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Dominik Andreas Breu

Frequency Detuning of ParametricRoll Resonance

Thesis for the degree of Philosophiae Doctor

Trondheim, August 2013

Norwegian University of Science and TechnologyFaculty of Information Technology, Mathematics and Electrical EngineeringDepartment of Engineering Cybernetics

NTNU

Norwegian University of Science and Technology

Thesis for the degree of Philosophiae Doctor

Faculty of Information Technology, Mathematics and Electrical EngineeringDepartment of Engineering Cybernetics

© 2013 Dominik Andreas Breu.

ISBN 978-82-471-4179-3 (printed version)ISBN 978-82-471-4180-9 (electronic version)ISSN 1503-8181ITK Report 2013-7-W

Doctoral theses at NTNU, 2013:43

Printed by Tapir Uttrykk

Summary

In this thesis, frequency detuning of parametric roll resonance is considered. Para-metric roll resonance is a nonlinear resonance phenomenon which may be dangerousto several types of ships, among those container ships, cruise ships, and shing ves-sels. It is characterized by a particular relationship of the wave encounter frequencyand the natural roll frequency and it is capable of causing roll amplitudes of up to50 degrees, causing danger to the ship, its crew and the cargo and possibly resultingin monetary losses in the order of tens of millions of US dollars.

There are several ways of controlling parametric roll resonance. They may beroughly classied as direct control and indirect control. Whereas the former con-siders regulation of parametrically excited roll motions by means of increasing thedamping in roll or creating an opposing roll moment, the later can be referred toas frequency detuning control. Frequency detuning control aims at detuning thewave encounter frequency the parametric excitation frequency and the naturalroll frequency. The wave encounter frequency is, among others, dependent on theship's surge speed and heading angle and can therefore be controlled. The maingoal of this thesis has been to develop active control methods using the frequencydetuning approach, exploiting the ship's surge speed and heading angle to alter theencounter frequency.

In addition to the frequency detuning control methods, this thesis oers contri-butions in modelling parametric roll resonance. Specically, a numerical 6-degree-of-freedom (DOF) computer model has been developed for the parametric roll res-onance for time-varying surge speed and heading angle. Reducing the 6-DOF shipmodel, a 1-DOF roll model is derived under the assumption of slowly time-varyingencounter frequency by considering the quasi-steady solutions for the heave andpitch motions which are the most important motions for parametric roll resonance.Commonly, the Mathieu equation is used to describe parametric roll resonance andit is shown that, while capable to describe parametric roll resonance for a constantencounter frequency, it is not adequate to model parametric roll resonance for atime-varying encounter frequency.

While changing the heading angle might have a benecial eect for the stabi-lization of parametric roll, it can result in directly induced roll motion when theship is no longer in sailing in longitudinal waves. In an attempt to investigate onthis eect, the 1-DOF roll model is expanded to incorporate time-varying speedand heading angle and thus directly induced roll.

Frequency detuning control approaches commonly depend on the knowledge ofthe considered frequencies, specically the natural roll frequency and the wave en-

i

counter frequency. In a practical situation, however, especially the wave encounterfrequency can hardly be assumed to be known. Therefore, a model-based extendedKalman lter (EKF) frequency observer and a signal-based nonlinear K exponen-tial stable frequency observer are presented and their feasibility with respect to theestimation of the considered frequencies is investigated.

ii

Contents

Contents iii

Preface v

I Preliminaries 1

1 Introduction 31.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . 31.2 Autoparametric Resonance . . . . . . . . . . . . . . . . . . . . . . 71.3 Parametric Resonance . . . . . . . . . . . . . . . . . . . . . . . . . 221.4 Parametric Roll Resonance in Ships . . . . . . . . . . . . . . . . . 251.5 Scope and Main Contributions . . . . . . . . . . . . . . . . . . . . 271.6 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.7 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Nomenclature 35

II Ship Modelling 45

3 A Six-Degree-Of-Freedom Ship Model 473.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Computer Implementation of the 6-DOF Model . . . . . . . . . . . 523.3 Encounter Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 1-DOF Roll Model for Time-Varying Speed 594.1 Model Verication . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 1-DOF Roll Model for Time-Varying Heading and Speed 695.1 Model Coecient Identication . . . . . . . . . . . . . . . . . . . . 705.2 Model Verication . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

III Frequency Detuning 79

6 Frequency Detuning 81

iii

Contents

6.1 Roll Model for Non-Constant Speed . . . . . . . . . . . . . . . . . 826.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7 Frequency Detuning by Optimal Speed and Heading Changes 977.1 3-DOF Ship Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.2 Encounter Frequency Revisited . . . . . . . . . . . . . . . . . . . . 997.3 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . 1017.4 Extremum Seeking Control . . . . . . . . . . . . . . . . . . . . . . 1057.5 L1 Adaptive and Extremum Seeking Control . . . . . . . . . . . . 115

IV Wave Frequency Estimation 131

8 Model-Based Frequency Estimator 1338.1 Design of the Extended Kalman Filter . . . . . . . . . . . . . . . . 1348.2 Frequency Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 136

9 Signal-Based Frequency Estimator 1439.1 Frequency Estimator Design . . . . . . . . . . . . . . . . . . . . . . 1449.2 Case Study: Wave Frequency Estimation . . . . . . . . . . . . . . . 1489.3 Verication in Irregular Waves . . . . . . . . . . . . . . . . . . . . 154

V Closing Remarks 157

10 Conclusions 159

Appendices 163

A Appendix 165

References 169

iv

Preface

This thesis is submitted in partial fullment of the requirements for the degree ofphilosophiae doctor (PhD) at the Norwegian University of Science and Technology(NTNU).

The presented work in this thesis has been carried out at the Department ofEngineering Cybernetics and the Centre for Ships and Ocean Structures at theNorwegian University of Science and Technology, in the period from August 2009to March 2013. My supervisor has been Professor Thor I. Fossen of the Departmentof Engineering Cybernetics and the Centre for Ships and Ocean Structures, NTNUand my co-supervisor Professor Mogens Blanke of the Department of ElectricalEngineering at the Technical University of Denmark and the Centre for Ships andOcean Structures, NTNU.

Acknowledgements

First and foremost, I would like to thank my supervisor, Professor Thor I. Fos-sen of the Department of Engineering Cybernetics and the Centre for Ships andOcean Structures, NTNU for giving me the possibility for my PhD studies and thecontinuous moral and academic support and guidance throughout them. I wouldalso like to show my gratitude to my co-supervisor Professor Mogens Blanke of theDepartment of Electrical Engineering at the Technical University of Denmark forhis support.

I would like to thank the sta at the Centre for Ships and Ocean Structuresand the Department of Engineering Cybernetics, NTNU for providing the enjoyableand stimulant environment which made the work on this thesis possible in the rstplace.

The Research Council of Norway has funded the rst three year of this PhDposition, whereas the funds for the remaining eight months have been providedby the Norwegian University of Science and Technology and the Department ofEngineering Cybernetics, for which I am very grateful.

During the course of my PhD work I have had the opportunity to make theacquaintance of many great fellow researchers and colleagues who have inuencedme in one way or another. A big thank you goes to my many friends at Tyholtand at the Department of Engineering Cybernetics. I would especially like to thankTasos for being the best oce mate ever and Christian for the good collaborationon some projects during my PhD studies.

v

Preface

During the last years I have had the chance to co-supervise two exchange stu-dents from the University of Eindhoven, Thijs and Dennis. It has truly been apleasure to work with them and I would like to thank them. Furthermore, I super-vised Gunnhild during her master thesis and I would like to thank her for the goodcollaboration.

In autumn 2010 and autumn 2011 I had the possibility to work as teachingassistance in the course Linear System Theory at the Department for EngineeringCybernetics, NTNU. I would like to thank Dr Esten Grøtli and Prof. Tor ArneJohansen for providing me the interesting opportunity.

This thesis would not have been possible without my family and friends. A spe-cial thanks goes to everybody who supported me during my adventure in Norway.I am especially thankful to my parents, Rita and Hans, and my brothers, Simonand Raphael. Furthermore, I would like to thank my friends in Switzerland forthe warm welcome home, the numerous visits in Trondheim and the spontaneousweekend trips in Europe.

Last but not least, I would like to thank Eriane for her love, her constantsupport and the patience during the last years.

vi

Part I

Preliminaries

1

Chapter 1

Introduction

This thesis deals with parametric roll resonance of ships. Contributions in math-ematical modelling and control of the phenomenon are presented where for thelatter, the focus in on frequency detuning control.

1.1 Motivation and Background

The possible threat that roll resonance poses to the ships' safety has been knownfor more than one and a half centuries. Froude [45] investigated the phenomenonof the gradual accumulation of [the roll] angle during several successive rolls[45, p. 180], resulting in large roll amplitudes of the vessel and he pointed out theundesirable eect some relationships of the natural roll frequency and the frequencyof the waves might have on the seakeeping characteristics. By considering regularsinusoidal waves and computing the righting moment of the vessel dependent onthe inclination of the wave surface in undulating water, the work explained theoccurrence of roll resonance in waves.

Shortly afterwards, Froude [46] described also the eect of parametric resonancein longitudinal waves. He elaborated that, if the natural roll frequency is close tohalf the natural heave frequency, the ship's transverse oscillations may cause verti-cal oscillations, and if the transverse oscillations were maintained with unabatedrange, the vertical ones would continue to increase in range by equal steps, oscilla-tion by oscillation, were not the accumulation limited by the growing resistance ofthe water [. . .] [46, p. 236]. Furthermore, he identied that [d]uring each derivedvertical oscillation, there is [. . .] a xed amount of cumulative energy supplied bythe primary transverse oscillation [. . .] [46, p. 236], implying the coupling of theroll and the heave motion.

The above mentioned coupling of the vertical and the transverse oscillations canbe formalized by considering the heave/roll motions as an autoparametric system:

Autoparametric systems are vibrating systems that consist of at leasttwo constituting subsystems. One is the primary system that will gen-erally be in a vibrating state. This primary system can be externallyforced, self-excited, parametrically excited, or a combination of these.[. . .] The secondary system is coupled to the primary system in a non-

3

1. Introduction

linear way, but such that the secondary system can be at rest while theprimary system is vibrating. [145, pp. 23]

In the context of the study of Froude [46], the roll is the externally excited primarysystem which in turn supplies energy to the coupled secondary system, the heavemotion. Interestingly, the reverse, i.e. the transfer of energy from heave to roll,is also possible. This is termed parametrically excited roll or parametric roll, aperdious threat to ship stability especially in head seas1 and following seas2,underestimated for quite some time.

According to Paulling [118], the early research on parametric roll was pioneeredin Germany in the late 1930s in order to understand the capsizing of small vesselsin following seas, cf. [3, 56] and the references therein. Among the rst, Grim [56]provided a concise mathematical and experimental treatment of parametric roll inhead seas and following seas, comparing it to induced roll in other sea states, suchas bow, quartering and beam seas. In head or following regular seas, the wavesdo not induce a direct moment in roll due to transverse symmetry of the vessel.However, Grim [56] showed that, by assuming a sinusoidal change of the rightingmoment of the vessel and thus her transverse stability due to the passage of thewaves, the roll dynamics can be described by Mathieu's equation. Depending onits parameters, Mathieu's equation is known to have stable or unstable solutions[92], the latter in this context characterized by certain ratios of the natural rollfrequency and the wave frequency. Grim [56] seems also to be among the rst toderive mathematically and conrm experimentally the coupling of the yaw, heaveand the roll motion and recognized it as an explanation for the onset of parametricroll. He also stated that this coupling is dependent on the frequency ratios and themost eective coupling, i.e. the largest resulting roll amplitudes, arises for heaveperiods about half the natural roll period.

