Adaptive RBF -SMC control for multi-point mooring system

A stable adaptive control scheme for multi-point mooring system (MPMS) with uncertain dynamics is proposed in this paper. The control scheme is designed by a hybrid controller based on RBF (Radial Basis Function) NN (Neural Network) and SMC (Sliding Mode Control), which learns the MPMS dynamic changes, and the compensation of external disturbances is realized through adaptive RBFNN control. Meanwhile the RBF-SMC control parameters are adapted by the Lyapunov method to minimize squares dynamic positioning (DP) error. The convergence of the hybrid controller is proved theoretically, and the proposed mooring control scheme is applied to the “Kantan3” mooring simulation system. Finally, the simulation results are compared with the traditional PID controller and standard RBF controller to demonstrate the effective mooring positioning performance of the control scheme for the MPMS.


Introduction
Many kinds of offshore structures, such as mooring positioning system (MPMS), dynamic positioning (DP) system, and anchor auxiliary dynamic positioning system are being widely used in semi-submersible production platform. Among these platforms, the mooring system is characterized by less investment and convenience with maintenance; it is the main positioning system [1]. The mooring system provides the positioning restoring force for the semi-submersible platform by coordinating the mooring cable's retracting and releasing to generate resistance to environmental disturbance. At present, the research of deepwater mooring systems is mainly reflected in the dynamic analysis and the study of mooring damping. [2] considered the influence of nonlinear factors such as inertia, damping, and elastic deformation of anchor cable, the concentrated mass method is extended from the frequency domain to the time domain. [3] employed the time domain method to numerically analyze the dynamic effects of nonlinear coupling and uncoupling between the Spar platform's main body and the mooring system. For the design of the mooring system, the experimental results show that dynamic analysis is particularly important. [4] proposed a finite element technique for simulation of grounded and floating systems in the time domain, which assumed that the additional mass coefficient did not change significantly with frequency, and explained the dynamic characteristics of anchor chain reasonably.
A lot of research has been performed on the architecture of the DP control system relative to mooring control [5]. The initial DP control system added a low-pass filter to the traditional PID control model; nevertheless, the positioning accuracy of the controller is affected. To solve this problem, [6] proposed an approach that integrates optimal control with the principle of Kalman filters, and the actual scale was verified later by [7]. This recommendation has been amended, as presented in [8], which suppressed the highfrequency interference pulse but made the positioning error phase lag. [9] performed LQG control, which effectively eliminates high-frequency disturbance signals and reduces energy consumption, however, it has poor adaptability to the uncertainties of offshore platforms.
Robustness is the significant issue of the controller design because the vessel is suffered from wind, currents and waves in an extremely environment. Some scholars, including [10]- [11] applied the H∞ control theory to prove that the controller has good robustness under great changes in environmental conditions. However, it is a linear controller based on a linear model. The multi-point mooring positioning process is an extremely nonlinear dynamics and oscillator. To solve its nonlinear problems, some methods have been applied to the DP.
2 Mathematical modeling of multi-point mooring system

Kinematics
A Cartesian coordinate system describes the motion of the multi-point mooring platform in Fig. 1. The basic coordinate system can be divided into the Earth-fixed XEYEZE coordinate system and body-fixed coordinate system, the body frame is usually chosen at the platform's center of gravity conveniently.

Nonlinear low-frequency model
The nonlinear coupling low-frequency motion equation of the multi-point mooring platform with anchoring automatic positioning in the direction of the surge, sway, and yaw in the moving coordinate system can be described as: where M is the superposition of the additional mass matrix and inertia matrix, ( ) [ ] T e eu ev er τ τ τ τ = represents the environmental forces (moments) exerted by wind, waves and ocean currents. It is possible to find descriptions of these terms in [12].

