# Minimum-Information-Entropy-Based Control Performance Assessment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Minimum-Information-Entropy Benchmark

#### 2.1. Information Entropy

#### 2.2. MIE Benchmark

^{2}) with variance ${\delta}^{2}$ and mean $\mu $, and its PDF is:

**u**is the control variable vector,

**e**is the tracking error vector,

**r**is the set-point vector. Letting ${V}_{k}={\displaystyle \sum _{i=1}^{n}{\alpha}_{i}{y}_{k-i}}+{\displaystyle \sum _{j=1}^{m}{b}_{j}{u}_{k-d-j}}-{r}_{k}$ and ${f}^{-1}(y,u,r,x)=x-{V}_{k}-{b}_{0}{u}_{k}$, the PDF of the tracking error can be obtained from (4) as:

#### 2.3. Upper Bound of the MIE Benchmark

#### 2.4. Extension to Nonlinear Processes and Non-Gaussian Disturbance Case

## 3. MIE Performance Assessment Index

_{act}is the information entropy of the tracking error.

- (1)
- If the upper bound of the MIE is $\mathrm{ln}(\sqrt{2\pi \mathrm{exp}(1)}{\delta}_{\text{mv}})$, which is selected as the performance benchmark to assess the control performance, the new CPA index will have the similar computational complexity and assessment result with the Harris index.
- (2)
- If the MIE is $\mathrm{ln}(\sqrt{2\pi \mathrm{exp}(1)}\delta )$, which is selected as the performance benchmark to assess the control performance, the delay need not be obtained as a prior knowledge. So, it is easier to be used than the Harris index in this case.
- (3)
- The new CPA index can be used in non-linear processes and non-Gaussian disturbances case.

#### 3.1. Information Entropy Calculation

^{th}basic function, ${\rho}_{i}(u)$ is the i

^{th}weight, $\epsilon $ represents the approximation error.

#### 3.2. H_{min} Estimation

## 4. MIE Based CPA Procedure

#### 4.1. CPA under Steady State

_{min }from (12).

#### 4.2. Transient CPA

- Step1: Estimate the deterministic tracking error by trend extraction.
- Step2: Estimate the stochastic tracking error by the actual tracking error minus the deterministic tracking error.
- Step3: Using the stochastic tracking error, the CPA under steady state method is selected to calculate the performance index.

## 5. Case Study

#### 5.1. Case 1: CPA under Steady State

Method | Benchmark | Performance Index | ||
---|---|---|---|---|

Name | Value | Name | Value | |

MIE based CPA (Time delay is known) | Upper bound of MIE | -0.5602 | MIE index | 0.5632 |

MIE based CPA (Time delay is unknown) | MIE | -0.493 | MIE index | 0.6024 |

nimum-variance-based CPA | Minimum variance | 0.0362 | Harris index | 0.5639 |

#### 5.2. Case 2: CPA under Transient State

^{−2}in Figure 5, so the transient CPA results have little difference with the steady-state CPA results. It indicates the effectiveness of the transient CPA procedure. For example 2, the actual information entropy with different states is different from each other in Figure 3, and the upper bound of the MIE is also different under different states in Figure 4. Although different calculation processes with different states data, the transient CPA results have little difference from the steady-state CPA results in Figure 5. It indicates the effectiveness of the transient CPA procedure. From any other example, the same conclusion can be obtained.

