Event-triggered robust fuzzy path following control for underactuated ships with input saturation

Ocean Engineering 186 (2019) 106122

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Event-triggered robust fuzzy path following control for underactuated ships with input saturation Yingjie Deng a, Xianku Zhang a, *, Namkyun Im b, **, Guoqing Zhang a, Qiang Zhang c a

Navigation College, Dalian Maritime University, Dalian, 116026, Liaoning, China Division of Navigation, Mokpo National Maritime University, Mokpo, 530729, Jeollanam-do, South Korea c Navigation College, Shandong Jiaotong University, Weihai, 264200, Shandong, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Path following Underactuated ships Input saturation Robust fuzzy damping Event-triggered control

To promise the high fidelity of path following control for the underactuated ship, this paper develops an eventtriggered robust fuzzy control scheme, which also releases the constraint of input saturation. To solve the underactuated problem, we add an adaptive bounded term to the tracking error in the sway motion, which allocates the error to actuated motions. In the control scheme, the fuzzy logic systems (FLS) are employed to approximate the uncertainties, while the methodology of robust damping is adopted. The Gauss error functions are introduced to approximate the structure of input saturation, such that the backstepping frame can be applied, and the control commands in the rudder angle and the revolving rate of the main engine are derived. To avert the frequent acting of actuators, the event-triggered control (ETC) technique is adopted. The static triggered con­ dition is constructed with a flexible adjustable variable, which regulates the minimal inter-event times for ac­ tuators. Via the direct Lyapunov approach, we prove the existence of the minimal inter-event time and the uniform boundedness of all tracking errors in the closed-loop system. Finally, the feasibility of the scheme is validated in the platform of the simulated ocean environment.

1. Introduction As one of typical scenarios of ship’s manoeuvering, path following has been widely studied over the past decades. A potential future on this issue is expected in the context of worldwide recognition for MASS (marine autonomous surface ships), which was legally admitted by IMO (international maritime organization) in 2018. In common working condition, the MASS is an underactuated system with only the surge and the rolling motions are controllable. The path-following task for it is usually comprised of three components, namely the path planning, the guidance and the control. In the specific frame of planning and guid­ ance, the control part is the most challenging issue, which needs to reconcile the problems of the underactuated characteristic, the envi­ ronmental disturbances and the modeling uncertainties etc (Liu et al., 2016). Although plenties of outcomes have been achieved in this issue by resorting to advanced cybernetics, most of them were not compatible with the marine devices. That is because of both complexity of these algorithms and lack of practical considerations, which refers to the informational unbalance (Fossen, 2011).

It can be divided into two categories to address the path following control of underactuated ships with respect to diverse guidance ap­ proaches (Liu et al., 2017a). One is to reinterpret the error dynamics by frame transformation or geometrical rationale. Two typical cases are line-of-sight (LOS) guidance (Fossen et al., 2003) and Serret-Frenet (SF) frame (Encarnacao et al., 2000). These approaches usually ignore the control in surge motion and suppose the surge speed to be a constant. To solve the underactuated problem, the assumption of omissible sway speed is made (Li et al., 2009; Liu et al., 2017a), or the leeway angle is compensated (Lekkas and Fossen, 2014; Fossen et al., 2015) or esti­ mated (Liu et al., 2017b). The other train of thought is to structure an ideal virtual ship sailing along the reference path without inertia and damping, so as to transform the path following to the tracking of the virtual ship (Zhang and Zhang, 2015). The surging motion is also considered in this approach and it is applicable to fruitful outcomes towards tracking control. In view of the tracking control for under­ actuated ships, Lyapunov theory based backstepping control approaches were developed in Jiang (2002) and Do et al. (2002) by utilizing the interconnected structure of the ship. Via state transformation, Dong and

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (X. Zhang), [email protected] (N. Im). https://doi.org/10.1016/j.oceaneng.2019.106122 Received 6 January 2019; Received in revised form 12 June 2019; Accepted 17 June 2019 Available online 24 June 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Ocean Engineering 186 (2019) 106122

Guo (2005) proposed three stabilizing controller with different convergent rates. Ghommam et al. (2010) developed an united stabi­ lizing and tracking controller on the basis of the cascade structure of the ship. Besides, slowly varying bias of the disturbances can be counter­ acted. Concluded from previous outcomes, Li et al. (2008) only considered surge and yaw motions in controller design while the sway motion was self-constrained by passivity. Although the above methodologies succeeded to solve the tracking control of underactuated ships in mathematical deduction, they are not palatable to the nautical practice. To meet the marine reality, the consideration of engineering factors must be involved in the control scheme. Do and Pan (2005) first coped with off-diagonal inertia matrix of the ship’s model, which accords with the asymmetric geometrical feature of the ship. In Do (2010), the notion of practical control was proposed, which reveals the present mainstream in this field. In most of the previous researches, the constraints generated by the actuators are omitted, whereas the analysis of these constraint is indispensable to ensure the applicability of control schemes. One of these constraints is the input saturation. To solve this, one can refer to Zheng and Sun (2016) and Liu et al. (2017a) for the dynamic auxiliary system, Shojaei (2015) for the saturated filters, Jin (2016) for the barrier Lyapunov function, Park et al. (2017) for the additional control term and Li et al. (2015, 2016); Zhou et al. (2017) for the sigmoid function approxima­ tion. Nevertheless, all these researches finally derived the control inputs with the form of thrust and turning torque, which can not be referred to by actuators directly. In contrast, it is more practical to offer the com­ mands in the rudder angle and the revolving rate of the propeller. To this end, Liu et al. (2017a) rendered the input of rudder angles with known actuator gains, and an adaptive approach for uncertain actuator gains was proposed in Zhang et al. (2017, 2018). Due to the large overshoot of the adaptive schemes at the initial stage, the control performance will be degraded. It should be further investigated that only the bounds of gains are available. Another constraint lies in the limited acting frequency of the actua­ tors. In nautical practice, the digital computer with sampler and zeroorder holder (ZOH) is employed to emulate the continuous-time con­ trol scheme, and the control command will be updated periodically (Katayama and Aoki, 2014). Long-term high-frequency action will lead to the mechanical abrasion and shorten the service life of actuators. Thus, it is practical to lower the acting frequencies of actuators and increase the acting periods. As larger period will degrade the control performance, it is preferred that control only happens when it is necessary, that is, the event-triggered control (Tabuada, 2007). For the fully actuated ship, emulation-like backstepping approaches were pro­ posed in Jiao and Wang (2016a, b), where the controller was first designed by ignoring the communication constraint. Fu et al. (2017) integrated the active disturbance rejection control with the ETC, and devised a dynamic positioning controller. In Postoyan et al. (2015), the ETC tracking controller for the underactuated unicycle mobile robot was developed. To the best of the author’s knowledge, there are still no re­ searches investigating the ETC for underactuated ships. Besides, all the above methods premised the accurate model. To approximate both in­ ternal and external uncertainties of the model, we can refer to the neural networks or the FLS for their universal approximation ability, and yet we need to find the way to promote their good match with the ETC. Further to reduce the complexity of the control laws, the methodology of robust damping should be considered (Deng et al., 2019). Motivated by the above analysis, this paper embarks on developing a synthesized path following control law for the underactuated ship with input saturation and uncertainties. The contributions are mainly twofold.

