تعقیب مسیر شناورهای سطحی همراه با محدودیت های Rudder و Roll: روش MPC Path Following for Marine Surface Vessels with Rudder and Roll Constraints: an MPC Approach

Controlling of marine surface vessels to follow a prescribed path or track a given trajectory has been a representative control problem for marine applications and has attracted considerable attention from the control community [1]–[۹]. One challenge for path following of marine surface vessels stems from the fact that the system is often underactuated. Conventional ships are usually equipped with one or two main propellers for forward speed control, and rudders for course keeping of the ship. For ship maneuvering problems, such as path following and trajectory tracking, where we seek control for all three degrees of freedom (surge, sway and yaw), the two controls can not influence all three variables independently, thereby leading to under-actuated control problems. Recent development [2], [4]–[۶], [۸] in nonlinear control and control of under-actuated systems has offered new tools and promising solutions to deal with all 3-DoF using two independent controls. Another challenge in the path following of marine surface vessels is the inherent physical limitations in the control inputs, namely the rudder saturation and rudder rate limit. More recently, given that the roll motion produces the highest acceleration and is considered as the principal villain for the sailor seasickness and cargo damage [10], enforcing roll constraints while maneuvering in seaways becomes an important design consideration in surface vessel control. While typical nonlinear control methodologies such as those pursued in [1]–[۹] do not take these input and output constraints explicitly into account in the design process, the constraint enforcement is often achieved through numerical simulations and trial-and-error tuning of the controller parameters. Few other control methodologies, such as the model predictive control (MPC) [11], [12] and reference governor [13], have a clear advantage in addressing input and state constraints explicitly. [14] considers rudder saturation in its MPC controller for tracking control of marine surface vessels and [15] achieves the roll reduction for the heading control problem using an MPC approach. For the path following control problem considered in this paper where both the crosstracking error and heading error are controlled by the rudder angle as an under-actuated problem and rudder limitation and roll constraints need to be enforced simultaneously, MPC applications have not been found in the open literature, to the best knowledge of the authors. MPC, also known as the receding horizon control (RHC), is a control technique which embeds optimization within feedback to deal with systems subject to constraints on inputs and states [11], [12]. Over the last few decades, MPC has proven to be successful for a wide range of applications including chemical, food processing, automotive and aerospace systems [11]. Using an explicit model and the current state as the initial state to predict the future response of a plant, it determines the control action by solving a finite horizon open-loop optimal control problem on-line at each sampling interval. Furthermore, because of its natural appeal to multi-variable systems, MPC can handle underacuated problem gracefully by combining all the objectives into a single objective function. This paper presents an MPC design of the path following problem for an integrated model of the surface vessel dynamics and 2-DoF path following kinematics. Our focus is on satisfying all the inputs and state constraints while achieving satisfactory path following performance. A 3-DoF simplified linear container model is adopted in the controller design and a corresponding 4-DoF nonlinear container model is used in simulations in order to study interactions between the path following maneuvering control and seakeeping roll dynamics. The path following performance of the proposed MPC controller and its sensitivity to the major controller parameters, such as the sampling time, predictive horizon and weighting matrices in the cost-function, are analyzed by numerical simulations. Finally, the effectiveness of the MPC path following controller in the wave field is studied by simulation on a numerical test-bed which combines both ship dynamics and wave impacts on vessels. This paper is organized as follows: in Section II, the 4- DoF container model and the corresponding simplified 3- DoF linear model are presented along with the Serret-Frenet formulation to facilitate the path following control design. In Section III, the MPC algorithm is developed to address the path following problem with rudder and roll constraints. The simulation results in both calm water and wave fields are presented in Section IV together with some discussions on the controller parameter tuning, followed by the conclusions in Section V.