White Noise Input Research Articles

Sufficient condition for reliable logic operations in an over damped bistable system driven by Gaussian white noise.

In this paper, we investigate an overdamped bistable system subject to Gaussian white noise and logical inputs. By solving and estimating the steady distribution of the corresponding Fokker-Planck equation and considering the two essential features of the reliable logic operations (RLOs)-initial value independence and sign invariance-we establish the sufficient condition for RLO occurrence and identify parametric resonance phenomena in the system. Our numerical simulations confirm the reliability and accuracy of the theoretical results. This work offers insights into enhancing the accuracy of stochastic systems, particularly in the realm of logical stochastic resonance, thereby contributing to advancements in understanding and controlling stochastic dynamical systems.

Modeling and optimization of car suspension system in the presence of magnetorheological damper using Simulink-PSO hybrid technique

The suspension system is one of the important parts of the vehicle, which plays a substantial role in the steering and comfort characteristics of it. In the present research work, the optimization of the suspension system parameters has been carried out with the aim of reducing the vehicle vibrations. In this study, the magnetorheological damper (MR damper) is added to the one-half car model with four degrees of freedom and the effects of improving the efficiency of the suspension system have been investigated. In order to model the vehicle system, MATLAB/Simulink software is employed. The particle swarm optimization (PSO) algorithm has been applied to optimize the variables of the suspension system in order to decrease the vehicle vibrations and passenger comfort. The optimization of suspension system variables has been done for two bond-limited white noise and sinusoidal wave inputs. The outcomes revealed that the addition of MR damper in any mode reduces the vehicle vibrations. The maximum reduction in the vehicle vibrations was related to the state where all the variables of the suspension system were optimized and the amount of the objective function in that state was improved by about 32%.

A method for evaluation of the probability density function of white noise filtered non-Gaussian stochastic process

Passing non-Gaussian white noises through a pre-designed linear filter can generate non-Gaussian stochastic processes with desired probabilistic and spectral properties. This linear filtering methodology has been extensively employed in non-Gaussian signal processing and simulation. At present, the investigation of probability properties for the output non-Gaussian processes is primarily concentrated on the higher-order statistics, spectra, and correlation functions. However, evaluating the probability density function (PDF) of the output process remains a challenging task. In order to address this issue, this short communication commences with the theoretical derivation of the PDF formula for the white noise filtered non-Gaussian stochastic process. The derived PDF is the result of a convolution of multiple PDFs, each obtained by applying a scaling transformation to the input white noise PDF. Subsequently, a discretized numerical calculation method is proposed to overcome the analytical difficulties associated with the computation of multiple convolutions. Finally, the accuracy and efficiency of the proposed method are demonstrated through several numerical examples.

Analytical response and Markovianity of systems governed by fractional differential equations driven by [formula omitted]-stable white noise processes

This paper investigates the probabilistic response of systems governed by fractional differential equations and forced by α-stable white noise processes. The response in terms of scale of the response process is built starting from a finite differences version of the governing equation. An efficient formulation in terms of scale of the response process is obtained for both non-stationary and stationary white noise input process. Remarkably, the proposed numerical scheme has an analytical counterpart in terms of time evolution of the scale that is a generalization of the well known Duhamel integral. This analytical formulation permits to find closed form solutions in terms of the time evolution of the scale of the response process. Further, when the input white noise process is stationary, it is demonstrated that the system remains Markovian of order one also in presence of fractional operators in the governing equation. This is a very interesting result that makes the proposed numerical procedure very efficient from a computational point of view and that can be very useful for future developments. Numerical applications demonstrate the accuracy of the proposed numerical scheme and the validity of the analytical solution for some systems of interest in engineering.

Input-size dependence of the baroreflex neural arc transfer characteristics during Gaussian white noise inputs.