design, the control laws possess a succinct form, while there is no need to update the weights of FLS. Thirdly, we derive the control commands in the rudder angle and the revolving rate of the main engine, which are more acceptable to the actuators. Fourthly, we restructure the tracking error in the sway motion with the Gauss error function of an adaptive compensating variable, which allocates this error to the actuated motions and solves the underactuated problem. 2. The ideology of ETC is employed in the scheme to avoid the frequent acting of actuators. Creatively, the propeller and the rudder are separately discussed with the mechanism to adjust their acting fre­ quency flexibly. The existence of minimal inter-event time is proven such that the accumulation of triggering instants (namely the socalled “Zeno” phenomenon) is avoided. The remainder of this paper is organized as following. Preliminaries are deployed in section 1. The control scheme is designed in section 2. Section 3 analyzes the closed-loop stability and the existence of minimal inter-event time. The comparative experiment in the simulated sea environment is conducted in section 4. Section 5 concludes the whole paper. Notations: Uniformly in this paper, signð⋅Þ denotes the sign function, k ⋅ k implies the 2 norm of the variable, inff⋅g denotes the infimum of the discussed element. 2. Preliminaries 2.1. Mathematical model According to Fossen (2011), the motions of a surface ship can be described in two right-handed cartesian coordinates of the body and the earth. As for path following, only three motions (namely surge, sway and yaw) need to be considered. In the earth frame, the kinematic loop of Eq. (1) is fabricated, which depicts the attitudes of the ship. While in the body frame, the dynamic loop of Eq. (2) is fabricated and the kinestates are described. 8 > > < x_ ¼ u cosðψ Þ v sinðψ Þ y_ ¼ u sinðψ Þ þ v cosðψ Þ (1) > > : ψ_ ¼ r 8 > mu u_ ¼ > > > > > < mv v_ ¼ > > > mr r_ ¼ > > > :

� � � � Yr_r2� þ � Xu u þ X � ujuj � uu � � � � v v þ Y r v þ Yr r τwv mu ur þ Yv v þ Y jvjv jrjv � � � � þYjvjr �v�r þ Yjrjr �r�r � � � � τr þ τwr þ ðm � �u mv Þuv þ Yr_ur � � þ Nv v þ� N� jvjv v v þNjrjv �r�v þ Nr r þ Njvjr �v�r þ Njrjr �r�r

τu þ τwu þ mv vr

(2)

where x; y imply the horizontal coordinates of the ship, ψ the heading angle. u; v; r denote the translational and the angular speeds in surge, sway and yaw motions respectively. mi ; i ¼ u; v; r denote the ship inertia including the additional mass or moment respectively. X; Y; N with subscripts denote hydraulic derivatives generated by the damping effect. τwi denotes the environmental disturbance of each motion with the upper bound τwi � jτwi j, where τwi is a positive constant. τu and τr are the thrust and the turning torque. According to characteristics of the rudder and the propeller, they can be further written as Eq. (3). 8 < τu ¼ cu ðtÞN; N ¼ jNa jNa (3) : τr ¼ cr ðtÞδ; δ ¼ jδa jδa where Na is the actual revolving rate of the propeller, δa is the actual rudder angle. In a simple way, we can deem N and δ as the control in­ puts. cu ðtÞ and cr ðtÞ are the time-varying gains of the propeller and the þ rudder. We assume that their upper bounds of cþ u and cr and lower

1. The proposed control scheme addresses several practical factors together. First, the saturation nonlinearities of actuators are repre­ sented by the Gauss error functions, such that the backstepping design can be easily used. Secondly, by following the robust damping 2

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Ocean Engineering 186 (2019) 106122

b and b the law for ideal control commands N δ to impel the ship with ki­ netics Eq. (1) and Eq. (2) to converge to the position ðxd ; yd Þ and the heading angle ψ d of the virtual ship, namely stabilize the errors xe , ye and ψ e . 2.3. Approximation of FLS In our scheme, the FLS is employed to approximate nonlinear func­ tions. According to Wang (1994), the FLS is constructed based on the fuzzy rules base, and it is utilized by three steps, that is, fuzzification, fuzzy reasoning and defuzzification. A multi-input single output system can thereby described by a FLS with N rules as Eq. (7). PN wl sl ðxÞ yðxÞ ¼ Pl¼1 (7) N l¼1 sl ðxÞ where y is the output, x is the input vector. wl is the weight of each fuzzy

rule, sl ðxÞ is the product of membership functions. Let’s rewrite φl ¼ sl = P sl = Nl¼1 sl as the fuzzy basis function. Defining W ¼ ½w1 ; ⋯; wN �T and

ϕ ¼ ½φ1 ; ⋯; φN �T , it can be described that y ¼ W T ϕ. Thus, the approxi­ mation of FLS can be concluded as the following lemma.

