Stochastic control of hysteretic structural systems

A method of stochastic optimal control of hysteretic structural systems under earthquake excitations is presented. Stochastic estimation and control problems are formulated in the form of Itô stochastic differential equations on the basis of the theory of continuous Markov processes. The conditional moment equations given observation data are derived for nonlinear filtering, and are closed by introducing appropriate analytical form of the conditional probability density functions of the state variables. Under the assumption that the admissible controls are expressed as functions of the conditional moment functions the Bellman equation is derived. If the spatial variables of the Bellman equation are defined by a part of the full set of conditional moment functions appearing in the closed moment equations, the resulting Bellman equation is coupled with conditional moment equations both for filtering and for prediction. The Gaussian and non-Gaussian stochastic linearization techniques combined with simple solution techniques to the Bellman equation are examined to solve the Bellman equation or extended Riccati equations without prediction procedures.     

Monte Carlo filters for identification of nonlinear structural dynamical systems

The problem of identification of parameters of nonlinear structures using dynamic state estimation techniques is considered. The process equations are derived based on principles of mechanics and are augmented by mathematical models that relate a set of noisy observations to state variables of the system. The set of structural parameters to be identified is declared as an additional set of state variables. Both the process equation and the measurement equations are taken to be nonlinear in the state variables and contaminated by additive and (or) multiplicative Gaussian white noise processes. The problem of determining the posterior probability density function of the state variables conditioned on all available information is considered. The utility of three recursive Monte Carlo simulation-based filters, namely, a probability density function-based Monte Carlo filter, a Bayesian bootstrap filter and a filter based on sequential importance sampling, to solve this problem is explored. The state equations are discretized using certain variations of stochastic Taylor expansions enabling the incorporation of a class of non-smooth functions within the process equations. Illustrative examples on identification of the nonlinear stiffness parameter of a Duffing oscillator and the friction parameter in a Coulomb oscillator are presented. This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

Asymptotic Lyapunov stability with probability one of Duffing oscillator subject to time-delayed feedback control and bounded noise excitation

The asymptotic Lyapunov stability with probability one of a Duffing system with time-delayed feedback control under bounded noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original system, and the asymptotic Lyapunov stability with probability one of the original system can be determined approximately by using the Lyapunov exponent. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. The theoretical results are well verified through digital simulation.     

Stochastic stability of Duffing oscillator with fractional derivative damping under combined harmonic and white noise parametric excitations

The stochastic stability of a Duffing oscillator with fractional derivative damping of order α (0 < α < 1) under parametric excitation of both harmonic and white noise is studied. First, the averaged Itô equations are derived by using the stochastic averaging method for an SDOF strongly nonlinear stochastic system with fractional derivative damping under combined harmonic and white noise excitations. Then, the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is obtained and the asymptotic Lyapunov stability with probability one of the original system is determined approximately by using the largest Lyapunov exponent. Finally, the analytical results are confirmed by using those from a Monte Carlo simulation of the original system.     

Application of stochastic differential equations to seismic reliability analysis of hysteretic structures

An analytical method of stochastic seismic response and reliability analysis of hysteretic structures based on the theory of Markov vector process is presented, especially from the methodological aspect. To formulate the above analysis in the form of stochastic differential equations, the differential formulations of general constitutive laws for a class of hysteretic characteristics are derived. The differential forms of the seismic safety measures such as the maximum ductility ratio, cumulative plastic deformation, low-cycle fatigue damage are also derived. The state equation governing the whole nonlinear dynamical system which is composed of the shaping filter generating seismic excitations, hysteretic structural system and safety measures is determined as the Itô stochastic differential equations. By introducing an appropriate non-Gaussian joint probability density function, the statistics and joint probability density function of the state variables can be evaluated numerically under nonstationary state. The merit of the proposed method is in systematically unifying the conventional response and reliability analyses into an analysis which requires knowledge of only first order (single-time) statistics or probability distributions.     

横浪中船舶的随机混沌运动

采用概率密度函数和数值模拟的方法研究随机横浪中船舶的混沌运动特性和发生混沌运动的临界参数条件。综合考虑非线性阻尼、非线性恢复力矩以及白噪声横浪激励,建立了船舶的横摇非线性随机微分方程。用随机Melnikov均方准则确定混沌运动的系统参数域后,应用路径积分法求解随机微分方程得到了响应的概率密度函数。研究发现:当噪声强度大于混沌临界值时,船舶出现随机混沌运动;对于高的白噪声激励强度,系统响应有两种较大可能的状态并在这两个状态间随机跳跃,这时船舶的运动不稳定并可能发生倾覆。     

A stochastic Galerkin expansion for nonlinear random vibration analysis

An approach is developed for the numerical solution of random vibration problems. It is based on treating random variables as functions in a certain Hilbert space. Stochastic processes are described as curves defined in this space, and concepts from deterministic approximation theory are applied to represent the solution as a series involving a known basis of stochastic processes, and a set of unknown coefficients which are deterministic functions of time. Then, a Galerkin projection procedure is utilized to derive a set of ordinary differential equations which can be solved numerically to determine the coefficients in the series. The versatility of the proposed approach is demonstrated by its application to a nonlinear vibration problem involving the probability density of a non-Markovian oscillator response.     

