# Design of Neural Network Quantizers for Networked Control Systems

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## Abstract

**:**

## 1. Introduction

## 2. Neural Network Quantizers

#### 2.1. System Description

#### 2.2. Regression Based Neural Network Quantizer

#### 2.3. Classification Based Neural Network Quantizer

## 3. Quantizer Design Problem

**Problem**

**1.**

Algorithm 1: DE (DE/best/1/bin strategy) |

Initialization: Given $N\in \mathbb{N}$, ${t}_{max}\in \mathbb{N}$, $F\in [0,2]$, $H\in [0,1]$ and the initial search space ${S}_{0}={[{\theta}_{min},{\theta}_{max}]}^{n}$. Set $t=0$ then select randomly N individuals $\{{\mathbf{\theta}}_{1},{\mathbf{\theta}}_{2},\dots ,{\mathbf{\theta}}_{N}\}$ in the search space.Step 1: The cost function $J\left(\mathbf{\theta}\right)$ is evaluated for each ${\mathbf{\theta}}_{i}$ and ${\mathbf{\theta}}_{base}={\mathbf{\theta}}_{l}$ is calculated by:
$$\begin{array}{cc}\hfill l& {\displaystyle =\underset{i\in \{1,2,\dots ,N\}}{arg\; min}J\left({\mathbf{\theta}}_{i}\right).}\end{array}$$
Step 2.Step 2 (Mutation): For each ${\mathbf{\theta}}_{i}$ a mutant vector ${\mathcal{M}}_{i}$ is generated by:
$$\begin{array}{c}\hfill {\mathcal{M}}_{i}={\mathbf{\theta}}_{base}+F({\mathbf{\theta}}_{{\tau}_{1,i}}-{\mathbf{\theta}}_{{\tau}_{2,i}}),\end{array}$$
Step 3 (Crossover): For each ${\mathbf{\theta}}_{i}$ and ${\mathcal{M}}_{i}$ a trial vector ${\mathcal{T}}_{i}$ is generated by:
$${\mathcal{T}}_{i,j}=\left\{\begin{array}{ll}{\mathcal{M}}_{i,j}\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}{\rho}_{i,j}\le H\phantom{\rule{4.pt}{0ex}}\mathrm{or}\phantom{\rule{4.pt}{0ex}}j={j}_{rand},\hfill \\ \phantom{\rule{4.pt}{0ex}}{\theta}_{i,j}& \phantom{\rule{1.em}{0ex}}\mathrm{otherwise},\end{array}\right)$$
Step 4 (Selection): The members of the next generation $k+1$ are selected by:
$${\mathbf{\theta}}_{i}\leftarrow \left\{\begin{array}{ll}{\mathcal{T}}_{i}\hfill & \phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}J\left({\mathcal{T}}_{i}\right)\le J\left({\theta}_{i}\right),\hfill \\ \phantom{\rule{4.pt}{0ex}}{\mathbf{\theta}}_{i}& \phantom{\rule{1.em}{0ex}}\mathrm{otherwise},\phantom{\rule{4.pt}{0ex}}\end{array}\right)$$
Step 1. |

## 4. Numerical Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Difference between the neural network quantizer based on regression and the one based on classification. (

**a**) Regression based approach. The neural net has one output and shapes the original signal; (

**b**) Classification based approach. The number of the neural network output is same as that of quantization levels, and each output correponds to the probability that a original signal is classified into a specific quantization level.

**Figure 7.**Signals resulting from applying $u\left(k\right)$ to the system with the ${Q}_{NN}$ designed for $M=2$ and ${n}_{L}=2$. The black lines represent the signals without quantization ($u\left(k\right)$, $y\left(k\right)$) and the blues ones are the signals when quantization is applied ($v\left(k\right)$, ${u}_{q}\left(k\right)$, ${y}_{q}\left(k\right)$).

**Figure 8.**Output signals ${y}_{q}\left(k\right)$ (blue) and $y\left(k\right)$ (black) resulting from applying $u\left(k\right)$ to the error system with ${Q}_{NN}$ designed for $M=2$ and ${n}_{L}=4$.

**Figure 9.**Output signals ${y}_{q}\left(k\right)$ (blue) and $y\left(k\right)$ (black) resulting from applying $u\left(k\right)$ to the error system with ${Q}_{NN}$ designed for $M=8$, ${n}_{L}=2$ (

**upper figure**) and ${n}_{L}=4$ (

**lower figure**).

**Figure 10.**Signals resulting from applying $u\left(k\right)$ to the systems with the static quantizer $\mathrm{q}$ in Figure 3 and the optimal dynamic quantizer proposed in [5]. The black lines represent the signals without quantization ($u\left(k\right)$, $y\left(k\right)$) and the blues ones are the signals when quantization is applied ($v\left(k\right)$, ${u}_{q}\left(k\right)$, ${y}_{q}\left(k\right)$).