Although recognized as being able to endanger the safety of some ships, para-metric roll was long considered merely a threat for small vessels of low stability,such as shing vessels or small carriers, in following seas. In the last two decadeshowever, other types of vessels, such as container ships, have experienced paramet-ric roll in head seas. This is due to the particular hull shape of those vessels withtheir pronounced bow are and stern overhangs, resulting in a signicant changeof the righting moment as the wave crest moves along the hull and consequentlymaking them vulnerable to parametric roll. Those incidents of parametric roll inhead seas attracted considerable attention of the research community and the navalauthorities. [118]

1.1.1 Incidents

Even though parametric roll of vessels occurs quite rarely, its occurrence may bedevastating. Only since the middle of the last century, sucient knowledge to iden-tify parametric roll correctly has been established. Furthermore, due to a lack ofawareness and/or detailed records of the ship motions and the environmental condi-tions it is often impossible to distinguish conclusively between large roll amplitudes

1The waves are approaching from the stern of the vessel.2The waves are coming towards the bow of the vessel.

4

1.1. Motivation and Background

due to harsh weather and parametrically excited roll oscillations [130]. Nevertheless,some of the more catastrophic maritime incidents could be attributed to paramet-ric roll after thorough investigations. Others are still unsolved and speculative, seefor example Ünsalan [150].

The following descriptions of incidents exemplify parametric rolling. Especiallythe incidents with the container ships are also widely recognized as a turning-point considering the research interest in the phenomenon and spurred furtherinvestigations.

APL China

In October 1998, the post-Panamax C113 containership APL China experiencedwhat is considered the largest container disaster in history. While sailing eastwardsin the North Pacic Ocean from Kaohsiung, Taiwan to Seattle, USA, she encoun-tered a violent and long-lasting storm.

As a common procedure, the master decelerated and attempted to steer thevessel into the prevailing seas as well as possible. Signicant pitching motions werereported and simultaneously, port and starboard roll amplitudes as large as 35to 40 degrees during the worst of the storm were experienced. According to themaster, the vessel was at times uncontrollable.

The APL China had 1300 containers stored on deck. Of those, almost a thirdhad been lost overboard and additionally, another third had been damaged. Theremaining third of the containers and cargo hung over both sides of the vessel.Figure 1.1 depicts the APL China after the incident.

(Account from France et al. [44])According to Ginsberg [55], it was estimated that the value of the lost cargo

exceeded the total value of the vessel herself, a staggering $50 million.

Mærsk Carolina

In January 2003 the Panamax G-class4 containership Mærsk Carolina was sailingwestwards in the North Atlantic from Algeciras, Spain, to Halifax, Nova Scotia,Canada when she encountered a severe storm.

The weather was worsening soon after the departure and the master decreasedthe speed of the vessel and steered her towards the incoming waves, followingprevailing practise. The vessel experienced heavy roll angles, in excess of 47 degrees,the maximum of the measurement device, in combination with violent pitch andlarge heave motions. Reports from the crew indicate that the heavy roll motionoccurred unexpectedly and suddenly, building to the maximum in only a few rollcycles.

As a consequence of the incident, of the laden 2180 containers, 133 were lostoverboard and additional 50 containers were damaged. The cargo claims exceeded

3A post-Panamax C11 has a length of 262 m, a breadth of 40 m and a maximum draft ofabout 14 m; she has a total container capacity of 4832 TEU.

4A Panamax G-class has a length of 292 m, a breadth of 32 m and a draft of about 13.5 m;she has a total container capacity of 4306 TEU.

5

1. Introduction

Figure 1.1: The APL China after the incident. (Source: http://www.cargolaw.com)

$4 million and the vessel sustained moderate structural damage. Figure 1.2 showsthe Mærsk Carolina after the incident.

(Account from Carmel [28])

Figure 1.2: The Mærsk Carolina after the incident. (Source: http://www.

cargolaw.com)

Wallenius PCTC M/V Aida

TheWallenius M/V Aida5, a pure car and truck carrier (PCTC) experienced para-metric rolling twice. In February 2003, she experienced roll angles as large as 50degrees while sailing in head seas in rough conditions southwest of the Azores.Partly due to that incident the Wallenius M/V Aida was consequently equipped

5The PCTC M/V AIDA has a length of 199 m, a breadth of 32 m and a maximum draft of11.3 m; she has a total capacity of 6700 cars.

6

http://www.cargolaw.com




1.2. Autoparametric Resonance

with an on-board operational decision support system, able to record, among oth-ers the ship motions and the environmental conditions. In February 2004, the vesselwas exposed to head sea parametrically excited roll motions up to 17 degrees ona passage from Southampton, UK to New York, USA. This incident is signicantbecause consequently, all the parameters have been recorded for the rst time everduring a normal ship service.

(Account from Review of the intact stability code Recordings of head sea para-metric rolling on a PCTC [124])

1.2 Autoparametric Resonance

Already Froude [46] remarked on the coupling between the heave and roll and alater investigation by Grim [56] veried mathematically and experimentally thatparametric roll may be explained by the coupling of the pitch motion to the rollmotion. The (nonlinear) coupling between the motions in two or more DOF sys-tems is of importance in order to understand the phenomenon of parametric rollresonance, since it captures the energy transfer between the motions, responsible ofthe onset of parametric roll resonance. According to Oh, Nayfeh, and Mook [116],the two basic mechanisms for parametric roll resonance to develop are parametricresonance and autoparametric resonance. They are basically explaining the samephenomenon from dierent points of view.

In the following, a simple autoparametric system is used to explain the basicproperties and features which lead to parametric roll. The subsequent analysis ismainly based on material presented in Tondl et al. [145], Cartmell [29], Nayfeh andMook [100], Dohnal [32], Oh, Nayfeh, and Mook [116], and the references therein.

Consider a 2-DOF autoparametric system given by a pendulum attached to amass-damper-spring system, as depicted in Figure 1.3. The mass mp > 0 ∈ R ismounted on a spring with linear coecient kp > 0 ∈ R. It represents the primarysystem, excited by a sinusoidal force fe = a sin (ωt), with amplitude a > 0 ∈ Rand frequency ω > 0 ∈ R . The secondary system is a pendulum with mass ms >0 ∈ R and length ls > 0 ∈ R which is attached to the primary system. Both thetranslational motion of the primary system zp ∈ R and the rotational motion of thesecondary system φs ∈ R are subject to viscous damping with damping coecientsdp > 0 ∈ R and ds > 0 ∈ R, respectively.

By using the translational displacement zp of the primary system and the angleφs of the secondary system as generalized coordinates of the autoparametric systemin Figure 1.3, the kinetic energy T > 0 ∈ R and the potential energy V > 0 ∈ Rare given by

T =1

2mpz

2p +

1

2msz

2p +

1

2msl

2s φ

2s +mszplsφs sin (φs) (1.1)

V =1

2kpz

2p +msgls (1− cos (φs)) (1.2)

where g > 0 ∈ R denotes the acceleration of gravity. The equations of motion can

7

1. Introduction

Mass mp

φs

ls

ms

zp

kp dp

fe

Figure 1.3: An autoparametric system The primary system with mass mp,mounted on a spring with coecient kp, is externally excited by a sinusoidal forcefe = a sin (ωt); a pendulum with mass ms and length ls is attached to the primarysystem and it represents the secondary system. Both primary and secondary sys-tems are subject to viscous damping with coecients dp and ds. (Figure adaptedfrom Tondl et al. [145].)

then be computed from the Lagrangian L = T − V by Lagrange's equations [100]:

d

dt

(∂L

∂zp

)− ∂L

∂zp= 0 (1.3)

d

dt

(∂L

∂φs

)− ∂L

∂φs= 0 (1.4)

Substituting (1.1) and (1.2) into (1.3) and (1.4) and including the viscous dampingterms in both systems and the external forcing in the primary system yields thecoupled equations of motion for the autoparametric system:

(mp +ms) zp + dpzp + kpzp +msls

(φs sin (φs) + φ2

s cos (φs))

= a sin (ωt) (1.5)

msl2s φs + dsφs +msgls sin (φs) +mslszp sin (φs) = 0 . (1.6)

For the analysis of (1.5) and (1.6) it is convenient to write the equations ofmotion in dimensionless form by scaling the horizontal displacement by zp = zp/lsand the time by t = ωt. Furthermore, the following coecients are dened:

κp ,dp

(mp +ms)ωωp ,

√kp

mp +msµ ,

ms

(mp +ms)

κs ,ds

msl2sωωs ,

√g

lsα ,

a

lsω2 (mp +ms).

8


The dimensionless equations of motions are then

z′′p + κpz′p +

ω2p

ω2zp + µ

(φ′′s sin (φs) + φ′s

2cos (φs)

)= α sin (t) (1.7)

φ′′s + κsφ′s +

(ω2s

ω2+ z′′p

)sin (φs) = 0 . (1.8)

Here, the prime denotes the derivative with respect to the new (dimensionless) timevariable t. Was it not for the coupling between (1.7) and (1.8), the primary sys-tem (1.7) would be a linear, externally forced system with dimensionless dampingcoecient κp > 0 ∈ R and spring coecient ω2

p/ω2, where ωp > 0 ∈ R is the nat-

ural frequency of the primary system. Similarly, the secondary system (1.8) wouldrepresent a nonlinear system with dimensionless damping coecient κs > 0 ∈ Rand nonlinear restoring moment with coecient ω2

s/ω2, where ωs > 0 ∈ R is the

natural frequency of the secondary system. Note that the nonlinearity is due tolarge angular deections φs. The secondary system (1.8) could be reduced to alinear system by approximating sin (φs) ≈ φs. However, this would imply that theangular deections φs be restricted to angles between ±20 degrees. Alternatively,a Taylor expansion for sin (φs) may be used:

sin (φs) =

∞∑

n=0

(−1)nφ1+2ns

(1 + 2n)!= φs −

φ3s

3!+φ5s

5!− · · ·+ . . . (1.9)

where it is rather common to neglect quintic and higher order terms and truncatethe series (1.9) after O

(φ3s

)such that sin (φs) ∼ φs − φ3

s/6 + O(φ3s

). Here O (·)

and O (·) are the Landau, or order symbols. The Taylor expansion (1.9) producesan asymptotic expansion of sin (φs) of cubic order with the Taylor polynomials asthe scale functions. The following denitions give some insight.

Denition 1.1 (Rephrased from Holmes [67, p. 5]).

1. A function f (ε) is of order O (ϕ) (big Oh of ϕ) if there exist constants δ1and ε1 (independent of ε) such that, as ε→ ε0

|f (ε)| ≤ δ1|ϕ (ε)| for ε0 < ε < ε1 .

2. A function f (ε) is of order O (ϕ) (little oh of ϕ) if there exist constants δ2and ε2 (independent of ε) such that, as ε→ ε0

|f (ε)| ≤ δ2|ϕ (ε)| for ε0 < ε < ε2 .

