# 统计代写|随机控制代写Stochastic Control代考|ELEC6410

## 统计代写|随机控制代写Stochastic Control代考|Polynomial function approximation

Consider a nonlinear function defined by
\begin{aligned} &u_k=u_k^i+v_k \ &y_k^i=1-2 u_k+u_k^2 \ &y_k=y_k^i+c_k-0.25 c_{k-1} \end{aligned}
In Fig. 2 are shown the true system $\left(u_k^i \in[0,2], y_k^i\right)$ and the noisy $\left(u_k, y_k\right)$ input-output observations with measurements corrupted by normal noise conditions of $\sigma_c=\sigma_v=0.2$. The results for the TS fuzzy models obtained by applying the proposed FIV algorithm as well as the LS estimation to tune the consequent parameters are shown in Fig. 3. It can be seen, clearly, that the curves for the polynomial function and for the proposed FIV based identification almost cover each other. The fuzzy c-means clustering algorithm was used to criate the antecedent membership functions of the TS fuzzy models, which are shown in Fig. 4. The FIV was based on the filtered output from a “fuzzy auxiliar model” with the same structure of the TS fuzzy model used to identify the nonlinear function. The clusters centers of the membership functions for the LS and FIV estimations were $\mathrm{c}=[-0.0983,0.2404,0.6909,1.1611]^T$ and $\mathbf{c}=[0.1022,0.4075,0.7830,1.1906]^T$, respectively. The TS fuzzy models have the following structure:
$R^i: \mathrm{IF}{y_k}$ is $F_i$ THEN $\hat{y}_k=a_0+a_1 u_k+a_2 u_k^2$ where $i=1,2, \ldots, 4$. For the FIV approach, the “fuzzy auxiliar model” has the following structure: $$R^i: \text { IF } y{\text {filt }} \text { is } F_i \text { THEN } y_{\text {filt }}=a_0+a_1 u_k+a_2 y_{\text {filt }}^2$$
where $y_{\text {filt }}$ is the filtered output, based on the consequent parameters LS estimation, and used to criate the membership functions, as shown in Fig. 4, as well as the instrumental variable matrix.

## 统计代写|随机控制代写Stochastic Control代考|Main results

Consider the following uncertain stochastic system with time-varying delay and nonlinear stochastic perturbations:

$$\left{\begin{array}{l} d x(t)=[(A+\Delta A(t)) x(t)+(B+\Delta B(t)) x(t-\tau(t))+f(t, x(t), x(t-\tau(t)))] d t+g(t, x(t), x(t-\tau(t))) d \omega(t), \ x(t)=\phi(t), \quad t \in[-\tau, 0], \end{array}\right.$$
where $x(t) \in \mathbb{R}^n$ is the state vector, $A, B, C, D$ are known real constant matrices with appropriate dimensions, $\omega(t)$ is a scalar Brownian motion defined on a complete probability space $(\Omega, F, P)$ with a nature filtration $\left{F_t\right}_{t \geq 0} . \phi(t)$ is any given initial data in $L_{F_0}^2\left([-\tau, 0] ; \mathbb{R}^n\right)$. $\tau(t)$ denotes the time-varying delay and is assumed to satisfy either (2a) or (2b):
\begin{aligned} 0 \leq \tau(t) & \leq \tau, \dot{\tau}(t) \leq d<1 \ 0 & \leq \tau(t) \leq \tau \end{aligned}
where $\tau$ and $d$ are constants and the upper bound of $\tau(t)$ and $\dot{\tau}(t)$, respectively. $\Delta A(t)$, $\Delta B(t)$ are all unknown time-varying matrices with appropriate dimensions which represent the system uncertainty and stochastic perturbation uncertainty, respectively. We assume that the uncertainties are norm-bounded and can be described as follows:
$$[\Delta A(t) \quad \Delta B(t)]=E F(t)\left[\begin{array}{ll} G_1 & G_2 \end{array}\right],$$
where $E, G_1, G_2$ are known real constant matrices with appropriate dimensions, $F(t)$ are unknown real matrices with Lebesgue measurable elements bounded by:
$$F^{\mathrm{T}}(t) F(t) \leq I .$$
$f(;, ;): R_{+} \times R^n \times R^n \rightarrow R^n$ and $g(\cdot, ;): R_{+} \times R^n \times R^n \rightarrow R^{n \times m}$ denote the nonlinear uncertainties which is locally Lipschitz continuous and satisfies the following linear growth conditions
$$|f(t, x(t), x(t-\tau(t)))| \leq\left|F_1 x(t)\right|+\left|F_2 x(t-\tau(t))\right|,$$
and
$$\text { Trace }\left[g^{\mathrm{T}}(t, x(t), x(t-\tau(t))) g(t, x(t), x(t-\tau(t)))\right] \leq\left|H_1 x(t)\right|^2+\left|H_2 x(t-\tau(t))\right|^2,$$
Throughout this paper, we shall use the following definition for the system (1).

