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CONCORDIA UNIVERSITY, School of Graduate Studies, This is to certify that the thesis proposal prepared. By Esmaeil Naderi, Entitled Fault Diagnosis and Estimation of Dynamical Systems with. Application to Gas Turbines, and submitted in partial fulfilment of the requirements for the degree of. Doctor of Philosophy, complies with the regulations of this University and meets the accepted standards. with respect to originality and quality ,Signed by the final examining committee .

Dr Jiayuan Yu Chair, Dr Farid Golnaraghi External examiner. Dr Fariborz Haghighat External examiner, Dr Shahin Hashtrudi Zad Examiner. Dr Wei Ping Zhu Examiner, Dr Khashayar Khorasani Supervisor. Approved by, Chair of the ECE Department, Dean of Engineering. ABSTRACT, Fault Diagnosis and Estimation of Dynamical Systems with Application to Gas Tur .

bines,Esmaeil Naderi Ph D ,Concordia Unviersity 2016. This thesis contributes and provides solutions to the problem of fault diagnosis and esti . mation from three different perspectives which are i fault diagnosis of nonlinear systems using. nonlinear multiple model approach ii inversion based fault estimation in linear systems and iii . data driven fault diagnosis and estimation in linear systems The above contributions have been. demonstrated to the gas turbines as one of the most important engineering systems in the power. and aerospace industries , The proposed multiple model approach is essentially a hierarchy of nonlinear Kalman filters. utilized as detection filters A nonlinear mathematical model for a gas turbines is developed and. verified The fault vector is defined using the Gas Path Analysis approach The nonlinear Kalman. filters that correspond to the defined single or concurrent fault modes provide the conditional. probabilities associated with each fault mode using the Bayes law The current fault mode is. then determined based on the maximum probability criteria The performance of both Extended. Kalman Filters EKF and Unscented Kalman Filters UKF are investigated and compared which. demonstrates that the UKF outperforms the EKF for this particular application . The problem of fault estimation is increasingly receiving more attention due to its practical. importance Fault estimation is closely related to the problem of linear systems inversion This. thesis includes two contributions for the stable inversion of non minimum phase systems First . a novel methodology is proposed for direct estimation of unknown inputs by using only measure . ments of either minimum or non minimum phase systems as well as systems with transmission. zeros on the unit circle A dynamic filter is then identified whose poles coincide with the transmis . sion zeros of the system A feedback is then introduced to stabilize the above filter dynamics as. well as provide an unbiased estimation of the unknown input The methodology is then applied to. iii, the problem of fault estimation and has been shown that the proposed inversion filter is unbiased. for certain categories of faults Second a solution for unbiased reconstruction of general inputs is. proposed It is based on designing an unknown input observer UIO that provides unbiased esti . mation of the minimum phase states of the system The reconstructed minimum phase states serve. then as inputs for reconstruction of the non minimum phase states The reconstruction error for. non minimum phase states exponentially decrease as the estimation delay is increased Therefore . an almost perfect reconstruction can be achieved by selecting the delay to be sufficiently large . The proposed inversion scheme is then applied to the output tracking control problem . An important practical challenge is the fact that engineers rarely have a detailed and accu . rate mathematical model of complex engineering systems such as gas turbines Consequently one. can find a trend towards data driven approaches in many disciplines including fault diagnosis In. this thesis explicit state space based fault detection isolation and estimation filters are proposed. that are directly identified from only the system input output I O measurements and through. the system Markov parameters The proposed procedures do not involve a reduction step and. do not require identification of the system extended observability matrix or its left null space . Therefore the performance of the proposed filters is directly connected to and linearly dependent. on the errors in the Markov parameters estimation process The estimation error dynamics is then. derived in terms of the Markov parameters identification errors and directly synthesized from the. healthy system I O data Consequently the estimation errors have been effectively compensated. for The proposed data driven scheme requires the persistently exciting condition for healthy in . put data which is not practical for certain real life applications and in particular to gas turbine. engines To address this issue a robust methodology for Markov parameters estimation using fre . quency response data is developed Finally the performance of the proposed data driven approach. is comprehensively evaluated for the fault diagnosis and estimation problems in the gas turbine. engines , iv, ACKNOWLEDGEMENTS, I would like to greatly thank my supervisor Prof Khorasani for the patient. guidance encouragement and advice he has provided throughout my time as his. student I have been extremely lucky to have a supervisor who cared so much about. my work and who devotes astonishing effort to improve his students works . v, TABLE OF CONTENTS,List of Figures x,List of Tables xvi.

