### 统计代写|随机控制代写Stochastic Control代考|Theoretical Principles

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机控制代写Stochastic Control代考|Preliminary Remarks

Stochastic optimal control is a subfield of control theory, which focuses upon the stochastic systems and develops into a cross-discipline between the stochastic process theory and the optimal control theory. The associated theories and technologies with the electronics and information engineering, mechanical engineering, and aerospace engineering, were flourished since $1960 \mathrm{~s}$, and just concerned the state adjustment of systems under random disturbances such as random excitations and measurement noise. The development in the field of civil engineering began after the seventies of twentieth century. Different from the requirements of the fields of mechanical engineering and aerospace engineering, the civil engineering structures exhibit a large size and experience a complicated external excitation. They have to encounter a series of challenging issues in regard to the safety, the durability, and the comfortability. These issues become more serious in the case of hazardous actions with uncertainties inherent in the occurring time, occurring space, and occurring intensity. The conventional stochastic optimal control theory, however, originated from the random process theory assumes the white Gaussian noise as the random disturbance, which is obviously far from the hazardous actions of engineering structures. Therefore, it is necessary to explore a logical theory and pertinent methods for the stochastic optimal control of civil engineering structures which circumvents the dilemma encountered by the conventional stochastic optimal control theory.

This chapter aims at addressing the theoretical principles relevant to the succeeding chapters in this book. The remaining sections included in this chapter include the classical stochastic optimal control, the random vibration of structures, and its advances that underlies the solution methods for controlled stochastic dynamical systems, the dynamic reliability of structures that underlies the design basis for probabilistic criteria of stochastic optimal control of structures, and the modeling of random dynamic excitations that underlies the uncertainty quantification and simulation of hazardous actions of engineering structures. Through integrating the involved sections, the principle for the theory and methods of stochastic optimal control of structures are provided.

## 统计代写|随机控制代写Stochastic Control代考|Classical Stochastic Optimal Control

The stochastic optimal control aims at attaining the optimal control law that promotes the stochastic system to an expected state through minimizing a certain cost function by the celebrated optimal control schemes. It is well recognized that the pioneering work on the optimal control theory is the proposal of calculus of variations. In history, Pierre and Fermat introduced firstly the so-called Fermat’s least action principle to explore the minimum path of ray propagating through the optical media in 1662. In 1755, Lagrange introduced the delta calculus, and then Euler proposed the elementary definition of variation method. In $1930 \mathrm{~s}$, the Hamilton-Jacobi equation was derived in the framework of the variation method owing to Hamilton and Jacobi’s contributions. Till the mid-twentieth century, the classical variation theory was completely established. The research of modern optimal control theory began from the late period of World War II. Its theoretical milestones consist of the maximum principle proposed by Pontryagin in 1956 , the dynamic programming proposed by Bellman in 1957 , the state-space method, and linear filtering theory developed by Kalman in 1960 (Yong and Zhou 1999). In early of 1960 s, owing to the developments of the stochastic maximum principle (Kushner 1962) and the stochastic dynamic programming (Florentin 1961 ), the research of stochastic optimal control theory was marked as the beginning.

In state space, the equation of motion of a controlled stochastic dynamical system can be written as
$$\dot{\mathbf{Z}}(t)=\mathbf{g}[\mathbf{Z}(t), \mathbf{U}(t), \mathbf{w}(t), t], \mathbf{Z}\left(t_{0}\right)=\mathbf{Z}_{0}$$
The output equation of the system is given by
$$\hat{\mathbf{Z}}(t)=\mathbf{h}[\mathbf{Z}(t), \mathbf{U}(t), \mathbf{w}(t), t]$$
The measure equation of the system is then given by
$$\mathbf{Y}(t)=\mathbf{j}[\hat{\mathbf{Z}}(t), \mathbf{n}(t), t]$$
where $\mathbf{Z}(t)$ is the $2 n$-dimensional column vector denoting system state; $\hat{\mathbf{Z}}(t)$ is the $m$ dimensional vector denoting system output; $\mathbf{U}(t)$ is the $r$-dimensional vector denoting control force; $\mathbf{w}(t)$ is the $s$-dimensional vector denoting random excitations; $\mathbf{n}(t)$ is the $m$-dimensional vector denoting measurement noise; $\mathbf{Y}(t)$ is the $m$-dimensional measured vector denoting system state; $\mathbf{g}(\cdot)$ is the $2 n$-dimensional functional vector denoting system state evolution; $\mathbf{h}(\cdot), \mathbf{j}(\cdot)$ are the $m$-dimensional functional vectors denoting the output and measurement of systems, respectively, which both rely upon the number of sensors.

## 统计代写|随机控制代写Stochastic Control代考|Spectral Transfer Matrix Method

A linear stochastic dynamical system is considered as follows:
$$\mathbf{M} \ddot{\mathbf{X}}(t)+\mathbf{C} \dot{\mathbf{X}}(t)+\mathbf{K X}(t)=\mathbf{F}(\boldsymbol{\Theta}, t)$$
where $\mathbf{M}, \mathbf{C}$, and $\mathbf{K}$ are the $n \times n$ mass, damping, and stiffness matrices, respectively; $\ddot{\mathbf{X}}(t), \dot{\mathbf{X}}(t), \mathbf{X}(t)$ are the $n$-dimensional column vectors denoting system acceleration, velocity, and displacement, respectively; $\mathbf{F}(\boldsymbol{\Theta}, t)$ is the $n$-dimensional column vector denoting random excitations, and $\boldsymbol{\Theta}$ is an $n_{\boldsymbol{\Theta} \text {-dimensional vector denoting random }}$ parameters of system which exhibits the joint probability density function $p_{\boldsymbol{\Theta}}(\boldsymbol{\theta})$.
Defining the $n \times n$ unit impulse response function matrices $\mathbf{h}(t)$, where the component $h_{i j}(t)$ denotes the response of the $i$ th degree in the case that the unit impulse acts on the $j$ th degree of the system, one can attain the system response $\mathbf{h}{j}(t)$ from the equation of motion as follows: $$\mathbf{M} \ddot{\mathbf{h}}{j}(t)+\mathbf{C} \dot{\mathbf{h}}{j}(t)+\mathbf{K h}{j}(t)=\mathbf{I}{j} \delta(\boldsymbol{\Theta}, t)$$ where $\mathbf{I}{j}=(\underbrace{0,0, \ldots, 0,1}_{j}, 0, \ldots, 0)^{\mathrm{T}}$ is $n$-dimensional column vectors denoting the location of the unit impulse $\delta(\boldsymbol{\Theta}, t)$ acting on the $j$ th degree of the system.

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## MATLAB代写

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