Number of Example | Plant | Controller | Disturbance |
---|---|---|---|

1 | $\frac{0.1{q}^{-3}}{1-0.8{q}^{-1}}$ | $\frac{6.53-9.236{q}^{-1}+3.357{q}^{-2}}{1-{q}^{-1}}$ | ${\alpha}_{k}$ |

2 | $\frac{0.1{q}^{-6}}{1-0.8{q}^{-1}}$ | $\frac{8.21-13.7{q}^{-1}+5.95{q}^{-2}}{1-{q}^{-1}}$ | ${\alpha}_{k}$ |

3 | $\frac{0.0891{q}^{-12}}{1-0.8669{q}^{-1}}$ | $\frac{1.84-3.38{q}^{-1}+1.54{q}^{-2}}{1-{q}^{-1}}$ | ${\alpha}_{k}$ |

4 | $\frac{0.5108{q}^{-28}}{1-0.9601{q}^{-1}}$ | $0.068+\frac{0.0021{q}^{-1}}{1-{q}^{-1}}-0.44\frac{0.03}{1+\frac{0.03{q}^{-1}}{1-{q}^{-1}}}$ | ${\alpha}_{k}$ |

5 | $\frac{0.2412{e}^{-28.6s}}{25.6s+1}$ | $1.89+\frac{0.075}{s}$ | ${\alpha}_{k}$ |

6 | $\frac{0.1859{e}^{-43.6s}}{20.58s+1}$ | $1.89+\frac{0.075}{s}$ | ${\alpha}_{k}$ |

7 | $\frac{0.1608{e}^{-46.8s}}{17.85s+1}$ | $1.89+\frac{0.075}{s}$ | ${\alpha}_{k}$ |

8 | $\frac{0.1507{e}^{-61.5s}}{17.3s+1}$ | $1.89+\frac{0.075}{s}$ | ${\alpha}_{k}$ |

#### 5.3. Case3: CPA for Nonlinear Non-Gaussian Case

Method | Item | Disturbution of Disturbance ${\omega}_{t}$ | ||
---|---|---|---|---|

Gaussian Distribution with Mean 0 and Variance 0.1 | Weibull Distribution with A=0.6 B=2 | Beta Distribution with $\lambda =3$ | ||

Entropy-based | Actual Entropy | 0.3493 | 0.3164 | -0.3655 |

Benchmark (MIE) | 0.1498 | 0.1484 | -0.1566 | |

MIE index | 0.8191 | 0.8453 | 0.8115 | |

Variance-based | Actual variance | 0.1239 | - | - |

Minimum variance | 0.0994 | - | - | |

Harris index | 0.8023 | - | - |

## 6. MIE-Based CPA of an Industrial Example

^{−5}by “whitening”. The disturbance distribution is plotted in Figure 6(d). From Figure 6d, we can find the disturbance obeys non-Gaussian distribution, so the entropy-based transient CPA should be chosen. With (12), the MIE is −3.926. Then, the stochastic performance index is obtained as 0.6589 by (40). Because the disturbance obeys non-Gaussian distribution, strictly speaking, the variance-based CPA method can be used. But, the disturbance distribution is approximately symmetrical, the minimum-variance-based CPA result can be given for reference. Using the minimum variance benchmark (the delay is 34 sample intervals) with the estimated stochastic error, the Harris index is 0.6612. The two performance assessment results are similar, with the two methods. When the CAP uses the minimum variance index, the delay as a priori is necessary to be estimated. However, the delay is not the necessary priori-knowledge for the MIE-based CPA. This is a significant advantage. It will make the MIE index being widely applied in the actual industrial process by the engineers.

## 7. Conclusions

## Acknowledgments

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Meng, Q.-W.; Fang, F.; Liu, J.-Z.
Minimum-Information-Entropy-Based Control Performance Assessment. *Entropy* **2013**, *15*, 943-959.
https://doi.org/10.3390/e15030943

**AMA Style**

Meng Q-W, Fang F, Liu J-Z.
Minimum-Information-Entropy-Based Control Performance Assessment. *Entropy*. 2013; 15(3):943-959.
https://doi.org/10.3390/e15030943

**Chicago/Turabian Style**

Meng, Qing-Wei, Fang Fang, and Ji-Zhen Liu.
2013. "Minimum-Information-Entropy-Based Control Performance Assessment" *Entropy* 15, no. 3: 943-959.
https://doi.org/10.3390/e15030943