A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning

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A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning Guoqing Zhang 1, Xianku Zhang n Navigation College, Dalian Maritime University, 1 Linghai Road, Dalian 116026, People's Republic of China

art ic l e i nf o

a b s t r a c t

Article history: Received 30 October 2014 Received in revised form 27 November 2014 Accepted 12 December 2014 This paper was recommended for publication by Dr. Jeff Pieper.

Around the waypoint-based path-following control for marine ships, a novel dynamic virtual ship (DVS) guidance principle is developed to implement the assumption “the reference path is generated using a virtual ship”, which is critical for applying the current theoretical studies in practice. Taking two steerable variables as control inputs, the robust adaptive scheme is proposed by virtue of the robust neural damping and dynamic surface control (DSC) techniques. The derived controller is with the advantages of concise structure and being easy-to-implement for its burdensome superiority. Furthermore, the low frequency learning method improves the applicability of the algorithm. Finally, the comparison experiments with the line-of-sight (LOS) based fuzzy scheme are presented to demonstrate the effectiveness of our results. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Control system design Underactuated ships Path-following control Guidance principle Low frequency gain-learning

1. Introduction The use of marine surface vessels for various mission is increasing globally because of its superiorities in terms of capacity and economy [1,2]. Arising from nonholonomic system theory [3,4], path-following and trajectory-tracking of underactuated ships have been actively studied for the last few years [5,6]. Actually, most marine ships are underactuated, because of that they are equipped with propellers and rudders for surge and yaw motions only, without any actuators for direct regulation of sway motion. Although state-of-the-art actuation systems, such as tunnel thrusters and azipods, are equipped, they would be ineffective in providing control actuation in the sway degree of freedom (DOF) at high speed [1,7,8]. Therefore, the path-following mission of underactuated ships requires much attention and is meaningful in practical engineering [9]. Traditionally, the path-following control for marine ships is functionally divided into three subsystems: navigation, guidance, and control [10], see Fig. 1. Navigation is with the function of directing a vehicle, e.g., measuring its attitude variables. These are commonly implemented by the nautical instrument onboard. Guidance is the principle or the system that provides the desired position, velocity, and acceleration of the ship to be used in the n

Corresponding author. Tel.: þ 86 41184729572. E-mail addresses: [email protected] (G. Zhang), [email protected] (X. Zhang). 1 Tel.: þ86 18940816403.

control derivation. Finally, control is to derive the necessary orders provided by actuators onboard to perform the predefined objective. This note focuses on the latter two terms to complete the control system design. For guidance principle in case of straight-line paths, the line-of-sight (LOS) guidance law is often incorporated to set the desired yaw angle ψd to the vessel's relative bearing to the upcoming waypoint. Then the control objective ψ-ψ d is guaranteed by the course control design [1,2]. In [11,12], the conventional LOS guidance is detailed and incorporated with the stability theory of cascade interconnected systems to achieve the path-following objective. Furthermore, [13] presents a variable radius LOS algorithm to achieve a fast convergence rate of the ship to its desired trajectory. In these LOS-related works, the concept “circle of acceptance” is incorporated to monitor the desired route segment (i.e., the consecutive pair of waypoints) and a self-governed controller regulates the vehicle's speed. And there exists no control action in an auto-turning process. In the past decades, a variety of theoretical studies around the path-following mission have been developed for underactuated ships. In [14], a continuous time invariant control law is derived. However, the orientation of the underactuated ship was not stabilized to avoid undesired whirling around in the vicinity of the reference path. In addition, [15] developed a high gain-based local exponential stabilization algorithm to attend the problem. By using Lyapunov's direct method, [16] developed two time-varying control laws, which were the typical research results of previous studies. As there exists the assumption of persistent excitation

http://dx.doi.org/10.1016/j.isatra.2014.12.002 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

G. Zhang, X. Zhang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

Operator input

Waves, wind and ocean currents

Way-point generator Reference path generator

Kinematics relations

Control law

Adaptive law

Guidance

Ship model

Estimated states

GNSS, Gyro-compass

Observer

Navigation

Control

Fig. 1. Conceptual signal flow box diagram for navigation, guidance, and control.

(i.e. the desired yaw velocity is nonzero), the proposed algorithm is only applied to the case of nonstraight reference trajectory. Furthermore, in [17], a nonlinear adaptive control strategy was developed with the classic parameter projection technique to settle the path-following problem of uncertain underactuated ships. The conclusion is extended to a more practical case, where the ship dynamics included linear/nonlinear off-diagonal damping terms and the unmodeled uncertainties [18,19]. In parallel, the discontinuous feedback control is another characteristic strategy to implement the path-following task, for example, [20,21]. To tackle the more practical uncertainty, a class of approximation-based adaptive scheme is well developed via neural networks (NNs) or fuzzy logic system (FLS) in the literatures [22–24]. Although few studies have been reported about its application in underactuated marine vehicles, relevant results for fully actuated ocean ships include [25] and [26]. In addition, [27,28] present a self-organizing adaptive fuzzy neural control for the general uncertain system, which obtains excellent efficacy in respect of compensating the structure and parameter uncertainties. In our previous study [7], a concise robust adaptive NNbased scheme was proposed with the DSC and minimal learning parameter (MLP) techniques. The actual control law is with a concise form and is easy to implement in practice because of a smaller computational burden. In the above studies, the surge force and the yaw moment are selected as the control actions of interest. However, the output force (or moment) provided by the actuating device is impossible to be measured for the microcontroller. That is, the force (or moment) order is not executable in practical engineering. In [12,13,29], the rudder angle is directly considered as the control input. That is a real measurable variable for the actuating hardware device and it more fits the engineering requirement to drive the rudder angle. Based on the above observations, the LOS steering law is not applicable to the practical condition “waypoints-based pathfollowing control for underactuated ships”. In addition, one drawback of the LOS principle is that there exists a jump in the desired yaw rate during transition from the straight line to the pivoting waypoint [1]. In this note, our attention is devoted to developing a control system that lends itself for easy tuning and implementation, which is one of the key considerations of industrial pathfollowing task. For the guidance term, a novel DVS-based guidance principle is originally developed for the practical path-following control. For the control term, one explores the merit of NNs by resorting to the DSC and robust neural damping technique, and two gain-related adaptive parameters are derived to compensate model uncertainties and the unknown control gain function. The path-following capability of the proposed design and the robustness against model uncertainties are demonstrated through theoretical analysis and numerical experiments. The outstanding merits of the proposed scheme can be summarized as follows:

(1) A DVS guidance principle is developed for underactuated ships. The marine practice for path-following control, that is, the reference path generated by waypoints, is first dealt with in this design. That is critical and important for applying the current theoretical ideas in the control engineering. (2) The robust adaptive neural controller is derived by virtue of robust neural damping and low frequency learning technique, focusing on the rudder angle δ and the main engine speed n. Only two gain-related learning parameters are updated online in the algorithm, although NNs are incorporated to attack model uncertainties and suppress the external interference. That leads to an easy-to-implement controller with smaller computational burden.

