research article semiphysical modelling of the nonlinear...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 890120 13 pageshttpdxdoiorg1011552013890120

Research ArticleSemiphysical Modelling of the Nonlinear Dynamics ofa Surface Craft with LS-SVM

David Moreno-Salinas1 Dictino Chaos1 Eva Besada-Portas2 Joseacute Antonio Loacutepez-Orozco2

Jesuacutes M de la Cruz2 and Joaquiacuten Aranda1

1 Departament of Computer Science and Automatic Control National University Distance Education (UNED) Madrid Spain2Department of Computer Architecture and Automatic Control Universidad Complutense de Madrid (UCM) Madrid Spain

Correspondence should be addressed to David Moreno-Salinas dmorenodiaunedes

Received 12 July 2013 Accepted 31 October 2013

Academic Editor Siddhivinayak Kulkarni

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

One of the most important problems in many research fields is the development of reliable mathematical models with goodpredictive ability to simulate experimental systems accurately Moreover in some of these fields as marine systems these modelsplay a key role due to the changing environmental conditions and the complexity and high cost of the infrastructure needed to carryout experimental tests In this paper a semiphysicalmodelling technique based on least-squares support vectormachines (LS-SVM)is proposed to determine a nonlinear mathematical model of a surface craft The speed and steering equations of the nonlinearmodel of Blanke are determined analysing the rudder angle surge and sway speeds and yaw rate from real experimental datameasured from a zig-zag manoeuvre made by a scale ship The predictive ability of the model is tested with different manoeuvringexperimental tests to show the good performance and prediction ability of the model computed

1 Introduction

The availability of tools and methods to compute mathemat-ical models for simulation purposes is of key importancemaking the system identification field one of the highlightsamong the research topics in engineering and one of themost important stages in the control research areaMoreoveraspects that go from the high cost of practical implementa-tions and tests in an open air environment to the complexityof the infrastructure needed to carry out experimental testsand even the changing environmental conditions call forthe availability of these mathematical models with whichnew designs and ideas can be tested in simulation withhigh accuracy In addition the importance of the modellingstage is crucial since an inadequate model identification mayyield large prediction errors The literature on linear andnonlinear system identification is extensive and covers manyareas of engineering research For a short survey on someessential features in the identification area and a classificationof methods the reader is referred to [1 2]

Among the number of techniques andmethods on systemidentification semiphysical modelling presents some inter-esting own characteristics In this method the prior knowl-edge about the application is used to develop a good modelstructure to be defined with raw measurements The modelstructures defined are not physically complete but allow forvery suitable models to describe the behaviour of the systemsinvolved [3] There exist several tools to be applied on semi-physical modelling [4] and potential tools that can supportthe process of semiphysical modelling are neural networksand support vector machines (SVM) The work at hand isfocused on the semiphysical modelling (based on SVM) of asurface craft to describe its dynamical behaviour in an experi-mental environmentThe computation of themodel is carriedout with the least-squares support vector machines (LS-SVM) technique [5] which is one of the different types ofSVM algorithms available in the literature

One of the most popular and widely used techniques inthe artificial intelligence (AI) field for system identificationis the artificial neural networks such asmultilayer perceptron

2 Mathematical Problems in Engineering

(MLP) see for example [6]This kind of techniques is robustand effective in many problems in identification and controlDespite this they present some disadvantages such as thelocalminima overfitting large computation time to convergeto the solution and so forth to name but a few Some ofthese problems can be solved effectively using SVM since itprovides a larger generalisation performance offering amoreattractive alternative for the system identification problem asit is not based on the empirical error implemented in neuralnetworks but on the structural risk minimization (SRM)The basic idea of SVM is Vapnik-Chervonenkis (VC) theorywhich defines ameasure of the capacity of a learningmachine[7] The idea is to map input data into a high-dimensionalfeature Hilbert space using a nonlinear mapping techniquethat is the kernel dot product trick [8] and to carry outlinear classification or regression in feature spaceThe Kernelfunctions replace a possibly very high-dimensional Hilbertspace without explicitly increasing the feature space [9]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) [10] Moreover SVM can be designed to dealwith sparse data where we have many variables but few dataFurthermore the solution of SVM is globally optimal Theformulation of SVM for regression that is support vectorregression (SVR) is very similar to the formulation of SVMfor classification For a survey on SVR the reader is referred to[11] and the references thereinThe formulation of SVR showshow this technique is suitable to be used as a semiphysicalmodelling tool to obtain the parameters of a mathematicalmodel In this sense it is of practical interest to describe anonlinear system from a finite number of input and outputmeasurements

Although there are not many results for system identifi-cation using SVM we can find some interesting works suchas the work in [12] where the authors make a study of thepossible use of SVM for system identification in [13] wherean identificationmethod based on SVR is proposed for linearregressionmodels or in [14] in which the application of SVMto time series modelling is considered by means of simulateddata from an autocatalytic reactor Other interesting workscan be found in [15ndash18] It is important to remark that mostof the papers that study the problem of system identificationusing SVM deal only with simulation data As mentionedabove among the different SVM techniques we can find LS-SVM [5] This technique allows a nice simplification of theproblem making it more tractable as will be commented inhigher detail in Section 2 We can also find some interestingworks that deal with this problem for example [19] whereLS-SVM is used for nonlinear system identification for somesimple examples of nonlinear autoregressive with exogenousinput (NARX) input-output models There also exist someother representative examples that deal with identificationusing LS-SVM see for example [20ndash22]

In this paper LS-SVM is used for the semiphysicalmodelling of a surface marine vessel System identification ofmarine vehicles starts in the 70s with the works in [23] wherean adaptive autopilot with reference model was presentedand in [24] where parametric linear identification techniques

were used to define the guidance dynamics of a ship using themaximum likelihood method There exist many algorithmsand tools to compute mathematical models that describe thedynamics of marine vehicles for different applications andscenarios For instance in [25] several parametric identifi-cation algorithms are used to design autopilots for differentkinds of ships in [26] the hydrodynamic characteristics of aship are determined by a Kalman filter (KF) and in [27] anextended kalman filter (EKF) is used for the identification ofthe ship dynamics for dynamic positioningThe computationof these models usually needs a lot of time and practicaltests to obtain enough information about the hydrodynamiccharacteristics of the vehicle and an important computationaleffort to define an accurate model so it is clear that theidentification task may become a complex and tedious taskMoreover the operational conditions may affect the vehicleproviding different models depending on these conditions asstudied in [28] For some other interesting related works thereader is referred to [29ndash31] and the references therein

For the above reason in some practical situations it isusual to employ simple vehicle models that although theyreproducewith less accuracy the dynamics of the vehicle theyshow very good results and prediction ability for most of thestandard operations see for example [32] where the authorsobtain a linear second-order Nomoto heading model with anadded autoregressive moving average (ARMA) disturbancemodel for an autonomous in-scale physical model of a fast-ferry They use a turning circle manoeuvre for the systemidentification See also [33] where a nonlinear ship modelis identified in towing tests in a marine control laboratoryfor automatic control purposes Following this trend in thispaper the nonlinear Blanke model is identified [30 34] Thismodel has a large prediction ability in the experimentalsetup for standard operations as will be seen throughout thiswork although the model is less precise than other modelsavailable in the literature such as the Abkowitz model [35]Furthermore it can be obtained with semiphysical modellingtechniques based on SVM in a fast manner with relatively fewdata

We can find some works that employ neural networksto define the dynamics of a surface marine vehicle such as[36ndash38] or [39] We can also find some interesting worksthat deal with the identification of marine vehicles by usingSVM for example [40] where an Abkowitz model forship manoeuvring is identified by using LS-SVM and [41]where 120598-SVM is employed for the computation of the samemodel These two above works search to determine thehydrodynamic coefficients of a mariner class vessel withsimple training manoeuvres however the identification ofthe mathematical model is made with data obtained fromsimulation and the prediction ability of the model is alsotested only in simulation These works do not deal with realdata Furthermore as far as the authors know most of theworks that deal with system identification using some SVRtechnique employ simulation data and numerical exampleswhere the models obtained are not tested on an experimentalsetup Two exceptions are the works [42] in which the steer-ing equations of a Nomoto second-order linear model withconstant surge speed are identified using LS-SVM and tested

Mathematical Problems in Engineering 3

in an experimental setup with a scale ship model and [43] inwhich an identification method based on SVM is proposedfor modelling nonlinear dynamics of a torpedo AUV In thisreference the authors determine the hydrodynamic modelwith a series of captive model tests and based on this experi-mentalmodel manoeuvring simulations are developedThenSVM is used to identify the damping terms and Coriolisand centripetal terms by analysing the simulation data Inthe work at hand following a similar methodology to thatexplained in [40 42] we seek to determine the nonlinearmodel of Blanke from raw data obtained from a physical scaleship and to validate themodelwith several experimental tests

