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  • Research ArticleParameter Identifiability of Ship ManoeuvringModeling Using System Identification

    Weilin Luo1,2

    1School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China2Fujian Province Key Laboratory of Structural Performances in Ship and Marine Engineering, Fuzhou 350116, China

    Correspondence should be addressed to Weilin Luo; wlluofzu@gmail.com

    Received 9 June 2016; Revised 17 October 2016; Accepted 19 October 2016

    Academic Editor: Laurent Mevel

    Copyright 2016 Weilin Luo. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

    To improve the feasibility of system identification in the prediction of ship manoeuvrability, several measures are presented to dealwith the parameter identifiability in the parametric modeling of ship manoeuvring motion based on system identification. Drift ofnonlinear hydrodynamic coefficients is explained from the point of view of regression analysis. To diminish the multicollinearityin a complicated manoeuvring model, difference method and additional signal method are employed to reconstruct the samples.Moreover, the structure of manoeuvring model is simplified based on correlation analysis. Manoeuvring simulation is performedto demonstrate the validity of the measures proposed.

    1. Introduction

    Prediction of ship manoeuvrability at the ship design stageis an important method for evaluating the manoeuvringperformances of ships [1]. In general, two ways are availableto obtain the manoeuvrability parameters (e.g., advance,transfer, tactical diameter, and overshoots), which includesimulation-free and simulation-based methods. For thesimulation-free method, one can obtain the manoeuvrabilityparameters directly from database, full-scale trials, or free-runningmodel tests. For the simulation-basedmethod, eithersystem based manoeuvring simulation or computationalfluid dynamics (CFD) based manoeuvring simulation can beemployed [2]. No matter which kind of simulation is used,usually the ship manoeuvring model is indispensable, whichalso determines the accuracy of the prediction ofmanoeuvra-bility. Generally, there are two ways to obtain a ship manoeu-vringmodel.Themost usedway is determination of hydrody-namic derivatives (coefficients) in the mathematical modelof ship manoeuvring. The second way is where artificialintelligence techniques are employed to obtain an implicitmodel (or so-called a black-box model) that describes theinput-output mapping characteristics of ship dynamics.Examples are the uses of artificial neural networks (ANN)[3, 4] and support vector machines (SVM) [5, 6].

    To determine the hydrodynamic coefficients in the math-ematical model of shipmanoeuvring, several methods can beused, including database, empirical formula, captive modeltest, CFD calculation, and system identification (SI) com-binedwith free-runningmodel test or full-scale trials. Amongthem, the SI basedmethod provides an effective and practicalway to identify the hydrodynamic parameters in the shipmanoeuvring models. When combined with free-runningmodel tests, this method requires relatively low cost becauseit is easy to generate more manoeuvres after a first set offree-running model tests [2]. When combined with full-scaletrials, the so-called scale effects due to different Reynoldsnumbers between that of the ship model and full-scale can beavoided. During last decades, the SI based method has madegoodprogresses andmany techniques have beendeveloped inthe application to ship manoeuvring modeling. Examples areleast squares (LS) regression [7, 8], model reference method(MRM) [9, 10], extend Kalman filter (EKF) [11, 12], maximumlikelihood (ML) estimation [13, 14], recursive predictionerror (RPE) method [15, 16], frequency spectrum analysis(FSA) method [17, 18], particle swarm optimization (PSO)[19, 20], genetic algorithm (GA) [21, 22], and support vectorregression (SVR) [23, 24]. Generally, there have beenmany SIapplications to the parameter identification of ship manoeu-vring models.

    Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 8909170, 10 pageshttp://dx.doi.org/10.1155/2016/8909170

