RBF network based integral backstepping sliding mode control for USV

A kind of USV course RBF network control algorithm is putted forward, which is on the basis of integral backstepping sliding mode. First of all, an integrator sliding surface were designed with the sliding mode variable structure control technology. Secondly, radial basis function neural network was applied to approximate the system nonlinear function and uncertain parameters. Furthermore, a nonlinear damping law was introduced to overcome the bounded outside interference. Finally, on the basis of the above, the system control law was deduced by using the backstepping method. The simulation results show that the neural network can accurately approximate the nonlinear function and uncertain parameters, and the controller output is smooth and the output is not sensitive to perturbation of parameters. Therefore, the proposed algorithm is effective for USV course control.


I I I Introduction ntroduction ntroduction ntroduction
Maritime shipping and production activities are becoming more and more busy, the Unmanned Surface Vehicle (USV) due to the advantages of high nautical rate, high efficiency, low cost, severe sea condition that can instead people to the various activities has become an important development trend of the future of the ship. Therefore, USV is playing increasing roles in commercial, scientific and military applications [1]. Due to the disturbances of wind, wave, current and other marine environment, the USV will inevitably deviate from the original given course when it is sailing. Precise course control is a prerequisite to solve the trajectory tracking, autonomous navigation and collision avoidance and other issues. The design of ship course controller is the issues of common concern for the control theory and control engineering [2].
With the development of computer technology and modern control theory, a variety of new control algorithms, such as the Lyapunov's direct method [3], backstepping control [4][5][6], sliding mode control [7][8][9], fuzzy control [10][11][12] and neural network control [13][14][15][16] have been applied to the research of ship course control. It is well known that the most advantage of sliding mode control is able to overcome uncertainty of the system, which has highly robustness for uncertain dynamic system with outside disturbance, especially effectiveness for control of modelless system. It is an important method for suppressing high frequency switching that causes chattering of control input. And the variable structure control system algorithm is simple, fast response, and robustness to external noise and parameter perturbation. Therefore, it is useful for ship steering control. The integral link joined into the algorithm of sliding mode control can also effectively eliminate the bounded interference outside so that the original controller has a stronger anti-jamming capability.
In this paper, by taking the "Lanxin" USV of Dalian maritime university as the research object, aiming at its motion control system, the nonlinear mathematical model for the USV planar motion is established by the method of responsive integrated backstepping modeling. Based on the Nomoto mathematical model and the combination of integral backstepping control, sliding mode control and neural network control, a kind of USV course RBF network control algorithm of USV course control is proposed. With the disturbances of wind, wave and current, simulation results show that the designed course controller can be properly adopted to the "Lanxin" USV course keeping with good effectiveness.

P P P Problem roblem roblem roblem description description description description
In actual voyages, ship movements usually exhibit a nonlinear state. Therefore, in the mathematical model of USV plane motion, the nonlinear model is considered. Based on the linear mathematical model of plane three degrees of freedom, the nonlinear model is described by adding nonlinear term as following.

( )
Where, K and T represent ship turning and follow index, Then formula (1) can be converted into the following format: Where, x and 2 x are system variable, y is the actual output of the system, 2 ( ) f x is unknown function of system, g is known gain of control input, ( ) d t represents ship external interference, and u is the control input of system.
Based on above, the goal of this article is to seek control law u which is able to make the system output y asymptotically tracking expected course r ϕ , the tracking error will be infinitely close to zero, that is 0 r e y ϕ = − → , and the stabilization time will be reduced as much as possible.

D D D Dteering teering teering teering control control control control design design design design based based based based on on on on b b b backstepping ackstepping ackstepping ackstepping
The design steps of the ship course controller contain the following steps.
Step Step Step Step 1. 1. 1. 1. According to the characteristics of system (2), the following sliding surfaces are defined.
Where, ϕ r is the expected course, σ 1 is virtual stabilization, 1 z is the heading error. ξ is the integral term which can be able to eliminate the static error caused by the uncertain interference term in the control process.
Step Step Step Step 2. 2. 2. 2. Lyapunov function is constructed to prove the asymptotic stability control system by using backstepping method. The first Lyapunov functionis constructed.
( ) The second Lyapunov function is constructed.
And the third Lyapunov function is constructed.

