# Predefined-Time Fault-Tolerant Trajectory Tracking Control for Autonomous Underwater Vehicles Considering Actuator Saturation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- A predefined-time sliding mode controller is developed based on a predefined-time disturbance observer. This is achieved by incorporating a novel predefined-time auxiliary system to prevent the control input from surpassing the actuator’s physical limitations;
- A non-singular backstepping approach is designed to avoid potential singularities in the predefined-time sliding mode controller, ensuring that the trajectory tracking error remains uniformly ultimately bounded (UUB) within the predefined time.

## 2. Preliminaries

#### 2.1. AUV Dynamic Model

**Remark**

**1.**

**η**are measurable in practice. It is clear that ultra-short baselines (USBL) can provide the position information; the electronic compass and gauge can provide the depth and attitude information; and the DVL and IMU can provide the linear velocity and angular velocity, respectively, which can be transformed into inertial frame by $\mathit{J}\left(\eta \right)$.

**Hypothesis**

**1.**

**Remark**

**2.**

**Hypothesis**

**2.**

**Remark**

**3.**

**Hypothesis**

**3.**

**Remark**

**4.**

#### 2.2. Definitions and Lemmas

**Lemma**

**1**

**Lemma**

**2**

**Lemma**

**3**

**Lemma**

**4**

## 3. Main Results

#### 3.1. Predefined-Time Sliding Mode Controller

**Theorem**

**1.**

**Remark**

**5.**

**Proof**

**of**

**Theorem**

**1.**

#### 3.2. Design of Predefined-Time Sliding Mode Controller

**Theorem**

**2.**

**Remark**

**6.**

**Proof**

**of**

**Theorem**

**2.**

**S**within a predefined time ${T}_{s}$. Additionally, considering that the actuator saturation constraints may have an adverse effect on the actuator, an auxiliary system is incorporated to avoid the control input going beyond the physical limitation of the actuator. Then, the predefined-time sliding mode controller is designed as follows:

**Theorem**

**3.**

**Remark**

**7.**

**Remark**

**8**

**Proof**

**of**

**Theorem**

**3.**

#### 3.3. Design of Nonsingular, Practical Predefined-Time Controller

**Hypothesis**

**4.**

**Theorem**

**4.**

**Remark**

**9**.

**Proof**

**of**

**Theorem**

**4.**

## 4. Simulation Cases

#### 4.1. Case 1: Disturbance-Observer-Based Predefined-Time Control

#### 4.2. Case 2: Disturbance-Observer-Based Practical Predefined-Time Control

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Schematic of the proposed adaptive practical prescribed-time fault-tolerant control architecture for AUVs.

**Figure 3.**Motion of the AUV trajectory tracking error ensured by the proposed predefined-time sliding mode controller in Equation (22).

**Figure 4.**The AUV trajectory tracking result with predefined time as 15 s under initial condition 1: (

**a**) AUV position tracking error; (

**b**) AUV attitude tracking error.

**Figure 5.**The AUV trajectory tracking result with predefined time as 15 s under initial condition 2: (

**a**) AUV position tracking error; (

**b**) AUV attitude tracking error.

**Figure 6.**The AUV trajectory tracking result with predefined time as 15 s: (

**a**) AUV position tracking error; (

**b**) AUV attitude tracking error.

**Figure 7.**Control input with auxiliary system with predefined time as 15 s under initial state 1: (

**a**) Control force under initial state 1; (

**b**) control moment under initial state 1.

**Figure 8.**The AUV trajectory tracking result with predefined time as 15 s: (

**a**) AUV position tracking error; (

**b**) AUV attitude tracking error.

**Figure 9.**Control input without auxiliary system with predefined time as 15 s under initial state 1: (

**a**) Control force under initial state 1; (

**b**) control moment under initial state 1.

**Figure 10.**The AUV trajectory tracking result with predefined time as ${T}_{c1}=15/\sqrt{2}\mathrm{s}$ under initial condition 1: (

**a**) AUV position tracking error; (

**b**) AUV attitude tracking error.

**Figure 11.**The AUV trajectory tracking result with predefined time as ${T}_{c1}=15/\sqrt{2}\mathrm{s}$ under initial condition 1: (

**a**) AUV position tracking error; (

**b**) AUV attitude tracking error.

**Figure 12.**The AUV trajectory tracking result with predefined time as ${T}_{c1}=15/\sqrt{2}\mathrm{s}$ under initial condition 1: (

**a**) AUV position tracking error; (

**b**) AUV attitude tracking error.

**Figure 13.**Lumped disturbance reconstruction with predefined-time observer in Equation (12): (

**a**) Lumped disturbance forces reconstruction result; (

**b**) lumped disturbance moments reconstruction result.

Parameter | $\mathit{M}$ | ${\mathit{I}}_{\mathit{x}\mathit{x}}$ | ${\mathit{I}}_{\mathit{y}\mathit{y}}$ | ${\mathit{I}}_{\mathit{z}\mathit{z}}$ |
---|---|---|---|---|

Value | 30 kg | 0.1215 kgm^{2} | 5.468 kgm^{2} | 5.468 kgm^{2} |

Parameters | Value | Parameters | Value |
---|---|---|---|

${X}_{\dot{u}}$ | −7.14 kg | ${Y}_{\dot{v}}$ | −67.7 kg |

${X}_{u}$ | −5.8 kg/s | ${Y}_{v}$ | −49.15 kg/s |

${X}_{\left|u\right|u}$ | −9.29 kg/m | ${Y}_{v\left|v\right|}$ | −79.71 kg/s |

${Z}_{\dot{w}}$ | −60.63 kg | ${N}_{\dot{r}}$ | −0.48 kgm^{2} |

${Z}_{w}$ | −49.52 kg/s | ${N}_{r}$ | −0.56 kgm^{2}/s |

${Z}_{\left|w\right|w}$ | −80.15 kg/m | ${N}_{r\left|r\right|}$ | −115.06 kgm^{2} |

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**MDPI and ACS Style**

Li, Y.; He, J.; Zhang, Q.; Zhang, W.; Li, Y.
Predefined-Time Fault-Tolerant Trajectory Tracking Control for Autonomous Underwater Vehicles Considering Actuator Saturation. *Actuators* **2023**, *12*, 171.
https://doi.org/10.3390/act12040171

**AMA Style**

Li Y, He J, Zhang Q, Zhang W, Li Y.
Predefined-Time Fault-Tolerant Trajectory Tracking Control for Autonomous Underwater Vehicles Considering Actuator Saturation. *Actuators*. 2023; 12(4):171.
https://doi.org/10.3390/act12040171

**Chicago/Turabian Style**

Li, Ye, Jiayu He, Qiang Zhang, Wenjun Zhang, and Yanying Li.
2023. "Predefined-Time Fault-Tolerant Trajectory Tracking Control for Autonomous Underwater Vehicles Considering Actuator Saturation" *Actuators* 12, no. 4: 171.
https://doi.org/10.3390/act12040171