.5)

0

According to Formulas (12.3, 12.4 and 12.5), the expected failure times including failure modes can be estimated on the basis of historical performance degradation amount within [t, tk+1 ], namely Nˆ i,k (t) = − ln

φi,th

−∞

f (φi (tk+1 )|φi (τ ), tk ≤ τ ≤ t)dφi (tk+1 ).

(12.6)

250

12 Cooperative Predictive Maintenance of Two-Component …

It should be noted that reliability prediction is not the focus of this chapter. Please refer to refs. [31–33] for details of real-time reliability prediction and performance reliability prediction steps based on exponential smoothing method. In addition, it is difficult to quantify the impact of planned preventive maintenance on performance variables, so the expected failure times occurred in the time period (tk+ j , tk+ j+1 ](1 ≤ j ≤ N (t) − 1) can only be calculated by failure rate function h 0,i (s) rather than the performance degradation data.

12.3.2 Cost Rate Model This subsection will reasonably arrange the maintenance operation after the current time t by minimizing the expected cost rate in the remaining time of the current replacement cycle. According to the description of CPdM model in Subsection 12.2, the expected length of remaining running time in the current replacement cycle at t = tk + t L is M(t) = N (t)T (t) − t L

(12.7)

The total expected cost C(t) in the remaining running time is the sum of the total replacement cost Cr , all preventive maintenance costs C p (t) and all minimal repair costs; that is, C(t) = Cr + C p (t) + Cm (t)

(12.8)

Next, each term on the right side of the equal sign in Formula (12.8) will be deduced in detail one by one, and its corresponding expression will be given as follows: (1) (2) (3)

Cr = cr,1 + cr,2 ; C p (t) = (N (t) − 1)c p ; Cm (t) includes two parts:

(a)

Expected maintenance cost obtained by performance reliability prediction in the time period of [t, tk+1 ]

Cm1 (t) =

2  i=1

(b)

T (t)−t L

h i (t + τ |t)dτ =

cm,i 0

2 

cm,i Nˆ i,k (t).

i=1

Expected maintenance cost in the time period of (tk+1 , tk+N (t) ] calculated by the failure rate function

12.3 Maintenance Decision Modeling and Optimization

251

⎛ Cm2 (t) =

2 

⎜ cm,i ⎝

N (t)−1

i=1

=

2 

ai

j=1

⎛ cm,i ⎝

⎞ ⎟ h 0,i (τ )dτ ⎠

yi,k+ j+1 (t)−T (t)

N (t)−1

i=1

yi,k+  j+1 (t)

z Ni,k+ (t) ¯ j

⎞

Ni,k+ j (t)⎠

j=1

N¯z (t)  yi,k+ j+1 (t) where Ni,k+ j (t) = ai i,k+ j yi,k+ j+1 (t)−T (t) h 0,i (τ )dτ . Therefore, the total expected minimal repair cost is

Cm (t) = Cm1 (t) + Cm2 (t) Then, according to the above derivation process, the total expected cost in the remaining running time can be obtained as follows. Ca (t) =

2 

cm,i Nˆ i,k (t) +

i=1

2 

⎛ cm,i ⎝

i=1

N (t)−1

⎞ Ni,k+ j (t)⎠ + cr,1 + cr,2

(12.9)

j=1

Therefore, from Formulas (12.7) and (12.9), the expected cost per unit time of the system in the remaining time of the current replacement cycle can be obtained by C(N (t), T (t), η(t)) =

Ca (t) M(t) (N (t) − 1)c p +

=



2 

   N (t)−1  cm,i Nˆ i,k (t) + Ni,k+ j (t) + cr,i

i=1

j=1

(12.10)

N (t)T (t) − t L

where N (t) ∈ N, T (t) ≥ t L , and η(t) ∈ η are all decision variables. When N (t) = 1, the cost rate model (12.10) degenerated into the following formula 2 

C(T (t)) =

cm,i Nˆ i,k (t) + cr,i

i=1

T (t) − t L

.

(12.11)

Accordingly, T (t) only depends on the failure times Nˆ i,k (t) obtained by prediction. In this case, only the replacement operation is optional without considering the arrangement of preventive maintenance operation. Therefore, there is no need to determine variables η(t). From the above analysis and CPdM model description, it can be seen that the cost rate C(N (t), T (t), η(t)) at t depends on the prediction information, such as the expected failure times in a time period of [t, tk+N (t) ). As time goes by, the cost rate

252

12 Cooperative Predictive Maintenance of Two-Component …

C(N (t), T (t), η(t)) will change accordingly. Therefore, the maintenance decision based on the cost rate reflects the real-time change of equipment health state.

