Dynamic Positioning of Ships with Unknown Parameters And

Control Engineering Practice 76 (2018) 22–30

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Control Engineering Practice

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Dynamic positioning of ships with unknown parameters and disturbances Jialu Du a,*,XinHua,MiroslavKrsti¢ b,YuqingSunc a School of Information Science and Technology, Dalian Maritime University, Dalian, Liaoning, 116026, China b Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA c School of Marine Engineering, Dalian Maritime University, Dalian, Liaoning, 116026, China

ARTICLE INFO ABSTRACT

Keywords: Robust adaptive control is proposed for the dynamic positioning (DP) of ships with unknown model parameters Dynamic positioning of ships and unknown time-varying disturbances. Through representing the parameter uncertain ship motion mathe- Unknown model parameters matical model and the unknown time-varying disturbances as parametric forms, respectively, constructing an Unknown time-varying disturbances observer for handling disturbances, and using the adaptive vectorial backstepping, the DP robust adaptive control Observer law is designed. The proposed DP control law achieves the global asymptotic regulation of positioning errors, Vectorial backstepping Global asymptotic regulation while guaranteeing the global uniform ultimate boundedness of all signals in the DP closed-loop control system. Simulation results involving a supply vessel validate the proposed DP control law.

1. Introduction in Pettersen and Fossen (2000). The environmental disturbances were not considered in Chang et al. (2002), Fossen and Grøvlen (1998), Due to the depletion of land and shallow water resources, the ocean Grøvlen and Fossen (1996), Mazenc et al. (2002)andPettersen and exploration and exploitation are gradually moving to the deep and Fossen (2000). Considering unknown constant disturbances, Veksler, distant ocean, which is inseparable from the support of dynamic posi- Johansen, Borrelli, and Realfsen (2016) developed a model-predictive tioning (DP) technology. The DP technology allows a floating structure controller for the DP of ships to realize positioning control and thrust such as a ship or a drilling platform to maintain its position and heading allocation. In the case that a priori knowledge of sea states is known, Lo- at the fixed location or along the pre-determined track exclusively using ria, Fossen, and Panteley (2000) developed a proportional–derivative its own propellers and thrusters and it is insensitive to the depth of type DP control scheme based on the nonlinear passive observer with water (Du, Hu, Liu, & Chen, 2015; Sørensen, 2011). It is essential in wave filtering capability in Fossen and Strand (1999) estimating slowly many offshore operations, such as marine rescue, submarine pipeline varying disturbances and reconstructing ship low-frequency positions laying and deep-sea oil drilling (Hassani, Sørensen, & Pascoal, 2013). and velocities from the noisy position measurements. In the pres- The ship operating in the ocean inevitably suffers the disturbances ence of unknown time-varying disturbances, Nguyen, Sørensen, and due to waves, wind and currents, which will cause the ship to deviate Quek (2007) presented a DP hybrid control scheme with supervisory from the desired position and heading. The disturbance suppression becomes the crucial problem of the DP control design. A globally switching logic so that different DP control laws and wave filters can uniformly asymptotically stable DP nonlinear control law was designed be automatically switched according to the peak frequency of the using the observer backstepping method (Grøvlen & Fossen, 1996) waves estimated from spectral analysis of the position and heading and was reformed in the vector settings (Fossen & Grøvlen, 1998). measurements; Du, Yang, Hu, and Chen (2014) presented a DP adaptive Agloballyuniformlyasymptoticallystabilizingcontrollawforthe robust control scheme incorporating the adaptive technique into the DP of underactuated vessels was proposed applying the backstepping dynamic surface control, where the unknown bounds of disturbances method in Mazenc, Pettersen, and Nijmeijer (2002). An asymptotically were online estimated using the adaptive laws with -modification stable DP nonlinear fuzzy controller was designed utilizing the Takagi– leakage terms; Hu, Du, and Sun (2017) developed a DP robust adaptive Sugeno type fuzzy model and the linear matrix inequality methodology control law incorporating a model-based disturbance observer into the in Chang, Chen, and Yeh (2002). An exponentially stable time-varying vectorial backstepping method. Simultaneously considering the input feedback control law with integral control was developed for the under- saturation, Du, Hu, Krsti¢, and Sun (2016) proposed a DP robust actuated DP of ships using averaging theory and homogeneity properties control scheme combining a disturbance observer and an auxiliary