Therefore the key contributions of the present paper aretwofold (i) the mathematical nonlinear Blanke model of ascale ship is computed from experimental data collected froma 2020 degree zig-zag manoeuvre with the LS-SVM tech-nique (ii) the prediction ability of the mathematical model istested on an open air environment with differentmanoeuvrescarried out with the scale shipThese tests allow checking theconnection between the mathematical model and the shipshowing how this nonlinear model predicts with large accu-racy the actual behaviour of the surface vessel In this sensethe model can be used to design control strategies to predictthe ship behaviour on a simulation environment before itsimplementation on the real vehicle It is important to keepin mind for the experimental results obtained in this paperthat the analytical properties of SVM can be compromisedin stochastic problems because the noise generates additionalsupport vectors However if the noise ratio is good and theamplitude is limited the SVM can solve the problem as if itwas deterministic [12]

The document is organized as follows In Section 2 LS-SVM is introduced The nonlinear model of Blanke andthe input and output data for the LS-SVM algorithm arestated in Section 3 In Section 4 the Blanke model obtainedfrom the semiphysical modelling is explicitly defined and itsprediction ability is tested with some manoeuvres namelyevolution circles and zig-zags Finally the conclusions and abrief discussion of topics for further research are included inSection 5

2 Least Squares Support VectorMachines for Regression

For the sake of completeness and clarity in this section LS-SVM is briefly introduced The notation and concepts of thissection follow the explanation in [5] The interested readercan see also [11] for a report on support vector regressionThebasic idea behind SVM is that using nonlinearmapping tech-niques the input data are mapped into a high-dimensionalfeature space where linear classification or regression iscarried out 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 output data 119887 isa bias term for the regression model 120596 is a matrix of weightsand 120593(sdot) R rarr R119899ℎ is the mapping to a high-dimensionalHilbert space where 119899

ℎcan be infinite The optimization

problem in the primal weight space for a given training set119909119894 119910119894119873119904

119894=1yields

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 119873119904is the number of samples 119890

119894are regression error

variables and 120574 is the regularisation parameter that deter-mines the deviation tolerated from the desired accuracy Theparameter 120574 must be always positive The minimization of120596119879120596 is closely related to the use of a weight decay term in the

training of neural networks and the second term of the right-hand side of (2) controls the tradeoff between the empiricalerror and the model complexity

In the above problem formulation 120596may become infinitedimensional and then the problem in the primalweight spacecannot be solved In this situation the Lagrangian must becomputed to derive the dual problem

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

Now the derivatives of (4) with respect to120596 119887 119890119894 and 120572

119894must

be computed to define the optimality conditions

120597L (120596 119887 119890 120572)

120597120596119887119890120572

997888rarr

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)

After straightforward computations variables 120596 and 119890 areeliminated from (5) and then the kernel trick is applied Thekernel trick allows us to work in large dimensional featurespaces without explicit computations on them [8] Thus theproblem formulation yields

119910 (119909) =

119873119904

sum

119894=1

120572119894119870(119909 119909

119894) + 119887 (6)

In (6) the term 119870(sdot sdot) represents the kernel function whichinvolves an inner product between its operands This kernelmust be positive definite and must satisfy the Mercer condi-tion [44] The equation defined in (6) may be applied now tocompute the regression model


Equation (6) is very similar to that which would beobtained for a standard SVM formulation The main differ-ences between both formulations are the equality constraintsin (3) and the squared error term of the second term in theright-hand side of (2) implying a significant simplificationof the problem

3 Semiphysical Modelling of the NonlinearModel of Blanke

In marine systems the experimental tests can become costlyin time and money due to the need of deployment cali-bration and recovery of complex systems at sea Thereforethe number of experimental tests that may be carried outare partially constrained by this reason among others likeenvironmental conditions transportation of equipment andso forth to name but a few In this sense the availability ofmathematical models which describe the real systems accu-rately is of utmost importance because most of these experi-mental testsmay be carried out in simulation predicting withhigh accuracy the real behaviour of the real systems andsaving a number of practical tests

There exists a wide range of different marine systems thatrequire mathematical models The problem arises when adetailed and trustworthymathematical shipmodel is neededsince it requires the identification of a multitude of hydrody-namic parameters see [35] This task can become hard andcomplex with the need of multiple experimental tests [30]

In many practical scenarios it is very usual to employsimple models that predict the behaviour of real ships withlarge accuracy in most of the standard operations like theNomoto models [45] For example in [42] the identificationof a second-order linear model of Nomoto for controlpurposes is defined although this model assumes linear shipdynamics anddescribes only the steering equationsThis kindof model may be insufficient accurate for some scenarios dueto its simplicity and thus its use would be seriously limited Itis necessary to compute a more general model to be appliedin a wider variety of situations and control actions and thiswork tries to overcome this limitation

Therefore in the present work a nonlinear manoeu-vring model based on second-order modulus functions isemployed The model used is the one proposed by Blanke[34] which is a simplification of the Norrbinrsquos nonlinearmodel but with the most important terms for steering andpropulsion loss assignmentThis 3-degree-of-freedom (DOF)manoeuvring model is defined following the definition in[30] as

(119898 minus 119883) = 119883

|119906|119906 |119906| 119906 + (119898 + 119883

120592119903) 120592119903

+ (119898119909119866+ 119883119903119903) 1199032+ (1 minus 119905) 119879 + 119883

1205751205751205752+ 119883ext

(119898 minus 119884 120592) 120592 + (119898119909

119866minus 119884 119903

) 119903 = minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592

+ 119884|120592|120592 |

120592| 120592 + 119884|120592|119903 |

120592| 119903

+ 119884120575120575 + 119884ext

(119898119909119866minus 119873 120592

) 120592 + (119868119911minus 119873 119903

) 119903 = minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592

+ 119873|120592|120592 |

120592| 120592 + 119873|120592|119903 |

120592| 119903

+ 119873120575120575 + 119873ext

(7)where 119906 is surge speed 119903 is yaw rate 120592 is sway velocity 120575 isthe rudder angle 119868

119911is moment of inertia about the 119911-axis119898

is mass 119909119866is the 119909-axis coordinate of the centre of gravity

119905 is the thrust deduction number 119879 is propeller thrust 119883120575120575

is resistance due to rudder deflection and 119883 119883|119906|119906

119883120592119903

119883119903119903 119883ext 119884 120592 119884119906119903 119884 119903 119884119906120592 119884|120592|120592 119884|120592|119903 119884120575 119884ext 119873 120592 119873 119903 119873119906119903

119873119906120592119873|120592|120592

119873|120592|119903

119873120575 and119873ext are added inertia hydrodynamic

coefficients For more details the reader is referred to [30]The interest of this particular model resides in that

despite its relative simplicity the most important nonlinearterms of the ship dynamics are taken into account Further-more it is possible to compute a dynamic model for controlpurposes from the experimental data without the need ofcomputing the hydrodynamic derivatives that define all theship characteristics and its complete behaviourTherefore (7)may be rewritten as

=

1

119898 minus 119883

(119883|119906|119906 |

119906| 119906 + (119898 + 119883120592119903) 120592119903

+ (119898119909119866+ 119883119903119903) 1199032+ (1 minus 119905) 119879

+ 1198831205751205751205752+ 119883ext)

120592 =

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592 + 119884

|120592|120592 |120592| 120592

+ 119884|120592|119903 |

120592| 119903 + 119884120575120575 + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592

+ 119873|120592|120592 |

120592| 120592 + 119873|120592|119903 |

120592| 119903 + 119873120575120575 + 119873ext)

119903 =

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592 + 119873

|120592|120592 |120592| 120592

+ 119873|120592|119903 |

120592| 119903 + 119873120575120575 + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592 + 119884

|120592|120592 |120592| 120592

+ 119884|120592|119903 |

120592| 119903 + 119884120575120575 + 119884ext)

(8)with Θ = (119868

119911minus 119873 119903

)(119898 minus 119884 120592) minus (119898119909

119866minus 119884 119903

)(119898119909119866minus 119873 120592

)Now we can proceed with the derivation of the semiphysicalmodel For simplicity reasons (8) is discretized with Eulerrsquosstepping method using a forward-difference approximationon the derivative119906 (119896 + 1) minus 119906 (119896)

Δ119896

=

1

119898 minus 119883

(119883|119906|119906 |

119906 (119896)| 119906 (119896)

+ (119898 + 119883120592119903) 120592 (119896) 119903 (119896)


+ (119898119909119866+ 119883119903119903) 119903(119896)2+ (1 minus 119905) 119879 (119896)

+ 119883120575120575120575(119896)2+ 119883ext)

120592 (119896 + 1) minus 120592 (119896)

Δ119896

=

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

119903 (119896 + 1) minus 119903 (119896)

