Hybrid Subset Simulation method for reliability estimation of dynamical systems subject to stochastic excitation

A hybrid Subset Simulation approach is proposed for reliability estimation for general dynamical systems subject to stochastic excitation. This new stochastic simulation approach combines the advantages of the two previously proposed Subset Simulation methods, Subset Simulation with Markov Chain Monte Carlo (MCMC) algorithm and Subset Simulation with splitting. The new method employs the MCMC algorithm before reaching an intermediate failure level and splitting after reaching the level to exploit the causality of dynamical systems. The statistical properties of the failure probability estimators are derived. Two examples are presented to demonstrate the effectiveness of the new approach and to compare with the previous two Subset Simulation methods. The results show that the new method is robust to the choice of proposal distribution for the MCMC algorithm and to the intermediate failure events selected for Subset Simulation.     

Bayesian inference in probabilistic risk assessment—The current state of the art

Markov chain Monte Carlo (MCMC) approaches to sampling directly from the joint posterior distribution of aleatory model parameters have led to tremendous advances in Bayesian inference capability in a wide variety of fields, including probabilistic risk analysis. The advent of freely available software coupled with inexpensive computing power has catalyzed this advance. This paper examines where the risk assessment community is with respect to implementing modern computational-based Bayesian approaches to inference. Through a series of examples in different topical areas, it introduces salient concepts and illustrates the practical application of Bayesian inference via MCMC sampling to a variety of important problems.     

Real-time reliability estimation for serviceability limit states in structures with uncertain dynamic excitation and incomplete output data

A novel technique is presented for indirectly monitoring threshold exceedance in a sparsely-instrumented structure represented by a linear dynamic model subject to uncertain excitation modeled as a Gaussian process. The goal is to answer the following question: given incomplete output data from a structure excited by uncertain dynamic loading, what is the probability that any particular unobserved response of the structure exceeds a prescribed threshold? It is assumed that a good linear dynamic model of the target structure has previously been identified using dynamic test data. The technique is useful for monitoring the serviceability limit states of a structure subject to unmeasured “small-amplitude” ambient excitation (e.g. wind excitation or non-damaging earthquake ground motions), or for monitoring the damage status of equipment housed in the structure that is vulnerable to such excitation. The ISEE algorithm developed by Au and Beck in 2000 is used to efficiently estimate the threshold exceedance (first-passage) probability by stochastic simulation. To improve computational efficiency for the monitoring problem, a new state-space version of ISEE is developed that incorporates state-estimation and a newly-developed state-sampling technique. The computational efficiency of the proposed technique is demonstrated through two numerical examples that show that it is vastly superior to Monte Carlo simulation in estimating the first-passage probability. Moreover, the approach produces useful by-products, including estimates for the model state and the uncertain excitation.     

A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.     

Likelihood ratio gradient estimation for dynamic reliability applications

This paper investigates the issue of performing a first-order sensitivity analysis in the setting of dynamic reliability. The likelihood ratio (LR) derivative/gradient estimation method is chosen to fulfill the mission. Its formulation and implementation in the system-based Monte Carlo approach that is commonly used in dynamic reliability applications is first given. To speed up the simulation, we then apply the LR method within the framework of Z-VISA, a biasing (or importance sampling) method we have developed recently. A widely discussed dynamic reliability example (a holdup tank) is studied to test the effectiveness and behaviors of the LR method when applied to dynamic reliability problems and also the effectiveness of the Z-VISA biasing technique for reducing the variance of LR derivative estimators.     

