### 统计代写|随机控制代写Stochastic Control代考| Polynomial Chaos Expansion

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## 统计代写|随机控制代写Stochastic Control代考|Polynomial Chaos Expansion

The solution of Eq. (2.3.35) can be represented by a truncated polynomial chaos expansion (PCE) (Ghanem and Spanos 1991), i.e.,
$$x_{i}(t)=\sum_{l=0}^{P} x_{i l}(t) \Psi_{l}(\xi)$$
where $\xi$ is the $M$-dimensional row vector of Gaussian random variables; $P$ denotes the highest order of the polynomial chaos expansion; $\Psi_{l}(\xi)$ denotes the polynomial chaos with parameter of Gaussian random variables; $x_{i l}(t)$ denotes the deterministic coefficient pertaining to the polynomial chaos which is often referred to as the random mode.
Substituting Eq. (2.3.37) into Eq. (2.3.35), then yields
\begin{aligned} &\sum_{i=1}^{n} \sum_{l=0}^{P} m_{j i} \ddot{x}{i l}(t) \Psi{l}(\xi)+\sum_{i=1}^{n} \sum_{k=0}^{q} \alpha_{j i, k}\left(\sum_{l=0}^{P} \dot{x}{i l}(t) \Psi{l}(\xi)\right)^{q-k}\left(\sum_{l=0}^{P} x_{i l}(t) \Psi_{l}(\xi)\right)^{k} \ &\quad=\sum_{l=0}^{P} F_{j l}(t) \Psi_{l}(\xi) \end{aligned}
Introducing a Galerkin projection technique (Ghanem and Spanos 1991), the polynomial chaos arises to pairwise orthogonal with respect to Gaussian measure, i.e.,
$$\left\langle\Psi_{i} \Psi_{j}\right\rangle=\left\langle\Psi_{i}^{2}\right\rangle \delta_{i j}$$
where $\langle\cdot\rangle$ denotes the inner product; $\delta_{i j}$ denotes the Kronecker delta function with two variables, which is 1 if the variables are equal, and 0 otherwise:
$$\delta_{i j}=\left{\begin{array}{l} 1, i=j \ 0, i \neq j \end{array}\right.$$
Equation (2.3.38) is thus discretized into an equation set:

\begin{aligned} &\sum_{i=1}^{n} m_{j i} \ddot{x}{i m}(t)+\sum{i=1}^{n} \sum_{k=0}^{q} \sum_{l_{1}=0}^{P} \cdots \sum_{l_{q-l}=0}^{P} \sum_{l_{q-k+1}=0}^{P} \cdots \sum_{l_{q}=0}^{P} \frac{c_{l_{1} \cdots l_{q-\lambda} l_{q-\lambda+1} \cdots l_{q} m}}{\left(\Psi_{m}^{2}\right)} \ &\alpha_{j i, k} \dot{x}{i l{1}}(t) \cdots \dot{x}{i l{q-k}}(t) x_{i l_{q-k+1}}(t) \cdots x_{i l_{q}}(t)=F_{j m}(t) \end{aligned}
where $c_{l_{1} \cdots l_{q-k} l_{q-k+1} \cdots l_{q} m}=\left\langle\Psi_{l_{1}} \ldots \Psi_{l_{q-k}} \Psi_{l_{q-k+1}} \ldots \Psi_{l_{q}} \Psi_{m}\right\rangle, m=0,1,2, \ldots, P$. The coefficient $c_{l_{1} \cdots l_{4-k} l_{4-k+1} \cdots l_{q} m}$ and $\left\langle\Psi_{m}^{2}\right\rangle$ can be derived from a multiple-dimensional integral (Ghanem and Spanos 1991).

## 统计代写|随机控制代写Stochastic Control代考|Statistical Linearization Technique

An alternative method for random vibration analysis of nonlinear systems is the statistical linearization technique (Roberts and Spanos 1990). This method exhibits

a hypothesis that the structural response is viewed as a stationary Gaussian process, thereby the equivalence between the linearized system and the original nonlinear system is attained by minimizing their differences in the sense of mean square. The random vibration analysis of nonlinear systems can then be carried out by the pertinent theory and methods to the random vibration of linear systems.

Therefore, the nonlinear multiple-degree-of-freedom system, shown in Eq. (2.3.34), can be substituted by a linearized system with equation of motion as follows:
$$\mathbf{M} \ddot{\mathbf{X}}(t)+\mathbf{C}{\mathrm{eq}} \dot{\mathbf{X}}(t)+\mathbf{K}{\mathrm{eq}} \mathbf{X}(t)=\mathbf{F}(\boldsymbol{\Theta}, t)$$
where $\mathbf{C}{e q}, \mathbf{K}{\text {eq }}$ are the $n \times n$ equivalent damping and equivalent stiffness matrices, respectively.