Quantizer Type | M | K | ${\mathit{n}}_{\mathit{L}}$ | ${\mathit{n}}_{\mathsf{w}}$ | n |
---|---|---|---|---|---|

R: ${Q}_{NNR}$ | $\{2,\phantom{\rule{4pt}{0ex}}8\}$ | $[10,\phantom{\rule{4pt}{0ex}}1]$ | 2 | 132 | 133 |

$[10,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}1]$ | 4 | 352 | 353 | ||

C: ${Q}_{NNC}$ | 2 | $[10,\phantom{\rule{4pt}{0ex}}2]$ | 2 | 142 | 143 |

$[10,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}2]$ | 4 | 362 | 363 | ||

8 | $[10,\phantom{\rule{4pt}{0ex}}8]$ | 2 | 208 | 209 | |

$[10,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}10,\phantom{\rule{4pt}{0ex}}8]$ | 4 | 428 | 429 |

M | Init. | ${\mathit{n}}_{\mathit{L}}$ | Type | Min. | Avg. | Std. Dev. |
---|---|---|---|---|---|---|

2 | Urand | 2 | R | 3.73724 | 4.32987 | 0.48300 |

C | 3.66946 | 4.27038 | 0.34762 | |||

4 | R | 3.66764 | 4.22516 | 0.45819 | ||

C | 3.54773 | 4.30158 | 0.49296 | |||

Xavier | 2 | R | 3.42879 | 4.15830 | 0.36054 | |

C | 3.53307 | 4.09696 | 0.35038 | |||

4 | R | 3.65081 | 4.10635 | 0.35909 | ||

C | 3.63822 | 4.04066 | 0.29597 | |||

8 | Urand | 2 | R | 0.17825 | 0.20622 | 0.01612 |

C | 0.91201 | 2.91111 | 1.54333 | |||

4 | R | 0.22911 | 0.32243 | 0.14041 | ||

C | 0.81667 | 2.81630 | 1.43382 | |||

Xavier | 2 | R | 0.20424 | 1.90045 | 1.22825 | |

C | 0.93284 | 2.85580 | 1.44467 | |||

4 | R | 0.24762 | 0.85188 | 1.03043 | ||

C | 1.00016 | 2.61201 | 1.06714 |

**Table 3.**Tukey pairwise comparison for $\mathsf{h}=\mathrm{sigmoid}$ and single factors]Tukey pairwise comparison 3-way ANOVA for $\mathsf{h}=\mathrm{sigmoid}$ and single factors. Grouping information using the Tukey test and $95\%$ confidence. Means that do not share a letter are significantly different.

M | Factor | N | Mean | Grouping | ||
---|---|---|---|---|---|---|

2 | Type | R | 200 | 4.20492 | A | |

C | 200 | 4.17739 | A | |||

Init | Urand | 200 | 4.28175 | A | ||

Xavier | 200 | 4.10057 | B | |||

${n}_{L}$ | L2 | 200 | 4.21388 | A | ||

L4 | 200 | 4.16844 | A | |||

8 | Type | C | 200 | 2.79881 | A | |

R | 200 | 0.82024 | B | |||

Init | Xavier | 200 | 2.05504 | A | ||

Urand | 200 | 1.56401 | B | |||

${n}_{L}$ | L2 | 200 | 1.96839 | A | ||

L4 | 200 | 1.65066 | B |

**Table 4.**$E\left({Q}_{NN}\right)$ results summary for $\mathsf{h}=tanh$ and $\mathsf{h}=\mathrm{ReLU}$ ($M=8$).

$\mathsf{h}$ | Init. | ${\mathit{n}}_{\mathit{L}}$ | Type | Min. | Avg. | Std. Dev. |
---|---|---|---|---|---|---|

tanh | Urand | 2 | R | 0.17945 | 0.57343 | 0.73814 |

C | 0.85707 | 2.17040 | 1.21598 | |||

4 | R | 0.24340 | 1.83621 | 1.37026 | ||

C | 0.69604 | 1.62132 | 0.68768 | |||

Xavier | 2 | R | 0.17010 | 0.20761 | 0.02103 | |

C | 0.92943 | 1.97048 | 0.78436 | |||

4 | R | 0.19868 | 0.25325 | 0.03900 | ||

C | 0.66041 | 1.77138 | 0.99609 | |||

$\mathrm{ReLU}$ | Urand | 2 | R | 0.16532 | 0.75854 | 1.21252 |

C | 0.70494 | 1.93226 | 0.98668 | |||

4 | R | 0.19175 | 3.04153 | 1.69148 | ||

C | 0.71718 | 1.93823 | 1.27409 | |||

Xavier | 2 | R | 0.22438 | 2.97202 | 1.24661 | |

C | 0.72398 | 2.12915 | 1.12578 | |||

4 | R | 0.22062 | 3.01670 | 1.70763 | ||

C | 0.63188 | 2.14319 | 1.03974 |

**Table 5.**Tukey pairwise comparison 3-way ANOVA for the activation functions comparison ($M=8$). Grouping information using the Tukey test and $95\%$ confidence. Means that do not share a letter are significantly different.

Factor | N | Mean | Grouping | ||||
---|---|---|---|---|---|---|---|

Type | C | 600 | 2.23930 | A | |||

R | 600 | 1.32836 | B | ||||

Init | Xavier | 600 | 1.89033 | A | |||

Urand | 600 | 1.67733 | B | ||||

$\mathsf{h}$ | ReLU | 400 | 2.24145 | A | |||

sigm | 400 | 1.80953 | B | ||||

tanh | 400 | 1.30051 | C |

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**MDPI and ACS Style**

Rodriguez Ramirez, J.E.; Minami, Y.
Design of Neural Network Quantizers for Networked Control Systems. *Electronics* **2019**, *8*, 318.
https://doi.org/10.3390/electronics8030318

**AMA Style**

Rodriguez Ramirez JE, Minami Y.
Design of Neural Network Quantizers for Networked Control Systems. *Electronics*. 2019; 8(3):318.
https://doi.org/10.3390/electronics8030318

**Chicago/Turabian Style**

Rodriguez Ramirez, Juan Esteban, and Yuki Minami.
2019. "Design of Neural Network Quantizers for Networked Control Systems" *Electronics* 8, no. 3: 318.
https://doi.org/10.3390/electronics8030318