A Hilbert-Huang transform based identification method for general linear time-varying systems and weakly nonlinear systems

This paper proposes an identification method for general linear time-varying (LTV) MDOF systems and weakly nonlinear systems based on the Hilbert-Huang Transform (HHT)[1]. The proposed method uses Empirical mode decomposition (EMD) to decompose the response signals of systems into intrinsic mode functions (IMFs) and residues, and then analyzes the IMFs and the residues by Hilbert transform (HT) to obtain the analytical IMFs and analytical residues. After that, the above signals are synthesized to form new response signals and new analytical response signals. Finally, the new synthesized signals are used to identify the stiffness and damping coefficients of the systems. Three types of variation: smooth, abrupt and periodical variations are considered in the numerical simulations of LTV chainlike[2] and nonchainlike systems as well as weakly nonlinear systems such as Duffing oscillators and Van der Pol oscillators with white noise added in the system responses to demonstrate the effectiveness, accuracy and robustness of the proposed method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

小参数时变非线性系统的技术稳定性

分析一类含小参数的时变非线性系统关于给定状态约束集合的技术稳定性.根据向量微分比较原理和基本的单调性准则,利用向量V函数方法给出由系统系数表达的技术稳定性判据.并讨论了基于派生系统和线性化方法研究非线性系统技术稳定性的条件.另外,对于派生时变线性系统的指数稳定性给出了简单的代数判据.最后给出示例说明文中方法.     

Target torque estimation for gearshift in dual clutch transmission with uncertain parameters

The target torque of engaging clutches during gearshift is a key factor that affects the dynamic response of powertrains equipped with the dual clutch transmissions (DCT). This paper investigates a method to estimate the target torque of engaging clutches under conditions where engine torque and measurement signals contain white noise and some vehicle parameters (the radius of wheel and rolling friction coefficient) are uncertain. To compute the target torque accurately, the state of system should be estimated when the uncertain parameters exist. The vehicle powertrain is modeled as the 3DOF system when one clutch is closed and the 4DOF system when two clutches are open, while the measured signals include speeds of the engine, transmission, and vehicle (rotational speed of wheels). In addition to traditional extended Kalman filter (EKF), both the joint extended Kalman filter (JEKF) and dual extended Kalman filter (DEKF) are used to estimate the target torque. The simulation results show that DEKF and JEKF provide much higher accuracy in the estimation of target torque than EKF when some parameters of the model are uncertain, so as to produce a better ride performance of the transmission during gearshift, i.e. reduction of power interruption and compressed shifting time. Furthermore, the DEKF provides higher accuracy than the JEKF in estimating uncertain parameters.     

Stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise

The stochastic response of a class of self-excited systems with Caputo-type fractional derivative driven by Gaussian white noise is considered. Firstly, the generalized harmonic function technique is applied to the fractional self-excited systems. Based on this approach, the original fractional self-excited systems are reduced to equivalent stochastic systems without fractional derivative. Then, the analytical solutions of the equivalent stochastic systems are obtained by using the stochastic averaging method. Finally, in order to verify the theoretical results, the two most typical self-excited systems with fractional derivative, namely the fractional van der Pol oscillator and fractional Rayleigh oscillator, are discussed in detail. Comparing the analytical and numerical results, a very satisfactory agreement can be found. Meanwhile, the effects of the fractional order, the fractional coefficient, and the intensity of Gaussian white noise on the self-excited fractional systems are also discussed in detail.     

On Generalized Regular Stochastic Differential Delay Systems with Time Invariant Coefficients

Abstract

In this article, we consider the generalized linear regular stochastic differential delay system with constant coefficients and two simultaneous external differentiable and non differentiable perturbations. These kinds of systems are inherent in many application fields; among them we mention fluid dynamics, the modeling of multi body mechanisms, finance and the problem of protein folding. Using the regular Matrix Pencil theory, we decompose it into two subsystems, whose solutions are obtained as generalized processes (in the sense of distributions). Moreover, the form of the initial function is given, so the corresponding initial value problem is uniquely solvable. Finally, two illustrative applications are presented using white noise and fractional white noise, respectively.     