12.3.3 Maintenance Optimization This section will study the optimization of objective function (12.10). For convenience, the following marks are simplified accordingly: N (t) = N , T (t) = T , Nˆ i,k (t) = Nˆ i,k , η(t) = η, Ni,k+ j (t) = Ni,k+ j and yi,k (t) = yi,k . Then, using these simplified labels, the cost rate function (12.10) is reorganized as

C(N , T, η) =

(N − 1)c p +

2 i=1

     −1 cm,i Nˆ i,k + Nj=1 Ni,k+ j + cr,i N T − tL

(12.12)

where N , T, η are decision variables. When N (t) = 1, the degraded cost rate function (12.11) is 2 C(T ) =

i=1

[cm,i Nˆ i,k + cr,i ] T − tL

(12.13)

where T is the decision variable. Note that there is no need to make decisions on variables η in this case. According to Formula (12.12) and its degraded function (12.13), it can be seen that the optimal maintenance times that can minimize the expected cost rate in the remaining running time must be found in the following two situations: 1) N = 1, 2) N ≥ 2. It has been pointed out above that, if N = 1, it is only necessary to decide when to implement the replacement operation, instead of deciding the number and time interval of preventive maintenance actions. Then, take the derivative of C(T ) for T in Formula (12.12) and take the derivative result 0, then there exists 2 

  cm,i (T − t L )h i (tk + T |t) − Nˆ i,k = cr,1 + cr,2

(12.14)

i=1

In order to find the optimal T value that can satisfy Formula (12.14), the following theorem is introduced: Theorem 12.1 Let εk (τ ) (0 ≤ τ ≤ T − t L ) represent the error between the actual hazard rate h¯ i (t + τ ) and its predicted value h i (t + τ |t), namely εk (τ ) = h i (t + τ |t)− h¯ i (t +τ ). Under the condition that |εk (τ )| < εup < ∞, there exists an optimal value satisfying Eq. (12.14).

12.3 Maintenance Decision Modeling and Optimization

253

Proof Let D(T ) denote the l.h.s of (12.14). Because D(t L ) = 0 < cr,1 + cr,2 , t L cannot be the solution to (12.14). So we have to consider the case that T > t L . Then we have. ⎤ ⎡ T−t L 2  D(T ) = cm,i ⎣(T − t L )h i (tk + T |t) − h i (t + τ |t)dτ ⎦ i=1

0

⎫  ⎬ = cm,i (T − t L ) h¯ i (tk + T ) + εk (T ) − h¯ i (t + τ ) + εk (τ ) dτ ⎭ ⎩ i=1 0 ⎫ ⎧ T−t L 2 ⎨     ⎬  ¯ ≥ h¯ i (t + τ ) + εup dτ cm,i (T − t L ) h i (tk + T ) − εup − ⎭ ⎩ i=1 0 ⎧ ⎡ ⎤⎫ T−t L 2 ⎬ ⎨  1 = cm,i (T − t L )⎣h¯ i (tk + T ) − h¯ i (t + τ )dτ − 2εup ⎦ ⎭ ⎩ T − tL 2 

⎧ ⎨





i=1

T−t L



0

(12.15)  T −t Take D(T ) = h¯ i (tk + T ) − 1/(T − t L ) 0 L h¯ i (t + τ )dτ . Considering that the real failure rate function h¯ i (·) is strictly convex in terms of time tk , the following conclusions are drawn T−t L

D(T ) =

[h¯ i (tk + T ) − h¯ i (t + τ )]dτ

0 T−t L !

>

T − tL h¯ i (tk + T ) − [h¯ i (t) +

0

h¯ i (tk + T ) − h¯ i (t) = 2

"

h¯ i (tk +T )−h¯ i (t) τ] T −t L

dτ

T − tL (12.16)

As the right part of Formula (12.16) tends to be ∞ when T → ∞, so there is D(T ) → ∞ as T → ∞. Furthermore, it can be found that there is D(T ) → ∞ when T → ∞. In addition, considering there is D(t L ) = 0 < cr,1 + cr,2 , we can get that there exists an optimal value T1∗ satisfying (12.14). Q.E.D. The condition of Theorem 12.1 can be satisfied by choosing the appropriate reliability prediction method. However, the reliability prediction method is not the focus of this chapter. Please refer to refs. [31–33] to understand the related work of real-time reliability prediction. If N ≥ 2, it is necessary to minimize the cost rate function C(N , T, η) determined by Formula (12.12) and find N ∗ , T ∗ and η∗ satisfying

254

12 Cooperative Predictive Maintenance of Two-Component …

C(N ∗ , T ∗ , η∗ ) = inf{C(N , T, η), N ≥ 2, T ≥ t L , η ∈ η }.

