Nonlinear control is an important subject drawing much attention. When a nonlinear system undergoes route-to-chaos, its response is naturally bounded in the time-domain while in the meantime becoming unstably broadband in the frequency-domain. Control scheme facilitated either in the time- or frequency-domain alone is insufficient in controlling route-to-chaos, where the corresponding response deteriorates in the time and frequency domains simultaneously. It is necessary to facilitate nonlinear control in both the time and frequency domains without obscuring or misinterpreting the true dynamics. The objective of the dissertation is to formulate a novel nonlinear control theory that addresses the fundamental characteristics inherent of all nonlinear systems undergoing route-to-chaos, one that requires no linearization or closed-form solution so that the genuine underlying features of the system being considered are preserved. The theory developed herein is able to identify the dynamic state of the system in real-time and restrain time-varying spectrum from becoming broadband. Applications of the theory are demonstrated using several engineering examples including the control of a non-stationary Duffing oscillator, a 1-DOF time-delayed milling model, a 2-DOF micro-milling system, unsynchronized chaotic circuits, and a friction-excited vibrating disk. Not subject to all the mathematical constraint conditions and assumptions upon which common nonlinear control theories are based and derived, the novel theory has its philosophical basis established in the simultaneous time-frequency control, on-line system identification, and feedforward adaptive control. It adopts multi-rate control, hence enabling control over nonstationary, nonlinear response with increasing bandwidth ? a physical condition oftentimes fails the contemporary control theories. The applicability of the theory to complex multi-input-multi-output (MIMO) systems without resorting to mathematical manipulation and extensive computation is demonstrated through the multi-variable control of a micro-milling system. The research is of a broad impact on the control of a wide range of nonlinear and chaotic systems. The implications of the nonlinear time-frequency control theory in cutting, micro-machining, communication security, and the mitigation of friction-induced vibrations are both significant and immediate.

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A scaling model based on an effective Gol'dberg number is proposed for predicting the presence of cumulative nonlinear distortions in the acoustic waveforms produced by high-speed jets. Two acoustic length scales, the shock formation distance and the absorption length are expressed in terms of jet exit parameters. This approach allows one to compute the degree of cumulative nonlinear distortion in a full-scale scenario, from laboratory-scale observations, or vice versa. Surveys of the acoustic pressure waveforms emitted by a laboratory-scale, shock-free and unheated Mach 3 jet are used to support the findings of the model. These acoustic waveforms are acquired on a planar grid in an acoustically treated and range-restricted environment. Various statistical metrics are employed to examine the degree of local and cumulative nonlinearity in the measured waveforms and their temporal derivatives. This includes skewness, kurtosis, the number of zero crossings in the waveform, a wave steepening factor, the Morfey-Howell nonlinearity indicator and an application of the generalized Burgers equation. It is advocated that in order for the Morfey-Howell indicator to be used as an investigative tool for the presence of cumulative nonlinear waveform distortion, that it be applied as a multi-point indicator. Based on findings of the model and the spatial topography of the metrics, it is concluded that cumulative nonlinear steepening effects are absent in the current data set. This implies that acoustic shock-structures in the waveforms are generated by local mechanisms in, or in close vicinity to, the jet's hydrodynamic region. Furthermore, these shock-structures induce the crackle noise component. The research aims to quantify crackle in a temporal and spectral fashion, and is motivated by the fact that (1) it is perceived as the most annoying component of jet noise, (2) no unique measures of crackle exist, and (3) significant reductions in jet noise will be achieved when crackle can be controlled. A unique detection algorithm is introduced which isolates the shock-structures in the temporal waveform that are responsible for crackle. Ensemble-averages of the identified waveform sections are employed to gain an in-depth understanding of the crackling structures. Moreover, PDF's of the temporal intermittence of these shocks reveal modal trends and show evidence that crackling shock-structures are present in groups of multiple shocks. A spectral measure of crackle is considered by using wavelet-based time-frequency analyses. The increase in sound energy is computed by considering the global pressure spectra of the waveforms and the ones that represent the spectral behavior during instances of crackle. This energy-based metric is postulated to be an appropriate metric for the level of crackle. / text

