# Application of Disturbance Observer-Based Fast Terminal Sliding Mode Control for Asynchronous Motors in Remote Electrical Conductivity Control of Fertigation Systems

^{1}

^{2}

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## Abstract

**:**

^{−1}over a pipeline distance of 120 m. The FTSMC-DO control consistently achieves the desired EC levels with minimal fluctuation and overshoot, outperforming traditional PID and SMC methods. This high level of precision is crucial for ensuring optimal nutrient delivery and efficient water usage in agricultural irrigation systems, highlighting the system’s potential as a valuable tool in modern, sustainable farming practices.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Remote Conductivity Control System for the Fertigation Machine

_{T}represents the volume of fertilizer liquid stored in the pipeline, in liters (L). The flow rate in the main drip irrigation pipe, q

_{m}, and the maximum fertilizer absorption rate of the fertigation machine, q

_{w}, are measured in liters per hour (L/h). The flow rate at the end of the drip irrigation tape, q

_{2}, is also in L/h. E

_{1}denotes the EC of the fertilizer solution in the tank, while E

_{0}represents the EC of the clean water in the inlet pipe.

#### 2.2. Asynchronous Motor Mathematical Model

_{sd}, i

_{sq}, u

_{sd}, and u

_{sq}represent the current and voltage in the dq axis, respectively; L

_{m}is the mutual inductance between the stator and rotor windings in the dq axis; L

_{s}, L

_{r}, are the self-inductances of the stator and rotor windings in the dq axis, respectively; R

_{s}, R

_{r}are the resistances of the stator and rotor, respectively; σ is the motor’s leakage magnetic coefficient, and σ = (L

_{m}

^{2}/L

_{s}L

_{r}); T

_{r}is the rotor electromagnetic time constant, T

_{r}= (L

_{r}/R); ω

_{1}, ω are the stator frequency synchronous speed and the rotor speed, respectively; and ω

_{m}is the mechanical speed.

#### 2.3. Internal Model Controller Design

_{1}(s) and Q

_{2}(s) represent the two-degree-of-freedom internal model controllers, while G(s) and G

_{n}(s) are the controlled object and the reference model of the controlled object, respectively. A comparison between Figure 2 and Figure 3 yields:

_{1}(s) and f

_{2}(s) are the introduced low-pass filters.

_{1}and λ

_{2}are the adjustable time constants of the filter.

#### 2.4. Fuzzy Sliding Mode Speed Regulator Design

_{rq}= 0. By combining the voltage equation and magnetic chain equation in the dq coordinate system, we get:

#### 2.5. Fast Terminal Sliding Mode Control Method Based on Asynchronous Motor Disturbance Observer

#### 2.5.1. Load Disturbance Observer Design

_{m}and load disturbance d(t) are chosen as the state variables for the load disturbance observer, where d(t) = (n

_{p}/J)T

_{L}.

_{L}= 0, d(t) = 0. Combining this with Equation (5), the state equation is given as follows:

**u**= T

_{e}.

_{ω}|. As indicated by Equations (15) and (16), when |e

_{ω}| > δ, tanh|e

_{ω}|

^{μ}sgn(e

_{ω}) enables the system to rapidly approach the actual values, leading e

_{ω}to converge to δ. On the other hand, when |e

_{ω}| < δ, tanh(e

_{ω}/δ

^{1−μ}) acts like a low-pass filter. Therefore, the proposed tanhFal(e

_{ω}, μ, δ) function possesses fast convergence characteristics, facilitating the convergence of observation errors.

#### 2.5.2. Fuzzy Sliding Mode Speed Controller Design

_{1}represents the motor speed error, with ω* and ω being the set speed and actual speed of the motor, respectively.

_{1}, c

_{2}> 0, 1 < p/q < 2 and p, q are positive odd integers.

_{1}| > δ, and the terms ${c}_{1}{{x}_{1}}^{p/q}$ and |x

_{1}|

^{μ}sgn(x

_{1}) work together to accelerate the error’s convergence toward the sliding mode surface. As the error approaches the equilibrium point along the sliding surface, the convergence speed is determined by the introduced term x

_{1}/$\delta $

^{1−}

^{μ}. This design ensures that the error moves towards the sliding surface more efficiently, especially as it nears the equilibrium point.