2. Problem formulation and preliminaries Throughout this note, jj indicates the absolute operator of a scalar. J  J is the Euclidean norm of a vector.
J  J F denotes the Frobenius For a given matrix A ¼ ai;j A Rmn , J A J 2F ¼ n o norm. P Pn 2 ^ tr AT A ¼ m i¼1 j ¼ 1 ai;j . ðÞ is the estimate of ðÞ and the estimation error ð~Þ ¼ ð^Þ  ðÞ. sgn is the sign function. 2.1. Dynamic model of a marine ship According to physical reasoning in the ship model [30,31], the nonlinear kinetic equations of three DOF (degree of freedom) marine ships can be expressed as Eq. (1). Eq. (2) presents briefly the generation mechanism of the propeller and rudder forces (or moments) T E ; F N , where the controllable inputs include the main engine speed n and the rudder angle δ: 8 Surge : ðm  X u_ Þu_  ðm  Y v_ Þvr ¼ X juju juju þ X vr vr > > > > > þ X vv v2 þ X rr r 2 þ ð1 t P ÞT E þ ð1  t R ÞF N sin δ > > > > < Sway : ðm  Y v_ Þv_ þ ðm  X u_ Þur ¼ Y v v þ Y r r þ Y jvjv jvjv ð1Þ þ Y jrjr jr jr þ Y vvr v2 r þ Y vrr vr 2 þ ð1 þ aH ÞF N cos δ > > > > > Yaw : ðI z  N r_ Þr_ þ ðX u_  Y v_ Þuv ¼ N v v þ N r r þ N jvjv jvjv > > > > : þ N jrjr jr jr þN vvr v2 r þ N vrr vr 2 þ ðxR þ aH xH ÞF N cos δ   T E ¼ ρjnjnD4P kT J P FN ¼ 

 6:13Λ AR  2  u þ v2R sin αR Λ þ 2:25 L2 R

ð2Þ

where v ¼ ½u; v; r T A R3 denote the surge, sway, and yaw velocities, respectively. m and Iz are ship mass and z axial moment of inertia, respectively. X; Y; and N are forces or moments with respect to surge, sway, and yaw motions, which are expressed by the hydrodynamic derivatives (i.e., X juju ; X vr ; Y v ; N v , etc). These parameters could be estimated using the empirical formulas or system identification based on the trial data. Parameters X u_ ; Y v_ ; and Nr_ are used to describe the Coriolis-centripetal forces, which are proportional to the acceleration of the marine ship. Thus, they

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

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also are called as “added-mass”. In addition, aH ; t P ; and tR are used to describe the interference extents between the hydrodynamic forces (or moments) acting on hull, propeller, and rudder, respectively. xR and xH are the dimension parameters, which are related to the distance between the ship's center of gravity and the rudder blade. For more details, refer to [1,8]. For the common marine ship (1), the prominent control inputs are allocated at the propeller and the rudder. The corresponding actuating forces (or moments) are ð1  t P ÞT E for surge and ðxR þ aH xH ÞF N cos δ for yaw motions. For the other DOFs, the impact forces (e.g. ð1  t R ÞF N sin δ and ð1 þ aH ÞF N cos δ) are slight and could be dealt with as the interferences or uncertainties. In Eq. (2), the effective attack angle of the rudder αR is a small value with the unit “rad”. Thus, we have sin αR  αR  δ, where the rudder angle
 δ A 0:5236 rad; 0:5236 rad . Based on the above analysis, one resets the kinetics equations by neglecting ð1  t R ÞF N sin δ and ð1 þ aH ÞF N cos δ. The actual underactuated ship model is presented by combining with kinematical equations, and it is the basis for the following design: 8 _ > < x ¼ u cos ψ  v sin ψ y_ ¼ u sin ψ þv cos ψ ð3Þ > : ψ_ ¼ r 8 mv f ðvÞ T u ðÞ > > u_ ¼ vr  u þ jnjn þ dwu > > mu mu m > u > > < mu f ðv Þ v_ ¼  ur  v þ dwv mv mv > > > > > ðmu  mv Þ f ðvÞ F r ðÞ > > uv  r þ δ þ dwr : r_ ¼ mr mr mr

ð4Þ

2

f u ðvÞ ¼  X juju juju þ X vr vr þX vv v þ X rr r

In marine navigation practice, the reference path is usually generated by waypoints, which guides the marine ship moving in the open sea. This note concerns with this practical scenario. The objective is to develop a novel DVS guidance principle and a practical robust adaptive controller with the output orders for the main engine speed n and the rudder angle δ, such that (1) the reference path could be logically programmed by a virtual ship with a desired speed ur; (2) the error dynamic system between the real ship and the virtual ship could be stabilized effectively, and all state variables of the underactuated ship are uniformly ultimately bounded stable. By combining points (1) and (2), a real ship could sail along the waypoints-based planning trajectory with a desired speed, positioning itself to a small neighborhood of the reference path. 2.2. Preliminaries Neural Networks are usually used as a tool to tackle system uncertainties for its good capabilities in unknown function approximation [7,33]. In this note, the Radial Basis Function (RBF) NNs are introduced in the proposed robust adaptive control to deal with the unstructured uncertainties. In addition, no adaptive parameters are derived for NNs by resorting to the robust neural damping technique, where Lemma 1 is useful for the design. Lemma 1 (Zhang and Zhang, and Yang and Zhou [7,34]). For any given real continuous function f ðxÞ with f ð0Þ ¼ 0 defined on a compact set Ωx , when the continuous function separation technique and RBF NNs approximation technique are used, the f ðxÞ can be written as follows: f ðxÞ ¼ S ðxÞAx þ εðxÞ;

2

8 x A Ωx

ð7Þ

where S ðxÞ ¼ ½s1 ðxÞ; s2 ðxÞ; …; sl ðxÞ is a vector of RBF basis functions with the form of Gaussian function (8), μi ; ξi are the center and width of the receptive field.  T  ! x  μi x  μi 1 si ðxÞ ¼ pffiffiffiffiffiffi exp  ; i ¼ 1; 2; …; l ð8Þ 2ξ2i 2π ξi

with 

3



 f v ðvÞ ¼  Y v v þ Y r r þ Y jvjv jvjv þ Y jrjr jr jr  þ Y vvr v2 r þ Y vrr vr 2  f r ðvÞ ¼  N v v þ N r r þ N jvjv jvjv þ N jrjr jr jr  þ Nvvr v2 r þ N vrr vr 2