Fig. 1. The schematic diagram of the virtual ship guidance.

bounds of cu and cr are known positive constants. In view of the satu­ ration constraint, we presume that N and δ follow Eq. (4). 8 8 > > > > b > Nþ < N þ ; if N < δþ ; if b δ > δþ b �N b ; if N þ � N N¼ N δ¼ b (4) δ; if δþ � b δ�δ > > > > b
Lemma 1. (Li et al., 2015, 2016, 2019) For any continuous function fðxÞ defined on a compact set, we can always find a FLS satisfying Eq. (8) with an arbitrarily small constant ε. � � �f ðxÞ W T ϕðxÞ� � ε (8)

b and b where N δ are the ideal control commands, Nþ and δþ denote positive upper limits, N and δ lower negative limits.

3. Controller design b and b In this section, the control laws for N δ are devised to resolve the tracking problem, namely stabilize xe , ye and ψ e . It is divided into three steps. In the first step, the virtual control laws in three motions are devised. Our scheme adopts the ideology of emulation-based ETC. Thus b and b in the second step, the continuous control laws for N δ denoted as Nc

2.2. The virtual ship guidance The virtual ship guidance principle is adopted to proceed the following design, which has been delineated in Zhang and Zhang (2015); Deng et al. (2019). This principle can transform the path following of the reference path to the tracking of the virtual ship, and its rationale is shown in Fig. 1. While the waypoints are set in the electronic chart, the reference path is first generated by connecting waypoints and interpolating arcs in the waypoints. A virtual ship without inertia and damping is employed to generate the time series of positional coordinates ðxd ; yd Þ and heading angles ψ d , its kinetics can be described as x_d ¼ ud cosðψ d Þ, y_d ¼ ud sinðψ d Þ and ψ_ d ¼ rd . To guarantee the asymptotic consensus of ψ to ψ d , we introduce the approaching angle as Eq. (5). � � � � qffiffiffiffiffiffiffiffiffiffiffiffiffiffi �� � qffiffiffiffiffiffiffiffiffiffiffiffiffiffi � y ψ da ¼ erf γ1 x2e þ y2e arctan e þ 1 erf γ1 x2e þ y2e ψ d (5) xe

and δc are devised in advance and the underactuated problem is resolved. Then in the third step, the triggering condition is devised. Step 1: According to kinematic loops of the Eq. (1) and the virtual ship, the error system of Eq. (9) is first obtained. 8 > > < x_e ¼ ðue þ αu Þcosðψ Þ ud cosðψ d Þ ðve þ αv Þsinðψ Þ y_e ¼ ðue þ αu Þsinðψ Þ ud sinðψ d Þ þ ðve þ αv Þcosðψ Þ (9) > > : ψ_ e ¼ re þ αr ψ_ da where αi is the virtual control laws of i ¼ u;v;r, and ie ¼ i αi . According to the backstepping approach, the virtual control laws are devised as Eq. (10). 8 > > > αu ¼ kx xe cosðψ Þ ky ye sinðψ Þ þ ud cosðψ d ψ Þ > > > > > αv ¼ kx xe sinðψ Þ ky ye cosðψ Þ þ ud sinðψ d ψ Þ > > > < ∂ψ (10) αr ¼ kψ ψ e þ da ðu sinðψ Þ þ v cosðψ Þ ud sinðψ d ÞÞ > > ∂ye > > > > > ∂ψ ∂ψ > > > þ da ðu cosðψ Þ v sinðψ Þ ud cosðψ d ÞÞ þ da rd > : ∂xe ∂ψ d

where xe ¼ x xd and ye ¼ y yd , γ1 is a parameter determining the transitional speed from arctanðye =xe Þ to ψ d . erfð⋅Þ is the Gauss error function of the sigmoid shape defined by Eq. (6). Z p2πffi x 2 2 erfðxÞ ¼ pffiffiffi e t dt (6)

π

0

where Eq. (6) is continuously differentiable with the upper and the lower bounds of �1. Defining ψ e ¼ ψ ψ da and from Eq. (5) and Eq. (6), we can know that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ψ da →ψ d while x2e þ y2e →0 and x→xd , y→yd , ψ →ψ d while xe → 0, ye → 0, ψ e →0. Then, the control objective is concluded as following. Control Objective: Under the saturation constraint of Eq. (4), design

where kx ; ky ; kψ are tuning parameter, the partial derivatives of ψ da can be acquired by synthesizing Eq. (5) and Eq. (9) as

3

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Ocean Engineering 186 (2019) 106122

8 � � � � π �� ∂ψ da ye ye > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan ¼ γ 1 exp γ 21 x2e þ y2e > > > ∂ ye 4 xe 2 2 > xe þ ye > > > > > > > > xe erfð⋅Þ > > þ 2 > 2 > > xe þ ye > > > � � � < ∂ψ � π �� xe ye da qffiffiffiffiffiffiffiffiffiffiffiffiffiffi arctan ¼ γ 1 exp γ 21 x2e þ y2e > ∂ xe 4 xe 2 2 > xe þ ye > > > > > > > > ye erfð⋅Þ > > > 2 2 > > xe þ ye > > > > � qffiffiffiffiffiffiffiffiffiffiffiffiffiffi � > ∂ψ da > > ¼ 1 erf γ 1 x2e þ y2e :

�

8 > > > > > u_e ¼ > > > > > < v_e ¼ > > > > > > > > r_e ¼ > > :

ψd

�

ψd

(11)

Then, by substituting Eq. (10) and Eq. (11) to Eq. (9), it can be rendered that 8 > > kx xe þ ue cosðψ Þ ve sinðψ Þ < x_e ¼ y_e ¼ ky ye þ ue sinðψ Þ þ ve cosðψ Þ (12) > > kψ ψ e þ re : ψ_ e ¼

V_ 1 ¼ xe x_e þ ye y_e þ ψ e ψ_ e ky y2e kψ ψ 2e þ ue xe cosðψ Þ ve xe sinðψ Þ þue ye sinðψ Þ þ ve ye cosðψ Þ þ ψ e re

