Hilbert Transform applications in signal analysis and non-parametric identification of linear and nonlinear systems

Hilbert Huang Transform faces several challenges in dealing with closely-spaced frequency components, short-time and weak disturbances, and interrelationships between two time-varying modes of nonlinear vibration due to its mixed mode problem associated with empirical mode decomposition (EMD). To address these challenges, analytical mode decomposition (AMD) based on Hilbert Transform is proposed and developed for an adaptive data analysis of both stationary and non-stationary responses. With a suite of predetermined bisecting frequencies, AMD can analytically extract the individual components of a structural response between any two bisecting frequencies and function like an adaptive bandpass filter that can deal with frequency-modulated responses with significant frequency overlapping. It is simple in concept, rigorous in mathematics, and reliable in signal processing. In this dissertation, AMD is studied for various effects of bisecting frequency selection, response sampling rate, and noise. Its robustness, accuracy, efficiency, and adaptability in signal analysis and system identification of structures are compared with other time-frequency analysis techniques such as EMD and wavelet analysis. Numerical examples and experimental validations are extensively conducted for structures under impulsive, harmonic, and earthquake loads, respectively. They consistently demonstrate AMD\u27s superiority to other time-frequency analysis techniques. In addition, to identify time-varying structural properties with a narrow band excitation, a recursive Hilbert Huang Transform method is also developed. Its effectiveness and accuracy are illustrated by both numerical examples and shake table tests of a power station structure --Abstract, page iii

Novel MARG-Sensor Orientation Estimation Algorithm Using Fast Kalman Filter

Orientation estimation from magnetic, angular rate, and gravity (MARG) sensor array is a key problem in mechatronic-related applications. This paper proposes a new method in which a quaternion-based Kalman filter scheme is designed. The quaternion kinematic equation is employed as the process model. With our previous contributions, we establish the measurement model of attitude quaternion from accelerometer and magnetometer, which is later proved to be the fastest (computationally) one among representative attitude determination algorithms of such sensor combination. Variance analysis is later given enabling the optimal updating of the proposed filter. The algorithm is implemented on real-world hardware where experiments are carried out to reveal the advantages of the proposed method with respect to conventional ones. The proposed approach is also validated on an unmanned aerial vehicle during a real flight. Results show that the proposed one is faster than any other Kalman-based ones and even faster than some complementary ones while the attitude estimation accuracy is maintained

A Signal Decomposition Theorem with Hilbert Transform and Its Application to Narrowband Time Series with Closely Spaced Frequency Components

Empirical mode decomposition and Hilbert spectral analysis have been extensively studied in recent years for the system identification of structures. It often encounters three challenges: (1) unable to decompose a signal with closely spaced frequency components such as wave groups in ocean engineering and beating responses in structural and mechanical systems, (2) difficult to distinguish the frequency components in a narrowband signal that is commonly seen in the free vibration of structures, and (3) unable to separate small intermittent fluctuations from a large wave. In this paper, a new analytical mode decomposition theorem based on the Hilbert Transform of a harmonics multiplicative time series is developed to address the challenges. The theorem can be applied with two procedures based on the decomposition of the original signal only or the previously decomposed (modified) signals in sequence. Numerical examples for four representative engineering applications indicate that the new theorem is superior to existing methods in decomposing a time series into many signals whose Fourier spectra are non-vanishing over mutually exclusive frequency ranges separated by bisecting frequencies. It is simple in concept, efficient in computation, consistent in performance, and reliable in signal processing. The accuracy of the new theorem is insensitive to noise. The discernable frequency spacing between the dominant frequencies of decomposed signals is theoretically near zero but practically equal to twice the frequency resolution of a finite length time series. Each bisecting frequency can be selected as an average of its two nearby frequencies of interest and is insensitive to other choices between 80% and 120% of the average value. The modified signal decomposition procedure can be less accurate than the original signal decomposition procedure due to potentially accumulated numerical errors in Hilbert transforms

Recursive Hilbert-Huang Transform Method for Time-Varying Property Identification of Linear Shear-Type Buildings under Base Excitations

This paper presents a recursive Hilbert-Huang transform method for the time-varying property identification of shear-type buildings under base excitations. To overcome nonorthogonality and modal perturbation issues, all significant intrinsic mode functions of each signal and their Hilbert transforms were summed to track any variation of structural parameters of a multistory building over time. Given floor masses, both the stiffness and damping coefficients of the building were identified one by one from the top to bottom story. The overall accuracy of the identified parameters was measured by an index of accuracy based on the weighted root-mean-squared evaluation proposed in this study. One- and two-story shear buildings with abruptly, gradually, and periodically varying parameters were used as examples. The numerical results indicated that the proposed method is efficient, robust, and accurate in tracking variations of the properties of multistory buildings. Finally, the proposed method was applied into the identification of the time-varying natural frequency of a real-world high-voltage switch structure due to the friction mechanism used in the switch. The range of the identified frequency by the proposed method was in good agreement with that attained by the conventional least-squares method

A Moving-Window Least Squares Fitting Method for Crack Detection and Rigidity Identification of Multispan Bridges

In this study, a moving-window least squares fitting method is proposed for rapid identification of cracks and flexural rigidities in multispan bridges. First, the dynamic deflections of a continuous bridge were locally measured under a dynamic point load. Their integrations over time, referred to as \u27integration-over-time deflections\u27, were used to derive \u27integration-over-time slopes\u27. These virtually static measurements over a short segment of the bridge were then fitted into a cubic curve in the least squares sense. Finally, the coefficient of the square term of the fitted curve was used to determine both the magnitude and location of local flexibility because of cracking and/or changing in flexural rigidity of the bridge. For multispan continuous bridges, an iterative procedure was developed to ensure that the end moments of various spans are compatible with the identified cracks and rigidity changes. To illustrate the proposed method, prismatic girder bridges with multiple cracks of various depths or non-prismatic girder bridges were analyzed. Sensitivity analysis was conducted on the effects of weighting factor, noise level, load type, window length, and bridge discretization. Numerical results demonstrated that the proposed method can accurately detect cracks and identify the change in flexural rigidity. The five-point equally weighted algorithm is recommended for practical applications. The spacing of two discernible cracks is equal to the window length. The identified results are insensitive to noise because of integration of the dynamic measurements