Although Gaussian white noise (GWN) inputs offer a theoretical framework for identifying higher-order nonlinearity, an actual application to the data of the neural arc of the carotid sinus baroreflex did not succeed in fully predicting the well-known sigmoidal nonlinearity. In the present study, we assumed that the neural arc can be approximated by a cascade of a linear dynamic (LD) component and a nonlinear static (NS) component. We analyzed the data obtained using GWN inputs with a mean of 120 mmHg and standard deviations (SDs) of 10, 20, and 30 mmHg for 15 min each in anesthetized rats (n = 7). We first estimated the linear transfer function from carotid sinus pressure to sympathetic nerve activity (SNA) and then plotted the measured SNA against the linearly predicted SNA. The predicted and measured data pairs exhibited an inverse sigmoidal distribution when grouped into 10 bins based on the size of the linearly predicted SNA. The sigmoidal nonlinearity estimated via the LD-NS model showed a midpoint pressure (104.1 ± 4.4 mmHg for SD of 30 mmHg) lower than that estimated by a conventional stepwise input (135.8 ± 3.9 mmHg, P < 0.001). This suggests that the NS component is more likely to reflect the nonlinearity observed during pulsatile inputs that are physiological to baroreceptors. Furthermore, the LD-NS model yielded higher R2 values compared with the linear model and the previously suggested second-order Uryson model in the testing dataset.NEW & NOTEWORTHY We examined the input-size dependence of the baroreflex neural arc transfer characteristics during Gaussian white noise inputs. A linear dynamic-static nonlinear model yielded higher R2 values compared with a linear model and captured the well-known sigmoidal nonlinearity of the neural arc, indicating that the nonlinear dynamics contributed to determining sympathetic nerve activity. Ignoring such nonlinear dynamics might reduce our ability to explain underlying physiology and significantly limit the interpretation of experimental data.

Asymptotic Theory for Regularized System Identification Part I: Empirical Bayes Hyperparameter Estimator

Regularized techniques, also named as kernel-based techniques, are the major advances in system identification in the last decade. Although many promising results have been achieved, their theoretical analysis is far from complete and there are still many key problems to be solved. One of them is the asymptotic theory, which is about convergence properties of the model estimators as the sample size goes to infinity. The existing related results for regularized system identification are about the almost sure convergence of various hyper-parameter estimators. A common problem of those results is that they do not contain information on the factors that affect the convergence properties of those hyper-parameter estimators, e.g., the regression matrix. In this paper, we tackle problems of this kind for the regularized finite impulse response model estimation with the empirical Bayes (EB) hyper-parameter estimator and filtered white noise input. In order to expose and find those factors, we study the convergence in distribution of the EB hyper-parameter estimator, and the asymptotic distribution of its corresponding model estimator. For illustration, we run Monte Carlo simulations to show the efficacy of our obtained theoretical results.

Vibration Signal Evaluation Based on K-Means Clustering as a Pre-Stage of Operational Modal Analysis for Structural Health Monitoring of Rotating Machines

Rotating machines are key components in energy generation processes, and faults can lead to shutdowns or catastrophes encompassing economic and social losses. Structural Health Monitoring (SHM) of structures in operation is successfully performed via Operational Modal Analysis (OMA), which has advantages over traditional methods. In OMA, white noise inputs lead to the accurate extraction of modal parameters without taking the system out of operation. However, this excitation condition is not easy to attain for rotating machines used in power generation, and OMA can provide inaccurate information. This research investigates the applicability of machine learning as a pre-stage of OMA to differentiate adequate from inadequate excitations and prevent inaccurate extraction of modal parameters. Data from a rotor system was collected under different conditions and OMA was applied. In a training stage, measurements were characterized by statistical features and K-means was used to determine which features provided information about the excitation condition, that is, which excitation was adequate to extract the rotor’s modal parameters via OMA. In a testing stage, data were successfully classified as adequate or not adequate for OMA, achieving 100% accuracy and revealing the technique’s potential to support SHM of rotating machines. The technique is extendable to other monitoring systems based on OMA.