Noise-induced vegetation transitions in the Grazing Ecosystem

The Grazing Ecosystem is a special case of predator-prey systems, which has attracted widespread attention since a long time ago. Because of the ubiquity of noise, there is a growing need to research the influences of noise on the Grazing Ecosystem. This paper is devoted to investigating the transition behaviors of the high vegetation biomass in the Grazing Ecosystem subjected to Gaussian noise and Lévy noise, respectively. Firstly, the original system is translated into the Itô stochastic differential equation, which is utilized to derive the analytical expression of the escape probability through the Dirichlet boundary value problems. Then the transitions between the two vegetation potential wells are explored by calculating the size of the stochastic basin of attraction based on the escape probability. The comparison between the analytical results and the ones through Monte Carlo simulations shows that the proposed method works very well. It turns out that the Gaussian white noise intensity, Lévy noise stability parameter and herbivore density have different impact mechanisms on the basin stability of high density vegetation in the stochastic Grazing Ecosystem.     

分析非线性系统随机响应的一种等效非线性化方法*

本文提出了分析非线性系统随机响应的一种新的等效非线性化方法。文中阐述了该方法的基本思想和处理方法,并对几种常见类型的非线性系统进行了分析和计算,结果表明,利用本文提出方法所得的均方位移响应与精确解或者Monte Carlc模拟解之间具有较好的一致性,并比等效线性化方法有更高的精度。     

Computation of the solutions of the Fokker–Planck equation for one and two DOF systems

Uncertainty in structures may come from unknowns in the modelisation and in the properties of the materials, from variability with time, external noise, etc. This leads to uncertainty in the dynamic response. Moreover, the consequences are issues in safety, reliability, efficiency, etc. of the structure. So an issue is the gain of information on the response of the system taking into account the uncertainties [Mace BR, Worden K, Manson G. Uncertainty in structural dynamics. J Sound Vib 2005;288(3):423–9].If the forcing or the uncertainty can be modelled through a white noise, the Fokker–Planck (or Kolmogorov forward) equation exists. It is a partial differential linear equation with unknown p(X, t), where p(X, t) is the probability density function of the state X at time t.In this article, we solve this equation using the finite differences method, for one and two DOF systems. The numerical solutions obtained are proved to be nearly correct.     

Gauss色噪声激励下含黏弹力摩擦系统的随机响应分析

研究了Gauss色噪声激励下含黏弹力、弱非线性阻尼的摩擦振子的随机响应.将适用于光滑系统的随机平均法推广到了非光滑摩擦系统,进而得到系统振幅、位移及速度的稳态概率密度函数.同时结合材料的黏弹性,研究了摩擦力和Gauss色噪声对系统响应的影响.研究表明,摩擦力、黏弹力及噪声项的相关参数均可引起随机P-分岔,并且在一定范围内系统响应对摩擦力极为敏感.此外,理论结果与Monte Carlo 模拟结果吻合较好,验证了方法的有效性.     

广义维纳泛函,白噪声分布及实Schwartz广义函数空间上的分布

在实Ｓｃｈｗａｒｔｚ广义函数空间上，证明了复值广义维纳泛函，由Ｋｏｎｄｒａｔｅｖ－Ｓｔｒｅｉｔ及Ｈｉｄａ构造的复值白噪声分布都是由Ｋｈｒｅｎｎｉｋｏｖ构造的分布。利用上述结果进而证明了，一类无穷维伪微分算子是由复值广义维纳泛函空间上的连续线性算子族扩张而成。更进一步，还证明了由Ｋｈｒｅｎｎｉｋｏｖ构造的关于分布的试验函数空间是关于白噪声泛函的Ｍｅｙｅｒ－Ｙａｎ试验函数空间的子空间。     

On the posterior median estimators of possibly sparse sequences

We adopt the Bayesian paradigm and discuss certain properties of posterior median estimators of possibly sparse sequences. The prior distribution considered is a mixture of an atom of probability at zero and a symmetric unimodal distribution, and the noise distribution is taken as another symmetric unimodal distribution. We derive an explicit form of the corresponding posterior median and show that it is an antisymmetric function and, under some conditions, a shrinkage and a thresholding rule. Furthermore we show that, as long as the tails of the nonzero part of the prior distribution are heavier than the tails of the noise distribution, the posterior median, under some constraints on the involved parameters, has the bounded shrinkage property, extending thus recent results to larger families of prior and noise distributions. Expressions of posterior distributions and posterior medians in particular cases of interest are obtained. The asymptotes of the derived posterior medians, which provide valuable information of how the corresponding estimators treat large coefficients, are also given. These results could be particularly useful for studying frequentist optimality properties and developing statistical techniques of the resulting posterior median estimators of possibly sparse sequences for a wider set of prior and noise distributions.     