Comparing Eqs. (2.3.34) and (2.3.45), and assuming that the linearized system and the original system have a same response, one can define the error vector between the internal forces of the two systems as follows:
$$\mathbf{e}=\mathbf{f}(\mathbf{X}(t), \dot{\mathbf{X}}(t))-\mathbf{C}{\mathrm{eq}} \dot{\mathbf{X}}(t)-\mathbf{K}{\mathrm{eq}} \mathbf{X}(t)$$
Minimization of the covariance matrix of the error vector, i.e.,
\begin{aligned} &\frac{\partial E\left[\mathbf{e e}^{\mathrm{T}}\right]}{\partial \mathbf{C}{\mathrm{eq}}}=0 \ &\frac{\partial E\left[\mathbf{e}^{\mathrm{T}}\right]}{\partial \mathbf{K}{\mathrm{eq}}}=0 \end{aligned}
yields the basic equations:
\begin{aligned} &\mathbf{C}{\mathrm{eq}} E\left[\dot{\mathbf{X}} \dot{\mathbf{X}}^{\mathrm{T}}\right]+\mathbf{K}{\mathrm{eq}} E\left[\mathbf{X} \dot{\mathbf{X}}^{\mathrm{T}}\right]=E\left[\mathbf{f}(\mathbf{X}, \dot{\mathbf{X}}) \dot{\mathbf{X}}^{\mathrm{T}}\right] \ &\mathbf{C}{\mathrm{eq}} E\left[\dot{\mathbf{X}} \mathbf{X}^{\mathrm{T}}\right]+\mathbf{K}{\mathrm{eq}} E\left[\mathbf{X} \mathbf{X}^{\mathrm{T}}\right]=E\left[\mathbf{f}(\mathbf{X}, \dot{\mathbf{X}}) \mathbf{X}^{\mathrm{T}}\right] \end{aligned}
Given the joint probability density functions for solving the mathematical expectation of responses, shown in Eqs. (2.3.48a) and (2.3.48b), the equivalent damping and equivalent stiffness matrices can be readily attained. This treatment, however, often refers to an iteration procedure, as shown in Fig. 2.3, where the tolerant error can be set as the difference of response vectors or as the norm of the difference of mean-square response vectors between the sequential steps.

As to a single-degree-of-freedom system, the basic equations with respect to the equivalent damping and equivalent stiffness matrices are given as follows:
$$C_{\mathrm{eq}} E\left[\dot{X}^{2}\right]+K_{\mathrm{eq}} E[X \dot{X}]=E[f(X, \dot{X}) \dot{X}]$$

## 统计代写|随机控制代写Stochastic Control代考|Fokker–Planck–Kolmogorov Equation

The mean-square solution of system response under random vibration just includes the former two-order moments information of stochastic dynamical systems, which is insufficient to represent the stochastic response as a complete probabilistic density function, especially for the nonlinear system, of which the probabilistic distribution is distinguished from the normal distribution. Therefore, seeking for the probability density of stochastic dynamical system has received extensive attention. Owing to the contributions from Fokker, Planck, and Kolmogorov, the probability density evolution equation related to random excitations were established in 1930 s. This is the celebrated Fokker-Planck-Kolmogorov equation, i.e., FPK equation, in the classical random vibration theory.

Considering a random process $\mathbf{Z}(t)$, one has the Itô stochastic differential equation as follows:
$$\mathrm{d} \mathbf{Z}(t)=\mathbf{A}(\mathbf{Z}, t) \mathrm{d} t+\mathbf{B}(\mathbf{Z}, t) \mathrm{d} \mathbf{w}(t)$$
As for a random function $f(\mathbf{Z})$ in terms of random process $\mathbf{Z}(t)$, the Taylor series expansion is given by
\begin{aligned} \mathrm{d} f(\mathbf{Z}) &=\sum_{i=1}^{m} \frac{\partial f}{\partial z_{i}} \mathrm{~d} z_{i}+\frac{1}{2} \sum_{i=1}^{m} \sum_{j=1}^{m} \frac{\partial^{2} f}{\partial z_{i} \partial z_{j}} \mathrm{~d} z_{i} \mathrm{~d} z_{j}+\cdots \ &=\sum_{i=1}^{m} \frac{\partial f}{\partial z_{i}}\left[A_{i} \mathrm{~d} t+\sum_{k=1}^{r} B_{i k} \mathrm{~d} w_{k}(t)\right]+\frac{1}{2} \sum_{i=1}^{m} \sum_{j=1}^{m}\left[\frac{\partial^{2} f}{\partial z_{i} \partial z_{j}} \sum_{k=1}^{r} B_{i k} \mathrm{~d} w_{k}(t) \sum_{s=1}^{r} B_{j s} \mathrm{~d} w_{s}(t)\right]+\cdots \end{aligned}
Taking mathematical expectation on both sides of Eq. (2.3.53), and utilizing the product $E\left[(\mathrm{~d} w(t))^{2}\right]=W \mathrm{~d} t$, the Taylor series expansion has a truncated formulation with respect to $\mathrm{d} t$ :
$$E[\mathrm{~d} f(\mathbf{Z})]=E\left{\left[\sum_{i=1}^{m} A_{i} \frac{\partial f}{\partial z_{i}}+\frac{1}{2} \sum_{i=1}^{m} \sum_{j=1}^{m}\left(\mathbf{B} \mathbf{W B}^{\mathrm{T}}\right){i j} \frac{\partial^{2} f}{\partial z{i} \partial z_{j}}\right] \mathrm{d} t\right}$$
where $\mathbf{W}(t)$ is the $s \times s$ symmetric, and semi-positive spectral density matrix, shown in Eq. (2.2.4). It is noted as well $E\left[\mathrm{~d} w_{k}(t)\right]=0$.