ð5Þ T

3

where mu ¼ m  X u_ ; mv ¼ m  Y v_ ; mr ¼ I z  N r_ . η ¼ ½x; y; ψ  A R are the surge, sway displacement, and yaw angle in the earth fixed coordinate, respectively. dwu ; dwv ; and dwr denote the unmeasurable environmental disturbance forces or moments and model uncertainties. In (4), T u ðÞ; F r ðÞ are with the form of (6). In practical engineering, the actuators' energy gain is certainly finite. Therefore, there must exist positive constants T u0 ; T u ; F r0 ; and F r such that 0 o T u0 r T u ðÞ r T u and 0 o F r0 r F r ðÞ r F r are satisfied. Note, T u0 ; T u ; F r0 ; and F r are not actually known and only for the stability analysis. In the controller design, two parameters are required and updated online to compensate uncertainties of the unknown control gain function T u ðÞ=mu and F r ðÞ=mr :   T u ðÞ ¼ ð1  t P ÞρD4P kT J P    6:13Λ AR  2  2 uR þv2R cos δ F r ðÞ ¼ ðxR þ aH xH Þ  ð6Þ Λ þ2:25 L Assumption 1. There exist unknown positive constants d wu ; d wv ; and d wr , such that the disturbance terms dwu ; dwv ; and dwr because of ocean and model   environment    uncertainties are bounded, that is, dwu  rd wu ; dwv  r d wv ; and dwr Þ r d wr . Assumption 2. The sway motion is passive-bounded stable. This conclusion has been systematically analyzed in [6,32], considering different cases.

εðxÞ is the approximation error with unknown upper bound ε, l is the node number of NNs, m is the dimension number of x, and 2 3 w11 w12 ⋯ w1m 6 7 6 w21 w22 ⋯ w2m 7 7 A¼6 6 ⋮ ⋮ ⋱ ⋮ 7 4 5 wl1 wl2 ⋯ wlm is an optimal weight matrix. 3. DVS guidance As shown in Fig. 1, the purpose of the guidance principle is to provide a desired reference signal for the control derivation such that the marine ship would converge to the desired waypointsbased planning trajectory. The relevant details around the guidance steering law have been reported in [1]. For path-following in case of straight-line paths, LOS guidance principle is often incorporated to implement the trajectory tracking control, as well as the course-keeping autopilot. However, no one guidance principle is applicable to the current theoretical studies related to the topic “path-following control for underactuated ships”, for example, the control algorithm in Section 4. In marine practice, the reference path is usually generated by   waypoints W 1 ; W 2 ; …; W n with W i ¼ xi ; yi , which guides the real ship moving in the open sea. The objective of the DVS guidance is to provide a virtual ship as the reference for the control design, and the virtual ship sails automatically following the reference path with the desired speed. Fig. 2 presents the general framework

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

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4

YE

Real ship

W i +1

Dynamical virtual ship Guidance virtual ship

vr

yr ud

Δ φ i = φ i ,i +1 − φ i −1,i ur

Wi

ψr

Wi

ψd

xb

W i −1

u

yb

v

y

ψe ψ ldb

Δφ i PoutWi arcW

P inWi

ze

i

φ i ,i +1 Wi+1

Ri

φ i −1,i

x

OE

xr

XE GuidanceVirtual Ship

Fig. 2. General framework of DVS guidance principle.

of the DVS guidance principle. There are two key elements in the proposed guidance law, that is, the guidance virtual ship (GVS) and the dynamical virtual ship (DVS). They are both in the form of Eq. (9). The dynamical assigned characteristic of the ship attitude distinguishes the DVS from the GVS, which is the eponymy of DVS guidance. GVS focuses on the mission of generating the smooth reference path, whereas DVS prompts the path following aim as an auxiliary model, including the practice problems when the real ship deviates largely from the GVS: 8 _ > < x i ¼ ui cos ψ i y_ i ¼ ui sin ψ i ; > : ψ_ ¼ r ; v ¼ 0 i i i

i ¼ r; d

ð9Þ

Referring to Fig. 2, the dynamical assigned strategy of DVS is interpreted as follows. Define Z e ; ldb to represent the following distances of GVS and DVS, respectively. If Z e Z ldbset , ldbset is a threshold parameter, the attitude of DVS is assigned following Eq. (10). In the transient time Z e o ldbset , coordinates of the DVS are set equating to that of GVS, that is, xd ¼ xr ; yd ¼ yr . The yaw angle is still calculated using Eq. (10) and ud ¼ ur ; r d ¼ r r . Then the attitude in next control time is resolved by fusing of the dynamical Eq. (9), i¼d:

   y y ψ d ¼ 0:5½1  sgnðxr  xÞsgn yr  y  π þ arctan r xr  x xd ¼ x þ ldbset cos ψ d ;

yd ¼ y þldbset sin ψ d

ð10Þ

In general, as in most control system, the performance can be improved by trying a few simulation runs and adjusting the related parameters to obtain good behavior, e.g. ldbset . The intention of setting ldbset is to classify the DVS's dynamics into two phases: the following error Ze is small or large. When Ze is small ðZ e o ldbset Þ, the real ship could be controlled to follow the GVS directly. Thus, one set the coordinates of DVS equating to that of GVS. When Ze is large ðZ e Z ldbset Þ, DVS desires to be slowed down to wait for the real ship. The consideration is to avoid that the large main engine speed input is required to compensate the large following error. Whereas the practical machinery is limited, existing the input saturation. Thus, the coordinates of DVS are assigned based on that of the real ship, i.e. Eq. (10). According to the above principle, the parameter ldbset is selected as follows. The simulation was first performed with ldbset ¼ 0:5L. Then ldbset ¼ 0:4L was used and it was found that the path following or the control input performance degraded. On the other hand, a simulation using ldbset ¼ 0:6L obtained the rational result. Therefore, ldbset ¼ 0:6L is selected.

Wi-1 Fig. 3. Basic principle for generating the smooth reference path with GVS.

After the above analysis, it is critical and important for implementing the DVS guidance to generate the desired reference path by fusing of GVS. Fig. 3 presents the basic principle for generating the smooth reference path by GVS. In the principle, the waypoints-based reference path is split into regular straight lines and smooth arcs. The objective is to obtain the input orders ur and rr and the corresponding time series tr. In Fig. 3, the smooth reference path is W i  1 P inW i -arcW i -P outW i W i þ 1 . Normally, ur is a positive constant determined by the ship's operator or based on the planned time of arrival, whereas rr is an order signal changing with time. In the straight lines W i  1 P inW i and P outW i W i þ 1 , r r ¼ 0; t r ¼ distance=ur (distance is the length of the corresponding route). In the smooth arcs P inW i P outW i , r r a 0 is a constant that could be calculated according to the geometric principle. Firstly, the bearing angle of W i  1 W i and W i W i þ 1 can be obtained by Eq. (11), and Δϕi is defined   as Δϕi ¼ ϕi;i þ 1  ϕi  1;i . The turning radius Ri, resting with Δϕi  in 0; π=2 , is determined by interpolation in ½Rmin ; Rmax . It is noted that the parameter setting of Rmin ; Rmax depends on the ship maneuvering performance. Especially the appropriate selection of Rmin could avoid or reduce the saturation effect of the rudder device. Then r r ; t r in arcs P inW i P outW i can be easily obtained by r r ¼ ur =Ri ; t r ¼ Δϕi =r r . By employing a similar procedure, generation of the smooth reference path is completed:

 y  yi  1 ϕi  1;i ¼ arctan i ð11Þ xi  xi  1 Remark 1. In current literatures, little attention is paid to the practical condition. It is often assumed that the reference path could be generated using a virtual ship (no details about how to implement this assumption in practical engineering), and then the control design only tackles the problem of following the virtual ship. The DVS guidance is just for this assumption, to deal with the correlation between the waypoints generator and the smooth reference path. As shown in Fig. 2, the DVS plays the part of an auxiliary reference model. It coordinates the control of the marine ship's following the GVS, and ultimately achieves its convergence to the waypoints-based planning trajectory. To guarantee that the