Monitoring of engineered systems consists of characterizing the normal behavior of the system and tracking departures from it. Techniques to monitor a system can be split into two classes based on their use of inputs and outputs of the system. Systems-based monitoring refers to the case when both inputs and outputs of a system are available and utilized. Conversely, symptomatic monitoring refers to the case when only outputs of the system are available. This thesis extended symptomatic and systems-based monitoring of biomedical systems via the use of non-stationary signal processing and advanced monitoring methods. Monitoring of various systems of the human body is encumbered by several key hurdles. First, current biomedical knowledge may not fully comprehend the extent of inputs and outputs of a particular system. In addition, regardless of current knowledge, inputs may not be accessible and outputs may be, at best, indirect measurements of the underlying biological process. Finally, even if inputs and outputs are measurable, their relationship may be highly nonlinear and convoluted. These hurdles require the use of advanced signal processing and monitoring approaches. Regardless of the pursuit of symptomatic or system-based monitoring, the aforementioned hurdles can be partially overcome by using non-stationary signal analysis to reveal the way frequency content of biomedical signals change over time. Furthermore, the use of advanced classification and monitoring methods facilitated reliable differentiation between various conditions of the monitored system based on the information from non-stationary signal analysis. The human brain was targeted for advancement of symptomatic monitoring, as it is a system responding to a plethora internal and external stimuli. The complexity of the brain makes it unfeasible to realize system-based monitoring to utilize all the relevant inputs and outputs for the brain. Further, measurement of brain activity (outputs), in the indirect form of electroencephalogram (EEG), remains a workhorse of brain disorder diagnosis. In this thesis, advanced signal processing and pattern recognition methods are employed to devise and study an epilepsy detection and localization algorithm that outperforms those reported in literature. This thesis also extended systems-based monitoring of human biomedical systems via advanced input-output modeling and sophisticated monitoring techniques based on the information from non-stationary signal analysis. Explorations of system-based monitoring in the NMS system were driven by the fact that joint velocities and torques can be seen NMS responses to electrical inputs provided by the central nervous system (CNS) and the electromyograph (EMG) provides an indirect measurement of CNS excitations delivered to the muscles. Thus, both inputs and outputs of this system are more or less available and one can approach its monitoring via the use of system-based approaches. / text

This dissertation has been motivated by the increasing application of sensing technologies in structural health monitoring. Many wireless sensor techniques exist for structural health monitoring while a challenge faced is the finite lifetime of batteries. The objective of this dissertation is to develop passive wireless technology to provide early warning of conditions that damage the structure. In this dissertation, sensing mechanism is proposed based on time and frequency domain characteristics of magneto-inductive (MI) waves. Experimental results are also presented to demonstrate the sensing mechanism. MI waves are predominantly magnetic waves that are supported in periodic arrays of magnetically coupled resonators and propagate within a narrow frequency band around the resonant frequency. The array is to be embedded in a structure and different types of transducers can be integrated for different sensing applications. With the onset of structure defect, the transducer introduces an impedance discontinuity that generates reflected MI waves along the array, which are monitored and processed by Smoothed Wigner-Ville distribution (WVD) to extract time-of-flight for frequency components in the narrow passband. The transmission and reflection coefficients of MI waves are also investigated based on the lumped-element circuit model of the array. Based on MI waves travel time, amplitude and group velocity, the position and severity of structure defect are decided. The sensing mechanisms for different distribution of defects are proposed. The validity of the sensing mechanism is examined in experiments. The guided wave testing is implemented in one-dimensional square-shaped printed spiral resonators with Q-factor of 161 at 13.6 MHz. It demonstrates that low MI waves propagation loss is achieved with value of 0.098 dB per element at mid-band with center-to-center distance of half an inch. A pitch-catch measurement system is built to capture traveling MI signal in resonant element and extract group velocity, and a pulse-echo measurement system is designed to monitor reflected MI signal and locate structure discontinuity. In both measurement systems, MI waves are excited with wide bandwidth voltage pulse, and a digitizer is attached to sense the MI signal in a specific resonant element circuit. A baseline signal is obtained from the healthy state to use as reference and comparison with the test case using pitch-catch system. The test signal subtracted from baseline signal infers the structure damage information with time and frequency domain characteristics. It can offer an effective method to estimate the structure discontinuity location, severity and type of damage. The experimental results are consistent with the theoretical predictions. At the end, future directions for the research to integrate with other technologies are suggested. / text