* Corresponding author. E-mail addresses: [email protected] (J. Du), [email protected] (X. Hu), [email protected] (M. Krsti¢), [email protected] (Y. Sun). https://doi.org/10.1016/j.conengprac.2018.03.015 Received 31 July 2017; Received in revised form 12 March 2018; Accepted 18 March 2018 0967-0661/© 2018 Elsevier Ltd. All rights reserved. J. Du et al. Control Engineering Practice 76 (2018) 22–30 dynamic system with the dynamic surface control, where the unknown time-varying disturbances were online estimated using the disturbance observer and the input saturation was handled employing the auxiliary dynamic system. All the aforementioned works on DP control design require the ship dynamic model parameters to be exactly known. In practice, the ship dynamics is related to the ship’s own characteristics and the operating conditions, thus there evidently exist parameter uncertainties in the ship motion mathematical model. Considering unknown model parameters and unknown constant disturbances, Do (2011) designed an adaptive observer to estimate the ship’s velocities and parameters from the ship position measurements and, based on these estimate signals and Lyapunov’s direct method, developed a DP robust adaptive output feedback control law. In the presence of variations in environmental and loading conditions, Tannuri, Agostinho, Morishita, and Moratelli (2010) derived a sliding mode controller, whose effectiveness was evaluated numerically and experimentally. Considering unknown model parameters and unknown time-varying disturbances, Du, Yang, Wang, and Guo (2013) presented a DP robust adaptive control scheme combining Fig. 1. North-east-down frame and ship-fixed frame (Fossen, 2011). radial basis function (RBF) neural networks with the adaptive vectorial backstepping method, where the RBF neural networks were employed to approximate the uncertainties of ship dynamics and time-varying T disturbances. The DP controllers in Du et al. (2013)andTannuri et al. where ⌘ =[x, y, ] represents the ship position vector in the north- (2010) ensure only the uniform ultimate boundedness of the positioning east-down frame, consisting of the position (x, y) and heading À [0, 2⇡] T errors of ships. being positive in the clockwise direction. ⌫ =[u, v, r] represents the ship Simultaneously considering unknown model parameters and un- velocity vector in the ship-fixed frame, consisting of the surge velocity known time-varying disturbances, this paper proposes a DP robust u,swayvelocityv and yaw rate r.TherotationmatrixJ( ) is given by adaptive control law which achieves the global asymptotic regulation cos( ) * sin( )0 of the positioning errors. The environmental disturbance compensation J( )= sin( ) cos( )0 (3) for the DP of ships is first converted into an adaptive control problem b 001c through rewriting ship motion mathematical model as the parametric f g f J T( )=gJ *1( ) M À R3 form, representing the environmental disturbances in the form of a with thed property e . is the inertia matrix multivariate linear regression model as the unknown outputs of a including the added mass, which is positive definite, symmetric and 3 T canonical linear model with uncertain disturbance term being input invertible. D À R is the damping matrix. ⌧ =[⌧1, ⌧2, ⌧3] represents and constructing an observer, then the DP robust adaptive control the control vector produced by the ship’s own propellers and thrusters, law is designed using the adaptive vectorial backstepping based on consisting of surge force ⌧1,swayforce⌧2 and yaw moment ⌧3. d(t)= the projection algorithm. The main contributions in this paper are as T d1(t), d2(t), d3(t) represents the low-frequency (LF) disturbance vector, follows: consisting of disturbance force d1(t) in surge, disturbance force d2(t) in (1) The adaptive laws together with the observer provide online esti- ⌅ ⇧ sway and disturbance moment d3(t) in yaw. mates of unknown time-varying disturbances, especially in the absence of a priori knowledge of the ship dynamic model parameters. Remark 1. The ocean disturbances can be separated into the LF (2) In the simultaneous presence of ship unknown model parameters disturbances due to the second-order waves, currents and wind as well and unknown time-varying disturbances, it is the first time that the as the wave-frequency (WF) disturbances due to the first-order waves. global asymptotic regulation of the positioning errors is achieved. The LF disturbances cause the ship to drift, while the WF disturbances (3) The proposed DP control law has both the adaptability to the cause the ship to oscillate. Compensating for the WF disturbances ship dynamic model parameter perturbations and the robustness against would induce the wear and tear of ship actuators and increase the fuel unknown time-varying disturbances. consumption. On the other hand, it is not necessary to compensate for the WF disturbances since they cause the ship’s back-and-forth rocking 2. Problem formulation motions (Veksler et al., 2016). These motions should be filtered out The ship’s motions are described in two right-hand coordinate frames from ship position and heading measurements before they are sent to the DP control system through the wave filtering. Several wave as shown in Fig. 1.Theearth-fixedframeindicatedbyOX0Y0Z0 is an inertial frame and the ship-fixed frame indicated by AXY Z is a non- filtering techniques have been proposed (Fossen & Strand, 1999; Fung inertial frame. The origin O of the earth-fixed frame can be chosen as & Grimble, 1983; Hassani, Sørensen, Pascoal, & Aguiar, 2012). In this paper, therefore the only LF disturbances are considered in the DP any point on the earth’s surface. The axis OX0 is directed to the north, the axis OY0 is directed to the east, and the axis OZ0 points towards control design. the center of the earth. When the ship is port-starboard symmetric, the origin A of the ship-fixed frame is located at the gravity center of the Assumption 1. (1) Model parameter matrices M and D in (2) are ( ) =12 3 ship. The axis AX is directed from aft to fore, the axis AY is directed to unknown; (2) The LF disturbances di t , i , , are unknown time- starboard, and the axis AZ is directed from top to bottom. The planes varying yet bounded and can be expressed as the superposition of a X0Y0 and XY are parallel to the still water surface. series of sinusoidal components as follows The mathematical model that describes the ship motions in DP mode q is (Fossen & Strand, 1999) di(t)= ai,j sin(!i,j t + "i,j ) =1 …j ⌘Ü = J( )⌫ (1) where q denotes the known number of the sinusoidal components; ai,j , !i,j and "i,j denote the unknown amplitudes, unknown frequencies and M⌫Ü = *D⌫ + ⌧ + d(t) (2) unknown phases, respectively.

