NONLINEAR ADAPTIVE HEADING CONTROL FOR AN UNDERACTUATED SURFACE VESSEL WITH CONSTRAINED INPUT AND SIDESLIP ANGLE COMPENSATION

In this paper, a nonlinear adaptive heading controller is developed for an underactuated surface vessel with constrained input and sideslip angle compensation. The controller design is accomplished in a framework of backstepping technique. First, to amend the irrationality of the traditional definition of the desired heading, the desired heading is compensated by the sideslip angle. Considering the actuator physical constrain, a hyperbolic tangent function and a Nussbaum function are introduced to handle the nonlinear part of control input. The error and the disturbance are estimated and compensated by an adaptive control law. In addition, to avoid the complicated calculation of time derivatives of the virtual control, the command filter is introduced to integrate with the control law. It is analysed by the Lyapunov theory that the closed loop system is guaranteed to be uniformly ultimately bounded stability. Finally, the simulation studies illustrate the effectiveness of the proposed control method.


Introduction
Marine mechatronic systems play an important role in modern industrial manufacturing.As a basic problem in the field of the vessel control, the heading control has always been a research focus.Many advanced control methods have been applied to the vessel control, but there are still many disadvantages.An adaptive output feedback control method was proposed to address uncertainties and external disturbances of a vessel without hydrodynamic structure information, but it ran with a slow calculation speed and expensive online equipment [1,2].Zhang and Zhang used a feedback linearization algorithm to avoid complicated proofs, but the input saturation was not considered and a high precision model was required [3].A fuzzy adaptive method was developed to investigate disturbances and uncertainty attenuation, but the control rules were usually difficult to determine and optimize [4,5].Because of its reliable and stable, PID control method has been widely used.With the progress of technology, traditional PID controller cannot satisfy the improvement of control quality requirements.So intelligent control algorithms were introduced and combined with PID in the field of the vessel control [6].Backstepping control introduces additional nonlinearity into the control process to eliminate undesirable nonlinearity in the system.It is one of the most popular control design techniques.The backstepping technique was more commonly used in fullactuated vessels rather than underactuated vessels, however, traditional surface vessels are usually underactuated [7][8][9][10].
Whether vessels are full-actuated or underactuated, input saturation is a common phenomenon in practical systems.The Input of actuator will enter a saturated state, when it reaches a certain limit.The saturation of actuator will reduce the system performance and even lead to the instability of the closed loop system.Therefore, it is necessary to consider saturation in the design of marine vehicle control systems.The adaptive neural network (NN) control method was presented by using a Gaussian error function [11].It tackled the discontinuous and differentiable problem of non-smooth asymmetric saturation effectively.The interval type-2 fuzzy system and an observer based on strictly positive real (SPR) theory were employed to handle the effect of unknown asymmetric saturations nonlinearity [12].The auxiliary system was adopted to handle the influence of input saturation and states of the system were used to design the controller [13].In order to tackle the potential unstable behaviour caused by the saturation of rudders, an adaptive fuzzy compensator was introduced in an autonomous underwater vehicle (AUV) bottom following control system [14].To satisfy the input saturation of the control system, Huang and Sharma built a new state-dependent coefficient (SDC) matrix to express the relationship between different variables [7].This method could maintain the heading accurately and was used for the vessel dynamic positioning control system [15,16].Note that the influence of nonzero sideslip angle was not considered in the above references.
The sideslip angle plays an important role for marine surface vessel control.The desired heading is usually chosen by the tangent direction of the path, with the assumption that the nonzero sideslip angle is ignored, the actual sailing direction of the vessel is not consistent with the tangent direction.Therefore, the traditional definition of the desired heading may lead to the deviation of the heading.However, the sideslip angle is time-varying which can be influenced by environmental disturbances, such as the currents, waves and winds.To handle the sideslip angle problem, an extended state observer was designed to estimate the sideslip angle timely and exactly [17,18].It was contributed to track the desired path accurately regardless of constant ocean disturbances.An adaptation law was adopted to estimate and compensate the unknown sideslip angle considered as a constant parameter [19,20].Nie and Lin proposed the fuzzy adaptive integral line-of-sight (FAILOS) guidance law which was established with the adaptive fuzzy logic system.This method calculated the desired heading and compensated the sideslip angle effectively [21].In [22], the ship course-keeping control was achieved with the sideslip angle compensation, but the global positioning system (GPS) and rate gyroscope sensors were needed.For nonlinear characteristics, input saturation and external disturbances of an underactuated vessel motion control system, a robust adaptive backstepping heading controller with constrained input and sideslip angle compensation is designed.The main contributions in this paper are summarized as follows.
• For input saturation, the hyperbolic tangent function is used to approximate the constraints.An adaptive control law is designed to handle the approximation error along with the external disturbance in the yaw direction; • By introducing the command filter, the tedious analytical calculation of the time derivative of the virtual control law is avoided, which simplifies the backstepping procedure.
• Combining the above points, the heading controller is developed based on the backstepping method.The heading error can be uniformly ultimately bounded (UUB) and all states of the closed-loop system are bounded.

