A novel active disturbance rejection control (ADRC) controller is proposed based on support vector regression (SVR). The SVRADRC is designed to force an underactuated autonomous underwater vehicle (AUV) to follow a path in the horizontal plane with the ocean current disturbance. It is derived using SVR algorithm to adjust the coefficients of the nonlinear state error feedback (ELSEF) part in ADRC to deal with nonlinear variations at different operating points. The trend of change about ELSEF coefficients in the simulation proves that the designed SVR algorithm maintains the characteristics of astringency and stability. Furthermore, the path following errors under current in simulation has proved the high accuracy, strong robustness, and stability of the proposed SVRADRC. The contributions of the proposed controller are to improve the characteristics of ADRC considering the changing parameters in operating environment which make the controller more adaptive for the situation.
AUVs are unmanned submarines that carry their own power source and a computer unit, running software, and control solutions that allow the execution of a mission without human intervention [
The control problem of underactuated AUV has been one of the active research areas because of its intrinsic nonlinear feature and practical requirements [
At present, some certain theoretical study results have been achieved on the AUV tracking control, for example, the fuzzy slid mode control, the neural network adaptive PID, the backstepping control, and so on [
Considering the particular steering scheme, obvious nonlinearity, and disturbance sensitivity, a new adaptive ADRC control algorithm using the support vector regression is designed to improve the control performance. In the case of path following, the interior and exterior disturbances can be estimated by a designed ESO observer [
The idea of ADRC technique, which originated from PID control algorithm, is proved to be very effective because it does not entirely depend on mathematical model of the plant and it can compensate the internal and external disturbances dynamically [
In recent years, Support Vector Machines (SVMs) have been proposed as learningfromsamples tools for a number of problems, including classification and regression. Since Vapnik [
In the traditional ADRC method, the adaptability in different operating environments cannot fulfill the accuracy required, because the parameters of NLSEF in ADRC method will be obtained as a fixed constant which is achieved based on experience or experimental data. Even in different operating conditions, the vehicle will also move under these set parameters, where some unexpected tracking error happens. If the parameters in NLSEF are changed by some method according to different conditions which mean different parameters and nonlinear fitting functions, the effect of controller will be improved obviously. In practical implementation, this method can be realized easily.
This paper makes an effort to apply SVRADRC to the path following for an underactuated AUV with the disturbances of ocean current and model uncertainty. Firstly, the desired course angle for the steering control is derived by using lineofsight (LOS) guidance law. Secondly, the ultimate control command is computed with ADRC, where the coefficients of NLSEF in SVRADRC are adjusted along with output of SVR to deal with nonlinear variations at different operating points. Finally, the computer simulation proves that the controller has satisfying path following characteristics, including high accuracy and strong robustness.
It is well known that establishing an accurate dynamic model of AUV is of prime importance for their maneuvering prediction and control application. The notation of math model according to SNAME is mentioned in Table
The notation of math model according to SNAME.
DOF  Forces and moments  Linear and angular velocities  Positions and Euler angles 

Motions in the 



Motions in the 



Motions in the 



Rotation about the 



Rotation about the 



Rotation about the 



From the control viewpoint, the 6degreeoffreedom (DOF) nonlinear dynamics of AUV, together with hydrodynamic coefficients’ uncertainties, makes underwater vehicles a challenging system to be accurately modeled and controlled [
The modeling method chosen in this work is a geometricalbased analysis. It consists mainly of finding the parameters of the model, which have been well defined through physical laws describing the motion of a rigid body in a liquid environment.
The dynamic model of AUV can be simplified as
The kinematic equation is
In this mathematical model,
The SVR maps the input space to the highdimensional feature space, and, in the feature space, the optimal linear regression function is built; then it can infer the output of any input. So, it has described the nonlinear relationship between the input and output space. With the training sample set, the regression function, which is used to describe the nonlinear relationship between the input and output space, is built by the machine learning; then the regression estimation of the output is finished.
Given a set of data points,
Based on the theory of SVM, the optimized goal can be achieved, and the standard form of SVR is
Then, an optimization problem has been formed:
In order to solve the dual problem of (
Then, the solution of original problem can be described as follows:
Considering the stability of the system, we design
Combined with (
ADRC is a relatively new control design concept and method. It is well known that the primary reason for using feedback control is to deal with the variation and uncertainties of the plant dynamics and unknown disturbance from the outside [
The controller contains a tracking differentiator (TD), an extended state observer (ESO), and a nonlinear state error feedback (NLSEF) [
The TD is functioned as below. The input signal passes through it and there are two or more outputs; one output signal is tracking the input signal, and other signals are the
The input signal is assumed as
With the parameter
Usually (
The ESO was first proposed by Han [
In the ESO,
The disturbance is assumed as
The architecture of ADRC.
If
Considering
Complicated structure, multiparameter, and lacking adaptivity are the disadvantages of ADRC. It is known that the parameters of TD and ESO own a big range of adaptability, but the coefficients of ELSEF need to be adjusted according to different system work plot.
The quadratic performance index is defined as
The relationship between the input and output can be expressed as
Define
Then, (
With the help of practical linearization, a new formula can be obtained as follows:
Considering the form of
Thus, (
Considering the relationship between the max tracking velocity and limit turning rate, a coordinated control scheme can be designed for the steering control system.
The function of steering control is to adjust the rudder angle of AUV based on the system input [
According to Figure
The algorithm of SVRADRC for steering control.
Finally, the control input can be calculated as
Based on the steering controller designed above, the path following control architecture can be obtained as in Figure
The architecture of path following controller.
The inner loop controller is the steering controller which is introduced above, and the outer loop can provide the actual heading command
Like the steering controller, the path error controller also has four parts, which are, respectively, TD (
Finally, combined with the azimuth angle
Two simulations are designed to demonstrate the path following control of the AUV using the SVRADRC and ADRC, respectively. In the simulation, the depth of AUV is assumed to have no change and keep moving in the horizontal plane with the ocean current disturbance, while the vehicle implements the searching and patrolling mission. Furthermore, there is a current interference around the AUV during mission. The current speed is 0.2 m/s, and its angle is 0 degree (NED).
It is assumed that the speed of AUV is 2 knot, initial position is at (0, 0) m, and initial heading is 50 degree. The planned path, which consists of three semicircles and four straight lines, starts at the point of (10, 100) m and the end is at (460, 100) m.
Figure
The result of AUV path following.
Figure
The heading angle of AUV.
Figure
The position error.
The response of rudder angle.
Compared with sliding mode control method from Figure
In Figure
The regulated parameters of
In Figure
The regulated parameters of
In this paper, a nonlinear path following in the horizontal plane for an underactuated AUV in the presence of ocean current and uncertain parameters of ADRC was proposed by using the simulation result. The designed SVRADRC path following controller is robust to nonlinear motion, the model parameter perturbations, and the external disturbances. The continuous changing parameters of NLSEF by SVR in the simulation indicated its stabilization and a better control effect. Finally, the simulation experiments had also verified good following result. And it was demonstrated that the proposed algorithm had high accuracy, strong robustness, and stability. In the future work, the SVRADRC will be applied to path following in 3D. In addition, more attention should be paid to the development of SVRADRC.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (NSFC51179038), Program for New Century Excellent Talents in University in China (NCET100053), and the Fundamental Research Funds for the Central Universities (HEUCFX041401).