Updating/Filtering of Nonlinear Dynamical Systems for Structural Health Monitoring

由于土木工程结构涉及许多不确定性，它们通常被建模为随机场。工程知识和系统或类似系统的过去数据有助于建立模型，即随机场。如果在现场对系统进行观测，则可以利用系统上新获得的信息对模型进行更新。这个过程可以解释为一个贝叶斯更新或过滤先验模型到后验模型的过程。本研究将处理一个随机场，它被建模为一个离散状态向量方程。x_n = F(x__ <n-1>,v_n)(1)，以及将状态向量x_n与观测向量byy_n = H(x_n)+w_n(2)联系起来的观测向量方程，其中x_n = k^*1的向量，v_n = r^*1的系统噪声向量，pdf q(v_n), y_n = s^*1的观测向量，w_n = s^*1的观测噪声向量，pdf r（w_n）。对于由式(1)和式(2)给出的随机场，首先讨论了更新/滤波先验随机场的数学工具的基本公式。在过去，高斯线性场被成功地处理，例如，卡尔曼滤波。然而，工程系统如结构或地基系统可能是非高斯和非线性的，由于其非高斯和非线性的性质，在处理概率分布时可能会遇到一些困难。Kitagawa的蒙特卡罗滤波作为一种通用的更新系统的工具，是一种顺序算法，它分别对预测的状态向量和滤波的状态向量生成一组样本实现。为了阐明该方法的潜力，首先讨论了非线性系统动态参数的识别，并利用数值模拟数据证明了非高斯空间随机场的随机插值。

Since civil engineering structures are involved in many uncertainties, they are often modeled as stochastic fields. Engineering knowledge and past data on a system or system of similar kinds are helpful to setting up the model, that is, a stochastic field. If observation is carried out on a system at site, then the model may be updated by the newly obtained information on the system. This procedure may be interpreted as a procedure of Bayesian updating or filtering of a prior model into a posterior model.This research will deal with a stochastic field, which is modeled as a discrete state vector equation.x_n = F(x_<n-1>,v_n) (1)and an observation vector equation which relates the state vector X_n to the observation vector byy_n = H(x_n)+w_n (2)where x_n = vector of k^*1 v_n = system noise vector of r^*1 with pdf q(v_n), y_n = observation vector of s^*1, w_n = observation noise vector of s^*1 with pdf r(w_n).For stochastic fields given by eqs(1) and (2), basic formulas are first discussed from which mathematical tools stem for updating/filtering of prior stochastic fields. In the past, Gaussian linear fields were successfully treated by, for example, the Kalman filter. However, engineering systems such as structural or soil foundation systems might be non-Gaussian and nonlinear in the nature and we might encounter with some difficulty due to the non-Gaussian and nonlinear properties in the dealing with probabilistic distributions. As a versatile tool to update such systems, Monte Carlo filter by Kitagawa is focused that is a sequential algorithm of generating a set of sample realizations of a predicted state vector and a filtered state vector respectively. In order to clarify the potential of this method, identification of dynamic parameters of a nonlinear system is first discussed, and stochastic interpolation of a non-Gaussian spatial random field is also demonstrated by using numerically simulated data.