In case that ϕ is not zero near ε0, the following statements hold:

Theorem 1.1 (Holmes [67, p. 5], without proof.).

1. If

limε→ε0

f (ε)

ϕ (ε)= L

where −∞ < L <∞, then f = O (ϕ) as ε→ ε0.

9

1. Introduction

2. If

limε→ε0

f (ε)

ϕ (ε)= 0 ,

then f = O (ϕ) as ε→ ε0.

Denition 1.2 (Holmes [67, p. 8]). Given f (ε) and ϕ (ε), ϕ (ε) is an asymptoticapproximation to f (ε) as ε→ ε0 whenever f = ϕ+O (ϕ) as ε→ ε0. This is denotedby f ∼ ϕ.

Consequently, for ϕ (ε) not zero near ε0, it holds that f ∼ ϕ as ε→ ε0 if [67]

limε→ε0

f (ε)

ϕ (ε)= 1 .

Denition 1.3 (Holmes [67, p. 10]).

1. The functions ϕ1 (ε) , ϕ2 (ε) , . . . form an asymptotic sequence as ε → ε0 ifand only if ϕm+1 = O (ϕm) as ε→ ε0 ∀m.

2. If ϕ1 (ε) , ϕ2 (ε) , . . . is an asymptotic sequence, then f (ε) has an asymptoticexpansion to n terms if and only if

f =

m∑

k=1

akϕk + O (ϕm) for m = 1, 2, . . . , n as ε→ ε0

where the ak are independent of ε and the ϕk are called the scale or basisfunctions. The asymptotic expansion is denoted by

f ∼ a1ϕ1 (ε) + a2ϕ2 (ε) + · · ·+ anϕn (ε) as ε→ ε0 .

The autoparametric system (1.5) and (1.6) and its dimensionless formulation(1.7) and (1.8), respectively, are characterized by three distinct frequencies whichare of importance for the stability of the system. The two natural frequencies asso-ciated with the primary and the secondary system are ωp and ωs. The excitationfrequency of the external forcing nally is ω. The relationship of those frequencieslargely determines whether the system is stable or unstable and how the motiontrajectories evolve with time.

1.2.1 Uncoupled Solution

Consider the uncoupled equations of the autoparametric system (1.7) and (1.8),that is

z′′p + κpz′p +

ω2p

ω2zp = α sin (t) (1.10)

φ′′s + κsφ′s +

ω2s

ω2sin (φs) = 0 . (1.11)

10


Primary System

The complementary solution zp,c ∈ R of (1.10) can be calculated from the un-forced primary system. Assume that the primary system is underdamped, that isκp < 2ωp/ω, or dp < 2ωp (mp +ms) for the dimensional system (1.5). Thus, thecomplementary solution is

zp,c (t) = cp,ce−0.5κp t cos

(√ω2p

ω2− κ2

p

4t+ ϕp,c

)(1.12)

where cp,c ∈ R and ϕp,c ∈ R are constants which are determined from the initialconditions. Note, that the complementary solution (1.12) dies out, that is zp (t)→ 0as t→∞.

The particular solution zp,p ∈ R of (1.10) is

zp,p =α√(

ω2p

ω2 − 1)2

+ κ2p

sin (t+ ϕp,p) (1.13)

where ϕp,p ∈ R is a constant phase angle which is determined from the initialconditions. The amplitude of (1.13) becomes a maximum for

d

dω

α√(ω2p

ω2 − 1)2

+ κ2p

= 0 ⇒ ω =

√ω2p −

1

2

(dp

mp +ms

)2

. (1.14)

For lightly damped systems in resonance, (1.14) reduces to ω ≈ ωp, that is, themaximum amplitude occurs when the frequency of the external excitation is closeto the natural frequency of the primary system.

Secondary System

To construct an asymptotic expansion of the solution to the equations describingthe uncoupled secondary system (1.11), a second-order Taylor expansion as in (1.9)is used for the sin (φs) nonlinearity. Additionally, it is assumed that the secondarysystem is lightly damped only, that is κs = εκs, where κs > 0 ∈ R represents thescaled dimensionless damping coecient, γ1 > 0 ∈ R the scaled coecient of thecubic nonlinearity, and ε > 0 ∈ R is a small, positive parameter. Consequently,(1.11) becomes

φ′′s + εκsφ′s +

ω2s

ω2

(φs − εγ1φ

3s

)= 0 . (1.15)

It is convenient to incorporate multiple time scales to approximate the solution to(1.15):

t1 = t (1.16)

t2 = εt . (1.17)

11

1. Introduction

The fast time scale is t1 > 0 ∈ R and the slow time scale is t2 > 0 ∈ R; they aretreated as independent. By employing the chain rule, the original time derivativewith respect to t > 0 ∈ R translates to

d

dt=

dt1dt

∂

∂t1+

dt2dt

∂

∂t2=

∂

∂t1+ ε

∂

∂t2(1.18)

d2

dt2=

∂2

∂t21+ 2ε

∂

∂t1

∂

∂t2+ ε2

∂2

∂t22. (1.19)

The method of multiple scales assumes that the solution φs can be approximatedby a power series expansion

φs (t; ε) ∼ φs,1 (t1, t2) + εφs,2 (t1, t2) + . . . . (1.20)

Substituting (1.18), (1.19), and (1.20) into (1.15) yields the following problemsfor the two time scales:

O (1)∂2

∂t21φs,1 +

ω2s

ω2φs,1 = 0 (1.21)

O (ε)∂2

∂t21φs,2 +

ω2s

ω2φs,2 + 2

∂

∂t1

∂

∂t2φs,1 + κs

∂

∂t1φs,1 −

ω2s

ω2γ1φ

3s,1 = 0 (1.22)

The general solution of (1.21) is found to be

φs,1 = as,1 (t2) et1ωs/ω + cc (1.23)

where cc denotes the complex conjugate and as,1 ∈ C is a coecient. Inserting(1.23) into (1.22) results in

∂2

∂t21φs,2 +

ω2s

ω2φs,2 + 2

ωsω

∂

∂t2as,1 (t2) et1ωs/ω + κs

ωsωas,1 (t2) et1ωs/ω

−ω2s

ω2γ1

(a3s,1 (t2) e3t1ωs/ω + 3a2

s,1 (t2) a∗s,1 (t2) et1ωs/ω)

+ cc = 0 . (1.24)

Here a∗s,1 denotes the complex conjugate of as,1. In order to avoid secular termsfor the solution of φs,2 ∈ R, the following condition must hold:

2ωsω

∂

∂t2as,1 (t2) + κs

ωsωas,1 (t2)− 3

ω2s

ω2γ1a

2s,1 (t2) a∗s,1 (t2) = 0 . (1.25)

Rewriting as,1 in polar notation

as,1 (t2) =1

2αs,1 (t2) eβs,1(t2) (1.26)

where αs,1 ∈ R and βs,1 ∈ R are the absolute value and the argument of as,1 andinserting (1.26) into the condition for a well-behaved expansion (1.25) results inthe dierential equations for the absolute value and the argument of the coecientfor the rst-order expansion of the solution to (1.15)

d

dt2αs,1 (t2) = −1

2κsαs,1 (t2) (1.27)

d

dt2βs,1 (t2) = −3

8

ωsωγ1α

2s,1 (t2) . (1.28)

12


This yields

αs,1 (t2) = αs,0e−0.5κs t2 (1.29)

βs,1 (t2) =3α2

s,0γ1

8κs

ωsωe−κs t2 + βs,0 (1.30)

where αs,0 ∈ R and βs,0 ∈ R are constants. Finally, the rst-order approximationof the solution to (1.15), derived with the two time scales t1 = t and t2 = εt, isgiven by

φs (t; ε) = αs,0e−0.5κsεt cos

(ωsωt+

3α2s,0γ1

8κs

ωsωe−κsεt + βs,0

)+ O (ε) . (1.31)

From (1.31) it is obvious that the angular deection φs of the secondary sys-tem (1.11) decays in absence of the coupling to the primary system. The primarysystem (1.10) on the other hand represents a externally excited damped oscillator.The non-transient vertical displacement of the primary system zp (1.13) may reachhigh amplitudes when the driving frequency ω is close to the natural frequency ωpof the system and resonance is present.

1.2.2 Coupled Solution

Here, the coupled autoparametric system (1.7) and (1.8) is considered. Similar toSection 1.2.1, a second-order Taylor expansion as in (1.9) is used for the sin (φs)nonlinearity in (1.7). The cos (φs) nonlinearity in (1.7) is approximated by

cos (φs) =

∞∑

n=0

(−1)nφ2ns

(2n)!= 1− φ2

s

2!+ · · ·+ . . . (1.32)

where quartic and higher order terms are neglected and the series (1.32) is truncatedafter O

(φ2s

). The Taylor expansion (1.32) produces an asymptotic expansion of

cos (φs) of quadratic order with the Taylor polynomials as the scale functions.Additionally, it is assumed that both the primary and secondary systems are

lightly damped only, that is κp = εκp and κs = εκs, where κp > 0 ∈ R and κsrepresents the scaled dimensionless damping coecients; γ1 and γ2 > 0 ∈ R arethe scaled coecients of the quadratic and cubic nonlinearity. Furthermore, thecoupling of the system is considered weak in a sense that it does not appear ina rst-order approximation; this is to obtain a valid approximate solution. Thus,(1.7) and (1.8) become

z′′p + εκpz′p +

ω2p

ω2zp + εµ

(φ′′s(φs − εγ1φ

3s

)+ φ′s

2 (1− εγ2φ

2s

))= α sin (t) (1.33)


ω2s

ω2

(φs − εγ1φ

3s

)+ εz′′p

(φs − εγ1φ

3s

)= 0 . (1.34)

Dening the two time scales as in (1.16) and (1.17), and assuming a power seriesexpansion for the vertical displacement of the primary system zp ∈ R:

zp (t; ε) ∼ zp,1 (t1, t2) + εzp,2 (t1, t2) + . . . . (1.35)

13

1. Introduction

Substituting (1.18), (1.19), and (1.35) into (1.33) and (1.34) yields the followingproblems for the two time scales:

O (1)∂2

∂t21zp,1 +

ω2p

ω2zp,1 = α sin (t) (1.36)

O (1)∂2

∂t21φs,1 +

ω2s

ω2φs,1 = 0 (1.37)

O (ε)∂2

∂t21zp,2 +

ω2p

ω2zp,2 + 2

∂

∂t1

∂

∂t2zp,1 + κp

∂

∂t1zp,1

+µφs,1∂2

∂t21φs,1 + µ

(∂

∂t1φs,1

)2

= 0 (1.38)

O (ε)∂2

∂t21φs,2 +

ω2s

ω2φs,2 + 2

∂

∂t1

∂

∂t2φs,1 + κs

∂

∂t1φs,1

−ω2s

ω2γ1φ

3s,1 + φs,1

∂2

∂t21zp,1 = 0 . (1.39)

The general solutions to (1.36) and (1.37) are given by

zp,1 = ap,1 (t2) et1ωp/ω − α

2(ω2p/ω

2 − 1)et1 + cc (1.40)

φs,1 = as,1 (t2) et1ωs/ω + cc (1.41)

where cc denotes the complex conjugate and ap,1 ∈ C and as,1 are coecients.Inserting (1.40) and (1.41) into (1.38) results in

∂2

∂t21zp,2 +

ω2p

ω2zp,2 + 2

∂

∂t2ap,1

ωpωet1ωp/ω + κpap,1

ωpωet1ωp/ω

+κpα

2(ω2p/ω

2 − 1)et1 − 2µ

ω2s

ω2a2s,1e

2t1ωs/ω + cc = 0 . (1.42)

Similarly, inserting (1.40) and (1.41) into (1.39) results in

∂2

∂t21φs,2 +

ω2s

ω2φs,2 + 2

∂

∂t2as,1

ωsωet1ωs/ω + κsas,1

ωsωet1ωs/ω

− ω2s

ω2γ1

(a3s,1e

3t1ωs/ω + 3a2s,1a∗s,1e

t1ωs/ω)− ap,1as,1

ω2p

ω2et1(ωp+ωs)/ω

+as,1α

2(ω2p/ω

2 − 1)(et1(1+ωs/ω) − et1(ωs/ω−1)

)

− ap,1a∗s,1ω2p

ω2et1(ωp−ωs)/ω + cc = 0 . (1.43)

Here a∗s,1 denotes the complex conjugate of as,1.