结构物理参数识别的贝叶斯估计马尔可夫蒙特卡罗方法

从结构动力特征方程出发,以结构主模态参数为观测量,推得结构物理参数线性回归模型。对该模型应用贝叶斯估计理论得到物理参数后验联合分布,再结合马尔可夫蒙特卡罗抽样方法给出各个物理参数的边缘概率分布和最优估计值,而提出了基于结构主模态参数的结构物理参数识别贝叶斯估计马尔可夫蒙特卡罗方法。对五层剪切型结构的数值研究表明,此方法能够利用少数主模态参数给出结构质量和刚度参数的概率分布和最优识别值,而且在主模态参数较准确时识别误差很小。     

Complex system reliability estimation methodology in the absence of failure data

This paper presents a comprehensive system reliability estimation methodology for cases when failure data are unavailable, at least initially. In this methodology, the laws of physics and thermal fundamentals are used to establish a mathematical model that relates the influential input operating characteristics, such as material properties and geometry, to system performance measures. Probability distributions for each influential operating characteristic, identified from the available manufacturing data, information found in instruction manuals and related technical journals, and expert knowledge, are used to simulate the system behavior with Monte Carlo simulation. An initial reliability estimate is obtained by comparing the simulated system performance with the permissible system performance. Fuzzy logic is used to incorporate the impact of environmental factors on the performance of the simulated system performance and hence the system reliability. Finally, with the use of Bayesian analysis, initial system reliability is updated to take into account the effect of environmental factors. The proposed methodology is applied to estimate the reliability of the hazardous gas detection system used in aerospace shuttles for the timely detection of explosive gases. Copyright © 2008 John Wiley & Sons, Ltd.     

Local estimation of failure probability function by weighted approach

In the reliability-based design of engineering systems, it is often required to evaluate the failure probability for different values of distribution parameters involved in the specification of design configuration. The failure probability as a function of the distribution parameters is referred as the ‘failure probability function (FPF)’ in this work. From first principles, this problem requires repeated reliability analyses to estimate the failure probability for different distribution parameter values, which is a computationally expensive task. A “weighted approach” is proposed in this work to locally evaluate the FPF efficiently by means of a single simulation. The basic idea is to rewrite the failure probability estimate for a given set of random samples in simulation as a function of the distribution parameters. It is shown that the FPF can be written as a weighted sum of sample values. The latter must be evaluated by system analysis (the most time-consuming task) but they do not depend on the distribution. Direct Monte Carlo simulation, importance sampling and Subset Simulation are incorporated under the proposed approach. Examples are given to illustrate their application.     

Fault detection and identification in dynamic systems with noisy data and parameter/modeling uncertainties

A probabilistic approach is presented which can be used for the estimation of system parameters and unmonitored state variables towards model-based fault diagnosis in dynamic systems. The method can be used with any type of input–output model and can accommodate noisy data and/or parameter/modeling uncertainties. The methodology is based on Markovian representation of system dynamics in discretized state space. The example system used for the illustration of the methodology focuses on the intake, fueling, combustion and exhaust components of internal combustion engines. The results show that the methodology is capable of estimating the system parameters and tracking the unmonitored dynamic variables within user-specified magnitude intervals (which may reflect noise in the monitored data, random changes in the parameters or modeling uncertainties in general) within data collection time and hence has potential for on-line implementation.     

First excursion probabilities for linear systems by very efficient importance sampling

An analytical study of the failure region of the first excursion reliability problem for linear dynamical systems subjected to Gaussian white noise excitation is carried out with a view to constructing a suitable importance sampling density for computing the first excursion failure probability. Central to the study are ‘elementary failure regions’, which are defined as the failure region in the load space corresponding to the failure of a particular output response at a particular instant. Each elementary failure region is completely characterized by its design point, which can be computed readily using impulse response functions of the system. It is noted that the complexity of the first excursion problem stems from the structure of the union of the elementary failure regions. One important consequence of this union structure is that, in addition to the global design point, a large number of neighboring design points are important in accounting for the failure probability. Using information from the analytical study, an importance sampling density is proposed. Numerical examples are presented, which demonstrate that the efficiency of using the proposed importance sampling density to calculate system reliability is remarkable.     