Nomenclature xx,1 Introduction 1, 1 1 Introduction 1. 1 2 Multiple Model Based Fault Detection and Isolation 7. 1 3 Inversion Based Fault Estimation 9, 1 4 Inversion Based Reconstruction of System States and General Un . known Input 11, 1 5 Data Driven Fault Detection Isolation and Estimation 13. 1 6 Fault Diagnosis of Gas Turbines 17, 1 7 Contributions of the Thesis 20. 1 8 Structure of the Thesis 24,2 Background 25, 2 1 Physics of the Gas Turbines 25.

2 2 The Gas Turbine Mathematical Model 27, 2 3 Inversion Based Fault Estimation for Minimum Phase Systems 33. 2 3 1 General Form of the Sain Massey Algorithm 34. 2 3 2 Application to Gas Turbine 39, 3 Nonlinear Multiple Model Based FDI of Gas Turbines 46. 3 1 MM Based FDI Algorithm 47, 3 2 Multiple Model Based Fault Diagnosis Design 51. 3 2 1 Fault Modeling and Detection Filter Design 52. vi, 3 3 Simulation Results 54, 3 3 1 Single Fault Scenarios 55. 3 3 2 Concurrent Fault Scenarios 58, 3 3 3 Operational Condition Variations 61.

3 3 4 A Comparison Between the Performance of the UKF and the. EKF Detection Filters 64, 3 4 Conclusions 67,4 Inversion Based Fault Estimation 69. 4 1 Problem Statement 70, 4 1 1 Problem 1 Inversion Based Input Estimation of Discrete . Time Linear Systems 70, 4 1 2 Problem 2 Inversion Based Fault Estimation of Discrete Time. Linear Systems 71, 4 2 Notations 72, 4 3 The Proposed Inversion Based Input Estimation of Linear Systems 73. 4 3 1 Linear Systems With No Invariant Zeros 74, 4 3 2 Minimum Phase Linear Systems 77.

4 3 3 Non Minimum Phase Systems 82, 4 4 The Proposed Inversion Based Fault Estimation for Non Minimum. Phase Fault to Output Systems 91, 4 5 Four Case Studies 93. 4 6 Conclusion 99, 5 Reconstruction of System States and General Unknown Inputs and. Faults 100, 5 1 Problem Statement 101, 5 2 State and Unknown Input Reconstruction 103. vii, 5 2 1 Partial Or Full Estimation of the System States 103.

5 2 2 Partitioning of the States 108, 5 2 3 Dynamics of the MP and NMP States 110. 5 2 4 Estimation of the NMP States 115, 5 3 Inversion Based Output Tracking 120. 5 4 Extension to Fault Estimation 123, 5 5 Numerical Case Study Simulations 127. 5 6 Conclusion 132, 6 Data Driven Fault Detection Isolation and Estimation 135. 6 1 Preliminaries 136, 6 2 Proposed Fault Estimation Scheme Using Exact Markov Parameters.

and Observability Matrix 140, 6 3 Data Driven Fault Detection and Isolation FDI Scheme 144. 6 3 1 Data Driven Estimation of the Filter Parameters 145. 6 3 2 Fault Detection and Isolation Filters 149, 6 3 3 Residual Dynamics In Presence of A Fault 152. 6 4 The Proposed Fault Estimation Scheme 153, 6 4 1 Sensor Fault Estimation Filters 154. 6 4 2 Actuator Fault Estimation Filters 158, 6 4 3 A Methodology for Solving the Minimization Problems 159. 6 5 Simulation Results 160, 6 6 Conclusion 170, 7 Implementation and Application of the FDI E Methodologies to.