(12.17)

In order to achieve this goal, the following theorems are given. Theorem 12.2 If Assumption 12.7 is satisfied, the first derivative of Ni,k+ j with respect to T is positive for all j ∈ N, η ∈ η and T ≥ t L . Proof if j = 1, there is Ni,k+1 =

yi,k+2

N¯z ai i,k+1

h 0,i (τ )dτ,

(12.18)

yi,k+2 −T + + (1 + ρi )T ≥ T . Then, where yi,k+2 = yi,k N¯z ∂ Ni,k+1 = ai i,k+1 [h 0,i (yi,k+2 )(1 + ρi ) − h 0,i (yi,k+2 − T )ρi ] ∂T yi,k+2 z Ni,k+1 ¯ h i¯ (tk + T |t) ln ai h 0,i (τ )dτ + ai

# N¯z ai i,k+1 [h 0,i (yi,k+2 )

yi,k+2 −T

− h 0,i (yi,k+2 − T )]ρi + h 0,i (yi,k+2 ) yi,k+2 z Ni,k+1 ¯ + ai h i¯ (tk + T |t) ln ai h 0,i (τ )dτ =

$

yi,k+2 −T

> h 0,i (yi,k+2 ) ≥ h 0,i (T ) ≥ h 0,i (t L ) > 0

(12.19)

+ + [1 + ( j − 1)ρi ]T , and If j ≥ 2, then there is yi,k+ j = yi,k

∂ Ni,k+ N¯z ∂ Ni,k+ j ¯ j = ai i,k+ j ln ai ∂T ∂T z

z Ni,k+ ¯ j

+ ai

yi,k+  j+1

h 0,i (τ )dτ yi,k+ j+1 −T

[h 0,i (yi,k+ j+1 )(1 + jρi ) − h 0,i (yi,k+ j+1 − T ) · jρi ].

Then, replace the j in Formula (12.20) with 2, and we can get

(12.20)

12.3 Maintenance Decision Modeling and Optimization

∂ Ni,k+2 N¯z ∂ Ni,k+2 ¯ = ai i,k+2 ln ai ∂T ∂T z

yi,k+3

h 0,i (τ )dτ yi,k+3 −T

 N¯z + ai i,k+2 h 0,i (yi,k+3 )(1 + 2ρi ) − h 0,i (yi,k+3 yi,k+3 z ∂ Ni,k+2 Ni,k+2 ¯ ¯ = ai ln ai h 0,i (τ )dτ ∂T

+

255

− T ) · 2ρi

yi,k+3 −T

 N¯z ai i,k+2 h 0,i (yi,k+3 )(1

+ 2ρi ) − h 0,i (yi,k+3 − T ) · 2ρi

>0



 (12.21)

z Suppose that ∂ Ni,k+ j /∂ T > 0 holds for 2 ≤ j ≤ m. Hence, ∂ Ni,k+ j /∂ T > 0, which implies. the two first term in (12.20) is greater than 0. In addition, since the hazard rate h 0,i (s) is strictly increasing, the second term in (12.20) is also positive. Therefore, ∂ Ni,k+m+1 /∂ T > 0. By complete induction, the first derivative of Ni,k+ j with respect to T is greater than 0, i.e. ~ ∂ Ni,k+ j /∂ T > 0 for j ≥ 2. Together with (12.19), we get ∂ Ni,k+ j /∂ T > 0 for j ∈ N. Q.E.D. To find the optimal N minimizing C(N , T, η) for fixed T and η, the approach developed by Nakagawa in Reference [35] is adopted here. The inequalities C(N +1, T, η) ≥ C(N , T, η) and C(N , T, η) < C(N −1, T, η) imply

A(N , T, η) ≥ cr,1 + cr,2 − c p and A(N − 1, T, η) < cr,1 + cr,2 − c p ,

(12.22)

where A(N , T, η) =

2  i=1

⎡

⎤ % & N −1  c p tL tL ˆ ⎣ N− Ni,k+N − Ni,k − , N ∈ N, N ≥ 2 cm,i Ni,k+ j ⎦ − T T j=1

(12.23) There are the following theorems for A(N , T, η). Theorem 12.3 If Assumption 12.7 is satisfied, then there is a unique finiteness ∗ N T,η ≥ 2 satisfying (12.22) for fixed T ≥ t L and η ∈ η . Proof It is easy to obtain (12.24)

256

12 Cooperative Predictive Maintenance of Two-Component …

A(N + 1, T, η) − A(N , T, η) =

2 

cm,i

i=1

⎡

% & N  ⎣ N + 1 − t L Ni,k+N +1 − Nˆ i,k − Ni,k+ j T j=1 ⎤ & % N −1  tL Ni,k+N + Nˆ i,k + − N− Ni,k+ j ⎦ T j=1 % =

N +1−

tL T

& 2

cm,i (Ni,k+N +1 − Ni,k+N ),

i=1

(12.24) where N + 1 − t L /T > 0. z z Since Ni,k+ j+1 ≥ Ni,k+ j ≥ 0, yi,k+ j+2 − ρT = yi,k+ j+1 ≥ T , and h 0,i (t) is continuous and strictly increasing, we have yi,k+  j+2

z Ni,k+ ¯ j+1

Ni,k+ j+1 − Ni,k+ j = ai

h 0,i (τ )dτ − ai yi,k+ j+2 −T

⎛ z Ni,k+ ¯ j⎜

> ai

yi,k+  j+1

h 0,i (τ )dτ −

yi,k+ j+2 −T

⎛ > ai

⎜ ⎝

yi,k+ j+1 −T

(1+ρ)T 

T h 0,i (τ )dτ −

ρT (1+ρ)T 

ρT

which implies

2 

⎞

⎟ h 0,i (τ )dτ ⎠ ⎞

⎟ h 0,i (τ )dτ ⎠

0

T h 0,i (τ )dτ −

≥

h 0,i (τ )dτ yi,k+ j+1 −T

yi,k+  j+2

⎝

z Ni,k+ ¯ j

yi,k+  j+1

z Ni,k+ ¯ j

h 0,i (τ )dτ = m 0 > 0,

(12.25)