Δ119896

=

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

(9)

whereΔ119896 is considered to be the sampling time of the inertialmeasurement unit (IMU) on board the ship and 119896 and 119896 + 1

define two successive data measurements from the IMU Wehave to rearrange the terms of (9) in a similar way as (6)Doing so (9) leads to

119906 (119896 + 1) = 119906 (119896) +

Δ119896 sdot 119883|119906|119906

119898 minus 119883

|119906 (119896)| 119906 (119896)

+

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

120592 (119896) 119903 (119896)

+

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

119903(119896)2

+

Δ119896

119898 minus 119883

(1 minus 119905) 119879 (119896) +

Δ119896 sdot 119883120575120575

119898 minus 119883

120575(119896)2

+

Δ119896 sdot 119883ext119898 minus 119883

120592 (119896 + 1)

= 120592 (119896) minus Δ119896Θminus1((119868119911minus 119873 119903

) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575) 120575 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)

119903 (119896 + 1)

= 119903 (119896) minus Δ119896Θminus1((119898 minus 119884 120592

) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575) 120575 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)

(10)

Equation (10) following the notation introduced in [5] and inSection 2 can be rewritten in compact form as

119910119896120585= Γ120585119909119896120585 (11)

for 119896 = 1 119873119904minus 1 where 120585 = 119906 120592 119903 119910

119896119906= 119906(119896 + 1) 119910

119896120592=

120592(119896 + 1) and 119910119896119903

= 119903(119896 + 1) are the output training data forthe sampling time 119896 and where the input training data are

119909119896119906= [119906 (119896) |119906 (119896)| 119906 (119896) 120592 (119896) 119903 (119896) 119903(119896)

2 119879 (119896) 120575(119896)

2 1]

119879


119909119896120592= [120592 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

119909119896119903= [119903 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

(12)

and with

Γ119906= [1

Δ119896 sdot 119883|119906|119906

119898 minus 119883

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

Δ119896

119898 minus 119883

(1 minus 119905)

Δ119896 sdot 119883120575120575

119898 minus 119883

Δ119896 sdot 119883ext119898 minus 119883

]

119879

Γ120592= [1 minusΔ119896Θ

minus1((119868119911minus 119873 119903

) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)]119879

Γ119903= [1 minusΔ119896Θ

minus1((119898 minus 119884 120592

) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575)

Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)]119879

(13)

The estimates of the elements in vectors (13) are obtainedfrom LS-SVM regression In fact we are interested in thevalues of the vectors Γ

120585themselves regardless of the values

of the different hydrodynamic coefficients that appear in (13)because these vectors will define the equations of motion ofthe ship and we look for a model for control purposes How-ever if we want to know the values of these hydrodynamiccoefficients additional manoeuvres should be carried outto identify some of them independently and then togetherwith the model obtained from LS-SVM those remainingwould be computed

The structure of the mathematical model is known inadvance and elements in vectors (13) are linear in theparameters so linear regression can be applied and a linear

Table 1 Main parameters and dimensions of the real and the scaleships

Parameter Real ship Scale shipLength between perpendiculars (Lpp) 74400m 4389mMaximum beam (B) 14200m 0838mMean depth to the top deck (H) 9050m 0534mDesign draught (Tm) 6300m 0372m

Figure 1 Scale ship used in the experimental tests

kernel 119870(119909119894 119909119895) = (119909

119894sdot 119909119895)may be used for the semiphysical

modelling

119910119896120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119896120585+ 119887120585

(14)

for 120585 = 119906 120592 119903 and 119896 = 1 119873119904 Comparing (14) with (11)

after the training process we have

Γ120585=

119873119904

sum

119894=1

120572119894120585119909119894120585 (15)

where the bias terms 119887120585must be equal to or approximately 0

The support vectors obtained allow to define the parametersof the Blanke model immediately from (13)

4 Experimental Results

The data used for the training of the LS-SVM algorithm wereobtained by carrying out a 2020 degree zig-zag manoeuvresince it is a simple manoeuvre but enough to define themain characteristics of the ship dynamics Once the modelis defined with the above zig-zag data its prediction abilitymust be compared with the real behaviour of the ship for thesame commanded input data namely surge speed and rudderangle

The vehicle used for the experimental tests is a scalemodel in a 11695 scale see Figure 1 The scale shiphereinafter referred to as the ship has the dimensions shownin Table 1 where the dimensions of the real ship that itrepresents are also shown

41 Semiphysical Modelling of the Surface Craft The 2020degree zig-zag manoeuvre to obtain the training data iscarried out with a commanded surge speed of 2ms during


08

06

04

02

0

minus02

minus04

minus06

minus080 10 20 30 40 50 60 70 80

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)

Yaw angleRudder angle

Figure 2 2020 degree zig-zag manoeuvre Yaw angle (solid line)and rudder angle (dashed line)

90 secondsThe sampling time is 02 seconds so 450 samplesare measured Figure 2 shows the commanded rudder angle(dashed line) and the corresponding yaw angle (solid line)defined by the vehicle during the 2020 degree zig-zagmanoeuvre The training data are the commanded controlsignals or inputs (rudder angle and surge speed) and the datameasured from the IMU on board the ship or outputs (effec-tive surge speed sway speed and yaw rate) For the sake ofclarity on the results shown the one sigma confidence levelsof the measured data from the IMU are heading 005 degattitude 0025 deg position 05m and velocity 004ms

Now the LS-SVMalgorithm for regressionmay be trainedwith these input and output data to compute the vectorsdefined in (13) Different values of the regularisation param-eter 120574 were tested and 120574 = 10

4 was selected as the bestcandidate Following the comments made by Blanke [34] theterm (119898119909

119866+ 119883119903119903) is considered to be zero since it will be

very small for most ships thus the nonlinear model after thetraining process yields

= minus00321 |119906| 119906 minus 27053120592119903 + 00600 minus 022571205752+ 006

120592 = minus04531119906119903 minus 05284119906120592 + 05354 |120592| 120592

minus 04121 |120592| 119903 + 00520120575 + 00007

119903 = minus11699119906119903 minus 06696119906120592 + 23001 |120592| 120592

+ 39335 |120592| 119903 minus 05503120575 minus 00054

(16)

Note that the term (1 minus 119905)119879(119896) of (8) is constant in (16)since the commanded surge speed is constant for all theexperiences carried out in the present work Once the modelis well defined we must check if it fits correctly the trainingdata that is it is necessary to compare the training datawith the results obtained with (16) for the same input signalsIn Figure 3 the comparison of the semiphysical modelling

17

16

15

14

13

12

11

1

Experimental dataSimulation data

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)

Figure 3 Surge speed measured in the zig-zag manoeuvre with theship (solid line) and in simulation (dashed line)

0 10 20 30 40 50 60 70 80

Time (s)

02

015

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)


Figure 4 Sway speed measured in the zig-zag manoeuvre with theship (solid line) and in simulation (dashed line)

results with the experimental data for the surge speed isshown It is important to notice that the scale used in Figure 3has been chosen to show clearly the difference between thesimulated and real surge speeds but we can see how themaximum error between both speeds is less than 01ms andhence the simulation results are very similar to the real ones

Similarly in Figure 4 the sway speed measured from theIMU on board the ship is shown together with the swayspeed obtained from the Blankemodel defined in (16) Noticehow the results are also very similar Moreover the largesimilarity between the real and simulated sway speeds is evenmore interesting because the sway speed cannot be directlycontrolled due to the fact that the ship studied is an under-actuated vehicle that is we have more degrees of freedom


0 10 20 30 40 50 60 70 80

Time (s)

015

01

005

0

minus005

minus01


Yaw

rate

(rad

s)

Figure 5 Yaw rate obtained in the zig-zag manoeuvre with the ship(solid line) and in simulation (dashed line)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

Error in surge speedError in sway speedError in yaw rate

0 10 20 30 40 50 60 70 80

Time (s)

Figure 6 Approximation errors in the surge speed (dashed line)sway speed (dotted line) and yaw rate (solid line)

(DOF) than control actions the latter acting directly on thesurge speed and the rudder angle

Finally in Figure 5 the IMU and simulated yaw rates areshown Notice how both curves are practically the sameshowing that the model has a dynamical behaviour very closeto the actual one of the ship

For comparison purposes in Figure 6 the approximationerrors for the surge speed the sway speed and the yaw rate areshown It can be seen how the errors are very small and theiraverage values are very close to zero The standard deviationof the error in the surge speed is 00486ms and in the swayspeed is 00171ms For the yaw rate the standard deviation

is 00066 rads so it is clear that the simulation model has abehaviour very close to the real one

42 Predictive Ability of the Model The predictive ability ofthemodelmust be testedwith different tests andmanoeuvresFor this purpose two different manoeuvres are now under-taken These tests are some turning manoeuvres (evolutioncircles) and a 1010 degree zig-zag manoeuvre The initialvalues of the effective surge speed sway speed and yaw rateused in the simulation tests are the same as those of the realones to show clearly the connection between the real and thesimulated systems