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T 9ù1 Remark 2. The ship dynamic model parameters are related to the ship’s ✓⌧ =[✓⌧1 , ✓⌧2 , … , ✓⌧9 ] À R (9) mass, the moment of inertia about the yaw rotation, the hydrodynamic À R3 =12 … 9 [ 0 0]T [ 0 0]T [ 0 0]T [0 0]T Here ⌫i , i , , , are u, , , v, , , r, , , , u, , derivatives and the operating conditions, which are not constants, and T T T T T 3 [0, v, 0] , [0, r, 0] , [0, 0, u] , [0, 0, v] and [0, 0, r] ,respectively; À R , are difficult to be accurately determined. The ocean environment is ⌧i i =1, 2, … , 9 [⌧ , 0, 0]T [⌧ , 0, 0]T [⌧ , 0, 0]T [0, ⌧ , 0]T [0, ⌧ , 0]T constantly changing and has finite energy. On the other hand, the LF are 1 , 2 , 3 , 1 , 2 , [0, ⌧ , 0]T [0, 0, ⌧ ]T [0, 0, ⌧ ]T [0, 0, ⌧ ]T disturbances can be decomposed into a series of sinusoidal components 3 , 1 , 2 and 3 ,respectively;thevectors [✓⌫1 , ✓⌫2 , ✓⌫3 ], [✓⌫4 , ✓⌫5 , ✓⌫6 ] and [✓⌫7 , ✓⌫8 , ✓⌫9 ] are the first, second and third with different frequencies, amplitudes and phases by the Fourier analy- *1 rows of *M D,respectively;thevectors[✓⌧1 , ✓⌧2 , ✓⌧3], [✓⌧4 , ✓⌧5 , ✓⌧6] and sis. Therefore, Assumption 1 is reasonable. *1 [✓⌧7 , ✓⌧8 , ✓⌧9] are the first, second and third rows of M ,respectively. Remark 3. For the positioning control purposes of a vessel, the only According to Assumption 1(2), the uncertain disturbance term can available signals are the rough position and heading sensor signals. be expressed as the output of the linear exosystem: Thus, the LF motions and the velocities of the vessel should be obtained Ü = (10) from those signals using feasible techniques, like passive observer (Fos- sen & Strand, 1999), Kalman filter (Fung & Grimble, 1983; Hassani et al., 2012), or computation from the filtered values of position and = H (11) heading measurements, etc. In this paper, we assume that the clean 6 6 ù6 3ù6 where À R q is the state vector, À R q q and H À R q are ship velocities are available for feedback and we mainly focus on unknown constant matrices. All eigenvalues of matrix lie on the achieving the global asymptotic regulation of the positioning errors imaginary axis. The matrices and H are given as follows. in the simultaneous presence of ship unknown model parameters and unknown time-varying disturbances. 1 02qù2q 02qù2q H1 01ù2q 01ù2q = 02qù2q 2 02qù2q , H = 01ù2q H2 01ù2q The control objective in this paper is to design a DP robust adaptive b02qù2q 02qù2q 3 c b01ù2q 01ù2q H3 c control law for the ship with unknown model parameters and unknown f g f g withf g f g time-varying disturbances under Assumption 1,sothattheshipactual d e d e position (x, y) and heading globally asymptotically converge to the 0 !i,1 5 00 T desired value ⌘d =[xd , yd , d ] ,whileallsignalsintheDPclosed-loop *!i,1 0 5 00 control system are globally uniformly ultimately bounded. i = b 44744c , f 005 0 ! g f i,qg 3. DP robust adaptive asymptotic regulating control design f 005 *!i,q 0 g f g Hi = f10105 10 g In this section, the DP robust adaptive control design is presented d e ⌅ ⇧ consisting of two parts. First, the ship motion mathematical model It is obviously seen that (H, ) is an observable pair when !i,j ë 0 T T T T 6 *1 T (1)–(2) with unknown model parameters and unknown time-varying (i =1, 2, 3; j =1, 2, … , q) since rank[H , H , 5 , ( ) q H ]=6q. 6 ù6 disturbances is rewritten as another parametric form with disturbance Let G À R q q be a Hurwitz matrix and (G, L) be a controllable 6 ù3 uncertainties. The uncertain disturbance term in this parametric model pair where L À R q .Thus,G and have no common eigenvalues. of the ship motion is expressed as the output of a linear exosystem Then, according to Proposition 1 (see Appendix), the following Sylvester 6 ù6 with unknown parameters and all eigenvalues of system matrix lying matrix equation has a unique nonsingular solution F À R q q. on the imaginary axis. By means of this exosystem, the environmental disturbance d is expressed in the form of a multivariate linear regression F * GF = LH (12) model as the unknown outputs of a canonical linear model with un- Then, we introduce the change of coordinate certain disturbance term being input. The regressor of the multivariate linear regression model is the state vector of the canonical linear & = F (13) model and is unavailable. An observer is constructed to estimate the unavailable regressor. Based on the regressor estimate, the disturbances which transforms the linear exosystem (10)–(11) into a canonical linear can be further represented as the linear parametric equation with an ex- model as shown in Theorem 1. ponentially decaying error function vector. Accordingly, the disturbance compensation for the DP of the ship is converted into an adaptive control Theorem 1. The unknown environmental disturbance d can be expressed problem. Second, using the adaptive vectorial backstepping based on the as the output of the canonical linear model: projection algorithm, a DP robust adaptive control law is designed. &Ü = G& + L (14) 3.1. Disturbance representation and observer design T d = ⌥ & (15) For the convenience of designing an observer later, the ship motion ⌥ =(MHF*1)T À R6qù3 mathematical model (1)–(2) is rewritten as the following parametric where is the unknown constant matrix. form with disturbance uncertainties: Proof. In the light of (10), (12) and (11),thetimederivativeof(13) is ⌘Ü = J( )⌫ (4) &Ü =F =GF + LH ⌫Ü = ⌫ (⌫)✓⌫ + ⌧ (⌧)✓⌧ + (t) (5) =G& + L (16) *1 *1 *1 where = M d(t), ⌫ (⌫)✓⌫ = *M D⌫, ⌧ (⌧)✓⌧ = M ⌧ with Thus, (14) holds. ( )=[ … ] À R3ù9 ⌫ ⌫ ⌫1 , ⌫2 , , ⌫9 (6) It follows from (13) that

*1 T 9ù1 = F & (17) ✓⌫ =[✓⌫1 , ✓⌫2 , … , ✓⌫9 ] À R (7) *1 Substituting (17) into (11) gives = HF &.Then,accordingto = *1 ( )=[ … ] À R3ù9 M d, (15) is obtained. Thus, Theorem 1 has been proved. ⌧ ⌧ ⌧1 , ⌧2 , , ⌧9 (8)

