A ship, as an object of course control, is characterized by a nonlinear function describing the static maneuvering characteristics. The backstepping method is one of the methods that can be used during the designing p...A ship, as an object of course control, is characterized by a nonlinear function describing the static maneuvering characteristics. The backstepping method is one of the methods that can be used during the designing process of a nonlinear course controller for ships. The method has been used for the purpose of designing two configurations of nonlinear controllers, which were then used to control the ship course. One of the configurations took dynamic characteristic of a steering gear into account during the designing stage. The parameters of the obtained nonlinear control structures have been tuned to optimise the operation of the control system. The optimisation process has been performed by means of genetic algorithms. The quality of operation of the designed control algorithms has been checked in simulation tests performed on the mathematical model of a tanker. The results of simulation experiments have been compared with the performance of the system containing a conventional proportional-derivative (PD) controller.展开更多
An adaptive robust control algorithm for ship straight path control system in the presence of both modeling uncertainties and the bounded disturbances is proposed. Motivated by the backstepping approach, the algorithm...An adaptive robust control algorithm for ship straight path control system in the presence of both modeling uncertainties and the bounded disturbances is proposed. Motivated by the backstepping approach, the algorithm is developed by using the dissipation theory, such that the resulting dosed-loop system is both strictly dissipative and asymptotically adaptively stable for all admissible uncertainties. Also, it is able to steer an underactuated ship along a prescribed straight path with ultimate bounds under external disturbances induced by wave, wind and ocean current. When there are no disturbances, the straight path control can be implemented in a locally asymptotically stable manner. Simulation results on an ocean-going training ship ‘YULONG' are presented to validate the effectiveness of the algorithm.展开更多
A path following controller is developed for underactuated ships with only surge force and yaw moment available to follow a predefined path.The proposed controller is based on nonswitch analytic model predictive contr...A path following controller is developed for underactuated ships with only surge force and yaw moment available to follow a predefined path.The proposed controller is based on nonswitch analytic model predictive control.It is shown that the optimal control law for a nonlinear path following system with ill-defined relative degree is continuous and nonsingular.The problem of ill-defined relative degree is solved.The path-following ability of the nonlinear system is guaranteed.Numerical simulations are provided to demonstrate the effectiveness of the proposed control law.展开更多
The drawbacks of common nonlinear Filtered-ε adaptive inverse control (AIC) method, such as the unreliability due to the change of delay time and the faultiness existing in its disturbance control loop, are discuss...The drawbacks of common nonlinear Filtered-ε adaptive inverse control (AIC) method, such as the unreliability due to the change of delay time and the faultiness existing in its disturbance control loop, are discussed. Based on it, the diagram of AIC is amended to accommodate with the characteristic of nonlinear object with time delay. The corresponding Filtered-ε adaptive algorithm based on RTRL is presented to identify the parameters and design the controller. The simulation results on a nonlinear ship model of "The R.O.V Zeefakker" show that compared with the previous scheme and adaptive PID control, the improved method not only keeps the same dynamic response performance, but also owns higher robustness and disturbance rejection ability, and it is suitable for the control of nonlinear objects which have higher requirement to the maneuverability under complex disturbance environment.展开更多
The aim of this paper is to investigate an adaptive sensorless direct voltage control(DVC)strategy for the stand-alone ship shaft brushless doubly-fed induction generators(BDFIGs).The proposed new rotor position obser...The aim of this paper is to investigate an adaptive sensorless direct voltage control(DVC)strategy for the stand-alone ship shaft brushless doubly-fed induction generators(BDFIGs).The proposed new rotor position observer using the space vector flux relations of BDFIG may achieve the desired voltage control of the power winding(PW)in terms of magnitude and frequency,without any speed/position sensors.The proposed algorithm does not require any additional observers for obtaining the generator speed.The proposed technique can directly achieve the desired DVC based on the estimated rotor position,which may reduce the overall system cost.The stability analysis of the proposed observer is investigated and confirmed with the concept of quadratic Lyapunov function and using the multi-model representation.In addition,the sensitivity analysis of the presented method is confirmed under different issues of parameter uncertainties.Comprehensive results from both simulation and experiments are realized with a prototype wound-rotor BDFIG,which demonstrate the capability and efficacy of the proposed sensorless DVC strategy with good transient behavior under different operating conditions.Furthermore,the analysis confirms the robustness of the proposed observer via the machine parameter changes.展开更多
