Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 803548 11 pageshttpdxdoiorg1011552013803548

Research ArticleIdentification of a Surface Marine Vessel Using LS-SVM

David Moreno-Salinas1 Dictino Chaos1 Jesuacutes Manuel de la Cruz2 and Joaquiacuten Aranda1

1 Department of Computer Science and Automatic Control UNED Madrid 28040 Spain2Department of Computer Architecture and Automatic Control Complutense University of Madrid (UCM) Madrid 28040 Spain

Correspondence should be addressed to David Moreno-Salinas dmorenodiaunedes

Received 25 January 2013 Accepted 15 March 2013

Academic Editor Mamdouh M El Kady

Copyright copy 2013 David Moreno-Salinas et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The availability of adequate system models to reproduce as faithfully as possible the actual behaviour of the experimental systemsis of key importance In marine systems the changing environmental conditions and the complexity of the infrastructure neededto carry out experimental tests call for mathematical models for accurate simulations There exist a wide number of techniques todefinemathematical models from experimental data Support VectorMachines (SVMs) have shown a great performance in patternrecognition and classification research areas having an inherent potential ability for linear and nonlinear system identification Inthis paper this ability is demonstrated through the identification of the Nomoto second-order ship model with real experimentaldata obtained from a zig-zag manoeuvre made by a scale ship The mathematical model of the ship is identified using LeastSquares Support VectorMachines (LS-SVMs) for regression by analysing the rudder angle surge and sway speed and yaw rateThecoefficients of the Nomotomodel are obtained with a linear kernel functionThemodel obtained is validated through experimentaltests that illustrate the potential of SVM for system identification

1 Introduction

Nowadays the breakthroughs in computer science and themany research works on control engineering and auto-nomous vehicles call for the availability of adequate math-ematical models with which new designs and ideas can betested in simulation to predict the behaviour of the realsystems with high accuracy Furthermore due to the highcost of practical implementations and tests it is of higherimportance to have available tools and methods to computethese mathematical models for simulation purposes In thissense system identification is one of the highlights among theresearch topics in engineering and one of the most importantstages in control of autonomous vehicles In the present workwe will focus on the computation of a mathematical modelthat describes the dynamical behaviour of a surface marinevessel bymeans of experimental dataThismodel is computedwith a Support Vector Machines (SVMs) technique theLeast Squares Support Vector Machines (LS-SVMs) SVMhas shown a great performance in pattern recognition andclassification research areas and it has a potential ability for

linear and nonlinear system identification that will be shownthroughout this work

The literature on linear and nonlinear system identifi-cation is extensive and covers many areas of engineeringresearch For a short survey on some essential features in theidentification area and a classification of methods the readeris referred to [1] The state-space identification methods areone of the most common and intuitive techniques to deter-mine mathematical models in the engineering practice Thestate-space identification methods for linear time-invariant(LTI) systems are well established [2] In the nonlinearcase the input and output signals of the system can beused to estimate the state-space representation Thus it is ofpractical interest to be able to determine the description of anonlinear system from a finite number of input and outputmeasurements in which the use of Support Vector Machines(SVMs) can provide an interesting new method to solve thisproblem

In surface and underwater marine robotics the systemsand vehicles employed can have a high degree of complexityMoreover the changing environmental conditions and the

2 Journal of Applied Mathematics

complexity of the infrastructure needed to carry out exper-imental tests call for mathematical models for accuratesimulations System identification of marine vehicles starts inthe 70swith theworks of [3] where an adaptive autopilot withreference model was presented and [4] where parametriclinear identification techniques were used to define theguidance dynamics of a ship using the maximum likelihoodmethod We can find in the literature on maritime researchseveral mathematical models to describe the dynamics ofsurface vessels and underwater vehicles and algorithms tocompute these models see for example [5] where sev-eral parametric identification algorithms are used to designautopilots for different kinds of ships [6] in which thehydrodynamic characteristics of a ship are determined bya Kalman Filter (KF) or [7] where an extended Kalmanfilter (EKF) is used for the identification of a ship dynamicsfor dynamic positioning Moreover different operationalconditions thatmay affect the vessel and that provide differentmodels have been analysed see [8] where an overview ofthe motion dynamics topic of surface marine vessels andmodels is given The procedure to obtain these models isnot an easy and simple task it usually requires a lot of timeon practical tests and an important computational effortIn [9] the authors define a practical ship model and theidentification process for simulation guidance and controlfor small and low-cost vehicles The paper shows the lengthyprocedure to obtain a correct identification of the shipmodelFor some more interesting works the reader is referred to[10 11] and the references therein

Despite this in most cases and practical situations itis possible to employ a simple model to reproduce thedynamics of a vessel in standard operational conditions Forthis reason in this paper the Nomoto second-order modelis used [12] This approach provides a less precise model ofthe vehicle however this model shows very good results inthe experimental setup for standard operations asmentionedabove and it can be obtained with SVM techniques in afast manner with relatively few data as it will be shown inSection 4

Artificial-neural-networks-based techniques have beenemployed for system identification such asmultilayer percep-tron (MLP) see for example [13] We can find some worksthat employ neural networks to define the dynamics of a sur-face marine vehicle such as [14ndash16] or [17] These techniqueshave shown to be robust and effective in identification andcontrol although they also show some inherent drawbackssuch as the multiple local minima problem or overfittingto name but a few These problems can be avoided usingSVM since it is not based on the empirical error implementedin neural networks rather it is based on the structural riskminimization (SRM) so SVM provides a larger generalisa-tion performance An estimate of generalisation ability maybe obtained from a trained network without using test dataassuming the training data to be representative Vapnik-Chervonenkis (VC) theory is the basic idea behind SVMwhich defines ameasure of the capacity of a learningmachine[18] The idea is to map input data into a high-dimensionalfeature Hilbert space using a nonlinear mapping techniquethat is the kernel dot product trick [19] and to carry out

linear classification or regression in feature spaceThe Kernelfunctions replace a possibly very-high-dimensional Hilbertspace without explicitly increasing the feature space [20]SVM both for regression and classification has the ability tosimultaneously minimize the estimation error in the trainingdata (the empirical risk) and the model complexity (thestructural risk) [21] Moreover SVM can be designed to dealwith sparse data where we have many variables but few dataFurthermore the solution of SVM is globally optimal

In support vector regression (SVR) a nonlinear model isrepresented by an expansion in terms of nonlinear mappingsof the model input so that they define a feature space thatmay be of infinite dimension A convenient feature in SVR isthat the optimal model complexity is obtained as part of thesolution and does not have to be determined separately For asurvey and a tutorial on support vector regression the readeris referred to [22] and the references therein The goal is tofind a set of parameters for a given model which are chosento fit measured values of output and input data

An inadequate model identification may yield large pre-diction errors It is important to remark that even thoughthe problem of system identification is of great importanceand the use of SVM has experienced a great boost anddiversification in the last few years not many results areavailable on the topic of system identification with SVMfor regression yet Exceptions include a series of interestingresults such as the work of [23] where the authors make astudy of the possible use of SVM for system identificationin particular for parameter estimation in discrete-time linearmodels and model structure identification for nonlinearmodelsmore specificallyNonlinearAuto-RegressiveMovingAverage with eXogeneous Input (NARMAX) models Thepotential of SVM for system identification is remarked butsome significant issues to be addressed are shown In [24]an identification method based on SVR is proposed forlinear regressionmodels such as finite impulse response (FIR)and Auto-Regressive with eXogeneous input (ARX) modelsThe paper shows how the method is robust for simulateddata and it is even applied to actual data from rolling millshowing a good learning performance In the work of [25]the application of SVM to time seriesmodelling is consideredby means of simulated data from an autocatalytic reactorIn particular a state-space model is defined for the processwith reconstruction methods from nonlinear dynamics Asmentioned above the identification process in [25] is doneby means of simulation data as most of the papers thatdeal with identification using SVM In [26] a SVM withlinear-kernel-function-based nonparametric model identi-fication and dynamic matrix control (DMC) technique isintroduced The paper [27] shows how the kernel canonicalcorrelation analysis (KCCA) can be used to construct astate sequence of an unknown nonlinear dynamic systemfrom delay vectors of inputs and outputs This sequence isthen used together with input and output data to identifya nonlinear state-space model In [28] a design procedureof SVM-based model identification and control strategy isproposed The proposed design procedure and the SVM-based model identification are illustrated with the controlof a simulated continuous-stirred tank reactor In [29] SVR

Journal of Applied Mathematics 3

is applied to the identification of a nonlinear system thatcorresponds to a Wiener model that consists of a lineardynamic system with a static nonlinear block The staticnonlinear block is determined by SVR The model employedis function of the input only not depending on the measuredsystem outputs The identification procedure is tested withnumerical simulations

In this work a variation of SVM the Least Squares Sup-port Vector Machines (LS-SVMs) [30] is used for systemidentification with regression This technique allows a nicesimplification of the problem making it more tractable Astudy on this subject can be found in [31] where LS-SVMis used for nonlinear system identification for some simpleexamples of Nonlinear Auto-Regressive with eXogenousinput (NARX) input-output models There also exist someother representative examples that deal with identificationusing LS-SVM For example in [32] the identification ofMIMO Hammerstein ARX models based on LS-SVM isstudied The results are compared with two other techniques(the Hermite polynomial expansion and a RBF network)with numerical simulations showing an improvement of theperformance with the LS-SVM approach In [33] a newsupport vector kernel is defined the wavelet kernel thatsatisfies wavelet frames This kernel is able to approximatearbitrary functions improving the generalisation ability ofthe SVM Based on this kernel and on LS-SVM the LS-WSVM is constructed and it is applied to a nonlinear systemidentification problem showing better accuracy than a Gaus-sian kernel for the problem studied Finally in [34] the LS-SVM is combined with the multiresolution analysis (MRA)to provide a new improved algorithm the multiresolutionleast squares support vector machines (MLS-SVMs) Thisalgorithm has the same theoretical framework as MRAbut it has a better approximation ability and can graduallyapproximate the target function at different scales

With respect to the identification of marine vehicles byusing SVM we can find two interesting references [35] and[36] where an Abkowitz model for ship manoeuvring isidentified by using LS-SVM and 120598-SVM respectively Thepurpose of these works is to determine the hydrodynamiccoefficients of a Mariner-class vessel although the identifica-tion of the mathematical model is made with data obtainedfrom simulation and then tested only in simulation Theseworks do not deal with real data Furthermore it is importantto point out that most of the works on system identificationwith SVR deal with simulation data and numerical exampleswhere the use of SVR is not tested on an experimental setupas far as the authors know In striking contrast to what iscommonly reported in the literature in the paper at hand themodel identification is done with experimental data obtainedfrom a scale surface marine vessel and the mathematicalmodel obtained is validated with experimental tests carriedout with the scale ship showing how the model predicts withlarge accuracy the actual behaviour of the surface vessel Itis important to keep in mind for the experimental resultsobtained in this paper that the analytical properties of SVMcan be compromised in stochastic problems because the noisegenerates additional support vectors However if the noise

ratio is good and the amplitude is limited the SVM can solvethe problem as if it was deterministic [23]

The key contributions of the present paper are twofold (i)themathematicalmodel of a surfacemarine vessel is obtainedwith the LS-SVM technique from experimental data collectedfrom a 2020 degree zig-zag manoeuvre and (ii) in strikingcontrast to what is customary in the literature the predictionability of the mathematical model computed with LS-SVMregression is tested on an open air environment with thereal ship This allows us to explicitly address the connectionbetween the mathematical model identified and the ship sothat the model can be used to design control strategies to beimplemented on the real vehicle and to predict its behaviour

The document is organized as follows In Section 2 a briefintroduction on LS-SVM is givenTheNomoto second-ordermodel and the training input and output data to be usedare defined in Section 3 In Section 4 the Nomoto modelcomputed from the training data is well defined and itsbehaviour is tested with some turning manoeuvres Finallythe conclusions and a brief discussion of topics for furtherresearch are included in Section 5

2 Least Squares Support VectorMachines for Regression

In this section we briefly introduce LS-SVM for systemidentification As mentioned in Section 1 for both regressionand classification SVM has the ability to simultaneouslyminimize the estimation error in the training data (theempirical risk) and the model complexity (the structuralrisk) SVM can also be designed to deal with sparse datawhere we may have many variables but few data The inputdata are mapped into a high dimensional feature space usingnonlinear mapping techniques and then linear classificationor regression is carried out in feature space

The use of LS-SVM for regression is very similar to its usefor classification Following the notation and definitions in[22 30] consider a model in the primal weight space

119910 (119909) = 120596119879120593 (119909) + 119887 (1)

where 119909 isin R119899 is the input data 119910 isin R is the outputdata 119887 is a bias term 120596 is a matrix of weights and 120593(sdot)

R rarr R119899ℎ is the mapping to a high-dimensional spacewhere 119899

ℎcan be infinite The optimization problem in the

primal weight space for a given training set 119909119894 119910119894119873119904

119894=1 with

119873119904as the number of samples becomes

min120596119887119890

J (120596 119890) =

1

2

120596119879120596 + 120574

1

2

119873119904

sum

119894=1

1198902

119894(2)

subject to

119910119894= 120596119879120593 (119909119894) + 119887 + 119890

119894 (3)

where 119894 = 1 119873119904 119890119894are error variables and 120574 is the

regularisation parameter that determines the deviation tol-erated from the desired accuracy which must be positiveThe minimization of 120596119879120596 is closely related to the use of a

4 Journal of Applied Mathematics

weight decay term in the training of neural networks and thesecond term of the right-hand side of (2) controls the tradeoffbetween the empirical error and the model complexity Themain differences between the formulation of LS-SVM withthe standard SVM are the equality constraints (3) and thesquared error term of the second term in the right-hand sideof (2) implying a significant simplification of the problem

The above primal problem cannot be solved when 120596

becomes infinite dimensional Thus the Lagrangian must becomputed and the dual problem derived yielding

L (120596 119887 119890 120572) = J (120596 119890) minus

119873119904

sum

119894=1

120572119894120596119879120593 (119909119894) + 119887 + 119890

119894minus 119910119894

(4)

where 120572119894 with 119894 = 1 119873

119904 are the Lagrange multipliers

The optimality conditions are defined by computing thederivatives of (4) with respect to 120596 119887 119890

119894 and 120572

119894

120597L

120597120596

= 0 997888rarr 120596 =

119873119904

sum

119894=1

120572119894120593 (119909119894)

120597L

120597119887

= 0 997888rarr

119873119904

sum

119894=1

120572119894= 0

120597L

120597119890119894

= 0 997888rarr 120572119894= 120574119890119894

120597L

120597120572119894

= 0 997888rarr 120596119879120593 (119909119894) + 119887 + 119890

119894minus 119910119894= 0

(5)

for 119894 = 1 119873119904 After elimination of variables 120596 and 119890

from the above equations and applying the kernel trickwhichallows us to work in large-dimensional feature spaces withoutexplicit computations on them [19] the LS-SVM model forfunction estimation yields

119910 (119909) =

119873119904

sum

119894=1

120572119894119870(119909 119909

119894) + 119887 (6)

where 119870(sdot sdot) is the kernel function that represents an innerproduct between its operands This kernel must be a positivedefinite kernel and must satisfy the Mercer condition [37]

3 Identification of a NomotorsquosSecond-Order Model

In marine systems as in a multitude of engineering areasan adequate mathematical model is essential to simulate andpredict the behaviour of a real system with the best possibleaccuracy If reliable mathematical models are available andthey predict the behaviour of the real systems accuratelythen control actions can be planned and tested in simulationavoiding the costly in time and money practical tests

A reliable mathematical ship model like the Abkowitzmodel [38] requires the identification of a multitude ofhydrodynamic parameters The identification task for this

large number of variables can be hard and may take a longtimeMoreover different tests for the different hydrodynamicparameters must be carried out [9 10]

For the above reason simpler vehicle models are com-monly used so that although without modelling all thedynamic features of the vehicle each of these models repro-duces the real behaviour of ships with large accuracy in mostof practical scenarios For the problem at hand we considera constant surge speed and we determine a mathematicalmodel that defines the ship steering equations Among thesekinds of models we can find the Nomoto models [12] thatare an alternative representation of the model of Davidsonand Schiff [39] We now briefly comment these two models

Given the linear steering dynamics

119898( + 1199060119903 + 119909119866119903) = 119884

119868119911119903 + 119898119909

119866( + 119906

0119903) = 119873

(7)

the hydrodynamic force and moment can be modelled as in[39]

119884 = 119884 + 119884 119903

119903 + 119884120592120592 + 119884119903119903 + 119884120575120575

119873 = 119873 + 119873 119903

119903 + 119873120592120592 + 119873

119903119903 + 119873120575120575

(8)

where 119884 is sway force 119873 is moment (rotation about the 119911-axis with the 119911-axis pointing downwards as is customary inmarine systems) 119903 is yaw rate 120592 is sway velocity 119906

