Abstract-In this paper an output feedback controller for tracking control of surface ships based on Euler-Lagrange equations has been proposed. It has been assumed that a surface ship is moving in a horizontal plane and under-actuated in sway direction. The change of coordinate's method is applied to overcome the third order component that arises in the Lyapunov function derivatives due to Coriolis and centripetal forces term. The design of the controller is based on the backstepping control technique and Lyapunov stability theory. Firstly, the observer is derived using the change of coordinate method. Next, backstepping control technique is employed to derive the control law. Finally, a global asymptotic convergence is proven using Lyapunov stability theorems. Simulations are provided to demonstrate the performance of the designed controller and prove tracking error of the controller convergence.
Index Terms-Change of coordinates, euler-lagrange equations, output feedback controller, under-actuated surface ship, third order component, backstepping control technique.
I. INTRODUCTIONA conventional ship considers the motion in surge (forward), sway (sideways), and yaw (heading). Normally, we have surge and sway control forces and yaw control moment available for navigating the ship. However, this assumption is not practical for all ships. For example, some ships are either equipped with two autonomous aft-thrusters or with one main aft-thruster and a rudder, but there is no any bow or side thrusters, like, for example, many supply ships. As a result, there is control force in the sway axis. In this paper, we deal with the tracking control for ships with only surge (x-axis) and yaw (z-axis) control moment available. Since we need to control 3DOF (three degrees of freedom) with only two inputs available, thus we are dealing with an under-actuated problem. Since we want to control the ship motion in the horizontal plane, therefore, we neglect the dynamics related to the motion in heave, roll, and pitch [1], [2].The tracking control of surface ships is compulsory in order to achieve offshore exploitation, activities and various applications such as the drilling, pipe laying, diving support, etc. [3]. Several control strategies have been proposed for surface ships such as global uniform asymptotic stabilization of an under-actuated surface vessel was presented [4]. A Robust adaptive ship autopilot with wave filter and integral action was proposed in [5]. Sliding mode control was presented and experimentally implemented for trajectory tracking of under-actuated autonomous surface vessels [6]. An approximation based control was developed to handle model uncertainties and unknown disturbances for fully actuated ocean surface vessels [7]. Another approach to design a global tracking controller for under-actuated ships with the sway axis unactuated has been introduced [8]. In the proposed work, the assumption that the mass and damping matrices are diagonal is not used as required globally to track the reference trajectory. ...