This study presents an adaptive fuzzy neural network (FNN) control system for the ship steering autopilot. For the Norrbin ship steering mathematical model with the nonlinear and uncertain dynamic characteristics, an ...This study presents an adaptive fuzzy neural network (FNN) control system for the ship steering autopilot. For the Norrbin ship steering mathematical model with the nonlinear and uncertain dynamic characteristics, an adaptive FNN control system is designed to achieve high-precision track control via the backstepping approach. In the adaptive FNN control system, a FNN backstepping controller is a principal controller which includes a FNN estimator used to estimate the uncertainties, and a robust controller is designed to compensate the shortcoming of the FNN backstepping controller. All adaptive learning algorithms in the adaptive FNN control system are derived from the sense of Lyapunov stability analysis, so that system-tracking stability can be guaranteed in the closed-loop system. The effectiveness of the proposed adaptive FNN control system is verified by simulation results.展开更多
This paper presents a method to design a control scheme for nonlinear systems using fuzzy optimal control.In the design process,the nonlinear system is first converted into local subsystems using sector non linearity ...This paper presents a method to design a control scheme for nonlinear systems using fuzzy optimal control.In the design process,the nonlinear system is first converted into local subsystems using sector non linearity approach of Takagi Sugeno(T S)fuzzy modeling.For each local subsystem,an optimal control is designed.Then,the parameters of local controllers are defuzzified to construct a global optimal controller.To prove the effectiveness of this control scheme,simulations are performed using the mathematical model of Esso Osaka tanker ship for set point regulation with and without disturbance and reference tracking.In addition,the simulation results are compared with that of a PID controller for further verification and validation.It has been shown that the proposed optimal controller can be used for the nonlinear ship steering with good rise time,zero steady state error and fast settling time.展开更多
Given the uncertainty of parameters and the random nature of disturbances that effect a ships course, a robust course controller should be designed on the basis of rudder/flap vector control. This paper analyzes syste...Given the uncertainty of parameters and the random nature of disturbances that effect a ships course, a robust course controller should be designed on the basis of rudder/flap vector control. This paper analyzes system uncertainty, and the choice of weighting functions is also discussed. When sea waves operate on a ship, the energy-concentrating frequency varies with the angle of encounter. For different angles of encounter, different weighting functions are designed. For the pole of a nominal model existing in an imaginary axis, the bilinear-transform method is used. The "2-Riccati" equation is adopted to solve the H∞ controller. A system simulation is given, and the results show that, compared with a PID controller, this system has higher course precision and more robust performance. This research has significant engineering value.展开更多
Among the promising application of autonomous surface vessels(ASVs)is the utilization of multiple autonomous tugs for manipulating a floating object such as an oil platform,a broken ship,or a ship in port areas.Consid...Among the promising application of autonomous surface vessels(ASVs)is the utilization of multiple autonomous tugs for manipulating a floating object such as an oil platform,a broken ship,or a ship in port areas.Considering the real conditions and operations of maritime practice,this paper proposes a multi-agent control algorithm to manipulate a ship to a desired position with a desired heading and velocity under the environmental disturbances.The control architecture consists of a supervisory controller in the higher layer and tug controllers in the lower layer.The supervisory controller allocates the towing forces and angles between the tugs and the ship by minimizing the error in the position and velocity of the ship.The weight coefficients in the cost function are designed to be adaptive to guarantee that the towing system functions well under environmental disturbances,and to enhance the efficiency of the towing system.The tug controller provides the forces to tow the ship and tracks the reference trajectory that is computed online based on the towing angles calculated by the supervisory controller.Simulation results show that the proposed algorithm can make the two autonomous tugs cooperatively tow a ship to a desired position with a desired heading and velocity under the(even harsh)environmental disturbances.展开更多
The fuzzy switched PID controller which combines fuzzy PD and conventional PI controller is proposed for ship track-keeping autopilot In this paper. By using rudder angle, the whole voyage is divided into two operatin...The fuzzy switched PID controller which combines fuzzy PD and conventional PI controller is proposed for ship track-keeping autopilot In this paper. By using rudder angle, the whole voyage is divided into two operating regimes which named transient operating regime and steady operating regime respectively. The fuzzy PD controller is employed in transient operating regime for increasing response, reducing overshoot and shorting transition time. And conventional PI controller is used to improve the stable accuracy in steady operating regime. The global controller is achieved by fuzzy blending of all local controllers. Routh stability criterion is utilized to obtain the stability condition of closed-loop system. The simulation results show the effectiveness of proposed method.展开更多
A linear quadratic optimal direct track-keeping control law was proposed based on first-order Nomoto nominal model. Furthermore, based on Lyapunov stabilized theory, considering parametric uncertainty from variations ...A linear quadratic optimal direct track-keeping control law was proposed based on first-order Nomoto nominal model. Furthermore, based on Lyapunov stabilized theory, considering parametric uncertainty from variations of ship speed and disturbances uncertain from wind, wave and sea current, a direct compensative robust optimal control (DCROC) law was developed. It can guarantee closed-loop system globally and uniformly converge to a remained set. High accuracy and robustness were achieved. By introducing some nonlinear blocks, closed-loop system achieves global and uniform asymptotical stableness. Numerical simulations on a Mariner Class ship are presented to validate the control law.展开更多
