## 1 Introduction  
In recent years, the maneuverability of underactuated surface vessels (USVs) has become a hot topic of ship control [1; 2; 3; 4; 5]. An underactuated system is one in which the number of degrees of freedom of the system is greater than the number of control inputs. An underactuated system refers to a situation where the dimension of the system control input vector space is smaller than the dimension of the system's generalized coordinate vector space [6]. In the study of ship maneuverability, the effectiveness of ship course control has a direct impact on ship maneuverability and the safety of navigation at sea. For vessels at sea, to keep the ship on the expected course, the rudder must be controlled by a control system to correct for the yaw caused by disturbances. Digital PID rudders are widely used as course controllers for ships at sea, but they are too sensitive to disturbances such as wind and waves, which can lead to frequent steering of the rudder. A great deal of research has been carried out on automatic rudder steering for ships in order to achieve satisfactory maneuvering performance, and many results have been obtained [7; 8; 9; 10; 11; 12]. In [7],[PERSON] proposed to combine fuzzy logic control with linear control theory for the design of a ship autopilot controller and used the controller for ship course control, achieving good control results. In [8], optimal control based on the linear quadratic optimal was proposed and applied to a ship's autopilot. In [9], the PSO method for an optimal autopilot PID controller was designed. In [10], the \(H\infty\) controller was solved by the \"2-Riccati\" equation, and it was used for the rudder and flap of a ship's course under system uncertainty. In [11], a new robust adaptive nonlinear feedback algorithm based on error-driven function was proposed, and it was used for ship course-keeping. In [12], an indirect adaptive robust controller was proposed. First, the parameter estimation law under system uncertainty was studied for a ship model. Secondly, an indirect adaptive robust controller was applied for a ship's heading. With the continuous improvement of precision control theory, fractional order calculus theory [13; 14; 15] and fractional order \(\text{PI}^{\lambda}\text{D}^{\mu}\) controllers [16; 17; 18] have been proposed one after another. In 1999, [PERSON] proposed fractional order \(\text{PI}^{\lambda}\text{D}^{\mu}\) controllers on the basis of fractional order calculus theory [16]. Compared with the traditional PID controller, the introduction of integral order \(\lambda\) and differential order \(\mu\) makes the PID controller have two more control parameters, expanding the original \"point\" control to \"surface\" control, further improving the control accuracy and making the controller more flexible [19; 20; 21; 22; 23; 24; 25; 26; 27]. In [19], a two-degrees-of-freedom planar robot was controlled by an FO PID controller tuned with particle swarm optimization (PSO) and a genetic algorithm (GA). In [20], an FO \(\text{PI}^{\lambda}\text{D}^{\mu}\) controller was applied to the speed control of subway trains, and optimization studies were carried out to make the train achieve better speed control and stability. In [21], a new FO \(\text{PI}^{\lambda}\text{D}^{\mu}\) controller was proposed, and a structural transformation method was designed to obtain the controller parameters, which were used to improve the anti-interference of the controller. In [22], a special case of fractional-order systems was first investigated, and on this basis, the FOPD controller and the FOPID controller were designed, which were used in the control of fractional-order systems to achieve a closed-loop system with a monotonic amplitude-frequency response. In [23], for a three-link rigid manipulator system with coupled multiple-input multiple-output characteristics and high nonlinearity, a fractional-order self-tuned fuzzy PID controller was proposed, which was applied to a robotic manipulator system and achieved good control results.