# 统计代写|随机控制代写Stochastic Control代考|MATH4406

## 统计代写|随机控制代写Stochastic Control代考|Fuzzy Structure Model

The nonlinear input-output representation is often used for building TS fuzzy models from data, where the regression vector is represented by a finite number of past inputs and outputs of the system. In this work, the nonlinear autoregressive with exogenous input (NARX) structure model is used. This model is applied in most nonlinear identification methods such as neural networks, radial basis functions, cerebellar model articulation controller (CMAC), and also fuzzy logic (Brown \& Harris (1994)). The NARX model establishes a relation between the collection of past scalar input-output data and the predicted output
$$y(k+1)=F\left[y(k), \ldots, y\left(k-n_y+1\right), u(k), \ldots, u\left(k-n_u+1\right)\right]$$
where $k$ denotes discrete time samples, $n_y$ and $n_u$ are integers related to the system’s order. In terms of rules, the model is given by
$$R^i: \operatorname{IF} y(k) \text { is } F_1^i \text { AND } \cdots \text { AND } y\left(k-n_y+1\right) \text { is } F_{n_y}^i$$
AND $u(k)$ is $G_1^i$ AND $\cdots$ AND $u\left(k-n_u+1\right)$ is $G_{n_u}^i$
$$\operatorname{THEN} \hat{y}i(k+1)=\sum{j=1}^{n_y} a_{i, j} y(k-j+1)+\sum_{j=1}^{n_\mu} b_{i, j} u(k-j+1)+c_i$$
where $a_{i, j}, b_{i, j}$ and $c_i$ are the consequent parameters to be determined. The inference formula of the TS fuzzy model is a straightforward extension of $(G)$ and is given by
$$y(k+1)=\frac{\sum_{i=1}^l h_i(\mathbf{x}) \hat{y}i(k+1)}{\sum{i=1}^l h_i(\mathbf{x})}$$
or
$$y(k+1)=\sum_{i=1}^l \gamma_i(\mathbf{x}) \hat{y}_i(k+1)$$
with
$$\mathbf{x}=\left[y(k), \ldots, y\left(k-n_y+1\right), u(k), \ldots, u\left(k-n_u+1\right)\right]$$
and $h_i(\mathbf{x})$ is given as (3). This NARX model represents multiple input and single output (MISO) systems directly and multiple input and multiple output (MIMO) systems in a decomposed form as a set of coupled MISO models.

## 统计代写|随机控制代写Stochastic Control代考|Off-line scheme

The FIV normal equations are formulated as
$$\sum_{j=1}^k\left[\beta_j^1 \mathbf{z}j, \ldots, \beta_j^l \mathbf{z}_j\right]\left[\gamma_j^1\left(\mathbf{x}_j+\xi_j\right), \ldots, \gamma_j^l\left(\mathbf{x}_j+\xi_j\right)\right]^T \hat{\theta}_k-\sum{j=1}^k\left[\beta_j^1 \mathbf{z}j, \ldots, \beta_j^l \mathbf{z}_j\right] y_j=0$$ or, with $\zeta_j=\left[\beta_j^1 \mathbf{z}_j, \ldots, \beta_j^l \mathbf{z}_j\right]$ $$\left[\sum{j=1}^k \zeta_j \chi_j^T\right] \hat{\theta}k-\sum{j=1}^k \zeta_j y_j=0$$
so that the FIV estimate is obtained as
$$\hat{\theta}k=\left{\sum{j=1}^k\left[\beta_j^1 \mathbf{z}j, \ldots, \beta_j^l \mathbf{z}_j\right]\left[\gamma_j^1\left(\mathbf{x}_j+\xi_j\right), \ldots, \gamma_j^l\left(\mathbf{x}_j+\xi_j\right)\right]^T\right}^{-1} \sum{j=1}^k\left[\beta_j^1 \mathbf{z}_j, \ldots, \beta_j^l \mathbf{z}_j\right] y_j$$
and, in vectorial form, the interest problem may be placed as
$$\hat{\theta}=\left(\Gamma^T \Sigma\right)^{-1} \Gamma^T \mathbf{Y}$$
where $\Gamma^T \in \Re^{l\left(n_y+n_u+1\right) \times N}$ is the fuzzy extended instrumental variable matrix with rows given by $\zeta_j, \Sigma \in \Re^{N \times l\left(n_y+n_u+1\right)}$ is the fuzzy extended data matrix with rows given by $\chi_j$ and $\mathbf{Y} \in \Re^{N \times 1}$ is the output vector and $\hat{\theta} \in \Re^{l\left(n_y+n_u+1\right) \times 1}$ is the parameters vector. The models can be obtained by the following two approaches:

• Global approach : In this approach all linear consequent parameters are estimated simultaneously, minimizing the criterion:
$$\hat{\theta}=\arg \min \left|\Gamma^T \Sigma \theta-\Gamma^T \mathbf{Y}\right|_2^2$$
• Local approach: In this approach the consequent parameters are estimated for each rule $i$, and hence independently of each other, minimizing a set of weighted local criteria $(i=1,2, \ldots, l)$ :
$$\hat{\theta}_i=\arg \min \left|\mathbf{Z}^T \Psi_i \mathbf{X} \theta_i-\mathbf{Z}^T \Psi_i \mathbf{Y}\right|_2^2$$
where $\mathbf{Z}^T$ has rows given by $\mathbf{z}_j$ and $\Psi_i$ is the normalized membership degree diagonal matrix according to $\mathbf{z}_j$

## 统计代写|随机控制代写随机控制代考|模糊结构模型

$$y(k+1)=F\left[y(k), \ldots, y\left(k-n_y+1\right), u(k), \ldots, u\left(k-n_u+1\right)\right]$$

$$R^i: \operatorname{IF} y(k) \text { is } F_1^i \text { AND } \cdots \text { AND } y\left(k-n_y+1\right) \text { is } F_{n_y}^i$$
AND $u(k)$ is $G_1^i$ AND $\cdots$ AND $u\left(k-n_u+1\right)$ is $G_{n_u}^i$
$$\operatorname{THEN} \hat{y}i(k+1)=\sum{j=1}^{n_y} a_{i, j} y(k-j+1)+\sum_{j=1}^{n_\mu} b_{i, j} u(k-j+1)+c_i$$

$$y(k+1)=\frac{\sum_{i=1}^l h_i(\mathbf{x}) \hat{y}i(k+1)}{\sum{i=1}^l h_i(\mathbf{x})}$$

$$y(k+1)=\sum_{i=1}^l \gamma_i(\mathbf{x}) \hat{y}_i(k+1)$$
with
$$\mathbf{x}=\left[y(k), \ldots, y\left(k-n_y+1\right), u(k), \ldots, u\left(k-n_u+1\right)\right]$$

## 统计代写|随机控制代写随机控制代考|离线方案

.

FIV的正常方程表述为
$$\sum_{j=1}^k\left[\beta_j^1 \mathbf{z}j, \ldots, \beta_j^l \mathbf{z}_j\right]\left[\gamma_j^1\left(\mathbf{x}_j+\xi_j\right), \ldots, \gamma_j^l\left(\mathbf{x}_j+\xi_j\right)\right]^T \hat{\theta}_k-\sum{j=1}^k\left[\beta_j^1 \mathbf{z}j, \ldots, \beta_j^l \mathbf{z}_j\right] y_j=0$$或$\zeta_j=\left[\beta_j^1 \mathbf{z}_j, \ldots, \beta_j^l \mathbf{z}_j\right]$$\left[\sum{j=1}^k \zeta_j \chi_j^T\right] \hat{\theta}k-\sum{j=1}^k \zeta_j y_j=0$$ ，因此FIV的估计得到为 $$\hat{\theta}k=\left{\sum{j=1}^k\left[\beta_j^1 \mathbf{z}j, \ldots, \beta_j^l \mathbf{z}_j\right]\left[\gamma_j^1\left(\mathbf{x}_j+\xi_j\right), \ldots, \gamma_j^l\left(\mathbf{x}_j+\xi_j\right)\right]^T\right}^{-1} \sum{j=1}^k\left[\beta_j^1 \mathbf{z}_j, \ldots, \beta_j^l \mathbf{z}_j\right] y_j$$ ，并且，以矢量形式，兴趣问题可以放置为 $$\hat{\theta}=\left(\Gamma^T \Sigma\right)^{-1} \Gamma^T \mathbf{Y}$$ ，其中$\Gamma^T \in \Re^{l\left(n_y+n_u+1\right) \times N}$是模糊扩展工具变量矩阵，行由$\zeta_j, \Sigma \in \Re^{N \times l\left(n_y+n_u+1\right)}$给出，是模糊扩展数据矩阵，行由$\chi_j$给出，$\mathbf{Y} \in \Re^{N \times 1}$是输出向量，$\hat{\theta} \in \Re^{l\left(n_y+n_u+1\right) \times 1}$是参数向量。该模型可通过以下两种方法获得: • 全局方法:在这种方法中，所有线性结果参数同时估计，最小化准则: $$\hat{\theta}=\arg \min \left|\Gamma^T \Sigma \theta-\Gamma^T \mathbf{Y}\right|_2^2$$ • 局部方法:在这种方法中，对每个规则$i$估计相应的参数，因此彼此独立，最小化一组加权局部准则$(i=1,2, \ldots, l)$: $$\hat{\theta}_i=\arg \min \left|\mathbf{Z}^T \Psi_i \mathbf{X} \theta_i-\mathbf{Z}^T \Psi_i \mathbf{Y}\right|_2^2$$ ，其中$\mathbf{Z}^T$有$\mathbf{z}_j$给出的行，而$\Psi_i$是根据$\mathbf{z}_j\$

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