The visual system receives a series of fluctuating light patterns on the retina, yet visual perception is strikingly different from this unorganized and ambiguous input. Thus visual processes must organize the input into coherent units, or objects, and segregate them from others. These processes, collectively called perceptual organization, are fundamental to our ability to perceive and interact with objects in the world. Nevertheless, they are not yet understood, perhaps because serial, hierarchical assumptions that were long held impeded progress. In a series of experiments, this dissertation investigated the mechanisms that contribute to perceptual organization and ultimately to our ability to perceive objects. A new hypothesis is that during the course of object assignment potential objects on either side of a border are accessed on a fast pass of processing and engage in inhibitory competition for object status; the winner is perceived as the object and the loser is suppressed, leading that region to be seen as part of the shapeless background. Previous research suggested that at least shape level representations are accessed on the fast pass of processing before object assignment. In the first series of experiments (Chapter 1), we found that meaning (semantics) is also accessed on the fast pass of processing for regions that are ultimately perceived as shapeless grounds. This finding contradicts traditional feed-forward theories of perception that assumed that meaning is accessed only for figures after object assignment. The experiments in Chapter 2 examine activity in the alpha band of the EEG, which has been used as an index of inhibition. More alpha activity was observed when participants viewed stimuli designed such that there was more competition for figural status from the region ultimately perceived as the ground. The results support the proposal that inhibitory competition occurs during the course of object perception, and these results are the first online measure of competition during figure assignment. The final series of experiments (Chapter 3) investigated how quickly saccadic behaviors that required perceptual organization can be initiated. The experiments show that participants can initiate saccades that are based on perceptual organization approximately 200 ms after stimulus onset, much faster than was assumed on feed-forward models of perception. Collectively, these experiment support models of object perception that involve the mutual interaction and competition of objects properties via feedforward and iterative feedback processing, and the eventual suppression of the losing ground regions before object assignment.

Time-frequency analysis for non-linear and non-stationary signals is extraordinarily challenging. To capture the changes in these types of signals, it is necessary for the analysis methods to be local, adaptive and stable. In recent years, decomposition based analysis methods were developed by different researchers to deal with non-linear and non-stationary signals. These methods share the feature that a signal is decomposed into finite number of components on which the time-frequency analysis can be applied. Differences lie in the strategies to extract these components: by iteration or by optimization. However, considering the requirements of being local, adaptive and stable, neither of these decompositions are perfectly satisfactory. Motivated to find a local, adaptive and stable decomposition of a signal, this thesis presents Adaptive Local Iterative Filtering (ALIF) algorithm. The adaptivity is obtained having the filter lengths being determined by the signal itself. The locality is ensured by the filter we designed based on a PDE model. The stability of this algorithm is shown and the convergence is proved. Moreover, we also propose a local definition for the instantaneous frequency in order to achieve a completely local analysis for non-linear and non-stationary signals. Examples show that this decomposition really helps in both simulated data analysis and real world application.

Time-frequency analysis is a powerful tool for signal analysis and processing. The Fourier transform and wavelet transforms are used extensively as is the Short-Time Fourier Transform (or Gabor transform). In 1996 the Stockwell transform was introduced to maintain the phase of the Fourier transform, while also providing the progressive resolution of the wavelet transform. The discrete orthonormal Stockwell transform is a more efficient, less redundant transform with the same properties. There has been little work on mathematical properties of the Stockwell transform, particularly how it behaves under operations such as translation and modulation. Previous results do discuss a resolution of the identity, as well as some of the function spaces that may be associated with it [2]. We extend the resolution of the identity results, and behaviour under translation, modulation, convolution and differentiation. boundedness and continuity properties are also developed, but the function spaces associated with the transform are unrelated to the focus of this thesis. There has been some work on image processing using the Stockwell transform and discrete orthonormal Stockwell transform. The tests were quite preliminary. In this thesis, we explore some of the mathematics of the Stockwell transform, examining properties, and applying it to various continuous examples. The discrete orthonormal Stockwell transform is compared directly with Newland’s harmonic wavelet transform, and we extend the definition to include varitions, as well as develop the discrete cosine based Stockwell transform. All of these discrete transforms are tested against current methods for image compression.