Gas Turbines 171, 7 1 Preliminaries 172, 7 2 Identification of the Aircraft Gas Turbine Engine Markov Parameters 173. viii, 7 3 Simulation Case Studies 177, 7 4 Conclusion 185. 8 Summary 196,9 Conclusions and Future Work 200, 9 1 Conclusion 200. 9 2 Future Work 203,References 205,A 223, A 1 Validity of the Estimated Markov Parameters 223. A 2 Markov Parameter Estimation Error for Different PLA 224. A 3 Confusion Matrix Analysis for PLA 55 226, A 4 Analysis On the Selection of the Parameter E 230.

ix, List of Figures, 1 1 A schematic representation of the inverse based fault estimation scheme 5. 1 2 Historical development of FDI schemes 5,2 1 A schematic of gas turbine 26. 2 2 Information flow diagram in a modular modeling of the gas turbine. dynamics 28, 2 3 Steady state series at PLAs ranging from 0 4 to 1 32. 2 4 A schematic representation of the gas turbine inverse based fault es . timation scheme The inherent delay of the gas turbine model is 1. i e L 1 For a description of notations refer to Figure 1 1 42. 2 5 Residuals that are generated by the fault estimation scheme under. four distinct simulation scenarios 44, 2 6 Residuals generated by the fault estimation scheme in presence of. concurrent faults 45, 3 1 General architecture of our proposed MM based FDI scheme 51.

3 2 The mode probabilities corresponding to the injected 2 decrease in. the compressor efficiency that is applied at t 5 seconds Mode 3 . a the UKF is used in the MM based FDI scheme and b the EKF. is used in the MM based FDI scheme 56, 3 3 The output measurements corresponding to the injected 2 decrease. in the compressor efficiency that is applied at t 5 seconds Mode. 3 56, x, 3 4 The mode probabilities corresponding to the injected 3 decrease in. the turbine efficiency that is applied at t 5 seconds Mode 5 a . the UKF is used in the MM based FDI scheme and b the EKF is. used in the MM based FDI scheme 57, 3 5 The detection time for each mode of fault that is applied at t 5. seconds as a function of the fault severity a the UKF is used in the. MM based FDI scheme and b the EKF is used in the MM based. FDI scheme 58, 3 6 The mode probabilities corresponding to the injected 2 59. 3 7 The output measurements corresponding to the injected 2 59. 3 8 The mode probabilities corresponding to the injected 2 decrease in. the turbine mass flow rate that is applied at t 5 seconds Mode 4 . while the ambient temperature is varying a the UKF is used in the. MM based FDI scheme and b the EKF is used in the MM based. FDI scheme 62, 3 9 The output measurements corresponding to the injected 2 decrease.

in the turbine mass flow rate that is applied at t 5 seconds Mode. 4 while the ambient temperature is varying 62, 3 10 The mode probabilities corresponding to the injected 2 decrease in. the turbine mass flow rate that is applied at t 5 seconds Mode. 4 while the PLA is varying a the UKF is used in the MM based. FDI scheme and b the EKF is used in the MM based FDI scheme 63. 3 11 The output measurements corresponding to the injected 2 decrease. in the turbine mass flow rate that is applied at t 5 seconds Mode. 4 while the PLA is varying 63, xi, 3 12 The detection time for each mode of fault that is applied at t 5. seconds as a function of the noise power factor The empty places. indicates the unsuccessful detection or isolation of the corresponding. fault a the UKF is used in the MM based FDI scheme and b the. EKF is used in the MM based FDI scheme 66, 3 13 The detection time for each mode of fault that is applied at t 5. seconds as a function of the number of the measurements or sensors. that are employed a the UKF is used in the MM based FDI scheme . and b the EKF is used in the MM based FDI scheme 66. 4 1 A graphical illustration of the C2M and H2M spaces Note that pro . jections of Y2M D2M U2M and D2M Uaux, 2M onto the row space of H2M. are identical 77, 4 2 Input estimation for the system 4 67 using two different rotation.

5 45 , matrices a 180, and b 180, 96, 4 3 The LOE fault input estimation of the MIMO non minimum phase. system 4 70 97, 4 4 The LOE fault estimation for the system 4 71 98. 4 5 Input estimation for system 4 72 99, 5 1 Upper bound of the NMP state estimation error versus nd 128. 5 2 The estimation of the MP state by utilizing the filter 5 18 129. 5 3 The estimation of the NMP state the graphs are shifted by nd n. time steps to the left for the purpose of comparison by utilizing the. filter 5 39 129, 5 4 The estimation of the unknown input the graphs are shifted by nd n. Fault Diagnosis and Estimation of Dynamical Systems with Application to Gas Tur bines Esmaeil Naderi Ph D Concordia Unviersity 2016 This thesis contributes and provides solutions to the problem of fault diagnosis and esti mation from three di erent perspectives which are i fault diagnosis of nonlinear systems using

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