0is nominal

surge velocity 120575 is the rudder angle 119868119911is moment of inertia

about the 119911-axis 119898 is mass 119909119866is the 119909-axis coordinate of

the centre of gravity and 119884 119884 119903

119884120592 119884119903 119884120575 119873 119873 119903

119873120592 119873119903

and 119873120575are added inertia hydrodynamic coefficients for

more details the reader is referred to [10] See Figure 1 fora definition of the reference frames used in marine systems

Following the procedure of [10] (8) can be rewritten as

M] + N] = b120575 (9)

where ] = [120592 119903]119879 is the state vector and

M = [

119898 minus 119884

119898119909119866minus 119884 119903

119898119909119866minus 119873

119868119911minus 119873 119903

]

N = [

minus119884120592

1198981199060minus 119884119903

minus1198731205921198981199091198661199060minus 119873119903

]

b = [

119884120575

119873120575

]

(10)

From (9) the alternative representation of the second-order model of Nomoto [12] can be defined This modelprovides the transfer function that relates the yaw rate 119903 tothe rudder angle 120575 according to

119903

120575

(119904) =

119870 (1 + 1198793119904)

(1 + 1198791119904) (1 + 119879

2119904)

(11)

or if an integrator (a pole) is included the transfer functionrelates the yaw angle 120595 to the rudder angle 120575 that is

120595

120575

(119904) =

119870 (1 + 1198793119904)

119904 (1 + 1198791119904) (1 + 119879

2119904)

(12)

Journal of Applied Mathematics 5

119874

119874119884

119884

119873

119883

119883

119906119900

119906119900

119910119910

119900

119900

119909

119909

119911

120595

120592

120592

Figure 1 Body (O) and earth-fixed (119900) reference frames forces (119883 119884) moment (119873) and linear speeds (1199060 120592) used in marine systems for

3 degrees of freedom (3DOF) models

where 1198791 1198792 and 119879

3are time constants and 119870 is gain Now

rewrite (11) in the time domain

11987911198792120595(3)

+ (1198791+ 1198792) + = 119870 (120575 + 119879

3120575) (13)

where = 119903 and 120595(3) represents the third derivative of the

yaw angle with respect to time The parameters of (11) arerelated to the hydrodynamic derivatives as follows

11987911198792=

det (M)

det (N)

1198791+ 1198792=

1198991111989822+ 1198992211989811minus 1198991211989821minus 1198992111989812

det (N)

119870 =

119899211198871minus 119899111198872

det (N)

1198701198793=

119898211198871minus 119898111198872

det (N)

(14)

where 119898119894119895 119899119894119895 and 119887

119894 with 119894 = 1 2 and 119895 = 1 2 are the

elements of matricesM N and bIn addition we can express the sway velocity in a similar

manner

120592

120575

(119904) =

119870120592(1 + 119879

120592119904)

(1 + 1198791119904) (1 + 119879

2119904)

(15)

where 119870] and 119879] are the gain and time constants to describethe sway velocity In the time domain (15) yields

11987911198792 + (119879

1+ 1198792) + 120592 = 119870

120592(120575 + 119879

120592120575) (16)

The main advantage of the Nomoto second-order modelis its simplicity and that the hydrodynamic derivatives donot have to be computed explicitly it is possible to definethe parameters of the Nomoto model directly from theexperimental data Therefore we can compute in a fast and

simplemanner a shipmodel for control purposes because it isnot the aim of this work to know explicitly the hydrodynamicderivatives

To proceed with the reconstruction of the ship manoeu-vring model the continuous equations of motion (13) and(16) are discretized because the experimental data are takenwith a given regular interval of time The discretizationis done by the Euler stepping method so that the aboveequations can be rewritten as

11987911198792

Δ119903 (119896 + 1) minus 2Δ119903 (119896) + Δ119903 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

ℎ

minus Δ119903 (119896 minus 1)

+ 119870Δ120575 (119896 minus 1) + 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

11987911198792

Δ120592 (119896 + 1) minus 2Δ120592 (119896) + Δ120592 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

ℎ

minus Δ120592 (119896 minus 1)

+ 119870120592Δ120575 (119896 minus 1) + 119870

120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

(17)

where ℎ is the time between two consecutive measurementstaken from the IMU (inertial measurement unit) on boardthe ship and 119896 minus 1 119896 and 119896 + 1 are the indices of threesuccessive sampling times Now it is possible to apply theLS-SVM regression to obtain the parameters of the modelof Nomoto by constructing the training data following the

6 Journal of Applied Mathematics

equation structure of (17) Thus the discrete motion equa-tions given by (17) can be rewritten as

Δ119903 (119896 + 1) = 2Δ119903 (119896) minus Δ119903 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

11987911198792

ℎ

minus

Δ119903 (119896 minus 1)

11987911198792

ℎ2+

119870Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

Δ120592 (119896 + 1) = 2Δ120592 (119896) minus Δ120592 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

11987911198792

ℎ

minus

Δ120592 (119896 minus 1)

11987911198792

ℎ2+

119870120592Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 119870120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

(18)

and in a simplified manner

Δ119903 (119896 + 1) = 119860X119903

Δ120592 (119896 + 1) = 119861X120592

(19)

where Δ119903(119896 + 1) = 119910119895119903and Δ120592(119896 + 1) = 119910

119895120592are the output

training data and X119903= 119909119895119903

and X120592

= 119909119895120592

are the inputtraining data given by

X119903= [Δ119903 (119896) Δ119903 (119896 minus 1) minusΔ119903 (119896) + Δ119903 (119896 minus 1)

Δ119903 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

X120592= [Δ120592 (119896) Δ120592 (119896 minus 1) minusΔ120592 (119896) + Δ120592 (119896 minus 1)

Δ120592 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

(20)

for 119895 = 1 119873119904 and with

119860 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870

11987911198792

ℎ1198701198793

11987911198792

]

119861 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870120592

11987911198792

ℎ119870120592119879120592

11987911198792

]

(21)

The vectors (21) are now computed from LS-SVM regres-sion It is important to point out that the vectors (21) areseparately identified that is 119860 is defined first and after that119861 is defined so that 119879

1and 119879

2are already known for the

computation of 119861 The structure of the mathematical modelis known in advance so we can use a linear kernel119870(119909

119894 119909119895) =

(119909119894sdot 119909119895) The result of applying the LS-SVM technique for the

training data gives us

119910119895120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119895120585+ 119887120585

(22)

for 120585 = 119903 120592 and 119895 = 1 119873119904 Comparing the above equation

with (19) after the training process we have

119860 =

119873119904

sum

119894=1

120572119894119903119909119894119903

119861 =

119873119904

sum

119894=1

120572119894120592119909119894120592

(23)

if the object function has been well approximated by the LS-SVM algorithm where the bias term 119887

120585must be equal to or

approximately 0 Once the support vectors are defined by theabove process the parameters of the Nomoto model can beobtained immediately from (21)

4 Experimental Results

The vehicle used for the experimental tests is a scale modelof an operational vessel in a 11695 scale see Figure 2 Thedimensions of the vessel and the scale model are shown inTable 1

The scale ship model henceforth referred to as the shipis equipped with an electric motor to control the propellerturning speed and a servo to control the rudder angleThe desired rudder angle and surge speed are commandedthrough aWi-Fi connection between the ship and the controlstation at land

The training data used for system identification areobtained from a 2020 degree zig-zag manoeuvre with asample time of 02 seconds and a nominal surge speed of1ms The zig-zag manoeuvre was executed for 100 secondsso a set of 500 samples is trained In Figure 3 the commandedrudder angle (dashed line) and the yaw angle (solid line)made by the ship during the 2020 degree zig-zag manoeuvreare shown The training data are constructed with the com-manded control signal (rudder angle) and the data measuredfrom the IMU on board the ship (yaw rate and sway velocity)From the input and output data and with (19) the LS-SVMalgorithm is now trained to determine the vectors 119860 and 119861

defined in (21)Using the data shown in Figure 3 with the regularisation

parameter 120574 = 104 the Nomoto model parameters obtained

from the identification process with LS-SVM regression are1198791= 11918 119879

2= 14442 119879

3= 15711 119870 = 03619 119879

120592=

03457 and119870120592= 03813 Then the Nomoto model yields

119903

120575

(119904) =

03619 (1 + 15711119904)

(1 + 11918119904) (1 + 14442119904)

120592

120575

(119904) =

03813 (1 + 03457119904)

(1 + 11918119904) (1 + 14442119904)

(24)

Figure 4 shows the yaw rate obtained from the IMU onboard the ship (solid line) together with the yaw rate obtainedin simulation (dotted line) with the Nomoto second-ordermodel previously defined and with the same commandedinput signal than the ship It can be noticed how the behaviourof the model fits the real behaviour of the ship in a nice

Journal of Applied Mathematics 7

Figure 2 Scale ship model used in the experimental tests

Table 1Main parameters and dimensions of the vessel and the scaleship model

Parameter Vessel Scale ship modelLength betweenperpendiculars (119871pp)

74400m 4389m

Maximum beam (119861) 14200m 0838mMean depth to the topdeck (119867) 9050m 0534m

Design draught (119879119898) 6300m 0372m

manner for the training data used in the identificationprocess

In the same way in Figure 5 the sway velocity measuredfrom the IMU on board the ship is shown together withthe sway velocity obtained from the Nomoto model Againthe results are very similar showing a good behaviour of themodel for the training data Despite the above the model forthe sway velocity is less precise than themodel for the yaw ratebecause the sway velocity cannot be directly controlled dueto the fact that the ship studied is an underactuated vehicleMoreover the environmental conditions like currents andwinds particularly affect the sway velocity In any case themodel is satisfactory furthermore the yaw rate is the com-ponent that defines in a higher degree the actual behaviour ofthe ship

Once the Nomoto model is well defined and it fits wellwith the training data as shown in Figures 4 and 5 themodelmust be tested with different data and manoeuvres to verifythe generalisation performance of the model obtained In theexample shown below two consecutive turning manoeuvres(evolution circles) are performed with the simulation modeland the ship for the same input signal The input signalcommanded is the rudder angle that takes the value of 10degrees during 255 seconds then a rudder angle of 0 degreesis commanded for 70 seconds and finally a rudder angleof minus10 degrees is commanded during 275 seconds with anominal surge speed of 1ms

In Figures 6 and 7 the yaw rate and sway velocity forthe ship (solid line) and the Nomoto model (dotted line) areshown respectively It is clear that the predicted behaviouris very similar to the actual one of the ship so that LS-SVM

0 20 40 60 80 100minus08

minus06

minus04

minus02

0

02

04

06

08

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)

Rudder angleYaw angle

Figure 3 2020 degree zig-zag manoeuvre Yaw angle (solid line)and rudder angle (dashed line)

0 20 40 60 80 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 4 Yaw rate obtained in the zig-zag manoeuvre with the ship(solid line) and in simulation (dotted line)

provides a good mathematical model to test different controlstrategies to be applied on the ship

We can notice in Figure 6 how the real system deviatesslightly in turnings with negative rudder angles because theactual yaw rate behaviour is not exactly symmetric it isslightly larger for negative rudder angles Despite this smalldeviation on the supposed symmetrical behaviour of the shipthe result obtained from the simulated model is very similarto the actual one and the difference is not significant Thisdeviation could be produced by environmental conditionslike currents or winds or by structural characteristics ofthe ship like the trimming that can make the ship to nothave exactly a symmetrical behaviour while this symmetrical

8 Journal of Applied Mathematics

0 20 40 60 80 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 5 Sway velocitymeasured in the zig-zagmanoeuvrewith theship (solid line) and in simulation (dotted line)

1000 200 300 400 500 600minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 6 Yaw rate obtained in two consecutive turningmanoeuvreswith the ship (solid line) and in simulation (dotted line)

behaviour is implicit on the Nomoto models This possiblesource of errors in the fidelity of the mathematical modelshould be further studied to be able to define more complexmathematical models although in our practical case as canbe deduced from Figures 6 and 7 these errors are negligible

In Figures 8 and 9 the error between the real and thepredicted yaw rate and sway velocity are shown respectivelyWe can notice from the yaw rate in Figure 8 how the erroralthough very small is larger for negative rudder anglesas mentioned above Despite the commented deviation theerror has a small magnitude and its average value is very closeto 0 providing a more than satisfactory prediction of the real

0 100 200 300 400 500 600

minus01

minus005

0

005

01

015

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 7 Sway velocity obtained in two consecutive turningmanoeuvres with the ship (solid line) and in simulation (dottedline)

0 100 200 300 400 500 600minus002

minus001

0

001

Time (s)

Erro

r in

yaw

rate

estim

atio

n (r

ads

)

Figure 8 Error in the identification of the yaw rate that isdifference between the yaw rate obtained in simulation and in thereal setup

dynamical behaviour of the ship The same comments applyto Figure 9 and the sway velocity instead of the yaw rate

Therefore it is clear that the mathematical model definedfor a surface marine vehicle with LS-SVMprovides a satisfac-tory result which predicts with large accuracy the dynamicsof the experimental system

Now it is interesting to compare the result of LS-SVMidentification with the result that can be obtained by standardidentification methods The latter model is computed usingthe identification toolbox of MATLAB called Ident for the

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

complexity of the infrastructure needed to carry out exper-imental tests call for mathematical models for accuratesimulations System identification of marine vehicles starts inthe 70swith theworks of [3] where an adaptive autopilot withreference model was presented and [4] where parametriclinear identification techniques were used to define theguidance dynamics of a ship using the maximum likelihoodmethod We can find in the literature on maritime researchseveral mathematical models to describe the dynamics ofsurface vessels and underwater vehicles and algorithms tocompute these models see for example [5] where sev-eral parametric identification algorithms are used to designautopilots for different kinds of ships [6] in which thehydrodynamic characteristics of a ship are determined bya Kalman Filter (KF) or [7] where an extended Kalmanfilter (EKF) is used for the identification of a ship dynamicsfor dynamic positioning Moreover different operationalconditions thatmay affect the vessel and that provide differentmodels have been analysed see [8] where an overview ofthe motion dynamics topic of surface marine vessels andmodels is given The procedure to obtain these models isnot an easy and simple task it usually requires a lot of timeon practical tests and an important computational effortIn [9] the authors define a practical ship model and theidentification process for simulation guidance and controlfor small and low-cost vehicles The paper shows the lengthyprocedure to obtain a correct identification of the shipmodelFor some more interesting works the reader is referred to[10 11] and the references therein

Despite this in most cases and practical situations itis possible to employ a simple model to reproduce thedynamics of a vessel in standard operational conditions Forthis reason in this paper the Nomoto second-order modelis used [12] This approach provides a less precise model ofthe vehicle however this model shows very good results inthe experimental setup for standard operations asmentionedabove and it can be obtained with SVM techniques in afast manner with relatively few data as it will be shown inSection 4

Artificial-neural-networks-based techniques have beenemployed for system identification such asmultilayer percep-tron (MLP) see for example [13] We can find some worksthat employ neural networks to define the dynamics of a sur-face marine vehicle such as [14ndash16] or [17] These techniqueshave shown to be robust and effective in identification andcontrol although they also show some inherent drawbackssuch as the multiple local minima problem or overfittingto name but a few These problems can be avoided usingSVM since it is not based on the empirical error implementedin neural networks rather it is based on the structural riskminimization (SRM) so SVM provides a larger generalisa-tion performance An estimate of generalisation ability maybe obtained from a trained network without using test dataassuming the training data to be representative Vapnik-Chervonenkis (VC) theory is the basic idea behind SVMwhich defines ameasure of the capacity of a learningmachine[18] The idea is to map input data into a high-dimensionalfeature Hilbert space using a nonlinear mapping techniquethat is the kernel dot product trick [19] and to carry out

linear classification or regression in feature spaceThe Kernelfunctions replace a possibly very-high-dimensional Hilbertspace without explicitly increasing the feature space [20]SVM both for regression and classification has the ability tosimultaneously minimize the estimation error in the trainingdata (the empirical risk) and the model complexity (thestructural risk) [21] Moreover SVM can be designed to dealwith sparse data where we have many variables but few dataFurthermore the solution of SVM is globally optimal

In support vector regression (SVR) a nonlinear model isrepresented by an expansion in terms of nonlinear mappingsof the model input so that they define a feature space thatmay be of infinite dimension A convenient feature in SVR isthat the optimal model complexity is obtained as part of thesolution and does not have to be determined separately For asurvey and a tutorial on support vector regression the readeris referred to [22] and the references therein The goal is tofind a set of parameters for a given model which are chosento fit measured values of output and input data