V_ 2 ¼ S_v Sv � � � � kv S2v þ γu ue Sv þ γr re Sv þ θv Φv ��Sv ��

� �

kv S2v þ γu ue Sv þ γr re Sv

x2e

þ 4ε

y2e 2 ve

x2e þ y2e 2 S 2ε v þ

x2e

y2e 2

λΦ2v S2v

(21)

θ2v

þ β þ 2ε 4λ

Next, we embark on designing the control laws of Nc and δc . Simi­

larly, define θu ¼ maxfjjW u jj; dN =mu þ τwu =mu þ εu g, Φu ¼ kϕu k þ 1 and

(15)

θr ¼ maxfkW r k; dδ =mr þ τwr =mr þ εr g, Φr ¼ kϕr k þ 1. Then, the continuous control laws of Nc and δc are designed as Eq. (22). 8 � � > > > γ2u S2v þ x2e þ y2e > N ¼ mu > u k u u k > c u e fu e e < cu hN 4ε (22) � � 2 2 > > mr γ r Sv þ ψ 2e > > δ ¼ r k r r k c r e fr e e > > cr hδ 4ε :

��

4δ2m �t2½0;bδ�

(20)

By synthesizing Eq. 18 and 19 and Eq. (20), the time derivative of V2 can be rendered as Eq. (21).

upper bounds jdN j � dN and jdδ j � dδ . Observing Eq. (14), it can be transformed to Eq. (15) by Lagrangian mean value theorem.

πt2 ��

cr hδ cr hδ dδ τwr δc þ eδ þ þ þ εr þ W Tr ϕr mr mr mr mr

v2e � 2S2v þ 2β2

b and δm ¼ ðδþ þ δ Þ= where Nm ¼ ðNþ þ N Þ=2 þ ðNþ N Þ=2signð NÞ þ 2 þ ðδ δ Þ=2signðb δÞ, dN and dδ are approximating errors with the

where hN and hδ are expressed by � � �� π t2 �� ; h ¼ exp hN ¼ exp δ 4N 2m �t2½0;b N�

(18)

þ εv þ W Tv ϕv

Then, we define the Lyapunov function as V2 ¼ 12S2v . From the structure of Sv , the following Young’s inequality is first constructed for the sub­ sequent simplification.

(13)

Step 2: According to Ma et al. (2015), the saturation constraint of Eq. (4) can be rewritten as the Gauss error function plus the approximating error of Eq. (14). � � �b� b N δ N ¼ Nm erf (14) þ dN ; δ ¼ δm erf þ dδ Nm δm

b þ dN ; δ ¼ hδ b N ¼ hN N δ þ dδ

mv

where kv is the tuning parameter, γu and γr are the distributing param­ eters which distributes the Sv to surge and yaw motions. ε denotes a small constant and kfv ¼ λΦ2v , where λ is a positive tuning parameter.

Defining the Lyapunov function as V1 ¼ 12x2e þ 12y2e þ 12ψ 2e , one can render its time derivative of Eq. (13) by differentiating it with Eq. (12). kx x2e

τmv

Aiming at the underactuated problem, we define the combined error function as Sv ¼ ve þ βerfðϑv Þ, where β is a small constant, ϑv is the adaptive compensating variable. As the Gauss error function erfðϑv Þ is bounded, the stabilization of ve equates to the stabilization of Sv . Ac­ cording to the robust damping technique of Zhang and Zhang (2017); Zhang et al. (2017), we define the damping term as Φv ¼ jjϕv jj þ 1 and the compressed weight as θv ¼ maxfkW v k; τwv =mv þ εv g. Then, the updating law of ϑv is devised as Eq. (19). � � � exp π4ϑ2v � x2e þ y2e ϑ_ v ¼ kv Sv þ γu ue þ γ r re kfv Sv Sv (19) β 2ε

∂ψ d

¼

cu hN cu hN dN τwu Nc þ eN þ þ þ εu þ W Tu ϕu mu mu mu mu

(16)

b and b As our scheme will not render an infinite value of N δ, it is tenable to presume that hN and hδ are bounded with hN < hN < hþ N and hδ < hδ < hþ . δ By employing the Lemma 1, the nonlinear terms in the dynamic loop Eq. (2) are expressed as Eq. (17). 8 � � �� > W Tu ϕu þ εu ¼ mv vr Yr_r2 þ Xu u þ� X�ujuj �u�u � α > > �_ u mu > > T � � � � > mu ur þ Yv�v �þ Yjvjv v �v �þ Yjrjv �r v þ Yr r < W v ϕv þ εv ¼ þYjvjr �v�r þ Yjrjr �r�r α_ v � � (17) > T > � � > W ϕ þ ε ¼ ðm m α_ r r u v �Þuv þ Yr_ur þ N�v v�þ Njvjv v� v� > r r � � > > : þNjrjv �r�v þ Nr r þ Njvjr �v�r þ Njrjr �r�r

where ku and kr are tuning parameters, the damping parameters are selected as kfu � λΦ2u and kfr � λΦ2r . Define the Lyapunov function as V3 ¼ 12u2e þ 12r2e . By synthesizing Eq. (18) and Eq. (22), we can render the time derivative of V3 as Eq. (23). V_ 3 ¼ u_e ue þ r_e re � � � � γ2u S2v þ x2e þ y2e 2 cu hN � ku u2e λΦ2u u2e þ θu Φu ��ue �� ue þ eN ue 4ε mu � � � � γ 2r S2v þ ψ 2e 2 cr hδ kr r2e λΦ2r r2e þ θr Φr ��re �� re þ eδ re 4ε mr � � � � þ2 þ2 cþ2 cþ2 γ2u S2v þ x2e þ y2e 2 u hN r hδ � ku kr ue u2e r2e 2 2 4ε 2mu 2mr

where ε denotes the approximating error, the subscripts u; v; r denote the discussing motions. As the continuous control law Nc and δc will be first designed, here b Nc and eδ ¼ we define the sampling-induced control errors as eN ¼ N

b δ δc . By synthesizing Eq. (2), Eq. (3), Eq. (15) and Eq. (17), one can get the error system of the dynamic loop as Eq. (18).

