Non-parametric linear time-invariant system identification by discrete wavelet transforms

We describe the use of the discrete wavelet transform (DWT) for non-parametric linear time-invariant system identification. Identification is achieved by using a test excitation to the system under test (SUT) that also acts as the analyzing function for the DWT of the SUT's output, so as to recover the impulse response. The method uses as excitation any signal that gives an orthogonal inner product in the DWT at some step size (that cannot be 1). We favor wavelet scaling coefficients as excitations, with a step size of 2. However, the system impulse or frequency response can then only be estimated at half the available number of points of the sampled output sequence, introducing a multirate problem that means we have to 'oversample' the SUT output. The method has several advantages over existing techniques, e.g., it uses a simple, easy to generate excitation, and avoids the singularity problems and the (unbounded) accumulation of round-off errors that can occur with standard techniques. In extensive simulations, identification of a variety of finite and infinite impulse response systems is shown to be considerably better than with conventional system identification methods.Department of Computin

Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition

In this paper, the adaptive chirplet decomposition combined with the Wigner-Ville transform and the empirical mode decomposition combined with the Hilbert transform are employed to process various non-stationary signals (strong ground motions and structural responses). The efficacy of these two adaptive techniques for capturing the temporal evolution of the frequency content of specific seismic signals is assessed. In this respect, two near-field and two far-field seismic accelerograms are analyzed. Further, a similar analysis is performed for records pertaining to the response of a 20-story steel frame benchmark building excited by one of the four accelerograms scaled by appropriate factors to simulate undamaged and severely damaged conditions for the structure. It is shown that the derived joint time–frequency representations of the response time histories capture quite effectively the influence of non-linearity on the variation of the effective natural frequencies of a structural system during the evolution of a seismic event; in this context, tracing the mean instantaneous frequency of records of critical structural responses is adopted. The study suggests, overall, that the aforementioned techniques are quite viable tools for detecting and monitoring damage to constructed facilities exposed to seismic excitations

Wavelet-Based High-Order Adaptive Modeling of Lossy Interconnects

Abstract—This paper presents a numerical-modeling strategy for simulation of fast transients in lossy electrical interconnects. The proposed algorithm makes use of wavelet representations of voltages and currents along the structure, with the aim of reducing the computational complexity of standard time-domain solvers. A special weak procedure for the implementation of possibly dynamic and nonlinear boundary conditions allows to preserve stability as well as a high approximation order, thus leading to very accurate schemes. On the other hand, the wavelet expansion allows the computation of the solution by using few significant coefficients which are automatically determined at each time step. A dynamically refinable mesh is then used to perform a sparse time-stepping. Several numerical results illustrate the high efficiency of the proposed algorithm, which has been tuned and optimized for best performance in fast digital applications typically found on modern PCB structures. Index Terms—Finite difference methods, time-domain analysis, transmission lines, wavelet transforms. I

An efficient nonlinear circuit simulation technique

This paper proposes a novel method for the analysis and simulation of integrated circuits (ICs) with the potential to greatly shorten the IC design cycle. The circuits are assumed to be subjected to input signals that have widely separated rates of variation, e.g., in communication systems, an RF carrier modulated by a low-frequency information signal. The proposed technique involves two stages. Initially, a particular order result for the circuit response is obtained using a multiresolution collocation scheme involving cubic spline wavelet decomposition. A more accurate solution is then obtained by adding another layer to the wavelet series approximation. However, the novel technique presented here enables the reuse of results acquired in the first stage to obtain the second-stage result. Therefore, vast gains in efficiency are obtained. Furthermore, a nonlinear model-order reduction technique can readily be used in both stages making the calculations even more efficient. Results will highlight the efficacy of the proposed approac

Feasibility of Using Nonlinear Time-Frequency Control for Magnetorheological Dampers in Vehicle Suspension

Semi-active vehicle suspensions that use magnetorheological (MR) dampers are able to better dissipate vibrations compared to conventional dampers because of their controllable damping characteristics. The performance of current MR damper control methods is often hindered by incorrect assumptions and linearized models. Therefore, a need exists to design an adaptive controller with improved accuracy and reliability. The objective of this research is to design an improved controller for MR dampers in vehicle suspension using the nonlinear time-frequency control approach and evaluate its feasibility by numerically employing MATLAB Simulink. Simulations in this research are performed using a simplified quarter car suspension model and modified Bouc-Wen damper model. The proposed control method is evaluated based on its ability to reduce the amplitude of vibrations and minimize acceleration of the car body for various test cases. Simulations are also performed using the skyhook controller and passive suspension to assess the performance of the proposed controller. The results of the simulations show that the proposed nonlinear time-frequency controller can successfully be applied to an MR damper suspensions system for vibration control. The proposed controller outperforms the skyhook controller in terms of reducing acceleration of the car body in each of the tested scenarios. The proposed controller also shows the ability to more efficiently manage the current input to the system. In general, the skyhook controller gives more improved vibration amplitude responses but is prone to generate large spikes in car body acceleration at higher frequency road profile inputs. Simulations performed with the passive system show large displacement amplitudes and inability to prevent oscillation. The feed-forward aspect and adaptive nature of the proposed controller gives it the ability to better compensate for the time-delay in the operation of the MR damper. The proposed controller shows sensitivity to controller parameters when pursuing the best response for different road profile input cases