0

cm,i (Ni,k+N +1 − Ni,k+N ) > 0. Thus, A(N + 1, T, η) −

i=1

A(N , T, η) > 0, ~ i.e. ~ A(N , T, η) is strictly increasing in N. According to Formulas (12.23) and (12.25), A(N , T, η) >

  c t p L cm,i (N − 1)Ni,k+N − Nˆ i,k − (N − 1)Ni,k+N −1 − T i=1

2 

= (N − 1)

2  i=1

cm,i (Ni,k+N − Ni,k+N −1 ) −

2  i=1

cm,i Nˆ i,k −

c p tL T

12.3 Maintenance Decision Modeling and Optimization

> (N − 1)m 0 −

2  i=1

cm,i Nˆ i,k −

257

c p tL . T

(12.26)

The right half of the greater than sign in Formula (12.26) also tends to be ∞ when N → ∞, which implies A(N , T, η) → ∞ when N → ∞. ∗ ≥ 2 making Formula (12.22) tenable Therefore, there is a unique finiteness N T,η when T ≥ t L and η ∈ η are fixed. Q.E.D. Derive C(N , T, η) with respect to T and let the result be 0, then there is ∂C(N , T, η) (N T − t L )2 = B(N , T, η) − cr,1 − cr,2 − (N − 1)c p = 0. (12.27) ∂T N where B(N , T, η) =

2 

cm,i

i=1

⎧ N −1 '% ⎨ ⎩

T−

j=1

tL N

&

∂ Ni,k+ j − Ni,k+ j ∂T

(

⎫ & ˆ ⎬ t L ∂ Ni,k − Nˆ i,k + T− ⎭ N ∂T %

Theorem 12.4 If Assumption 12.7 is satisfied, there exists a finite TN∗ ,η minimizing (12.12), namely C(N , TN∗,η , η) = min T ≥tL {C(N , T, η}) for fixed N ≥ 2 and η ∈ η . Proof First of all, demonstrate that B(N , T, η) tends to ∞ when T → ∞. %

tL T− N

&

⎧ &⎪ z ∂ N ¯z ∂ Ni,k+ j t L ⎨ Ni,k+ ¯ i,k+ j j = T− ai ln ai ∂T N ⎪ ∂T ⎩ %

N ¯z

+ ai %

i,k+ j

yi,k+ j+1



h 0,i (τ )dτ yi,k+ j+1 −T

*

[h 0,i (yi,k+ j+1 )(1 + jρi ) − h 0,i (yi,k+ j+1 − T ) jρi ]

& Nz ∂ N ¯z tL ¯ i,k+ j i,k+ j > T− ai ln ai N ∂T

yi,k+ j+1



h 0,i (τ )dτ yi,k+ j+1 −T

% & Nz t ¯ i,k+ j + T − L ai h 0,i (yi,k+ j+1 ) N % & & Nz % ∂ N ¯z tL tL ¯ i,k+ j i,k+ j = T− ln ai Ni,k+ j + T − a h 0,i (yi,k+ j+1 ) N ∂T N i % & & Nz % ∂ Ni,k+1 ¯ tL tL ¯ i,k+ j ≥ T− ln ai Ni,k+ j + T − a h 0,i (yi,k+ j+1 ) N ∂T N i & & Nz % % tL tL ¯ i,k+ j ln ai h 0,i¯ (t L )Ni,k+ j + T − a ≥ T− h 0,i (yi,k+ j+1 ) N N i

(12.28)

258

12 Cooperative Predictive Maintenance of Two-Component …

Considering that T tends to infinity, ai > 1, and h 0,i¯ (t L ) is a constant greater than 0, it is possible to find a certain value T making (T0 − t L /N ) ln ai h 0,i¯ (t L ) > 1. After T > T0 , Formula (12.28) changes to %

tL T− N

&

& z % N¯ ∂ Ni,k+ j tL − Ni,k+ j > T − a i,k+ j h 0,i (yi,k+ j+1 ), ∂T N i

(12.29)