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

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attitude variables of DVS and their derivatives exist and are bounded, that is, x2d þ x_ 2d þ y2d þ y_ 2d þ ψ 2d þ ψ_ 2d r B0 for any B0 4 0, they may be filtered by the first (or second) order filter with a small time constant. To prepare for the control design, the following is introduced: 2 3 2 3 2 xe xd  x cos ψ sin ψ 6y 7 6 y y 7 6 T T 4 e 5 ¼ J ðψ Þ4 d 5; J ðψ Þ ¼ 4  sin ψ cos ψ ψe ψd ψ 0 0

error operator 0

3

7 05

ð12Þ

1

where J T ðψ Þ is the rotation matrix derived from (3). Applying the derivation operation to (12), the path-following error dynamics is obtained by combining (13) and (4): 8 _ > < x e ¼  u þud cos ψ e þ rye y_ e ¼  v þ ud sin ψ e  rxe ð13Þ > : ψ_ ¼  r þ r d e

4. Formulation of the robust adaptive path-following control Motivated by the ideas of the DSC [35] and robust neural damping techniques, we develop a robust adaptive neural controller for the error dynamic system (4) and (13). The control design procedure consists of two steps for the kinematic and kinetic parts, and is detailed in Section 4.1. The stability of the closed system is discussed in Section 4.2. Then the conclusion is further extended by the merit of the low frequency learning. 4.1. Controller design Step 1: Referring to the dynamics of xe and ye in Eq. (13), one can directly choose αu and αψ e as the virtual controls of u and ψe. kxe and kye are two positive design parameters: αu ¼ kxe xe þ ud cos ψ e

 kye ye v αψ e ¼  arctan ud0

ð14Þ

surge Remark 2. In Eq. (14), ud0 is an immediate reference   speed for the design. It is defined as Eq. (15). Actually, kye ye  v r ud is often satisfied if the design parameters are chosen reasonably. For very few extreme cases kye ye v 4 ud , one chooses ud0 ¼ ud based on the ship manoeuvring experience and its feasibility is easy to understand by referring to Fig. 2: 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < u2  k y  v2 ; k y  v r u ye e ye e d d ud0 ¼ ð15Þ   :u ;   k y  v 4 u y d d e

e

Since there exists complicated quadrant conversion operations in derivation of the arcsin function, the arctan function is employed for calculating α_ ψ e instead of the arcsin function. To avoid redifferentiating the virtual controls in next step, which leads to the so-called “explosion of complexity”, the DSC technique [35,36] is employed here. Introducing two first-order filters (16) with time constants τu ; τψ e , and letting αu and αψ e pass through them to obtain the immediate variables βu ; βψ e for the next step τi β_ i þ βi ¼ αi ;

βi ð0Þ ¼ αi ð0Þ;

i ¼ u; ψ e

ð16Þ

Define qi ¼ αi  βi and ue ¼ βu u; ψ~ e ¼ βψ e  ψ e , then one has   x_ e ¼ αu þ ud cos ψ e þ rye þ qu þue ð17Þ y_ e ¼  v þ ud sin αψ e þ ud Ψ y  rxe

5





where Ψ y ¼ cos qψ e þ ψ~ e  1 sin αψ e  sin qψ e þ ψ~ e cos αψ e , and it is obvious that the variable Ψy is bounded. From (14) and (16), q_ u ¼  β_ u þ α_ u q ∂αu ∂αu ψ_ ¼  uþ x_ e þ τu ∂xe ∂ψ e e   q ¼  u þ Bu xe ; x_ e ; ψ e ; ψ_ e τu q_ ψ e ¼  β_ ψ e þ α_ ψ e qψ ∂αψ e ∂αψ e v_ y_ þ ¼  eþ τψ e ∂ye e v   qψ ¼  e þ Bψ e ye ; y_ e ; v; v_ τψ e

ð18Þ

The above description has not completed the design for the kinematic motion, as the virtual control βψ e is not required in Step 2. Thus, a similar operation is applied to the dynamics ψ_ e in (13). The dynamic surface (19) is incorporated by defining the variables qr ¼ αr βr and r e ¼ βr  r, where αr and βr are the virtual control and the filtered variables of r: τr β_ r þ βr ¼ αr ;

βr ð0Þ ¼ αr ð0Þ

ð19Þ

Furthermore, the time derivative of ψ~ e is obtained in Eq. (20) along (13):   ψ~_ e ¼ β_ ψ e r d þ αr  qr þ r e ð20Þ From Eq. (20), the variable αr is chosen as (21) and kψ~ e is a positive design parameter. Moreover, the derivative of qr is presented as (22): αr ¼  kψ~ e ψ~ e þ r d  β_ ψ e q_ r ¼ 

 qr _ ψ~ e ; ψ~_ e Þ þ Br ye ; y_ e ; v; v; τr

ð21Þ ð22Þ

In Eqs. (18) and (22), Bu ðÞ; Bψ e and Br ðÞ are all bounded continuous functions, which will be detailed in Section 4.2. Step 2: Taking the time derivative of ue and re along Eqs. (4), (16) and (19), the following dynamic Eq. (23) can be obtained with N2 ¼ jnjn:  1 _ mu β u  mv vr þ f u ðvÞ  T u ðÞN 2  mu dwu mu  1 _ mr β r  ðmu mv Þuv þ f r ðvÞ  F r ðÞδ  mr dwr r_ e ¼ mr

u_ e ¼

ð23Þ

In Eq. (23), the unknown function f i ðvÞ; i ¼ u; r is approximated by NNs as (24): f i ðvÞ ¼ S ðvÞAi v þ εi ðvÞ ¼ S ðvÞAi βv  S ðvÞAi ve þ εi ðvÞ ¼ S ðvÞAÞiβv  bi S ðvÞωi þ εi ðvÞ ð24Þ
T T where i ¼ u; r, βv ¼ βu ; v; βr ; ve ¼ ½ue ; 0; r e  . Define bi ¼ J Ai J F and m Am i ¼ Ai = J Ai J F , thus ωi ¼ Ai v e , bi ωi ¼ Ai ve . Then, one can construct the robust neural damping terms, as shown in Eqs. (25) and (26): νu ¼ S ðvÞAu βv þ εu  mu dwu  mv vr r S ðvÞAu βv þ ε u þmu d wu þ du ζ u ðvÞ r θu φu ðvÞ

ð25Þ

νr r θr φr ðvÞ

ð26Þ n o In (25) and (26), θi ¼ max J Ai J F ; ε i þ mi d wi ; di is the unknown parameter, and the damping term φi ðvÞ ¼ 1 þ J S ðvÞ J J βv J þ ζ i ðvÞ, where i ¼ u; r. In addition, di is the positive unknown constant, ζ u ðvÞ ¼ r 2 =4 þ v2 ; ζ r ðvÞ ¼ u2 =4 þ v2 .