A nonlinear robust control strategy is proposed to force an underactuated surface ship to follow a predefined path with uncertain environmental disturbance and parameters.In the controller design,a high-gain observer ...A nonlinear robust control strategy is proposed to force an underactuated surface ship to follow a predefined path with uncertain environmental disturbance and parameters.In the controller design,a high-gain observer is used to estimate velocities,thus only position and yaw angle measurements are required.The control problem of underactuated system is transformed into a control of fully actuated system through adopting an improved line-of-sight(LOS) guidance law.A sliding-mode controller is designed to eliminate the yaw angle error,and provide the control system robustness.The control law is proved semi-globally exponentially stable(SGES) by applying Lyapunov stability theory,and numerical simulation using real data of a monohull ship illustrates the effectiveness and robustness of the proposed methodology.展开更多
In order to damp the heave motion of surface effect ships(SESs),a sliding mode controller with a disturbance observer was designed.At first,a disturbance observer was proposed to estimate the unknown time-varying dist...In order to damp the heave motion of surface effect ships(SESs),a sliding mode controller with a disturbance observer was designed.At first,a disturbance observer was proposed to estimate the unknown time-varying disturbance acting on SESs due to waves.Then,based on the disturbance,a slide mode controller was designed to minimize the magnitude of SES's heave motion position.It was theoretically proved that the designed sliding mode controller with the disturbance observer could guarantee the stability of the closed-loop heave motion control system of SESs.Simulations on a Norwegian Navy's SES were carried out and the simulation results illustrated the effectiveness of the proposed controller with the disturbance observer.展开更多
Motion state of ship out of control in bridge area was analyzed.Motion procedure after losing control was divided into two steps.One is drift step within stopping period.The other is drift step without inertia,which i...Motion state of ship out of control in bridge area was analyzed.Motion procedure after losing control was divided into two steps.One is drift step within stopping period.The other is drift step without inertia,which is induced by wind and current.Mathematical model for motion of ship out of control,considering wind-induced drift,current-induced drift,stopping ability,etc.,was established.Dangerous collision areas for main pier and auxiliary piers were analyzed according to different calculation conditions.展开更多
The cascade systems which stabilize the transverse deviation of the ship in relation to the set path is presented. The ship's path is determined as a broken line with specified coordinates of way points. Three con...The cascade systems which stabilize the transverse deviation of the ship in relation to the set path is presented. The ship's path is determined as a broken line with specified coordinates of way points. Three controllers are used in the system. The main primary controller is the trajectory controller. The set value of heading for the course control system or angular velocity for the turning control system is generated. The course control system is used on the straight line of the set trajectory while the turning controller is used during a change of the set trajectory segment. The characteristics of the non-linear controllers are selected in such a way that the properties of the control system with the rate of turn controller are modelled by the first-order inertia, while the system with the course keeping controller is modelled by a second-order linear term. The presented control system is tested in computer simulation. Some results of simulation tests are presented and discussed.展开更多
The existence of linear quadratic optimal control of ship automatic steering instruments is studied. Firstly, the sufficient conditions for the quadratic integrability of the solutions of linear second order time-vari...The existence of linear quadratic optimal control of ship automatic steering instruments is studied. Firstly, the sufficient conditions for the quadratic integrability of the solutions of linear second order time-variant differential equations are developed. Secondly, the optimal control form of the ship automatic steering instrument is obtained by using the dynamic programming method, which guarantees a minimal ship sway range, during long-distance navigation, by using as little energy as possible. Finally, based on the above mentioned sufficient conditions, the conditions for the realization of optimal control are obtained, which provides a foundation for choosing the weighted coefficients for optimal control in engineering.展开更多
基金supported by Polish Ministry of Science and Higher Education (No. N514 015 32/1712)
文摘A ship, as an object of course control, is characterized by a nonlinear function describing the static maneuvering characteristics. The backstepping method is one of the methods that can be used during the designing process of a nonlinear course controller for ships. The method has been used for the purpose of designing two configurations of nonlinear controllers, which were then used to control the ship course. One of the configurations took dynamic characteristic of a steering gear into account during the designing stage. The parameters of the obtained nonlinear control structures have been tuned to optimise the operation of the control system. The optimisation process has been performed by means of genetic algorithms. The quality of operation of the designed control algorithms has been checked in simulation tests performed on the mathematical model of a tanker. The results of simulation experiments have been compared with the performance of the system containing a conventional proportional-derivative (PD) controller.
