# Adaptive Command-Filtered Fuzzy Nonsingular Terminal Sliding Mode Backstepping Control for Linear Induction Motor

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## Abstract

**:**

## 1. Introduction

- The nonsingular terminal sliding mode control method is integrated with the adaptive fuzzy backstepping control to enhance the robustness of the system and ensure to reach the equilibrium point within a limited time.
- The fuzzy logic technique is introduced to estimate the nonlinear part of the system model to make the controller design process more clear and easy, and the stability proof of the proposed PACFTB control strategy is provided concretely.
- The simulations and experiments are both carried out and discussed to further prove the effectiveness of the proposed PACFTB control strategy.

## 2. Establishment of LIM Dynamic Model and Preliminaries

#### 2.1. Establishment of LIM Dynamic Model

#### 2.2. Fuzzy Logic System (FLS)

**Lemma**

**1.**

#### 2.3. Projection Operator

**Property**

**1.**

**Property**

**2.**

## 3. Design Process of the PACFTB Controller

**Step 1:**So as to get the virtual control formula for ${i}_{qs}^{d}$, the following Lyapunov function is designed:

**Step 2:**In order to obtain the control law in $q-axis$, the Lyapunov function ${V}_{2}$ is selected as:

**Step 3:**In order to design the control law of d-axis, the Lyapunov function ${V}_{3}$ is chosen as:

**Step 4:**This step is used to get the precise adaptive law for FLS, then select the following Lyapunov function V as:

## 4. Stability Analysis

## 5. Simulation and Experiment Study

#### 5.1. Simulation Study

#### 5.2. Experiment Study

## 6. Conclusions

- The problem of insufficient modeling, unknown nonlinear components and uncertain parameters in the LIM is solved by the FLS combined with an adaptive law.
- The introduction of the command filter solves the differential expansion problem in the conventional backstepping algorithm, and the inherent filter error is compensated via the proposed compensation algorithm.
- The introduction of projection operator guarantees the boundedness of estimated parameters and FLS.
- The simulation results and experimental results indicate that the proposed PACFTB control strategy has remarkable speed tracking performance of the LIM with end effects.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**Adaptive estimation for unknown parameters. (

**a**) The estimation of ${f}_{1}\left(X\right)$; (

**b**) The estimation of ${f}_{2}\left(X\right)$; (

**c**) The estimation of ${f}_{3}\left(X\right)$; (

**d**) The estimation of load disturbance.

Parameter | Value | Parameter | Value |
---|---|---|---|

${R}_{s}(\Omega )$ | 0.0709 | ${R}_{r}(\Omega )$ | 0.1311 |

${L}_{s}$(mH) | 4.8 | ${L}_{r}$(mH) | 4.8 |

${L}_{m}$(mH) | 3.9 | M(kg) | 351.264 |

D(kg/s) | 40.95 | h(m) | 0.2 |

P | 4 | l(m) | 2 |

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**MDPI and ACS Style**

Zhang, L.; Xia, Y.; Zhang, W.; Yang, W.; Xu, D.
Adaptive Command-Filtered Fuzzy Nonsingular Terminal Sliding Mode Backstepping Control for Linear Induction Motor. *Appl. Sci.* **2020**, *10*, 7405.
https://doi.org/10.3390/app10217405

**AMA Style**

Zhang L, Xia Y, Zhang W, Yang W, Xu D.
Adaptive Command-Filtered Fuzzy Nonsingular Terminal Sliding Mode Backstepping Control for Linear Induction Motor. *Applied Sciences*. 2020; 10(21):7405.
https://doi.org/10.3390/app10217405

**Chicago/Turabian Style**

Zhang, Li, Yan Xia, Weiming Zhang, Weilin Yang, and Dezhi Xu.
2020. "Adaptive Command-Filtered Fuzzy Nonsingular Terminal Sliding Mode Backstepping Control for Linear Induction Motor" *Applied Sciences* 10, no. 21: 7405.
https://doi.org/10.3390/app10217405