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Singular Value Decomposition Based Model Order Reduction Techniques

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Singular Value Decomposition Based Model Order Reduction Techniques Singular Value Decomposition Based Model Order Reduction Techniques by Ahmad Jazlan Bin Haja Mohideen A thesis submitted to the School of Electrical, Electronic and Computer Engineering in partial fulfilment of the requirements for the degree of Doctor of Philosophy Faculty of Engineering, Computing and Mathematics University of Western Australia September 2016 Statement of Originality The content of this thesis are the results of original research and have not been submitted for a higher degree at any other institution. The content of Chapter 3 are currently under review for publication in an international refereed journal. The content of Chapter 4 and Chapter 5 have been published in international ref- ereed journals. The content of Chapter 6 and Chapter 8 have been published in international refereed conference proceedings. The content of Chapter 7 have been presented at an international refereed conference and will appear in the conference proceedings. Published International Refereed Journal Papers 1. Jazlan, A., Sreeram, V., Shaker, H.R.,Togneri, R. and Minh, H.B., Frequency Interval Cross Gramians for Linear and Bilinear Systems, Asian Journal of Control, Vol. 19, No. 1, pp. 22-34, (2016) - Chapter 4 2. Jazlan, A., Sreeram, V., Shaker, H.R. and Togneri, R., Frequency interval balanced truncation of discrete-time bilinear systems, Cogent Engineering, Vol. 3, No. 1, pp. 1-15, (2016) - Chapter 5 Journal Paper Submission Currently Under Review 1. Jazlan, A., Houlis, P., Sreeram, V. and Togneri, R., Comments on \A Parametrized Controller Reduction Technique via a New Frequency Weighted Model Reduc- tion Formulation \- Under Review by Asian Journal of Control - Chapter 3 International Refereed Conference Papers 1. Jazlan, A., Sreeram, V., Togneri, R. and Bettayeb, M., Time weighted model reduction of flat plate solar collectors, In: Proceedings of 4th Australian Control Conference, pp. 107-111, Canberra, ACT (2014) - Chapter 6 2 2. Jazlan, A., Sreeram, V., Togneri, R. and Minh, H.B., Generalized Gramian Based Frequency Interval Model Reduction for Unstable Systems, In: Pro- ceedings of 6th Australian Control Conference, Newcastle (2016) - Chapter 7 3. Kumar, D., Jazlan, A., Sreeram, V. and Togneri, R., Partial Fraction Expan- sion Based Frequency Weighted Balanced Singular Perturbation Approxima- tion Model Reduction Technique with Error Bounds, In: Proceedings of the 6th IFAC Symposium on System Structure and Control, Vol. 49, No. 9, pp. 45-50, Istanbul (2016) - Chapter 8 The following papers represent related work which has been carried out to comple- ment the work presented in the chapters of this thesis, however these papers have not been included as part of this thesis for brevity. Related Journal and Conference Papers 1. Kumar, D., Jazlan, A., Sreeram, V. and Togneri, R., Partial Fraction Expan- sion based Frequency Weighted Model Reduction for Discrete Time Systems, Numerical Algebra, Control and Optimization, Vol. 6, No. 3, pp. 329-337 - Further work which complements Chapter 8. 2. Jazlan, A., Sreeram, V. and Togneri, R., Cross gramian based time interval model reduction, In: Proceedings of the 5th Australian Control Conference, pp. 274-276, Gold Coast, Queensland (2015) - Further work which complements Chapter 4. 3. Du, X., Jazlan, A., Sreeram, V., Togneri, R., Ghafoor, A. and Sahlan, S., A frequency limited model reduction technique for linear discrete systems, In: Proceedings of the 3rd Australian Control Conference, pp.421-426, Fremantle, Western Australia (2013) - Further work which complements Chapter 2. 3 4. Jazlan, A., Sreeram, V., Togneri, R., Mousa, W., A review on reduced order approximation for digital filters with complex coefficients using model reduc- tion, In Proceedings of the 3rd Australian Control Conference, pp. 79-84, Fre- mantle, Western Australia (2013) - Further work which complements Chapter 2. My doctoral studies were conducted under the guidance of Professor Dr. Victor Sreeram as my main supervisor and Professor Dr. Roberto Togneri as my co- supervisor. This research described in this thesis is the result of a collaborative effort with Professor Dr. Victor Sreeram, Professor Dr. Roberto Togneri, Associate Professor Dr. Hamid Reza Shaker from the University of Southern Denmark, Dr Pantazis Houlis, Dr. Ha Binh Minh from Banking University in Vietnam, Professor Dr. Maamar Bettayeb from Sharjah University in the United Arab Emirates and Dr Deepak Kumar from Motilal Nehru National Institute of Technology in India. The collaborators provided expertise and guidance and the main contribution to this work is my own. Ahmad Jazlan Bin Haja Mohideen School of Electrical, Electronic and Computer Engineering The University of Western Australia Crawley, W.A. 6009 Australia 4 Acknowledgements First and foremost I would like to thank my main supervisor, Professor Dr. Victor Sreeram and co-supervisor Professor Dr. Roberto Togneri for giving me the opportunity to pursue my PhD studies under their supervision. Without their guidance and support I would