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

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6

Based on the above analysis, the errors system (23) can be rewritten as  1 _ mu β u þ νu bu S ðvÞωu  T u ðÞN 2 mu  1 _ mr β r þ νr  br S ðvÞωr  F r ðÞδ r_ e ¼ mr

u_ e ¼

ð27Þ

In the control design, λ^ T u and λ^ F r are estimations of λT u ¼ 1=T u ðÞ, λF r ¼ 1=F r ðÞ, respectively. They are updated online to compensate uncertainties of the nonlinear gain functions T u ðÞ; F r ðÞ. The actual control for n and δ is derived in Eq. (28), where αN2 and αδ are the desired immediate control for T u ðÞN 2 and F r ðÞδ, respectively. Moreover, the corresponding adaptive laws are presented as Eq. (29), and the detailed synthesis analysis will be given in Section 4.2: pffiffiffiffiffiffiffiffiffi n ¼ sgnðN 2 Þ jN 2 j; N 2 ¼ λ^ T u αN2 ; δ ¼ λ^ F r αδ αN ¼ kue ue þ β_ u þkun Φu ðÞue 2

αδ ¼ kre r e þ β_ r þ krn Φr ðÞr e

ð28Þ

h

i _ λ^ T u ¼ γ T u αN2 ue σ T u λ^ T u  λ^ T u ð0Þ h

i _ λ^ F r ¼ γ F r αδ r e  σ F r λ^ F r  λ^ F r ð0Þ ð29Þ h i In (28) and (29), Φi ðÞ ¼ 14 φ2i ðvÞ þS ðvÞS T ðvÞ , ði ¼ u; r Þ. kue ; kre ; kun ; krn ; γ T u ; σ T u ; γ F r ; σ F r are positive design parameters. λ^ i ð0Þ denotes the initial value of λ^ i . Remark 3. It can be observed that the control designs (28) and (29) are with a concise form and are easy to implement in ship engineering, which benefits from the following two points. (1) Although RBF NNs are incorporated in the control design to tackle the unstructured uncertainty and the external disturbance, no NNs related adaptive parameter requires to be updated online because of the merit of the robust neural damping technique. (2) To improve the availability of the developed algorithm in practice, this study considers two real actuator steerable variables (the main engine speed n and the rudder angle δ) as the control inputs. That generates more uncertainties around the nonlinear gain function, which are just tuned and compensated online by the adaptive parameters λ^ T u and λ^ F r .

For all initial conditions satisfying x2e ð0Þ þq2u ð0Þ þ y2e ð0Þ þ q2ψ e ð0Þ þ 2 2 ψ~ 2e ð0Þ þ q2r ð0Þ þu2e ð0Þ þ r 2e ð0Þ þ λ~ T u ð0Þ þ λ~ F r ð0Þ r 2Δ, with any Δ 4 0, one can tune the control parameters kxe ; kye ; kψ~ e ; kue ; kre ; kun ; krn ; γ T u ; γ F r ; σ T u ; σ F r appropriately, such that all the signals in the closed-loop system are semi-global uniformly ultimately bounded (SGUUB). Proof. Based on the control design process, the Lyapunov function candidate is constructed as 1 1 1 1 1 1 V ¼ x2e þ q2u þ y2e þ q2ψ e þ ψ~ 2e þ q2r 2 2 2 2 2 2 1 T u ðÞ ~ 2 1 F r ðÞ ~ 2 2 2 λ þ mr r e þ λ þ mu ue þ 2 2γ T u T u 2 2γ F r F r

It should be mentioned that the NNs used in (24) always guarantee good approximation capability on the suitable compact set. That is, the stable performance cannot be held once the input vector is out of the compact set. It is also why the stability of the closed-loop system obtained in this note is only semiglobal instead of global. Actually, it is still an open problem, that is, how to identify the compact set and obtain the global stability by fusing of the approximation-based schemes [22,37]. As to the DVS guidance and the discussions in Remark 1, it is known that Π 1 ≔fðxd ; x_ d ; yd ; y_ d ; ψ d ; ψ_ d Þ : x2d þ x_ 2d þ y2d þ y_ 2d þψ 2d þ ψ_ 2d r B0 g is compact in Euclidean space with any B0 4 0. Therefore, the conclusion can be obtained that Π≔fðxe ; x_ e ; ye _ is compact in R10 , and there exist positive ; y_ e ; ψ e ; ψ_ e ; ψ~ e ; ψ~_ e ; v; vÞg     M u ; M ψ e and Mr satisfying Bu ðÞ rM u ; Bψ e ðÞ r M ψ e and constants  Br ðÞ rM r . The time derivative V_ can be derived along (17), (18), (20), (22), and (27), where b is a small positive constant: !  X

u2 1 V_ r  ki  i2  kye  d y2e 2 4 i ¼ xe ;ψ~ e " # ! ! 2 2 X 1 Mi 1 Mψ e 2 2 2  1  qi  ie  qψ e τi τψ e 2b 2b i ¼ u;r
   þ ue mu β_ u þ νu  bu S ðvÞωu  T u ðÞ λT u þ λ~ T u αN2
   þ r e mr β_ r þ νr  br S ðvÞωr  F r ðÞ λF r þ λ~ F r αδ þ

4.2. Main results In this section, the main result is stated as follows. Theorem 1. Consider the closed-loop system consisting of the error dynamic system (4) and (13) satisfying Assumptions 1 and 2, the robust adaptive control (28), and the gain-related adaptive law (29).

ð30Þ

T u ðÞ ~ ^_ F r ðÞ ~ ^_ 3b λT λ T þ λ F λ F þ Ψ 2y þ γT u u u γFr r r 2

ð31Þ

From Young's inequality, the following (32) and (33) would be incorporated in the further derivation, i ¼ u; r. In Eq. (35), wi;a denotes the ath column vector of the NNs weight matrix Ai : νi ie  bi S ðvÞωi ie

Fig. 4. The scientific research vessel YUKUN of DMU, China, with single propeller and rudder.