14


Non-Resonance Condition

In order to avoid secular terms for the solution of zp,2 ∈ R, the following conditionmust hold:

2ωpω

∂

∂t2ap,1 (t2) + κp

ωpωap,1 (t2) = 0 . (1.44)

Rewriting ap,1 in polar notation

ap,1 (t2) =1

2αp,1 (t2) eβp,1(t2) (1.45)

where αp,1 ∈ R and βp,1 ∈ R are the absolute value and the argument of ap,1 andinserting (1.45) into the condition for a well-behaved expansion (1.44) results inthe dierential equations for the absolute value and the argument of the coecientfor the rst-order expansion of the solution to (1.33)

d

dt2αp,1 (t2) = −1

2κpαp,1 (t2) (1.46)

d

dt2βp,1 (t2) = 0 . (1.47)

This results in

αp,1 (t2) = αp,0e−0.5κp t2 (1.48)

βp,1 (t2) = βp,0 (1.49)

where αp,0 ∈ R and βp,0 ∈ R are constants. Finally, the rst-order approximationof the solution to (1.34) is given by

zp (t; ε) =α(

ω2p/ω

2 − 1) sin (t) + αp,0e

−0.5κpεt cos(ωpωt+ βp,0

)+ O (ε) . (1.50)

To avoid secular terms for the solution of φs,2, it must hold that

2ωsω

∂

∂t2as,1 (t2) + κs

ωsωas,1 (t2)− 3

ω2s

ω2γ1a

2s,1 (t2) a∗s,1 (t2) = 0 . (1.51)

This is exactly identical to the condition given in (1.25), so that the rst orderapproximation of the coupled secondary system (1.34) is given by (1.31).

Resonance Condition

From (1.42) and (1.43) it is evident that the autoparametric system is in resonancefor certain combinations of the excitation frequency ω and the natural frequencies ofthe primary and secondary systems, ωp and ωs, respectively; among those primary,superharmonic, subharmonic, and combination resonance. Of particular interest isthe situation when the primary system is excited at its natural frequency, thatis ω ≈ ωp, and the natural frequency of the secondary system ωs is about halfthe excitation frequency or equivalently ω ≈ 2ωs. This specic relationship will be

15

1. Introduction

considered here where it is furthermore assumed that the excitation amplitude issmall, that is α = εα with α ∈ R the scaled version of the dimensionless externalexcitation amplitude α ∈ R. This is to prevent unbounded oscillations as predictedby the linear theory in the expansion. To that matter, the right-hand side of (1.36)is set to zero while the term α sin (t) is included on the right-hand side of (1.38).For convenience, detuning parameters σp ∈ R and σs ∈ R for the primary andsecondary system are introduced. They describe the closeness to resonance, that isexact resonance happens when σp = σs = 0.

ωpω

= 1 + εσp (1.52)

ωsω

=1

2+ εσs . (1.53)

With those modications, the equations for the coupled autoparametric systemfor the two time scales (1.36)-(1.39) are modied to

O (1)∂2

∂t21zp,1 + zp,1 = 0 (1.54)

O (1)∂2

∂t21φs,1 +

1

4φs,1 = 0 (1.55)

O (ε)∂2

∂t21zp,2 + zp,2 + 2σpzp,1 + 2

∂

∂t1

∂

∂t2zp,1 + κp

∂

∂t1zp,1

+µφs,1∂2

∂t21φs,1 + µ

(∂

∂t1φs,1

)2

= α sin (t) (1.56)

O (ε)∂2

∂t21φs,2 +

1

4φs,2 + 2

∂

∂t1

∂

∂t2φs,1 + κs

∂

∂t1φs,1 + σsφs,1

+φs,1∂2

∂t21zp,1 −

1

4γ1φ

3s,1 = 0 . (1.57)

The general solutions to (1.54) and (1.55) are

zp,1 = ap,1 (t2) et1 + cc (1.58)

φs,1 = as,1 (t2) e0.5t1 + cc . (1.59)

Comparing (1.58) and (1.59) to (1.40) and (1.41), respectively, it is noteworthythat the second term on the right-hand side of (1.40) is not present in (1.58), while(1.59) is structurally the same as (1.41); in both (1.58) and (1.59), the detuningas in (1.52) and (1.53) was introduced but the detuning parameters σp and σs arenot present in the O (1) approximation.

With the general solutions (1.58) and (1.59), (1.56) can be evaluated, yieldingfor the primary system:

∂2

∂t21zp,2 + zp,2 + 2σpap,1e

t1 + 2∂

∂t2ap,1e

t1 + κpap,1et1

− 1

2µa2

s,1et1 = −

2αet1 . (1.60)

16


Inserting (1.58) and (1.59) into (1.57) for the secondary system results in

∂2

∂t21φs,2 +

1

4φs,2 +

∂

∂t2as,1e

0.5t1 +

2κsas,1e

0.5t1 + σsas,1e0.5t1

− as,1ap,1e3/2t1 − a∗s,1ap,1e0.5t1 − 1

4γ1a

3s,1e

3/2t1 − 3

4γ1a

2s,1a∗s,1e

0.5t1 = 0 (1.61)

with a∗s,1 the complex conjugate of as,1.To construct a uniformly valid approximate solution to (1.33) and (1.34), the

secular terms in the higher-order level approximation, that is O (ε) in (1.56) and(1.57), have to be eliminated. From (1.60) it follows

2σpap,1 + 2∂

∂t2ap,1 + κpap,1 −

1

2µa2

s,1 = − 2α . (1.62)

Similarly, from (1.61), the solvability condition is

∂

∂t2as,1 +

2κsas,1 + σsas,1 − a∗s,1ap,1 −

3

4γ1a

2s,1a∗s,1 = 0 . (1.63)

Invoking again the polar notation for ap,1 and as,1 as in (1.45) and (1.26) andinserting them into (1.62) and (1.63), the dierential equations for the absolutevalues and arguments αp,1, αs,1, βp,1, and βs,1 can be found:

d

dt2αp,1 +

1

2κpαp,1 +

1

8µα2

s,1 sin (βp,1 − 2βs,1) +α

2cos (βp,1) = 0 (1.64)

αp,1d

dt2βp,1 − σpαp,1 +

1

8µα2

s,1 cos (βp,1 − 2βs,1)− α

2sin (βp,1) = 0 (1.65)

d

dt2αs,1 +

1

2κsαs,1 −

1

2αs,1αp,1 sin (βp,1 − 2βs,1) = 0 (1.66)

d

dt2βs,1 − σs +

1

2αp,1 cos (βp,1 − 2βs,1) +

3

16γ1α

2s,1 = 0 . (1.67)

For the steady-state response it holds that the derivative with respect to t2vanishes for αp,1, αs,1, βp,1, and βs,1. Then, the system (1.64)(1.67) can be solvedanalytically to obtain the rst-order approximation to the coupled autoparametricsystem (1.33) and (1.34). However, the last term in (1.67) which is due to the cubicnonlinearity results in a rather complicated expression for the solution. To easethe readability and to obtain a simpler rst-order approximation, it is thereforeneglected for the moment. Consequently, under this assumption, the solution to(1.64)(1.67) for the steady-state amplitudes is given by

αp,1 =√κ2s + 4σ2

s (1.68)

αs,1 = 2

√√√√ 1

µ

(4σpσs − κpκs ±

√(α2 − 4 (κpσs + κsσp)

2))

. (1.69)

17

1. Introduction

Therefore, the rst-order approximation of the autoparametric system (1.33)and (1.34) is given by

zp = αp,1 cos (t+ βp,0) + O (ε) (1.70)

φs = αs,1 cos (0.5t+ βs,0) + O (ε) (1.71)

where αp,1 and αs,1 are given by (1.68) and (1.69); βp,0 and βs,0 are constants.It is noteworthy that the amplitude of the displacement of the primary sys-

tem zp is not dependent on the external forcing when in resonance condition; incontrary, to a rst approximation, it depends only on the parameters of the sec-ondary system. As (1.68) and (1.70) indicate, the amplitude of the primary systemis directly proportional to the damping of the secondary system κs and the de-tuning parameter σs. An increase of the amplitude of the external forcing α doesnot increase the amplitude of the primary system but results in an increase of theamplitude of the secondary system, as can be seen in (1.69).

Depending on the dimensionless amplitude of the external forcing α, (1.69) hasdierent positive solutions. There are two positive solutions to (1.69) if

4 (κpσs + κsσp)2< α2 <

(κ2p + 4σ2

p

) (κ2s + 4σ2

s

). (1.72)

and κpκs > 4σpσs. Note, that the initial conditions determine the appropriatesolution. On the other hand, there is one positive solution if the dimensionlessamplitude of the external forcing satises

α2 >(κ2p + 4σ2

p

) (κ2s + 4σ2

s

)(1.73)

and there is no positive solution otherwise. The secondary system is not directlyexcited by the external force. As long as the amplitude of the external excitationof the primary system is small enough such that (1.72) and (1.73) are not satised,the secondary system does not oscillate. Any angular disturbances of the secondarysystem do normally decay rapidly, as seen when the systems are uncoupled inSection 1.2.1 and when the systems are coupled, but the coupling between theprimary system and the secondary system is inactive, see Section 1.2.2. However,when the excitation amplitude is increased and eventually satises (1.72) or (1.73),the coupling between the primary system and the secondary system becomes activeand triggers an autoparametric resonance of the secondary system. If that happens,energy is fed from the primary to the secondary system and large oscillations ofthe secondary system may be the result.

As (1.72) and (1.73) illustrate, the conditions for autoparametric resonance aredependent on the detuning parameters which represent the relationship of the threemain frequencies in the autoparametric system; the frequency of the excitation ofthe primary system ω and the natural frequencies of the primary and the secondarysystem, ωp and ωs, respectively. Additionally, the instability region, that is theparameter space when the coupling between the systems is active, depends alsoon the damping of the primary and the secondary system, κp and κs, and onthe amplitude of the excitation α. It can be seen from (1.72) and (1.73) thatincreased damping can raise the threshold for the excitation amplitude to triggerautoparametric resonance.

18


The cubic nonlinearity is neglected in (1.68) and (1.69). Inspecting (1.67), itcan be seen that it does not directly aect the amplitude αs,1 but rather indirectlythrough the phase angle βs,1.

