## 1 Introduction  
With the development of automation, ship autopilot has been used to keep course, change course, follow path and roll reduction for the purpose of safety and economy. Most merchant ship autopilots are underactuated mechanical systems, especially for the path following autopilots with rudder roll reduction (PFA-RRR), which have become an active research field [1; 2]. For the PFA-RRR, some theoretical challenges hinder its widespread use on board surface ships, such as underactuated mechanism, non-minimum phase, performances trade-offs, the engineering significance of the control schemes and the uncertainties caused by the model perturbation and disturbances [3; 4]. Underactuated ships are only equipped with rudders and propellers for yawing and surging motions, while there are no actuators for direct control of roll and sway motions. In accordance with the Brockett necessary condition, it is impossible to stabilize the underactuated ships by time invariant continuous control laws [5]. Several novel control strategies have been proposed to solve the path following of underactuated ships [6; 7; 8; 9]. Although the effectiveness of the PFA has been proven, the marine surface ships with PFA are still negatively influenced by the wave-induced roll motions, such as cargo damage, personal seasickness, equipment operation failures and even ship capsizing. Therefore, PFA-RRR is of prominent safety significance for ship navigation in heavy seas, because most ships now are only equipped with rudders rather than other roll reduction devices, especially for merchant ships. The mechanism of RRR is to compensate the wave-induced rolling by rudder-induced initial inverse response [9; 10]. [PERSON] [3] and [PERSON] and [PERSON] [1] gave brief reviews of the developments of RRR from the earliest in 1970. However, these research developments and practical applications were mostly limited to naval vessels. Due to loading on upper deck, container ships have increased windward area and higher center of gravity, which result in higher risk in rough seas. Therefore, the PFA-RRR of a container ship is taken into consideration in this paper. As reported by World Shipping Council, container ships around the world lost 2683 containers annually from 2011 to 2013. In addition, due to the rough sea conditions, MOL Comfort broke apart and sank on 17 June 2013, Svendborg Maersk lost 520 containers on 14 February 2014, and so on [10]. Why are these container ships not equipped with anti-roll fins? For merchant ships, a more decisive factor is cost vs. performance. Although the roll fin stabilizer performs well, it requires great installation costs, machine running consumption and hull space. In the case of common merchant ship equipped with autopilot, when the heavy sea states affect the safety of the ship or cargo, the ship can only stop the autopilot steering and give up the planned route and then adopt the method of manual steering to head the wind and wave for the roll reduction, while the ship equipped with RRR can move on with reduced rolling impact, resulting in better safety and economic performance. All that is required is the upgrade of the steering gear's quick performance and the addition of the controller module. The selection of a suitable controller for PFA-RRR is an important technology. For ship motion control, much literature has studied the control of course, track and roll separately or jointly. [PERSON] et al. [11; 12] used linear reduction of Backstepping and closed-loop gain shaping algorithm (CGSA) for ship course control. [PERSON] et al. [13] presented adaptive neural path-following control for underactuated ships in fields of marine practice. [PERSON] and [PERSON] [1] gave a complete review of the research on ship roll reduction. The above studies performed well on the separate control of course and path following. In recent years, the study of rudder roll reduction has attracted more and more research on the development of full mission autopilot.