## 统计代写|随机控制代写随机控制代考|多项式函数逼近

\begin{aligned} &u_k=u_k^i+v_k \ &y_k^i=1-2 u_k+u_k^2 \ &y_k=y_k^i+c_k-0.25 c_{k-1} \end{aligned}

$R^i: \mathrm{IF}{y_k}$ is $F_i$ THEN $\hat{y}k=a_0+a_1 u_k+a_2 u_k^2$其中$i=1,2, \ldots, 4$。对于FIV方法，“模糊辅助模型”具有如下结构:$$R^i: \text { IF } y{\text {filt }} \text { is } F_i \text { THEN } y{\text {filt }}=a_0+a_1 u_k+a_2 y_{\text {filt }}^2$$
，其中$y_{\text {filt }}$是经过过滤的输出，基于随后的参数LS估计，并用于创建隶属函数，如图4所示，以及工具变量矩阵。

## 统计代写|随机控制代写随机控制代考|主要结果

$$\left{\begin{array}{l} d x(t)=[(A+\Delta A(t)) x(t)+(B+\Delta B(t)) x(t-\tau(t))+f(t, x(t), x(t-\tau(t)))] d t+g(t, x(t), x(t-\tau(t))) d \omega(t), \ x(t)=\phi(t), \quad t \in[-\tau, 0], \end{array}\right.$$

\begin{aligned} 0 \leq \tau(t) & \leq \tau, \dot{\tau}(t) \leq d<1 \ 0 & \leq \tau(t) \leq \tau \end{aligned}
，其中$\tau$和$d$分别为常数和$\tau(t)$和$\dot{\tau}(t)$的上界。$\Delta A(t)$、$\Delta B(t)$均为具有适当维数的未知时变矩阵，分别表示系统不确定性和随机摄动不确定性。我们假设不确定性是范数有界的，可以描述如下:
$$[\Delta A(t) \quad \Delta B(t)]=E F(t)\left[\begin{array}{ll} G_1 & G_2 \end{array}\right],$$

$$F^{\mathrm{T}}(t) F(t) \leq I .$$
$f(;, ;): R_{+} \times R^n \times R^n \rightarrow R^n$和$g(\cdot, ;): R_{+} \times R^n \times R^n \rightarrow R^{n \times m}$表示局部Lipschitz连续的非线性不确定性，满足以下线性增长条件
$$|f(t, x(t), x(t-\tau(t)))| \leq\left|F_1 x(t)\right|+\left|F_2 x(t-\tau(t))\right|,$$

$$\text { Trace }\left[g^{\mathrm{T}}(t, x(t), x(t-\tau(t))) g(t, x(t), x(t-\tau(t)))\right] \leq\left|H_1 x(t)\right|^2+\left|H_2 x(t-\tau(t))\right|^2,$$

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