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Remark 4. It is noted from Theorem 1 that the disturbance d is Since G is Hurwitz, the estimation error vector &É exponentially decays T expressed in the form of a multivariate linear regression model (15) and hence the error function vector "d = ⌥ &É exponentially decays. as the unknown output of the canonical linear model (14)–(15) with Thus, Theorem 2 has been proved. uncertain disturbance term being input. The regressor & is the state vector in the canonical linear model (14)–(15).Sincetheinputofthe Remark 5. Theorem 2 indicates that the disturbance uncertainties of canonical linear model is the uncertain disturbance term ,theregressor the ship are transformed into the parametric uncertainties. As such, the & is unavailable. disturbance compensation for the DP of the ship is converted into the adaptive control problem. Let the estimate vector expression &Ç of the unavailable regressor & in (15) be 9 9 3.2. DP robust adaptive asymptotic regulating control design based on = + + + observer &Ç ⌘Ç 0 ⌘Ç ⌫i ✓⌫i ⌘Ç ⌧i ✓⌧i L⌫ (18) =1 =1 …i …i 6 6 6 In this subsection, based on the observer (19)–(21),theDProbust where the estimate vectors ⌘Ç À R q, ⌘Ç À R q and ⌘Ç À R q(i = 0 ⌫i ⌧i adaptive asymptotic regulating control law for the ship (1)–(2) is de- 1, 2, … , 9) are generated by the following observer (Basturk, 2017; signed using the adaptive vectorial backstepping based on the projection Basturk & Krsti¢, 2013, 2014, 2015; Basturk, Rosenthal, & Krsti¢, 2015; algorithm (Krsti¢, Kanellakopoulos, & Kokotovi¢, 1995). Nikiforov, 2004): The following change of coordinates is taken: Ü ⌘Ç 0 = G⌘Ç 0 + GL⌫ (19) z1 = ⌘ * ⌘d (26) Ü = * =1 2 … 9 ⌘Ç ⌫i G⌘Ç ⌫i L⌫i , i , , , (20) z2 = ⌫ * ↵1 (27) ⌘ÇÜ = G⌘Ç * L , i =1, 2, … , 9 3 ⌧i ⌧i ⌧i (21) where ↵1 À R is the intermediate control function vector which will be designed later. The whole design process consists of the following two Theorem 2. Consider the rewritten parametric form (4)–(5) of ship motion steps. mathematical model, the linear exosystem (10)–(11),thecanonicallinear Step 1: According to (1),takingthetimederivativeof(26) yields model (14)–(15) and the observer (19)–(21). The time-varying environmental disturbance d acting on the ship can be represented as the following linear zÜ 1 = J( )⌫ (28) parametric equation: where ⌫ is viewed as the virtual control vector. T Select the Lyapunov function candidate for the subsystem (28): d = K !Ç + "d (22) T T T T T T 114qù3 1 T where K =[⌥ , ✓ ⌥ , … , ✓ ⌥ , ✓ ⌥ , … , ✓ ⌥ ] À R , !Ç = V1 = z z (29) ⌫1 ⌫9 ⌧1 ⌧9 2 1 1 T T T T T T 114 ù1 3 [(⌘Ç + L⌫) , ⌘Ç , … , ⌘Ç , ⌘Ç , … , ⌘Ç ] À R q , " À R is the error 0 ⌫1 ⌫9 ⌧1 ⌧9 d In terms of (28) and (27),thetimederivativeof(29) is function vector which exponentially decays. T VÜ1 = z J( )(z2 + ↵1) (30) 6 1 Proof. Define the estimation error vector &É À R q Design the intermediate control function vector = * &É & &Ç (23) T ↵1 = *J ( )K1z1 (31) According to (23) and (18),itisimpliedthat = T À 3ù3 where K1 K1 R is a positive definite design matrix. 9 9 T *1 Substituting (31) into (30) and using the property J ( )=J ( ) = + = + + + + & &Ç &É ⌘Ç 0 ⌘Ç ⌫i ✓⌫i ⌘Ç ⌧i ✓⌧i L⌫ &É (24) =1 =1 yield …i …i T " = ⌥ &É Ü = * T + T ( ) Substituting (24) into (15) results in (22) with d . V1 z1 K1z1 z1 J z2 (32) In the light of (14), (18)–(21) and (5)–(9),thetimederivativeof(23) Step 2: According to (2),takingthetimederivativeof(27) yields is *1 9 9 zÜ 2 = M [*D⌫ + ⌧ + d * M↵Ü 1] (33) &ÉÜ = &Ü * ⌘ÇÜ * ⌘ÇÜ ✓ * ⌘ÇÜ ✓ * L⌫Ü 0 ⌫i ⌫i ⌧i ⌧i ↵Ü i=1 i=1 In the light of (31),theexpressionof 1 is … … 9 T T = + * * * ( * ) ↵Ü = *JÜ ( )K z * J ( )K zÜ (34) G& L G⌘Ç 0 GL⌫ G⌘Ç ⌫i L⌫i ✓⌫i 1 1 1 1 1 i=1 9 … From (3),itfollowsthat * (G⌘Ç * L )✓ * L[ (⌫)✓ + (⌧)✓ + (t)] ⌧i ⌧i ⌧i ⌫ ⌫ ⌧ ⌧ *r sin( ) *r cos( )0 i=1 … 9 9 JÜ ( )= r cos( ) *r sin( )0 = G& + L * G⌘Ç * GL⌫ * G⌘Ç ✓ + L ✓ b 000c 0 ⌫i ⌫i ⌫i ⌫i f g i=1 i=1 cos( ) * sin( )00 *r 0 9 9 … 9 … f g = dsin( ) cos( )0r e 00 * G⌘Ç ✓ + L ✓ * L ✓ ⌧i ⌧i ⌧i ⌧i ⌫i ⌫i b 001c b000c i=1 i=1 i=1 …9 … … = f ( ) ( ) g f g fJ S r g f g (35) * L ✓ * L d e d e ⌧i ⌧i 0 *r 0 =1 i r 00 … 9 9 where S(r)= . L000M = * * * * G & ⌘Ç 0 L⌫ ⌘Ç ⌫i ✓⌫i ⌘Ç ⌧i ✓⌧i Substituting (35) and (28) into (34) gives H =1 =1 I …i …i = G&É (25) T T T ↵Ü 1 = *S (r)J ( )K1z1 * J ( )K1J( )⌫ (36)