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.     

A reduced polynomial chaos expansion method for the stochastic finite element analysis

The stochastic finite element analysis of elliptic type partial differential equations is considered. A reduced method of the spectral stochastic finite element method using polynomial chaos is proposed. The method is based on the spectral decomposition of the deterministic system matrix. The reduction is achieved by retaining only the dominant eigenvalues and eigenvectors. The response of the reduced system is expanded as a series of Hermite polynomials, and a Galerkin error minimization approach is applied to obtain the deterministic coefficients of the expansion. The moments and probability density function of the solution are obtained by a process similar to the classical spectral stochastic finite element method. The method is illustrated using three carefully selected numerical examples, namely, bending of a stochastic beam, flow through porous media with stochastic permeability and transverse bending of a plate with stochastic properties. The results obtained from the proposed method are compared with classical polynomial chaos and direct Monte Carlo simulation results.     

Joint distribution of peaks and valleys in a stochastic process

General expressions and numerical results are presented pertaining to the occurrence of two local extrema of a stochastic process at prescribed time values. The extrema may be either peaks or valleys and the process may be either stationary or nonstationary. General formulas are presented for the rates of occurrence, the joint and conditional probability distributions, and the moments of the extreme values. These formulas are relatively simple multiple-integral expressions, but the integrands involve the joint probability density function for six random variables. The procedures are then applied for the special case of a stationary mean-zero Gaussian process for which the calculations are greatly simplified. Numerical results for three different spectral density functions demonstrate that conditioning on either only the existence or both the existence and the value of one peak can have a very significant effect on both the rate of occurrence and the probability distribution of a second peak.     

Probabilistic nonconvex constrained optimization with fixed number of function evaluations

A methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain. Statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations. A diffusion manifold over the initial set of data points is first identified and an associated basis computed. The joint probability density function of this initial set is estimated using a kernel density model and an Itô stochastic differential equation (ISDE) constructed with this model as its invariant measure. This ISDE is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters, design variables, and function values, which are obtained as solutions of the ISDE. The expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations. The methodology brings together novel ideas from manifold learning and stochastic Hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization. Three examples are presented to highlight different aspects of the proposed methodology.     

一种测量光学滤波器参数的低相干方法

在光纤通信网络中, 以微环为代表的光学滤波器是实现在波长频段选择功能的重要器件。测量该器件的物理参数不仅是实际应用的需求更有利于设计更加复杂的滤波器结构。本文基于低相干干涉测量技术, 给出一种通用数学方法, 可获得复杂结构微环滤波器的物理参数。该方法引入数字滤波器概念以及 Z变换分析法来对光学滤波器进行建模, 并用简洁的线性方程组将干涉图中峰值以及传输函数的系数关联起来。对于一个给定的滤波器结构, 根据传输函数与物理参数的内在关联, 物理参数就能被准确地计算出来, 从而为器件的激光修正提供关键参数。实验结果表明, 该方法实现了低成本并且准确可行。     

Stochastic stability of quasi non-integrable Hamiltonian systems under parametric excitations of Gaussian and Poisson white noises

The asymptotic Lyapunov stability with probability one of n-degree-of-freedom (n-DOF) quasi non-integrable Hamiltonian systems subject to weakly parametric excitations of combined Gaussian and Poisson white noises is studied by using the largest Lyapunov exponent. First, an n-DOF quasi non-integrable Hamiltonian system subject to weakly parametric excitations of combined Gaussian and Poisson white noises is reduced to a one-dimensional averaged Itô stochastic differential equation (SDE) for Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. Then, the expression for the Lyapunov exponent of the averaged Itô SDE is derived and the approximately necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained. Finally, one example is worked out to illustrate the proposed procedure and its effectiveness is confirmed by comparing with Monte Carlo simulation. It is found that analytical and simulation results agree well.     

A harmonic-wavelet-based representation for non-stationary stochastic excitations

Representation of nonstationary stochastic excitations is crucial for stochastic response analyses of (time-varying) linear and nonlinear structural systems. This paper proposes a new representation method of non-stationary stochastic excitations based on the generalized harmonic wavelet (GHW) that takes the phase angles and frequencies as basic random variables. The orthogonal properties of the discrete-form spectral process increments describing non-stationary stochastic processes are formulated. Then the GHW-based representation is derived by using the orthogonal properties. This method can be used to accurately reproduce non-stationary stochastic excitations with the target asymptotic Gaussianity and evolutionary power spectrum density. The effectiveness and accuracy of the proposed method have been validated via numerical examples. This study provides a novel way for the representation of non-stationary processes and deserves to be applied in the stochastic response analyses of structures.     