421 Test 1 Evolution Circles Thefirst validation test consistsin two turning manoeuvres (evolution circles) for com-manded rudder angles of plusmn20 deg The test was run during240 seconds for each of the turning manoeuvres In Figures7(a) and 7(b) we can check the effective surge speed forthe ship (solid line) and for the simulation model (dashedline) during these experimental tests for commanded rudderangles ofminus20 deg and+20 deg respectivelyNotice the similarbehaviour of both speeds and how the simulated surge speedis smoother than the real one because the simulated model isnot affected by noise or disturbances

In Figures 8(a) and 8(b) the sway speeds for the ship andthe simulation model are shown for the commanded rudderangles of minus20 deg and +20 deg respectively It can be seenhow both speeds are very similar although as mentionedabove the simulated one is free of noise and disturbances

The yaw rate for the simulated and the actual systems canbe studied in Figures 9(a) and 9(b) for the two manoeuvreswhere it is shown that the simulationmodel and the ship havea similar behaviour In Figures 9 and 7 we can also noticethat the real system behaviour is not exactly symmetric theturnings are slightly larger for negative rudder angles Thisnonsymmetrical behaviour is possibly also the reason forthe different (small) errors in surge and sway speeds whichvary depending on the turning angle Despite the abovementioned the results obtained from the simulated modelare very similar to the actual ones and their difference is notsignificant Moreover the nonsymmetrical behaviour may becaused by environmental conditions like currents or windsor by structural characteristics like the trimming of the shipThis problem does not arise with the semiphysical modelsince it does not incorporate environmental disturbancesthat are always present in an experimental setup Includingthe possibility of modelling the environmental disturbanceswould be some interesting future work

In Figures 10(a) and 10(b) the approximation errorsbetween the real and the predicted surge speed sway speedand yaw rate are shown Notice how the yaw rate erroris larger for negative rudder angles as mentioned aboveDespite the commented deviation the errors are small andtheir average values are close to zero providing a more thansatisfactory prediction of the real dynamical behaviour of theship In this sense the standard deviation of the predictedsurge speed with respect to the real one is 01786ms in thefirst manoeuvre and 004ms in the second one For the swayspeed the standard deviations are 00303ms and 00236ms


185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)

Figure 7 Surge speed obtained in two turning manoeuvres with the ship (solid line) and in simulation (dashed line) (a) minus20 deg and (b)+20 deg

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)

Figure 8 Sway speed obtained in two turning manoeuvres with the ship (solid line) and in simulation (dashed line) (a) minus20 deg and (b)+20 deg


01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)

Figure 9 Yaw rate obtained in two turningmanoeuvres with the ship (solid line) and in simulation (dashed line) (a) minus20 deg and (b) +20 deg


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)

Figure 10 Approximation errors in the surge speed (dashed line) sway speed (dotted line) and yaw rate (solid line) for the turningmanoeuvres (a) minus20 deg and (b) +20 deg



196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)

Figure 11 Surge speed obtained in a 1010 degree zig-zagmanoeuvrewith the ship (solid line) and in simulation (dashed line)

respectively Finally the yaw rate standard deviations are00160 rads and 00129 rads respectively These standarddeviations give us a clear image of the slightly nonsymmetri-cal behaviour of the real ship showing also how the simulatedmodel has a dynamical behaviour very close to that seen inthe real ship

422 Test 2 1010 Degree Zig-Zag Manoeuvre In this secondtest a 1010 degree zig-zag manoeuvre is carried out to provethe prediction ability of the model The manoeuvre is runduring 90 seconds In Figure 11 the surge speed during thezig-zag test is shown for both the simulated model and theship Notice again that the scale used in Figure 11 has beenchosen to show the difference between both speeds andthat the maximum error is around 006ms Therefore bothspeeds are very similar and the approximation error is verysmall as the details in Figure 14 show

In Figure 12 the sway speed for both systems is shown andthe similarity between both outputs is again easy to checkFinally in Figure 13 the yaw rate shows that the simulationmodel obtained with LS-SVM regression has a dynamicalbehaviour very close to that of the real ship

In Figure 14 the approximation errors in the surge speedsway speed and yaw rate are shown The standard deviationof the error in the surge speed for this case is 00466msin the sway speed is 00239ms and in the yaw rate is00097 rads Hence the model predicts again the behaviourof the real ship with large accuracy validating the modelobtained with the LS-SVM regression algorithm

Therefore it is clear that the nonlinear mathematicalmodel defined for a surface marine vehicle with LS-SVMprovides a satisfactory result which predicts with large accu-racy the nonlinear dynamics of the experimental system andthat it is suitable to be used for control purposes Thus thistechnique has the potential to be implemented for differentkinds ofmarine vehicles in a simple and fastmanner avoiding


0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)

Figure 12 Sway speed obtained in a 1010 degree zig-zagmanoeuvrewith the ship (solid line) and in simulation (dashed line)


01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)

Figure 13 Yaw rate obtained in a 1010 degree zig-zag manoeuvrewith the ship (solid line) and in simulation (dashed line)

many practical tests to define a reliable mathematical modeland providing a very large prediction ability

It would be interesting as future research to compare theresults obtained in this work with the results that would beobtained using extreme learning machines (ELM) [46] asthis technique overcomes some drawbacks that neural net-works present and it also reduces significantly the computa-tion time [47]

5 Conclusions and Future Work

In this work the nonlinear ship model of Blanke has beencomputed using experimental data obtained from a zig-zag


008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)


manoeuvre test A semiphysical modelling technique basedon a least squares support vector machines algorithm hasbeen applied to determine the parameters of the nonlinearmodel using the rudder angle surge and sway speeds andyaw rate as training data It was shown that the modelobtained fits the training data in a nice manner showing thesimulated system a behaviour very similar to that of the realship Furthermore the prediction ability of the model wasvalidated carrying out several experimental tests like turningmanoeuvres and zig-zags demonstrating that the mathemat-ical model can reproduce the actual ship dynamics with largeaccuracy in different manoeuvres In addition the modelcomputed is suitable to be used for testing control algorithmsin simulation avoiding the execution of a large number ofexperimental tests

Future work will aim at (i) extending the methodologydeveloped to deal with models whose structures are notknown in advance to capture all the features of the real shipincorporating disturbances and environmental conditions(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 and(iii) comparing the results obtained in this work with otherdifferent identification techniques like the extreme learningmachines (ELM)

Acknowledgments

The authors wish to thank the Spanish Ministry of Scienceand Innovation (MICINN) for support under ProjectsDPI2009-14552-C02-01 and DPI2009-14552-C02-02 Theauthors wish to thank also the National University DistanceEducation (UNED) for support under Project 2012VPUNED0003

References

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

[2] L Ljung ldquoIdentification of Nonlinear Systemsrdquo in Proceedingsof the International Conference onControl Automation Roboticsand Vision 2006

[3] D E Rivera ldquoTeaching semiphysical modeling to ChE studentsusing a brine-water mixing tank experimentrdquo Chemical Engi-neering Education vol 39 no 4 pp 308ndash315 2005

[4] P Lindskog and L Ljung ldquoTools for semiphysical modellingrdquoInternational Journal of Adaptive Control and Signal Processingvol 9 no 6 pp 509ndash523 1995

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

[6] 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

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

[8] M Aizerman E Braverman and L Rozonoer ldquoTheoreticalfoundations of the potential function method in pattern recog-nition learningrdquo Automation and Remote Control vol 25 pp821ndash837 1964

[9] B Scholkopf and A J Smola LearningWith Kernels MIT pressCambridge Mass USA 2002

[10] V Vapnik Statistical Learning Theory John Wiley amp Sons NewYork NY USA 1998

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

[12] P M L Drezet and R F Harrison ldquoSupport vector machinesfor system identificationrdquo in Proceedings of the InternationalConference on Control pp 688ndash692 September 1998

[13] 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 LrsquoAquila Italy 2001

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

[15] W Zhong D Pi and Y Sun ldquoSVM based nonparametric modelidentification and dynamicmodel controlrdquo in Proceedings of theFirst International Conference on Natural Computation (ICNCrsquo05) pp 706ndash709 August 2005

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

[17] W ZhongHGe and FQian ldquoModel identification and controlfor nonlinear discrete-time systems with time delay a supportvector machine approachrdquo in Proceedings of International Con-ference on Intelligent Systems and Knowledge Engineering (ISKErsquo07) Chengdu China October 2007

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


[19] X-D Wang and M-Y Ye ldquoNonlinear dynamic system identifi-cation using least squares support vector machine regressionrdquoin Proceedings of International Conference on Machine Learningand Cybernetics pp 941ndash945 Shanghai China August 2004