14)
Step Step Step Step 3. 3. 3. 3. The RBF network is used to adapt the system function ( ) f x , and its algorithm is (15).
Where x x x x is the network input, i represents one of the first input network input layer, j is a hidden layer of the network the first network input, Where, The global Lyapunov function is constructed. 3 It can be inferred formula (18) by formula (17).
In order to ensure system asymptotically stable, the following adaptive law is designed.
It is easy obtained formula (23) by theorem Where, The fitness law of the network weights is still adopted formula (22), therefore, the formula (24) can be acquired. In order to ensure that the system (2) is asymptotically stable, it is only need to prove the establishment which is . Proof slightly, see the literature [4].

S S S Simulation imulation imulation imulation
USV course tracking control simulation is excuted by applying the above control algorithm.

Simulation Simulation Simulation Simulation Object Object Object Object and and and and Conditions Conditions Conditions Conditions Setting Setting Setting Setting
The establishment of USV planar motion mathematical model requires 8 known USV parameters, as shown in the table 1. Table Table Table Table 1  In this paper, the planer motion mathematical model for the "Lanxin" USV adopts Nomoto model from the nonlinear mathematical model of formula (1).
There is the formula of matrix parameters of the mathematical model of the ship to calculate the above parameters, When the speed is 8.5kn, the gain constant and time constant are obtained through MATLAB programming, an They are the Nomoto model parameters of the "Lanxin" USV [17] . In the actual simulation, the influence of wind and wave for USV motion is obtained with white noise and a second-order wave ' s transfer function [18,19]. The formula can be written as Where 0 ω is the dominant frequency of sea waves, ξ is the damping coefficient, K ω is the gain constant. m σ is the constant describing wave intensity.
Based on USV motion model, the controller adds the model of the rudder servo system as formula (27). 1 1 Where r T is time constant, for USV is generally about 0.2s. r δ is Command rudder angle. There is the actually limit of rudder angle that is the formula (28). 35 δ°≤ (28).

Course changes experiment
(1) experiment of 20°change of course Protocol initial heading as 000°, expected course 020°, the simulation results without external interference and model parameter perturb are shown in Fig.1. It can be seen from Figure 1, the control effect is good,and the output of course will reach the expected value faster, costing 30s without overshoot nearly; and rudder control horn is reasonable which can meet the characteristics requirements of steering system.  Fig. Fig. Fig.1. 1. 1. 1 Fig. Fig. Fig.2. 2. 2. 2

Interference experiment
(1) experiments of white noise interference Protocol initial heading as 000°, desired track as 030°, apply amplitude as 0.1 white noiseinterference to course angle, the experiment results are shown in Fig.3.   Fig. Fig. Fig.3.

the the the the output output output output of of of of course course course course and and and and rudder rudder rudder rudder angle angle angle angle under under under under white white white white noise noise noise noise interference interference interference interference
It can be seen from Figure 3, when the ship was disturbed by the outside world, the designed course controller will overcome outside interference to a certain degree, and will track the expected course with no shock of rudder controlling.
(2) experiments of ship parameter perturbation and white noise interference Setting the protocol initial heading as 000°, desired track as 030°, propose maneuvering ship indices K、T perturb at 40% ， apply amplitude as 0.1 and make white noise interference to the course, the experiment results are shown in    Fig. Fig. Fig.4 It can be seen from Figure 4, when the ship model parameters occur perturbat with interference from the outside world, the course output from the new controller has smaller fluctuation around expect course, and has smaller shock amplitude of rudder controlling, thereby, it is illustrated that the anti-interference ability of the new controller is stronger.

Conclusion Conclusion Conclusion Conclusion
An integral sliding mode controller with neural network was investigated for USV navigation, which realizes the accurate and stable tracking of the ship in the steering process, on the basis of backstepping method. With disturbances of wind, wave and current, the RBF network based integral backstepping sliding mode control system is simulated and the results of simulation show that the proposed algorithm is effective for USV course control.