# Robust Adaptive Full-Order TSM Control Based on Neural Network

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Assumption**

**1.**

**Definition**

**1.**

**Remark**

**1.**

## 3. Model-Based FOTSM Controller

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

## 4. Model-Free FOTSM Controller

**Assumption**

**2.**

**Remark**

**4.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

## 5. Simulation Studies

#### 5.1. Control of a Second-Order System

**Case 1**. The test of the model-based FOTSM controller for the second-order nonlinear system.

**Case 2.**The test of the model-free FOTSM controller for the second-order nonlinear system.

#### 5.2. Control of a Third-Order System

**Case 3**. The test of the model-based FOTSM controller for the third-order nonlinear system.

**Case 4**. The test of the model-free FOTSM controller for the third-order nonlinear system.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**System state vector $x={[{x}_{1},\text{\hspace{0.05em}}{x}_{2},\text{\hspace{0.05em}}{x}_{3}]}^{\mathrm{T}}$ of Case 3.

**Figure 11.**System state vector $x={[{x}_{1},\text{\hspace{0.05em}}{x}_{2},\text{\hspace{0.05em}}{x}_{3}]}^{\mathrm{T}}$ of Case 4.

**Table 1.**Parameters of the full-order terminal sliding mode (FOTSM) manifold and the proposed controller.

Parameters | Values |
---|---|

${\alpha}_{1},\text{}{\alpha}_{2}$ | ${\alpha}_{1}\text{\hspace{0.17em}}=\text{}9/23$, ${\alpha}_{2}=\text{}9/16$ |

${\beta}_{1},\text{}{\beta}_{2}$ | ${\beta}_{1}\text{\hspace{0.17em}}=\text{}10$, ${\beta}_{2}\text{\hspace{0.17em}}=\text{}7$ |

$k,\text{}\eta ,\text{\hspace{0.17em}}{\gamma}_{1}$ | $k+\eta =3$, ${\gamma}_{1}=50$ |

Parameters | Values |
---|---|

${\alpha}_{1},\text{}{\alpha}_{2}$ | ${\alpha}_{1}\text{\hspace{0.17em}}=\text{}9/23$, ${\alpha}_{2}\text{\hspace{0.17em}}=\text{}9/16$ |

${\beta}_{1},\text{}{\beta}_{2}$ | ${\beta}_{1}\text{\hspace{0.17em}}=\text{}10$, ${\beta}_{2}\text{\hspace{0.17em}}=\text{}7$ |

$k,\text{}\eta ,\text{\hspace{0.17em}}{\gamma}_{2},\chi $ | $k+\eta =40$, ${\gamma}_{2}=50$,$\chi =5$ |

Parameters | Values |
---|---|

${\alpha}_{1},\text{}{\alpha}_{2},\text{\hspace{0.17em}}{\alpha}_{3}$ | ${\alpha}_{1}\text{\hspace{0.17em}}=\text{}4/7$, ${\alpha}_{2}\text{\hspace{0.17em}}=\text{}2/3$, ${\alpha}_{3}\text{\hspace{0.17em}}=\text{}4/5$ |

${\beta}_{1},\text{}{\beta}_{2},\text{}{\beta}_{3}$ | ${\beta}_{1}\text{}=\text{}80$, ${\beta}_{2}\text{\hspace{0.17em}}=\text{}66$, ${\beta}_{3}\text{\hspace{0.17em}}=\text{}15$ |

$k,\text{}\eta ,\text{\hspace{0.17em}}{\gamma}_{1}$ | $k+\eta =3$, ${\gamma}_{1}=50$ |

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## Share and Cite

**MDPI and ACS Style**

Cao, Q.; Cao, C.; Wang, F.; Liu, D.; Sun, H.
Robust Adaptive Full-Order TSM Control Based on Neural Network. *Symmetry* **2018**, *10*, 726.
https://doi.org/10.3390/sym10120726

**AMA Style**

Cao Q, Cao C, Wang F, Liu D, Sun H.
Robust Adaptive Full-Order TSM Control Based on Neural Network. *Symmetry*. 2018; 10(12):726.
https://doi.org/10.3390/sym10120726

**Chicago/Turabian Style**

Cao, Qianlei, Chongzhen Cao, Fengqin Wang, Dan Liu, and Hui Sun.
2018. "Robust Adaptive Full-Order TSM Control Based on Neural Network" *Symmetry* 10, no. 12: 726.
https://doi.org/10.3390/sym10120726