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r

1 M2 m þ 1 ; ¼ aq þ 1 þ i þ i τi 4 2b

kin φ2i i2e θ2i kin S ðvÞS T ðvÞi2e bi ωTi ωi þ þ þ 4 kin 4 kin 2

mi β_ i ie  β_ i ie r

bi ωTi ωi θ2i þ kin kin

mi þ 1 2 mi þ 1 2 qi i þ 4 τ2i e



2 1 2 1  λ~ i λ^ i  λ^ i ð0Þ r  λ~ i þ λi  λ^ i ð0Þ ; 2 2

i ¼ T u; Fr

2

2

ð33Þ

kre ¼ ae þ1 þ

ð34Þ

Then, (36) can be rewritten as

bu br mr þ 1 þ þ kun krn τ2r

V_ r  2aV þ ϱ

wTi;1 wi;1 þwTi;2 wi;2 þ ⋯ þ wTi;n wi;n J Ai J F 2

vTe ve

¼ u2e þ r 2e

ð35Þ

With the actual control law (28), (29) and (32)–(35), V_ yields !  X

u2d 2 1 2 _ ki  i  kye  ye V r 2 4 i ¼ xe ;ψ~ e ! ! 2 X 1 M2 m þ 1 2 1 Mψ e 2  1 i  i  qi  qψ e τi 4 τψ e 2b 2b i ¼ u;r ! 2 2 b b mu þ1 2 T u ðÞ ~ 2  kue  1  u  r  ue  σ u γ T u λ 2γ T u T u kun krn τ2u ! 2 2 b b mr þ 1 2 F r ðÞ ~ 2  kre 1  u  r  ð36Þ λ þϱ re  σ r γ F r 2γ F r F r kun krn τ2r

2 2 3b=2 þ θ2u =kun þ θ2r =krn þ ðσT u =2Þ λT u  λ^ T u ð0Þ þ ðσF where ϱ ¼ Ψ y þ 2 ^ . r =2Þ λF r  λ F r ð0Þ For the UUB stability, the design parameters are selected appropriately as follows. aη ; aq ; and ae are positive constants: 1 ki ¼ aη þ ; 2

2

kue ¼ ae þ 1 þ

2 ωTi ωi ¼ J Am i ve J

¼

2

bu br mu þ1 þ þ ; kun krn τ2u

ð32Þ

kye ¼ aη þ

u2d 4

;

ð37Þ   where a ¼ min aη ; aq ; ae ; σ u γ T u =2; σ r γ F r =2 . One can integrate (37) to get V ðtÞ r ϱ=2a þ Vð0Þ  ϱ=2a expð 2atÞ. Based on the closedloop gain shaping algorithm [38], V(t) is bounded satisfying limt-1 V ðtÞ ¼ ϱ=2a. In addition, Assumption 2 is satisfied. Thus,

2000

Pm6 Pm5

1800 1600 1400

Waypoints reference The proposed scheme (LF) The proposed scheme LOS based adaptive fuzzy sliding mode scheme

Pm4

1200

y (m)

rkin Φi ðÞi2e þ

7

Pm3

1000

M 2ψ e

1 ; ¼ aq þ τψ e 2b

800

Table 1 The main data and particulars of vessel YUKUN.

Pm2

600

Elements

Values

Length between perpendiculars Breadth Mean draft Displacement volume Height of the initial stability Block coefficient Rudder area Aspect ratio Max. rudder angle Max. rudder rate Propeller diameter Max. shaft velocity

105 m 18 m 5.4 m 5710.2 m3 1.3 m 0.5595 11.8 m2 1.9525 301 2.81/s 3.8 m 150 rpm

400

Pm1

200 0 -200

0

200

400

600

800

1000

x (m) Fig. 6. Comparison of path-following trajectory in two-dimensional (2D) plane: the proposed scheme (LF), the proposed scheme, and LOS-based adaptive fuzzy sliding mode scheme.

Fig. 5. The JONSWAP spectrum and the corresponding wind-generated waves with the seventh level sea state.

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

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8

all the error signals in the closed-loop control system as well as the control law (28) are SGUUB. □ The practical environmental disturbances are highly nonlinear and have both low frequency and high frequency effects to the ship's motion. To improve the robust performance of the proposed algorithm, the low frequency learning technique is used and Corollary 1 is presented directly.

Corollary 1. Under Assumptions 1 and 2, one considers the error dynamic system (4), (13) with initial conditions satisfying x2e ð0Þ þq2u ð0Þ þ y2e ð0Þ þ 2 2 2 2 q2ψ e ð0Þ þ ψ~ 2e ð0Þ þ q2r ð0Þ þ u2e ð0Þ þ r 2e ð0Þ þ λ~ T u ð0Þ þ λ~ F r ð0Þ þ λ~ T uf ð0Þ þ λ~ F rf ð0Þ r 2Δ. The actual control (28) and the low frequency gain-learning law (38) can guarantee that the solution of the closed-loop system is SGUUB, by setting the design parameters appropriately: n

h

i h i _ _ λ^ T u ¼ γ T u αN2 ue  σ T u λ^ T u  λ^ T uf λ^ T uf ¼ γ T uf λ^ T u  λ^ T uf :

n

h

i h i _ _ λ^ F r ¼ γ F r αδ r e σ F r λ^ F r  λ^ F rf λ^ F rf ¼ γ F rf λ^ F r  λ^ F rf :

ð38Þ

where λ^ T uf ; λ^ F rf are the filtered estimations of λT u and λF r , and λ~ T uf ¼ λ^ T uf  λT u ; λ~ F rf ¼ λ^ F rf  λF r . Proof. Corollary 1 follows from applying Theorem 1 to the proposed scheme. As to the low frequency adaptive law (38), the regional Lyapunov candidate can be constructed as 1 ~2 σT 2 1 ~2 σF 2 λ þ u λ~ þ λ þ r λ~ 2γ T u T u 2γ T uf T uf 2γ F r F r 2γ F rf F rf

i ¼ T u ;F r

r λ~ T u αN2 ue þ λ~ F r αδ r e

5. Illustrative comparison experiments In this section, the comparison experiments are presented to verify the performance of the proposed control scheme, comparing with the result in [13]. It is a LOS guidance scheme and implements the course-keeping task using a adaptive fuzzy sliding mode controller. For this purpose, one considers the scientific research vessel YUKUN of Dalian Maritime University (DMU), China as the plant, see Fig. 4. There exists a set of attitudemeasuring devices for research, including the GPS-based matrix antenna attitude-measuring instrument (type No. ADU5), the Fiber-Optic Gyrocompass and Attitude Reference system (type No. NAVIGAT 2100), and so on. Table 1 gives the main parameters of vessel YUKUN. As it is cost-expensive and difficult to test the comparison onboard the practical YUKUN vessel, the test confirmation is implemented by resorting to the ship motion math-

ð39Þ

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

400

0

50

100

150

200

250

300

350

400

20 0

-20

0

-40

-1 150

0

50

100

150

200

250

300

350

20

n (RPM )

u (m/s) v (m/s)

δ (deg )

0

-2

φ (deg )

50

40

8

1

0 -20

100 50 0

0

50

100

150

200

250

300

350

100

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

30

δ (deg )

200

ψ (deg )

100

0

10

0

The proposed scheme (LF) The proposed scheme LOS based adaptive fuzzy sliding mode scheme

150

12

6

ð40Þ

In (40), the remainder λ~ T u αN2 ue þ λ~ F r αδ r e can be canceled by the error terms in (31). Thus, the proof of Corollary 1 is completed. □

n (RPM )

Vλ ¼

The corresponding time derivative can be deduced as follows, along (38): 2

X 2 V_ λ ¼ λ~ T u αN2 ue þ λ~ F r αδ r e  σ i λ~ i  2λ~ i λ~ if þ λ~ if

20 10 0

-10

0

50

100

150

200

250

300

350

Time (s) Fig. 7. The attitude variables of vessel YUKUN under the proposed scheme (LF).