γ2r S2v þ ψ 2e 2 θ2u þ θ2r re þ þ WN þ Wδ 4ε 4λ

4

(23)

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Y. Deng et al.

4. Stability analysis

where WN ¼ 12e2N and Wδ ¼ 12e2δ are the Lyapunov functions of the sampleinduced errors. Let V ¼ V1 þ V2 þ V3 , by combining Eq. (13), Eq. (21) and Eq. (23), it renders that V_ ¼ V_ 1 þ V_ 2 þ V_ 3 � � � � β2 2 β2 2 � kx ky kψ ψ 2e kv S2v xe y 2ε 2ε e � � þ2 � þ2 � cþ2 cþ2 u hN r hδ u2e r2e þ WN þ Wδ ku kr 2 2 2mu 2mr þ7ε þ

The proposed scheme in this paper is concluded as the following theorem. Theorem 1. Assume that all the following variables are within a compact � set Ω ¼ fðxe ; ye ; ψ e ; u; v; rÞ�x2e þy2e þψ 2e þu2 þv2 þr2 þu2d þr2d � ΔΩ g, where ΔΩ is a constant. For the underactuated ship with kinetics Eqs. (1-3), the semi-global uniformly ultimate boundedness(SGUUB) of all the error signals is guaranteed via the virtual control law Eq. (10), the adaptive law Eq. (19) and the control law Eq. (22) with the triggering condition Eq. (26). By tuning parameters appropriately, all the tracking errors can be regulated to be arbitrarily small. Furthermore, the minimal inter-event times of both the rudder and the propeller can be ensured, the Zeno phenomenon is averted. Proof of SGUUB: According to Eq. (26), it can be known that ei ; i ¼ N; N; δ is reset to zero once the triggering condition is satisfied, thus WN < að1 σ ÞζV and Wδ < að1 σÞð1 ζÞV always hold for V > μ along the control process. Synthesizing with Eq. (25), it renders

(24)

θ2u þ θ2v þ θ2r 4λ (

Here

we

define

a ¼ 2min kx

β2 ;k 2ε y

β2 ; k ; kv ; ku 2ε ψ

þ2 cþ2 u hN ; kr 2m2u

) þ2 cþ2 r hδ 2m2r

V_ �

and b ¼ 7ε þ

θ2u þθ2v þθ2r , 4λ

aV þ b þ WN þ Wδ

Eq. (24) can be rewritten as (25)

V_ �

Step 3: In the common continuous path-following control schemes, the design has already been complete in the last step. Here, the ideology of ETC is employed and the triggering condition will be established in b and b this step. We require that the signals of Nc and δc are only sent to N δ

(27)

aσ V þ b

which can be further transformed to VðtÞ � Vðt0 Þexpð aσtÞ þ abσ ð1 expð aσtÞÞ. According to the triggering condition of Eq. (26), we know that VðtÞ will finally converge to μ. As all the components of the squared errors in VðtÞ are less than VðtÞ, it is clear that all the errors finally pffiffiffiffiffi converge to the bound of 2μ. By increasing a, this bound can be tuned to be arbitrarily small. Thus, the Proof of SGUUB for all the errors is complete. Proof of minimal inter-event times: The minimal inter-event times of the rudder and the propeller are first defined as tδ and tN respectively. In

at the triggering instants. During the flow period between two successive b and b triggering instants, the ZOHs are employed to keep N δ unchanged.

Observed from Eq. (25), we are inspired with the further design. Suppose that the current time point lands in the flow time of ðtli ; tli þ1 � with i ¼ N; δ respectively, and the current triggering instant is denoted as tli . It should be noted that the common initial triggering time t0 is also deemed as the initial time of the voyage. Then, the next triggering instant tli þ1 is determined by satisfying the following triggering condition. 8 < tlN þ1 ¼ infft 2 Rjt > tlN ^ WN � að1 σ ÞζV ^ V > μg (26) : tlδ þ1 ¼ infft 2 Rjt > tlδ ^ Wδ � að1 σ Þð1 ζÞV ^ V > μg

brief expression, we define kδ ¼ kr þ

γu S2v þx2e þy2e

x2e þy2e

kfr þ

γ2r S2v þψ 2e 4ε ,

kN ¼ ku þ kfu þ

and kSv ¼ kv þ kfv þ 2ε . Then, we differentiate Wδ along the 4ε following flow time of the triggering instant tlδ , it renders Eq. (28) by synthesizing Eq. (12), Eq. (18), Eq. (19) and Eq. (22). mr _ mr k δ re e δ þ δ_ c eδ ¼ kδ r_e eδ cr hδ cr hδ � � � � cr hδ 2 mr cr hδ 2 mr kδ � � � eδ kδ k r e þ θ Φ e δ r r �eδ � cr hδ cr hδ cr 2 hδ 2 δ � � � � m r re e δ mr γ 2r kψ ψ 2e þ ψ e re þ θv Φv ��Sv re eδ �� þ 2εcr hδ 2εcr hδ � γ2r kSv S2v þ γ2r γu ue Sv þ γ 3r re Sv ! cr hδ kδ mr cr hδ k2δ mr kδ kfr mr γ2r kfv S2v r2e 2 eδ � þ þ þ cr hδ cr hδ 2εcr hδ 2cr 2 hδ 2 � �� � 3 � � mr r2e þ e2δ 1 γ 1 þ kψ þ ψ 2e þ r þ r2e 2 2 2 4εcr hδ � 2 3� � 1 γ γ γ mr kδ θ2r þ γ2r γu u2e þ γ 2r kSv þ r u þ r S2v þ 2 2 2 4λcr hδ W_ δ ¼

where 0 < σ < 1 is the tuning parameter, 0 < ζ < 1 is the adjustable variable concerning separating triggering conditions, which determines minimal inter-event times of the rudder and the propeller respectively. μ is a positive constant concerning the ultimate bound of VðtÞ, which is selected to ensure Vðt0 Þ > μ > b =ðaσÞ. Remark 1. According to Tabuada (2007), σ should be selected in a trade-off way. It can be inferred from Eq. (26) that a larger σ implies a smaller inter-event time, higher convergence speed and precision, and vice versa. Different with Jiao and Wang (2016a, b), triggering conditions are separately set for the rudder and the propeller in our scheme. It will be proved subsequently that the larger ζ implies the longer inter-event time of the propeller and the shorter of the rudder, and vice versa. Following the nautical practice, ζ should select to be slightly larger so as to guarantee the slow changing of the revolving rate of the main engine.