When the conditions for a steady-state solution (1.72) or (1.73) are met, that isthe amplitude of the external forcing α is above the threshold to trigger autopara-metric resonance, energy is transferred from the primary system to the secondarysystem. Consequently, the amplitude αs,1 of the secondary system increases. Notenow that an increase of αs,1 changes the phase βs,1 through (1.67) which in turnalters the rate at which energy is pumped from the primary system to the sec-ondary system by the cubic nonlinearity. From (1.66) and (1.67) the steady-stateamplitude αs,1 can be found:

αs,1 =

√16σs3γ1

± 8

3γ1

√α2p,1 − κ2

s (1.74)

where αp,1 is also dependent on γ1.It is noteworthy from (1.74) that the cubic nonlinearity limits the amplitude of

the secondary system αs,1. Steady-state motion occurs when the derivatives withrespect to t2 vanishes for αp,1, αs,1, βp,1, and βs,1; the rate at which energy ispumped from the primary system to the secondary system is then balanced by therate at which energy is dissipated by viscous eects.

1.2.3 Energy Flow

It is revealing to study the energy ow in the autoparametric system (1.5) and(1.6). Generally, dierential work is dened as the product of force and dierentialtranslation [32]:

w12 =

∫ s2

s1

fds (1.75)

=

∫ t

0

f (τ)ds (τ)

dτdτ (1.76)

where w12 is the work done by the applied force f along a path s with startingpoint s (0) = s1 and end point s (t) = s2 which represents the motion. Dierentialwork is then

dw = f (t)ds (t)

dtdt (1.77)

where (1.77) might be used to compute the work done by integration of the forcemultiplied by the velocity. To study the energy ow in system (1.5) and (1.6),the dierent locations in the primary and the secondary systems where energy isconcentrated, are considered. A negative energy quantity indicates that energy istransferred to the respective subsystem. Consequently, energy fed from the primarysystem to the secondary system would appear as a positive quantity in the balanceof the former and as a negative quantity for the latter. The energy due to the

19

1. Introduction

inertia forces is

wmpp =

∫ t

0

mpzpdzpdτ

dτ (1.78)

wmss =

∫ t

0

msl2s φs

dφsdτ

dτ . (1.79)

Similarly the energy dissipated due to viscous damping is computed by

wdpp =

∫ t

0

dpzpdzpdτ

dτ (1.80)

wdss =

∫ t

0

dsφsdφsdτ

dτ . (1.81)

The energy due to the linear spring in the primary system and the energy due tothe gravity eld is

wkpp =

∫ t

0

kpzpdzpdτ

dτ (1.82)

wkss =

∫ t

0

msgls sin (φs)dφsdτ

dτ . (1.83)

The energy portions due to the coupling of the primary system and the secondarysystem represent the energy which is transferred from the primary system to thesecondary system and the other way around:

wcps =

∫ t

0

msls

(φs sin (φs) + φ2

s cos (φs)) dzp

dτdτ (1.84)

wcsp =

∫ t

0

mslszp sin (φs)dφsdτ

dτ . (1.85)

The energy fed to the primary system due to the external forcing is

wextp = −∫ t

0

a sin (ωτ)dzpdτ

dτ (1.86)

where a negative sign represents that the energy is fed into the autoparametricsystem. The total energy at all time instants is the sum of the above energy portionsand is constant.

Figures 1.4-1.6 depict the result of a simulation study of the autoparametricsystem (1.5) and (1.6). Figure 1.4(a) shows the vertical displacement zp of the pri-mary system when externally excited by a sinusoidal force. The natural frequencyof the primary system ωp and the excitation frequency ω are identical, resulting inthe primary system resonating. Figure 1.4(b) depicts the angular displacement ofthe secondary system φs. The rst 50 s, the natural frequencies of the excitationforce and the secondary system, ω and ωs, are detuned, that is ω 6≈ 2ωs. Thisimplies that the coupling between the primary system and the secondary system isnot active and hence no energy is transferred in that direction to enable a sustained

20


oscillation of the secondary system. It is noteworthy that the initial displacementreadily decays, see also Section 1.2.1. At t = 50 s, the excitation frequency ω andthe natural frequency of the secondary system ωs are tuned in, that is ω ≈ 2ωs.Although that the coupling is enabled and parametric resonance possible, it takessome time to see the eect on φs. In Figure 1.4(a), the amplitude of the verti-cal oscillation of the primary system is shown. However, once enough energy istransferred from the primary to the secondary system, the build-up of the angulardisplacement φs is quick. The amplitude of the oscillation of the secondary systemis increased until a steady-state value, mainly characterized by the damping κs andthe restoring nonlinearity in the secondary system. After 300 s, the excitation fre-quency ω and the natural frequency of the secondary system ωs are detuned again.The immediate eect is visible on the displacement of the primary system zp. Sincethere is no more energy transferred to the secondary system, the amplitude of zpincreases again.

Time (s)

Displacementz p

(m)

0 100 200 300 400 500-10

-5

0

5

10

(a) Displacement of the primary system.

Time (s)

Angulardeflectionφs()

0 100 200 300 400 500

-50

0

50

(b) Angular deection of the pendulum.

Figure 1.4: Simulation results for the autoparametric system.

Figure 1.5 depicts the energy portions due to the inertia and damping. Theenergy due to the viscous damping shows its proportionality to the velocity of themotions of the primary and secondary system.

Figure 1.6(b) shows the transfer of energy from the primary to the secondaryand the other way around. Before 50 s, the two systems are de facto decoupledand there is no energy transfer then. However, once the excitation frequency ωand the natural frequency of the secondary system ωs are tuned to a parametricresonance condition, the energy starts to ow from the primary system to the

21

1. Introduction

Time (s)

Energydueto

theinertiaforce

Primary systemSecondary system

0 100 200 300 400 500-100

0

100

200

(a) Energy due to the inertia force.

Time (s)

Energydueto

thedampingforce


0 100 200 300 400 5000

5000

10000

15000

(b) Energy due to the damping force.


secondary system. It is interesting to note that there seems to be no, or negligiblylittle energy ow from the secondary system to the primary system as indicated inFigure 1.6(b). The sum of the energy portions due to the coupling of the primarysystem and the secondary system is zero, since the transfer of energy from onesubsystem to the other appears as positive quantity for the transferring subsystemand as negative quantity for the receiving subsystem. Soon after about 300 s, whenthe coupling is not active anymore, there is no further energy transferred in betweenthe systems.

1.3 Parametric Resonance

Although the formalism of autoparametric resonance as in Section 1.2 oers in away a more complete characterization of the underlying physical phenomenon, thereare reasons to prefer the mathematical description of parametric resonance. For itsdisadvantage it oers a simpler formulation, perhaps more preferable for analysispurposes. Since the main purpose of this thesis is the research of control methodsof (auto-)parametric resonance of ship roll motions, reducing the complexity of theautoparametric system to a simpler parametric system oers enhanced possibilityfor control synthesis. One should, however, keep in mind that from reformulatingan autoparametric system as parametric system, some insights are lost. The energy

22

1.3. Parametric Resonance

Time (s)

Energydueto

springforce


0 100 200 300 400 500-100

0

100

200

(a) Energy due to the spring force.

Time (s)

Energydueto

thecoupling


0 100 200 300 400 500-2000

-1000

0

1000

2000

(b) Energy due to the coupling between primary and secondary system.


ow formulation as in Section 1.2.3 is implicitly only in an autoparametric system.However, as visible in Figure 1.6, depending on the situation, there might be littleenergy transfer from the secondary system to the primary system, and thus theinformation might be discarded safely. It is mainly a matter of what the purposesof the analysis is when deciding for a autoparametric or a parametric formulation.After all, the underlying physic they try to capture, is basically the same.

Whereas an autoparametric formulation is characterized by constant (or at leastslowly time-varying) coecients in the describing system equations, the parametricmodelling introduces, in general periodically, time-varying coecients. This is oflittle surprise as it incorporates the primary system into the secondary system'sdierential equations.

The main distinction between principal parametric resonance and ordinary mainresonance is that the former is characterized by a possible large response for a para-metric excitation of about twice the natural frequency, whereas the latter resonatesat an excitation with a frequency close to the natural frequency of the parametricsystem.

Parametrically excited systems have been known for centuries. Earliest men-tions date back to Faraday in the beginning of the 19th century who discoveredthat when a water glass is excited vertically by a force, the surface waves oscillateby half the frequency of the excitation [38]. Other early observations of parametricresonance were done by Melde [93] and Mathieu [92] pioneered the mathemati-

23

1. Introduction

cal description of the phenomenon. The following introductory discussion followsmainly the work published by Nayfeh and Mook [100], Cartmell [29], Dohnal [32],and Tondl et al. [145].

A large class of systems that can parametrically resonate can be written as [100]

xm + dm (t) xm + km (t)xm = 0 (1.87)

where xm ∈ R is the state of the system; dm ∈ R and km ∈ R are time-dependentperiodic coecients.

Under the assumption that dm is dierentiable, a change of variables can beused to eliminate xm in (1.87):

xm (t) = xm (t) e12

∫dm(τ)dτ . (1.88)

Inserting (1.88) into (1.87) allows the reformulation into the standard form ofthe arche-typical parametric system

¨xm + km (t)xm = 0 (1.89)

with the transformed coecient km (t) given by

km (t) = km (t)− 1

4d2m (t)− 1

2dm (t) . (1.90)

Equation (1.89) is also known as Hill's equation. A special case of Hill's equation(1.89) is when km (t) = δ + 2ε cos (2t) and (1.89) becomes

¨xm + (δ + 2ε cos (2t)) xm = 0 (1.91)

which is known as Mathieu's equation.Consider again the coupled autoparametric system of Section 1.2 given by (1.33)

and (1.34):

z′′p + εκpz′p +

ω2p

ω2zp + εµ

(φ′′s(φs − εγ1φ

3s

)+ φ′s

2 (1− εγ2φ

2s

))= α sin (t) (1.92)


ω2s

ω2

(φs − εγ1φ

3s

)+ εz′′p

(φs − εγ1φ

3s

)= 0 . (1.93)

The semi-trivial solution is dened by φs = dφs/dt = 0. Then the particu-lar periodic solution to (1.92) can be found as zp,p in (1.13). Inserting zp,p =ˆzp,p sin (t+ ϕp,p) and φs (t) = 0 + φs in (1.93) and linearising around the semi-trivial solution then yields for the secondary system:

φ′′s +1

4φs + ε

(κsφ

′s + σsφs − ˆzp,p sin (t+ ϕp,p)φs

)= 0 (1.94)

which is recognized as a Mathieu-type equation with a additional damping term.Thus, by the above simplications, the analysis of the autoparametric system ofSection 1.2 can be reduced to the analysis of a Mathieu-type equation. However,it is important to note that the simplication in the analysis may come at the costof lower accuracy.

24

1.4. Parametric Roll Resonance in Ships

Floquet's theorem allows to analyse the solutions of a linear ordinary dierentialequation with time-varying periodic coecients with known period. It does so byrelating the periodic system to a linear system with real, constant coecients.