An inadequate model identification may yield large pre-diction errors It is important to remark that even thoughthe problem of system identification is of great importanceand the use of SVM has experienced a great boost anddiversification in the last few years not many results areavailable on the topic of system identification with SVMfor regression yet Exceptions include a series of interestingresults such as the work of [23] where the authors make astudy of the possible use of SVM for system identificationin particular for parameter estimation in discrete-time linearmodels and model structure identification for nonlinearmodelsmore specificallyNonlinearAuto-RegressiveMovingAverage with eXogeneous Input (NARMAX) models Thepotential of SVM for system identification is remarked butsome significant issues to be addressed are shown In [24]an identification method based on SVR is proposed forlinear regressionmodels such as finite impulse response (FIR)and Auto-Regressive with eXogeneous input (ARX) modelsThe paper shows how the method is robust for simulateddata and it is even applied to actual data from rolling millshowing a good learning performance In the work of [25]the application of SVM to time seriesmodelling is consideredby means of simulated data from an autocatalytic reactorIn particular a state-space model is defined for the processwith reconstruction methods from nonlinear dynamics Asmentioned above the identification process in [25] is doneby means of simulation data as most of the papers thatdeal with identification using SVM In [26] a SVM withlinear-kernel-function-based nonparametric model identi-fication and dynamic matrix control (DMC) technique isintroduced The paper [27] shows how the kernel canonicalcorrelation analysis (KCCA) can be used to construct astate sequence of an unknown nonlinear dynamic systemfrom delay vectors of inputs and outputs This sequence isthen used together with input and output data to identifya nonlinear state-space model In [28] a design procedureof SVM-based model identification and control strategy isproposed The proposed design procedure and the SVM-based model identification are illustrated with the controlof a simulated continuous-stirred tank reactor In [29] SVR

Journal of Applied Mathematics 3

is applied to the identification of a nonlinear system thatcorresponds to a Wiener model that consists of a lineardynamic system with a static nonlinear block The staticnonlinear block is determined by SVR The model employedis function of the input only not depending on the measuredsystem outputs The identification procedure is tested withnumerical simulations

In this work a variation of SVM the Least Squares Sup-port Vector Machines (LS-SVMs) [30] is used for systemidentification with regression This technique allows a nicesimplification of the problem making it more tractable Astudy on this subject can be found in [31] where LS-SVMis used for nonlinear system identification for some simpleexamples of Nonlinear Auto-Regressive with eXogenousinput (NARX) input-output models There also exist someother representative examples that deal with identificationusing LS-SVM For example in [32] the identification ofMIMO Hammerstein ARX models based on LS-SVM isstudied The results are compared with two other techniques(the Hermite polynomial expansion and a RBF network)with numerical simulations showing an improvement of theperformance with the LS-SVM approach In [33] a newsupport vector kernel is defined the wavelet kernel thatsatisfies wavelet frames This kernel is able to approximatearbitrary functions improving the generalisation ability ofthe SVM Based on this kernel and on LS-SVM the LS-WSVM is constructed and it is applied to a nonlinear systemidentification problem showing better accuracy than a Gaus-sian kernel for the problem studied Finally in [34] the LS-SVM is combined with the multiresolution analysis (MRA)to provide a new improved algorithm the multiresolutionleast squares support vector machines (MLS-SVMs) Thisalgorithm has the same theoretical framework as MRAbut it has a better approximation ability and can graduallyapproximate the target function at different scales

With respect to the identification of marine vehicles byusing SVM we can find two interesting references [35] and[36] where an Abkowitz model for ship manoeuvring isidentified by using LS-SVM and 120598-SVM respectively Thepurpose of these works is to determine the hydrodynamiccoefficients of a Mariner-class vessel although the identifica-tion of the mathematical model is made with data obtainedfrom simulation and then tested only in simulation Theseworks do not deal with real data Furthermore it is importantto point out that most of the works on system identificationwith SVR deal with simulation data and numerical exampleswhere the use of SVR is not tested on an experimental setupas far as the authors know In striking contrast to what iscommonly reported in the literature in the paper at hand themodel identification is done with experimental data obtainedfrom a scale surface marine vessel and the mathematicalmodel obtained is validated with experimental tests carriedout with the scale ship showing how the model predicts withlarge accuracy the actual behaviour of the surface vessel Itis important to keep in mind for the experimental resultsobtained in this paper that the analytical properties of SVMcan be compromised in stochastic problems because the noisegenerates additional support vectors However if the noise

ratio is good and the amplitude is limited the SVM can solvethe problem as if it was deterministic [23]

The key contributions of the present paper are twofold (i)themathematicalmodel of a surfacemarine vessel is obtainedwith the LS-SVM technique from experimental data collectedfrom a 2020 degree zig-zag manoeuvre and (ii) in strikingcontrast to what is customary in the literature the predictionability of the mathematical model computed with LS-SVMregression is tested on an open air environment with thereal ship This allows us to explicitly address the connectionbetween the mathematical model identified and the ship sothat the model can be used to design control strategies to beimplemented on the real vehicle and to predict its behaviour

The document is organized as follows In Section 2 a briefintroduction on LS-SVM is givenTheNomoto second-ordermodel and the training input and output data to be usedare defined in Section 3 In Section 4 the Nomoto modelcomputed from the training data is well defined and itsbehaviour is tested with some turning manoeuvres Finallythe conclusions and a brief discussion of topics for furtherresearch are included in Section 5

2 Least Squares Support VectorMachines for Regression

In this section we briefly introduce LS-SVM for systemidentification As mentioned in Section 1 for both regressionand classification SVM has the ability to simultaneouslyminimize the estimation error in the training data (theempirical risk) and the model complexity (the structuralrisk) SVM can also be designed to deal with sparse datawhere we may have many variables but few data The inputdata are mapped into a high dimensional feature space usingnonlinear mapping techniques and then linear classificationor regression is carried out in feature space

The use of LS-SVM for regression is very similar to its usefor classification Following the notation and definitions in[22 30] consider a model in the primal weight space

119910 (119909) = 120596119879120593 (119909) + 119887 (1)

where 119909 isin R119899 is the input data 119910 isin R is the outputdata 119887 is a bias term 120596 is a matrix of weights and 120593(sdot)

R rarr R119899ℎ is the mapping to a high-dimensional spacewhere 119899

ℎcan be infinite The optimization problem in the

primal weight space for a given training set 119909119894 119910119894119873119904

119894=1 with

119873119904as the number of samples becomes

min120596119887119890

J (120596 119890) =

1

2

120596119879120596 + 120574

1

2

119873119904

sum

119894=1

1198902

119894(2)

subject to

119910119894= 120596119879120593 (119909119894) + 119887 + 119890

119894 (3)

where 119894 = 1 119873119904 119890119894are error variables and 120574 is the

regularisation parameter that determines the deviation tol-erated from the desired accuracy which must be positiveThe minimization of 120596119879120596 is closely related to the use of a

4 Journal of Applied Mathematics

weight decay term in the training of neural networks and thesecond term of the right-hand side of (2) controls the tradeoffbetween the empirical error and the model complexity Themain differences between the formulation of LS-SVM withthe standard SVM are the equality constraints (3) and thesquared error term of the second term in the right-hand sideof (2) implying a significant simplification of the problem

The above primal problem cannot be solved when 120596

becomes infinite dimensional Thus the Lagrangian must becomputed and the dual problem derived yielding

L (120596 119887 119890 120572) = J (120596 119890) minus

119873119904

sum

119894=1

120572119894120596119879120593 (119909119894) + 119887 + 119890

119894minus 119910119894

(4)

where 120572119894 with 119894 = 1 119873

119904 are the Lagrange multipliers

The optimality conditions are defined by computing thederivatives of (4) with respect to 120596 119887 119890

119894 and 120572

119894

120597L

120597120596

= 0 997888rarr 120596 =

119873119904

sum

119894=1

120572119894120593 (119909119894)

120597L

120597119887

= 0 997888rarr

119873119904

sum

119894=1

120572119894= 0

120597L

120597119890119894

= 0 997888rarr 120572119894= 120574119890119894

120597L

120597120572119894

= 0 997888rarr 120596119879120593 (119909119894) + 119887 + 119890

119894minus 119910119894= 0

(5)

for 119894 = 1 119873119904 After elimination of variables 120596 and 119890

from the above equations and applying the kernel trickwhichallows us to work in large-dimensional feature spaces withoutexplicit computations on them [19] the LS-SVM model forfunction estimation yields

119910 (119909) =

119873119904

sum

119894=1

120572119894119870(119909 119909

119894) + 119887 (6)

where 119870(sdot sdot) is the kernel function that represents an innerproduct between its operands This kernel must be a positivedefinite kernel and must satisfy the Mercer condition [37]

3 Identification of a NomotorsquosSecond-Order Model

In marine systems as in a multitude of engineering areasan adequate mathematical model is essential to simulate andpredict the behaviour of a real system with the best possibleaccuracy If reliable mathematical models are available andthey predict the behaviour of the real systems accuratelythen control actions can be planned and tested in simulationavoiding the costly in time and money practical tests

A reliable mathematical ship model like the Abkowitzmodel [38] requires the identification of a multitude ofhydrodynamic parameters The identification task for this

large number of variables can be hard and may take a longtimeMoreover different tests for the different hydrodynamicparameters must be carried out [9 10]

For the above reason simpler vehicle models are com-monly used so that although without modelling all thedynamic features of the vehicle each of these models repro-duces the real behaviour of ships with large accuracy in mostof practical scenarios For the problem at hand we considera constant surge speed and we determine a mathematicalmodel that defines the ship steering equations Among thesekinds of models we can find the Nomoto models [12] thatare an alternative representation of the model of Davidsonand Schiff [39] We now briefly comment these two models

Given the linear steering dynamics

119898( + 1199060119903 + 119909119866119903) = 119884

119868119911119903 + 119898119909

119866( + 119906

0119903) = 119873

(7)

the hydrodynamic force and moment can be modelled as in[39]

119884 = 119884 + 119884 119903

119903 + 119884120592120592 + 119884119903119903 + 119884120575120575

119873 = 119873 + 119873 119903

119903 + 119873120592120592 + 119873

119903119903 + 119873120575120575

(8)

where 119884 is sway force 119873 is moment (rotation about the 119911-axis with the 119911-axis pointing downwards as is customary inmarine systems) 119903 is yaw rate 120592 is sway velocity 119906

0is nominal

surge velocity 120575 is the rudder angle 119868119911is moment of inertia

about the 119911-axis 119898 is mass 119909119866is the 119909-axis coordinate of

the centre of gravity and 119884 119884 119903

119884120592 119884119903 119884120575 119873 119873 119903

119873120592 119873119903

and 119873120575are added inertia hydrodynamic coefficients for

more details the reader is referred to [10] See Figure 1 fora definition of the reference frames used in marine systems

Following the procedure of [10] (8) can be rewritten as

M] + N] = b120575 (9)

where ] = [120592 119903]119879 is the state vector and

M = [

119898 minus 119884

119898119909119866minus 119884 119903

119898119909119866minus 119873

119868119911minus 119873 119903

]

N = [

minus119884120592

1198981199060minus 119884119903

minus1198731205921198981199091198661199060minus 119873119903

]

b = [

119884120575

119873120575

]

(10)

From (9) the alternative representation of the second-order model of Nomoto [12] can be defined This modelprovides the transfer function that relates the yaw rate 119903 tothe rudder angle 120575 according to

119903

120575

(119904) =

119870 (1 + 1198793119904)

(1 + 1198791119904) (1 + 119879

2119904)

(11)

or if an integrator (a pole) is included the transfer functionrelates the yaw angle 120595 to the rudder angle 120575 that is

120595

120575

(119904) =

119870 (1 + 1198793119904)

119904 (1 + 1198791119904) (1 + 119879

2119904)

(12)

Journal of Applied Mathematics 5

119874

119874119884

119884

119873

119883

119883

119906119900

119906119900

119910119910

119900

119900

119909

119909

119911

120595

120592

120592

Figure 1 Body (O) and earth-fixed (119900) reference frames forces (119883 119884) moment (119873) and linear speeds (1199060 120592) used in marine systems for

3 degrees of freedom (3DOF) models

where 1198791 1198792 and 119879

3are time constants and 119870 is gain Now

rewrite (11) in the time domain

11987911198792120595(3)

+ (1198791+ 1198792) + = 119870 (120575 + 119879

3120575) (13)

where = 119903 and 120595(3) represents the third derivative of the

yaw angle with respect to time The parameters of (11) arerelated to the hydrodynamic derivatives as follows

11987911198792=

det (M)

det (N)

1198791+ 1198792=

1198991111989822+ 1198992211989811minus 1198991211989821minus 1198992111989812

det (N)

119870 =

119899211198871minus 119899111198872

det (N)

1198701198793=

119898211198871minus 119898111198872

det (N)

(14)

where 119898119894119895 119899119894119895 and 119887

119894 with 119894 = 1 2 and 119895 = 1 2 are the

elements of matricesM N and bIn addition we can express the sway velocity in a similar

manner

120592

120575

(119904) =

119870120592(1 + 119879

120592119904)

(1 + 1198791119904) (1 + 119879

2119904)

(15)

where 119870] and 119879] are the gain and time constants to describethe sway velocity In the time domain (15) yields

11987911198792 + (119879

1+ 1198792) + 120592 = 119870

120592(120575 + 119879

120592120575) (16)

The main advantage of the Nomoto second-order modelis its simplicity and that the hydrodynamic derivatives donot have to be computed explicitly it is possible to definethe parameters of the Nomoto model directly from theexperimental data Therefore we can compute in a fast and

simplemanner a shipmodel for control purposes because it isnot the aim of this work to know explicitly the hydrodynamicderivatives

To proceed with the reconstruction of the ship manoeu-vring model the continuous equations of motion (13) and(16) are discretized because the experimental data are takenwith a given regular interval of time The discretizationis done by the Euler stepping method so that the aboveequations can be rewritten as

11987911198792

Δ119903 (119896 + 1) minus 2Δ119903 (119896) + Δ119903 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

ℎ

minus Δ119903 (119896 minus 1)

+ 119870Δ120575 (119896 minus 1) + 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

11987911198792

Δ120592 (119896 + 1) minus 2Δ120592 (119896) + Δ120592 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

ℎ

minus Δ120592 (119896 minus 1)

+ 119870120592Δ120575 (119896 minus 1) + 119870

120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

(17)

where ℎ is the time between two consecutive measurementstaken from the IMU (inertial measurement unit) on boardthe ship and 119896 minus 1 119896 and 119896 + 1 are the indices of threesuccessive sampling times Now it is possible to apply theLS-SVM regression to obtain the parameters of the modelof Nomoto by constructing the training data following the

6 Journal of Applied Mathematics

equation structure of (17) Thus the discrete motion equa-tions given by (17) can be rewritten as

Δ119903 (119896 + 1) = 2Δ119903 (119896) minus Δ119903 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

11987911198792

ℎ

minus

Δ119903 (119896 minus 1)

11987911198792

ℎ2+

119870Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

Δ120592 (119896 + 1) = 2Δ120592 (119896) minus Δ120592 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

11987911198792

ℎ

minus

Δ120592 (119896 minus 1)

11987911198792

ℎ2+

119870120592Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 119870120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

(18)

and in a simplified manner

Δ119903 (119896 + 1) = 119860X119903

Δ120592 (119896 + 1) = 119861X120592

(19)

where Δ119903(119896 + 1) = 119910119895119903and Δ120592(119896 + 1) = 119910

119895120592are the output

training data and X119903= 119909119895119903

and X120592

= 119909119895120592

are the inputtraining data given by

X119903= [Δ119903 (119896) Δ119903 (119896 minus 1) minusΔ119903 (119896) + Δ119903 (119896 minus 1)

Δ119903 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

X120592= [Δ120592 (119896) Δ120592 (119896 minus 1) minusΔ120592 (119896) + Δ120592 (119896 minus 1)

Δ120592 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

(20)

for 119895 = 1 119873119904 and with

119860 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870

11987911198792

ℎ1198701198793

11987911198792

]

119861 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870120592

11987911198792

ℎ119870120592119879120592

11987911198792

]

(21)

The vectors (21) are now computed from LS-SVM regres-sion It is important to point out that the vectors (21) areseparately identified that is 119860 is defined first and after that119861 is defined so that 119879

1and 119879

2are already known for the

computation of 119861 The structure of the mathematical modelis known in advance so we can use a linear kernel119870(119909

119894 119909119895) =

(119909119894sdot 119909119895) The result of applying the LS-SVM technique for the

training data gives us

119910119895120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119895120585+ 119887120585