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This paper is organized as follows.The section 2 describes the vessel dynamics and related disturbances.The section 3 gives the control objectives.The section 4 amends the desired heading and designs the nonlinear robust adaptive heading controller.The simulation results are explained in the section 5 and the conclusion is expressed in the section 6.

Dynamics of the underactuated surface vessel
Assuming the vessel has a xz plane of symmetry; the heave, roll and pitch motions can be neglected.To simplify the vessel model and facilitate the design of controller, the surge velocity u is assumed to be a constant and the sway velocity v is assumed to passive bounded [23,24].Based on these assumptions, the dynamic model of the underactuated surface vessel (USV) can be simplified to a 2 degree of freedom (DOF) model (i.e. in the direction of sway and yaw) and it is described as follows [25]: where  denotes the heading angle of the vessel in the earth fixed frame where ri d , vi d are the hydrodynamic damping coefficients in the yaw and sway axes for 1, 2,3 i = , respectively. is a small positive constant.The rudder moment  is the only control input of the USV, which is calculated actually.Considering the input constraint [26], the new input () r  is the output of the saturator and described by sign( ) , () , ) denote unknown but bounded time varying disturbances in the yaw and sway direction.vessel with constrained input and sideslip angle compensation  stand for the actual velocity and relative velocity of the vessel with respect to the body fixed frame, respectively.The relationship between them is . Thus, considering the effect of the current on the vessel without using GPS equipment, the sideslip angle can be obtained by sin( ) arctan arctan cos( ) Fig. 1 The traditional desired heading 2.2 Wave model Among many possible external disturbances acting on vessels, the waves are extremely important, which have an impact on the heading control performance.Therefore, the wave is considered as the disturbance r  in the yaw direction.The wave model is given as  stands for a zero mean Gauss white noise process with a power spectral density equal to 0.1.And () hs is a transfer function of the second order wave, described by 22 00 () 2

Ks hs ss
 and m  denote the wave frequency and the wave intensity, respectively. represents a damping coefficient, K  is a constant that expresses gain.

Control Objective
Backstepping method requires that functions can be differentiated in the process of derivation of virtual control variables, while saturation function () r  is not a smooth curve.
To use the backstepping method to develop the heading controller, we divide () r  into two parts, which can be given as , and a smooth function () g  with hyperbolic tangent function is defined to approximate the saturation nonlinearity.
( ) sign( ) tanh( ) sign( ) We define b as the bound of the function ()  and it can be expressed as From Equations (4, 8, 9), the plant ( 1) is redefined as  are bounded. is bounded and we assume   .c is a positive constant.In the new plant (10), it is hard to design the actual input  directly.Therefore the auxiliary signal  is introduced to develop the controller and analysis the system stability.
The objective is to design a heading control law for (10) with input saturation (8) such that: • The heading error converges to the neighborhood of a small value; • The control input moment keeps a small value.

Preliminaries
This section gives several definitions and lemmas used in the controller development and the system stability analysis later.Definition 1: The following Young's inequality holds for any x  , y  , where 0   and the positive constants p , q satisfy 11 1 pq += with 1 p  , 1 q  .And the Young's inequality used in this paper is presented as , then the following inequality holds [28]:  is a Nussbaum type function.

Control design
To deal with the influence of nonzero sideslip angle and the input saturation on the heading control, the following two aspects are shown in this section: • The desired heading is compensated by the sideslip angle; • The input saturation function is handled by hyperbolic tangent function and Nussbaum function.The approximation error and disturbances are estimated and compensated by the adaptive law.The command filter is introduced to avoid the complicated calculation.The heading controller is designed based on the backstepping method and the stability of the whole control system is proved.