The right part of Formula (12.29) also tends to infinity when T → ∞. In addition, %

tL T− N

T−t L & ˆ & % ∂ Ni,k tL ˆ − Ni,k = T − h i (tk + T |t) − h i (t + τ |t)dτ ∂T N 0

> (T − t L )h i (tk + T |t) − (T − t L )h i (tk + T |t) = 0. (12.30) According to Formulas (12.27, 12.29 and 12.30), the following conclusions can be easily obtained: When T tends to infinity, B(N , T, η) also tends to infinity. Next, demonstrate the existence of TN∗ ,η . If there is Tˆ > t L making the inequality B(N , Tˆ , η) ≤ cr,1 + cr,2 + (N − 1)c p satisfied, then there is a finite TN∗ ,η making Formula (12.27) tenable when N ≥ 2 and η ∈ η are fixed. If the inequality B(N , Tˆ , η) > cr,1 + cr,2 + (N − 1)c p is always tenable for the fixed N ≥ 2 and η ∈ η , it means that ∂C(N , T, η)/∂ T > 0 is always tenable as well, and then we can draw a conclusion that C(N , T, η) is strictly increasing in T. In this case, TN∗ ,η = t L . Therefore, it can be concluded that there is a limited TN∗ ,η to minimize the value of Formula (12.12) when N ≥ 2 and η ∈ η are fixed. Q.E.D. Theorem 12.5 If Assumption 12.7 is satisfied, (1)

A(N , T, η) is strictly increasing in T for all N > N L and η ∈ η , where N L is defined as ⎧ ⎨

tL

N L = arg min N0 ln ai N0 ≥2⎩

h 0,i (τ )dτ ln ai¯ 0

(2)

tL

⎫ ⎬ h 0,i¯ (τ )dτ > 1 , ⎭

(12.31)

0

∗

Let N T,η denote the optimal solution to min N >N L {C(N , T, η)} for all T ≥ t L ∗ ∗ and η ∈ η . If T1 ≤ T2 , then N T1 ,η ≥ N T2 ,η for fixed η ∈ η .

Proof After the derivative of a function A(N , T, η) with respect to a variable T, we can get

12.3 Maintenance Decision Modeling and Optimization

259

⎡ ⎤ % & 2 N −1 ∂ A(N , T, η)  ⎣ t L t L ∂ Ni,k+N ∂ Nˆ i,k  ∂ Ni,k+ j ⎦ Ni,k+N + N − = − − ∂T T2 T ∂T ∂T ∂T i=1 j=1 & ( % 2 '  c p tL tL t L ∂ Ni,k+N = N + 1 − i,k+N T2 T2 T ∂T i=1 ⎤  2 N −1   ∂ Ni,k+ j ⎦ c p t L ∂ Nˆ i,k ∂ Ni,k+N − − + (N − 1) + 2 . ∂T ∂T ∂T T i=1 j=1

+

(12.32) According to Theorem 12.1, it can be easily proved that the first term on the right side of the equal sign in Formula (12.32) is a positive number. Next, it will be proved that the second term is also positive. Take Bi ( j) = ∂ Ni,k+ j+1 /∂ T − ∂ Ni,k+ j /∂ T, j = 1, · · · , N − 2. Then, for each n ≥ 1,there are Bi (n) = =

∂ Ni,k+n ∂ Ni,k+n+1 − ∂T ∂T N¯z ai i,k+n+1

ln ai

yi,k+n+2 

z ∂ Ni,k+n+1 ¯

∂T

z  Ni,k+n+1 ¯

h 0,i (τ )dτ yi,k+n+2 −T

h 0,i (yi,k+n+2 )[1 + (n + 1)ρi ] − h 0,i (yi,k+n+2 − T ) · (n + 1)ρi yi,k+n+1 z  z ∂ Ni,k+n Ni,k+n ¯ ¯ − ai ln ai h 0,i (τ )dτ ∂T + ai



yi,k+n+1 −T

−

 N¯z ai i,k+n h 0,i (yi,k+n+1 )(1

 + nρi ) − h 0,i (yi,k+n+1 − T ) · nρi .

(12.33)

Since h 0,i (t) is strictly increasing and yi,k+n+2 > yi,k+n+1 , there exists yi,k+n+2 

yi,k+n+1 

h 0,i (τ )dτ > yi,k+n+2 −T

h 0,i (τ )dτ,

(12.34)

yi,k+n+1 −T

and h 0,i (yi,k+n+2 )[1 + (n + 1)ρi ] − h 0,i (yi,k+n+2 − T ) · (n + 1)ρi   − h 0,i (yi,k+n+1 )(1 + nρi ) − h 0,i (yi,k+n+1 − T ) · nρi > 0. So Formula (12.33) changes to

(12.35)

260

12 Cooperative Predictive Maintenance of Two-Component …

Bi (n) >

−

N¯z ai i,k+n

N¯z ai i,k+n

ln ai

ln ai

yi,k+n+1 

z ∂ Ni,k+n+1 ¯

∂T

yi,k+n+1 −T yi,k+n+1 

z ∂ Ni,k+n ¯

∂T

z  Ni,k+n ¯

h 0,i (τ )dτ

h 0,i (τ )dτ yi,k+n+1 −T

h 0,i (yi,k+n+2 )[1 + (n + 1)ρi ] − h 0,i (yi,k+n+2 − T ) · (n + 1)ρi z   Ni,k+n ¯ − ai h 0,i (yi,k+n+1 )(1 + nρi ) − h 0,i (yi,k+n+1 − T ) · nρi yi,k+n+1  z ∂ Ni,k+n Ni,k+n ¯ ¯ > ai ln ai h 0,i (τ )dτ ∂T

