# Performance Assessment of Cascade Control System with Non-Gaussian Disturbance Based on Minimum Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Routinely MVC for Cascade Control System

#### 2.1. Introduction of Cascade System

#### 2.2. MVC Strategy of Cascade Control System

## 3. Minimum Entropy Control

#### 3.1. Entropy Information Analysis

**Lemma**

**2**

**.**The entropy of joint distribution can be represented by conditional entropy which can be expressed as $H\left[(X,Y)\right]=H(X|Y)+H(Y)=H(Y|X)+H(X)$, where $X$ and $Y$ are random variables.

**Lemma**

**3**

**.**On a common probability space, ${H}_{\alpha}(X,Y)\ge \mathrm{max}({H}_{\alpha}(X),{H}_{\alpha}(Y))\ge 0$ while $X$ and $Y$ are the random variables.

#### 3.2. Minimum Entropy Based on EDA

- (1)
- Firstly, generate $N$ individuals ${A}^{(l)}=\left\{{\theta}_{1}^{(l)},\dots ,{\theta}_{N}^{(l)}\right\}$ as a seed by recursive extended least squares algorithm (RELS), where $l=0$ is the count of generation.
- (2)
- From the seed in $l-th$ generation, select R $(N\ge R)$ group individuals to calculate the mean value of residuals according to Equations (23)–(25). Remove those seeds which is less than the threshold value $\epsilon $. Re-sampling to make the number of parameter space is not less than $N$.
- (3)
- Add some seeds from the new parameter space ${A}^{(l)}$ and establish new probability model where the probability model can be expressed as,$$\begin{array}{l}f({\theta}_{i,j}^{(l+1)})=\frac{1}{\sqrt{2\pi}\sigma j}{\mathrm{exp}}^{-\frac{({({\theta}_{i,j}^{(l+1)}-{\mu}_{j})}^{2})}{2{\sigma}_{j}^{2}}},\\ {\mu}_{j}=\frac{1}{R}{\displaystyle \sum _{i=1}^{R}{\theta}_{i,j}^{(l)}},{\sigma}_{j}^{2}=\frac{1}{R}{\displaystyle \sum _{i=1}^{R}({\theta}_{i,j}^{(l)}}-{\mu}_{j});\end{array}$$
- (4)
- Set $l=l+1$ and add N-R new parameter into space based on the probability model which is set upon step (3).
- (5)
- Repeat step (2) to step (5) until meeting the stopping criterion.

## 4. Simulation Case

#### 4.1. Cascade Control System with Gaussian Disturbance

#### 4.2. Cascade Control System with Non-Gaussian Disturbance

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The actual and estimated distribution of (

**a**) ${a}_{1}$ and (

**b**) ${a}_{2}$ which are non-Gaussian disturbances.

**Figure 3.**Normal probability plots of the actual disturbance (

**a**) ${a}_{1}$ and (

**b**) ${a}_{2}$ which are non-Gaussian disturbance.

**Figure 8.**The actual and estimated distribution plots from different disturbances (

**a**) ${a}_{1}$ and (

**b**) ${a}_{2}$ based on minimum variance control (MVC).

**Figure 9.**The distribution plots of two different disturbances (

**a**) ${a}_{1}$ and (

**b**) ${a}_{2}$ based on minimum entropy control (MEC).

**Figure 10.**The actual and estimated distribution plots of two disturbances (

**a**) ${a}_{1}$ and (

**b**) ${a}_{2}$ based on MEC.

Index | Estimated Index | Theoretical Index |
---|---|---|

${\eta}_{mv}$ | 0.7010 | 0.7291 |

${\eta}_{me}$ | 0.9562 | 0.9757 |

Index | Estimated Index | Theoretical Index |
---|---|---|

${\eta}_{mv}$ | 0.5290 | 0.6238 |

${\eta}_{me}$ | 0.8122 | 0.8212 |

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**MDPI and ACS Style**

Zhang, Q.; Wang, Y.-G.; Lee, F.-F.; Zhang, W.; Chen, Q.
Performance Assessment of Cascade Control System with Non-Gaussian Disturbance Based on Minimum Entropy. *Symmetry* **2019**, *11*, 379.
https://doi.org/10.3390/sym11030379

**AMA Style**

Zhang Q, Wang Y-G, Lee F-F, Zhang W, Chen Q.
Performance Assessment of Cascade Control System with Non-Gaussian Disturbance Based on Minimum Entropy. *Symmetry*. 2019; 11(3):379.
https://doi.org/10.3390/sym11030379

**Chicago/Turabian Style**

Zhang, Qian, Ya-Gang Wang, Fei-Fei Lee, Wei Zhang, and Qiu Chen.
2019. "Performance Assessment of Cascade Control System with Non-Gaussian Disturbance Based on Minimum Entropy" *Symmetry* 11, no. 3: 379.
https://doi.org/10.3390/sym11030379