Separation of random noise models from seasonal time series data

The autocorrelation function of seasonal time series data is shown to have peaks which occur at the correlation lags equal to the integer multiples of the fundamental period that is present in the series. This property is shown to be valid even if some of the harmonics including the fundamental are removed from the time series data. Using this property, an analytical procedure is presented for estimating the variance of the white noise generating the low frequency random walk model present in the data. The procedure is similarly extended to estimate the variance of white noise generating the autoregressive (AR) and moving average (MA) noise models. The method is validated on several seasonal time series data whose components are known a priori.     

Collapse of attractors for ODEs under small random perturbations

This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to ‘collapse’. Unlike existing results of this type, no order preserving property is necessary.      

Optimal stabilization of linear stochastic systems : PMM vol.40, n6, 1976, pp. 1034–1039

The problem of optimal stabilization is solved for controlled linear systems with white noise. The optimal solution is obtained by the method of successive approximations each of which represents the optimal solution of the related determinate problem. Necessary and sufficient conditions of stabilizability are given.     

Asymptotic properties of posterior distributions in nonparametric regression with non-Gaussian errors

We investigate the asymptotic behavior of posterior distributions in nonparametric regression problems when the distribution of noise structure of the regression model is assumed to be non-Gaussian but symmetric such as the Laplace distribution. Given prior distributions for the unknown regression function and the scale parameter of noise distribution, we show that the posterior distribution concentrates around the true values of parameters. Following the approach by Choi and Schervish (Journal of Multivariate Analysis, 98, 1969–1987, 2007) and extending their results, we prove consistency of the posterior distribution of the parameters for the nonparametric regression when errors are symmetric non-Gaussian with suitable assumptions.     

Noise and coupling induced synchronization in a network of chaotic neurons

The synchronization in four forced FitzHugh–Nagumo (FHN) systems is studied, both experimentally and by numerical simulations of a model. We show that synchronization may be achieved either by coupling of systems through bidirectional diffusive interactions, by introducing a common noise to all systems or by combining both ingredients, noise and coupling together. Here we consider white and colored noises, showing that the colored noise is more efficient in synchronizing the systems respect to white noise. Moreover, a small addition of common noise allows the synchronization to occur at smaller values of the coupling strength. When the diffusive coupling in the absence of noise is considered, the system undergoes the transition to subthreshold oscillations, giving a spike suppression regime. We show that noise destroys the appearance of this dynamical regime induced by coupling.     

A new method for chaos control in communication systems

With the increasing needs of global communication, the improvement of secure communication is of vital importance. This study proposes a new scheme for establishing secure communication systems. The new scheme separates white Gaussian noises from the chaotic signals with modified Independent Component Analysis (ICA) and then controls each chaotic signal. This scheme is able to deal with white Gaussian noises in the natural world. However, the signals separated by traditional ICA shows opposite phase and unequal amplitude, making chaos control impossible. Our study proposed a modified ICA, which can calculate accurately the phase and amplitude and ensure control of the chaotic systems. The result indicates that our proposed system can successfully separate white Gaussian noise and stabilize all the chaotic signals.     

A principal–agent problem with heterogeneous demand distributions for a carbon capture and storage system

Mechanism design problems optimize contract offerings from a principal to different types of agents who have private information about their demands for a product or a service. We study the implications of uncertainty in agents’ demands on the principal’s contracts. Specifically, we consider the setting where agents’ demands follow heterogeneous distributions and the principal offers a menu of contracts stipulating quantities and transfer payments for each demand distribution. We present analytical solutions for the special case when there are two distributions each taking two discrete values, as well as a method for deriving analytical solutions from numerical solutions. We describe one application of the model in carbon capture and storage systems to demonstrate various types of optimal solutions and to obtain managerial insights.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

Non-linear responses of a one-sided constrained beam with base excitation

The paper discussed the non-linear responses of a buckled beamunder base excitation and constrained by a one-sided motionrestraint. The geometric non-linearity due to axial extensionis taken into account. We apply the Galerkin method to the governingpartial differential equation of the transverse motion to obtaina general model of n degrees of freedom (nDOF). The resultsof dynamic response for the 1DOF mode of a pinned-pinned beamare presented for the system both with and without one-sidedmotion restraint. The regular and irregular motions of the 1DOFmodel for the beam are represented in the forms of time trace,phase plot, bifurcation diagram and power spectra. An in-depthstudy based on an energy approach is done to illustrate thenon-linear responses resulting from the multiplicity of resonantsolutions. It is shown that the effect of the motion restrainton the dynamics of the system is significant.