(22)

for 120585 = 119903 120592 and 119895 = 1 119873119904 Comparing the above equation

with (19) after the training process we have

119860 =

119873119904

sum

119894=1

120572119894119903119909119894119903

119861 =

119873119904

sum

119894=1

120572119894120592119909119894120592

(23)

if the object function has been well approximated by the LS-SVM algorithm where the bias term 119887

120585must be equal to or

approximately 0 Once the support vectors are defined by theabove process the parameters of the Nomoto model can beobtained immediately from (21)

4 Experimental Results

The vehicle used for the experimental tests is a scale modelof an operational vessel in a 11695 scale see Figure 2 Thedimensions of the vessel and the scale model are shown inTable 1

The scale ship model henceforth referred to as the shipis equipped with an electric motor to control the propellerturning speed and a servo to control the rudder angleThe desired rudder angle and surge speed are commandedthrough aWi-Fi connection between the ship and the controlstation at land

The training data used for system identification areobtained from a 2020 degree zig-zag manoeuvre with asample time of 02 seconds and a nominal surge speed of1ms The zig-zag manoeuvre was executed for 100 secondsso a set of 500 samples is trained In Figure 3 the commandedrudder angle (dashed line) and the yaw angle (solid line)made by the ship during the 2020 degree zig-zag manoeuvreare shown The training data are constructed with the com-manded control signal (rudder angle) and the data measuredfrom the IMU on board the ship (yaw rate and sway velocity)From the input and output data and with (19) the LS-SVMalgorithm is now trained to determine the vectors 119860 and 119861

defined in (21)Using the data shown in Figure 3 with the regularisation

parameter 120574 = 104 the Nomoto model parameters obtained

from the identification process with LS-SVM regression are1198791= 11918 119879

2= 14442 119879

3= 15711 119870 = 03619 119879

120592=

03457 and119870120592= 03813 Then the Nomoto model yields

119903

120575

(119904) =

03619 (1 + 15711119904)

(1 + 11918119904) (1 + 14442119904)

120592

120575

(119904) =

03813 (1 + 03457119904)

(1 + 11918119904) (1 + 14442119904)

(24)

Figure 4 shows the yaw rate obtained from the IMU onboard the ship (solid line) together with the yaw rate obtainedin simulation (dotted line) with the Nomoto second-ordermodel previously defined and with the same commandedinput signal than the ship It can be noticed how the behaviourof the model fits the real behaviour of the ship in a nice

Journal of Applied Mathematics 7

Figure 2 Scale ship model used in the experimental tests

Table 1Main parameters and dimensions of the vessel and the scaleship model

Parameter Vessel Scale ship modelLength betweenperpendiculars (119871pp)

74400m 4389m

Maximum beam (119861) 14200m 0838mMean depth to the topdeck (119867) 9050m 0534m

Design draught (119879119898) 6300m 0372m

manner for the training data used in the identificationprocess

In the same way in Figure 5 the sway velocity measuredfrom the IMU on board the ship is shown together withthe sway velocity obtained from the Nomoto model Againthe results are very similar showing a good behaviour of themodel for the training data Despite the above the model forthe sway velocity is less precise than themodel for the yaw ratebecause the sway velocity cannot be directly controlled dueto the fact that the ship studied is an underactuated vehicleMoreover the environmental conditions like currents andwinds particularly affect the sway velocity In any case themodel is satisfactory furthermore the yaw rate is the com-ponent that defines in a higher degree the actual behaviour ofthe ship

Once the Nomoto model is well defined and it fits wellwith the training data as shown in Figures 4 and 5 themodelmust be tested with different data and manoeuvres to verifythe generalisation performance of the model obtained In theexample shown below two consecutive turning manoeuvres(evolution circles) are performed with the simulation modeland the ship for the same input signal The input signalcommanded is the rudder angle that takes the value of 10degrees during 255 seconds then a rudder angle of 0 degreesis commanded for 70 seconds and finally a rudder angleof minus10 degrees is commanded during 275 seconds with anominal surge speed of 1ms

In Figures 6 and 7 the yaw rate and sway velocity forthe ship (solid line) and the Nomoto model (dotted line) areshown respectively It is clear that the predicted behaviouris very similar to the actual one of the ship so that LS-SVM

0 20 40 60 80 100minus08

minus06

minus04

minus02

0

02

04

06

08

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)

Rudder angleYaw angle

Figure 3 2020 degree zig-zag manoeuvre Yaw angle (solid line)and rudder angle (dashed line)

0 20 40 60 80 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 4 Yaw rate obtained in the zig-zag manoeuvre with the ship(solid line) and in simulation (dotted line)

provides a good mathematical model to test different controlstrategies to be applied on the ship

We can notice in Figure 6 how the real system deviatesslightly in turnings with negative rudder angles because theactual yaw rate behaviour is not exactly symmetric it isslightly larger for negative rudder angles Despite this smalldeviation on the supposed symmetrical behaviour of the shipthe result obtained from the simulated model is very similarto the actual one and the difference is not significant Thisdeviation could be produced by environmental conditionslike currents or winds or by structural characteristics ofthe ship like the trimming that can make the ship to nothave exactly a symmetrical behaviour while this symmetrical

8 Journal of Applied Mathematics

0 20 40 60 80 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 5 Sway velocitymeasured in the zig-zagmanoeuvrewith theship (solid line) and in simulation (dotted line)

1000 200 300 400 500 600minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 6 Yaw rate obtained in two consecutive turningmanoeuvreswith the ship (solid line) and in simulation (dotted line)

behaviour is implicit on the Nomoto models This possiblesource of errors in the fidelity of the mathematical modelshould be further studied to be able to define more complexmathematical models although in our practical case as canbe deduced from Figures 6 and 7 these errors are negligible

In Figures 8 and 9 the error between the real and thepredicted yaw rate and sway velocity are shown respectivelyWe can notice from the yaw rate in Figure 8 how the erroralthough very small is larger for negative rudder anglesas mentioned above Despite the commented deviation theerror has a small magnitude and its average value is very closeto 0 providing a more than satisfactory prediction of the real

0 100 200 300 400 500 600

minus01

minus005

0

005

01

015

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 7 Sway velocity obtained in two consecutive turningmanoeuvres with the ship (solid line) and in simulation (dottedline)

0 100 200 300 400 500 600minus002

minus001

0

001

Time (s)

Erro

r in

yaw

rate

estim

atio

n (r

ads

)

Figure 8 Error in the identification of the yaw rate that isdifference between the yaw rate obtained in simulation and in thereal setup

dynamical behaviour of the ship The same comments applyto Figure 9 and the sway velocity instead of the yaw rate

Therefore it is clear that the mathematical model definedfor a surface marine vehicle with LS-SVMprovides a satisfac-tory result which predicts with large accuracy the dynamicsof the experimental system

Now it is interesting to compare the result of LS-SVMidentification with the result that can be obtained by standardidentification methods The latter model is computed usingthe identification toolbox of MATLAB called Ident for the

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

is applied to the identification of a nonlinear system thatcorresponds to a Wiener model that consists of a lineardynamic system with a static nonlinear block The staticnonlinear block is determined by SVR The model employedis function of the input only not depending on the measuredsystem outputs The identification procedure is tested withnumerical simulations

In this work a variation of SVM the Least Squares Sup-port Vector Machines (LS-SVMs) [30] is used for systemidentification with regression This technique allows a nicesimplification of the problem making it more tractable Astudy on this subject can be found in [31] where LS-SVMis used for nonlinear system identification for some simpleexamples of Nonlinear Auto-Regressive with eXogenousinput (NARX) input-output models There also exist someother representative examples that deal with identificationusing LS-SVM For example in [32] the identification ofMIMO Hammerstein ARX models based on LS-SVM isstudied The results are compared with two other techniques(the Hermite polynomial expansion and a RBF network)with numerical simulations showing an improvement of theperformance with the LS-SVM approach In [33] a newsupport vector kernel is defined the wavelet kernel thatsatisfies wavelet frames This kernel is able to approximatearbitrary functions improving the generalisation ability ofthe SVM Based on this kernel and on LS-SVM the LS-WSVM is constructed and it is applied to a nonlinear systemidentification problem showing better accuracy than a Gaus-sian kernel for the problem studied Finally in [34] the LS-SVM is combined with the multiresolution analysis (MRA)to provide a new improved algorithm the multiresolutionleast squares support vector machines (MLS-SVMs) Thisalgorithm has the same theoretical framework as MRAbut it has a better approximation ability and can graduallyapproximate the target function at different scales

With respect to the identification of marine vehicles byusing SVM we can find two interesting references [35] and[36] where an Abkowitz model for ship manoeuvring isidentified by using LS-SVM and 120598-SVM respectively Thepurpose of these works is to determine the hydrodynamiccoefficients of a Mariner-class vessel although the identifica-tion of the mathematical model is made with data obtainedfrom simulation and then tested only in simulation Theseworks do not deal with real data Furthermore it is importantto point out that most of the works on system identificationwith SVR deal with simulation data and numerical exampleswhere the use of SVR is not tested on an experimental setupas far as the authors know In striking contrast to what iscommonly reported in the literature in the paper at hand themodel identification is done with experimental data obtainedfrom a scale surface marine vessel and the mathematicalmodel obtained is validated with experimental tests carriedout with the scale ship showing how the model predicts withlarge accuracy the actual behaviour of the surface vessel Itis important to keep in mind for the experimental resultsobtained in this paper that the analytical properties of SVMcan be compromised in stochastic problems because the noisegenerates additional support vectors However if the noise

ratio is good and the amplitude is limited the SVM can solvethe problem as if it was deterministic [23]

The key contributions of the present paper are twofold (i)themathematicalmodel of a surfacemarine vessel is obtainedwith the LS-SVM technique from experimental data collectedfrom a 2020 degree zig-zag manoeuvre and (ii) in strikingcontrast to what is customary in the literature the predictionability of the mathematical model computed with LS-SVMregression is tested on an open air environment with thereal ship This allows us to explicitly address the connectionbetween the mathematical model identified and the ship sothat the model can be used to design control strategies to beimplemented on the real vehicle and to predict its behaviour

The document is organized as follows In Section 2 a briefintroduction on LS-SVM is givenTheNomoto second-ordermodel and the training input and output data to be usedare defined in Section 3 In Section 4 the Nomoto modelcomputed from the training data is well defined and itsbehaviour is tested with some turning manoeuvres Finallythe conclusions and a brief discussion of topics for furtherresearch are included in Section 5

2 Least Squares Support VectorMachines for Regression

In this section we briefly introduce LS-SVM for systemidentification As mentioned in Section 1 for both regressionand classification SVM has the ability to simultaneouslyminimize the estimation error in the training data (theempirical risk) and the model complexity (the structuralrisk) SVM can also be designed to deal with sparse datawhere we may have many variables but few data The inputdata are mapped into a high dimensional feature space usingnonlinear mapping techniques and then linear classificationor regression is carried out in feature space

The use of LS-SVM for regression is very similar to its usefor classification Following the notation and definitions in[22 30] consider a model in the primal weight space

119910 (119909) = 120596119879120593 (119909) + 119887 (1)

where 119909 isin R119899 is the input data 119910 isin R is the outputdata 119887 is a bias term 120596 is a matrix of weights and 120593(sdot)

R rarr R119899ℎ is the mapping to a high-dimensional spacewhere 119899

ℎcan be infinite The optimization problem in the

primal weight space for a given training set 119909119894 119910119894119873119904

119894=1 with

119873119904as the number of samples becomes

min120596119887119890

J (120596 119890) =

1

2

120596119879120596 + 120574

1

2

119873119904

sum

119894=1

1198902

119894(2)

subject to

119910119894= 120596119879120593 (119909119894) + 119887 + 119890

119894 (3)

where 119894 = 1 119873119904 119890119894are error variables and 120574 is the

regularisation parameter that determines the deviation tol-erated from the desired accuracy which must be positiveThe minimization of 120596119879120596 is closely related to the use of a

4 Journal of Applied Mathematics

weight decay term in the training of neural networks and thesecond term of the right-hand side of (2) controls the tradeoffbetween the empirical error and the model complexity Themain differences between the formulation of LS-SVM withthe standard SVM are the equality constraints (3) and thesquared error term of the second term in the right-hand sideof (2) implying a significant simplification of the problem

The above primal problem cannot be solved when 120596

becomes infinite dimensional Thus the Lagrangian must becomputed and the dual problem derived yielding

L (120596 119887 119890 120572) = J (120596 119890) minus

119873119904

sum

119894=1

120572119894120596119879120593 (119909119894) + 119887 + 119890

119894minus 119910119894

(4)

where 120572119894 with 119894 = 1 119873

119904 are the Lagrange multipliers

The optimality conditions are defined by computing thederivatives of (4) with respect to 120596 119887 119890

119894 and 120572

119894

120597L

120597120596

= 0 997888rarr 120596 =

119873119904

sum

119894=1

120572119894120593 (119909119894)

120597L

120597119887

= 0 997888rarr

119873119904

sum

119894=1

120572119894= 0

120597L

120597119890119894

= 0 997888rarr 120572119894= 120574119890119894

120597L

120597120572119894

= 0 997888rarr 120596119879120593 (119909119894) + 119887 + 119890

119894minus 119910119894= 0

(5)

for 119894 = 1 119873119904 After elimination of variables 120596 and 119890

from the above equations and applying the kernel trickwhichallows us to work in large-dimensional feature spaces withoutexplicit computations on them [19] the LS-SVM model forfunction estimation yields

119910 (119909) =

119873119904

sum

119894=1

120572119894119870(119909 119909

119894) + 119887 (6)

where 119870(sdot sdot) is the kernel function that represents an innerproduct between its operands This kernel must be a positivedefinite kernel and must satisfy the Mercer condition [37]

3 Identification of a NomotorsquosSecond-Order Model

In marine systems as in a multitude of engineering areasan adequate mathematical model is essential to simulate andpredict the behaviour of a real system with the best possibleaccuracy If reliable mathematical models are available andthey predict the behaviour of the real systems accuratelythen control actions can be planned and tested in simulationavoiding the costly in time and money practical tests

A reliable mathematical ship model like the Abkowitzmodel [38] requires the identification of a multitude ofhydrodynamic parameters The identification task for this

large number of variables can be hard and may take a longtimeMoreover different tests for the different hydrodynamicparameters must be carried out [9 10]

For the above reason simpler vehicle models are com-monly used so that although without modelling all thedynamic features of the vehicle each of these models repro-duces the real behaviour of ships with large accuracy in mostof practical scenarios For the problem at hand we considera constant surge speed and we determine a mathematicalmodel that defines the ship steering equations Among thesekinds of models we can find the Nomoto models [12] thatare an alternative representation of the model of Davidsonand Schiff [39] We now briefly comment these two models

Given the linear steering dynamics

119898( + 1199060119903 + 119909119866119903) = 119884

119868119911119903 + 119898119909

119866( + 119906

0119903) = 119873

(7)

the hydrodynamic force and moment can be modelled as in[39]

119884 = 119884 + 119884 119903

119903 + 119884120592120592 + 119884119903119903 + 119884120575120575

119873 = 119873 + 119873 119903

119903 + 119873120592120592 + 119873

119903119903 + 119873120575120575

(8)

where 119884 is sway force 119873 is moment (rotation about the 119911-axis with the 119911-axis pointing downwards as is customary inmarine systems) 119903 is yaw rate 120592 is sway velocity 119906

0is nominal

surge velocity 120575 is the rudder angle 119868119911is moment of inertia

about the 119911-axis 119898 is mass 119909119866is the 119909-axis coordinate of

the centre of gravity and 119884 119884 119903

119884120592 119884119903 119884120575 119873 119873 119903

119873120592 119873119903

and 119873120575are added inertia hydrodynamic coefficients for

more details the reader is referred to [10] See Figure 1 fora definition of the reference frames used in marine systems

Following the procedure of [10] (8) can be rewritten as

M] + N] = b120575 (9)

where ] = [120592 119903]119879 is the state vector and

M = [

119898 minus 119884

119898119909119866minus 119884 119903

119898119909119866minus 119873

119868119911minus 119873 119903

]

N = [

minus119884120592

1198981199060minus 119884119903

minus1198731205921198981199091198661199060minus 119873119903

]

b = [

119884120575

119873120575

]

(10)

From (9) the alternative representation of the second-order model of Nomoto [12] can be defined This modelprovides the transfer function that relates the yaw rate 119903 tothe rudder angle 120575 according to

119903

120575

(119904) =

119870 (1 + 1198793119904)

(1 + 1198791119904) (1 + 119879

2119904)

(11)

or if an integrator (a pole) is included the transfer functionrelates the yaw angle 120595 to the rudder angle 120575 that is

120595

120575

(119904) =

119870 (1 + 1198793119904)

119904 (1 + 1198791119904) (1 + 119879

2119904)

(12)