(28)

mr cr hδ k2δ r2e mr γ 2r θ2v þ þ 2 2 8λεcr hδ 2cr hδ

Remark 2. In the scheme, the so-called static triggering condition is constructed, which is not the state-of-art ETC technology. There are some recent outcomes towards setting of the triggering condition with superb control performances, such as the dynamic triggering condition (Girard, 2015), the combined time-event triggering condition(Abdelra­ him et al., 2016). Nevertheless, these methods are with strict re­ quirements for the model, which does not accords with the ship. Furthermore, it is convenient to prove the existence of minimal inter-event time in our scheme. More advanced ETC will be addressed in our future works.

m c h k2

2m k k

Define two class K functions as αδ ðVÞ ¼ 2cc r hhδ kδ þ cr 2r hδ 2δ þ c r hδ fr þ r δ r δ r δ � � �� � � 2 2 2 3 2mr kδ kfr mγ k S r þ r εrc fvh v e þ 2εcmrh kψ þ 12 ψ 2e þ γ2r þ 12 r2e þ 12γ 2r γu u2e þ γ 2r kSv þ c r hδ r δ r δ � � �� � � � mr cr hδ k2δ r2e mr r2e γ2r γu γ3r γ3r 2 1 2 1 S and β k r2e þ þ ðVÞ ¼ þ þ ψ þ þ ψ δ e v 2 2 2 2 4εcr hδ 2 2cr 2 hδ 2 � � � 2 2 2 3 mr kδ θ2r 1 2 γ γ u2 þ γ 2r kSv þ γr2γu þ γ2r S2v , the constant as bδ ¼ 4λc þ 8λmεr cγr θhv . 2 r u e h r

δ

r

δ

Then, Eq. (28) can be rewritten as W_ δ � αδ ðVÞWδ þ βδ ðVÞ þ bδ

(29)

It can be observed from Eq. (27) that VðtÞ is bounded by the 5

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Ocean Engineering 186 (2019) 106122

Table 1 Basic parameters of “YU PENG” in ballast draft. Indexes

Values

Length between perpendiculars (Lpp )

189 m

Breadth (B) Moulded depth Displacement volume (r)

27.8 m 15.5 m 22036.7m3

Mean draft (d) Height of initial stability (GM)

6.1 m 3.7 m

Block coefficient (CB )

0.661

Rudder area (AR )

31.6m2

Rudder aspect ratio (λr )

2.5

Maximum rudder angle (δrmax )

Fig. 2. The research and training ship “YU PENG”.

decreasing function from Vðt0 Þ to μ. Thus, it has Vðt0 Þ > VðtÞ in the discussing time interval t 2 ðtlδ ;tlδ þ1 �. According to Eq. (29), the following inequality holds. β ðVðt0 ÞÞ þ bδ αδ ðVðt0 ÞÞðt Wδ ðtÞ � δ e αδ ðVðt0 ÞÞ

tlδ Þ

βδ ðVðt0 ÞÞ þ bδ αδ ðVðt0 ÞÞ

tN �

(30)

mu _ mu k N ue eN þ N_ c eN ¼ kN u_e eN cu hN cu hN � � � � cu hN 2 mu cu hN 2 mu kN � eN kN k ue þ θu Φu ��eN �� 2 N e N 2 cu hN cu hN cu hN � � � � mu ue eN mu γ2u kx x2e ky y2e þ θv Φv ��Sv ue eN �� þ 2εcu hN 2εcu hN

(32)

(33)

(34)

To guarantee the high fidelity of simulation, the experiment is con­ ducted in the platform of simulated sea environment, which was also adopted in our recent work Deng et al. (2019). It assumes a non-fully developed sea state with the wave described by the JONSWAP spec­ trum and the wind by Davenport spectrum. Via the mechanism model of wave and wind, the environmental disturbances τwu , τwv and τwr can be further deduced. The platform is representative to describe the wind-generated waves with finite water depth. As only limited infor­ mation is needed here, the formulaic discussion is omitted. In the simulation, an usual sea state is presumed with Beaufort wind scale No. 5 of the northwest wind, the average speed of which is approximately 9:8m =s, seeing Fig. 3. Set six waypoints in the electronic chart with the coordinates as ½0m; 0m�, ½0m;2500m�, ½2500m;3500m�, ½2500m;6500m�, ½ 500m;7500m�,

m c h k2

Here we define two class K functions as αN ðVÞ ¼ 2ccu hhN kN þ cu 2u hN 2 N þ u N u N � � � 3 2mu kN kfu mu γ2u kfv S2v u2e γ mu 2 2 2 2 u þ þ þ 1Þx þ ðk þ 1Þy þ þ 1 u þ v ðk x y e e e e þ c u hN εcu hN 2εcu hN 2 � � � � 2 2 2 m c h k2 u2 γ3u þ γu2γr þ γ 2u kSv S2v þ γu2γr r2e and βN ðVÞ ¼ u2cu 2Nh N2 e þ 4mεcu uhe ðkx þ 1Þx2e þ 2 u N u N � � � � � 3 3 2 2 ðky þ 1Þy2e þ γ2u þ 1 u2e þ v2e þ γ2u þ γu2γr þ γ 2u kSv S2v þ γu2γr r2e , the con­ 2 2