Time (s) Fig. 8. Control efforts n; δ: the proposed scheme (LF), the proposed scheme, and the LOS-based adaptive fuzzy sliding mode scheme.

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

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ematical model, which is with 4 DOFs and identified from the fullscale trial data. For the detailed model data and particulars, refer to Appendix A. The mathematical model could accurately describe 4 DOF dynamical response of YUKUN, and the studies related to modeling are being considered by the relevant journal (Fig. 5). In the experiment, the marine navigation practice is considered. The reference path is generated by waypoints W 1 ð0; 0Þ, W 2 ð500; 100Þ, W 3 ð800; 900Þ, W 4 ð800; 1800Þ, and W 5 ð100; 2000Þ with units m, ur ¼ 8:5 m=s. As to the marine environmental disturbances, sea wind, irregular wind-generated wave, and ocean currents are taken into consideration in the simulation. They are all simulated by fusing of the physical-based mathematical model.

λˆTu

50

0

-50

0

50

100

150

200

250

300

350

λˆFr

11.8 11.6 11.4 11.2

Refer to [1,10] for more details. To illustrate the robustness of the algorithm, the wind speed (Beaufort No. 7) V wind ¼ 15:25 m=s, wind direction ψ wind ¼ 701. The Joint North Sea Wave Observation Project (JONSWAP) spectrum is adapted to produce the corresponding wind-generated waves, which has been defined as an International Towing Tank Conference (ITTC) standard. The ocean current speed V current ¼ 1 kn and current direction βcurrent ¼ 2301. The initial states of YUKUN are ½xð0Þ; yð0Þ; ϕð0Þ; ψð0Þ; uð0Þ; vð0Þ; pð0Þ; rð0Þ; nð0Þ; δð0Þ ¼ ½  50 m; 20 m, 01, 01, 8 m/s, 0 m/s, 01/s, 01/s, 90 RPM, 01, where ϕ; p are the roll angle and the rolling rate, respectively. In the following comparison, “the proposed scheme” generally represents the proposed algorithm (28) and (29) without low frequency gain-learning consideration. Whereas, “the proposed scheme (LF)” is on behalf of the scheme with the low frequency gainlearning. For the proposed control algorithm (LF), the parameters setting follows Eq. (41). L is the length between perpendiculars. Note, the adaptive law (38) is equal to (29) when set γ T uf ¼ γ F rf ¼ 0. That is the scheme without the low frequency gain-learning consideration. As to the LOS based adaptive fuzzy sliding mode controller, the parameters setting refers to the example in [13]. Moreover, one needs to tune the parameters λs ¼ 6; Φs ¼ 0:04; λh ¼ 0:1; Φh ¼ 0:01 because of that the manoeuvrability of YUKUN is different from that of the plant in [13]: kxe ¼ 2:95;

kye ¼ 0:15;

kre ¼ 0:05;

kun ¼ 0:045;

τu ¼ τψ e ¼ τr ¼ 0:1; 0

50

100

150

200

250

300

350

9

σ T u ¼ σ F r ¼ 1:0;

kψ~ e ¼ 0:05; krn ¼ 0:075;

γ T u ¼ 0:03;

kue ¼ 0:055; ldbset ¼ 0:6L

γ F r ¼ 0:01;

γ T uf ¼ γ F rf ¼ 0:1

ð41Þ

Time (s )

λˆTu

500

0

-500

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

λˆFr

14 13 12 11

Time (s ) Fig. 9. Adaptive adjusting parameters λ^ T u and λ^ F r : (a) the proposed scheme (LF), (b) the proposed scheme.

Fig. 6 gives the comparison of path-following trajectory using the proposed scheme (LF), the proposed scheme and LOS-based adaptive fuzzy sliding mode scheme. Comparing with the LOSbased scheme, the two robust adaptive path-following controllers show obviously better performance. Especially, they could effectively control the marine ship for the path-following mission around the pivoting waypoint, whereas the LOS-based adaptive fuzzy sliding mode scheme only achieves the automatic turn manoeuvering for this condition. Fig. 7 presents the attitude variables of YUKUN u; v; ϕ; and ψ under the proposed scheme (LF), and the ship attitudes under the proposed scheme are similar to Fig. 7. It is noted that these states are UUB in the experiment. In Fig. 8, the actual control efforts n and δ are presented by using the proposed scheme (LF), the proposed scheme and the LOSbased adaptive fuzzy sliding mode one. The one significant merit is that the proposed scheme (LF) requires more practical and smoother control efforts (especially for the rudder angle δ), which benefits from the low frequency gain-learning technique. That is very meaningful and critical for the algorithm to be applied in practice. Fig. 9 gives the corresponding adaptive parameters by fusing of the two proposed

Table 2 Comparison valuation of the performances using the two proposed schemes and LOS-based adaptive fuzzy sliding mode one [13]. Indexes

ME

Segments

P m1 P m2

P m3 P m4

P m5 P m6

P m1 P m2

P m3 P m4

P m5 P m6

The proposed scheme (LF) The proposed scheme LOS-based adaptive fuzzy sliding mode scheme

0.6608  1.6102  2.6892

 0.2606 0.1448 1.7028

0.0303 0.3704 10.5322

1.0772 1.7540 2.6892

0.4218 0.3996 1.8181

0.7902 1.7883 11.0331

Indexes

Time of arrival (s)

The proposed scheme (LF) The proposed scheme LOS-based adaptive fuzzy sliding mode scheme

MIA

Actual distance of all (m)

P m1

P m2

P m3

P m4

P m5

P m6

99.2 100.2 115.5

141.4 137.7 164.8

194.6 194.1 218.7

228.9 228.4 251.8

296.4 298.0 325.3

329.6 328.8 355.1

2902.8 2899.1 3030.90

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

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robust adaptive control algorithms: Z t1 1 eðτÞ dτ; ME ¼ t 1  t 0 t0 Z t1   1 eðτÞ dτ MIA ¼ t 1 t 0 t 0