_{rq}= Ψ

_{r}, and Ψ

_{rq}= 0. From Equation (2), we obtain:

^{2}/2 is chosen, and its derivative is:

_{1}> 0, 0 < (p/q) – 1 < 1, and p, q are positive odd integers, η > 0. Therefore, it can be concluded:

_{p})$\widehat{d}\left(t\right)$.The sliding mode control parameter k is then produced as the output. The schematic diagram of the fuzzy sliding mode control is illustrated in Figure 5.

_{p}= 8, k

_{i}= 0.6, for the traditional SMC η = 10, k = 200, and for the fuzzy sliding mode-internal model control p = 15, q = 11, c

_{1}= 12, c

_{2}= 0.5. The FTSMC-DO is depicted in Figure 6.

#### 2.6. Test Platform

## 3. Results and Discussion

#### 3.1. Control Results of the Asynchronous Motor

#### 3.1.1. Motor Speed Control

- During the start-up phase of the asynchronous motor, FTSMC-DO control reaches the set speed faster than PID and SMC controls, with virtually no overshoot.
- Under sudden load conditions, the speed under PID control significantly drops, while the FTSMC-DO control demonstrates enhanced speed regulation capability, maintaining the set speed effectively.
- During acceleration and deceleration, PID and SMC controls show noticeable delays in adjustment time, whereas FTSMC-DO control exhibits rapid and accurate convergence speed.

#### 3.1.2. Motor Torque Control

- During steady operation, traditional PID control exhibits significant fluctuations, SMC control still shows noticeable fluctuations, while FTSMC-DO control has substantially less fluctuation.
- At 25 s under sudden load, traditional SMC control takes longer to stabilize and still displays some fluctuation. FTSMC-DO control smoothly transitions and quickly reaches a steady state with minimal fluctuation.

#### 3.2. Experimental Validation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**The fast terminal sliding mode control based on the asynchronous motor disturbance observer.

**Figure 10.**Acceleration, deceleration, and sudden unload speed response of the three control methods.

$\mathit{\omega}$$\backslash {\widehat{\mathit{T}}}_{\mathit{L}}$ | NB | NS | ZO | PS | PB |
---|---|---|---|---|---|

NB | ZO | NS | NS | NS | NB |

NS | PS | ZO | NS | ZO | NS |

ZO | PS | PS | ZO | ZO | NS |

PS | PB | PS | PS | PS | ZO |

PB | PB | PS | PS | PS | PS |

Control Method | Target EC (mS·cm^{−1}) | Steady State EC (mS·cm^{−1}) | Fluctuation Range (mS·cm^{−1}) | Steady State Time (s) | Overshoot (%) |
---|---|---|---|---|---|

PID | 1.4 | 1.25~1.56 | 0.42 | 155 | 19.2 |

1.8 | 1.65~1.93 | 0.38 | 175 | 21.7 | |

2.2 | 2.10~2.30 | 0.29 | 190 | 24.9 | |

SMC | 1.4 | 1.30~1.50 | 0.31 | 115 | 15.4 |

1.8 | 1.67~1.90 | 0.28 | 120 | 16.3 | |

2.2 | 2.12~2.28 | 0.20 | 135 | 17.1 | |

FTSMC-DO | 1.4 | 1.18~1.60 | 0.20 | 95 | 14.5 |

1.8 | 1.60~1.98 | 0.18 | 100 | 15.7 | |

2.2 | 2.06~2.35 | 0.16 | 120 | 16.1 |

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**MDPI and ACS Style**

Wang, H.; Zhao, J.; Zhang, L.; Yu, S.
Application of Disturbance Observer-Based Fast Terminal Sliding Mode Control for Asynchronous Motors in Remote Electrical Conductivity Control of Fertigation Systems. *Agriculture* **2024**, *14*, 168.
https://doi.org/10.3390/agriculture14020168

**AMA Style**

Wang H, Zhao J, Zhang L, Yu S.
Application of Disturbance Observer-Based Fast Terminal Sliding Mode Control for Asynchronous Motors in Remote Electrical Conductivity Control of Fertigation Systems. *Agriculture*. 2024; 14(2):168.
https://doi.org/10.3390/agriculture14020168

**Chicago/Turabian Style**

Wang, Huan, Jiawei Zhao, Lixin Zhang, and Siyao Yu.
2024. "Application of Disturbance Observer-Based Fast Terminal Sliding Mode Control for Asynchronous Motors in Remote Electrical Conductivity Control of Fertigation Systems" *Agriculture* 14, no. 2: 168.
https://doi.org/10.3390/agriculture14020168