� � 1 aμð1 σ ÞζαN ðVðt0 ÞÞ ln 1 þ βN ðVðt0 ÞÞ þ bN αN ðVðt0 ÞÞ

mu ¼ 2:33 � 107 ; mv ¼ 3:60 � 107 ; mr ¼ 5:04 � 1010 ; Yr_ ¼ 1:10 � 107 ; Xu ¼ 4:06 � 106 ; Xjuju ¼ 9:25 � 104 ; Yv ¼ 1:50 � 106 ; Yjvjv ¼ 1:50 � 106 ; Yjrjv ¼ 2:20 � 105 ; Yr ¼ 3:26 � 107 ; Yjvjr ¼ 2:20 � 105 ; Yjrjr ¼ 4:52 � 108 ; Nv ¼ 2:96 � 107 ; Njvjv ¼ 6:32 � 105 ; Njrjv ¼ 5:78 � 105 ; Nr ¼ 8:90 � 109 ; Njvjr ¼ 5:63 � 107 ; Njrjr ¼ 2:50 � 1011 :

mu cu hN k2N u2e mu γ 2u θ2v þ þ 2 2 8λ εcu hN 2cu hN

u

9765 kW 17.8knot

10000rpm2 , Nþ ¼ 22500rpm2 , δ ¼ 1225deg2 and δþ ¼ 1225deg2 respectively. The bounds of uncertain gains of the actuators are set as 5 þ 5 cu ¼ 1000, cþ u ¼ 2000, cr ¼ 1:5� 10 and cr ¼ 2:5 � 10 respectively. The actual values of these gains are slowly changing within the bounds according to the mechanic characteristics of actuators (Fossen, 2011). It should be noted that the bounds of actuators can be roughly estimated in reality as our scheme is robust.

þue xe cosðψ Þ

2

101.4 rpm

Mean power of the main engine Ship speed through water (U)

To verify the effectiveness of the proposed scheme and outstand the superiority of it with common methods, three cases are compared in the simulation with identical sea states. The training ship in Dalian Maritime University called “YU PENG” is selected as the control objective, seeing Fig. 2. Here we assume the ballast draft of “YU PENG” in the simulation, the basic coefficients in this condition are presented in Table 1. Ac­ cording to Eq. (2), the inertial parameters are calculated by empirical equations referring to Fossen (2011) and the hydrodynamic parameters of “YU PENG” are preliminarily identified by regressive least square method from the data of real sea trial, seeing the Eq. (34). To comply with the practice, we set the limits of the actuators as N ¼

W_ N ¼

N

Revolving rate of main shaft (Nm )

5. Simulation

The right side of Eq. (31) is a positive, the Proof of the rudder is complete. Similarly, if we differentiate WN along the flow time of the triggering instant tlN , it yields

stant as bN ¼ m4λcu kNhθu þ 8λmεucγu θhv . Similar to Eq. 29–31, it can be inferred that u

6.7 m

The right side of Eq. (33) is a positive constant, the Proof of the propeller is complete.

From the triggering condition of Eq. (26), we know that V > μ holds at triggering instants. Thus, we can infer from Eq. (30) that the interevent time of the rudder satisfy � � 1 aμð1 σ Þð1 ζÞαδ ðVðt0 ÞÞ tδ � (31) ln 1 þ βδ ðVðt0 ÞÞ þ bδ αδ ðVðt0 ÞÞ

ve xe sinðψ Þ þ ue ye sinðψ Þ þ γ 3u ue Sv � þve ye cosðψ Þ γ 2u kSv S2v þ γ 2u γ r re Sv ! cu hN kN mu cu hN k2N mu kN kfu mu γ 2u kfv S2v u2e 2 � eN n þ þ þ cu hN cu hN 2εcu hN 2cu 2 hN 2 � � 3 � � mu u2e þ e2N γ þ ððkx þ 1Þx2e þ ky þ 1 y2e þ u þ 1 u2e 2 4εcu hN � 3 � � 2 2 2 γ γ γ γ γ m u kN θu þv2e þ u þ u r þ γ 2u kSv S2v þ u r r2e þ 2 2 2 4λcu hN

35∘

Propeller diameter (Dp )

N

the inter-event time of the propeller satisfy

6

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Ocean Engineering 186 (2019) 106122

Fig. 3. Views of the wind field and the wave surface.

Fig. 4. Planar trajectories of the entire course.

½ 500m; 10000m�. Considering the turning radius of “YU PENG”, we set the interpolated radiuses of the virtual ship guidance around these waypoints as 500m. The initial attitudes of “YU PENG” are set as ½x;y; ψ � ¼ ½ 500m;0m;90∘ �, the initial kinestates and compensating variable ϑv are all set to be zeros. The surge speed ud of the virtual ship is set to be 15knot. The control parameters of the proposed scheme in the simula­ tion are set as Eq. (35). γ1 ¼ 0:1; kx ¼ 2; ky ¼ 2; kψ ¼ 0:1; ku ¼ 0:1; kr ¼ 0:1; β ¼ 1; kv ¼ 0:1; γu ¼ 1; γ r ¼ 0:01; ε ¼ 2; kfu ¼ 10; kfr ¼ 15; σ ¼ 0:5; μ ¼ 0:1:

ships are used to reveal the superiority of the developed scheme and illustrate the adjustment of the event-triggered mechanism here. They are both outfitted with the continuous control law of Eq. (22), but the No.2 ship is set with ζ ¼ 0:8 while the No. 3 ship ζ ¼ 0:2. Then, we proceed the simulation and launch out all three ships simultaneously. After all ships arrive the terminal waypoint, the planar trajectories of them are observed in Fig. 4. It shows that all the ships could accomplish final convergence to the reference path. Whereas in the local enlarged region, our scheme has higher precision than the LOS based approach. As only the ζ is changed, there is little difference in trajectories between the No. 2 and the No. 3 ships. It can also be learnt from Fig. 5 and Fig. 6 that attitudes and kinestates of them are very similar. The position errors of xe and ye in Fig. 5 are stabilized to the vicinity of zero after 400s. Due to the large inertia of “YU PENG”, overshoots of the heading angle error ψ e occur at every turning manoeuvring, but the proposed scheme can ensure the reversion of overshoots soon afterwards. It shows in Fig. 6 that u of the ship finally converges to ud of the virtual ship and v to the neighborhood of zero, but overshoots of v still happen. Because of the environmental disturbances, it is observed from Fig. 6 that small high-frequency chattering accom­ panies the entire voyage, especially in the yaw motion. In fact, by the merit of robust damping design, the chattering phenomenon has already been largely mitigated here compared with the adaptive scheme. In Fig. 7, the evolution of the adaptive compensating variable ϑv of the No.