New method for efficient Monte Carlo–Neumann solution of linear stochastic systems

The Neumann series is a well-known technique to aid the solution of uncertainty propagation problems. However, convergence of the Neumann series can be very slow, often turning its use highly inefficient. In this article, a λ convergence parameter is introduced, which yields accurate and efficient Monte Carlo–Neumann solutions of linear stochastic systems using first order Neumann expansions. The λ convergence parameter is found as solution to a distance minimization problem, for an approximation of the inverse of the system matrix using the Neumann series. The method presented herein is called Monte Carlo–Neumann with λ convergence, or simply MC–N λ method. The accuracy and efficiency of the MC–N λ method is demonstrated in application to stochastic beam bending problems.     

An experiment of state estimation for predictive maintenance using Kalman filter on a DC motor

Preventive maintenance (PM) is an effective approach to promoting reliability. Time-based and condition-based maintenance are two major approaches for PM. No matter which approach is adopted for PM, whether a failure can be early detected or even predicted is the key point. This paper presents the experimental results of a failure prediction method for preventive maintenance by state estimation using the Kalman filter on a DC motor. The rotating speed of the motor was uninterruptedly measured and recorded every 5 min from 1 April until 20 June 2001. The measured data are used to execute Kalman prediction and to verify the prediction accuracy. The resultant prediction errors are acceptable. Futhermore, the shorter the increment time for every step used in Kalman prediction, the higher prediction accuracy it achieves. Failure can be prevented in time so as to promote reliability by state estimation for predictive maintenance using the Kalman filter.     

Bayesian system identification of dynamical systems using large sets of training data: A MCMC solution

In the last 20 years the applicability of Bayesian inference to the system identification of structurally dynamical systems has been helped considerably by the emergence of Markov chain Monte Carlo (MCMC) algorithms – stochastic simulation methods which alleviate the need to evaluate the intractable integrals which often arise during Bayesian analysis. In this paper specific attention is given to the situation where, with the aim of performing Bayesian system identification, one is presented with very large sets of training data. Building on previous work by the author, an MCMC algorithm is presented which, through combing Data Annealing with the concept of ‘highly informative training data’, can be used to analyse large sets of data in a computationally cheap manner. The new algorithm is called Smooth Data Annealing.     

基于改进SCAM算法的结构物理参数识别贝叶斯方法

针对贝叶斯估计中逐分量自适应Metropolis(single component adaptive Metropolis,SCAM)算法易生成重复性样本,导致抽样效率低、结果误差大等问题,重新定义了提议分布方差的表达式,提出了改进的SCAM算法,使得抽样样本序列构成的马尔可夫链相对稳定。进而将贝叶斯理论与改进的SCAM算法相结合,求解结构物理参数的后验边缘概率分布、最优估计值以识别和估计结构损伤,通过理论分析和结构数值模拟算例验证了改进的SCAM算法的有效性。结果表明,改进的SCAM算法既提高了抽样效率,又提高了计算结果准确性,可应用于物理参数识别及损伤识别与评估等工作。     

A Bayesian evaluation of simulation models of multiple-site fatigue cracks

The rivet holes along the longitudinal top row of the outer skin of the fuselage over a two-bay length are considered as the independent structural unit for the simulated multiple-site fatigue cracks. Models of multiple-site fatigue cracks are proposed. The models are divided into several phases with some uncertain parameters. These phases are incorporated sequentially into a computer code with the Monte Carlo simulation method. The Bayesian estimation of uncertain parameters in the model can be identified on visual inspections by the Bayesian procedure from in-service inspection data measuring crack lengths of each rivet hole. In summary, this study evaluates effects of differences in the simulation models for the crack coalescence and failure phase for the distribution of inspection data measuring crack lengths with the Bayesian estimation of uncertain parameters from simulated in-service inspection data.     