文摘An adaptive robust control algorithm for ship straight path control system in the presence of both modeling uncertainties and the bounded disturbances is proposed. Motivated by the backstepping approach, the algorithm is developed by using the dissipation theory, such that the resulting dosed-loop system is both strictly dissipative and asymptotically adaptively stable for all admissible uncertainties. Also, it is able to steer an underactuated ship along a prescribed straight path with ultimate bounds under external disturbances induced by wave, wind and ocean current. When there are no disturbances, the straight path control can be implemented in a locally asymptotically stable manner. Simulation results on an ocean-going training ship ‘YULONG' are presented to validate the effectiveness of the algorithm.
基金supported by the National Natural Science Foundation of China(No.50779033)the National High Technology Research and Development Program(863 Program)of China(No.2007AA11Z250)
文摘A path following controller is developed for underactuated ships with only surge force and yaw moment available to follow a predefined path.The proposed controller is based on nonswitch analytic model predictive control.It is shown that the optimal control law for a nonlinear path following system with ill-defined relative degree is continuous and nonsingular.The problem of ill-defined relative degree is solved.The path-following ability of the nonlinear system is guaranteed.Numerical simulations are provided to demonstrate the effectiveness of the proposed control law.
基金This project was supported by the National Defence Pre-research Foundation of Shipbuilding Industry (01J1.50) and theWeapon & Equipment Pre-research Foundation of General Armament Department (51414030204JW0322).
文摘The drawbacks of common nonlinear Filtered-ε adaptive inverse control (AIC) method, such as the unreliability due to the change of delay time and the faultiness existing in its disturbance control loop, are discussed. Based on it, the diagram of AIC is amended to accommodate with the characteristic of nonlinear object with time delay. The corresponding Filtered-ε adaptive algorithm based on RTRL is presented to identify the parameters and design the controller. The simulation results on a nonlinear ship model of "The R.O.V Zeefakker" show that compared with the previous scheme and adaptive PID control, the improved method not only keeps the same dynamic response performance, but also owns higher robustness and disturbance rejection ability, and it is suitable for the control of nonlinear objects which have higher requirement to the maneuverability under complex disturbance environment.
基金This work was supported in part by the National Natural Science Foundation of China(NSFC)under Grants 51707079 and 51877093in part by the National Key Research and Development Program of China(Project ID:YS2018YFGH000200)in part by the Fundamental Research Funds for the Central Universities(Project ID:2019kfyXMBZ031).
文摘The aim of this paper is to investigate an adaptive sensorless direct voltage control(DVC)strategy for the stand-alone ship shaft brushless doubly-fed induction generators(BDFIGs).The proposed new rotor position observer using the space vector flux relations of BDFIG may achieve the desired voltage control of the power winding(PW)in terms of magnitude and frequency,without any speed/position sensors.The proposed algorithm does not require any additional observers for obtaining the generator speed.The proposed technique can directly achieve the desired DVC based on the estimated rotor position,which may reduce the overall system cost.The stability analysis of the proposed observer is investigated and confirmed with the concept of quadratic Lyapunov function and using the multi-model representation.In addition,the sensitivity analysis of the presented method is confirmed under different issues of parameter uncertainties.Comprehensive results from both simulation and experiments are realized with a prototype wound-rotor BDFIG,which demonstrate the capability and efficacy of the proposed sensorless DVC strategy with good transient behavior under different operating conditions.Furthermore,the analysis confirms the robustness of the proposed observer via the machine parameter changes.