(35)

For convenience, “YU PENG” with three cases are denoted as the No. 1, the No. 2 and the No. 3 ships separately. The No. 1 ship is selected as the comparing example by adopting the ideology of discrete control (Katayama and Aoki, 2014). In that scheme, the LOS guidance is employed with the constant reference surge speed of the ship, also set as 15knot in the simulation. Accordingly, the model Eq. (1) and Eq. (2) is first transformed to the discrete form by Euler’s approximation and the discrete controller is ulteriorly derived with the sampling period T involved as a design factor. As the larger T will degrade the control performance, it is selected to be 0:1s for the No. 1 ship. To keep uni­ formity in the control inputs, the actuator gains are assumed to be known for that ship. With no more tautology, one can refer to Katayama and Aoki (2014, 2011) for its formulaic details. The No. 2 and the No 3. 7

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Ocean Engineering 186 (2019) 106122

Fig. 7. Time evolution of adaptive compensating variable.

Fig. 5. Time evolution of attitude errors.

Fig. 6. Time evolution of kinestates.

2 ship is given. We can observe that steps of ϑv occur with large over­ shoots of v, namely at the time of turning manoeuvering. It tends to be moderate in the straight-line of path following. Then, we analyze control inputs of the three cases. In Fig. 8(a), the evolution of terminal control inputs of Na and δa are exhibited from the 0s–1700s. It can be observed that the input saturation happens in the initial stage and the turning manoeuvring due to the large attitude er­ rors. For the propeller, the proposed scheme has larger overshoots than

Fig. 8. Time evolution of control inputs. 8

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Ocean Engineering 186 (2019) 106122

the time-triggered scheme which is proportional to time.

Table 2 Comparison of the event-triggered performances. Number

Indexes

Tmin

Tmax

Tmean

Triggered times

No. 2

propeller rudder propeller rudder

0.03s 0.01s 0.01s 0.01s

14.32s 25.88s 8.29s 30.68s

0.41s 0.15s 0.30s 0.18s

4609 11560 6218 9643

No. 3

6. Conclusion This paper developed a practical path following control scheme for the underactuated ship, which took into account the existence of un­ certainties, the input saturation and the limited acting frequency of actuators. The nonlinearities of input saturation were restructured as the Gauss error functions. The underactuated problem was solved by compensating the tracking error of the sway motion with an adaptive variable. With the FLS approximating the uncertainties, the robust damping control design was conducted in the backstepping frame. By following the ETC in the channel of controller-to-actuator, the control commands of actuators were finally rendered. Furthermore, the trig­ gering condition was designed separately. Comparing with the previous time-triggered schemes for the ship, the proposed one is more intellec­ tive and reduces the acting frequency in a general meaning. Comparing with the previous event-triggered schemes, the proposed one is able to tune the minimum inter-event times of actuators manually according to practical requirements. Nevertheless, the proposed scheme may suffer from the shortage of conservatism while possessing robustness to envi­ ronmental disturbances and saturation constraints. This issue will be addressed in the future. Acknowledgements This work is partially supported by the National Science Foundation of China (No.51679024 and No.51779029), the Fundamental Research Funds for the Central University (No.3132016315), the National High Technology Research and Development Program of China (No.2015AA016404), the University 111 Project of China (No.B08046), the Key Research and Development Plan of Shandong Province (No.2018GGX105014), the Project of Shandong Province Higher Educational Science and Technology (No.J18KA010), the Project of Shandong Province Transportation Science and Technology Program (No.2018B69), the Project of Development of Shiphandling and Pas­ senger Evacuation Support System funded by the Ministry of Oceans and Fisheries of Korea. The author would sincerely thank the anonymous reviewers for their pertinent and instructive suggestions to improve the quality of this paper.

Fig. 9. Total triggering times of the actuators.

the comparison. That is because the difference in guidance approaches. In the LOS guidance, the constant surge speed is tracked, whereas in the virtual ship guidance, the virtual control term αu is tracked which relies on the attitude errors. For the rudder, the three cases follow the same mainstream of evolution, but the amplitude of the proposed scheme in straight-line path following is greater than that of the comparison. It is reasonable since the higher tracking precision of the proposed scheme is guaranteed in Fig. 4, which means more energy is costed to cancel out the adverse effects of disturbances. In Fig. 8(b), the local enlarged dia­ grams of control inputs around 358s for the propeller and 1067s for the rudder are shown. Because the running time for the entire voyage is very long, only these two typical points are selected. We can observe from Fig. 8(b) that the inter-event time of the No. 2 ship is larger than that of the No. 3 ship in the propeller, and vice versa in the rudder. It conforms to the analysis in the Remark 1 that the inter-event time of actuators can be tuned by ζ. In a quantified way, we denote the minimum, the maximum and the average inter-event time as Tmin , Tmax and Tmean respectively. As calculated in the Table 2, Tmax and Tmean of the No. 2 in the propeller are larger than that of the No. 3, while the total triggered times of the entire voyage of the No. 2 in the propeller are smaller, and vice versa in the rudder. It should be noted that in the first figure of Fig. 8(b) both the inter-event time of the No.2 and the No. 3 ship are smaller than that of the No. 1 ship, namely 0.1s. That doesn’t mean the inferior performance of the event-triggered scheme. As the time around 358s accounts for the turning manoeuvring near the waypoint of ½0m; 2500m�, finer control process is preferred. It can also be seen in Table 2 that although all the Tmin s in actuators are smaller than 0.1s, all the Tmean s are larger. As is observed, the time-triggered ideology of the No. 1 ship acts every 0.1s even if the amplitude difference between two consecutive triggering instants is very small. In contrary, the event-triggered ideology of the No. 2 and the No. 3 ship acts in a more intellective way, namely fine controls in necessary manoeuvring and slack controls in straight-line courses. Reflected on Fig. 9, the evolution of total triggering times of the proposed scheme increases slowly in straight-line path following but rapidly in turning manoeuvring, and is significantly lower than that of

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