Desired heading amendment
In the heading control design, the tangent angle of the desired path is usually chosen as the desired heading angle, which is the traditional definition of heading angle.The sideslip angle is affected by not only steering, but also the ocean current on the vessel.However, the nonzero sideslip angle is seldom considered in the practical design.Especially for the underactuated vessel in this paper, one input is used to control both the yaw rate and sway velocity so that the direction of the vessel's gravity velocity is not coincide with the heading of the vessel, which gradually causes the vessel not following the right course.
To handle this problem, we can use the nonzero sideslip angle to amend the irrationality of the desired heading.Because the desired direction of gravity center velocity differs from the tangent angle of the desired path by a sideslip angle [29].The amended desired heading and the new heading error for underactuated vessel are proposed, respectively.
The amended desired heading da  shown in Fig. 2 is presented as With the nonzero sideslip angle, the new heading error a e is given as Fig. 2 The amended desired heading

Controller design
Before the controller is designed, since the backstepping method does not have a good solution to the problems caused by the expansion of terms in the derivative process of virtual control, a command filter is introduced to avoid the complex calculation [30].The filter shown in Fig. 3 can be expressed by 12 2 denotes the virtual control law, and we can get ) , chosen, the filter states can converge fast, which attain the input value and its derivative.
Based on the above new definitions, the backstepping method is summarized as the following steps: Step 1: start with the expression of 1   , from Equation (24), we have Define an auxiliary design system to solve the effect of the estimation error 1   :  f is given as Design this virtual control law as where Therefore, we can get Step 2: to handle 2   like an auxiliary design system in step1, define  For  in (10) with assumption   , we define where  ,  are the estimation and the estimation error of  , respectively.Considering the errors  and 2   , another CLF can be written as It is difficult to obtain the derivative of disturbance directly.Compare with the adaptive law dynamics, the frequency of disturbance changes slowly.Therefore, we have 0  = .
It leaves the derivative of 2 V from Equations (25, 32, 35, 36, 38)   tanh( ) At this time, the virtual control law 20  can be given as To settle the estimation error of  , the adaptive law is introduced.Moreover, the updating law of  is given as 0.5 0.5 Step 3: denote a bounded signal  as MM sign( ) 1, 0 Normally, calculating 1  − is unavoidable by using the usual method following step.To simplify design and analysis, the Nussbaum function is considered as [31] 3 where   is a positive constant.
Define the following Lyapunov candidate function: Differentiating Equation (51) and inserting (49) yields We design the auxiliary signal  as Substituting Equation (53) into (52) leads to Xiaoyang Lu, Zhiquan Liu Nonlinear adaptive heading control for an underactuated surface and Zhenzhong Chu.
vessel with constrained input and sideslip angle compensation where , , 2 Integrating Equation (47) directly, the following inequality has the following performance: With the Lemma 2, 3 () Vt and  are bounded.
Using the definition of UUB stability [32], which expresses final state of the control system is properly governed to converge to a small ball centered at the origin.It can be thus concluded from (55) that 3 V is UUB together with the states ( 1, 2,3) i zi = , which means for any 0   (  is a small constant), there exists a positive value of finite time T such that 83

Simulations
In this section, to verify the effectiveness of the proposed controller, several studies are carried out.We choose a 2-DOF nonlinear mathematical model of the underactuated vessel with yaw, surge and sway as the simulation object.The vessel parameters are taken from [33], as shown in Tab.1.The yaw motion disturbance model is given by ( 6) and (7), and the parameters are presented in Tab.2.The external disturbances affecting the sway velocity are chosen as The designed parameters of the heading and the command filter are specified in Tabs.3 and 4, respectively.The vessel is sailing with a speed of 10 m/ s and following the desired heading angle given as 30 , 60 and 120 , respectively.This paper compares the simulation results of the proposed controller with constrained input and sideslip angle compensation (A-BAWIC), the traditional backstepping controller (T-B) and the backstepping controller with sideslip angle compensation (A-B).And the latter two controllers do not deal with input saturation constraints.The corresponding initial conditions and related parameters are maintained with the same values in the simulation of three models.All simulation results are shown below which are divided into two parts: the heading angle and the rudder moment.T-desired represents the desired heading.

