# Nonlinear Optimal Control Law of Autonomous Unmanned Surface Vessels
## 1 Introduction  
The ocean is an unknown, challenging, and mysterious territory. In order to complete given missions, researchers explore the ocean using multiple methods and equipment, such as using unmanned aerial vehicles and research ships. However, these methods may have several drawbacks, such as huge financial pressure and time-consuming operation processes for ocean message searching. To resolve these mentioned difficulties, throughout the past few decades, researchers have developed new exploration vehicles, autonomous unmanned surface vessels (AUSVs), to navigate the ocean. Control-system design is a key function of AUSVs that allows them to successfully achieve tracking missions. The main design challenge for the tracking problem of AUSVs during the execution of sailing missions or the acquisition of sea resources is how to precisely guide AUSVs to designated locations while consuming low power. For collecting long-term data while consuming low power in an unpredictable ocean environment, designing an effective nonlinear optimal control law is essential for AUSVs. Until now, numerous researchers tried to overcome this challenge by developing proper and effective control methods. Many existing studies for the trajectory-tracking problem of AUSVs were proposed, such as [1, 2], which proposed linear control methods that are only capable of treating scenarios with certain operation points. These kinds of linear control designs usually work weakly in global ocean environments. Several studies for trajectory-tracking design have been proposed. Some of them are nonlinear control investigations based on sliding-control perspectives, which may possess global characteristics for the autonomous design of AUSVs [3, 4]. However, the conservation features(high control gain) of control forces and the applied torque, as well as an inevitable system-chattering character, always appear in AUSVs. In [5; 6], two adaptive-based control approaches with adjustable weights that should be optimized were investigated for AUSVs. In [7], a backstepping control method for treating the stabilization and tracking problem of AUSVs under the effect of constant-bias ocean-environment disturbances was studied. A backstepping Type 2 fuzzy control design with a series of decision-making procedures was proposed in [8]. The authors proposed nonlinear control designs for the trajectory-tracking problem of AUSVs [9; 10]. These control designs are usually too complicated in control structure, resulting in relatively huge power consumption for calculations in each mission. To reduce calculation-power consumption and develop an easy-to-implement control structure, two concise control laws were proposed based on the linear-algebra approach and experiment tests for AUSVs in [11; 12]. In these two control achievements, trajectory-tracking performance was well-considered, but the energy-saving problem of AUSVs was not considered in the control-design process. Closed-form nonlinear optimal control design of AUSVs remedies the aforementioned problems. However, an optimal closed-form solution of trajectory- and waypoint-tracking problems of AUSVs have not yet been solved because of the extremely complex dynamics between controlled AUSVs and random ocean environments. To simplify the control structure and precisely guide AUSVs to the desired destinations, an important contribution of this research is finding an optimal closed-form solution that has the simplest control structure for the trajectory- and waypoint-tracking problems of AUSVs. This paper is organized as follows: in Section 2, the mathematical expression of the marine-surface-vessel model and the design objective are given; in Section 3, the proposed optimal control law design is derived; in Section 4, the simulation results of the trajectory- and waypoint-tracking scenarios with modelling uncertainties and ocean-environment disturbances are shown and discussed; in the last section of the paper, the major conclusion of this investigation is stated.