According to Floquet's theorem the solutions of (1.91) take the following form:

xm =[xm (δ, ε, t) ˙xm (δ, ε, t)

]>= Φ (t) eΓt (1.95)

where Γ ∈ C2×2 is a constant matrix with eigenvalues γi, and Φ (t) is a matrixwith period π. In general Φ (t) is not sinusoidal.

The stability of xm = 0 is determined by the real part of the γ's. If the realparts of all γ's are negative or equal to zero, then xm ∈ R2 is bounded and xm = 0therefore stable. On the other hand, if one of the γ's has a positive real part, xmis unbounded, thus the equilibrium xm = 0 is unstable.

The corresponding values for ε and δ in (1.91) of the boundary between xm ∈ Rbeing growing in time and xm being decaying in time, determine the transitioncurves. It is not trivial to compute the transition curves but there are severalnumerical techniques.

Consider the two unique solutions of the Mathieu equation (1.91), which aredened for xed δ and ε, by

C (δ, ε, t) =xm (δ, ε, t) + xm (δ, ε,−t)

2xm (δ, ε, 0)(1.96)

S (δ, ε, t) =xm (δ, ε, t)− xm (δ, ε,−t)

2 ˙xm (δ, ε, 0)(1.97)

Equation (1.96) denes the Mathieu cosine which is an even function with theproperty that C (δ, ε, 0) = 1 and dC (δ, ε, 0) /dt = 0, whereas (1.97) represents theMathieu sine. The Mathieu sine is an odd function for which S (δ, ε, 0) = 0 anddS (δ, ε, 0) /dt = 1.

The general solution to the Mathieu equation (1.91) is then a linear combinationof the above dened Mathieu cosine and Mathieu sine functions.

Remarkably, a characteristic of parametrically excited systems, which are de-scribed by Mathieu's equation, is, that if the parameters vary at a frequency whichis close to 2/n (n = 1, 2, . . . ) times the natural frequency of the system, the systembegins to resonate and the oscillation amplitude grows exponentially in the absenceof damping. The case when the parametric excitation frequency is approximatelytwice the natural frequency is called principal parametric resonance.

Parametric resonance can occur in many real systems, such as ships [20], oatingwind turbines [113], oshore structures [85, 125], spar platform [129], and manyothers [31, 74, 81, 112]. A informative introduction is also given in Fossen andNijmeijer [40].

1.4 Parametric Roll Resonance in Ships

Large roll amplitudes in ships can be caused by several eects. Direct excitation ofthe roll motion in severe sea. This is equivalent to an external forcing as describedfor the primary system in Section 1.2 in absence to a coupling to a secondary

25

1. Introduction

system. The direct excitation of roll is only partly covered in this thesis and if soonly in combination with (auto-)parametric resonance.

Contrary to directly excited oscillation, the occurrence of large roll amplitudescan be accounted by two indirect mechanisms. According to Oh, Nayfeh, and Mook[116], large roll oscillations can be caused by the nonlinear coupling of the heaveand pitch modes to the roll mode. The energy of the waves which directly excitesthe pitch and heave modes, is transferred into the roll mode. As a consequence,the ship can suddenly experience dangerous conditions for both the ship as well asthe crew and the cargo.

This mechanism was studied by various scientists, and among those Oh, Nayfeh,and Mook [116] provided a solid theoretical background for the study of this phe-nomenon. This mechanism is referred to as parametric excitation. The nonlinearcoupling between heave, pitch and roll lead to time-varying coecients in the gov-erning dierential equation in the roll mode and thus results in a Hill or Mathieu'sequation. The parametrically excited oscillator was considered in Section 1.3.

The roll mode is assumed to be independent of the heave and pitch motionswhose amplitudes and frequencies represent the eective amplitude and frequencyof the parametric excitation. The time varying meta-centric height forms the para-metric term in (1.91). The fundamental and principal resonance occurs when thewave frequency is approximately the same or twice the natural roll frequency,respectively. According to Oh, Nayfeh, and Mook [116], the latter is the more dan-gerous.

The second indirect method is referred to in Oh, Nayfeh, and Mook [116] asautoparametric resonance. It considers the two nonlinear coupled motions in rolland pitch. Thus both the pitch motion and the roll motion are functions of theprescribed wave oscillation. The nonlinear coupling is due to hydrostatic terms.This autoparametric or internal resonance phenomenon occurs when the naturalfrequency in roll is approximately half of the pitch natural frequency and the shipapproaches the waves with the encounter frequency near to either the pitch or theroll natural frequency. This leads to saturation eects, i.e. although initially thepitch mode is excited, a saturation of it leads to an energy transfer to the rollmode. The coecients of the governing dierential equation for the roll motion areconstant.

In this context of denition of autoparametric resonance, the ship can be re-garded as a autoparametric system consisting of two subsystems. The primarysystem is composed of both the heave and pitch dynamics, while the secondarysubsystem considers the dynamics in roll motion. The primary system is then ex-ternally excited by the motion of the waves and energy is transferred from the wavesto the primary system. Through parametric excitation, the second subsystem, i.e.the roll mode, gets excited by the primary system.

Recently the research on the phenomenon of parametric resonance in shipshas intensied, surely partly caused by accidents in this regards [e.g. 44] whichreported signicant danger for the crew and the ship as well as considerable mon-etary damages. Especially susceptible to parametric roll resonance are containercarriers because of their design. The periodically varying parameters of the govern-ing dierential equations for the roll motions are caused by the variations in thegeometry of the submerged hull or more specically, by the intercepted water-plane

26

1.5. Scope and Main Contributions

area which varies with the exciting wave motions. Three main factors can be iden-tied to lead to parametric resonance: pitch motion, heave motion and variationsin hull geometry [20].

Shin et al. [132] illustrated the importance of researching parametric resonanceby stating, that the practice in heavy weather condition is to sail into head seasat reduced speed. This causes the ship to be even more prone to parametricallyexcited resonance of the roll motion.

France et al. [44] identied the criteria for parametric roll resonance

The encounter frequency is equal to approximately twice the natural rollfrequency, i.e. ωe ≈ 2ωφ.

The wave length is in the order of the ship length, λ ≈ Lpp. The wave height exceeds a critical threshold, which is ship dependent, i.e.ζ > hs.

The damping in roll is low.

There has been considerable research activities concerning parametric reso-nance. It has been shown that several types of vessels are prone to experienceparametric roll resonance, such as container ships [7, 12, 59, 60, 62, 88, 96, 97, 115,127, 136, 140, 142, 148], PCTC [60, 130], shing vessels [72, 102104, 106108, 120,135], multihull vehicles [22, 23, 82], passenger ships [98], and others [117].

Some major research has been done on the prediction of the roll response of thevessels [75, 95, 141, 152], on parametric roll in irregular waves [24, 25, 69, 7779,86, 89, 90, 146, 147, 151], on modelling parametric roll resonance [19, 21, 42, 43,101, 105, 111, 128, 137, 143, 144], on the detection of parametric roll resonance[5053, 65, 73, 76, 94, 119], and on the inuence of surge speed on parametric rollresonance [131, 138]. Probabilistic approaches to parametric roll resonance havealso been researched [46].

1.5 Scope and Main Contributions

6-DOF Ship Model

In Chapter 3, a 6-DOF ship model for parametric roll resonance under non-constantspeed is presented. The encounter frequency is the Doppler-shifted frequency ofthe waves as seen from the ship. It is dependent on the ship's surge speed andher heading angle and consequently, also for constant heading angles, variationsin the speed change the encounter frequency. Commonly, the ship dynamics inparametric roll resonance are described by a Mathieu-type 1-DOF roll equationwith the encounter frequency as the parametric excitation frequency, such as

m44φ+ d44φ+ [k44 + kφt cos(ωet+ αφ)]φ = 0 .

Here, m44 > 0 ∈ R is the sum of the moment of inertia and the added momentof inertia in roll, d44 > 0 ∈ R the linear hydrodynamic damping coecient, andk44 > 0 ∈ R the linear restoring moment coecient. The amplitude of the changein the linear restoring coecient is kφt > 0 ∈ R, ωe ∈ R is the encounter fre-quency, and αφ ∈ R is a phase angle. All the parameters are considered constant.

27

1. Introduction

A system described by the above Mathieu's equation parametrically resonates atωe ≈ 2

√k44/m44 [100].

However, modelling parametric roll resonance in that way assumes a constantparametric excitation frequency and therefore a constant ship speed. In Chapter 4it is shown both mathematically and in simulations that Mathieu-type equationsare not suitable to accurately capture the dynamics of the ship if the encounterfrequency is time-varying. Consequently, there is a need for a novel 1-DOF rollmodel applicable to control purposes. To that matter, in Chapter 3, a highly accu-rate 6-DOF computer model of the ship is derived and implemented for numericalsimulations. The 6-DOF ship model takes into account the external forces and mo-ments induced on the ship by the hydrostatic and hydrodynamic pressure eld ofthe surrounding ocean. For each instant in time, the pressure is integrated overthe instantaneous submerged hull, yielding the restoring forces. The 6-DOF modelis capable of handling complex sea states with non-steady ship motion, and wave-induced eects enter as rst-order forces via the pressure eld. However, this 6DOFmodel is not analytical, and can only be implemented on a computer. Its appli-cability to control purposes and mathematical analysis is limited, but it is highlysuitable for simulation purposes. The 6-DOF ship model has been published inBreu, Holden, and Fossen [16].

1-DOF Roll Model for Time-Varying Speed

The 6-DOF ship model presented in Chapter 3 is a numerical model. In Chap-ter 4, an attempt is therefore made to reduce the 6-DOF model to a 1-DOF rollmodel which is capable of describing the basic underlying phenomenon of para-metric roll resonance and at the same time is simple enough to oer the possibilityfor controller design and analysis. To achieve this, the 6-DOF model is in a rststep reduced to the three most important degrees of freedom for ships in paramet-ric roll resonance, that is the heave, roll, and pitch motions [62, 110]. The wavesare assumed to be steady and planar. The ship's surge speed is explicitly allowedto be slowly time-varying. In a second step and by employing a quasi-steady ap-proach to derive explicit time-domain solutions to the heave and pitch motions,the intermediate 3-DOF heave-pitch-roll model is further reduced to a 1-DOF rollmodel.

It is pointed out in Chapter 4 how the derived 1-DOF roll model is relatedto the 1-DOF Mathieu-type roll model with constant parametric excitation. Thederived 1-DOF roll model is analytical, and preserves the majority of the accuracyof the 6-DOF model.

The verication of the simplied 1-DOFmodel is also presented in Chapter 4. Tothat matter, numerical simulations of the 6-DOF computer model and the 1-DOFroll model are analysed. The simplied 1-DOF roll model is shown to qualitativelycapture the behaviour of the highly accurate 6-DOF model for constant but alsotime-varying encounter frequency. A comparison with the Mathieu-type 1-DOF rollmodel reveals that the Mathieu-type model fails to adequately describe the rollingof the ship in parametric roll resonance for non-constant encounter frequency. The1-DOF roll model and the verication study has been published in Breu, Holden,and Fossen [16].

28

1.5. Scope and Main Contributions

1-DOF Roll Model for Time-Varying Heading and Speed

The 1-DOF roll model presented in Chapter 4 is valid for slowly time-varying speedwhen the ship sails in head seas. That is, the waves are travelling directly againstthe course of the ship. As a consequence, although there might be parametricallyinduced roll oscillations, the wave do not excite rolling by a direct moment in roll.