基金Supported by Doctoral Bases Foundation of the Educational Committee of P. R. China under Grant No. 20030151005 and the Ministry of Communication of P. R. China under Grant No. 200332922505.
文摘This study presents an adaptive fuzzy neural network (FNN) control system for the ship steering autopilot. For the Norrbin ship steering mathematical model with the nonlinear and uncertain dynamic characteristics, an adaptive FNN control system is designed to achieve high-precision track control via the backstepping approach. In the adaptive FNN control system, a FNN backstepping controller is a principal controller which includes a FNN estimator used to estimate the uncertainties, and a robust controller is designed to compensate the shortcoming of the FNN backstepping controller. All adaptive learning algorithms in the adaptive FNN control system are derived from the sense of Lyapunov stability analysis, so that system-tracking stability can be guaranteed in the closed-loop system. The effectiveness of the proposed adaptive FNN control system is verified by simulation results.
基金supported in part by the National Natural Science Foundation of China (No. 61751210)the Jiangsu Natural Science Foundation of China (No. BK20171417)the Fundamental Research Funds for the Central Universities(No. NG2019002)
文摘This paper presents a method to design a control scheme for nonlinear systems using fuzzy optimal control.In the design process,the nonlinear system is first converted into local subsystems using sector non linearity approach of Takagi Sugeno(T S)fuzzy modeling.For each local subsystem,an optimal control is designed.Then,the parameters of local controllers are defuzzified to construct a global optimal controller.To prove the effectiveness of this control scheme,simulations are performed using the mathematical model of Esso Osaka tanker ship for set point regulation with and without disturbance and reference tracking.In addition,the simulation results are compared with that of a PID controller for further verification and validation.It has been shown that the proposed optimal controller can be used for the nonlinear ship steering with good rise time,zero steady state error and fast settling time.
文摘Given the uncertainty of parameters and the random nature of disturbances that effect a ships course, a robust course controller should be designed on the basis of rudder/flap vector control. This paper analyzes system uncertainty, and the choice of weighting functions is also discussed. When sea waves operate on a ship, the energy-concentrating frequency varies with the angle of encounter. For different angles of encounter, different weighting functions are designed. For the pole of a nominal model existing in an imaginary axis, the bilinear-transform method is used. The "2-Riccati" equation is adopted to solve the H∞ controller. A system simulation is given, and the results show that, compared with a PID controller, this system has higher course precision and more robust performance. This research has significant engineering value.
基金supported by the China Scholarship Council(201806950080)the Researchlab Autonomous Shipping(RAS)of Delft University of Technology,and the INTERREG North Sea Region Grant“AVATAR”funded by the European Regional Development Fund.
文摘Among the promising application of autonomous surface vessels(ASVs)is the utilization of multiple autonomous tugs for manipulating a floating object such as an oil platform,a broken ship,or a ship in port areas.Considering the real conditions and operations of maritime practice,this paper proposes a multi-agent control algorithm to manipulate a ship to a desired position with a desired heading and velocity under the environmental disturbances.The control architecture consists of a supervisory controller in the higher layer and tug controllers in the lower layer.The supervisory controller allocates the towing forces and angles between the tugs and the ship by minimizing the error in the position and velocity of the ship.The weight coefficients in the cost function are designed to be adaptive to guarantee that the towing system functions well under environmental disturbances,and to enhance the efficiency of the towing system.The tug controller provides the forces to tow the ship and tracks the reference trajectory that is computed online based on the towing angles calculated by the supervisory controller.Simulation results show that the proposed algorithm can make the two autonomous tugs cooperatively tow a ship to a desired position with a desired heading and velocity under the(even harsh)environmental disturbances.
文摘The fuzzy switched PID controller which combines fuzzy PD and conventional PI controller is proposed for ship track-keeping autopilot In this paper. By using rudder angle, the whole voyage is divided into two operating regimes which named transient operating regime and steady operating regime respectively. The fuzzy PD controller is employed in transient operating regime for increasing response, reducing overshoot and shorting transition time. And conventional PI controller is used to improve the stable accuracy in steady operating regime. The global controller is achieved by fuzzy blending of all local controllers. Routh stability criterion is utilized to obtain the stability condition of closed-loop system. The simulation results show the effectiveness of proposed method.