+ ai



yi,k+n+1 −T

≥ ln ai

∂ Ni,k+n ¯ ∂T

∂ Ni,k+n ¯ ≥ ln ai ∂T

yi,k+n+1 

h 0,i (τ )dτ yi,k+n+1 −T

tL h 0,i (τ )dτ

(12.36)

0

Further, ∂ Ni,k+n N¯z ∂ Ni,k+n ¯ = ai i,k+n ln ai ∂T ∂T

yi,k+n+1 

z

h 0,i (τ )dτ yi,k+n+1 −T

N¯z + ai i,k+n [h 0,i (yi,k+n+1 )(1 + nρi ) − yi,k+n+1 z  ∂ Ni,k+n ¯ > ln ai h 0,i (τ )dτ

∂T

∂ Nˆ i,k ¯ ≥ ln ai ∂T ∂ Nˆ i,k ¯ ≥ ln ai ∂T

h 0,i (yi,k+n+1 − T ) · nρi ]

yi,k+n+1 −T yi,k+n+1 

h 0,i (τ )dτ yi,k+n+1 −T

tL h 0,i (τ )dτ

(12.37)

0

¯ i and substitute the expression obtained Replace i, i¯ in Formula (12.37) with i, by replacement into Formula (12.36), and we can get

12.3 Maintenance Decision Modeling and Optimization

tL Bi (n) > ln ai

261

tL h 0,i (τ )dτ ln ai¯

0

h 0,i¯ (τ )dτ 0

∂ Nˆ i,k ∂T

 Combined tL

with

=

NL +

min N0 N0 ln ai

(12.38) tL

h 0,i (τ )dτ ln ai¯

0

h 0,i¯ (τ )dτ > 1, N0 ∈ N and Formula (12.38) for consideration, there is

0

tL N L Bi (n) > N L ln ai

tL h 0,i (τ )dτ ln ai¯

0

h 0,i¯ (τ )dτ 0

∂ Nˆ i,k ∂ Nˆ i,k > . ∂T ∂T

(12.39)

Now for all N ≥ N L + 1, the second term of the r.h.s of Formula (12.32) is ⎡

⎤ N −1  ˆ ∂ Ni,k+ j ⎦ ⎣(N − 1) ∂ Ni,k+N − ∂ Ni,k − ∂ T ∂ T ∂T i=1 j=1  N −1  2   ∂ Nˆ i,k ≥ Bi (n) − ∂T i=1 n=1   2  ∂ Nˆ i,k N − 1 ∂ Nˆ i,k − = NL ∂ T ∂T i=1

2 

≥ 0.

(12.40)

Therefore, the first derivative of A(N , T, η) for T is positive when N ≥ N L + 1. And A(N , T, η) is strictly separate increasing function when N ≥ N L + 1 and η ∈ η for T. Finally, the second part of the theorem is proved. According to Theorem 12.3, it is ∗ ∗ easy to know that there are N T1 ,η and N T2 ,η satisfying Formula (12.22) for different T1 and T2 . Then, ∗

A(N T1 ,η , T1 , η) ≥ cr,1 + cr,2 − c p . ∗

∗

Considering N T1 ,η > N L ,we have the monotonicity in T of A(N T1 ,η , T, η) for η ∈ η which implies ∗

A(N T1 ,η , T2 , η) ≥ A(N T∗1 ,η , T1 , η) ≥ cr,1 + cr,2 − c p . ∗

∗

where T1 ≤ T2 . Hence, we have N T2 ,η ≥ N T1 ,η which completes the proof of part 2. Q.E.D.

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12 Cooperative Predictive Maintenance of Two-Component …

" ! ∗ Theorem 12.6 Let Ta  c p / min T ≥tL {C(2, T, η1 )} and Na  max N Ta ,η , η ∈ η . If Assumption 12.7 is satisfied, then min{C(N , T, η), N ≥ 2, T ≥ t L , η ∈ η } has finite solutions and C(N ∗ , T ∗ , η∗ ) = min {min { inf {C(N , T, η)}}}

(12.41)

2≤N ≤Na η∈η T ≥t L

Proof The method proposed by Zhang and Jardine in Reference [36] is used here for demonstration. From (12.12) and cr,1 + cr,2 > c p , we conclude that, for every T such that t L ≤ T ≤ Ta , there exists. (N − 1)c p +

2 



 cm,i Nˆ i,k +

i=1

C(N , T, η) =

N −1

 Ni,k+ j

 + cr,i

j=1

N T − tL cr,1 + cr,2 + (N − 1)c p > NT cp cp ≥ > T Ta = min {C(2, T, η1 )} T ≥t L

≥ inf{C(N , T, η), N ≥ 2, T ≥ t L , η ∈ η } Hence, inf{C(N , T, η), N ≥ 2, T ≥ t L , η ∈ η } = inf{C(N , T, η), N ≥ 2, T ≥ Ta , η ∈ η } = inf {min{min {C(N , T, η)}}} T ≥Ta N ≥2 η∈η

= inf {min{ min {min {C(N , T, η)}}, min {min {C(N , T, η)}}}} T ≥Ta

2≤N ≤N L η∈η

N >N L η∈η

.