The heading angle and the corresponding error
The steady state error of the heading angle is affected by the sideslip angle, which is related to many reasons, such as the vessel sway speed, current velocity and so on.
Considering the above factors and the balance between the control input and output, the current velocity and azimuth with respect to the earth fixed frame are given as 1m/ s c V = and 10 c  = .
The simulation results of the output heading angle and corresponding errors of different controllers are shown in Figs. 4 and 5.When the course is changing, the output errors are all very as shown in Fig. 5.The overshoot is large, because of the sudden course changing.Due to the irrationality of desired heading, it can be seen that the heading error cannot converge to zero.A-BAWIC can ensure that the system tends to be stable within 3-5 seconds and the overshoot is reduced by more than 90%.In the stable state, the heading error of A-BAWIC is 12 more accurate than that of T-B, as shown in Fig. 4. In other words, by comparing with the other controllers, A-BAWIC can improve the performance of the vessel heading control and reduce the heading error effectively; it also can restrain the overshoot of the system in a shorter time., other parameters can also affect the control performance.In the process of simulation debug, the chosen of larger ( 1,2) ei ki = and   choose can lead to the smaller the heading error, but the control input is larger simultaneously.Excessive control input (rudder moment) will increase the wear and tear of the rudder servo system.Therefore, the control parameters are given carefully (i.e. == ) to cope with the trade-off between the heading performance and the saturation constraint.Fig. 6 presents the simulation results about the control input.It can be founded that rudder moments without constraints are large when the heading is changing.Especially, the input of A-B exceeds the operation condition of the rudder.To solve this problem, A-BAWIC is designed to limit the control input before the control commands are putted on the actuators.
In Fig. 7, The average of the rudder moment of A-BAWIC in the range of ( )N , which is 7 1.33 10 −  of the saturation limit.Even the maximum rudder moment of A-BAWIC does not exceed 8  Nm .And it is much smaller than these of T-B and A-B whether the desired heading is changing or not.In Tab.5, the maximum rudder moments are the absolute value, others are the average absolute values.

Conclusion
Based on the backstepping technique, a constructive method has been proposed to solve the heading control problem for an underactuated surface vessel with input saturation and sideslip angle in this paper.The undifferentiable input saturation was approximated by hyperbolic tangent function and Nussbaum function, which satisfied the requirement of the backstepping method for continuous functions.The command filter simplified the derivation calculation of virtual control law.Finally, the simulation studies verified the proposed controller can effectively improve the heading control performance of the underactuated vessel with a small rudder moment.However, it should be pointed out that the surface vessel model is simplified.The nonlinear dynamics and uncertain parameters of the vessel can be fully considered in the future research.Furthermore, the overshoot is large when the heading is changed, so the authors plan to use the pre-filter method to improve the tracking accuracy in the future work.

33 m
vessel surge velocity, sway velocity and yaw rate coordinated in the body fixed frame stand for the vessel inertia including added masses in the surge, sway and yaw; the damping functions


are the current speed and angle in the earth fixed frame oo OX Y , respectively.Then Xiaoyang Lu, Zhiquan Liu Nonlinear adaptive heading control for an underactuated surface and Zhenzhong Chu.
27]: For x  , the following inequality holds Xiaoyang Lu, Zhiquan Liu Nonlinear adaptive heading control for an underactuated surface and Zhenzhong Chu.vessel with constrained input and sideslip angle compensation


are limited by virtual control laws.

Fig. 3 2 
Fig. 3 Configuration of the command filter Introduce the following new variables to transform states:

1 e
is a small positive constant.

Tab. 1
Parameters of the ship model parameters kg

Tab. 3
Design parameters of the heading controller 1

Fig. 4 5
Fig.4 The heading angles of the controllers Fig.5The heading errors of the

Fig. 6 7
Fig. 6 The rudder moment of the controllers Fig. 7 The rudder moment of the controller A-BAWIC Tab.5 Control effect comparison between different control methods Type Rising time /s settling time /s as All signals are bounded in the closed loop system.And the adaptive law (43) can estimate and compensate the bounded external disturbances robustly.Under the proposed control techniques, the desired heading can be followed accurately with a smaller control input.