Approximate analysis of nonlinear stochastic differential equations using certain generalized quasi-moment functions

Summary The application and the advantages of the method of certain generalized quasi-moment functions are demonstrated by way of a simple mechanical example. The stress of the considered viscoelastic beam, subjected to a stochastically variable temperature, is described by a nonlinear (casea) or a linear (caseb) equation of first order with stochastic coefficients resulting by passing Gaussian white noise through a linear shaping filter. As a result, the final differential equation system is nonlinear also in caseb. The mean value, the variance (for the casea andb), the covariance function and the spectral density (for caseb only) of the stress are estimated by means of linear quasi-moment equations with good convergence. In contrast to this, the results which were obtained by the normal distribution method, here being used as the basic approximation, are affected with great deviations.With 6 Figures     

Feedback stabilization of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A procedure for designing a feedback control to asymptotic Lyapunov stability with probability one of quasi nonintegrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations is proposed. First, a one dimensional partially averaged Itô stochastic differential equation for controlled Hamiltonian is derived from the motion equations of the system by using the stochastic averaging method. Second, the dynamical programming equation for the ergodic control problem of the averaged system with undetermined cost function is set up based on the dynamical programming principle and the jump–diffusion chain stochastic differential rules. The optimal control law is obtained by solving the dynamical programming equation. Third, the analytical expression for the largest Lyapunov exponent of the averaged system is derived. Finally, the asymptotic Lyapunov stability with probability one of the originally controlled system is analyzed approximately by using the largest Lyapunov exponent. The cost function and optimal control forces are determined by the requirements of stabilizing the system. An example is worked out in detail to illustrate the effectiveness of the proposed method for stabilization control, and the control effect of the proposed feedback stabilization varies with the change of parameters is also studied in this paper, such as, the greater the excitation intensity of Gaussian and Poisson white noise, the better the stabilization control effect.     

Stability of low-frequency pulsations in a transient heat transfer regime upon phase transitions

Using the principle of maximum entropy, we investigate the stability of stochastic processes with the 1/f power spectrum in the system of two nonlinear stochastic differential equations when modeling pulsations in crisis and transient heat-mass transfer regimes with intensive phase transitions. An analysis of the stability of the resultant process, which appears upon the interaction of the stochastic process and the 1/f spectrum of external deterministic impact, is performed. Under the action of harmonic force, stable resultant processes are divided into two types depending on the amplitude of the harmonic force. We have experimentally studied the influence of harmonic impact on the stability of pulsations with the 1/f spectrum upon the crisis of water boiling on a heated wire. The results are in qualitative agreement with theoretical estimations.     

An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Itô’s stochastic differential equations

Probability densities for solutions of nonlinear Itô’s stochastic differential equations are described by the corresponding Kolmogorov-forward/Fokker-Planck equations. The densities provide the most complete information on the related probability distributions. This is an advantage crucial in many applications such as modelling floating structures under the stochastic-load due to wind or sea waves. Practical methods for numerical solution of the probability density equations are combined, analytical-numerical techniques. The present work develops a new analytical-numerical approach, the successive-transition (ST) method, which is a version of the path-integration (PI) method. The ST technique is based on an analytical approximation for the transition probability density. It enables the PI approach to explicitly allow for the damping matrix in the approximation. This is achieved by extending another method, introduced earlier for bistable nonlinear reaction-diffusion equations, to the probability density equations. The ST method also includes a control for the size of the time-step. The overall accuracy of the ST method can be tested on various nonlinear examples. One such example is proposed. It is one-dimensional nonlinear Itô’s equation describing the velocity of a ship maneuvering along a straight line under the action of the stochastic drag due to wind or sea waves. Another problem in marine engineering, the rolling of a ship up to its possible capsizing is also discussed in connection with the complicated damping matrix picture. The work suggests a few directions for future research.     

Construction of probability distributions in high dimension using the maximum entropy principle: Applications to stochastic processes,random fields and random matrices

The construction of probabilistic models in computational mechanics requires the effective construction of probability distributions of random variables in high dimension. This paper deals with the effective construction of the probability distribution in high dimension of a vector‐valued random variable using the maximum entropy principle. The integrals in high dimension are then calculated in constructing the stationary solution of an Itô stochastic differential equation associated with its invariant measure. A random generator of independent realizations is explicitly constructed in this paper. Three fundamental applications are presented. The first one is a new formulation of the stochastic inverse problem related to the construction of the probability distribution in high dimension of an unknown non‐stationary random time series (random accelerograms) for which the velocity response spectrum is given. The second one is also a new formulation related to the construction of the probability distribution of positive‐definite band random matrices. Finally, we present an extension of the theory when the support of the probability distribution is not all the space but is any part of the space. The third application is then a new formulation related to the construction of the probability distribution of the Karhunen–Loeve expansion of non‐Gaussian positive‐valued random fields. Copyright © 2008 John Wiley & Sons, Ltd.