Journal of Applied Mathematics 5

119874

119874119884

119884

119873

119883

119883

119906119900

119906119900

119910119910

119900

119900

119909

119909

119911

120595

120592

120592

Figure 1 Body (O) and earth-fixed (119900) reference frames forces (119883 119884) moment (119873) and linear speeds (1199060 120592) used in marine systems for

3 degrees of freedom (3DOF) models

where 1198791 1198792 and 119879

3are time constants and 119870 is gain Now

rewrite (11) in the time domain

11987911198792120595(3)

+ (1198791+ 1198792) + = 119870 (120575 + 119879

3120575) (13)

where = 119903 and 120595(3) represents the third derivative of the

yaw angle with respect to time The parameters of (11) arerelated to the hydrodynamic derivatives as follows

11987911198792=

det (M)

det (N)

1198791+ 1198792=

1198991111989822+ 1198992211989811minus 1198991211989821minus 1198992111989812

det (N)

119870 =

119899211198871minus 119899111198872

det (N)

1198701198793=

119898211198871minus 119898111198872

det (N)

(14)

where 119898119894119895 119899119894119895 and 119887

119894 with 119894 = 1 2 and 119895 = 1 2 are the

elements of matricesM N and bIn addition we can express the sway velocity in a similar

manner

120592

120575

(119904) =

119870120592(1 + 119879

120592119904)

(1 + 1198791119904) (1 + 119879

2119904)

(15)

where 119870] and 119879] are the gain and time constants to describethe sway velocity In the time domain (15) yields

11987911198792 + (119879

1+ 1198792) + 120592 = 119870

120592(120575 + 119879

120592120575) (16)

The main advantage of the Nomoto second-order modelis its simplicity and that the hydrodynamic derivatives donot have to be computed explicitly it is possible to definethe parameters of the Nomoto model directly from theexperimental data Therefore we can compute in a fast and

simplemanner a shipmodel for control purposes because it isnot the aim of this work to know explicitly the hydrodynamicderivatives

To proceed with the reconstruction of the ship manoeu-vring model the continuous equations of motion (13) and(16) are discretized because the experimental data are takenwith a given regular interval of time The discretizationis done by the Euler stepping method so that the aboveequations can be rewritten as

11987911198792

Δ119903 (119896 + 1) minus 2Δ119903 (119896) + Δ119903 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

ℎ

minus Δ119903 (119896 minus 1)

+ 119870Δ120575 (119896 minus 1) + 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

11987911198792

Δ120592 (119896 + 1) minus 2Δ120592 (119896) + Δ120592 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

ℎ

minus Δ120592 (119896 minus 1)

+ 119870120592Δ120575 (119896 minus 1) + 119870

120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

(17)

where ℎ is the time between two consecutive measurementstaken from the IMU (inertial measurement unit) on boardthe ship and 119896 minus 1 119896 and 119896 + 1 are the indices of threesuccessive sampling times Now it is possible to apply theLS-SVM regression to obtain the parameters of the modelof Nomoto by constructing the training data following the

6 Journal of Applied Mathematics

equation structure of (17) Thus the discrete motion equa-tions given by (17) can be rewritten as

Δ119903 (119896 + 1) = 2Δ119903 (119896) minus Δ119903 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

11987911198792

ℎ

minus

Δ119903 (119896 minus 1)

11987911198792

ℎ2+

119870Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

Δ120592 (119896 + 1) = 2Δ120592 (119896) minus Δ120592 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

11987911198792

ℎ

minus

Δ120592 (119896 minus 1)

11987911198792

ℎ2+

119870120592Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 119870120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

(18)

and in a simplified manner

Δ119903 (119896 + 1) = 119860X119903

Δ120592 (119896 + 1) = 119861X120592

(19)

where Δ119903(119896 + 1) = 119910119895119903and Δ120592(119896 + 1) = 119910

119895120592are the output

training data and X119903= 119909119895119903

and X120592

= 119909119895120592

are the inputtraining data given by

X119903= [Δ119903 (119896) Δ119903 (119896 minus 1) minusΔ119903 (119896) + Δ119903 (119896 minus 1)

Δ119903 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

X120592= [Δ120592 (119896) Δ120592 (119896 minus 1) minusΔ120592 (119896) + Δ120592 (119896 minus 1)

Δ120592 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

(20)

for 119895 = 1 119873119904 and with

119860 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870

11987911198792

ℎ1198701198793

11987911198792

]

119861 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870120592

11987911198792

ℎ119870120592119879120592

11987911198792

]

(21)

The vectors (21) are now computed from LS-SVM regres-sion It is important to point out that the vectors (21) areseparately identified that is 119860 is defined first and after that119861 is defined so that 119879

1and 119879

2are already known for the

computation of 119861 The structure of the mathematical modelis known in advance so we can use a linear kernel119870(119909

119894 119909119895) =

(119909119894sdot 119909119895) The result of applying the LS-SVM technique for the

training data gives us

119910119895120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119895120585+ 119887120585

(22)

for 120585 = 119903 120592 and 119895 = 1 119873119904 Comparing the above equation

with (19) after the training process we have

119860 =

119873119904

sum

119894=1

120572119894119903119909119894119903

119861 =

119873119904

sum

119894=1

120572119894120592119909119894120592

(23)

if the object function has been well approximated by the LS-SVM algorithm where the bias term 119887

120585must be equal to or

approximately 0 Once the support vectors are defined by theabove process the parameters of the Nomoto model can beobtained immediately from (21)

4 Experimental Results

The vehicle used for the experimental tests is a scale modelof an operational vessel in a 11695 scale see Figure 2 Thedimensions of the vessel and the scale model are shown inTable 1

The scale ship model henceforth referred to as the shipis equipped with an electric motor to control the propellerturning speed and a servo to control the rudder angleThe desired rudder angle and surge speed are commandedthrough aWi-Fi connection between the ship and the controlstation at land

The training data used for system identification areobtained from a 2020 degree zig-zag manoeuvre with asample time of 02 seconds and a nominal surge speed of1ms The zig-zag manoeuvre was executed for 100 secondsso a set of 500 samples is trained In Figure 3 the commandedrudder angle (dashed line) and the yaw angle (solid line)made by the ship during the 2020 degree zig-zag manoeuvreare shown The training data are constructed with the com-manded control signal (rudder angle) and the data measuredfrom the IMU on board the ship (yaw rate and sway velocity)From the input and output data and with (19) the LS-SVMalgorithm is now trained to determine the vectors 119860 and 119861

defined in (21)Using the data shown in Figure 3 with the regularisation

parameter 120574 = 104 the Nomoto model parameters obtained

from the identification process with LS-SVM regression are1198791= 11918 119879

2= 14442 119879

3= 15711 119870 = 03619 119879

120592=

03457 and119870120592= 03813 Then the Nomoto model yields

119903

120575

(119904) =

03619 (1 + 15711119904)

(1 + 11918119904) (1 + 14442119904)

120592

120575

(119904) =

03813 (1 + 03457119904)

(1 + 11918119904) (1 + 14442119904)

(24)

Figure 4 shows the yaw rate obtained from the IMU onboard the ship (solid line) together with the yaw rate obtainedin simulation (dotted line) with the Nomoto second-ordermodel previously defined and with the same commandedinput signal than the ship It can be noticed how the behaviourof the model fits the real behaviour of the ship in a nice

Journal of Applied Mathematics 7

Figure 2 Scale ship model used in the experimental tests

Table 1Main parameters and dimensions of the vessel and the scaleship model

Parameter Vessel Scale ship modelLength betweenperpendiculars (119871pp)

74400m 4389m

Maximum beam (119861) 14200m 0838mMean depth to the topdeck (119867) 9050m 0534m

Design draught (119879119898) 6300m 0372m

manner for the training data used in the identificationprocess

In the same way in Figure 5 the sway velocity measuredfrom the IMU on board the ship is shown together withthe sway velocity obtained from the Nomoto model Againthe results are very similar showing a good behaviour of themodel for the training data Despite the above the model forthe sway velocity is less precise than themodel for the yaw ratebecause the sway velocity cannot be directly controlled dueto the fact that the ship studied is an underactuated vehicleMoreover the environmental conditions like currents andwinds particularly affect the sway velocity In any case themodel is satisfactory furthermore the yaw rate is the com-ponent that defines in a higher degree the actual behaviour ofthe ship

Once the Nomoto model is well defined and it fits wellwith the training data as shown in Figures 4 and 5 themodelmust be tested with different data and manoeuvres to verifythe generalisation performance of the model obtained In theexample shown below two consecutive turning manoeuvres(evolution circles) are performed with the simulation modeland the ship for the same input signal The input signalcommanded is the rudder angle that takes the value of 10degrees during 255 seconds then a rudder angle of 0 degreesis commanded for 70 seconds and finally a rudder angleof minus10 degrees is commanded during 275 seconds with anominal surge speed of 1ms

In Figures 6 and 7 the yaw rate and sway velocity forthe ship (solid line) and the Nomoto model (dotted line) areshown respectively It is clear that the predicted behaviouris very similar to the actual one of the ship so that LS-SVM

0 20 40 60 80 100minus08

minus06

minus04

minus02

0

02

04

06

08

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)

Rudder angleYaw angle

Figure 3 2020 degree zig-zag manoeuvre Yaw angle (solid line)and rudder angle (dashed line)

0 20 40 60 80 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 4 Yaw rate obtained in the zig-zag manoeuvre with the ship(solid line) and in simulation (dotted line)

provides a good mathematical model to test different controlstrategies to be applied on the ship

We can notice in Figure 6 how the real system deviatesslightly in turnings with negative rudder angles because theactual yaw rate behaviour is not exactly symmetric it isslightly larger for negative rudder angles Despite this smalldeviation on the supposed symmetrical behaviour of the shipthe result obtained from the simulated model is very similarto the actual one and the difference is not significant Thisdeviation could be produced by environmental conditionslike currents or winds or by structural characteristics ofthe ship like the trimming that can make the ship to nothave exactly a symmetrical behaviour while this symmetrical

8 Journal of Applied Mathematics

0 20 40 60 80 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 5 Sway velocitymeasured in the zig-zagmanoeuvrewith theship (solid line) and in simulation (dotted line)

1000 200 300 400 500 600minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 6 Yaw rate obtained in two consecutive turningmanoeuvreswith the ship (solid line) and in simulation (dotted line)

behaviour is implicit on the Nomoto models This possiblesource of errors in the fidelity of the mathematical modelshould be further studied to be able to define more complexmathematical models although in our practical case as canbe deduced from Figures 6 and 7 these errors are negligible

In Figures 8 and 9 the error between the real and thepredicted yaw rate and sway velocity are shown respectivelyWe can notice from the yaw rate in Figure 8 how the erroralthough very small is larger for negative rudder anglesas mentioned above Despite the commented deviation theerror has a small magnitude and its average value is very closeto 0 providing a more than satisfactory prediction of the real

0 100 200 300 400 500 600

minus01

minus005

0

005

01

015

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 7 Sway velocity obtained in two consecutive turningmanoeuvres with the ship (solid line) and in simulation (dottedline)

0 100 200 300 400 500 600minus002

minus001

0

001

Time (s)

Erro

r in

yaw

rate

estim

atio

n (r

ads

)

Figure 8 Error in the identification of the yaw rate that isdifference between the yaw rate obtained in simulation and in thereal setup

dynamical behaviour of the ship The same comments applyto Figure 9 and the sway velocity instead of the yaw rate

Therefore it is clear that the mathematical model definedfor a surface marine vehicle with LS-SVMprovides a satisfac-tory result which predicts with large accuracy the dynamicsof the experimental system

Now it is interesting to compare the result of LS-SVMidentification with the result that can be obtained by standardidentification methods The latter model is computed usingthe identification toolbox of MATLAB called Ident for the

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

weight decay term in the training of neural networks and thesecond term of the right-hand side of (2) controls the tradeoffbetween the empirical error and the model complexity Themain differences between the formulation of LS-SVM withthe standard SVM are the equality constraints (3) and thesquared error term of the second term in the right-hand sideof (2) implying a significant simplification of the problem

The above primal problem cannot be solved when 120596

becomes infinite dimensional Thus the Lagrangian must becomputed and the dual problem derived yielding

L (120596 119887 119890 120572) = J (120596 119890) minus

119873119904

sum

119894=1

120572119894120596119879120593 (119909119894) + 119887 + 119890

119894minus 119910119894

(4)

where 120572119894 with 119894 = 1 119873

119904 are the Lagrange multipliers

The optimality conditions are defined by computing thederivatives of (4) with respect to 120596 119887 119890

119894 and 120572

119894

120597L

120597120596

= 0 997888rarr 120596 =

119873119904

sum

119894=1

120572119894120593 (119909119894)

120597L

120597119887

= 0 997888rarr

119873119904

sum

119894=1

120572119894= 0

120597L

120597119890119894

= 0 997888rarr 120572119894= 120574119890119894

120597L

120597120572119894

= 0 997888rarr 120596119879120593 (119909119894) + 119887 + 119890

119894minus 119910119894= 0

(5)

for 119894 = 1 119873119904 After elimination of variables 120596 and 119890

from the above equations and applying the kernel trickwhichallows us to work in large-dimensional feature spaces withoutexplicit computations on them [19] the LS-SVM model forfunction estimation yields

119910 (119909) =

119873119904

sum

119894=1

120572119894119870(119909 119909

119894) + 119887 (6)

where 119870(sdot sdot) is the kernel function that represents an innerproduct between its operands This kernel must be a positivedefinite kernel and must satisfy the Mercer condition [37]

3 Identification of a NomotorsquosSecond-Order Model

In marine systems as in a multitude of engineering areasan adequate mathematical model is essential to simulate andpredict the behaviour of a real system with the best possibleaccuracy If reliable mathematical models are available andthey predict the behaviour of the real systems accuratelythen control actions can be planned and tested in simulationavoiding the costly in time and money practical tests

A reliable mathematical ship model like the Abkowitzmodel [38] requires the identification of a multitude ofhydrodynamic parameters The identification task for this

large number of variables can be hard and may take a longtimeMoreover different tests for the different hydrodynamicparameters must be carried out [9 10]

For the above reason simpler vehicle models are com-monly used so that although without modelling all thedynamic features of the vehicle each of these models repro-duces the real behaviour of ships with large accuracy in mostof practical scenarios For the problem at hand we considera constant surge speed and we determine a mathematicalmodel that defines the ship steering equations Among thesekinds of models we can find the Nomoto models [12] thatare an alternative representation of the model of Davidsonand Schiff [39] We now briefly comment these two models

Given the linear steering dynamics

119898( + 1199060119903 + 119909119866119903) = 119884

119868119911119903 + 119898119909

119866( + 119906

0119903) = 119873

(7)

the hydrodynamic force and moment can be modelled as in[39]

119884 = 119884 + 119884 119903

119903 + 119884120592120592 + 119884119903119903 + 119884120575120575

119873 = 119873 + 119873 119903

119903 + 119873120592120592 + 119873

119903119903 + 119873120575120575

(8)

where 119884 is sway force 119873 is moment (rotation about the 119911-axis with the 119911-axis pointing downwards as is customary inmarine systems) 119903 is yaw rate 120592 is sway velocity 119906

0is nominal

surge velocity 120575 is the rudder angle 119868119911is moment of inertia

about the 119911-axis 119898 is mass 119909119866is the 119909-axis coordinate of

the centre of gravity and 119884 119884 119903

119884120592 119884119903 119884120575 119873 119873 119903

119873120592 119873119903

and 119873120575are added inertia hydrodynamic coefficients for

more details the reader is referred to [10] See Figure 1 fora definition of the reference frames used in marine systems

Following the procedure of [10] (8) can be rewritten as

M] + N] = b120575 (9)

where ] = [120592 119903]119879 is the state vector and

M = [

119898 minus 119884

119898119909119866minus 119884 119903

119898119909119866minus 119873

119868119911minus 119873 119903

]

N = [

minus119884120592

1198981199060minus 119884119903

minus1198731205921198981199091198661199060minus 119873119903

]

b = [

119884120575

119873120575

]

(10)

From (9) the alternative representation of the second-order model of Nomoto [12] can be defined This modelprovides the transfer function that relates the yaw rate 119903 tothe rudder angle 120575 according to

119903

120575

(119904) =

119870 (1 + 1198793119904)

(1 + 1198791119904) (1 + 119879

2119904)

(11)

or if an integrator (a pole) is included the transfer functionrelates the yaw angle 120595 to the rudder angle 120575 that is

120595

120575

(119904) =

119870 (1 + 1198793119904)

119904 (1 + 1198791119904) (1 + 119879

2119904)

(12)