An efficient framework for the reliability-based design optimization of large-scale uncertain and stochastic linear systems

This paper is focused on the development of an efficient reliability-based design optimization algorithm for solving problems posed on uncertain linear dynamic systems characterized by large design variable vectors and driven by non-stationary stochastic excitation. The interest in such problems lies in the desire to define a new generation of tools that can efficiently solve practical problems, such as the design of high-rise buildings in seismic zones, characterized by numerous free parameters in a rigorously probabilistic setting. To this end a novel decoupling approach is developed based on defining and solving a limited sequence of deterministic optimization sub-problems. In particular, each sub-problem is formulated from information pertaining to a single simulation carried out exclusively in the current design point. This characteristic drastically limits the number of simulations necessary to find a solution to the original problem while making the proposed approach practically insensitive to the size of the design variable vector. To demonstrate the efficiency and strong convergence properties of the proposed approach, the structural system of a high-rise building defined by over three hundred free parameters is optimized under non-stationary stochastic earthquake excitation.     

A Bayesian approach to modeling and predicting pitting flaws in steam generator tubes

Steam generators in nuclear power plants have experienced varying degrees of under-deposit pitting corrosion. A probabilistic model to accurately predict pitting damage is necessary for effective life-cycle management of steam generators. This paper presents an advanced probabilistic model of pitting corrosion characterizing the inherent randomness of the pitting process and measurement uncertainties of the in-service inspection (ISI) data obtained from eddy current (EC) inspections. A Markov chain Monte Carlo simulation-based Bayesian method, enhanced by a data augmentation technique, is developed for estimating the model parameters. The proposed model is able to predict the actual pit number, the actual pit depth as well as the maximum pit depth, which is the main interest of the pitting corrosion model. The study also reveals the significance of inspection uncertainties in the modeling of pitting flaws using the ISI data: Without considering the probability-of-detection issues and measurement errors, the leakage risk resulted from the pitting corrosion would be under-estimated, despite the fact that the actual pit depth would usually be over-estimated.     

Modeling and state of charge estimation of lithium-ion battery

Modeling and state of charge(SOC) estimation of lithium-ion(Li-ion) battery are the key techniques of battery pack management system(BMS) and critical to its reliability and safety operation.An auto-regressive with exogenous input(ARX) model is derived from RC equivalent circuit model(ECM) due to the discrete-time characteristics of BMS.For the time-varying environmental factors and the actual battery operating conditions,a variable forgetting factor recursive least square(VFFRLS)algorithm is adopted as an adaptive parameter identification method.Based on the designed model,an SOC estimator using cubature Kalman filter(CKF) algorithm is then employed to improve estimation performance and guarantee numerical stability in the computational procedure.In the battery tests,experimental results show that CKF SOC estimator has a more accuracy estimation than extended Kalman filter(EKF) algorithm,which is widely used for Li-ion battery SOC estimation,and the maximum estimation error is about 2.3%.     

利用贝叶斯推理估计二维含源对流扩散方程参数

为了克服观测数据的不确定性给参数反演带来的困难,利用贝叶斯推理建立了二维含源对流扩散方程参数估计的数学模型。通过贝叶斯定理,获得了模型参数的后验分布,从而获得反问题的解。对于多参数反演问题,基于数值解计算得到的参数后验分布很难直观地表现出来,采用马尔科夫链蒙特卡罗方法对参数的后验分布进行采样,获得了扩散系数和降解系数的估计值。研究了观测点位置对计算结果的影响;同时研究了似然函数的形式对估计结果的影响,结果表明在异常值可能出现时采用Laplace分布型的似然函数可以获得稳健估计。对不同观测点数目下的估计值进行了对比,认为对于二维稳态对流扩散方程的双参数估计问题,至少需要两个观测点才有可能得到合理的解。     

A combination of Monte Carlo simulation and cellular automata for computing the availability of complex network systems

This paper combines Monte Carlo simulation and cellular automata for computing the availability of a complex network system and the importance measures of its elements.