[20] 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

[21] Z Yu and Y Cai ldquoLeast squares wavelet support vectormachines for nonlinear system identificationrdquo in Proceedingsof the Second International Symposium on Neural NetworksAdvances in Neural Networks (ISNN rsquo05) pp 436ndash441 June2005

[22] LWang H Lai and T Zhang ldquoAn improved algorithm on leastsquares support vectormachinesrdquo Information Technology Jour-nal vol 7 no 2 pp 370ndash373 2008

[23] J van Amerongen and A J Udink Ten Cate ldquoModel referenceadaptive autopilots for shipsrdquo Original Research Article Auto-matica vol 11 no 5 pp 441ndash449 1975

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

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

[26] M A Abkowitz ldquoMeasurement of hydrodynamic character-istics from ship maneuvering trials by system identificationrdquoTransactions of Society of Naval Architects andMarine Engineersvol 88 pp 283ndash318 1981

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

[28] 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

[29] 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

[30] T I FossenMarine Control Systems Guidance Navigation andControl of Ships Rigs and Underwater Vehicles Marine Cyber-netics Trondheim Norway 2002

[31] 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

[32] F J Velasco E Revestido L Eopez and E Moyano ldquoIdentifi-cation for a heading autopilot of an autonomous in-scale fastferryrdquo IEEE Journal of Oceanic Engineering vol 38 no 2 pp263ndash274 2013

[33] R Skjetne Oslash N Smogeli and T I Fossen ldquoA nonlinear shipmanoeuvering model identification and adaptive control withexperiments for a model shiprdquo Modeling Identification andControl vol 25 no 1 pp 3ndash27 2004

[34] M Blanke Ship propulsion losses related to automated steeringand primemover control [PhD thesis]TheTechnical Universityof Denmark Lyngby Denmark 1981

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

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

[37] M R Haddara and J Xu ldquoOn the identification of ship coupledheave-pitch motions using neural networksrdquo Ocean Engineer-ing vol 26 no 5 pp 381ndash400 1998

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

[39] 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

[40] 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

[41] X-G Zhang and Z-J Zou ldquoIdentification of Abkowitz modelfor ship manoeuvring motion using 120598-support vector regres-sionrdquo Journal of Hydrodynamics vol 23 no 3 pp 353ndash360 2011

[42] D Moreno-Salinas D Chaos J M de la Cruz and J ArandaldquoIdentification of a surface marine vessel using LS-SVMrdquo Jour-nal of Applied Mathematics vol 2013 Article ID 803548 11pages 2013

[43] F Xu Z-J Zou J-C Yin and J Cao ldquoIdentification modelingof underwater vehiclesrsquononlinear dynamics based on supportvectormachinesrdquoOcean Engineering vol 67 Article ID 002980pp 68ndash76 2013

[44] 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

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

[46] G-B Huang Q-Y Zhu and C-K Siew ldquoExtreme learningmachine a new learning scheme of feedforward neural net-worksrdquo in Proceedings of the IEEE International Joint Conferenceon Neural Networks pp 985ndash990 July 2004

[47] R Rajesh and J Siva Prakash ldquoExtreme learning machinesmdasha review and state-of-the-artrdquo International Journal of WisdomBased Computing vol 1 no 1 2011

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(MLP) see for example [6]This kind of techniques is robustand effective in many problems in identification and controlDespite this they present some disadvantages such as thelocalminima overfitting large computation time to convergeto the solution and so forth to name but a few Some ofthese problems can be solved effectively using SVM since itprovides a larger generalisation performance offering amoreattractive alternative for the system identification problem asit is not based on the empirical error implemented in neuralnetworks but on the structural risk minimization (SRM)The basic idea of SVM is Vapnik-Chervonenkis (VC) theorywhich defines ameasure of the capacity of a learningmachine[7] The idea is to map input data into a high-dimensionalfeature Hilbert space using a nonlinear mapping techniquethat is the kernel dot product trick [8] and to carry outlinear classification or regression in feature spaceThe Kernelfunctions replace a possibly very high-dimensional Hilbertspace without explicitly increasing the feature space [9]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) [10] Moreover SVM can be designed to dealwith sparse data where we have many variables but few dataFurthermore the solution of SVM is globally optimal Theformulation of SVM for regression that is support vectorregression (SVR) is very similar to the formulation of SVMfor classification For a survey on SVR the reader is referred to[11] and the references thereinThe formulation of SVR showshow this technique is suitable to be used as a semiphysicalmodelling tool to obtain the parameters of a mathematicalmodel In this sense it is of practical interest to describe anonlinear system from a finite number of input and outputmeasurements

Although there are not many results for system identifi-cation using SVM we can find some interesting works suchas the work in [12] where the authors make a study of thepossible use of SVM for system identification in [13] wherean identificationmethod based on SVR is proposed for linearregressionmodels or in [14] in which the application of SVMto time series modelling is considered by means of simulateddata from an autocatalytic reactor Other interesting workscan be found in [15ndash18] It is important to remark that mostof the papers that study the problem of system identificationusing SVM deal only with simulation data As mentionedabove among the different SVM techniques we can find LS-SVM [5] This technique allows a nice simplification of theproblem making it more tractable as will be commented inhigher detail in Section 2 We can also find some interestingworks that deal with this problem for example [19] whereLS-SVM is used for nonlinear system identification for somesimple examples of nonlinear autoregressive with exogenousinput (NARX) input-output models There also exist someother representative examples that deal with identificationusing LS-SVM see for example [20ndash22]

In this paper LS-SVM is used for the semiphysicalmodelling of a surface marine vessel System identification ofmarine vehicles starts in the 70s with the works in [23] wherean adaptive autopilot with reference model was presentedand in [24] where parametric linear identification techniques

were used to define the guidance dynamics of a ship using themaximum likelihood method There exist many algorithmsand tools to compute mathematical models that describe thedynamics of marine vehicles for different applications andscenarios For instance in [25] several parametric identifi-cation algorithms are used to design autopilots for differentkinds of ships in [26] the hydrodynamic characteristics of aship are determined by a Kalman filter (KF) and in [27] anextended kalman filter (EKF) is used for the identification ofthe ship dynamics for dynamic positioningThe computationof these models usually needs a lot of time and practicaltests to obtain enough information about the hydrodynamiccharacteristics of the vehicle and an important computationaleffort to define an accurate model so it is clear that theidentification task may become a complex and tedious taskMoreover the operational conditions may affect the vehicleproviding different models depending on these conditions asstudied in [28] For some other interesting related works thereader is referred to [29ndash31] and the references therein

For the above reason in some practical situations it isusual to employ simple vehicle models that although theyreproducewith less accuracy the dynamics of the vehicle theyshow very good results and prediction ability for most of thestandard operations see for example [32] where the authorsobtain a linear second-order Nomoto heading model with anadded autoregressive moving average (ARMA) disturbancemodel for an autonomous in-scale physical model of a fast-ferry They use a turning circle manoeuvre for the systemidentification See also [33] where a nonlinear ship modelis identified in towing tests in a marine control laboratoryfor automatic control purposes Following this trend in thispaper the nonlinear Blanke model is identified [30 34] Thismodel has a large prediction ability in the experimentalsetup for standard operations as will be seen throughout thiswork although the model is less precise than other modelsavailable in the literature such as the Abkowitz model [35]Furthermore it can be obtained with semiphysical modellingtechniques based on SVM in a fast manner with relatively fewdata

We can find some works that employ neural networksto define the dynamics of a surface marine vehicle such as[36ndash38] or [39] We can also find some interesting worksthat deal with the identification of marine vehicles by usingSVM for example [40] where an Abkowitz model forship manoeuvring is identified by using LS-SVM and [41]where 120598-SVM is employed for the computation of the samemodel These two above works search to determine thehydrodynamic coefficients of a mariner class vessel withsimple training manoeuvres however the identification ofthe mathematical model is made with data obtained fromsimulation and the prediction ability of the model is alsotested only in simulation These works do not deal with realdata Furthermore as far as the authors know most of theworks that deal with system identification using some SVRtechnique employ simulation data and numerical exampleswhere the models obtained are not tested on an experimentalsetup Two exceptions are the works [42] in which the steer-ing equations of a Nomoto second-order linear model withconstant surge speed are identified using LS-SVM and tested







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




119894=1yields

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)






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



119894must


120597L (120596 119887 119890 120572)

120597120596119887119890120572

997888rarr

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)


119910 (119909) =

119873119904

sum

119894=1

120572119894119870(119909 119909

119894) + 119887 (6)









(119898 minus 119883) = 119883

|119906|119906 |119906| 119906 + (119898 + 119883

120592119903) 120592119903

+ (119898119909119866+ 119883119903119903) 1199032+ (1 minus 119905) 119879 + 119883