Journal of Applied Mathematics 5

119874

119874119884

119884

119873

119883

119883

119906119900

119906119900

119910119910

119900

119900

119909

119909

119911

120595

120592

120592

Figure 1 Body (O) and earth-fixed (119900) reference frames forces (119883 119884) moment (119873) and linear speeds (1199060 120592) used in marine systems for

3 degrees of freedom (3DOF) models

where 1198791 1198792 and 119879

3are time constants and 119870 is gain Now

rewrite (11) in the time domain

11987911198792120595(3)

+ (1198791+ 1198792) + = 119870 (120575 + 119879

3120575) (13)

where = 119903 and 120595(3) represents the third derivative of the

yaw angle with respect to time The parameters of (11) arerelated to the hydrodynamic derivatives as follows

11987911198792=

det (M)

det (N)

1198791+ 1198792=

1198991111989822+ 1198992211989811minus 1198991211989821minus 1198992111989812

det (N)

119870 =

119899211198871minus 119899111198872

det (N)

1198701198793=

119898211198871minus 119898111198872

det (N)

(14)

where 119898119894119895 119899119894119895 and 119887

119894 with 119894 = 1 2 and 119895 = 1 2 are the

elements of matricesM N and bIn addition we can express the sway velocity in a similar

manner

120592

120575

(119904) =

119870120592(1 + 119879

120592119904)

(1 + 1198791119904) (1 + 119879

2119904)

(15)

where 119870] and 119879] are the gain and time constants to describethe sway velocity In the time domain (15) yields

11987911198792 + (119879

1+ 1198792) + 120592 = 119870

120592(120575 + 119879

120592120575) (16)

The main advantage of the Nomoto second-order modelis its simplicity and that the hydrodynamic derivatives donot have to be computed explicitly it is possible to definethe parameters of the Nomoto model directly from theexperimental data Therefore we can compute in a fast and

simplemanner a shipmodel for control purposes because it isnot the aim of this work to know explicitly the hydrodynamicderivatives

To proceed with the reconstruction of the ship manoeu-vring model the continuous equations of motion (13) and(16) are discretized because the experimental data are takenwith a given regular interval of time The discretizationis done by the Euler stepping method so that the aboveequations can be rewritten as

11987911198792

Δ119903 (119896 + 1) minus 2Δ119903 (119896) + Δ119903 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

ℎ

minus Δ119903 (119896 minus 1)

+ 119870Δ120575 (119896 minus 1) + 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

11987911198792

Δ120592 (119896 + 1) minus 2Δ120592 (119896) + Δ120592 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

ℎ

minus Δ120592 (119896 minus 1)

+ 119870120592Δ120575 (119896 minus 1) + 119870

120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

(17)

where ℎ is the time between two consecutive measurementstaken from the IMU (inertial measurement unit) on boardthe ship and 119896 minus 1 119896 and 119896 + 1 are the indices of threesuccessive sampling times Now it is possible to apply theLS-SVM regression to obtain the parameters of the modelof Nomoto by constructing the training data following the

6 Journal of Applied Mathematics

equation structure of (17) Thus the discrete motion equa-tions given by (17) can be rewritten as

Δ119903 (119896 + 1) = 2Δ119903 (119896) minus Δ119903 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

11987911198792

ℎ

minus

Δ119903 (119896 minus 1)

11987911198792

ℎ2+

119870Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

Δ120592 (119896 + 1) = 2Δ120592 (119896) minus Δ120592 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

11987911198792

ℎ

minus

Δ120592 (119896 minus 1)

11987911198792

ℎ2+

119870120592Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 119870120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

(18)

and in a simplified manner

Δ119903 (119896 + 1) = 119860X119903

Δ120592 (119896 + 1) = 119861X120592

(19)

where Δ119903(119896 + 1) = 119910119895119903and Δ120592(119896 + 1) = 119910

119895120592are the output

training data and X119903= 119909119895119903

and X120592

= 119909119895120592

are the inputtraining data given by

X119903= [Δ119903 (119896) Δ119903 (119896 minus 1) minusΔ119903 (119896) + Δ119903 (119896 minus 1)

Δ119903 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

X120592= [Δ120592 (119896) Δ120592 (119896 minus 1) minusΔ120592 (119896) + Δ120592 (119896 minus 1)

Δ120592 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

(20)

for 119895 = 1 119873119904 and with

119860 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870

11987911198792

ℎ1198701198793

11987911198792

]

119861 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870120592

11987911198792

ℎ119870120592119879120592

11987911198792

]

(21)

The vectors (21) are now computed from LS-SVM regres-sion It is important to point out that the vectors (21) areseparately identified that is 119860 is defined first and after that119861 is defined so that 119879

1and 119879

2are already known for the

computation of 119861 The structure of the mathematical modelis known in advance so we can use a linear kernel119870(119909

119894 119909119895) =

(119909119894sdot 119909119895) The result of applying the LS-SVM technique for the

training data gives us

119910119895120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119895120585+ 119887120585

(22)

for 120585 = 119903 120592 and 119895 = 1 119873119904 Comparing the above equation

with (19) after the training process we have

119860 =

119873119904

sum

119894=1

120572119894119903119909119894119903

119861 =

119873119904

sum

119894=1

120572119894120592119909119894120592

(23)

if the object function has been well approximated by the LS-SVM algorithm where the bias term 119887

120585must be equal to or

approximately 0 Once the support vectors are defined by theabove process the parameters of the Nomoto model can beobtained immediately from (21)

4 Experimental Results

The vehicle used for the experimental tests is a scale modelof an operational vessel in a 11695 scale see Figure 2 Thedimensions of the vessel and the scale model are shown inTable 1

The scale ship model henceforth referred to as the shipis equipped with an electric motor to control the propellerturning speed and a servo to control the rudder angleThe desired rudder angle and surge speed are commandedthrough aWi-Fi connection between the ship and the controlstation at land

The training data used for system identification areobtained from a 2020 degree zig-zag manoeuvre with asample time of 02 seconds and a nominal surge speed of1ms The zig-zag manoeuvre was executed for 100 secondsso a set of 500 samples is trained In Figure 3 the commandedrudder angle (dashed line) and the yaw angle (solid line)made by the ship during the 2020 degree zig-zag manoeuvreare shown The training data are constructed with the com-manded control signal (rudder angle) and the data measuredfrom the IMU on board the ship (yaw rate and sway velocity)From the input and output data and with (19) the LS-SVMalgorithm is now trained to determine the vectors 119860 and 119861

defined in (21)Using the data shown in Figure 3 with the regularisation

parameter 120574 = 104 the Nomoto model parameters obtained

from the identification process with LS-SVM regression are1198791= 11918 119879

2= 14442 119879

3= 15711 119870 = 03619 119879

120592=

03457 and119870120592= 03813 Then the Nomoto model yields

119903

120575

(119904) =

03619 (1 + 15711119904)

(1 + 11918119904) (1 + 14442119904)

120592

120575

(119904) =

03813 (1 + 03457119904)

(1 + 11918119904) (1 + 14442119904)

(24)

Figure 4 shows the yaw rate obtained from the IMU onboard the ship (solid line) together with the yaw rate obtainedin simulation (dotted line) with the Nomoto second-ordermodel previously defined and with the same commandedinput signal than the ship It can be noticed how the behaviourof the model fits the real behaviour of the ship in a nice

Journal of Applied Mathematics 7

Figure 2 Scale ship model used in the experimental tests

Table 1Main parameters and dimensions of the vessel and the scaleship model

Parameter Vessel Scale ship modelLength betweenperpendiculars (119871pp)

74400m 4389m

Maximum beam (119861) 14200m 0838mMean depth to the topdeck (119867) 9050m 0534m

Design draught (119879119898) 6300m 0372m

manner for the training data used in the identificationprocess

In the same way in Figure 5 the sway velocity measuredfrom the IMU on board the ship is shown together withthe sway velocity obtained from the Nomoto model Againthe results are very similar showing a good behaviour of themodel for the training data Despite the above the model forthe sway velocity is less precise than themodel for the yaw ratebecause the sway velocity cannot be directly controlled dueto the fact that the ship studied is an underactuated vehicleMoreover the environmental conditions like currents andwinds particularly affect the sway velocity In any case themodel is satisfactory furthermore the yaw rate is the com-ponent that defines in a higher degree the actual behaviour ofthe ship

Once the Nomoto model is well defined and it fits wellwith the training data as shown in Figures 4 and 5 themodelmust be tested with different data and manoeuvres to verifythe generalisation performance of the model obtained In theexample shown below two consecutive turning manoeuvres(evolution circles) are performed with the simulation modeland the ship for the same input signal The input signalcommanded is the rudder angle that takes the value of 10degrees during 255 seconds then a rudder angle of 0 degreesis commanded for 70 seconds and finally a rudder angleof minus10 degrees is commanded during 275 seconds with anominal surge speed of 1ms

In Figures 6 and 7 the yaw rate and sway velocity forthe ship (solid line) and the Nomoto model (dotted line) areshown respectively It is clear that the predicted behaviouris very similar to the actual one of the ship so that LS-SVM

0 20 40 60 80 100minus08

minus06

minus04

minus02

0

02

04

06

08

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)

Rudder angleYaw angle

Figure 3 2020 degree zig-zag manoeuvre Yaw angle (solid line)and rudder angle (dashed line)

0 20 40 60 80 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 4 Yaw rate obtained in the zig-zag manoeuvre with the ship(solid line) and in simulation (dotted line)

provides a good mathematical model to test different controlstrategies to be applied on the ship

We can notice in Figure 6 how the real system deviatesslightly in turnings with negative rudder angles because theactual yaw rate behaviour is not exactly symmetric it isslightly larger for negative rudder angles Despite this smalldeviation on the supposed symmetrical behaviour of the shipthe result obtained from the simulated model is very similarto the actual one and the difference is not significant Thisdeviation could be produced by environmental conditionslike currents or winds or by structural characteristics ofthe ship like the trimming that can make the ship to nothave exactly a symmetrical behaviour while this symmetrical

8 Journal of Applied Mathematics

0 20 40 60 80 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 5 Sway velocitymeasured in the zig-zagmanoeuvrewith theship (solid line) and in simulation (dotted line)

1000 200 300 400 500 600minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 6 Yaw rate obtained in two consecutive turningmanoeuvreswith the ship (solid line) and in simulation (dotted line)

behaviour is implicit on the Nomoto models This possiblesource of errors in the fidelity of the mathematical modelshould be further studied to be able to define more complexmathematical models although in our practical case as canbe deduced from Figures 6 and 7 these errors are negligible

In Figures 8 and 9 the error between the real and thepredicted yaw rate and sway velocity are shown respectivelyWe can notice from the yaw rate in Figure 8 how the erroralthough very small is larger for negative rudder anglesas mentioned above Despite the commented deviation theerror has a small magnitude and its average value is very closeto 0 providing a more than satisfactory prediction of the real

0 100 200 300 400 500 600

minus01

minus005

0

005

01

015

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 7 Sway velocity obtained in two consecutive turningmanoeuvres with the ship (solid line) and in simulation (dottedline)

0 100 200 300 400 500 600minus002

minus001

0

001

Time (s)

Erro

r in

yaw

rate

estim

atio

n (r

ads

)

Figure 8 Error in the identification of the yaw rate that isdifference between the yaw rate obtained in simulation and in thereal setup

dynamical behaviour of the ship The same comments applyto Figure 9 and the sway velocity instead of the yaw rate

Therefore it is clear that the mathematical model definedfor a surface marine vehicle with LS-SVMprovides a satisfac-tory result which predicts with large accuracy the dynamicsof the experimental system

Now it is interesting to compare the result of LS-SVMidentification with the result that can be obtained by standardidentification methods The latter model is computed usingthe identification toolbox of MATLAB called Ident for the

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

119874

119874119884

119884

119873

119883

119883

119906119900

119906119900

119910119910

119900

119900

119909

119909

119911

120595

120592

120592

Figure 1 Body (O) and earth-fixed (119900) reference frames forces (119883 119884) moment (119873) and linear speeds (1199060 120592) used in marine systems for

3 degrees of freedom (3DOF) models

where 1198791 1198792 and 119879

3are time constants and 119870 is gain Now

rewrite (11) in the time domain

11987911198792120595(3)

+ (1198791+ 1198792) + = 119870 (120575 + 119879

3120575) (13)

where = 119903 and 120595(3) represents the third derivative of the

yaw angle with respect to time The parameters of (11) arerelated to the hydrodynamic derivatives as follows

11987911198792=

det (M)

det (N)

1198791+ 1198792=

1198991111989822+ 1198992211989811minus 1198991211989821minus 1198992111989812

det (N)

119870 =

119899211198871minus 119899111198872

det (N)

1198701198793=

119898211198871minus 119898111198872

det (N)

(14)

where 119898119894119895 119899119894119895 and 119887

119894 with 119894 = 1 2 and 119895 = 1 2 are the

elements of matricesM N and bIn addition we can express the sway velocity in a similar

manner

120592

120575

(119904) =

119870120592(1 + 119879

120592119904)

(1 + 1198791119904) (1 + 119879

2119904)

(15)

where 119870] and 119879] are the gain and time constants to describethe sway velocity In the time domain (15) yields

11987911198792 + (119879

1+ 1198792) + 120592 = 119870

120592(120575 + 119879

120592120575) (16)

The main advantage of the Nomoto second-order modelis its simplicity and that the hydrodynamic derivatives donot have to be computed explicitly it is possible to definethe parameters of the Nomoto model directly from theexperimental data Therefore we can compute in a fast and

simplemanner a shipmodel for control purposes because it isnot the aim of this work to know explicitly the hydrodynamicderivatives

To proceed with the reconstruction of the ship manoeu-vring model the continuous equations of motion (13) and(16) are discretized because the experimental data are takenwith a given regular interval of time The discretizationis done by the Euler stepping method so that the aboveequations can be rewritten as

11987911198792

Δ119903 (119896 + 1) minus 2Δ119903 (119896) + Δ119903 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

ℎ

minus Δ119903 (119896 minus 1)

+ 119870Δ120575 (119896 minus 1) + 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

11987911198792

Δ120592 (119896 + 1) minus 2Δ120592 (119896) + Δ120592 (119896 minus 1)

ℎ2

= (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

ℎ

minus Δ120592 (119896 minus 1)

+ 119870120592Δ120575 (119896 minus 1) + 119870

120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

ℎ

(17)

where ℎ is the time between two consecutive measurementstaken from the IMU (inertial measurement unit) on boardthe ship and 119896 minus 1 119896 and 119896 + 1 are the indices of threesuccessive sampling times Now it is possible to apply theLS-SVM regression to obtain the parameters of the modelof Nomoto by constructing the training data following the

6 Journal of Applied Mathematics

equation structure of (17) Thus the discrete motion equa-tions given by (17) can be rewritten as

Δ119903 (119896 + 1) = 2Δ119903 (119896) minus Δ119903 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

11987911198792

ℎ

minus

Δ119903 (119896 minus 1)

11987911198792

ℎ2+

119870Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

Δ120592 (119896 + 1) = 2Δ120592 (119896) minus Δ120592 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

11987911198792

ℎ

minus

Δ120592 (119896 minus 1)

11987911198792

ℎ2+

119870120592Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 119870120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

(18)

and in a simplified manner

Δ119903 (119896 + 1) = 119860X119903

Δ120592 (119896 + 1) = 119861X120592

(19)

where Δ119903(119896 + 1) = 119910119895119903and Δ120592(119896 + 1) = 119910

119895120592are the output

training data and X119903= 119909119895119903

and X120592

= 119909119895120592

are the inputtraining data given by

X119903= [Δ119903 (119896) Δ119903 (119896 minus 1) minusΔ119903 (119896) + Δ119903 (119896 minus 1)

Δ119903 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

X120592= [Δ120592 (119896) Δ120592 (119896 minus 1) minusΔ120592 (119896) + Δ120592 (119896 minus 1)

Δ120592 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

(20)

for 119895 = 1 119873119904 and with

119860 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870

11987911198792

ℎ1198701198793

11987911198792

]

119861 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870120592

11987911198792

ℎ119870120592119879120592

11987911198792

]

(21)

The vectors (21) are now computed from LS-SVM regres-sion It is important to point out that the vectors (21) areseparately identified that is 119860 is defined first and after that119861 is defined so that 119879

1and 119879

2are already known for the

computation of 119861 The structure of the mathematical modelis known in advance so we can use a linear kernel119870(119909

119894 119909119895) =

(119909119894sdot 119909119895) The result of applying the LS-SVM technique for the

training data gives us

119910119895120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119895120585+ 119887120585

(22)

for 120585 = 119903 120592 and 119895 = 1 119873119904 Comparing the above equation

with (19) after the training process we have

119860 =

119873119904

sum

119894=1

120572119894119903119909119894119903

119861 =

119873119904

sum

119894=1

120572119894120592119909119894120592

(23)

if the object function has been well approximated by the LS-SVM algorithm where the bias term 119887