Nonlinear resonant ultrasound spectroscopy using white-noise excitation

Nonlinear resonant ultrasound spectroscopy (NRUS) is a technique for non-destructive sample evaluation by which a strain-dependent nonlinear parameter is quantified that exhibits heightened sensitivity to defects as compared to monitoring of linear resonance attributes. As NRUS measurements have traditionally used a controlled sweep of sinusoidal inputs, applications of the method have been largely limited to laboratory experiments. Here we present NRUS results which utilized a white noise signal as input, in order to better approximate ambient excitation and facilitate efficient validation using time-domain models. Studied samples were chosen to reflect a range of nonlinear responses and underlying mechanisms (acrylic, sandstone, fractured steel). We excited the studied samples using bonded piezoelectric transducers which delivered white-noise inputs at increasing amplitudes, with the nonlinear response measured with a laser doppler vibrometer. By comparing results against those obtained from traditional NRUS, we find comparable trends in the nonlinear parameter under white-noise excitation, including similar relative differences in magnitude across the samples and consistent values across similar mode shapes. While we note that white-noise strain estimation is complicated by simultaneous excitation of multiple resonant modes, NRUS results from white-noise excitation demonstrate promising viability of the technique for (1) in-situ inspection, and (2) a more-tractable approach to model the physics of various NRUS signals.

Sampling Gaussian stationary random fields: A stochastic realization approach

Generating large-scale samples of stationary random fields is of great importance in the fields such as geomaterial modeling and uncertainty quantification. Traditional methodologies based on covariance matrix decomposition have the difficulty of being computationally expensive, which is even more serious when the dimension of the random field is large. This paper proposes an efficient stochastic realization approach for sampling Gaussian stationary random fields from a systems and control point of view. Specifically, we take the exponential and squared exponential covariance functions as examples and make a decoupling assumption when there are multiple dimensions. Then a rational spectral density is constructed in each dimension using techniques from covariance extension, and the corresponding autoregressive moving-average (ARMA) model is obtained via spectral factorization. As a result, samples of the random field with a specific covariance function can be generated very efficiently in the space domain by implementing the ARMA recursion using a Gaussian white noise input. Such a procedure is computationally cheap due to the fact that the constructed ARMA model has a low order. Furthermore, the same method is integrated to multiscale simulations where interpolations of the generated samples are achieved when one zooms into finer scales. Both theoretical analysis and simulation results show that our approach performs favorably compared with covariance matrix decomposition methods.

On the optimal design and placement of Tuned-Mass-Damper-Inerter for Multi-Degree-Of-Freedom structures

Even if Tuned-Mass-Damper-Inerter (TMDI) design for Multi-Degree-Of-Freedom (MDOF) structures is a topic largely addressed in the current literature, some aspects still deserving insight emerge from a literature review. The state of the art presented in the paper highlights that Inerter-Vibration-Absorber (IVA) design is typically conducted based on simplified models (e.g. generalized 2-Degree-Of-Freedom (2-DOF) models) or sophisticated ones but a comparison between the two approaches is rarely made. Moreover, the device placement along the structure is not often addressed in the design formulation; nevertheless, a proper location greatly influences the absorber performance and its physical dimensions. The paper aims to be a comprehensive study that investigates in a unique formulation the optimal design together with the optimal placement of a TMDI in a MDOF structure and aims to make a comparison between the use of a 2-DOF model and a complete MDOF model for its optimal design and placement. The overall design methodology and performance evaluation is formulated considering a stationary white noise input. An exemplificative literature case study of a 10-DOF structure is utilized to apply the design procedure, evaluate the performance and make the comparisons. It is demonstrated that a reduced order model is useful for a first estimate of the TMDI design parameters but some, especially the damping coefficient, need improvement adopting more refined models. Structural performance is well cached with reduced order models, especially for those quantities that mainly depend on the first natural mode. An optimal placement, tradeoff between structural performances and absorber physical dimensions, is proposed. TMDI effectiveness is assessed also considering non-stationary input with natural earthquakes and comparisons with conventional and non-conventional Tuned-Mass-Damper (TMD) are conducted.