RBF-SMC controller design
A MPMS consisting of multiple sensors information fusion, hybrid algorithms and many mooring winches. The sensors can detect the floating platform location and the mooring cable strain; the control algorithm calculates the force exerted on each mooring cable to counteract the environmental force. A simplified block diagram of a MPMS shown in Fig.2. Considering Eqn. (1) and (3), the following equilibrium equation is given by To simplify Eqn. (4), the definition is as follows The formula obtained from Eqn. (4), (5), (6) and (7) is as follows Eqn. (8) is simplified by 3-DOF as follows The sliding surface can be defined as: Take the derivative concerning S is given by The SMC control law is given by: To reduce vibration problems, the saturation function ( ) sat s is adopted to supersede the sign function ( ) sgn s in the controller.
where ∆ is the boundary layer.
The three-layer feedforward network of the RBF with good performance and the NN is adopted in this paper to approximate the uncertain function ( ) , F η η , and the network algorithm is as follows: Take the Gaussian ( ) h σ as follows: The corresponding adaptive law is designed as follows: where Ŵ is the estimation of W * and γ is a positive constant.
The Lyapunov function defined as:

Results and analysis
In order to verify the effects of the RBF-SMC controller, the semi-submersible offshore platform "Kantan3" is the research object and simulated. Table 2 gives the key dimensions and mass characteristics of the platform under operating conditions. According to the calculation formula (Fossen, 2011), the mass matrix and damping matrix of the platform can be written as:

Setting-point control test
The The setting point position tracking curve of surge, sway and yaw motion is shown in Fig.  3. Simulation results show that, compared with traditional PID control and RBF control, RBF-SMC control can significantly reduce the maximum overshoot time and rise time, suppress oscillation and finally converge to a given point. Fig. 4 is a setting point position tracking error curve of surge, sway and yaw motion. Compared with the other two controllers, the positioning error value of RBF-SMC control keeps the minimum, and the error is basically zero after reaching stability. In the yaw direction, the positioning error always exists because of the boundary value limitation in the NN controller. Fig. 5 shows the velocity variation curves of the three control modes in setting point control, Fig. 6 shows the setting point position tracking movement route of the mooring platform in the earth coordinate system. From the curve changes, it can be seen that the response speed of RBF-SMC control is faster than the other two controls, and it has the best movement path.

Trajectory tracking control test
The sinusoidal trajectory tracking process of the semi-submersible offshore mooring platform positioning is simulated; its parameters are the same as setting point control. The simulation time is 10 seconds and the expected trajectory is changed as follows: 10sin 5cos sin In the sinusoidal trajectory tracking control, the trajectory position tracking curves of surge, sway and yaw motions are shown in Fig. 8. The rising amplitude of horizontal displacement of PID control and RBF control is larger than that of RBF-SMC control. The maximum offset of horizontal movement of mooring platform with RBF-SMC controller is suppressed, and the dynamic positioning accuracy is higher. The error curve of sinusoidal track position tracking is shown in Fig. 9. The traditional PID control always has a large position tracking error. The position tracking error of RBF control is obvious in yaw motion, and the position tracking error of RBF-SMC control tends to zero after reaching stability. It shows that RBF-SMC controller has better adaptability and robustness under irregular wave conditions.

Conclusions
In this paper, a stable adaptive control scheme for MPMS with uncertain dynamics is proposed. The control scheme is designed by a hybrid controller based on RBF NN and SMC, which learns the MPMS dynamic changes, and the compensation of external disturbances is realized through adaptive RBFNN control. The proposed mooring control scheme is applied to the "Kantan3" mooring simulation system. Firstly, the setting point position tracking control experiment is carried out under the condition of irregular waves, and then the sinusoidal trajectory tracking control experiment is carried out. Comparing the simulation results with standard PID controller and traditional RBF controller, the following conclusions are drawn. 1) Compared with PID controller and RBF controller, RBF-SMC controller can significantly reduce the maximum overshoot time and rise time, suppress oscillation and finally converge to a given point. With the movement of the setting point, RBF-SMC controller has better stability and robustness under irregular wave conditions.
2) The RBF-SMC controller is adaptive to the evolution conditions, and can approach the uncertainty of the model dynamically, thus reducing the fuzzy advantage under the disturbance of variable environment. On the horizontal axis, the positioning accuracy is greatly improved. This project is supported by the National Natural Science Foundation of China (Grant No.51779136).