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Select the Lyapunov function candidate for the whole systems (28) From (25) and (45),itfollowsthat and (33): T &ÉÜ P &É + &É TP &ÉÜ =&É TGTP &É + &É TPG&É 1 T =&É T GTP + PG &É V2 = V1 + z Mz2 (37) 2 2 T T = * &É (⌥⌥ )&É (48) By virtue of (32), (33) and (22),thetimederivativeof(37) is Substituting (47) and (48) into (46) gives Ü = Ü + T V2 V1 z2 MzÜ 2 T T T T T 1 2 T 2 T T = * + ( ) + [* + + * ] VÜ2a f * z K1z1 * z K2z2 + z2 + ⌥ &É * &É (⌥⌥ )&É z1 K1z1 z1 J z2 z2 D⌫ ⌧ d M↵Ü 1 1 2 4 T T T T = * + ( ) + [* + + + T T 1 z1 K1z1 z1 J z2 z2 D⌫ ⌧ K !Ç "d = * * * z1 K1z1 z2 K2 I3ù3 z2 * ] 4 Ò Ò Ò Ò M↵Ü 1 (38) 2 ⇠ 2 ⇡ f * z1 + z2 (49) Since the matrices M and D are unknown, the term D⌫ + M↵Ü 1 in = min * 1 (38) is unknown and can be represented as the linear parametric form: where Ò Ò minÒ KÒ1 , min K2 4 I3ù3 with the design matrix K2 satisfying $ ⇠ ⇡% + = T D⌫ M↵Ü 1 ⇥↵↵ (39) 1 min K > (50) T 6ù3 2 4 where ⇥ =[✓ , ✓ , ✓ ]=[D 4 M] À R is the unknown matrix ↵ ↵1 ↵2 ↵3 =[ … ]T À R6( =12 3) th Therefore, there is the following theorem. with ✓↵i ✓1,↵i , , ✓6,↵i i , , being the i column vector =[ ]T =[ T T]T À 6 and ↵ u, v, r, Ü↵ 11, Ü↵ 12, Ü↵ 13 ⌫ , ↵Ü 1 R is the regressor. Substituting (39) into (38) results in Theorem 3. Consider the closed-loop system consisting of the ship (1)–(2) with unknown model parameters and unknown time-varying disturbances Ü = * T + T ( ) + T[ + T + * T ] V2 z1 K1z1 z1 J z2 z2 ⌧ K !Ç "d ⇥↵↵ (40) under Assumption 1 and the DP robust adaptive control law (41)–(43) based on the observer (19)–(21).Theship’sposition(x, y) and heading globally The following DP robust adaptive control law (41) with the adaptive T asymptotically converge to the desired value ⌘d =[xd , yd , d ] ,whileall laws (42)–(43) is designed: signals in the DP closed-loop control system are guaranteed to be globally T T Ç Ç T uniformly ultimately bounded through an appropriate choice of the design ⌧ = * J ( )z1 * K2z2 + ⇥↵↵ * K !Ç T parameters r↵ , "↵ , r , " , G, L, Q↵ , Q (i =1, 2, 3), K1 and K2 satisfying = * J T( ) ⌘ * ⌘ * K ⌫ * ↵ + ⇥Ç * KÇ T!Ç i i i i i i d 2 1 ↵ ↵ (41) (50). Ü VÜ2 V2 ✓Ç ↵ = Q↵ Proj *z2 ↵, ✓Ç ↵ , i =1, 2, 3 (42) Proof. Since a is negative semi-definite from (49) and a is positive i i i i 0 ( ) (0) ⇠ ⇡ definite from (44),itisknownthat f V2a t f V2a ,whichmeans ( ) É ( =12 3) V2a t is bounded. Thus, z1 , z2 , ✓↵i , É i i , , and &É are Ü = Proj =1 2 3 Ç Ç i Qi z2i !Ç , Ç i , i , , (43) all bounded; hence ✓↵i and Ç i are bounded. In the light of (26) Ò ⌘ Ò Ò Ò Ò Ò Ò Ò ⌘ Ò Ò ⇠ T 3ù⇡3 T 6ù6 T 114 ù114 and the boundedness of d ,itisknownthat is bounded. From where K = K À R , Q = Q À R ,andQ = Q À R q q 2 2 ↵i ↵i i i (31), ↵1 is boundedÒ andÒ henceÒ Ò⌫ is bounded from (27). Further, 6ù3 are positive definite design matrices; ⇥Ç =[✓Ç , ✓Ç , ✓Ç ] À R is the ( =12 … 9) ↵ ↵1 ↵2 ↵3 it is seen from (19)–(21)ÒthatÒ ⌘Ç 0 , ⌘Ç ⌫i , ⌘Ç ⌧i Òi Ò , , , are all Ç =[Ç … Ç ]T À R6( =12 3) estimate of ⇥↵ with ✓↵i ✓1,↵i , , ✓6,↵i i , , being the bounded.Ò Ò Therefore, all signals in theÒ Ò DP closed-loop control system are 114qù3 globally uniformly ultimatelyÒ bounded.Ò Ò Ò Furthermore,Ò Ò it follows from ith column vector; KÇ =[Ç 1, Ç 2, Ç 3] À R is the estimate of K with T 114q (28) and (33) that zÜ 1 and zÜ 2 are bounded, respectively, which Ç i =[Ç 1,i, … , Ç 114q,i] À R (i =1, 2, 3) being the ith column vector. sup zÜ 2 f c sup zÜ 2 f c c > 0 Select the following augmented Lyapunov function candidate: means tg0 1 Üz 1 and tg0 2 Üz 2 with the constants Üz 1 and c > 0. Ò Ò Ò Ò 3 3 Üz 2 1 T *1 1 T *1 T Define theÒ functionÒ Ò Ò V2 = V2 + ✓É Q ✓É + É Q É + &É P &É (44) a 2 ↵i ↵i ↵i 2 i i i i=1 i=1 2 2 … … ⇠(t)= z1 + z2 (51) É = * Ç = * ( =12 3) where ✓↵i ✓↵i ✓↵i , É i i Ç i i , , ,andthepositivedefinite T 6 ù6 In terms of (49),thereis matrix P = P À R q q is a solution of the following matrix equation Ò Ò Ò Ò Ü T T ⇠(t) f *V2a (52) G P + PG = *⌥⌥ (45) Integrating (52) over [0, t] obtains In terms of (40)–(43) and (66),takingthetimederivativeof(44) t obtains ⇠(⌧t)d⌧t f V2a(0) * V2a(t) (53) 3 3 0 T *1 Ü T *1 T T VÜ2 = VÜ2 + ✓É Q ✓É + É Q ÉÜ + &ÉÜ P &É + &É P &ÉÜ V2 a ↵i ↵i ↵i i i i Since a is positive definite from (44),then i=1 i=1 t T … T … T T T = *z K1z1 + z J( )z2 + z [⌧ + K !Ç + "d * ⇥ ↵] lim ( ) (0) * (ÿ) (0) ÿ 1 1 2 ↵ ô ⇠ ⌧t d⌧t f V2a V2a f V2a < (54) 3 3 t ÿ 0 T T * ✓É Q*1✓ÇÜ * É TQ*1ÇÜ + &ÉÜ P &É + &É TP &ÉÜ ÿ 2 ↵i ↵i ↵i i i i It is obviously seen from (54) and (51) that î0 z1 dt is bounded, i=1 i=1 … … 3 3 i.e., z1 is square integrable on [0, ÿ).ItisobtainedbasedonLemma 1 T T T T lim ( ) = lim [ ( )* ]= = *z K z * z K z * z ✓É + z É !Ç (Barbalat’s lemma) (see Appendix) that tôÿz1 t Ò Ò tôÿ ⌘ t ⌘d 1 1 1 2 2 2 2i ↵i ↵ 2i i i=1 i=1 03ù1,whichmeanstheship’spositionÒ Ò (x, y) and heading globally 3 … 3 … =[ ]T T asymptotically converge to the desired value ⌘d xd , yd , d .Thus, + zT" * ✓É Q*1✓ÇÜ * É TQ*1ÇÜ 2 d ↵i ↵i ↵i i i i Theorem 3 has been proved. i=1 i=1 T … … + &ÉÜ P &É + &É TP &ÉÜ * T * T + T + Ü T + T Ü 4. Simulations f z1 K1z1 z2 K2z2 z2 "d &É P &É &É P &É (46) T According to Peter–Paul inequality and "d = ⌥ &É,itisimpliedthat AMatlab/Simulink-basedmodularsimulatorincorporatinghigh- fidelity models, called Marine System Simulator (MSS) toolbox (Fos- T 1 2 T 2 z " f z + ⌥ &É (47) 2 d 4 2 sen, 2011), was developed at the Department of Marine Technology,

