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Estimation of Zeros from Response Measurements ESTIMATION OF ZEROS FROM RESPONSE MEASUREMENTS A Dissertation Presented by Omer Faruk Tigli to The Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Civil Engineering Northeastern University Boston, Massachusetts May, 2008 ABSTRACT This study examines conditions where one can estimate the zeros of transfer functions from response measurements. It is shown in the literature that the cepstrum can be exploited to estimate the poles and zeros of a system from its response measurements provided that the system is excited at a single point with an input which has a flat log- spectrum and the location of the input is known. Poles and zeros are estimated by curve- fitting an analytical expression of the transfer function to the region of the cepstrum where the contribution of the input is minimal. Since the source and input information cannot be separated completely anywhere in the cepstrum, the approach is always approximate even in the case of noise-free data. This dissertation improves the cepstrum technique such that the requirement on the characteristics of the input is completely eliminated provided that the number of measurements is more than one. Although the improved technique cannot estimate the poles, it can estimate the zeros exactly in the case of noise-free data. The main shortcoming of the cepstrum approach to the estimation of zeros in the unknown input case is the requirement of a single input. In many civil engineering applications systems are constantly excited by unknown forces such as wind, traffic and passengers that act at many locations so the cepstrum technique is not applicable. A second objective of the dissertation is to devise a technique which can estimate the zeros from response measurements in the case of multiple inputs. The technique that has been developed is for systems whose mass matrices can be approximated as diagonal. Assuming the measured responses are due to collocated forces, the technique makes use of state-space matrices A and C which are obtained from stochastic system identification and estimates a surrogate matrix for the missing input matrix (B) connected with a collocated distribution of forces. The approach works by forcing two constraints: (1) the off-diagonal terms of the mass matrix are zero, (2) the fact that there is no direct transmission term relating forces to velocity or displacement measurements, namely (CA-pB = 0). The technique yields exact results provided that mass matrix is diagonal and the system matrices A and C are not truncated and the number of sensors is larger than a certain threshold. Otherwise, the technique is approximate. i ACKNOWLEDGEMENTS I would like to express my sincere gratitude and appreciation to my advisor, Professor Dionisio Bernal for his continuous guidance during the course of this work. His technical knowledge, enthusiasm and imagination have been a constant source of encouragement for me. I would also like to thank Professors Shafai and Sznaier and Caracoglia for reading this dissertation and offering constructive comments. I would also like to extend my thanks to all Professors who have given me the education and supported me each in his own way: Prof. Yegian, Sasani, Furth, Sheahan and Wadia- Fascetti. I must also thank to my friends Yalcin, Eric, Ece, Marlon, Arash, Markus, Mustafa, Orcun, Hasan, Emrah especially to Serkan for their continuous support. Finally, I would like to thank my family for their life-long love and support. I especially thank to my wife, Gulhan, who made the most difficult times a joy for me. ii TABLE OF CONTENTS 1 INTRODUCTION..................................................................................................... 1 1.1 Research Context................................................................................................ 1 1.2 Motivation........................................................................................................... 3 1.3 Objectives ........................................................................................................... 7 1.4 Contributions of the Dissertation........................................................................ 8 1.5 Organization of the Dissertation ......................................................................... 8 2 PRELIMINARIES.................................................................................................. 10 2.1 Theoretical Background.................................................................................... 10 2.1.1 Continuous-Time State-Space Description............................................... 10 2.1.1.1 Decoupling and the Canonical Forms................................................... 12 2.1.2 Discrete-Time State-Space Description.................................................... 14 2.1.3 Transfer Function...................................................................................... 15 2.1.4 Transfer Function in Discrete Systems..................................................... 16 2.1.4.1 Z-Transform.......................................................................................... 16 2.1.4.2 Pulse Transfer Function ........................................................................ 17 2.1.4.3 Pulse Transfer Function from State-Space Matrices ............................ 18 2.1.4.4 Pulse Transfer Function Sampled from Continuous Transfer Function18 2.2 Zeros of Multivariable Systems........................................................................ 19 2.2.1 Invariant Zeros.......................................................................................... 20 2.2.1.1 A Comment on the Complex Input and Initial Condition..................... 22 2.2.1.2 Invariant Transformations..................................................................... 23 2.2.2 Transmission Zeros................................................................................... 25 2.2.3 Decoupling Zeros...................................................................................... 25 2.2.3.1 Input Decoupling Zeros ........................................................................ 26 2.2.3.2 Output Decoupling Zeros...................................................................... 26 2.2.3.3 Input-Output Decoupling Zeros............................................................ 27 2.2.4 Relationship between Invariant, Transmission and Decoupling Zeros .... 27 2.2.5 Invariant Zeros of Discrete-Time Systems ............................................... 30 2.2.6 Connection between the Continuous- and Discrete- Time Zeros ............. 30 2.3 Summary........................................................................................................... 38 3 ESTIMATION OF ZEROS FROM RESPONSE MEASUREMENTS IN SIMO SYSTEMS: CEPSTRUM TECHNIQUE...................................................................... 39 3.1 Introduction....................................................................................................... 39 3.2 Theoretical Background.................................................................................... 41 3.2.1 Complex Cepstrum................................................................................... 41 3.2.2 Power Cepstrum........................................................................................ 44 3.2.3 The Connection between Complex and Power Cepstra............................ 47 3.3 Practical Implementation of the Cepstrum Concept ......................................... 47 3.3.1 Complex and Power Cepstra of Sequences .............................................. 48 3.3.2 Differential Cepstrum............................................................................... 49 3.3.3 Pulse Transfer function and its Cepstrum................................................. 50 3.3.4 Curve-fitting the Cepstrum ....................................................................... 53 3.3.4.1 The Contribution of Discretization Zeros to the Cepstrum .................. 58 iii 3.3.4.2 Which Cepstrum is More Convenient For Curve-Fitting? ................... 60 3.3.4.3 Parameter Estimation from Differential Cepstrum............................... 65 3.3.5 The Cepstrum of Noisy Measurements..................................................... 70 3.3.5.1 Parameter Estimation from Cepstrum of the Noisy Measurements...... 77 3.4 Summary........................................................................................................... 82 4 MODIFIED CEPSTRUM TECHNIQUE............................................................. 84 4.1 Theoretical Background.................................................................................... 84 4.2 Implementation Issues...................................................................................... 87 4.2.1 De-coupling Zeros.................................................................................... 87 4.2.2 Zero-Zero Cancellation............................................................................. 87 4.2.3 Marginal Pole-Pole Cancellation .............................................................. 88 4.2.4 The effect of Noise to the Proposed Technique........................................ 89 4.3 Summary........................................................................................................... 99 5 ESTIMATION OF ZEROS FROM RESPONSE MEASUREMENTS OF MIMO SYSTEMS........................................................................................................
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