基金Navy Engineering University Natural Science Foundation (NoHGDJJ05013)
文摘A linear quadratic optimal direct track-keeping control law was proposed based on first-order Nomoto nominal model. Furthermore, based on Lyapunov stabilized theory, considering parametric uncertainty from variations of ship speed and disturbances uncertain from wind, wave and sea current, a direct compensative robust optimal control (DCROC) law was developed. It can guarantee closed-loop system globally and uniformly converge to a remained set. High accuracy and robustness were achieved. By introducing some nonlinear blocks, closed-loop system achieves global and uniform asymptotical stableness. Numerical simulations on a Mariner Class ship are presented to validate the control law.
基金Projects(61004008,51509055)supported by the National Natural Science Foundation of ChinaProject(61422230302162223013)supported by the Laboratory of Science and Technology on Water Jet Propulsion,China
文摘A nonlinear robust control strategy is proposed to force an underactuated surface ship to follow a predefined path with uncertain environmental disturbance and parameters.In the controller design,a high-gain observer is used to estimate velocities,thus only position and yaw angle measurements are required.The control problem of underactuated system is transformed into a control of fully actuated system through adopting an improved line-of-sight(LOS) guidance law.A sliding-mode controller is designed to eliminate the yaw angle error,and provide the control system robustness.The control law is proved semi-globally exponentially stable(SGES) by applying Lyapunov stability theory,and numerical simulation using real data of a monohull ship illustrates the effectiveness and robustness of the proposed methodology.
基金National Natural Science Foundations of China(Nos.51579026,51079013)Program for Excellent Talents in Universities of Liaoning Province,China(No.LR2015007)+1 种基金Project of Resource and Social Security of Ministry of Human Province,ChinaFundamental Research Funds for the Central Universities,China(No.3132016020)
文摘In order to damp the heave motion of surface effect ships(SESs),a sliding mode controller with a disturbance observer was designed.At first,a disturbance observer was proposed to estimate the unknown time-varying disturbance acting on SESs due to waves.Then,based on the disturbance,a slide mode controller was designed to minimize the magnitude of SES's heave motion position.It was theoretically proved that the designed sliding mode controller with the disturbance observer could guarantee the stability of the closed-loop heave motion control system of SESs.Simulations on a Norwegian Navy's SES were carried out and the simulation results illustrated the effectiveness of the proposed controller with the disturbance observer.
基金Supported by Disaster Reduction Foundation of Large Complex Project for Sutong Bridge(No.2006BAG04B06)
文摘Motion state of ship out of control in bridge area was analyzed.Motion procedure after losing control was divided into two steps.One is drift step within stopping period.The other is drift step without inertia,which is induced by wind and current.Mathematical model for motion of ship out of control,considering wind-induced drift,current-induced drift,stopping ability,etc.,was established.Dangerous collision areas for main pier and auxiliary piers were analyzed according to different calculation conditions.
文摘The cascade systems which stabilize the transverse deviation of the ship in relation to the set path is presented. The ship's path is determined as a broken line with specified coordinates of way points. Three controllers are used in the system. The main primary controller is the trajectory controller. The set value of heading for the course control system or angular velocity for the turning control system is generated. The course control system is used on the straight line of the set trajectory while the turning controller is used during a change of the set trajectory segment. The characteristics of the non-linear controllers are selected in such a way that the properties of the control system with the rate of turn controller are modelled by the first-order inertia, while the system with the course keeping controller is modelled by a second-order linear term. The presented control system is tested in computer simulation. Some results of simulation tests are presented and discussed.
基金supported by National Nature Science Foundation of P.R.China(No.69974032).
文摘The existence of linear quadratic optimal control of ship automatic steering instruments is studied. Firstly, the sufficient conditions for the quadratic integrability of the solutions of linear second order time-variant differential equations are developed. Secondly, the optimal control form of the ship automatic steering instrument is obtained by using the dynamic programming method, which guarantees a minimal ship sway range, during long-distance navigation, by using as little energy as possible. Finally, based on the above mentioned sufficient conditions, the conditions for the realization of optimal control are obtained, which provides a foundation for choosing the weighted coefficients for optimal control in engineering.