ð42Þ

For the further quantitative comparison, six measuring points are set in Fig. 6, that is, P m1 ð615:5; 408Þ; P m2 ð750:5; 768Þ; P m3 ð800; 1200Þ; P m4 ð800; 1496Þ; P m5 ð446:5; 1901Þ; P m6 ð166:5; 1981Þ. The following two popular performance specifications are used to evaluate the algorithm. They are the mean error (ME) value and the mean integral absolute (MIA) value, where eðτÞ denotes the cross track error (i.e. the perpendicular distance from the real ship to the planned route). “Times of arrival” and “Actual distance of all” are another two indexes, which focus on aspects of time and energy saving. By resorting to these indexes, the comparison valuation is measured and summarized in Table 2. In practice, the hardware of the autopilot is an industrial personal computer (IPC), that is, a computer suited to serve in marine engineering. Based on the consideration, one can incorporate the active control scanning time per period (ACST) as the evaluation index for the computational burden. That is, the time cost for scanning the control program at each execution period. Table 3 summarizes the comparison of computational burden for all the three control schemes. It is noted that the proposed scheme is with one order smaller computing load than the LOS-based scheme in [13]. Based on the above comparisons, the two proposed robust adaptive control schemes own the improved superiority around aspects of control performance, energy cost and computational burden, and could remedy the drawback of the LOS-based adaptive fuzzy sliding mode algorithm by the virtue of the novel DVS guidance. Especially, the scheme with low frequency gain-learning is more in accordance with the engineering requirements.

6. Conclusion This note has proposed a novel DVS guidance principle for the practical path-following control of underactuated ship. In addition, the robust adaptive path-following control scheme is developed with the rudder angle δ and the main engine speed n as control input variables, which is more in accordance with the engineering requirements. Comparing with the previous research, the marine practice “waypoint-based path-following control” is first dealt with by virtue of DVS guidance. It is critical and important for applying the current theoretical studies in the practical control engineering. As to the control part, the proposed algorithms are with some advantages such as concise form, robustness, and ease of implementation because of their smaller computational burden. Furthermore, the stability analysis guarantees UUB of all the signals in the closed-loop system by fuse of the Lyapunov theory, and the real ship sails within a small neighborhood of the waypoint-based reference path. The experiment comparisons have been presented to illustrate the effectiveness of the proposed strategy. In the application prospects, this research would provide more reliable algorithm for the autopilot equipment of marine ships, as Table 3 Comparison valuation of computational burden using the two proposed schemes and LOS-based adaptive fuzzy sliding mode one [13]. Indexes CPU frequency of processor The proposed scheme (LF) The proposed scheme LOS-based adaptive fuzzy sliding mode scheme

ACST value (s) 1.90 GHz

2.66 GHz 4

4.996  10 5.748  10  4 9.827  10  3

4

3.210  10 2.924  10  4 4.180  10  3

well as the automatic sailing of the remote operated vessel (ROV). Note, this work naturally cannot attend to every detail of the closed-loop system design, e.g. the nonsmooth nonlinearity existed in the control allocation. The authors are working on an extension of the proposed scheme to the following points. And the related works will be reported elsewhere:

 Control design with nonsmooth nonlinearity: The input satura-



tion is one of the most common nonsmooth nonlinearities existed in the control allocation. It severely limits system performance and needs to be conducted with in practical applications. Trajectory tracking: Combining with the DVS guidance principle, the finite time control [39,40] is a novel design idea for the trajectory tracking task, which has been applied effectively in the field of guided missile and spacecraft [41].

Acknowledgments This study is partially supported by the National Natural Science Foundation of China (Grant nos. 51109020, 51409033) and the Fundamental Research Funds for the Central University (Grant nos. 2014YB01; 3132014302). The authors would like to thank anonymous reviewers for their valuable comments to improve the quality of this article.

Appendix A. The identification parameters for the mathematic model of vessel YUKUN The mathematical model for vessel YUKUN includes four DOFs: surge, sway, roll, and yaw. The basic kinetic equations for surge, sway, and  yaw DOFs follow Eq. (1), whereas that for the roll DOF is I z þ J z r_ ¼ N. Table A1 presents the detailed model structure and parameters for vessel YUKUN. Table A1 The model structure and parameters for vessel YUKUN. The normalized mass and moments of inertia m0 ¼ 0:009865, X 0u_ ¼ 0:000384, Y 0v_ ¼ 0:007988, N 0r_ ¼ 0 I 0z ¼ 0:000617, J 0z ¼ 0:000469, I 0x þ J 0x ¼ 0:000004264 Model structure for the hydrodynamic force (or moment)   X : u3 ; uv2 ; uvr, ur 2 ; uvϕ2 , T J P ;  F N sin δ Y : uv; ur; up; uϕ, u2 r; v3 ; uvp2 ; urp2 , jvjv;  F N cos δ K : p; p3 ; ur; u2 r, r 3 ; vr2 ; v2 r, uvϕ2 ; urϕ2 ; jvjv;  F N cos δ N : uv; ur; up; uϕ, u2 r; u2 v; v3 ; r 3 ; vr 2 , v2 r; uvp2 ; urp2 ; jvjr; jr jr;  F N cos δ The hydrodynamic parameters normalized using the Prime-system I Y 0jvjv ¼  0:0086 N 0up ¼ 0:0002 X 0uuu ¼  0:0004 X 0uvv ¼  0:0179

C 0Rv ¼ 1:3925

N 0uϕ ¼  0:0005

X 0uvr ¼  0:0123

K 0p ¼ 1:35E  6

N 0uur ¼ 0:0683

X 0urr ¼  0:0022

K 0ppp ¼  3:17E  5

N 0uuv ¼ 0:0044

X 0uvϕϕ ¼  0:0194

K 0ur ¼  0:0059

N 0vvv ¼ 0:0008

C 0P ¼ 0:9415 C 0Ru ¼ 0:7715 Y 0uv ¼  0:0093 Y 0ur ¼ 0:3801

K 0uur ¼ 0:0057 K 0rrr ¼  0:0004 K 0vrr ¼ 0:0019 K 0vvr ¼ 0:0028

N 0rrr ¼  0:0046 N 0vrr ¼  0:0001 N 0vvr ¼ 0:0099 N 0uvpp ¼ 0:0008

Y 0up ¼ 1:35E  5

K 0uvϕϕ ¼ 0:0063

N 0urpp ¼ 0:0002

Y 0uϕ ¼  0:0013

K 0urϕϕ ¼ 0:0044

N 0jvjr ¼ 0:0048

Y 0uur ¼  0:3762

K 0jvjv ¼ 0:0006

N 0jrjr ¼ 0:0002

Y 0vvv ¼ 0:1786

C 0Rp ¼  0:052

C 0Rr ¼  0:5891

Y 0uvpp ¼ 0:0068

N 0uv ¼  0:0080

Y 0urpp ¼ 0:0008

N 0ur ¼  0:0703

Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i

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Please cite this article as: Zhang G, Zhang X. A novel DVS guidance principle and robust adaptive path-following control for underactuated ships using low frequency gain-learning. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2014.12.002i