Noting the conditional probability density of $\mathbf{Z}(t)$ as $\tilde{p}{\mathbf{Z}}\left(\mathbf{z}, t \mid \mathbf{z}{0}, t_{0}\right)$, the derivative of left side of Eq.

## 统计代写|随机控制代写Stochastic Control代考|Polynomial Chaos Expansion

X一世(吨)=∑l=0磷X一世l(吨)Ψl(X)

∑一世=1n∑l=0磷米j一世X¨一世l(吨)Ψl(X)+∑一世=1n∑ķ=0q一种j一世,ķ(∑l=0磷X˙一世l(吨)Ψl(X))q−ķ(∑l=0磷X一世l(吨)Ψl(X))ķ =∑l=0磷Fjl(吨)Ψl(X)

⟨Ψ一世Ψj⟩=⟨Ψ一世2⟩d一世j

$$\delta_{ij}=\left{ 1,一世=j 0,一世≠j\对。$$

∑一世=1n米j一世X¨一世米(吨)+∑一世=1n∑ķ=0q∑l1=0磷⋯∑lq−l=0磷∑lq−ķ+1=0磷⋯∑lq=0磷Cl1⋯lq−λlq−λ+1⋯lq米(Ψ米2) 一种j一世,ķX˙一世l1(吨)⋯X˙一世lq−ķ(吨)X一世lq−ķ+1(吨)⋯X一世lq(吨)=Fj米(吨)

## 统计代写|随机控制代写Stochastic Control代考|Statistical Linearization Technique

∂和[和和吨]∂C和q=0 ∂和[和吨]∂ķ和q=0

C和q和[X˙X˙吨]+ķ和q和[XX˙吨]=和[F(X,X˙)X˙吨] C和q和[X˙X吨]+ķ和q和[XX吨]=和[F(X,X˙)X吨]

C和q和[X˙2]+ķ和q和[XX˙]=和[F(X,X˙)X˙]

## 统计代写|随机控制代写Stochastic Control代考|Fokker–Planck–Kolmogorov Equation

d从(吨)=一种(从,吨)d吨+乙(从,吨)d在(吨)

dF(从)=∑一世=1米∂F∂和一世 d和一世+12∑一世=1米∑j=1米∂2F∂和一世∂和j d和一世 d和j+⋯ =∑一世=1米∂F∂和一世[一种一世 d吨+∑ķ=1r乙一世ķ d在ķ(吨)]+12∑一世=1米∑j=1米[∂2F∂和一世∂和j∑ķ=1r乙一世ķ d在ķ(吨)∑s=1r乙js d在s(吨)]+⋯

E[\mathrm{~d} f(\mathbf{Z})]=E\left{\left[\sum_{i=1}^{m} A_{i} \frac{\partial f}{\partial z_{i}}+\frac{1}{2} \sum_{i=1}^{m} \sum_{j=1}^{m}\left(\mathbf{B} \mathbf{WB}^ {\mathrm{T}}\right){i j} \frac{\partial^{2} f}{\partial z{i} \partial z_{j}}\right] \mathrm{d} t\right}E[\mathrm{~d} f(\mathbf{Z})]=E\left{\left[\sum_{i=1}^{m} A_{i} \frac{\partial f}{\partial z_{i}}+\frac{1}{2} \sum_{i=1}^{m} \sum_{j=1}^{m}\left(\mathbf{B} \mathbf{WB}^ {\mathrm{T}}\right){i j} \frac{\partial^{2} f}{\partial z{i} \partial z_{j}}\right] \mathrm{d} t\right}

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