1205751205751205752+ 119883ext

(119898 minus 119884 120592) 120592 + (119898119909

119866minus 119884 119903

) 119903 = minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592

+ 119884|120592|120592 |

120592| 120592 + 119884|120592|119903 |

120592| 119903

+ 119884120575120575 + 119884ext

(119898119909119866minus 119873 120592

) 120592 + (119868119911minus 119873 119903

) 119903 = minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592

+ 119873|120592|120592 |

120592| 120592 + 119873|120592|119903 |

120592| 119903

+ 119873120575120575 + 119873ext






119883120592119903

119883119903119903 119883ext 119884 120592 119884119906119903 119884 119903 119884119906120592 119884|120592|120592 119884|120592|119903 119884120575 119884ext 119873 120592 119873 119903 119873119906119903

119873119906120592119873|120592|120592

119873|120592|119903




=

1

119898 minus 119883

(119883|119906|119906 |

119906| 119906 + (119898 + 119883120592119903) 120592119903

+ (119898119909119866+ 119883119903119903) 1199032+ (1 minus 119905) 119879

+ 1198831205751205751205752+ 119883ext)

120592 =

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592 + 119884

|120592|120592 |120592| 120592

+ 119884|120592|119903 |

120592| 119903 + 119884120575120575 + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592

+ 119873|120592|120592 |

120592| 120592 + 119873|120592|119903 |

120592| 119903 + 119873120575120575 + 119873ext)

119903 =

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592 + 119873

|120592|120592 |120592| 120592

+ 119873|120592|119903 |

120592| 119903 + 119873120575120575 + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592 + 119884

|120592|120592 |120592| 120592

+ 119884|120592|119903 |

120592| 119903 + 119884120575120575 + 119884ext)

(8)with Θ = (119868

119911minus 119873 119903

)(119898 minus 119884 120592) minus (119898119909

119866minus 119884 119903

)(119898119909119866minus 119873 120592


Δ119896

=

1

119898 minus 119883

(119883|119906|119906 |

119906 (119896)| 119906 (119896)

+ (119898 + 119883120592119903) 120592 (119896) 119903 (119896)


+ (119898119909119866+ 119883119903119903) 119903(119896)2+ (1 minus 119905) 119879 (119896)

+ 119883120575120575120575(119896)2+ 119883ext)

120592 (119896 + 1) minus 120592 (119896)

Δ119896

=

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

119903 (119896 + 1) minus 119903 (119896)

Δ119896

=

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

(9)



119906 (119896 + 1) = 119906 (119896) +

Δ119896 sdot 119883|119906|119906

119898 minus 119883

|119906 (119896)| 119906 (119896)

+

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

120592 (119896) 119903 (119896)

+

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

119903(119896)2

+

Δ119896

119898 minus 119883

(1 minus 119905) 119879 (119896) +

Δ119896 sdot 119883120575120575

119898 minus 119883

120575(119896)2

+

Δ119896 sdot 119883ext119898 minus 119883

120592 (119896 + 1)


) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575) 120575 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)

119903 (119896 + 1)


) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575) 120575 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)

(10)


119910119896120585= Γ120585119909119896120585 (11)

for 119896 = 1 119873119904minus 1 where 120585 = 119906 120592 119903 119910

119896119906= 119906(119896 + 1) 119910

119896120592=

120592(119896 + 1) and 119910119896119903


119909119896119906= [119906 (119896) |119906 (119896)| 119906 (119896) 120592 (119896) 119903 (119896) 119903(119896)

2 119879 (119896) 120575(119896)

2 1]

119879


119909119896120592= [120592 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

119909119896119903= [119903 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

(12)

and with

Γ119906= [1

Δ119896 sdot 119883|119906|119906

119898 minus 119883

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

Δ119896

119898 minus 119883

(1 minus 119905)

Δ119896 sdot 119883120575120575

119898 minus 119883

Δ119896 sdot 119883ext119898 minus 119883

]

119879

Γ120592= [1 minusΔ119896Θ

minus1((119868119911minus 119873 119903

) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)]119879

Γ119903= [1 minusΔ119896Θ

minus1((119898 minus 119884 120592

) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575)

Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)]119879

(13)








kernel 119870(119909119894 119909119895) = (119909


modelling

119910119896120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119896120585+ 119887120585

(14)



Γ120585=

119873119904

sum

119894=1

120572119894120585119909119894120585 (15)








08

06

04

02

0

minus02

minus04

minus06

minus080 10 20 30 40 50 60 70 80

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)









120592 = minus04531119906119903 minus 05284119906120592 + 05354 |120592| 120592

minus 04121 |120592| 119903 + 00520120575 + 00007

119903 = minus11699119906119903 minus 06696119906120592 + 23001 |120592| 120592

+ 39335 |120592| 119903 minus 05503120575 minus 00054

(16)


17

16

15

14

13

12

11

1


Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)


0 10 20 30 40 50 60 70 80

Time (s)

02

015

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)






0 10 20 30 40 50 60 70 80

Time (s)

015

01

005

0

minus005

minus01


Yaw

rate

(rad

s)


008

006

004

002

0

minus002

minus004

minus006

minus008

minus01Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)


0 10 20 30 40 50 60 70 80

Time (s)












185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)


01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















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




119894=1yields

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)






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



119894must


120597L (120596 119887 119890 120572)

120597120596119887119890120572

997888rarr

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)


119910 (119909) =

119873119904

sum

119894=1

120572119894119870(119909 119909

119894) + 119887 (6)









(119898 minus 119883) = 119883

|119906|119906 |119906| 119906 + (119898 + 119883

120592119903) 120592119903

+ (119898119909119866+ 119883119903119903) 1199032+ (1 minus 119905) 119879 + 119883

1205751205751205752+ 119883ext

(119898 minus 119884 120592) 120592 + (119898119909

119866minus 119884 119903

) 119903 = minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592

+ 119884|120592|120592 |

120592| 120592 + 119884|120592|119903 |

120592| 119903

+ 119884120575120575 + 119884ext

(119898119909119866minus 119873 120592

) 120592 + (119868119911minus 119873 119903

) 119903 = minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592

+ 119873|120592|120592 |

120592| 120592 + 119873|120592|119903 |

120592| 119903

+ 119873120575120575 + 119873ext






119883120592119903

119883119903119903 119883ext 119884 120592 119884119906119903 119884 119903 119884119906120592 119884|120592|120592 119884|120592|119903 119884120575 119884ext 119873 120592 119873 119903 119873119906119903

119873119906120592119873|120592|120592

119873|120592|119903




=

1

119898 minus 119883

(119883|119906|119906 |

119906| 119906 + (119898 + 119883120592119903) 120592119903

+ (119898119909119866+ 119883119903119903) 1199032+ (1 minus 119905) 119879

+ 1198831205751205751205752+ 119883ext)

120592 =

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592 + 119884

|120592|120592 |120592| 120592

+ 119884|120592|119903 |

120592| 119903 + 119884120575120575 + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592

+ 119873|120592|120592 |

120592| 120592 + 119873|120592|119903 |

120592| 119903 + 119873120575120575 + 119873ext)

119903 =

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592 + 119873

|120592|120592 |120592| 120592

+ 119873|120592|119903 |

120592| 119903 + 119873120575120575 + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592 + 119884

|120592|120592 |120592| 120592

+ 119884|120592|119903 |

120592| 119903 + 119884120575120575 + 119884ext)

(8)with Θ = (119868

119911minus 119873 119903

)(119898 minus 119884 120592) minus (119898119909

119866minus 119884 119903

)(119898119909119866minus 119873 120592


Δ119896

=

1

119898 minus 119883

(119883|119906|119906 |

119906 (119896)| 119906 (119896)

+ (119898 + 119883120592119903) 120592 (119896) 119903 (119896)


+ (119898119909119866+ 119883119903119903) 119903(119896)2+ (1 minus 119905) 119879 (119896)

+ 119883120575120575120575(119896)2+ 119883ext)

120592 (119896 + 1) minus 120592 (119896)

Δ119896

=

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

119903 (119896 + 1) minus 119903 (119896)

Δ119896

=

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

(9)



119906 (119896 + 1) = 119906 (119896) +

Δ119896 sdot 119883|119906|119906

119898 minus 119883

|119906 (119896)| 119906 (119896)

+

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

120592 (119896) 119903 (119896)

+

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

119903(119896)2

+

Δ119896

119898 minus 119883

(1 minus 119905) 119879 (119896) +

Δ119896 sdot 119883120575120575

119898 minus 119883

120575(119896)2

+

Δ119896 sdot 119883ext119898 minus 119883

120592 (119896 + 1)


) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575) 120575 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)

119903 (119896 + 1)


) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575) 120575 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)

(10)


119910119896120585= Γ120585119909119896120585 (11)

for 119896 = 1 119873119904minus 1 where 120585 = 119906 120592 119903 119910

119896119906= 119906(119896 + 1) 119910

119896120592=

120592(119896 + 1) and 119910119896119903


119909119896119906= [119906 (119896) |119906 (119896)| 119906 (119896) 120592 (119896) 119903 (119896) 119903(119896)