= min{ inf { min {min {C(N , T, η)}}}, inf { min {min {C(N , T, η)}}}} T ≥Ta 2≤N ≤N L η∈η

T ≥Ta N >N L η∈η

= min{ inf { min {min {C(N , T, η)}}}, min { inf { min {C(N , T, η)}}}} T ≥Ta 2≤N ≤N L η∈η

η∈η T ≥Ta N >N L

∗ = min{ inf { min {min {C(N , T, η)}}}, min { inf {C( N¯ T,η , T, η)}}} T ≥Ta 2≤N ≤N L η∈η

η∈η T ≥Ta

Furthermore, according to Theorem 12.5 and the definition of Na , when T ≥ Ta , ∗ ∗ there is N T,η ≤ N Ta ,η ≤ Na , Thus, inf{C(N , T, η), N ≥ 2, T ≥ t L , η ∈ η } = min{ inf { min {min {C(N , T, η)}}}, inf { min {min {C(N , T, η)}}}} T ≥Ta 2≤N ≤N L η∈η

= min{ inf { min {min {C(N , T, η)}}}} T ≥Ta 2≤N ≤Na η∈η

T ≥Ta N L t, do not take any maintenance action, but let it run autonomously until the next decision time. The maintenance optimization algorithm will be adopted again when the measured value of the performance variable when t + n · t, n ∈ N, is obtained. The optimal values optimized at the last time t +(n−1)·t will be replaced by N ∗ (t +n·t), T ∗ (t +n·t) and η∗ (t +n·t), respectively. Repeat this process until T ∗ (t + n · t) − (t L + n · t) ≤ t. Once the

264

12 Cooperative Predictive Maintenance of Two-Component …

above termination conditions are met, an appropriate maintenance action (preventive maintenance or replacement) will be recommended according to the following rules: if N ∗ (t + n · t) > 1, the (k + 1)-th preventive maintenance will be carried out; if N ∗ (t + n · t) = 1, it is necessary to implement the replacement operation. If preventive maintenance actions are recommended, the maintenance resource for failure mode 1 is c p × η∗ (t + n · t). Moreover, the decision-making process after the implementation of preventive maintenance for the (k + 1)-th time is exactly the same as that after the maintenance for the k-th time. Please refer to the algorithm for detailed cooperative predictive maintenance decision algorithm. Algorithm 12.2 Cooperative Predictive Maintenance Decision Algorithm Step 1: After the maintenance for the k-th time, let the system run for t L unit time to obtain enough performance degradation amount; Step 2: At the current time t = tk + t L , calculate the number of times that the failure mode i has failed before t and perform the reliability prediction to obtain Ri (tk+1 |t) according to the performance degradation measurements within [tk , t] [33]; Step 3: Implement Algorithm 12.1 and output N ∗ (t), T ∗ (t) and η∗ (t). Step 4: Make a reasonable decision according to the obtained optimal value: (a) (b) (c)

If T ∗ (t) − t L ≤ t and N ∗ (t) = 1, implement the replacement measures, and take k = 0, then return to Step 1; If T ∗ (t) − t L ≤ t, and N ∗ (t) > 1, then carry out preventive maintenance for the (k + 1)-th time. Then take k = k + 1, and return to Step 1; If T ∗ (t) − t L > t, then set t L = t L + t, and return to Step 2.

12.4 Numerical Simulation This subsection will study the simulation application of CPdM model proposed in the previous section in a system with two interdependent failure modes. Without loss of generality, it is assumed that no preventive maintenance has been carried out on the equipment since its installation and operation, that is k = 0. Therefore, tk = 0 and t − t L . In this simulation, the performance degradation measurements of two failure modes are generated by the following simulation model: φi (t) = φ0 + μi t + ωi (t)

(12.44)

where μ1 = 0.02, μ2 = 0.019, φ0 = 0.3, ωi (t) are Gaussian white noises, with the mean value of 0 and the variance of 0.042 and 0.0382 , respectively. Then, the exponential smoothing algorithm [33] is used to analyze the performance degradation amount associated with each failure mode, and then the reliability is predicted. Here, the failure thresholds corresponding to the two failure modes are φ1,th = 0.6 and

12.4 Numerical Simulation

265

Fig. 12.2 Reliability prediction for each failure mode given survival to t

φ2,th = 0.9, respectively. Then, according to the two cases that the system has run safely to, t L and t L + nt, the reliability prediction values of each failure mode in these two cases are given, respectively, and the specific prediction results are shown in Fig. 12.2. Among them, the sampling interval t is 0.01 unit time, t L = 3 unit time, and there is n = 40. Once the predicted reliability Ri (tk+1 |t) is obtained, the expected failure times can be estimated within [t, tk+1 ), while the expected failure times after tk+1 can be estimated by the failure rate function h 0,i (s). Here, assuming that the failure rate function has the following form αi h 0,i (s) = βi

