Control Engineering Practice 76 (2018) 22–30 Contents lists available at ScienceDirect Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac 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 param- eters 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