Ò Ò Ò Ò 26 J. Du et al. Control Engineering Practice 76 (2018) 22–30

Table 1 Table 2 Main particulars of the supply vessel in DP_MotionRAO (Fossen, 2011). Performance indices in Case 1 and Case 2. Length between perpendiculars 82.8 m Performance indices Case 1 Case 2 Draft 6 m ⌧⌧com ⌧⌧com Breadth 19.2 m Mass 6.3622 ù 106 kg Settling time in north (s) 618 506 653 788 Displacement 6.2070 ù 103 m3 Settling time in east (s) 468 421 424 672 Settlingtimeinyaw(s) 456 291 454 545 Mean deviation in north (m) 1.2158 1.0044 1.3404 2.518 Meandeviationineast(m) 0.5994 0.5016 0.5352 0.9034 ˝ Norwegian University of Science and Technology. In this section, the Mean deviation in yaw ( )0.3890.19620.38060.6474 high-fidelity simulations are carried out using the module MSS‰MSS Hydro‰Vessel Templates‰DP_MotionRAO Model, which is a vessel motion simulator for low-speed maneuvering and station-keeping in 6 de- where !n is the natural frequency and &w is the relative damping factor, gree of freedom (DOF), to test the performance of the DP robust adaptive respectively. Here, Tf = 10, !n =0.83 and &w =0.1. control law developed in Section 3.AsupplyvesselinDP_MotionRAO The simulation results under the control law ⌧ are plotted using is taken as the simulation example, whose main particulars are listed the solid lines in Figs. 2(a)–2(f).Thequantitativeperformanceindices in Table 1.ThenominalvaluesM0 and D0 of its motion mathematical are summarized in Table 2,wherethesettlingtimesinnorth,east, model parameters are included in the MSS toolbox. Furthermore, the and yaw are taken as the times at which the position x in north, simulation comparison with the existing DP model-based control law position y in east, and heading in yaw enter and remain within the ˝ ˝ proposed in Hu et al. (2017) is presented. specified error bands 3 m, 3 m, and (1 _180 ) ù ⇡ rad,respectively. Fig. 2(a), Fig. 2(b) and Table 2 indicate that the actual position (x, y) 4.1. Performance of proposed DP control law and heading of the vessel asymptotically converge to the desired T value ⌘d = [0m, 0m, 0 rad] . Fig. 2(c) shows that the vessel velocities Ç In this subsection, to verify the adaptability of vessel to dynamic u, v, r are bounded. Fig. 2(d) demonstrates that the 2-norms ✓↵1 , Ç Ç model parameter perturbations and the robustness against unknown ✓↵2 , ✓↵3 of the parameter estimates are bounded, yet they get large time-varying disturbances, the simulations are carried out in the fol- because the true parameters are large and the estimates approachÒ thoseÒ lowing two cases. values.Ò Ò Ò It isÒ obvious from Fig. 2(e) that the control inputs ⌧1, ⌧2, ⌧3 are Case 1:theparametersofthevesselmotionmodelaresetasthe reasonable. It can be observed from Fig. 2(f) that the adaptive laws (43) nominal values M0 and D0. together with the observer (19)–(21) provide the satisfactory estimates T T T In simulations, the wave spectrum type is Jonswap, the significant Ç 1 !Ç , Ç 2 !Ç , Ç 3 !Ç for unknown environmental disturbances d1, d2, d3 ,as wave height is 0.8 m, the peak frequency is 0.87 rad/s, and the proved in Theorem 3.Thesimulationresultsshowtheeffectivenessof ˝ ˝ mean wave direction is (35 _180 ) ù ⇡ rad in the north-east frame, the proposed DP robust adaptive control law in different disturbance which are corresponding to the sea state 3. The wind speed is 5 m/s conditions. ˝ ˝ and the mean wind angle is (30 _180 ) ù ⇡ rad. The current speed Case 2:theparametersofthevesselmotionmodelareperturbedto ˝ ˝ is 0.1 m/s and the current direction is (350 _180 ) ù ⇡ rad. At and (1 + 50%)M0 and (1 + 50%)D0. after 2500 s, the significant wave height is changed to be 1.5 m, In simulations, the ocean disturbance conditions, the initial con- the peak frequency becomes 0.76 rad/s, the mean wave direction is ditions and the design parameters of control law are identical to the ˝ ˝ (35 _180 ) ù ⇡ rad, which are corresponding to the sea state 4. The counterparts in Case 1, which means that our proposed DP robust wind speed, mean wind angle, current speed, and current direction are adaptive control law is kept unchanged when the parameters of the kept unchanged. The coordinate origin of the north-east-down frame vessel motion model are perturbed. T is taken as the desired vessel position, that is ⌘d = [0m, 0m, 0 rad] . The simulation results under the control law ⌧ are plotted using the The following initial conditions and design parameters are adopted: solid lines in Figs. 3(a)–3(f) and the quantitative performance indices ˝ ˝ T T ⌘(0) = 5m, 5m, (5 _180 ) ù ⇡ rad , ⌫(0) = 0m_s, 0m_s, 0 rad_s , are summarized in Table 2.ItisclearlyobservedfromFigs. 2(a)–2(f), ⌘Ç (0) = 0 ⌘Ç (0) = 0 ⌘Ç (0) = 0 (i =1, 2, … , 9) ✓Ç (0) = Figs. 3(a)– 3(f) and Table 2 that our developed DP robust adaptive 0 ⌅ 12ù1, ⌫i 12ù1, ⌧i ⇧ 12ù1 ⌅ , ↵1 ⇧ Ç (0) = Ç (0) = 0 (0) = (0) = (0) = 0 = = control law exhibits almost identical satisfactory transient and steady- ✓↵2 ✓↵3 6ù1, Ç 1 Ç 2 Ç 3 228ù1 and r↵1 r↵2 6 10 7 10 state performance to that in Case 1. It is obviously shown that our 10 , r↵3 = 10 , "↵1 = "↵2 = "↵3 = 100, r1 = r2 = 10 , r3 = 10 , "1 = *3 *3 proposed DP robust adaptive control law has the adaptability to vessel " = " = 100, q =2, G = diag(*1.205 ù 10 , *1.260 ù 10 , *1.315 ù 2 3 dynamic model parameter perturbations and the robustness against 10*3, *1.370 ù 10*3, *1.425 ù 10*3, *1.480 ù 10*3, *1.535 ù 10*3, *1.590 ù *3 *3 *3 *3 *3 unknown time-varying disturbances. 10 , *1.645 ù 10 , *1.655 ù 10 , *1.710 ù 10 , *1.765 ù 10 ), L = [1 0 0; 1 0 0; 1 0 0; 1 0 0; 0 1 0; 0 1 0; 0 1 0; 0 1 0; 0 0 1; 0 0 1; 0 0 1; 0 0 1], 4.2. Comparison with existing DP model-based control law =8ù 105 =5ù 106 =1ù 109 = Q↵1 I6ù6, Q↵2 I6ù6, Q↵3 I6ù6, Q1 *13 *13 *10 2 ù 10 I228ù228, Q =3ù 10 I228ù228, Q =1ù 10 I228ù228, 2 3 In this subsection, the control performance comparison between our K = diag(3 ù 10*3, 3 ù 10*3, 5 ù 10*3) K = diag(1 ù 104, 2 ù 104, 1 ù 108) 1 , 2 . proposed DP robust adaptive control law and the following DP model- In addition, the zero-mean Gaussian white noises with the standard ˝ ˝ based control law (57) (Hu et al., 2017) with the intermediate control deviations (0.05 m, 0.05 m) and (0.1 _180 ) ù ⇡ rad are injected in function vector ↵mb1 is presented. position (x, y) and heading as the measurement noises. T T The low-pass filter and the notch filter are included in the simu- ⌧com = * J ( )z1 * Kmb2z2 + D0⌫ + M0↵Ü mb1 * ✓Ç &Ç (57) T lations to suppress the measurement noises and the first-order wave- ↵mb1 = * J ( )Kmb1z1 (58) induced motions in position and heading. The low-pass filter is chosen = T À 3ù3 = T À 3ù3 = T À 6qù6q where Kmb1 Kmb1 R , Kmb2 Kmb2 R and Qi Qi R , as 6qù3 (i =1, 2, 3) are positive-definite design matrices, ✓Ç =[✓Ç 1, ✓Ç 2, ✓Ç 3] À R 1 Ç Ç Ç T 6q 6q h (s)= (55) with ✓i =[✓i,1, … , ✓i,6q] À R (i =1, 2, 3) and &Ç À R are generated LP 1+ Tf s by the following model-based observer (59)–(60) and the adaptive law where Tf is the time constant. The notch filter is taken as (61),respectively. 2 2 = + s +2&w!ns + !n &Ç N⌫ (59) hNO(s)= (56) *1 *1 2 2 Ü = G + GN + NM D0 ⌫ * NM ⌧ (60) s +2!ns + !n 0 0 com 27 J. Du et al. Control Engineering Practice 76 (2018) 22–30

(a) (b) (c)

(d) (e) (f)

Fig. 2. Simulation results under control laws ⌧ and ⌧com in Case 1. (a) Filtered values of vessel position (x, y) and heading .(b)2-norms z1 of positioning error Ç Ç Ç Ç Ç Ç vector. (c) Vessel velocities u, v, r.(d)2-norms ✓↵1 , ✓↵2 , ✓↵3 of parameter estimates ✓↵1 , ✓↵2 , ✓↵3 .(e)Actualcontrolinputs⌧1, ⌧2, ⌧3 and ⌧com1, ⌧com2, ⌧com3.(f) T T T T T T Ò Ò LF disturbances d1, d2, d3,disturbanceestimatesÇ !Ç , Ç !Ç , Ç !Ç under ⌧,anddisturbanceestimates✓Ç 1 &Ç, ✓Ç 2 &Ç, ✓Ç 3 &Ç under ⌧com. Ò 1 Ò Ò2 Ò 3Ò Ò

(a) (b) (c)

(d) (e) (f)