2 119879 (119896) 120575(119896)

2 1]

119879


119909119896120592= [120592 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

119909119896119903= [119903 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

(12)

and with

Γ119906= [1

Δ119896 sdot 119883|119906|119906

119898 minus 119883

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

Δ119896

119898 minus 119883

(1 minus 119905)

Δ119896 sdot 119883120575120575

119898 minus 119883

Δ119896 sdot 119883ext119898 minus 119883

]

119879

Γ120592= [1 minusΔ119896Θ

minus1((119868119911minus 119873 119903

) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)]119879

Γ119903= [1 minusΔ119896Θ

minus1((119898 minus 119884 120592

) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575)

Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)]119879

(13)








kernel 119870(119909119894 119909119895) = (119909


modelling

119910119896120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119896120585+ 119887120585

(14)



Γ120585=

119873119904

sum

119894=1

120572119894120585119909119894120585 (15)








08

06

04

02

0

minus02

minus04

minus06

minus080 10 20 30 40 50 60 70 80

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)









120592 = minus04531119906119903 minus 05284119906120592 + 05354 |120592| 120592

minus 04121 |120592| 119903 + 00520120575 + 00007

119903 = minus11699119906119903 minus 06696119906120592 + 23001 |120592| 120592

+ 39335 |120592| 119903 minus 05503120575 minus 00054

(16)


17

16

15

14

13

12

11

1


Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)


0 10 20 30 40 50 60 70 80

Time (s)

02

015

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)






0 10 20 30 40 50 60 70 80

Time (s)

015

01

005

0

minus005

minus01


Yaw

rate

(rad

s)


008

006

004

002

0

minus002

minus004

minus006

minus008

minus01Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)


0 10 20 30 40 50 60 70 80

Time (s)












185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)


01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















(119898 minus 119883) = 119883

|119906|119906 |119906| 119906 + (119898 + 119883

120592119903) 120592119903

+ (119898119909119866+ 119883119903119903) 1199032+ (1 minus 119905) 119879 + 119883

1205751205751205752+ 119883ext

(119898 minus 119884 120592) 120592 + (119898119909

119866minus 119884 119903

) 119903 = minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592

+ 119884|120592|120592 |

120592| 120592 + 119884|120592|119903 |

120592| 119903

+ 119884120575120575 + 119884ext

(119898119909119866minus 119873 120592

) 120592 + (119868119911minus 119873 119903

) 119903 = minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592

+ 119873|120592|120592 |

120592| 120592 + 119873|120592|119903 |

120592| 119903

+ 119873120575120575 + 119873ext






119883120592119903

119883119903119903 119883ext 119884 120592 119884119906119903 119884 119903 119884119906120592 119884|120592|120592 119884|120592|119903 119884120575 119884ext 119873 120592 119873 119903 119873119906119903

119873119906120592119873|120592|120592

119873|120592|119903




=

1

119898 minus 119883

(119883|119906|119906 |

119906| 119906 + (119898 + 119883120592119903) 120592119903

+ (119898119909119866+ 119883119903119903) 1199032+ (1 minus 119905) 119879

+ 1198831205751205751205752+ 119883ext)

120592 =

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592 + 119884

|120592|120592 |120592| 120592

+ 119884|120592|119903 |

120592| 119903 + 119884120575120575 + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592

+ 119873|120592|120592 |

120592| 120592 + 119873|120592|119903 |

120592| 119903 + 119873120575120575 + 119873ext)

119903 =

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906119903 + 119873

119906120592119906120592 + 119873

|120592|120592 |120592| 120592

+ 119873|120592|119903 |

120592| 119903 + 119873120575120575 + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906119903 + 119884

119906120592119906120592 + 119884

|120592|120592 |120592| 120592

+ 119884|120592|119903 |

120592| 119903 + 119884120575120575 + 119884ext)

(8)with Θ = (119868

119911minus 119873 119903

)(119898 minus 119884 120592) minus (119898119909

119866minus 119884 119903

)(119898119909119866minus 119873 120592


Δ119896

=

1

119898 minus 119883

(119883|119906|119906 |

119906 (119896)| 119906 (119896)

+ (119898 + 119883120592119903) 120592 (119896) 119903 (119896)


+ (119898119909119866+ 119883119903119903) 119903(119896)2+ (1 minus 119905) 119879 (119896)

+ 119883120575120575120575(119896)2+ 119883ext)

120592 (119896 + 1) minus 120592 (119896)

Δ119896

=

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

119903 (119896 + 1) minus 119903 (119896)

Δ119896

=

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

(9)



119906 (119896 + 1) = 119906 (119896) +

Δ119896 sdot 119883|119906|119906

119898 minus 119883

|119906 (119896)| 119906 (119896)

+

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

120592 (119896) 119903 (119896)

+

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

119903(119896)2

+

Δ119896

119898 minus 119883

(1 minus 119905) 119879 (119896) +

Δ119896 sdot 119883120575120575

119898 minus 119883

120575(119896)2

+

Δ119896 sdot 119883ext119898 minus 119883

120592 (119896 + 1)


) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575) 120575 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)

119903 (119896 + 1)


) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575) 120575 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)

(10)


119910119896120585= Γ120585119909119896120585 (11)

for 119896 = 1 119873119904minus 1 where 120585 = 119906 120592 119903 119910

119896119906= 119906(119896 + 1) 119910

119896120592=

120592(119896 + 1) and 119910119896119903


119909119896119906= [119906 (119896) |119906 (119896)| 119906 (119896) 120592 (119896) 119903 (119896) 119903(119896)

2 119879 (119896) 120575(119896)

2 1]

119879


119909119896120592= [120592 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

119909119896119903= [119903 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

(12)

and with

Γ119906= [1

Δ119896 sdot 119883|119906|119906

119898 minus 119883

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

Δ119896

119898 minus 119883

(1 minus 119905)

Δ119896 sdot 119883120575120575

119898 minus 119883

Δ119896 sdot 119883ext119898 minus 119883

]

119879

Γ120592= [1 minusΔ119896Θ

minus1((119868119911minus 119873 119903

) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)]119879

Γ119903= [1 minusΔ119896Θ

minus1((119898 minus 119884 120592

) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575)

Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)]119879

(13)








kernel 119870(119909119894 119909119895) = (119909


modelling

119910119896120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119896120585+ 119887120585

(14)



Γ120585=

119873119904

sum

119894=1

120572119894120585119909119894120585 (15)








08

06

04

02

0

minus02

minus04

minus06

minus080 10 20 30 40 50 60 70 80

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)









120592 = minus04531119906119903 minus 05284119906120592 + 05354 |120592| 120592

minus 04121 |120592| 119903 + 00520120575 + 00007

119903 = minus11699119906119903 minus 06696119906120592 + 23001 |120592| 120592

+ 39335 |120592| 119903 minus 05503120575 minus 00054

(16)


17

16

15

14

13

12

11

1


Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)


0 10 20 30 40 50 60 70 80

Time (s)

02

015

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)






0 10 20 30 40 50 60 70 80

Time (s)

015

01

005

0

minus005

minus01


Yaw

rate

(rad

s)


008

006

004

002

0

minus002

minus004

minus006

minus008

minus01Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)


0 10 20 30 40 50 60 70 80

Time (s)












185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)


01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ (119898119909119866+ 119883119903119903) 119903(119896)2+ (1 minus 119905) 119879 (119896)

+ 119883120575120575120575(119896)2+ 119883ext)

120592 (119896 + 1) minus 120592 (119896)

Δ119896

=

119868119911minus 119873 119903

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

minus

119898119909119866minus 119884 119903

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

119903 (119896 + 1) minus 119903 (119896)

Δ119896

=

119898 minus 119884 120592

Θ

(minus (119898119909119866minus 119873119906119903) 119906 (119896) 119903 (119896)

+ 119873119906120592119906 (119896) 120592 (119896) + 119873

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119873|120592|119903 |

120592 (119896)| 119903 (119896) + 119873120575120575 (119896) + 119873ext)

minus

119898119909119866minus 119873 120592

Θ

(minus (119898 minus 119884119906119903) 119906 (119896) 119903 (119896)

+ 119884119906120592119906 (119896) 120592 (119896) + 119884

|120592|120592 |120592 (119896)| 120592 (119896)

+ 119884|120592|119903 |

120592 (119896)| 119903 (119896) + 119884120575120575 (119896) + 119884ext)

(9)



119906 (119896 + 1) = 119906 (119896) +

Δ119896 sdot 119883|119906|119906

119898 minus 119883

|119906 (119896)| 119906 (119896)

+

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

120592 (119896) 119903 (119896)