120585must be equal to or

approximately 0 Once the support vectors are defined by theabove process the parameters of the Nomoto model can beobtained immediately from (21)

4 Experimental Results

The vehicle used for the experimental tests is a scale modelof an operational vessel in a 11695 scale see Figure 2 Thedimensions of the vessel and the scale model are shown inTable 1

The scale ship model henceforth referred to as the shipis equipped with an electric motor to control the propellerturning speed and a servo to control the rudder angleThe desired rudder angle and surge speed are commandedthrough aWi-Fi connection between the ship and the controlstation at land

The training data used for system identification areobtained from a 2020 degree zig-zag manoeuvre with asample time of 02 seconds and a nominal surge speed of1ms The zig-zag manoeuvre was executed for 100 secondsso a set of 500 samples is trained In Figure 3 the commandedrudder angle (dashed line) and the yaw angle (solid line)made by the ship during the 2020 degree zig-zag manoeuvreare shown The training data are constructed with the com-manded control signal (rudder angle) and the data measuredfrom the IMU on board the ship (yaw rate and sway velocity)From the input and output data and with (19) the LS-SVMalgorithm is now trained to determine the vectors 119860 and 119861

defined in (21)Using the data shown in Figure 3 with the regularisation

parameter 120574 = 104 the Nomoto model parameters obtained

from the identification process with LS-SVM regression are1198791= 11918 119879

2= 14442 119879

3= 15711 119870 = 03619 119879

120592=

03457 and119870120592= 03813 Then the Nomoto model yields

119903

120575

(119904) =

03619 (1 + 15711119904)

(1 + 11918119904) (1 + 14442119904)

120592

120575

(119904) =

03813 (1 + 03457119904)

(1 + 11918119904) (1 + 14442119904)

(24)

Figure 4 shows the yaw rate obtained from the IMU onboard the ship (solid line) together with the yaw rate obtainedin simulation (dotted line) with the Nomoto second-ordermodel previously defined and with the same commandedinput signal than the ship It can be noticed how the behaviourof the model fits the real behaviour of the ship in a nice

Journal of Applied Mathematics 7

Figure 2 Scale ship model used in the experimental tests

Table 1Main parameters and dimensions of the vessel and the scaleship model

Parameter Vessel Scale ship modelLength betweenperpendiculars (119871pp)

74400m 4389m

Maximum beam (119861) 14200m 0838mMean depth to the topdeck (119867) 9050m 0534m

Design draught (119879119898) 6300m 0372m

manner for the training data used in the identificationprocess

In the same way in Figure 5 the sway velocity measuredfrom the IMU on board the ship is shown together withthe sway velocity obtained from the Nomoto model Againthe results are very similar showing a good behaviour of themodel for the training data Despite the above the model forthe sway velocity is less precise than themodel for the yaw ratebecause the sway velocity cannot be directly controlled dueto the fact that the ship studied is an underactuated vehicleMoreover the environmental conditions like currents andwinds particularly affect the sway velocity In any case themodel is satisfactory furthermore the yaw rate is the com-ponent that defines in a higher degree the actual behaviour ofthe ship

Once the Nomoto model is well defined and it fits wellwith the training data as shown in Figures 4 and 5 themodelmust be tested with different data and manoeuvres to verifythe generalisation performance of the model obtained In theexample shown below two consecutive turning manoeuvres(evolution circles) are performed with the simulation modeland the ship for the same input signal The input signalcommanded is the rudder angle that takes the value of 10degrees during 255 seconds then a rudder angle of 0 degreesis commanded for 70 seconds and finally a rudder angleof minus10 degrees is commanded during 275 seconds with anominal surge speed of 1ms

In Figures 6 and 7 the yaw rate and sway velocity forthe ship (solid line) and the Nomoto model (dotted line) areshown respectively It is clear that the predicted behaviouris very similar to the actual one of the ship so that LS-SVM

0 20 40 60 80 100minus08

minus06

minus04

minus02

0

02

04

06

08

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)

Rudder angleYaw angle

Figure 3 2020 degree zig-zag manoeuvre Yaw angle (solid line)and rudder angle (dashed line)

0 20 40 60 80 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 4 Yaw rate obtained in the zig-zag manoeuvre with the ship(solid line) and in simulation (dotted line)

provides a good mathematical model to test different controlstrategies to be applied on the ship

We can notice in Figure 6 how the real system deviatesslightly in turnings with negative rudder angles because theactual yaw rate behaviour is not exactly symmetric it isslightly larger for negative rudder angles Despite this smalldeviation on the supposed symmetrical behaviour of the shipthe result obtained from the simulated model is very similarto the actual one and the difference is not significant Thisdeviation could be produced by environmental conditionslike currents or winds or by structural characteristics ofthe ship like the trimming that can make the ship to nothave exactly a symmetrical behaviour while this symmetrical

8 Journal of Applied Mathematics

0 20 40 60 80 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 5 Sway velocitymeasured in the zig-zagmanoeuvrewith theship (solid line) and in simulation (dotted line)

1000 200 300 400 500 600minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 6 Yaw rate obtained in two consecutive turningmanoeuvreswith the ship (solid line) and in simulation (dotted line)

behaviour is implicit on the Nomoto models This possiblesource of errors in the fidelity of the mathematical modelshould be further studied to be able to define more complexmathematical models although in our practical case as canbe deduced from Figures 6 and 7 these errors are negligible

In Figures 8 and 9 the error between the real and thepredicted yaw rate and sway velocity are shown respectivelyWe can notice from the yaw rate in Figure 8 how the erroralthough very small is larger for negative rudder anglesas mentioned above Despite the commented deviation theerror has a small magnitude and its average value is very closeto 0 providing a more than satisfactory prediction of the real

0 100 200 300 400 500 600

minus01

minus005

0

005

01

015

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 7 Sway velocity obtained in two consecutive turningmanoeuvres with the ship (solid line) and in simulation (dottedline)

0 100 200 300 400 500 600minus002

minus001

0

001

Time (s)

Erro

r in

yaw

rate

estim

atio

n (r

ads

)

Figure 8 Error in the identification of the yaw rate that isdifference between the yaw rate obtained in simulation and in thereal setup

dynamical behaviour of the ship The same comments applyto Figure 9 and the sway velocity instead of the yaw rate

Therefore it is clear that the mathematical model definedfor a surface marine vehicle with LS-SVMprovides a satisfac-tory result which predicts with large accuracy the dynamicsof the experimental system

Now it is interesting to compare the result of LS-SVMidentification with the result that can be obtained by standardidentification methods The latter model is computed usingthe identification toolbox of MATLAB called Ident for the

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

equation structure of (17) Thus the discrete motion equa-tions given by (17) can be rewritten as

Δ119903 (119896 + 1) = 2Δ119903 (119896) minus Δ119903 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ119903 (119896) + Δ119903 (119896 minus 1)

11987911198792

ℎ

minus

Δ119903 (119896 minus 1)

11987911198792

ℎ2+

119870Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 1198701198793

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

Δ120592 (119896 + 1) = 2Δ120592 (119896) minus Δ120592 (119896 minus 1)

+ (1198791+ 1198792)

minusΔ120592 (119896) + Δ120592 (119896 minus 1)

11987911198792

ℎ

minus

Δ120592 (119896 minus 1)

11987911198792

ℎ2+

119870120592Δ120575 (119896 minus 1)

11987911198792

ℎ2

+ 119870120592119879120592

Δ120575 (119896) minus Δ120575 (119896 minus 1)

11987911198792

ℎ

(18)

and in a simplified manner

Δ119903 (119896 + 1) = 119860X119903

Δ120592 (119896 + 1) = 119861X120592

(19)

where Δ119903(119896 + 1) = 119910119895119903and Δ120592(119896 + 1) = 119910

119895120592are the output

training data and X119903= 119909119895119903

and X120592

= 119909119895120592

are the inputtraining data given by

X119903= [Δ119903 (119896) Δ119903 (119896 minus 1) minusΔ119903 (119896) + Δ119903 (119896 minus 1)

Δ119903 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

X120592= [Δ120592 (119896) Δ120592 (119896 minus 1) minusΔ120592 (119896) + Δ120592 (119896 minus 1)

Δ120592 (119896 minus 1) Δ120575 (119896 minus 1) Δ120575 (119896) minus Δ120575 (119896 minus 1)]119879

(20)

for 119895 = 1 119873119904 and with

119860 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870

11987911198792

ℎ1198701198793

11987911198792

]

119861 = [2 minus1

(1198791+ 1198792) ℎ

11987911198792

minusℎ2

11987911198792

ℎ2119870120592

11987911198792

ℎ119870120592119879120592

11987911198792

]

(21)

The vectors (21) are now computed from LS-SVM regres-sion It is important to point out that the vectors (21) areseparately identified that is 119860 is defined first and after that119861 is defined so that 119879

1and 119879

2are already known for the

computation of 119861 The structure of the mathematical modelis known in advance so we can use a linear kernel119870(119909

119894 119909119895) =

(119909119894sdot 119909119895) The result of applying the LS-SVM technique for the

training data gives us

119910119895120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119895120585+ 119887120585

(22)

for 120585 = 119903 120592 and 119895 = 1 119873119904 Comparing the above equation

with (19) after the training process we have

119860 =

119873119904

sum

119894=1

120572119894119903119909119894119903

119861 =

119873119904

sum

119894=1

120572119894120592119909119894120592

(23)

if the object function has been well approximated by the LS-SVM algorithm where the bias term 119887

120585must be equal to or

approximately 0 Once the support vectors are defined by theabove process the parameters of the Nomoto model can beobtained immediately from (21)

4 Experimental Results

The vehicle used for the experimental tests is a scale modelof an operational vessel in a 11695 scale see Figure 2 Thedimensions of the vessel and the scale model are shown inTable 1

The scale ship model henceforth referred to as the shipis equipped with an electric motor to control the propellerturning speed and a servo to control the rudder angleThe desired rudder angle and surge speed are commandedthrough aWi-Fi connection between the ship and the controlstation at land

The training data used for system identification areobtained from a 2020 degree zig-zag manoeuvre with asample time of 02 seconds and a nominal surge speed of1ms The zig-zag manoeuvre was executed for 100 secondsso a set of 500 samples is trained In Figure 3 the commandedrudder angle (dashed line) and the yaw angle (solid line)made by the ship during the 2020 degree zig-zag manoeuvreare shown The training data are constructed with the com-manded control signal (rudder angle) and the data measuredfrom the IMU on board the ship (yaw rate and sway velocity)From the input and output data and with (19) the LS-SVMalgorithm is now trained to determine the vectors 119860 and 119861

defined in (21)Using the data shown in Figure 3 with the regularisation

parameter 120574 = 104 the Nomoto model parameters obtained

from the identification process with LS-SVM regression are1198791= 11918 119879

2= 14442 119879

3= 15711 119870 = 03619 119879

120592=

03457 and119870120592= 03813 Then the Nomoto model yields

119903

120575

(119904) =

03619 (1 + 15711119904)

(1 + 11918119904) (1 + 14442119904)

120592

120575

(119904) =

03813 (1 + 03457119904)

(1 + 11918119904) (1 + 14442119904)

(24)

Figure 4 shows the yaw rate obtained from the IMU onboard the ship (solid line) together with the yaw rate obtainedin simulation (dotted line) with the Nomoto second-ordermodel previously defined and with the same commandedinput signal than the ship It can be noticed how the behaviourof the model fits the real behaviour of the ship in a nice

Journal of Applied Mathematics 7

Figure 2 Scale ship model used in the experimental tests

Table 1Main parameters and dimensions of the vessel and the scaleship model

Parameter Vessel Scale ship modelLength betweenperpendiculars (119871pp)

74400m 4389m

Maximum beam (119861) 14200m 0838mMean depth to the topdeck (119867) 9050m 0534m

Design draught (119879119898) 6300m 0372m

manner for the training data used in the identificationprocess

In the same way in Figure 5 the sway velocity measuredfrom the IMU on board the ship is shown together withthe sway velocity obtained from the Nomoto model Againthe results are very similar showing a good behaviour of themodel for the training data Despite the above the model forthe sway velocity is less precise than themodel for the yaw ratebecause the sway velocity cannot be directly controlled dueto the fact that the ship studied is an underactuated vehicleMoreover the environmental conditions like currents andwinds particularly affect the sway velocity In any case themodel is satisfactory furthermore the yaw rate is the com-ponent that defines in a higher degree the actual behaviour ofthe ship

Once the Nomoto model is well defined and it fits wellwith the training data as shown in Figures 4 and 5 themodelmust be tested with different data and manoeuvres to verifythe generalisation performance of the model obtained In theexample shown below two consecutive turning manoeuvres(evolution circles) are performed with the simulation modeland the ship for the same input signal The input signalcommanded is the rudder angle that takes the value of 10degrees during 255 seconds then a rudder angle of 0 degreesis commanded for 70 seconds and finally a rudder angleof minus10 degrees is commanded during 275 seconds with anominal surge speed of 1ms

In Figures 6 and 7 the yaw rate and sway velocity forthe ship (solid line) and the Nomoto model (dotted line) areshown respectively It is clear that the predicted behaviouris very similar to the actual one of the ship so that LS-SVM

0 20 40 60 80 100minus08

minus06

minus04

minus02

0

02

04

06

08

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)

Rudder angleYaw angle

Figure 3 2020 degree zig-zag manoeuvre Yaw angle (solid line)and rudder angle (dashed line)

0 20 40 60 80 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 4 Yaw rate obtained in the zig-zag manoeuvre with the ship(solid line) and in simulation (dotted line)

provides a good mathematical model to test different controlstrategies to be applied on the ship

We can notice in Figure 6 how the real system deviatesslightly in turnings with negative rudder angles because theactual yaw rate behaviour is not exactly symmetric it isslightly larger for negative rudder angles Despite this smalldeviation on the supposed symmetrical behaviour of the shipthe result obtained from the simulated model is very similarto the actual one and the difference is not significant Thisdeviation could be produced by environmental conditionslike currents or winds or by structural characteristics ofthe ship like the trimming that can make the ship to nothave exactly a symmetrical behaviour while this symmetrical

8 Journal of Applied Mathematics

0 20 40 60 80 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 5 Sway velocitymeasured in the zig-zagmanoeuvrewith theship (solid line) and in simulation (dotted line)

1000 200 300 400 500 600minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 6 Yaw rate obtained in two consecutive turningmanoeuvreswith the ship (solid line) and in simulation (dotted line)

behaviour is implicit on the Nomoto models This possiblesource of errors in the fidelity of the mathematical modelshould be further studied to be able to define more complexmathematical models although in our practical case as canbe deduced from Figures 6 and 7 these errors are negligible

In Figures 8 and 9 the error between the real and thepredicted yaw rate and sway velocity are shown respectivelyWe can notice from the yaw rate in Figure 8 how the erroralthough very small is larger for negative rudder anglesas mentioned above Despite the commented deviation theerror has a small magnitude and its average value is very closeto 0 providing a more than satisfactory prediction of the real

0 100 200 300 400 500 600

minus01

minus005

0

005

01

015

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 7 Sway velocity obtained in two consecutive turningmanoeuvres with the ship (solid line) and in simulation (dottedline)

0 100 200 300 400 500 600minus002

minus001

0

001

Time (s)

Erro

r in

yaw

rate

estim

atio

n (r

ads

)

Figure 8 Error in the identification of the yaw rate that isdifference between the yaw rate obtained in simulation and in thereal setup

dynamical behaviour of the ship The same comments applyto Figure 9 and the sway velocity instead of the yaw rate

Therefore it is clear that the mathematical model definedfor a surface marine vehicle with LS-SVMprovides a satisfac-tory result which predicts with large accuracy the dynamicsof the experimental system

Now it is interesting to compare the result of LS-SVMidentification with the result that can be obtained by standardidentification methods The latter model is computed usingthe identification toolbox of MATLAB called Ident for the

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Figure 2 Scale ship model used in the experimental tests

Table 1Main parameters and dimensions of the vessel and the scaleship model

Parameter Vessel Scale ship modelLength betweenperpendiculars (119871pp)

74400m 4389m

Maximum beam (119861) 14200m 0838mMean depth to the topdeck (119867) 9050m 0534m

Design draught (119879119898) 6300m 0372m

manner for the training data used in the identificationprocess

In the same way in Figure 5 the sway velocity measuredfrom the IMU on board the ship is shown together withthe sway velocity obtained from the Nomoto model Againthe results are very similar showing a good behaviour of themodel for the training data Despite the above the model forthe sway velocity is less precise than themodel for the yaw ratebecause the sway velocity cannot be directly controlled dueto the fact that the ship studied is an underactuated vehicleMoreover the environmental conditions like currents andwinds particularly affect the sway velocity In any case themodel is satisfactory furthermore the yaw rate is the com-ponent that defines in a higher degree the actual behaviour ofthe ship