not have been able to accomplish this major milestone. I would also like to thank the Malaysian Ministry of Higher Education and the International Islamic University Malaysia for giving me the opportunity to pursue my studies in one of the Group of Eight Australian universities. My sincere ap- preciation also goes to the University of Western Australia for providing me with a six month Scholarship for International Research Fees (SIRF) which has helped me greatly in completing this thesis. I would like to thank Associate Professor Dr. Asan Gani Abdul Muthalif, As- sociate Professor Dr. Shahrul Naim Sidek and all the staff of the Mechatronics Engineering Department at the International Islamic University Malaysia for sup- porting me throughout my stint as a graduate research student. I would also like to thank all the staff at the Kulliyah of Engineering, International Islamic University Malaysia. I look forward to producing high quality research and teaching upon my return. I am deeply thankful to both my parents Dr. Haja Mohideen Bin Mohamed Ali and Samsunisha Yusof who have always been the pillars of strength in my life and have supported me throughout the ups and downs in my life. This thesis is dedicated to both of them. To my siblings and their spouses - Shamimah, Kamal Bacha Ibrahim, Mohamed Zafran and Nur Haslinda Harun, thank you for motivating me to achieve more each day. To my adorable nephews Mohammad Fauzan Irfan Bin Kamal Bacha and Zahin Azfar Bin Mohamed Zafran thank you for adding more happiness in my life. I have gained a comprehensive understanding about model order reduction from many discussions with Dr. Shafishuhaza Sahlan and Dr. Wan Mariam Wan Muda. 5 Both of your ideas have inspired me to look at model order reduction from various angles and beyond theoretical aspects. I have received strong support and encouragement from the Malaysian commu- nity in Perth. Thank you Safwan Samsuddin and Sharlinda Talib for supporting me during my early days in Perth. Thank you to all members of the Malaysian Post- graduate Students Association (MYPSA). Thank you Hasan Firdaus, Khairuzzaman Saudi, Mohd Hafiz Kamaruddin, Ahmad Husaini, Mohd Hafizi Abdul Rahman, Ku Ashman Ku Aziz, Azrul Azmi, Zainol Izwan Zulkifli, Ahmad Syakir, Jaffri Nasir and Ming Fook Lim. 6 Abstract This thesis investigates the development of new singular value decomposition based model order reduction techniques. Firstly a solution to the controller reduction problem via a parameterized double- sided frequency weighted controller reduction technique is developed for the feedback control of MIMO discrete time systems particularly for non-unity feedback control system configurations which have the controller located in the feedback path. Secondly, new frequency interval cross gramians are developed for both linear and bilinear systems. New generalized Sylvester equations for calculating the fre- quency interval cross gramians are derived in order to be used to obtain information regarding controllability and observability of a system within a single matrix and therefore leading to increased computational efficiency. Thirdly, a new model reduction method for discrete-time bilinear systems is developed based on newly derived frequency interval controllability and observability gramians for discrete-time bilinear systems. Fourthly, time weighted cross gramians are developed and applied to a state space model derived from two partial differential equations representing the thermal dynamics of a flat plate solar collector. Fifthly, generalized frequency interval controllability and observability gramians are introduced in order to obtain the frequency interval controllability and observ- ability gramians for continuous-time linear systems which do not have a solution to the standard Lyapunov equation. Finally, a new frequency weighted partial fraction expansion based model re- duction technique is developed based on the partial fraction expansion approach. In order to further reduce the frequency weighted approximation error, singular perturbation approximation is incorporated into the algorithm. 7 Contents Acknowledgements 5 Abstract 7 List of Tables 15 List of Figures 16 1 Introduction 18 1.0.1 Organization and Contribution . 20 2 Singular Value Decomposition Based Model Order Reduction Tech- niques: A Review 25 2.1 Preliminaries . 25 2.1.1 Singular Value Decomposition . 25 2.1.2 Balanced Truncation Method . 26 2.1.3 Balanced Singular Perturbation Approximation . 29 2.1.4 Controller Reduction . 32 2.1.4.1 Closed-loop Transfer Functions Obtained by Consid- ering Frequency Weightings . 32 2.2 Frequency Weighted Model Reduction Technique . 34 2.2.1 Frequency Weighting . 34 2.2.2 Enns Method . 35 2.2.3 Lin and Chiu's Technique . 36 8 2.2.4 Varga and Anderson's modification to Lin and Chiu's Technique 38 2.2.5 Wang, Sreeram and Liu's Technique . 39 2.2.6 Varga and Anderson's modification to Wang, Sreeram and Liu's Technique . 40 2.2.7 Frequency Interval Gramian Based Model Order Reduction . 42 2.2.7.1 Gawronski and Juang's Method . 42 2.2.7.2 Gugercin and Antoulas's Method . 44 2.3 Time Weighted Model Reduction . 45 2.4 Bilinear Systems . 47 2.4.1 Balanced Truncation of Bilinear Systems .
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