%

s βi

&αi −1

where αi > 1. Take α1 = 2, α2 = 1.5, β1 = 1.05 and β2 = 1.02. Furthermore, the cost parameters associated with minimal repair, PM and the replacement are set, respectively, as cm,1 = 1.8, cm,2 = 2, c p = 45, cr,1 = 100 and cr,2 = 85. Parameters used to describe the correlation between failure modes are a1 = 1.006 and a2 = 1.007, respectively. The resource allocation ratio corresponding to failure mode 1 belongs to the set η = {0.1 × (l − 1), l = 1, · · · , 11}. Finally, other parameters are chosen to be b1 = 0.75 and b2 = 0.5. According to the predicted reliability corresponding to each failure mode, the optimal values at the current time t can be obtained by Algorithm 12.1 as N ∗ (t), T ∗ (t) and η∗ (t). Detailed results are shown in Figs. 12.3, 12.4, 12.5 and Table 12.1, respectively. Figures 12.3 and 12.4 show the results when N = 5, and the

266

12 Cooperative Predictive Maintenance of Two-Component …

Fig. 12.3 Curve of expected cost rate C(5, T, 0.5) versus T at t

Fig. 12.4 Curve of expected cost rate versus η at t

12.4 Numerical Simulation

267

Fig. 12.5 Curve of expected cost rate C(N , TN∗ ,η(N ) (t), η(N ) ) versus N at t

Table 12.1 Optimization results at different t t

N ∗ (t)

T ∗ (t)

η∗ (t)

C(N ∗ (t), T ∗ (t), η∗ (t))

3

5

5.01

0.7

32.1391

3.4

4

5.71

0.7

32.4618

3.8

4

5.67

0.7

34.3378

4.2

4

5.65

0.7

36.0363

5.33

4

5.34

0.6

46.6026

∗ minimum cost rate C(5, T5,η (t), η(5) ) can be found accordingly. Figure 12.3 shows (5) the trend of expected cost rate C(5, T, 0.5) changing with variables T when the current time is t = 3 and t = 3.4, respectively, while η = 0.5. It can be obtained from ∗ ∗ (t) = 5.01, C(5, T5,0.5 (t), 0.5) = 32.3313; Fig. 12.3: When t = 3, there are T5,0.5 ∗ ∗ when t = 3.4, there are T5,0.5 (t) = 5.4, C(5, T5,0.5 (t), 0.5) = 32.8099. Similarly, ∗ ∗ (t) and C(5, T5,η (t), η), which is shown in for other values of η, we can find T5,η ∗ Fig. 12.4. The minimum value C(5, T5,η(5) (t), η(5) )) can be obtained from Fig. 12.4. For other values of N, the corresponding minimum cost rate C(N , TN∗,η(N ) (t), η(N ) ) at time t can also be found. The details are as shown in Fig. 12.5. In this figure, the curve of C(N , TN∗ ,η(N ) (t), η(N ) ) changing with N (1 ≤ N ≤ 10) is given. Based on this, we can obtain the minimum cost rate C(N ∗ (t), T ∗ (t), η∗ (t)) at t and the corresponding optimal values N ∗ (t), T ∗ (t) and η∗ (t). The detailed optimal results for different time t are provided in Table 12.1. Through this table, it can be observed that the optimal value is constantly updated with the change of equipment health state. In addition, the frequency of maintenance will

268

12 Cooperative Predictive Maintenance of Two-Component …

accelerate with the increase of time. At t = 5.33, there exist T ∗ (t) − t L = 0.01 ≤ t, and N ∗ (t) = 4, which means that preventive maintenance must be carried out immediately. It can be seen from the above that the simulation results verify the effectiveness of the CPdM model proposed for equipment with interdependent failure modes.

12.5 Summary of This Chapter In this chapter, in view of the system with two-way influence failure mode, a cooperative predictive maintenance model is proposed to solve the problem of how to predict the expected failure times in the future according to the performance degradation amount obtained by real-time monitoring under the condition of limited maintenance resources, and on this basis, the limited maintenance resources are allocated reasonably, and the optimal preventive maintenance interval and the maximum number of times for preventive maintenance needed are determined, so as to minimize the expected cost rate in the remaining replacement cycle of the system. Different from the traditional model, this model is based on the framework of predictive maintenance and makes use of real-time reliability prediction information and system failure rate statistics information, which makes the decision result change with the actual health state of the system. It should be noted that the CPdM model proposed in this chapter has the assumption of periodic maintenance of equipment in the future. However, in fact, periodic maintenance itself has an essential disadvantage, that is, excessive and insufficient maintenance due to the fixed maintenance interval. Therefore, the next research direction is to cancel the limitation of equal preventive maintenance intervals and introduce the concept of sequential maintenance.

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