Time-frequency methods are widely used tools to diagnose planetary gearbox fault under non-stationary conditions. However, the existing time-frequency methods still have some problems, such as smearing effect and cross-term interference, and these problems limit the effectiveness of the existing time-frequency methods in planetary gearbox fault diagnosis under non-stationary conditions. To address the aforementioned problems, four time-frequency methods are proposed in this thesis. As nowadays a large portion of the industrial equipment is equipped with tachometers, the first three methods are for the cases that the shaft rotational speed is easily accessible and the last method is for the cases of shaft rotational speed is not easily accessible. The proposed methods are itemized as follows: (1) The velocity synchronous short-time Fourier transform (VSSTFT), which is a type of linear transform based on the domain mappings and short-time Fourier transform to address the smear effect of the existing linear transforms under known time-varying speed conditions; (2) The velocity synchrosqueezing transform (VST), which is a type of remapping method based on the domain mapping and synchrosqueezing transform to address the smear effect of existing remapping methods under known time-varying speed conditions; (3) The velocity synchronous bilinear distribution (VSBD), which is a type of bilinear distribution based on the generalized demodulation and Cohen’s class bilinear distribution to address the smear effect and cross-term interference of existing bilinear distributions under known time-varying speed conditions and (4) The velocity synchronous linear chirplet transform (VSLCT), which is a non-parametric combined approach of linear transform and concentration-index-guided parameter determination to provide a smear-free and cross-term-free TFR under unknown time-varying speed conditions. In this work, simple algorithms are developed to avoid the signal resampling process required by the domain mappings or demodulations of the first three methods (i.e., the VSSTFT, VST and VSBD). They are designed to have different resolutions, readabilities, noise tolerances and computational efficiencies. Therefore, they are capable to adapt different application conditions. The VSLCT, as a kind of linear transform, is designed for unknown rotational speed conditions. It utilizes a set of shaft-rotational-speed-synchronous bases to address the smear problem and it is capable to dynamically determine the signal processing parameters (i.e., window length and normalized angle) to provide a clear TFR with desirable time-frequency resolution in response to condition variations. All of the proposed methods in this work are smear-free and cross-term-free, the TFRs generated by the methods are clearer and more precise compared with the existing time-frequency methods. The faults of planetary gearboxes, if any, can be diagnosed by identifying the fault-induced components from the obtained TFRs. The four methods are all newly applied to fault diagnosis. The effectiveness of them has been validated using both simulated and experimental vibration signals of planetary gearboxes collected under non-stationary conditions.

This thesis presents results on the description of plasma waves in terms of wavepackets. The wave field is decomposed into a distribution of wavepackets in a space of position, wavevector, time, and frequency. A complex structure joining each pair of Fourier conjugate variables into a single complex coordinate allows the efficient derivation of equations of motion for the phase space distribution by exploiting its analytic properties. The Wick symbol calculus, a mathematical tool generalizing many convenient properties of the Fourier transform to a local setting, is used to derive new exact phase space equations which maintain full information on the phase of the waves and include effects nonlocal in phase space such as harmonic generation. A general purpose asymptotic expansion of the Wick symbol product formula is used to treat dispersion, refraction, photon acceleration, and ponderomotive forces. Examples studied include the nonlinear Schrödinger equation, mode conversion, and the Vlasov equation. The structure of partially coherent wave fields is understood in terms of zeros in the phase space distribution caused by dislocations in its complex phase which are shown to be correlated with the field entropy. Simulations of plasma heating by crossing electron beams are understood by representing the resulting plasma waves in phase space. The local coherence properties of the beam driven Langmuir waves are studied numerically.