Nonlinear Time-Frequency Control Theory with Applications

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

Self-organizing input space for control of structures

We propose a novel type of neural networks for structural control, which comprises an adaptive input space. This feature is purposefully designed for sequential input selection during adaptive identification and control of nonlinear systems, which allows the input space to be organized dynamically, while the excitation is occurring. The neural network has the main advantages of (1) automating the input selection process for time series that are not known a priori; (2) adapting the representation to nonstationarities; and (3) using limited observations. The algorithm designed for the adaptive input space assumes local quasi-stationarity of the time series, and embeds local maps sequentially in a delay vector using the embedding theorem. The input space of the representation, which in our case is a wavelet neural network, is subsequently updated. We demonstrate that the neural net has the potential to significantly improve convergence of a black-box model in adaptive tracking of a nonlinear system. Its performance is further assessed in a full-scale simulation of an existing civil structure subjected to nonstationary excitations (wind and earthquakes), and shows the superiority of the proposed method

Study of double-potential-well leaf spring system’s chaotic vibration

Chaotic vibration has become increasingly popular in the study of acoustic and vibration engineering. Many engineering designs have taken advantage of the special characteristics of chaos, and deliberately introduced it into the system to improve efficiency. As an important component, leaf springs have long been used in the suspension system of wheeled vehicles. Recent development is considering chaotic vibration in the design of leaf springs to improve the system’s reliability. However, little experimental research has been carried out to investigate the chaos characteristics of leaf springs. Meanwhile, a preliminary study showed that some of the conventional signal processing methods may not be able to successfully identify the chaos features from a leaf spring test rig due to the complexity of the practical signal. Therefore, in this paper, a leaf spring system’s chaotic vibration and relevant signal processing strategy were investigated in theory and experiment. Firstly, the relationship between the amplitude and frequency of the double potential well system is derived with averaging method. The stability is analyzed on the Vander pol plane and the global bifurcation diagram and Lyapunov exponent spectrum are applied to determine the chaotic regime accurately. Numerical simulation was conducted using a finite element method to give an idea of the leaf spring’s natural frequencies where chaotic vibration can be potentially generated. The experimental rig was then designed based on double potential well theory to generate stable and repeatable chaotic vibration, and an experimental study was carried out to investigate the system’s response characteristics under different excitation strengths and frequencies. An improved signal processing method, Wavelet-SG-EEMD (Wavelet, Savitzky-Golay (SG) and Ensemble Empirical Mode Decomposition (EEMD)), was used to reduce noise and beneficial to identify chaotic features of the vibration signal generated by the system. The nonlinear vibration response features of the system were carefully analyzed. Sub-harmonic phenomena, periodic modes and chaotic behavior were discovered during the experiment

Methods, Apparatus And Systems For Real Time Identification And Control Of Modes Of Oscillation

A system for real time identification of modes of oscillation includes a sensor, an observer, a controller and an actuator. The sensor senses a controlled system such as a combustor, and generates a signal indicative of the modes of oscillation in the controlled system. For example, these modes of oscillation can be combustion instabilities. The observer receives the signal from the sensor, and uses the signal to determine modal functions and frequencies of the modes of interest with a pair of integrals with changing time limits. The controller receives the modal functions and frequency for each mode of interest from the observer, and effects a gain and phase shift for each mode. Based on the modal functions, the frequency, the gain and the phase shift, the controller generates and outputs a control signal, that is supplied to the actuator. The actuator controls the modes of oscillation of the controlled system, based on the control signal. The system of this invention can be used to damp or enhance oscillation modes of the controlled system, depending upon whether the oscillation modes are beneficial or detrimental to system performance.Georgia Tech Research Corporatio

Sparse representations and harmonic wavelets for stochastic modeling and analysis of diverse structural systems and related excitations

In this thesis, novel analytical and computational approaches are proposed for addressing several topics in the field of random vibration. The first topic pertains to the stochastic response determination of systems with singular parameter matrices. Such systems appear, indicatively, when a redundant coordinate modeling scheme is adopted. This is often associated with computational cost-efficient solution frameworks and modeling flexibility for treating complex systems. Further, structures are subject to environmental excitations, such as ground motions, that typically exhibit non-stationary characteristics. In this regard, aiming at a joint time-frequency analysis of the system response a recently developed generalized harmonic wavelet (GHW)-based solution framework is employed in conjunction with tools originated form the generalized matrix inverse theory. This leads to a generalization of earlier excitation-response relationships of random vibration theory to account for systems with singular matrices. Harmonic wavelet-based statistical linearization techniques are also extended to nonlinear multi-degree-of-freedom (MDOF) systems with singular matrices. The accuracy of the herein proposed framework is further improved by circumventing previous “local stationarity” assumptions about the response. Furthermore, the applicability of the method is extended beyond redundant coordinate modeling applications. This is achieved by a formulation which accounts for generally constrained equations of motion pertaining to diverse engineering applications. These include, indicatively, energy harvesters with coupled electromechanical equations and oscillators subject to non-white excitations modeled via auxiliary filter equations. The second topic relates to the probabilistic modeling of excitation processes in the presence of missing data. In this regard, a compressive sampling methodology is developed for incomplete wind time-histories reconstruction and extrapolation in a single spatial dimension, as well as for related stochastic field statistics estimation. An alternative methodology based on low rank matrices and nuclear norm minimization is also developed for wind field extrapolation in two spatial dimensions. The proposed framework can be employed for monitoring of wind turbine systems utilizing information from a few measured locations as well as in the context of performance-based design optimization of structural systems. Lastly, the problem of with data-driven sparse identification methods of nonlinear dynamics is considered. In particular, utilizing measured responses a Bayesian compressive sampling technique is developed for determining the governing equations of stochastically excited structural systems exhibiting diverse nonlinear behaviors and also endowed with fractional derivative elements. Compared to alternative state-of-the-art schemes that yield deterministic estimates for the identified model, the herein developed methodology exhibits additional sparsity promoting features and is capable of quantifying the uncertainty associated with the model estimates. This provides a quantifiable degree of confidence when employing the proposed framework as a predictive tool