+

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

119903(119896)2

+

Δ119896

119898 minus 119883

(1 minus 119905) 119879 (119896) +

Δ119896 sdot 119883120575120575

119898 minus 119883

120575(119896)2

+

Δ119896 sdot 119883ext119898 minus 119883

120592 (119896 + 1)


) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575) 120575 (119896)

+ Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)

119903 (119896 + 1)


) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

sdot 119906 (119896) 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

sdot 119906 (119896) 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

sdot |120592 (119896)| 120592 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

sdot |120592 (119896)| 119903 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575) 120575 (119896)

+ Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)

(10)


119910119896120585= Γ120585119909119896120585 (11)

for 119896 = 1 119873119904minus 1 where 120585 = 119906 120592 119903 119910

119896119906= 119906(119896 + 1) 119910

119896120592=

120592(119896 + 1) and 119910119896119903


119909119896119906= [119906 (119896) |119906 (119896)| 119906 (119896) 120592 (119896) 119903 (119896) 119903(119896)

2 119879 (119896) 120575(119896)

2 1]

119879


119909119896120592= [120592 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

119909119896119903= [119903 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

(12)

and with

Γ119906= [1

Δ119896 sdot 119883|119906|119906

119898 minus 119883

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

Δ119896

119898 minus 119883

(1 minus 119905)

Δ119896 sdot 119883120575120575

119898 minus 119883

Δ119896 sdot 119883ext119898 minus 119883

]

119879

Γ120592= [1 minusΔ119896Θ

minus1((119868119911minus 119873 119903

) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)]119879

Γ119903= [1 minusΔ119896Θ

minus1((119898 minus 119884 120592

) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575)

Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)]119879

(13)








kernel 119870(119909119894 119909119895) = (119909


modelling

119910119896120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119896120585+ 119887120585

(14)



Γ120585=

119873119904

sum

119894=1

120572119894120585119909119894120585 (15)








08

06

04

02

0

minus02

minus04

minus06

minus080 10 20 30 40 50 60 70 80

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)









120592 = minus04531119906119903 minus 05284119906120592 + 05354 |120592| 120592

minus 04121 |120592| 119903 + 00520120575 + 00007

119903 = minus11699119906119903 minus 06696119906120592 + 23001 |120592| 120592

+ 39335 |120592| 119903 minus 05503120575 minus 00054

(16)


17

16

15

14

13

12

11

1


Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)


0 10 20 30 40 50 60 70 80

Time (s)

02

015

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)






0 10 20 30 40 50 60 70 80

Time (s)

015

01

005

0

minus005

minus01


Yaw

rate

(rad

s)


008

006

004

002

0

minus002

minus004

minus006

minus008

minus01Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)


0 10 20 30 40 50 60 70 80

Time (s)












185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)


01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119909119896120592= [120592 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

119909119896119903= [119903 (119896) 119906 (119896) 119903 (119896) 119906 (119896) 120592 (119896) |120592 (119896)| 120592 (119896)

|120592 (119896)| 119903 (119896) 120575 (119896) 1]119879

(12)

and with

Γ119906= [1

Δ119896 sdot 119883|119906|119906

119898 minus 119883

Δ119896 sdot (119898 + 119883120592119903)

119898 minus 119883

Δ119896 sdot (119898119909119866+ 119883119903119903)

119898 minus 119883

Δ119896

119898 minus 119883

(1 minus 119905)

Δ119896 sdot 119883120575120575

119898 minus 119883

Δ119896 sdot 119883ext119898 minus 119883

]

119879

Γ120592= [1 minusΔ119896Θ

minus1((119868119911minus 119873 119903

) (119898 minus 119884119906119903)

minus (119898119909119866minus 119884 119903

) (119898119909119866minus 119873119906119903))

Δ119896Θminus1((119868119911minus 119873 119903

) 119884119906120592minus (119898119909

119866minus 119884 119903

)119873119906120592)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|120592

minus (119898119909119866minus 119884 119903

)119873|120592|120592

)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884|120592|119903

minus (119898119909119866minus 119884 119903

)119873|120592|119903)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884120575minus (119898119909

119866minus 119884 119903

)119873120575)

Δ119896Θminus1((119868119911minus 119873 119903

) 119884ext minus (119898119909119866minus 119884 119903

)119873ext)]119879

Γ119903= [1 minusΔ119896Θ

minus1((119898 minus 119884 120592

) (119898119909119866minus 119873119906119903)

minus (119898119909119866minus 119873 120592

) (119898 minus 119884119906119903))

Δ119896Θminus1((119898 minus 119884 120592

)119873119906120592minus (119898119909

119866minus 119873 120592

) 119884119906120592)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|120592

minus (119898119909119866minus 119873 120592

) 119884|120592|120592

)

Δ119896Θminus1((119898 minus 119884 120592

)119873|120592|119903

minus (119898119909119866minus 119873 120592

) 119884|120592|119903)

Δ119896Θminus1((119898 minus 119884 120592

)119873120575minus (119898119909

119866minus 119873 120592

) 119884120575)

Δ119896Θminus1((119898 minus 119884 120592

)119873ext minus (119898119909119866minus 119873 120592

) 119884ext)]119879

(13)








kernel 119870(119909119894 119909119895) = (119909


modelling

119910119896120585= (

119873119904

sum

119894=1

120572119894120585119909119894120585) sdot 119909119896120585+ 119887120585

(14)



Γ120585=

119873119904

sum

119894=1

120572119894120585119909119894120585 (15)








08

06

04

02

0

minus02

minus04

minus06

minus080 10 20 30 40 50 60 70 80

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)









120592 = minus04531119906119903 minus 05284119906120592 + 05354 |120592| 120592

minus 04121 |120592| 119903 + 00520120575 + 00007

119903 = minus11699119906119903 minus 06696119906120592 + 23001 |120592| 120592

+ 39335 |120592| 119903 minus 05503120575 minus 00054

(16)


17

16

15

14

13

12

11

1


Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)


0 10 20 30 40 50 60 70 80

Time (s)

02

015

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)






0 10 20 30 40 50 60 70 80

Time (s)

015

01

005

0

minus005

minus01


Yaw

rate

(rad

s)


008

006

004

002

0

minus002

minus004

minus006

minus008

minus01Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)


0 10 20 30 40 50 60 70 80

Time (s)












185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)


01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










08

06

04

02

0

minus02

minus04

minus06

minus080 10 20 30 40 50 60 70 80

Time (s)

Rudd

er an

gle a

nd y

aw an

gle (

rad)









120592 = minus04531119906119903 minus 05284119906120592 + 05354 |120592| 120592

minus 04121 |120592| 119903 + 00520120575 + 00007

119903 = minus11699119906119903 minus 06696119906120592 + 23001 |120592| 120592

+ 39335 |120592| 119903 minus 05503120575 minus 00054

(16)


17

16

15

14

13

12

11

1


Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)


0 10 20 30 40 50 60 70 80

Time (s)

02

015

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)






0 10 20 30 40 50 60 70 80

Time (s)

015

01

005

0

minus005

minus01


Yaw

rate

(rad

s)


008

006

004

002

0

minus002

minus004

minus006

minus008

minus01Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)


0 10 20 30 40 50 60 70 80

Time (s)












185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)


01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80

Time (s)

015

01

005

0

minus005

minus01


Yaw

rate

(rad

s)


008

006

004

002

0

minus002

minus004

minus006

minus008

minus01Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)


0 10 20 30 40 50 60 70 80

Time (s)












185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)


01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










185

18

175

17

165

16

155

15

145

140 40 80 120 160 200 240

Time (s)

Surg

e spe

ed (m

s)


(a)

0 40 80 120 160 200 240

Time (s)

185

18

175

17

165

16

155

15

145

14

Surg

e spe

ed (m

s)


(b)


01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(a)

01

005

0

minus005

minus01

minus015

Sway

spee

d (m

s)

0 40 80 120 160 200 240

Time (s)


(b)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(a)

01

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

Yaw

rate

(rad

s)

0 40 80 120 160 200

Time (s)


(b)



006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)

(a)


006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

minus014

Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)

0 40 80 120 160 200

Time (s)240

(b)




196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











196

194

192

19

188

186

184

182

18

Surg

e spe

ed (m

s)

0 10 20 30 40 50 60 70 80

Time (s)








0 10 20 30 40 50 60 70 80

Time (s)

008

006

004

002

0

minus002

minus004

minus006

minus008

minus01

minus012

Sway

spee

d (m

s)



01

008

006

004

002

0

minus002

minus004

minus006

minus008

Yaw

rate

(rad

s)

0 10 20 30 40 50 60 70 80

Time (s)







008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










008

006

004

002

0

minus002

minus004

minus006

0 10 20 30 40 50 60 70 80

Time (s)90


Surg

e and

sway

spee

ds (m

s) a

nd y

aw ra

te (r

ads

)




Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article semiphysical modelling of the nonlinear...

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