Once the Nomoto model is well defined and it fits wellwith the training data as shown in Figures 4 and 5 themodelmust be tested with different data and manoeuvres to verifythe generalisation performance of the model obtained In theexample shown below two consecutive turning manoeuvres(evolution circles) are performed with the simulation modeland the ship for the same input signal The input signalcommanded is the rudder angle that takes the value of 10degrees during 255 seconds then a rudder angle of 0 degreesis commanded for 70 seconds and finally a rudder angleof minus10 degrees is commanded during 275 seconds with anominal surge speed of 1ms

In Figures 6 and 7 the yaw rate and sway velocity forthe ship (solid line) and the Nomoto model (dotted line) areshown respectively It is clear that the predicted behaviouris very similar to the actual one of the ship so that LS-SVM

0 20 40 60 80 100minus08

minus06

minus04

minus02

0

02

04

06

08

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)

Rudder angleYaw angle

Figure 3 2020 degree zig-zag manoeuvre Yaw angle (solid line)and rudder angle (dashed line)

0 20 40 60 80 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 4 Yaw rate obtained in the zig-zag manoeuvre with the ship(solid line) and in simulation (dotted line)

provides a good mathematical model to test different controlstrategies to be applied on the ship

We can notice in Figure 6 how the real system deviatesslightly in turnings with negative rudder angles because theactual yaw rate behaviour is not exactly symmetric it isslightly larger for negative rudder angles Despite this smalldeviation on the supposed symmetrical behaviour of the shipthe result obtained from the simulated model is very similarto the actual one and the difference is not significant Thisdeviation could be produced by environmental conditionslike currents or winds or by structural characteristics ofthe ship like the trimming that can make the ship to nothave exactly a symmetrical behaviour while this symmetrical

8 Journal of Applied Mathematics

0 20 40 60 80 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 5 Sway velocitymeasured in the zig-zagmanoeuvrewith theship (solid line) and in simulation (dotted line)

1000 200 300 400 500 600minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 6 Yaw rate obtained in two consecutive turningmanoeuvreswith the ship (solid line) and in simulation (dotted line)

behaviour is implicit on the Nomoto models This possiblesource of errors in the fidelity of the mathematical modelshould be further studied to be able to define more complexmathematical models although in our practical case as canbe deduced from Figures 6 and 7 these errors are negligible

In Figures 8 and 9 the error between the real and thepredicted yaw rate and sway velocity are shown respectivelyWe can notice from the yaw rate in Figure 8 how the erroralthough very small is larger for negative rudder anglesas mentioned above Despite the commented deviation theerror has a small magnitude and its average value is very closeto 0 providing a more than satisfactory prediction of the real

0 100 200 300 400 500 600

minus01

minus005

0

005

01

015

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 7 Sway velocity obtained in two consecutive turningmanoeuvres with the ship (solid line) and in simulation (dottedline)

0 100 200 300 400 500 600minus002

minus001

0

001

Time (s)

Erro

r in

yaw

rate

estim

atio

n (r

ads

)

Figure 8 Error in the identification of the yaw rate that isdifference between the yaw rate obtained in simulation and in thereal setup

dynamical behaviour of the ship The same comments applyto Figure 9 and the sway velocity instead of the yaw rate

Therefore it is clear that the mathematical model definedfor a surface marine vehicle with LS-SVMprovides a satisfac-tory result which predicts with large accuracy the dynamicsof the experimental system

Now it is interesting to compare the result of LS-SVMidentification with the result that can be obtained by standardidentification methods The latter model is computed usingthe identification toolbox of MATLAB called Ident for the

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

0 20 40 60 80 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 5 Sway velocitymeasured in the zig-zagmanoeuvrewith theship (solid line) and in simulation (dotted line)

1000 200 300 400 500 600minus01

minus008

minus006

minus004

minus002

0

002

004

006

008

01

Time (s)

Yaw

rate

(rad

s)

Experimental dataSimulation data

Figure 6 Yaw rate obtained in two consecutive turningmanoeuvreswith the ship (solid line) and in simulation (dotted line)

behaviour is implicit on the Nomoto models This possiblesource of errors in the fidelity of the mathematical modelshould be further studied to be able to define more complexmathematical models although in our practical case as canbe deduced from Figures 6 and 7 these errors are negligible

In Figures 8 and 9 the error between the real and thepredicted yaw rate and sway velocity are shown respectivelyWe can notice from the yaw rate in Figure 8 how the erroralthough very small is larger for negative rudder anglesas mentioned above Despite the commented deviation theerror has a small magnitude and its average value is very closeto 0 providing a more than satisfactory prediction of the real

0 100 200 300 400 500 600

minus01

minus005

0

005

01

015

Time (s)

Sway

velo

city

(ms

)

Experimental dataSimulation data

Figure 7 Sway velocity obtained in two consecutive turningmanoeuvres with the ship (solid line) and in simulation (dottedline)

0 100 200 300 400 500 600minus002

minus001

0

001

Time (s)

Erro

r in

yaw

rate

estim

atio

n (r

ads

)

Figure 8 Error in the identification of the yaw rate that isdifference between the yaw rate obtained in simulation and in thereal setup

dynamical behaviour of the ship The same comments applyto Figure 9 and the sway velocity instead of the yaw rate

Therefore it is clear that the mathematical model definedfor a surface marine vehicle with LS-SVMprovides a satisfac-tory result which predicts with large accuracy the dynamicsof the experimental system

Now it is interesting to compare the result of LS-SVMidentification with the result that can be obtained by standardidentification methods The latter model is computed usingthe identification toolbox of MATLAB called Ident for the

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

0 100 200 300 400 500 600minus008

minus006

minus004

minus002

0

002

004

006

Time (s)

Erro

r in

sway

velo

city

estim

atio

n (m

s)

Figure 9 Error in the identification of the sway velocity that isdifference between the sway velocity obtained in simulation and inthe real setup

documentation of this toolbox the reader is referred to [40]The model computed with MATLAB is defined by

119903

120575

(119904) =

035942 (1 + 15119119904)

(1 + 1201119904) (1 + 12312119904)

120592

120575

(119904) =

0396 (1 + 034984119904)

(1 + 1201119904) (1 + 12312119904)

(25)

In Figures 10 and 11 the yaw rate and sway velocityrespectively are shown for the experimental test (solid line)the LS-SVM identification model (dotted line) and theidentificationmodel obtainedwith the Ident tool ofMATLAB(dashed line) for the training data that is the 2020 degreezig-zag manoeuvre We can check from Figures 10 and 11 thatfor the practical case studied the behaviours obtained withboth identification methods are practically the same

The main difference between both methods lies in theimportant fact that the SVM approach can deal with alarge number of variables and few data as shown in [35]where multiple hydrodynamic variables are computed froma single manoeuvre in a simulation scenario In a practicalscenario and with a classical identification methodologyseveral practical tests should be carried out to define thesemultiple hydrodynamic variables with the correspondingtime cost Moreover if the model structure is not knownin advance some approach of black box identification maybe used as in [32] or [29] by means of nonlinear kernelsIn these systems a nonlinear block that is approximated byselection of a basis function is used In SVM the nonlinearmappings of this function can be unknown they may bedefined with a nonlinear kernel function Therefore thistechnique has the potential to be implemented for differentkinds of marine vehicles in a simple and fast manner Theseissues are a very interesting line for future research

0 10 20 30 40 50 60 70 80 90 100

minus01

minus005

0

005

01

015

Time (s)

Yaw

rate

(rad

s)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 10 Comparison between the experimental yaw rate (solidline) the yaw rate of the LS-SVM identification model (dotted line)and the one of the identification model obtained with the Ident toolof MATLAB (dashed line) for a 2020 degree zig-zag manoeuvre

0 10 20 30 40 50 60 70 80 90 100minus015

minus01

minus005

0

005

01

015

02

Time (s)

Sway

velo

city

(ms

)

Measured and simulated model outputs

SVM identificationMATLAB (Ident) identificationExperimental data

Figure 11 Comparison between the experimental sway velocity(solid line) the sway velocity of the LS-SVM identification model(dotted line) and the one of the identification model obtained withthe Ident tool of MATLAB (dashed line) for a 2020 degree zig-zagmanoeuvre

5 Conclusions and Future Work

In this work the Nomoto second-order model of a ship hasbeen determined through real experimental data obtainedfrom a zig-zag manoeuvre test This ship model has beenidentified by analysing the rudder angle surge and sway

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

speeds and yaw rate obtained from experimental tests Thisidentification has been done with a Least Squares SupportVectorMachines algorithmThe LS-SVM technique has beenwidely used for pattern recognition and classification withvery good results and in this work it has been shown that thistechnique has a high potential to be used in system identifi-cation The model obtained with LS-SVM has been validatedthrough experimental tests showing that the behaviour ofthe mathematical model is very similar to that of the realship Furthermore the model obtained is suitable to be usedfor testing control algorithms that can be implemented onthe ship It has been proved that LS-SVM is a powerfultool for system identification that can deal with relative fewexperimental data and low time cost

Future work will aim at (i) extending the methodologydeveloped to deal with more complex ship models in whichthe hydrodynamic coefficients can be well defined and todeal with models whose structure is not known in advanceand (ii) studying the performance of control algorithms forpath following and tracking with the ship model defined incomparison with the results obtained for the real vehicle

Acknowledgment

The authors wish to thank the Spanish Ministry of Sci-ence and Innovation (MICINN) for support under ProjectDPI2009-14552-C02-02

References

[1] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the 9th International Conference on Control AutomationRobotics and Vision (ICARCV rsquo06) December 2006

[2] L Ljung System Identification Theory for the User Prentice-Hall Upper Saddle River NJ USA 1999

[3] J vanAmerongen andA JU T Cate ldquoModel reference adaptiveautopilots for shipsrdquoAutomatica vol 11 no 5 pp 441ndash449 1975

[4] K J Astrom andCGKallstrom ldquoIdentification of ship steeringdynamicsrdquo Automatica vol 12 no 1 pp 9ndash22 1976

[5] C G Kallstrom and K J Astrom ldquoExperiences of systemidentification applied to ship steeringrdquo Automatica vol 17 no1 pp 187ndash198 1981

[6] M A Abkowitz ldquoMeasurements of hydrodynamic character-istic from ship maneuvering trials by system identificationrdquoTransactions of the Society of Naval Architects and MarineEngineers vol 88 pp 283ndash318 1980

[7] T I Fossen S I Sagatun and A J Soslashrensen ldquoIdentificationof dynamically positioned shipsrdquo Modeling Identification andControl vol 17 no 2 pp 153ndash165 1996

[8] T Perez A J Soslashrensen and M Blanke ldquoMarine vessel modelsin changing operational conditionsmdasha tutorialrdquo in Proceedingsof the 14th IFAC Symposium on System Identification NewcastleAustralia 2006

[9] M Caccia G Bruzzone and R Bono ldquoA practical approach tomodeling and identification of small autonomous surface craftrdquoIEEE Journal of Oceanic Engineering vol 33 no 2 pp 133ndash1452008

[10] T I Fossen Marine Control Systems Guidance Navigationand Control of Ships Rigs and Underwater Vehicles MarineCybernetics Trondheim Norway 2002

[11] J M De La Cruz J Aranda and J M Giron ldquoAutomaticaMarina una revision desde el punto de vista de controlrdquo RevistaIberoamericana de Automatica e Informatica Industrial vol 9pp 205ndash218 2012

[12] K Nomoto T Taguchi K Honda and S Hirano ldquoOn the steer-ing qualities of shipsrdquo Tech Rep 4 International ShipbuildingProgress 1957

[13] K S Narendra andK Parthasarathy ldquoIdentification and controlof dynamical systems using neural networksrdquo IEEE Transac-tions on Neural Networks vol 1 no 1 pp 4ndash27 1990

[14] M R Haddara and Y Wang ldquoParametric identification ofmanoeuvring models for shipsrdquo International ShipbuildingProgress vol 46 no 445 pp 5ndash27 1999

[15] M R Haddara and J Xu ldquoOn the identification of shipcoupled heave-pitch motions using neural networksrdquo OceanEngineering vol 26 no 5 pp 381ndash400 1999

[16] K Hornik M Stinchcombe and HWhite ldquoMultilayer feedfor-ward networks are universal approximatorsrdquo Neural Networksvol 2 no 5 pp 359ndash366 1989

[17] A B Mahfouz ldquoIdentification of the nonlinear ship rollingmotion equation using the measured response at seardquo OceanEngineering vol 31 no 17-18 pp 2139ndash2156 2004

[18] V Vapnik and Z Chervonenkis ldquoOn the uniform convergenceof relative frequencies of events to their probabilitiesrdquo DokladyAkademii Nauk USS vol 4 no 181 1968

[19] M A Aızerman E M Braverman and L I Rozonoer ldquoA prob-abilistic problem on automata learning by pattern recognitionand themethod of potential functionsrdquoAutomation andRemoteControl vol 25 pp 821ndash837 1964

[20] B Scholkopf and A J Smola LearningWith Kernels MIT PressCambridge Mass USA 2002

[21] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[22] A J Smola and B Scholkopf ldquoA tutorial on support vectorregressionrdquo Statistics and Computing vol 14 no 3 pp 199ndash2222004

[23] PM L Drezet and R F Harrison ldquoSupport vectormachines forsystem identificationrdquo in Proceedings of the 1998 InternationalConference on Control pp 688ndash692 September 1998

[24] S Adachi and T Ogawa ldquoA new system identification methodbased on support vector machinesrdquo in Proceedings of theIFAC Workshop Adaptation and Learning in Control and SignalProcessing Cernobbio-Como Italy 2001

[25] G T Jemwa and C Aldrich ldquoNon-linear system identificationof an autocatalytic reactor using least squares support vectormachinesrdquoThe Journal of The South African Institute of Miningand Metallurgy vol 103 no 2 pp 119ndash125 2003

[26] W Zhong D Pi andY Sun ldquoSVMbased non parametricmodelidentification and dynamicmodel controlrdquo in Proceedings of the1rst International Conference on Natural Computation (ICNCrsquo05) vol 3610 of Lecture Notes in Computer Science pp 706ndash709 2005

[27] V Verdult J A K Suykens J Boets I Goethals and B deMoorldquoLeast squares support vector machines for kernel CCA innon-linear state-space identificationrdquo in Proceedings of the 16thInternational Symposium on Mathematical Theory of Networksand Systems (MTNS rsquo04) Leuven Belgium July 2004

[28] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of the InternationalConference on Intelligent Systems and Knowledge Engineering(ISKE rsquo07) Chengdu China October 2007

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

[29] S Totterman and H T Toivonen ldquoSupport vector method foridentification ofWienermodelsrdquo Journal of Process Control vol19 no 7 pp 1174ndash1181 2009

[30] J A K Suykens T van Geste J de Brabanter B de Moor andJ Vandewalle Least Squares Support Vector Machines WorldScientific Singapore 2002

[31] X D Wang and M Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of the 2004 International Conference on MachineLearning andCybernetics pp 941ndash945 Shanghai China August2004

[32] I Goethals K Pelckmans J A K Suykens and B De MoorldquoIdentification of MIMO Hammerstein models using leastsquares support vector machinesrdquoAutomatica vol 41 no 7 pp1263ndash1272 2005

[33] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Advances inNeural Networks (ISNN rsquo05) vol 3497 of Lecture Notes inComputer Science pp 436ndash441 2005

[34] L Wang H Lai and T Zhang ldquoAn improved algorithm onleast squares support vector machinesrdquo Information TechnologyJournal vol 7 no 2 pp 370ndash373 2008

[35] W L Luo and Z J Zou ldquoParametric identification of shipmaneuvering models by using support vector machinesrdquo Jour-nal of Ship Research vol 53 no 1 pp 19ndash30 2009

[36] X G Zhang andZ J Zou ldquoIdentification of Abkowitzmodel forship manoeuvring motion using 120576-support vector regressionrdquoJournal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[37] J Mercer ldquoFunctions of positive and negative type and theirconnection with the theory of integral equationsrdquo PhilosophicalTransactions of the Royal Society A vol 209 pp 415ndash446 1909

[38] M A Abkowitz ldquoLectures on ship hydrodynamics steering andmanoeuvrabilityrdquo Tech Rep Hy-5 Hydro and AerodynamicsLaboratory Lyngby Denmark 1964

[39] K S M Davidson and L I Schiff ldquoTurning and course keepingqualitiesrdquo Transactions of SNAME vol 54 pp 189ndash190 1946

[40] Matlab Help Files 2012 Cambridge Mass USA httpwwwmathworkscomhelpident

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of