In Chapter 5, an analytical 1-DOF roll model which is able to describe thedynamics of a ship in parametric resonance for both time-varying heading angleand surge speed is presented. This is motivated by the possibility to change theencounter frequency by changes in the heading angle and the surge speed. Since achange in the course makes the ship potentially vulnerable to directly induced rollmotion, the 1-DOF roll model of Chapter 4 has to be extended to incorporate directwave excitation. The hydrostatic and hydrodynamic coecients of the 1-DOF rollmodel are identied based on simulations of the accurate 6-DOF ship model ofChapter 3 which accounts for rst-order generalized pressure forces and momentsby numerical integration of the instantaneous pressure eld of the surroundingocean over the instantaneous submerged hull. To arrive at an analytical modelwhich is suitable for mathematical analysis and control purposes, the functionalexpressions for the heading-dependent hydrodynamic coecients are determinedby curve tting. A verication of the resulting 1-DOF roll against the 6-DOF shipmodel in simulations is presented as well as an investigation of the stability regions.The contents of this chapter has been published in Breu, Holden, and Fossen [17].

Frequency Detuning

There are many ways of controlling parametric roll resonance. The control methodscan roughly be categorized as direct or indirect methods. The direct methods areaimed at directly controlling the roll motion by generating an opposing roll moment.Indirect strategies attempt to violate the empirical conditions necessary for theonset of parametric roll resonance. A hybrid approach, doing both at the sametime, is also possible.

Chapter 6 presents a control strategy for a ship in parametric roll resonance. Thestrategy is based on changing the frequency of the parametric excitation by varyingthe ship's surge speed via Doppler shift. The control philosophy considers the the 1-DOF roll model of Chapter 4 which is capable of describing the roll motion for time-varying encounter frequency. The stability properties of the closed-loop system areproven and the eectiveness of the control system is demonstrated by simulatingthe closed-loop system using both the 6-DOF ship model of Chapter 3 and thesimplied 1-DOF model of Chapter 4. The simulations show a good agreementwith the theoretical analysis.

The main purpose of this chapter is to show that it is feasible to control para-metric roll resonance by changing the encounter frequency to violate the conditionωe ≈ 2

√k44/m44. This control approach is termed frequency-detuning.

As shown in Chapter 4, the Mathieu-type equations are not valid for non-constant ωe. To design and analyse the control system, the simplied, 1-DOFModelpresented in Chapter 5, is therefore considered. The ship's forward speed is allowed

29

1. Introduction

to be slowly time-varying. For ships susceptible to parametric roll many of whichare large [28, 4244] this is not an unreasonable assumption.

Based on the 1-DOF roll model of Chapter 5, a simple controller based on alinear change of the encounter frequency by variation of the ship's forward speed isproposed. It is mathematically proven that the proposed controller is able to drivethe ship out of parametric resonance, driving the roll motion to zero. It is worthnoting that the controller is in fact simple enough that a helmsman can performthe necessary control action, rendering a speed controller unnecessary.

The controller is tested with the simplied 1-DOF model of Chapter 5 and thecomplex, numerical 6-DOF model presented in Chapter 3, and is shown to work asexpected in both cases.

Frequency Detuning by Optimal Speed and Heading Changes

In Chapter 7 three strategies to actively control ships experiencing parametric rollresonance are proposed. Those approaches aim at changing the frequency of theparametric excitation by controlling the Doppler shift, which, in recent results,has been shown to be eective to reduce the roll angle signicantly. However,exactly how to change the frequency of the parametric excitation to stabilize theparametric oscillations has remained an open research topic. Thus, three optimalcontrol philosophies that alter the ship's forward speed and heading angle, whichin turn changes the encounter frequency and consequently the frequency of theparametric excitation, are presented. This is referred to as frequency detuning.

As a rst approach, the use of a model predictive control (MPC) strategy isproposed. By addressing both states and input constraints explicitly, the MPCformulation is used to change the ship's forward speed and heading angle in anoptimal manner to reduce parametric roll oscillations.

As a second approach, the methodology of extremum seeking (ES) control isadapted for ships in parametric roll resonance to iteratively determine the optimalset-point of the encounter frequency in order to avoid one of the conditions forparametric roll. The encounter frequency is consequently mapped to the ship'sforward speed and heading angle by a control allocation (CA). This is formulatedas a constrained nonlinear optimization problem.

Finally, the ES speed and heading control strategy to stabilize roll paramet-ric resonance in ships is extended, aiming at robustness. Two major modicationsare suggested. Firstly, the speed and heading controllers are formulated in theframework of L1 adaptive control which guarantees robustness while still havingfast adaptation. By doing so, robustness is increased with respect to model uncer-tainty, lack of knowledge, and bounded external disturbances. Secondly, the ES loopis modied towards limit cycle minimization, which replaces the objective functionof the previously presented control approach, thus relaxing the assumption thatthe frequency range for roll parametric resonance is a priori known.

The eectiveness of the proposed control approaches to stabilize parametricoscillations, by simultaneously changing the ship's forward speed and heading angleoptimally, is illustrated by computer simulations. This clearly veries the concepton frequency detuning.

30

1.6. List of Publications

Frequency Estimation

The frequency detuning control approaches presented in Chapter 6 and 7 rely toa certain degree on the knowledge of the wave encounter frequency. In Chapter 8and 9 two attempts to estimate the wave encounter frequency and the modal wavefrequency are presented.

As a rst approach, the feasibility of a EKF based frequency estimation for aship in parametric roll resonance is investigated in Chapter 8. The EKF representsa type of model-based frequency estimation. The performance of the proposed EKFis veried in a simulation study.

Secondly, a signal based frequency estimator for the estimation of the waveencounter frequency at sea, is considered in Chapter 9. The frequency estima-tor under consideration is a second-order lter designed for the estimation of thefrequency of a sinusoid with unknown amplitude, frequency, and phase. The signal-based frequency estimator is used to estimate the wave encounter frequency usingmeasurements of the roll angle of the ship. The frequency estimator is tted witha switching gain in the frequency adaptation law. A Lyapunov stability proof isprovided for the frequency estimator with switching gain and the frequency esti-mator is shown to be K exponentially stable. The designed estimator is veried ina simulation study.

1.6 List of Publications

In the course of this PhD research, eight international publications have been pro-duced whereas one has been recently submitted.

1.6.1 Book Chapters

[14] D. A. Breu, L. Feng, and T. I. Fossen. Optimal Speed and Heading Con-trol for Stabilization of Parametric Oscillations in Ships. In: ParametricResonance in Dynamical Systems. Ed. by T. I. Fossen and H. Nijmeijer.Springer New York, 2012. Chap. 11, pp. 213238. isbn: 978-1-4614-1042-3.doi: 10.1007/978-1-4614-1043-0_11.

[16] D. A. Breu, C. Holden, and T. I. Fossen. Ship Model for Parametric RollIncorporating the Eects of Time-Varying Speed. In: Parametric Resonancein Dynamical Systems. Ed. by T. I. Fossen and H. Nijmeijer. Springer NewYork, 2012. Chap. 9, pp. 167189. isbn: 978-1-4614-1042-3. doi: 10.1007/978-1-4614-1043-0_9.

[64] C. Holden, D. A. Breu, and T. I. Fossen. Frequency Detuning Control byDoppler Shift. In: Parametric Resonance in Dynamical Systems. Ed. by T. I.Fossen and H. Nijmeijer. Springer New York, 2012. Chap. 10, pp. 193212.isbn: 978-1-4614-1042-3. doi: 10.1007/978-1-4614-1043-0_10.

31

http://dx.doi.org/10.1007/978-1-4614-1043-0_11

http://dx.doi.org/10.1007/978-1-4614-1043-0_9

http://dx.doi.org/10.1007/978-1-4614-1043-0_9

http://dx.doi.org/10.1007/978-1-4614-1043-0_10

1. Introduction

1.6.2 Conference Papers

[9] D. J. W. Belleter, D. A. Breu, T. I. Fossen, and H. Nijmeijer. NonlinearObserver Design for Parametric Roll Resonance. In: Proceedings of the 11thInternational Conference on the Stability of Ships and Ocean Vehicles. 2012,pp. 699705. isbn: 978-618-80163-0-9.

[10] D. J. W. Belleter, D. A. Breu, T. I. Fossen, and H. Nijmeijer. AK-ExponentialStable Nonlinear Observer for the Wave Encounter Frequency. In: Submit-ted to the 9th IFAC Conference on Control Applications in Marine Systems.2013.

[15] D. A. Breu and T. I. Fossen. Extremum Seeking Speed and Heading ControlApplied to Parametric Resonance. In: Proceedings of the 8th IFAC Confe-rence on Control Applications in Marine Systems. 2010, pp. 2833. isbn:978-3-902661-88-3. doi: 10.3182/20100915-3-DE-3008.00003.

[17] D. A. Breu, C. Holden, and T. I. Fossen. Stability of Ships in ParametricRoll Resonance Under Time-Varying Heading and Speed. In: Proceedingsof the 11th International Conference on the Stability of Ships and OceanVehicles. 2012, pp. 305313. isbn: 978-618-80163-0-9.

[18] D. A. Breu and T. I. Fossen. L1 Adaptive and Extremum Seeking ControlApplied to Roll Parametric Resonance in Ships. In: Proceedings of the 9thIEEE International Conference on Control and Automation. 2011, pp. 871876. isbn: 978-1-4577-1475-7. doi: 10.1109/ICCA.2011.6138047.

1.7 Thesis Outline

The remainder of the thesis is organized as follows:

Part I Chapter 2 This chapter contains the nomenclature which is used in thisthesis.

Part II This part contains the contributions in modelling of parametric roll reso-nance.

Chapter 3 This chapter presents the 6-DOF computer model. The materialof this chapter has been published in Breu, Holden, and Fossen [16].

Chapter 4 This chapter presents the derivation of the simplied 1-DOF rollmodel which is valid for time-varying speed. The material of this chapterhas been published in Breu, Holden, and Fossen [16].

Chapter 5 This chapter presents an extension to the model of Chapter 4,incorporating the eect of time-varying speed and heading angle. Thematerial of this chapter has been published in Breu, Holden, and Fossen[17].

Part III This part contains the contributions in frequency detuning of parametricroll resonance.

Chapter 6 This chapter presents and investigates the concept of frequencydetuning using a very simple speed change. The material presented inthis chapter has been published in Holden, Breu, and Fossen [64].

32

http://dx.doi.org/10.3182/20100915-3-DE-3008.00003

http://dx.doi.org/10.1109/ICCA.2011.6138047

1.7. Thesis Outline

Chapter 7 This chapter contains the contribution in frequency detuning bya combined change of the surge speed and the heading angle. Threeoptimal control approaches are considered. The work of this chapter hasbeen published in Breu, Feng, and Fossen [14], Breu and Fossen [18],and Breu and Fossen [15].

Part IV This part contains the contributions in the estimation of the consideredfrequencies.

Chapter 8 This chapter presents the frequency estimator formulated as anEKF. The material of this chapter has been published in Belleter et al.[9].

Chapter 9 This chapter presents a signal-based K exponentially stable fre-quency observer. The material of this chapter has been published inBelleter et al. [10].

Part V Chapter 10 This chapter contains the conclusions.

