随机动力学响应的高斯相关过程模拟
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摘要
动力学响应是描述系统振动状态、评估其性能、用于控制等的重要变量,复杂系统的不确定性、随机激励样本的不可测量性等导致传统随机响应时程与统计分析的计算困难,因此需要发展基于系统响应观测的、直接的随机过程概率模型与评估新方法。近年来,人工智能与数据处理技术等领域发展的无确定性系统模型的、直接随机过程概率模型,及其概率评估、系统状态预测等方法为动力学响应的概率分析提供了新思路,特别是具有很好普适性与可分析性的高斯相关过程已具有较完整的理论方法。鉴于此,提出针对动力学系统响应的、直接的随机过程概率模型与评估方法,并作探索性研究。先基于高斯白噪声激励动力学系统响应的统计特性分析,说明系统响应的高斯随机过程特性、响应在时间维度上的相关性、及其协方差随时间差的指数衰减特性等;再给出该系统响应的高斯相关过程概率建模与评估方法,包括由响应协方差计算,高斯过程协方差或核函数的拟合,到高斯相关过程概率模型的确定,响应样本过程的直接生成,及其统计评估等,并给出高斯相关过程的贝叶斯更新与系统状态预测有关基本公式。数值结果表明该高斯相关过程的概率建模与响应评估方法可行且有效。
The dynamic response is an important criterion to evaluate system vibration state and performance and control the system. The conventional stochastic response analysis is hampered by the system complexity and uncertainty and the measurement difficulty of random excitations. Thus, new methods for direct random process probability modeling and statistics analysis based on system response observation need to be studied and developed. Recently, the random process probability modeling, probability evaluation and system state prediction of uncertainty system model developed in artificial intelligence and data processing technology have provided different approaches for the probability analysis of dynamic response. Especially, the Gaussian correlation process with better applicability and mathematical expression has provided a complete theoretical method. Therefore, this paper proposes a direct method for the random process probability modeling and statistics analysis of dynamic system response and presents a preliminary research. Based on the statistics characteristics of dynamic system response to Gaussian white noise excitation, Gaussian random process characteristics of the system response, response correlation in time domain and exponential reduction of response covariance are analyzed. Then, the probability modeling and statistics analysis method of Gaussian correlation process is given. It includes response covariance calculation, Gaussian process covariance and kernel function fitting, probability modeling of Gaussian correlation process,direct response sample production and statistics evaluation. The basic expressions of Bayesian update and system state prediction of Gaussian correlation process are also given. Numerical results illustrate the feasibility and availability of the proposed probability modeling and statistics analysis method of Gaussian correlation process.
The dynamic response is an important criterion to evaluate system vibration state and performance and control the system. The conventional stochastic response analysis is hampered by the system complexity and uncertainty and the measurement difficulty of random excitations. Thus, new methods for direct random process probability modeling and statistics analysis based on system response observation need to be studied and developed. Recently, the random process probability modeling, probability evaluation and system state prediction of uncertainty system model developed in artificial intelligence and data processing technology have provided different approaches for the probability analysis of dynamic response. Especially, the Gaussian correlation process with better applicability and mathematical expression has provided a complete theoretical method. Therefore, this paper proposes a direct method for the random process probability modeling and statistics analysis of dynamic system response and presents a preliminary research. Based on the statistics characteristics of dynamic system response to Gaussian white noise excitation, Gaussian random process characteristics of the system response, response correlation in time domain and exponential reduction of response covariance are analyzed. Then, the probability modeling and statistics analysis method of Gaussian correlation process is given. It includes response covariance calculation, Gaussian process covariance and kernel function fitting, probability modeling of Gaussian correlation process,direct response sample production and statistics evaluation. The basic expressions of Bayesian update and system state prediction of Gaussian correlation process are also given. Numerical results illustrate the feasibility and availability of the proposed probability modeling and statistics analysis method of Gaussian correlation process.
引文
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[7]张巍,应祖光,叶淑琴.利用子结构模态的结构随机响应精确综合[J].噪声与振动控制,2010,30(6):75-78.
[2] MILLER S L, CHILDERS D G. Probability and random processes with applications to signal processing and communications[M]. Oxford:Academic Press, 2012.
[3] RASMUSSEN C E, WILLIAMS C K I. Gaussian processes for machine learning[M]. Cambridge:MIT Press, 2006.
[4] BERGER J O. Statistical decision theory and Bayesian analysis[M]. New York:Springer, 2010.
[5] DIAZDELAO F A, ADHIKARI S. Structural dynamic analysis using Gaussian process emulators[J].Engineering Computations, 2010, 27:580-605.
[6] XIA Z, TANG J. Characterization of dynamic response of structures with uncertainty by using Gaussian processes[J]. ASME Journal of Vibration and Acoustics, 2013,135:051006.
[7]张巍,应祖光,叶淑琴.利用子结构模态的结构随机响应精确综合[J].噪声与振动控制,2010,30(6):75-78.