# Robust adaptive finite-time course tracking control of vessel under actuator attacks

*Complex Eng Syst*2023;3:12.

## Abstract

This paper studies the course tracking control problem of unmanned surface vessels under the influence of uncertain dynamics, external unknown disturbances, constraints, and actuator attacks. In the design of the control scheme, adaptive technology is applied to approach the uncertain dynamics of the system, and a nonlinear finite-time disturbance observer is established to reconstruct the actuator attack signal and the unknown time-varying disturbances online. Combining disturbance compensation and adaptive technology, a finite-time course tracking control scheme is designed. The control scheme does not need to obtain the prior knowledge of the model in advance, and it has good robustness in the face of uncertain dynamics within the system, external disturbances, and actuator attacks. A rigorous stability analysis is provided for the control scheme based on the Lyapunov stability theory. Finally, the simulation shows that the proposed control scheme can effectively resist the influence of actuator attacks and external uncertain disturbances.

## Keywords

*,*course tracking

*,*unmanned surface vessel

*,*finite-time disturbance observer

## 1. INTRODUCTION

With the continuous development of marine economy, unmanned surface vessels (USV) have become the most economical and effective means of marine transportation and have received special attention in the field of marine engineering ^{[1, 2]}. At the same time, the course tracking control of USV is a classic basic research topic, and many researchers have published important research results in this field. The goal of course tracking control is to overcome various internal and external disturbances and realize the high-precision tracking target course of USV ^{[3, 4]}. In order to complete the control task, many control methods, such as neural networks (NNs), sliding modes, self-adaptation, event-triggered control (ETC), nonlinear feedback, and nonlinear decoration, are applied in the design of the control scheme ^{[5]}.

In course tracking control, PID has been widely used in engineering because of its simple control structure and good control effect ^{[6]}. Witkowska *et al.*^{[7]} combined backstepping and genetic algorithm to propose a course tracking control scheme. However, this scheme does not consider the disturbance of the environment. In addition, PID is in the face of disturbances, such as wind, waves, and currents. Its robustness also fails to meet further demands. In order to further solve the problem of external interference, Le *et al.*^{[8]} combined PID and fuzzy logic control to develop an automatic driving scheme for surface vessels and verified the feasibility of the control scheme through simulation. Annamalai *et al.*^{[9]} developed a robust USV adaptive course keeping control scheme combined with the gradient descent algorithm. Yang *et al.*^{[10]} proposed a robust adaptive nonlinear control algorithm for ship steering based on the projection method and Lyapunov stability theory, which simultaneously solved the uncertain dynamics inside and outside the system. Li *et al.*^{[11]} and Zhang *et al.*^{[12]} combine Radial Basis Function NNs (RBFNNS) and dynamic surface control (DSC) technology to further discuss the problem of "differential explosion" in backstepping.

In practice, there is often the challenge of controlling signal transmission, which can lead to channel overload ^{[13]}. This problem is more prominent in many control systems, especially when long-distance transmission is required or when operating under harsh environmental conditions^{[14]}. Factors such as bandwidth limitations, signal delays, and others all bring great challenges to the reliability of the control scheme. ETC adopts an event-driven approach, which triggers the controller to send a control signal only when the system state reaches a certain condition^{[15]}. This means that signals are only sent when adjustments or corrective control actions are required, saving communication bandwidth. Zhang *et al.*^{[16]} combined ETC technology and proposed a heading tracking fault-tolerant control (FTC) scheme. This scheme effectively improves bandwidth efficiency and saves computing resources.

In theoretical research, nonlinear feedback^{[17]} and nonlinear decoration^{[18]} have also been widely used in course tracking control. In Ref.^{[19]}, a course tracking control algorithm is designed by establishing the error driving function to address unknown time-varying disturbance, uncertain ship model parameters, and input saturation. Zhang *et al.*^{[20]} introduced a nonlinear function of heading error in the feedback loop to replace the heading error itself and designed an improved compact backstepping controller based on the Lyapunov candidate function. Zhang *et al.*^{[21]} adopted PID technology, introduced a bipolar sigmoid function, and designed a nonlinear feedback algorithm. Finally, the effect of the algorithm is analyzed using the theory of closed-loop gain shaping. Cao *et al.*^{[22]} proposed an active disturbance rejection control algorithm based on nonlinear feedback to solve the problems of external disturbance, internal model uncertainty, and rudder angle energy input in the process of ship course keeping.

Robustness is an important performance index of the ship control system, which plays a vital role in ensuring the safe and effective operation of the ship ^{[23]}. FTC technology is widely used in the field of ship control because of its good control effect. Compared with traditional control methods, FTC techniques focus on achieving the control objectives of the system within a predetermined finite time. Through the research of the above reference, this paper designs a USV robust adaptive finite-time course tracking control scheme under actuator attacks. The main contributions of this paper are as follows:

(1) Using the features of adaptive online approximation and high reconstruction accuracy of nonlinear finite-time disturbance observer (NFTDO), a robust adaptive tracking control scheme based on depth information robust adaptive method is developed. The scheme not only overcomes internal and external uncertainties but also effectively resists the impact of actuator attacks.

(2) The FTC technology is introduced to further improve the control performance of the course tracking system so that the errors of the system can be converged within a finite time. Compared with traditional control schemes, the steady-state performance and transient response of the system are improved.