Computation-efficient solution for fully-connected active noise control window: Analysis and implementation of multichannel adjoint least mean square algorithm

The multichannel active noise control (MCANC) system, in which multiple reference sensors and actuators are used to enlarge the noise-cancellation zone, is widely utilized in complex acoustic environments. However, as the number of channels increases, the practicality decreases due to the exponential rise in computational complexity. This paper, therefore, revisits the adjoint least mean square (ALMS) algorithm and its multichannel applications. The computational analysis reveals that the multichannel adjoint least mean square (McALMS) algorithm11The code is shared on https://www.mathworks.com/matlabcentral/fileexchange/127554-multichannel-adjoint-lms-algorithm-for-multichannel-anc. has a significantly lower computation cost when implementing the fully connected active noise control (ANC) structure. In addition to this advantage, the theoretical analysis presented in this paper demonstrates that the McALMS algorithm can achieve the same optimal solution as the standard adaptive algorithm without the assumptions of input independence and white Gaussian noise. In addition, a practical step-size estimation strategy based on the Golden-section search (GSS) method is proposed to predict the fast step size of the McALMS algorithm. The numerical simulations in a multichannel ANC system demonstrate the effectiveness of the McALMS algorithm and validate the derived theoretical analysis. Furthermore, the McALMS algorithm with proposed step-size approach is used to implement a multichannel noise cancellation window that achieves satisfactory global noise reduction performance for tonal, broadband, and even real-world noises.

Wind turbulence modeling for real-time simulation

This paper proposes a method to design a real-time wind turbulence simulator. The objective of such a simulator is to make dynamical models more accurate in order to develop more robust controls, especially in the case of mechanical systems operating outdoors such as tracking antennas. The models used to generate a random wind speed are based on the wind spectral characteristics rather than time domain ones. Indeed, turbulence is a stochastic and non-stationary process corresponding to the short-term component of wind and therefore is difficult to model in time domain. Wind spectral characteristics are described by the power spectral density whose approximation is used in the real-time simulator to reproduce wind behavior. The von Kármán model is the most commonly used model to approximate the power spectral density of wind turbulence. However, this model was originally designed for aircraft, and therefore for high altitudes and moving systems. In our case, tracking antennas are used under different conditions: low altitude and slow moving systems. This makes the von Kármán model less precise in middle and high range frequency. Consequently, there is a need in more accurately modeling wind turbulence under these specific conditions. As von Karman’s model expression uses fractional calculus, other models with the same range of parameter number are proposed to better approximate the wind power spectral density by using Cole-Cole and Davidson-Cole fractional models. The idea is to use shaping filters issued from these more precise fractional models to generate realistic random wind speed turbulence from a random white noise input. These filters are approximated by rational functions and their parameters are tuned in the same way as those of the von Kármán model with the same inputs (mean speed, standard deviation and length scale). If a rational model is sought, a very high number of parameters (more than six times) is needed to have the same precision as our proposed fractional Cole-Cole model which has only four parameters. Finally, an accurate turbulence wind speed generator is proposed.

Information Rates for Channels with Fading, Side Information and Adaptive Codewords.

Generalized mutual information (GMI) is used to compute achievable rates for fading channels with various types of channel state information at the transmitter (CSIT) and receiver (CSIR). The GMI is based on variations of auxiliary channel models with additive white Gaussian noise (AWGN) and circularly-symmetric complex Gaussian inputs. One variation uses reverse channel models with minimum mean square error (MMSE) estimates that give the largest rates but are challenging to optimize. A second variation uses forward channel models with linear MMSE estimates that are easier to optimize. Both model classes are applied to channels where the receiver is unaware of the CSIT and for which adaptive codewords achieve capacity. The forward model inputs are chosen as linear functions of the adaptive codeword's entries to simplify the analysis. For scalar channels, the maximum GMI is then achieved by a conventional codebook, where the amplitude and phase of each channel symbol are modified based on the CSIT. The GMI increases by partitioning the channel output alphabet and using a different auxiliary model for each partition subset. The partitioning also helps to determine the capacity scaling at high and low signal-to-noise ratios. A class of power control policies is described for partial CSIR, including a MMSE policy for full CSIT. Several examples of fading channels with AWGN illustrate the theory, focusing on on-off fading and Rayleigh fading. The capacity results generalize to block fading channels with in-block feedback, including capacity expressions in terms of mutual and directed information.

