# Control Synthesis as Machine Learning Control by Symbolic Regression Methods

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement of Control Synthesis as MLC

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 3. Small Variations of the Basic Solution

## 4. Symbolic Regression Methods

- -
- a set of arguments or a set of functions without arguments$${\mathrm{F}}_{0}=\{{f}_{0,1},\dots ,{f}_{0,n+p+v}\}=\{{x}_{1},\dots ,{x}_{n},{q}_{1},\dots ,{q}_{p},{e}_{1},\dots ,{e}_{v}\},$$
- -
- a set of functions with one argument$${\mathrm{F}}_{1}=\{{f}_{1,1}\left(z\right)=z,{f}_{1,2}\left(z\right),\dots ,{f}_{1,w}\left(z\right)\},$$
- -
- a set of functions with two arguments$${\mathrm{F}}_{2}=\{{f}_{2,1}({z}_{1},{z}_{2}),\dots ,{f}_{2,v}({z}_{1},{z}_{2})\}.$$All functions with two arguments have to possess the following properties:
- -
- be commutative$${f}_{2,i}({z}_{1},{z}_{2})={f}_{2,i}({z}_{2},{z}_{1}),\phantom{\rule{0.277778em}{0ex}}i=1,\dots ,v$$
- -
- be associative$${f}_{2,i}({z}_{1},f({z}_{2},{z}_{3}))={f}_{2,i}({f}_{2,i}({z}_{1},{z}_{2}),{z}_{3}),$$
- -
- have a unit element$${f}_{2,i}(z,{e}_{i})={f}_{2,i}({e}_{i},z)=z,\phantom{\rule{0.277778em}{0ex}}$$

- -
- the set of arguments of the mathematical expression (30)$${\mathrm{F}}_{0}=\{{f}_{0,1}={x}_{1},{f}_{0,2}={x}_{2},{f}_{0,3}={x}_{3},{f}_{0,4}={q}_{1},{f}_{0,5}={q}_{2},{f}_{0,6}=0,{f}_{0,7}=1\};$$
- -
- the set of functions with one argument$${\mathrm{F}}_{1}=\{{f}_{1,1}\left(z\right)=z,{f}_{1,2}\left(z\right)={z}^{2},{f}_{1,3}\left(z\right)=-z,{f}_{1,4}\left(z\right)=exp\left(z\right),{f}_{1,5}\left(z\right)=sin\left(z\right)\};$$
- -
- the set of functions with two arguments$${\mathrm{F}}_{2}=\{{f}_{2,1}({z}_{1},{z}_{2})={z}_{1}+{z}_{2},{f}_{2,2}({z}_{1},{z}_{2})={z}_{1}{z}_{2}\}.$$

#### 4.1. The Genetic Programming

#### 4.2. The Network Operator

#### 4.3. Cartesian Genetic Programming

#### 4.4. Binary Complete Genetic Programming

## 5. Computational Experiment

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The computational tree for the function (30).

**Figure 2.**The NOP graph of the function (30).

**Figure 3.**The BCGP graph of the function (30).

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Shmalko, E.; Diveev, A. Control Synthesis as Machine Learning Control by Symbolic Regression Methods. *Appl. Sci.* **2021**, *11*, 5468.
https://doi.org/10.3390/app11125468

**AMA Style**

Shmalko E, Diveev A. Control Synthesis as Machine Learning Control by Symbolic Regression Methods. *Applied Sciences*. 2021; 11(12):5468.
https://doi.org/10.3390/app11125468

**Chicago/Turabian Style**

Shmalko, Elizaveta, and Askhat Diveev. 2021. "Control Synthesis as Machine Learning Control by Symbolic Regression Methods" *Applied Sciences* 11, no. 12: 5468.
https://doi.org/10.3390/app11125468