Fig. 3. Simulation results under control laws ⌧ and ⌧com in Case 2. (a) Filtered values of vessel position (x, y) and heading . (b) 2-norms z1 of positioning error Ç Ç Ç Ç Ç Ç vector. (c) Vessel velocities u, v, r.(d)2-norms ✓↵1 , ✓↵2 , ✓↵3 of parameter estimates ✓↵1 , ✓↵2 , ✓↵3 .(e)Actualcontrolinputs⌧1, ⌧2, ⌧3 and ⌧com1, ⌧com2, ⌧com3.(f) T T T T T T Ò Ò LF disturbances d1, d2, d3,disturbanceestimatesÇ !Ç , Ç !Ç , Ç !Ç under ⌧,anddisturbanceestimates✓Ç 1 &Ç, ✓Ç 2 &Ç, ✓Ç 3 &Ç under ⌧com. Ò 1 Ò Ò2 Ò 3Ò Ò

28 J. Du et al. Control Engineering Practice 76 (2018) 22–30

Ü ✓Ç i = QiProj(z2,i&Ç, ✓Ç i), i =1, 2, 3 (61) Definition 1 (Krsti¢ et al., 1995). Let ✓(t) À ⌦ be an unknown time- varying parameter vector, ✓Ç be the estimate, and ⌦ œ Rp be a closed À 6q À 6qù3 where R is the auxiliary vector of the observer, N R is the Ç *1 6qù6q 6qù3 ball of known radius r⌦.TheprojectionalgorithmProj w, ✓ is given design matrix satisfying NM0 = L, G À R and L À R are *3 *3 *3 by design matrices. Here, q =2, Kmb1 = diag(6 ù 10 , 6 ù 10 , 8 ù 10 ), 5 5 9 *13 Ç Kmb2 = diag(2 ù 10 , 4 ù 10 , 2 ù 10 ), Q1 = Q2 =3ù 10 I12ù12,and w, ifp(✓) < 0 *10 T Q3 =6ù10 I12ù12. Furthermore, the design matrices G and L,desired )p(✓Ç) w, if p(✓Ç) g 0 and w f 0 vessel position and heading as well as initial conditions are identical h Ç Proj Ç = H )✓ I to the counterparts of our proposed DP control law in Section 4.1, w, ✓ n T (63) n (Ç) )p )p p ✓ Ç Ç respectively. n * )✓ )✓ otherwise w 2 w, The simulation results under the control law ⌧com in Case 1 and Case l )p n )✓Ç 2 of Section 4.1 are depicted using the dashed lines in Figs. 2(a)–2(c), n Ç 2* 2 2(e), 2(f) and Figs. 3(a)–3(c), 3(e), 3(f), respectively. Furthermore, the n✓ r⌦ Ù Ù where p(✓Ç)=j2 ,Ù" isÙ an arbitrary positive real. " +2"r⌦ Ù Ù performance indices are also summarized in Table 2.Itcanbeseenfrom Ò Ò Figs. 2(a)–2(c), 2(e), 2(f) and Table 2 that the DP model-based control If ✓Ç(0) À ⌦,thefollowingpropertiesfollowfromAppendixE law ⌧com exhibits some better control performance than our proposed in Krsti¢ et al. (1995): control law ⌧ in Case 1. This is because the DP model-based control law exploits the known parameters of the vessel dynamic model. However, it ✓Ç(t) f r⌦ + ", ≈t g 0 (64) can be seen from Figs. 3(a)–3(c), 3(e), 3(f) and Table 2 that ⌧com results Proj(w, ✓Ç) f w (65) T Ò Ò T in longer settling times and larger mean deviations in the position and * ✓É Proj(w, ✓Ç) f *✓É w, ✓É = ✓ * ✓Ç (66) heading of the vessel, which is enough to show our proposed DP control Ò Ò Ò Ò law has the adaptability and the robustness in the presence of model Lemma 1 (Barbalat’s Lemma (Gang, 1997)). For a vector signal '(t) À Rn, parameter perturbations and unknown time-varying disturbances, and ÿ 2 2 if î0 '(t) dt f c1 and suptg0 'Ü (t) f c2 for some constants c1 > 0 and is superior to the DP model-based control law. c2 > 0,thenlimtôÿ'(t)=0nù1. Ò Ò Ò Ò 5. Conclusion References In this paper, a new robust adaptive control scheme has been developed for the DP of ships based on the observer, the projection algorithm, Basturk, H. I. (2017). Cancellation of unmatched biased sinusoidal disturbances for and the vectorial backstepping. The disturbance compensation for the unknown LTI systems in the presence of state delay. Automatica, 76, 169–176. DP of ships has been first converted into an adaptive control problem by Basturk, H. I., & Krsti¢,M.(2013). Adaptive wave cancelation by acceleration feedback for ramp-connected air cushion-actuated surface effect ships. Automatica, 49(9), 2591– means of the parametric representations of ship motion mathematical 2602. model and environmental disturbances and the constructed observer. Basturk, H. I., & Krsti¢,M.(2014). State derivative feedback for adaptive cancellation of The adaptive laws based on the projection algorithm together with the unmatched disturbances in unknown strict-feedback LTI systems. Automatica, 50(10), observer provide the online estimates of the unknown time-varying 2539–2545. disturbances, where no a priori knowledge of ship dynamic model Basturk, H. 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L., Yang, Y., Hu, X., & Chen, H. Q. (2014). Control law design of dynamic positioning This work was supported by the National Natural Science Foundation for ship based on dynamic surface control. Journal of Traffic and Transportation of China [grant number 51079013]; the Program for Liaoning Excellent Engineering, 14(5), 36–50 (in Chinese). Du, J. L., Yang, Y., Wang, D. H., & Guo, C. (2013). Arobustadaptiveneuralnetworks Talents in University [grant number LR2015007]; the Technology Foun- controller for maritime dynamic positioning system. Neurocomputing, 110, 128–136. dation for Selected Overseas Chinese Scholar, the Ministry of Human Fossen, T. I. (2011). Handbook of marine craft hydrodynamics and motion control. Chichester, Resources and Social Security of the People’s Republic of China; and the UK: John Wiley & Sons, Inc. Fundamental Research Funds for the Central Universities [grant number Fossen, T. I., & Grøvlen, A. (1998). 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