Self-similarity and response of fractional differential equations under white noise input

Self-similarity, fractal behaviour and long-range dependence are observed in various branches of physical, biological, geological, socioeconomics and mechanical systems. Self-similarity, also termed self-affinity, is a concept that links the properties of a phenomenon at a certain scale with the same properties at different time scales as it happens in fractal geometry. The fractional Brownian motion (fBm), i.e. the Riemann–Liouville fractional integral of the Gaussian white noise, is self-similar; in fact by changing the temporal scale t→at (a>0), the statistics in the new time axis (at) remain proportional to those calculated in the previous axis (t). The proportionality coefficient is a2H being H>0 the Hurst index. In the practical applications, the phenomena are usually ruled by fractional differential equations involving more terms. In this paper it is shown that the response of a multi-term fractional differential equation is a linear combination of self-similar processes with increasing order of Hurst exponent. The consequences of self-similarity are discussed in detail, closed forms of correlation and variance are presented for the general case and particularized for the cases of engineering interest.

Determination of monopile offshore structure response to stochastic wave loads via analog filter approximation and GV-GDEE procedure

Efficient and accurate stochastic response analysis of offshore structures subjected to ocean wave loads has been a challenging problem for several decades. This is especially true for multi-degree-of-freedom nonlinear structures when information about the probability density is desired. How to account for the random field of the wave loads, and how to deal with the coupling of high-dimensional partial differential equation governing the joint probability density functions are among the major challenges. To this end, in the present paper an approach is proposed by incorporating the filter approximation of random fields of sea waves with the recently developed Globally-evolving based generalized density evolution equation (GV-GDEE). To deal with the first issue regarding the random wave filed, the analog filter technique is adopted. In particular, a filter approximation of the wave kinematics field distributed over a vertical line along the ocean domain is proposed. It comprises a linear differential equation set with white noise input. An augmented system is thereby constructed by incorporating the additional differential equation set yielding wave excitation simulated by an analog filter into the equation of motion of a monopile offshore structure. To circumvent the issue of coupling of high-dimensional equations for this augmented system with even higher dimension, the GV-GDEE approach is employed. Specifically, by constructing the effective drift coefficients based on the high-dimensional drift coefficients in the associated Fokker-Planck-Kolmogorov (FPK) equation, a two-dimensional GV-GDEE in terms of the response quantity of interest is derived. Further, this GV-GDEE is solved by the path integral method. Furthermore, two numerical examples are included. In particular, the nonstationary stochastic responses of a 5 megawatt (MW) fixed monopile offshore wind turbine structure under two operating conditions are studied. Comparisons with pertinent Monte Carlo simulations (MCS) demonstrate the accuracy and efficiency of the proposed approach.

An Optimized Zero-Attracting LMS Algorithm for the Identification of Sparse System

This paper introduces an optimized zero-attractor to improve the performance of least mean square (LMS)-based algorithms for the identification of sparse system. Compared with previous LMS-based algorithms for sparse system identification, the performance of the proposed optimized zero-attracting LMS (OZ-LMS) is much less sensitive to the tuning parameters and measurement noise power, and performs much better for sparse system. Comprehensive performance analysis of the mean-square deviation (MSD) of OZ-LMS is derived in detail. Moreover, the parameter selection rules for optimal steady-state MSD are discussed. Simulation results, using white Gaussian noise and speech input signals, show improved performance over existing methods. Furthermore, we show that the numerical results of OZ-LMS agree with the theoretical predictions.

Fractional modeling of wind speed turbulence

This paper proposes a method to design a wind turbulence model based on real wind spectral characteristics. It uses models based on Cole-Cole fractional functions to approximate the wind turbulence power spectral density. Von Kármán model is the most commonly used model but originally designed for aircraft in high altitude. Therefore, it is not suitable for systems operating at low altitude, hence the need for more accurate models in these specific conditions. Shaping filters issued from the fractional models are employed to generate realistic random wind speed turbulence from a random white noise input. Then, a grey box model is proposed from physical parameters such as the classic von Kármán ones (mean speed, turbulence intensity and length scale).