  • 2 Mathematical Problems in Engineering

    Like the other means to come to manoeuvring predic-tions, SI has its own advantages and disadvantages whenapplied to ship manoeuvring modeling. The main disadvan-tage is the problem of parameter identifiability. It can beexplained from two aspects. One kind of parameter identi-fiability is whether the parameters can be identified or not.When solving a state space equation with respect to the shipmanoeuvring motion, some parameters cannot be identifiedfor instance the added masses or acceleration derivatives, nomatterwhat input-output samples are provided andnomatterwhich SI technique is adopted. The other kind of parameteridentifiability is the effect of parameter drift. The identifiedcoefficients might be not physically correct but be mathe-matically correct [2], which implies that even if the identifiedparameters deviate from their true values, the manoeuvringmodel can predict the manoeuvring motion well, comparedwith the test results. However, this prediction ability is usuallyrestricted to the prediction of the manoeuvre which providesthe data samples for system identification. For example, amanoeuvring model for a ship based on the identification of10/10 zigzag manoeuvre can predict the same manoeuvrewell (i.e., 10/10 zigzag manoeuvre) but fails when predictingother manoeuvres, for instance, 20/20 zigzag manoeuvreor turning circle manoeuvres. The effect of parameter drifthappens more frequently and severely for a complicated shipmanoeuvring model in which a number of hydrodynamiccoefficients are contained, for instance, the Abkowitz model.

    SI application to the modeling of ship manoeuvring hadever been widely studied in the 1980s with the developmentof SI techniques for instance the applications of EKF,ML, andRPE. Correspondingly, the issues on parameter identifiabilitywere addressed and some measures had been proposed [11,25, 26]. In the 1990s, studies on the application of SI to the shipmanoeuvring had been paid relatively less attention to due toa new charming method (well known as the CFD method)and the limitations of test technology and SI techniques avail-able, including the system observability, the sensitivity to theinitial estimation of parameters andnoises, and the parameteridentifiability especially the effect of parameter drift. Withthe development ofmodernmeasurement equipment and theintroduction of advanced SI techniques, SI based modelingof ship manoeuvring received renewed attention during thepast decade. On one hand, some classical SI techniques havebeen continuously applied and developed, for instance, theLS method [8], the EKF method [12], the RPE method [16],and some hybridmethods [27, 28]. On the other hand, severalnovel and interesting SI methods have been proposed. Rep-resentative examples are the artificial intelligence techniquesincluding PSO [19, 20], GA [21, 22], SVM [23, 24], and ANN[2931]. However, it should be noted thatmuchwork (includ-ing the work on classical SI or the work on new SI) concernedlittle about the problem of parameter identifiability especiallyfor the effect of parameter drift or avoided dealing withthe problem by selecting a simple manoeuvring model (e.g.,the Nomoto model or the response model) in which theparameter drift is weak because few parameters are involved.

    In this paper, some issues on the parameter identifiabilityare summarized. Furthermore, several measures to deal withthe identifiability problem are presented especially for the

    diminishment of the parameter drift in the identificationof a complicated ship manoeuvring model, the Abkowitzmodel. Combined with free-runningmodel test results, SVMbased system identification is employed to obtain the hydro-dynamic coefficients. Based on the identified manoeuvringmodel, manoeuvring simulation is conducted and predictionresults are comparedwith the test results, which demonstratesthe validity of the proposed methods.

    2. Determination of Added Masses andAdded Moment of Inertia

    Three kinds of models are available for the prediction of shipmanoeuvring, including the Abkowitz model [32], the MMGmodel [33], and the response model [34]. Comparatively,the first two models are frequently used because they give amore comprehensive understanding of the shipmanoeuvringmotion. On the other hand, a number of hydrodynamiccoefficients are contained in themodels, whichmakes it chal-lenging to determine these coefficients accurately. From thepoint of view of system identification, one of the difficulties isthat not all coefficients can be identified.

    Taking a linear manoeuvring model as an example, thestate space equation can be written as

    [[[

    0 00 V 0 V

    ]]]

    [[[

    V

    ]]]

    = [[[

    0 00 V 00 V 0

    ]]]

    [[[

    V

    ]]]

    + [[[

    0

    ]]]

    ,(1)

    where is the surge speed, V is the sway speed, is the yawrate, is the rudder angle, is the mass of the ship, isthe moment of inertia of the ship, longitudinal coordinateof the ships center of gravity, 0 is the service speed of theship, et al. are the hydrodynamic derivatives amongwhichthe acceleration derivatives , V, , V are named asthe added masses, and is the added moment of inertia.Denote

    = [[[

    V

    ]]]

    ,

    = [[[

    0 00 V 0 V

    ]]]

    ,

    = [[[

    0 00 V 00 V 0

    ]]]

    ,

    = [[[

    0

    ]]]

    .

    (2)

  • Mathematical Problems in Engineering 3

    Table 1: Added masses and added moment of ine

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