# Command-Filter-Based Region-Tracking Control for Autonomous Underwater Vehicles with Measurement Noise

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- In order to ensure that the AUV tracking error meets the transient and steady-state performance constraints without converging to zero, an improved nonlinear tracking error transformation method is proposed in this paper. Different from previous studies [28,31], which used error transformation and the Lyapunov function in coordination to achieve this goal, we do not need to design a Lyapunov function but only to carry out nonlinear transformation of tracking errors based on allowable error variables, hyperbolic tangent functions, and performance functions. The objective can be realized by stabilizing the transformed error variable.
- (2)
- Aiming at the problem of strong fluctuation of the control signal when tracking an AUV region with measurement noise, a finite-time control method based on a two-stage command filter is proposed in this paper. Different from the traditional method, which directly uses the control law derived from the backstep method as the control signal, in this paper we adopt a finite-time sliding mode differentiator to filter the virtual and final control laws and design a finite-time compensator to compensate the filtering loss and stabilize the closed-loop system. Among them, the filtered final control law is used as the control signal to reduce the fluctuation of control signal caused by measurement noise.

## 2. Problem Statement and Preliminaries

#### 2.1. Analysis of the Existing Problems and Their Causes in Traditional Methods

#### 2.2. Preliminaries

- (1)
- Dynamic model

- (2)
- Lemma and assumption

**Lemma**

**1**

**.**For any given normal numbers $m$, $n$, and $w$, the following inequality holds.

**Lemma**

**2**

**.**For any $x\in R$ and $\omega >0$, the following inequality holds.

**Lemma**

**3**

**.**Consider the system $\dot{x}=f(x)$ . Suppose $V(x)$ is a smooth positive definite function of class C, if $V(x)$ satisfies $\dot{V}(x)\le -{\beta}_{1}V(x)-{\beta}_{2}{V}^{P}(x)+\chi $, where $V(0)=0$, ${\beta}_{1},{\beta}_{2}>0$, $0<P<1$, and $0<\chi <\infty $; then the equilibrium point of system $\dot{x}=f(x)$ is finite-time $T=\mathrm{max}\{\frac{1}{\gamma {\beta}_{1}(1-P)}\mathrm{ln}\frac{\gamma {\beta}_{1}{V}^{1-P}({x}_{0})+{\beta}_{2}}{{\beta}_{2}},\frac{1}{{\beta}_{1}(1-P)}\mathrm{ln}\frac{{\beta}_{1}{V}^{1-P}({x}_{0})+\gamma {\beta}_{2}}{\gamma {\beta}_{2}}\}$ uniform and finally bounded, and the system state converges to the set $\Omega =\mathrm{min}\{\frac{\chi}{(1-\gamma ){\beta}_{1}},{(\frac{\chi}{(1-\gamma ){\beta}_{2}})}^{\frac{1}{P}}\}$ of the equilibrium domain, where $0<\gamma <1$.

**Assumption**

**1**

**.**The interference force and torque on the AUV center of gravity are bounded:

## 3. AUV Finite-Time Region-Tracking Control Method Based on Command Filter

#### 3.1. The Idea of This Method and Difference between This Method and Traditional Method

#### 3.1.1. Idea of the Method in This Paper

- (1)
- Improved nonlinear tracking error transformation method

- (2)
- AUV finite-time control method based on two-stage command filter

#### 3.1.2. Difference between This Method and Traditional Method

- (1)
- Treatment of error variables

- (2)
- Smoothing of control signals

#### 3.2. Implementation Process of Proposed Method

#### 3.2.1. Improved Nonlinear Tracking Error Transformation Method

- (1)
- Construct the performance constraint inequality

- (2)
- Introduce tolerance error to rewrite performance constraint inequality

**Theorem**

**1.**

**Proof**

**1.**

- (3)
- Nonlinear transformation of constrained inequalities

**Theorem**

**2.**

**Proof**

**2.**

#### 3.2.2. AUV Finite-Time Control Method Based on Second-Stage Command Filter

#### 3.3. Stability Analysis of Closed Loop System

**Theorem**

**3.**

**Proof**

**3.**

## 4. Simulation Experiment

#### 4.1. Simulation Settings

- (1)
- Thrust distribution [28,31]: ODIN is an overdriven AUV composed of four horizontal thrusters and four verticals. In order to compare the input control signals of each thruster and subsequently conduct an energy consumption analysis, $\tau $ is written as $\tau =Eu$ in the dynamics model, where $E$ is a $6\times 8$ thrust distribution matrix, and $u$ is a $8\times 1$ vector, representing the thrust of eight thrusters. The expression is as follows:

- (2)
- Dead zone and saturation [31]: Dead zone and saturation are the actual constraints of thrust existence and were set as:

- (3)
- Ocean current [31]: In order to get close to the real marine environment, the first-order Gauss–Markov process with Gaussian white noise (mean 1, variance 2) was used to simulate the amplitude of ocean current, and the integral of Gaussian white noise (mean 0, variance 50) was used to represent the phase angle of ocean current.
- (4)
- Noise [31]: In practical applications, a low-pass filter is used to reduce the overamplification effect caused by measurement noise. In this experiment, Gaussian noise (mean 0, variance 0.05) after a low-pass filter (2 rad/s) was added to the original position signal as the noisy position signal, and Gaussian noise (mean 0, variance 0.01) after a low-pass filter (1 rad/s) was added to the original velocity signal as the noisy speed signal.
- (5)
- Fault [31]: Considering the loss of thrust due to possible faults caused by long-term use of the thruster, it is assumed that failure of thruster T2 occurs; that is, the actual output of thruster T2 is only $(1-{k}_{f}){u}_{s2}$, where

- (6)
- Expected trajectory [31] is described as

- (7)
- The initial state [31] of the AUV is: $\begin{array}{l}\eta (0)={[1,1,-1,\pi /18,\pi /18,\pi /9]}^{T}\\ \dot{\eta}(0)={[0.04,0.04,0.04,0.02,0.02,0.02]}^{T}\end{array}$.

#### 4.2. Experimental Results and Analysis

- (1)
- Experimental results and analysis in case 1

- (2)
- Experimental results and analysis of case 2

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Overall idea of AUV finite-time region tracking control method based on command filter in this paper.

**Figure 3.**Tracking error and control signal of proposed method in Case 1. (

**a**) Tracking error of the proposed method. (

**b**) control signal of the proposed method.

**Figure 6.**Tracking error and control signal of the proposed method in Case 2. (

**a**) Tracking error of the proposed method; (

**b**) control signal of the proposed method.

**Table 1.**Performance of control signals between the proposed method and the traditional method in Case 1.

SCIS (N) | SSCS (107 × N2) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | Total | ||

This paper | 2546 | 1244 | 2695 | 1466 | 1932 | 1546 | 1414 | 1778 | 14,621 | 6.41 |

[28] | 30,522 | 7107 | 18,611 | 5532 | 7202 | 9463 | 9319 | 8220 | 95,976 | 12.92 |

[31] | 6737 | 2887 | 6680 | 2932 | 1877 | 1918 | 1938 | 1902 | 26,871 | 6.13 |

**Table 2.**Performance of control signals between the proposed method and the traditional method in Case 2.

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## Share and Cite

**MDPI and ACS Style**

Lv, T.; Wang, Y.; Liu, X.; Zhang, M.
Command-Filter-Based Region-Tracking Control for Autonomous Underwater Vehicles with Measurement Noise. *J. Mar. Sci. Eng.* **2023**, *11*, 2119.
https://doi.org/10.3390/jmse11112119

**AMA Style**

Lv T, Wang Y, Liu X, Zhang M.
Command-Filter-Based Region-Tracking Control for Autonomous Underwater Vehicles with Measurement Noise. *Journal of Marine Science and Engineering*. 2023; 11(11):2119.
https://doi.org/10.3390/jmse11112119

**Chicago/Turabian Style**

Lv, Tu, Yujia Wang, Xing Liu, and Mingjun Zhang.
2023. "Command-Filter-Based Region-Tracking Control for Autonomous Underwater Vehicles with Measurement Noise" *Journal of Marine Science and Engineering* 11, no. 11: 2119.
https://doi.org/10.3390/jmse11112119