Seismic pounding mitigation of asymmetric-plan buildings by using viscoelastic links

Connection of adjacent buildings with viscoelastic links (VE) has been proved to be an efficient method for preventing or at least attenuating the structural collisions during earthquakes. The efficiency of this pounding mitigation method was extensively studied in the past, mostly based on assuming a two-dimensional problem without addressing the effects of plan irregularities. In this paper, the effect of VE links between adjacent asymmetric-plan buildings is studied in detail. First, two adjacent one-story models with various structural as well as link-wise dynamic parameters are analyzed under white-noise input. The inelastic behavior of the adjacent buildings is also taken into account. The response of two adjacent 5 story buildings is then evaluated under the actual earthquake record to validate the findings obtained under white-noise analysis. Results demonstrate that not only plan asymmetries do not necessarily lead to higher responses compared to the symmetric adjacent buildings, but the linked-to-unlinked response ratio would be efficiently reduced by providing supplemental damping in the link elements. As the link stiffness decreases, the reduction in the relative displacement of the buildings due to supplemental damping could be up to 80% and even more. Results also reveal that it is possible to obtain a dominant modal damping ratio up to twice than that in symmetric plans for certain plan eccentricities.

Robust CAWOF Kalman predictors for uncertain multi‐sensor generalized system

SummaryRobust centralized and weighted observation fusion (CAWOF) prediction algorithm is addressed in this article for an uncertain multi‐sensor generalized system with linear correlation between observation noises and an input white noise. This uncertainty in the generalized system primarily means that the variances of the aforementioned types of noise, as well as the multiplicative noise variances, are uncertain. Through singular value decomposition and virtual noise compensation, the original generalized system is changed to non‐generalized reduced‐order subsystems in which only noise variances are uncertain. Utilizing the minimax robustness estimation criterion, robust CAWOF Kalman predictors are put forward on account of the first subsystem with conservative upper bounds of noise variances. Eventually, robust observation fusion Kalman predictors of the original generalized system are proposed. The Lyapunov equation method is applied to verify two fusion predictors' robustness. With regard to all permissible uncertain practical noise variances, CAWOF predictors are robust, namely, the practical prediction error variances of two robust predictors will have minimum upper bounds. This equivalence between CAWOF Kalman predictors is proved by an information filter. In this article, the precision relationship of fusion predictors is given. Meanwhile, robust Kalman predictors for steady‐state case are proposed, and the astringency of robust time‐variant Kalman predictors is analyzed through the analysis of dynamic error system. The validity and correctness of proposed algorithm are proved by the simulation example of random dynamic input and output system in an economic system.

Flexural band gaps and vibration control of a periodic railway track

Periodic structures exhibit unique band gap characteristics by virtue of which they behave as vibro-acoustic filters thereby allowing only waves within a certain frequency range to pass through. In this paper, lateral and vertical flexural wave propagation and vibration control of a railway track periodically supported on rigid sleepers using fastenings are studied in depth. The dispersion relations in both lateral and vertical directions are obtained using the Floquet-Bloch theorem and the resulting dispersion curves are verified using finite element models. Afterwards, tuned mass dampers (TMDs) with different mass ratios are designed to control vibrations of the examined rail in both the directions. Moreover, the influence of damping of rail and resonators on band gap characteristics is investigated. As a replacement to the conventional TMD, a novel possibility to control vibration relies on using another existing rail as a lateral distributed resonator (LDR). Although the effectiveness of LDR is lower than that of localized resonators, the former represents a simple and promising way to control vibrations. Efficacy of the proposed control methods is finally verified by applying a random Gaussian white noise input. The study presented here is useful to understand the propagation and attenuation behavior of flexural waves and to develop efficient and novel vibration control strategies for track structures.