## 2. PROBLEM FORMULATION AND PRELIMINARIES

The nonlinear ship course tracking mathematical model can be expressed in the following form^{[24]}:

where

Let

where

**Assumption 1** The external environment disturbances

**Assumption 2** The reference course

**Assumption 3** Both model parameters

**Lemma 1**^{[25]} For system

where

**Lemma 2**^{[26]} Assuming that there is a positive definite Lyapunov function

where

## 3. CONTROL DESIGN AND STABILITY ANALYSIS

The designed procedure of control law is shown in Figure 1. First, define the error variable

where

Define the following virtual control variables

where

Then, using the following DSC technique, we can obtain the derivative of the virtual control

where

Taking the derivative of Eq. (6), we can get

where

According to Eq. (9), it can be further obtained

where

Define a new variable

where

The design control law is as follows

where

Construct the Lyapunov function as follows

where

Deriving Eq. (13) and substituting Eqs. (7)-(12), one can get

where ^{[27]}. According to Assumption 2 and Lemma 1, we can get A=B, where

Substituting Eq. (15) into Eq. (14), one can get

Using Young's inequality, we can get

^{[28]}. So

where

According to Eq. (17), it can be obtained

where

According to Lemma 2,

where

**Remark 1** The basic theory of control design is backstepping. As shown in Figure 1, both virtual control and controller design phases introduce finite-time techniques. At the same time, a finite-time disturbance observer is further introduced to compensate for actuator attacks and external disturbances. This ensures that the response speed and steady state of the system are improved compared to the traditional adaptive scheme.

## 4. SIMULATION

This paper takes the Dalian Maritime University practice ship "Yulong" as the test object for simulation research^{[19]}. The main parameters are shown in Table 1.

Ship particulars of the ship "Yu Long"

Length between perpendiculars (LBP) | 126 | |

Molded breadth | 20.8 | |

Molded draught | 8.0 | |

Rudder area | 18.8 | |

Block coefficient | 0.681 | |

Trial speed | 15 |

The relevant parameters used for simulation are

Figure 2, Figure 3, Figure 4 and Figure 5 show the USV under the influence of time-varying disturbances and actuator attacks, the NFTDO control scheme designed in this paper, the scheme in Ref.^{[19]}, and the ship course tracking effect of the traditional Backstepping control scheme.

The control scheme in Ref.^{[19]} is as follows:

where

It can be seen from Figure 3 that both the NFTDO control scheme designed in this paper and the comparative scheme have completed the heading tracking task, but the dynamic adjustment performance of the scheme in Ref.^{[19]} and the traditional Backstepping control scheme is not as good as that of the NFTDO scheme. It can be seen from the tracking error duration curve shown in Figure 4 that the tracking accuracy under the NFTDO control scheme is higher than that of the comparison scheme. Figure 5 shows the control input response for the three control schemes. The control inputs of the three control schemes tend to be stable over time. Figure 6 shows the reconstruction effect of the NFTDO scheme on the composite uncertain term composed of time-varying interference and actuator attack signals. Figure 7 is the change curve of uncertain items over time. In summary, compared with the control scheme designed in this paper, the tracking effect has been greatly improved.

## 5. CONCLUSIONS

In this paper, by combining FTC and disturbance compensation technology, the problems of actuator attacks, external unknown disturbances, and dynamic uncertainty in USV course tracking control are effectively solved. Without any prior knowledge, a finite-time course tracking control scheme is designed. Finally, the effectiveness is verified by simulation. The simulation results show that the steady-state performance and transient response of the USV are improved under the control scheme designed in this paper. In addition, since ships have unstable situations caused by uncertainties, such as mooring forces during offshore operations, it is necessary to systematically deal with such uncertainties and ensure stability. We will explore this area further in future research.

## DECLARATIONS

### Acknowledgments

The authors would like to acknowledge the National Natural Science Foundation of China (NSFC51779136), Science and Technology Commission of Shanghai Municipality (NO.20dz1206002), and the Natural Science Foundation of Fujian Province of China (2022J011128).

### Authors' contributions

Methodology, software, validation, writing- original draft: Meng X

Writing-reviewing and editing, investigation: Zhang G

Conceptualization, data curation, visualization: Han B

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

None.

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2023.

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## Cite This Article

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**OAE Style**

Meng X, Zhang G, Han B. Robust adaptive finite-time course tracking control of vessel under actuator attacks. *Complex Eng Syst* 2023;3:12. http://dx.doi.org/10.20517/ces.2023.18

**AMA Style**

Meng X, Zhang G, Han B. Robust adaptive finite-time course tracking control of vessel under actuator attacks. *Complex Engineering Systems*. 2023; 3(3): 12. http://dx.doi.org/10.20517/ces.2023.18

**Chicago/Turabian Style**

Xiangfei Meng, Guichen Zhang, Bing Han. 2023. "Robust adaptive finite-time course tracking control of vessel under actuator attacks" *Complex Engineering Systems*. 3, no.3: 12. http://dx.doi.org/10.20517/ces.2023.18

**ACS Style**

Meng, X.; Zhang G.; Han B. Robust adaptive finite-time course tracking control of vessel under actuator attacks. *Complex. Eng. Syst.* **2023**, *3*, 12. http://dx.doi.org/10.20517/ces.2023.18

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