33

Chapter 2

Nomenclature

This chapter lists the acronyms and variables used throughout this thesis. In gen-eral, scalars will be denoted by italic lowercase letters, vectors by bold face lowercaseletters and matrices by bold face capital letters.

Acronyms

CA control allocation. 30, 84, 107, 110, 111, 114, 124, 126, 161cc complex conjugate. 12, 14, 16DOF degree-of-freedom. i, 7, 2730, 32, 47, 48, 50, 52, 53, 5863, 65,

67, 6971, 74, 76, 78, 8183, 88, 89, 91, 94, 95, 98, 99, 102, 117,124, 126, 133, 148, 151, 159, 160

EKF extended Kalman lter. ii, 31, 33, 133137, 162ES extremum seeking. 30, 97, 98, 105107, 110, 112115, 117, 124

126, 161FFT fast Fourier transform. 143FL feedback linearization. 112GES globally exponentially stable. 112MPC model predictive control. 30, 97, 101105, 161MSE mean-squared-error. 151NMPC nonlinear model predictive control. 104PCTC pure car and truck carrier. 6, 27PID proportional integral derivative. 52, 76QCAT Quadratic Programming Control Allocation Toolbox. 111TEU twenty-foot equivalent unit. 5

Autoparametric and parametrically excited system

a > 0 ∈ R amplitude of the external excitation. 7α ∈ R scaled dimensionless excitation amplitude of the primary sys-

tem. 16, 18, 19

35


α ∈ R dimensionless excitation amplitude of the primary system. 8,16

αp,0 ∈ R constant of integration. 15αp,1 ∈ R absolute value of the coecient of the rst order expansion of

the displacement of the primary system. 15, 1719αs,0 ∈ R constant of integration. 13αs,1 ∈ R absolute value of the coecient of the rst order expansion of

the angular deection of the secondary system. 12, 13, 1719ap,1 ∈ C coecient of the rst order expansion of the vertical motion of

the primary system. 14, 15, 17as,1 ∈ C coecient of the rst order expansion of the angular deection

of the secondary system. 12, 14, 17βp,0 ∈ R constant of integration. 15, 18βp,1 ∈ R argument of the coecient of the rst order expansion of the

displacement of the primary system. 15, 17βs,0 ∈ R constant of integration. 13, 18βs,1 ∈ R argument of the coecient of the rst order expansion of the

angular deection of the secondary system. 12, 13, 17, 19cp,c ∈ R constant in the complementary solution of the primary system.

11dm ∈ R periodic coecient in the parametrically excited system. 24dp > 0 ∈ R damping coecient of the primary system. 7, 11ds > 0 ∈ R damping coecient of the secondary system. 7ε > 0 ∈ R small positive constant. 11, 84fe ∈ R external excitation force. 7Γ ∈ C2×2 constant matrix with eigenvalues γi. 25γ1 > 0 ∈ R scaled, dimensionless coecient of the quadratic nonlinearity

of the autoparametric system. 11, 13, 19γ2 > 0 ∈ R scaled, dimensionless coecient of the quadratic nonlinearity

of the autoparametric system. 13κp > 0 ∈ R scaled, dimensionless damping coecient of the primary sys-

tem. 13, 18κs > 0 ∈ R scaled, dimensionless damping coecient of the secondary sys-

tem. 11, 13, 18κp > 0 ∈ R dimensionless damping coecient of the primary system. 8, 9,

11, 13κs > 0 ∈ R dimensionless damping coecient of the secondary system. 8,

9, 11, 13, 21km ∈ R transformed periodic coecent in the parametrically excited

system. 24km ∈ R periodic coecent in the parametrically excited system. 24kp > 0 ∈ R spring coecient of the primary system. 7L ∈ R Lagrangian of the autoparametric system. 8ls > 0 ∈ R pendulum length of the secondary system. 7mp > 0 ∈ R mass of the primary system. 7ms > 0 ∈ R mass of the secondary system. 7

36


µ ∈ R dimensionless quantity, coupling of the primary system to thesecondary system. 8

ω > 0 ∈ R frequency of the external excitation. 7, 10, 13, 15, 18, 20, 21ωp > 0 ∈ R natural frequency of the primary system. 810, 13, 15, 18, 20ωs > 0 ∈ R natural frequency of the secondary system. 810, 15, 18, 20, 21Φ ∈ R2×2 constant matrix with eigenvalues γi. 25φs ∈ R angular deection of the secondary system. 7, 9, 12, 13, 20, 21φs,1 ∈ R rst order expansion of the angular deection of the secondary

system. 12, 14, 16φs,2 ∈ R second order expansion of the angular deection of the sec-

ondary system. 12, 15ϕp,c ∈ R constant phase angle in the complementary solution of the pri-

mary system. 11ϕp,p ∈ R constant phase angle in the particular solution of the primary

system. 11σp ∈ R small detuning parameter for the primary system. 16σs ∈ R small detuning parameter for the secondary system. 16, 18T > 0 ∈ R kinetic energy of the autoparametric system. 7t > 0 ∈ R dimensionless time variable. 8, 12t1 > 0 ∈ R fast time scale. 11, 12t2 > 0 ∈ R slow time scale. 11, 12V > 0 ∈ R potential energy of the autoparametric system. 7wdpp ∈ R energy due to the damping force in the primary system. 20

wextp ∈ R energy in the primary system due to external forcing. 20

wkpp ∈ R energy due to the spring force in the primary system. 20

wmpp ∈ R energy due to the inertia force in the primary system. 20

wcps ∈ R energy in the primary system due to the coupling of the primaryand secondary system. 20

wdss ∈ R energy due to the damping force in the secondary system. 20wkss ∈ R energy due to the gravity force in the secondary system. 20wmss ∈ R energy due to the inertia force in the secondary system. 20wcsp ∈ R energy in the secondary system due to the coupling of the pri-

mary and secondary system. 20xm ∈ R transformed state of the parametrically excited system. 25xm ∈ R2 state vector of Mathieu's equation. 25xm ∈ R state of the parametrically excited system. 24zp ∈ R dimensionless horizontal displacemnt of the primary system. 8,

13, 15, 18zp,1 ∈ R rst order expansion of the vertical displacement of the primary

system. 14, 16zp,2 ∈ R second order expansion of the angular deection of the sec-

ondary system. 15zp,c ∈ R complementary solution of the dimensionless primary system.

11zp,p ∈ R particular solution of the dimensionless primary system. 11zp ∈ R vertical translation of the primary system. 7, 13, 20, 21

37

General and rigid-body dynamics

General and rigid-body dynamics

β ∈ R sideslip angle. 100, 101βw ∈ R encounter angle expressed in the body frame. 54βnw ∈ R encounter angle expressed in the inertial reference frame. 100,

104, 114, 118, 126χ ∈ R course angle. 100∆ψ ∈ R (small) variations in heading angle. 111∆u ∈ R (small) variations in surge speed. 111η ∈ R6 generalized position vector. 50ηr2 ∈ R2 reduced-order generalized position vector; consiting of heave

and pitch. 60ηr3 ∈ R3 reduced-order generalized position vector; consiting of heave,

roll, and pitch. 59ex = [1, 0, 0]

> Unit vector in x-direction. 58ez = [0, 0, 1]

> Unit vector in z-direction. 51, 54G ∈ R3×3 rotational rotation matrix. 50J ∈ R6×6 rotation matrix. 50, 53ν ∈ R3 the generalized velocity vector. 50, 100νr3 ∈ R3 reduced-order generalized velocity vector, consisting of heave,

roll, and pitch velocities. 59O (·) Landau symbol, big Oh order symbol. 9, 13, 16, 17, 86O (·) Landau symbol, little oh order symbol. 9, 13, 15, 18ωb ∈ R3 the angular velocity of the ship observed in a reference frame

attached to the ship. 50, 82, 83ωn ∈ R3 the angular velocity of the ship observed in an (assumed iner-

tial) reference frame attached to the mean ocean surface. 82φc ∈ R roll angle based on the 1-DOF roll model. 62φm ∈ R roll angle based on the Mathieu model. 62, 76φ ∈ R rotation about the x-axis. 48, 76, 83, 88, 99, 104, 114, 125, 133,

148, 151pn ∈ R3 location of the body frame origin relative to the inertial frame.

48, 50, 99Ψ ∈ R local pressure eld. 51, 54ψ ∈ R rotation about the z-axis; heading angle. 49, 58, 59, 70, 71, 76,

100, 118, 123ψ0 ∈ R initial (nominal) heading angle. 103, 104, 110, 114, 126ψd ∈ R desired heading angle. 52, 107, 110, 111, 122124R ∈ SO(3) ⊂ R3×3 a rotation matrix rotating vectors from the body-xed to the

inertial frame. 4850, 82, 100S ∈ SS(3) ⊂ R3×3 a rotation matrix rotating vectors from the body-xed to the

inertial frame. 51SS group of all skew-symmetric matrices. 51Θ ∈ R3 the rollpitchyaw Euler angles. 49, 50, 82θ ∈ R rotation about the y-axis. 49

38

Ship

u ∈ R surge speed. 70, 71, 76, 98, 133, 148u0 ∈ R initial (nominal) surge speed. 103, 104, 110, 113, 126ud ∈ R desired surge speed. 52, 65, 107, 110, 111, 119, 120, 124, 136ur ∈ R reference surge speed. 136vb ∈ R3 the linear velocity of the ship observed in a reference frame

attached to the ship. 50, 82vn ∈ R3 The linear velocity of the ship observed in an (assumed inertial)

reference frame attached to the mean ocean surface. 82

Environment

αζ ∈ R phase shift of the ocean wave. 54, 100cw > 0 ∈ R phase velocity of the wave. 100ew ∈ R3 propagation vector of the wave. 100g > 0 ∈ R acceleration of gravity. 7, 51, 54, 99hw > 0 ∈ R water depth. 100kw ∈ R3 the wave vector as seen by an observer in the inertial frame

attached to the mean ocean surface. 100kw > 0 ∈ R the wave number as seen by an observer in the inertial frame

attached to the mean ocean surface. 54, 61, 82, 83, 100λ > 0 ∈ R wave length. 27, 83, 100ω0 > 0 ∈ R the frequency of the waves as seen by an observer in the inertial

frame attached to the mean ocean surface. 54, 58, 82, 83, 100,118, 133, 134

ρ > 0 ∈ R density of sea water. 54, 98, 117σw > 0 ∈ R constant describing the wave intensity in second-order wave

model. 136tw > 0 ∈ R period of the wave. 100ζ0 ∈ R wave amplitude. 54, 99, 100ζ ∈ R wave height. 27, 54, 58, 60, 99ζw > 0 ∈ R damping coecient in the second-order wave model. 136

Ship

a ∈ R parameter. 51αφ ∈ R phase of the time-varying change in the linear restoring coe-

cient in roll. 27, 81b ∈ R parameter. 51C > 0 ∈ R6×6 coriolis/centripetal matrix of the ship. 51Cr3 > 0 ∈ R3×3 reduced-order coriolis/centripetal matrix of the ship. 60D > 0 ∈ R6×6 damping matrix of the ship. 5053, 60, 62d44 > 0 ∈ R linear damping coecient in roll. 27, 69, 81, 83δr ∈ R rudder angle. 99, 107, 112δr,d ∈ R desired rudder deection. 118, 120Dp > 0 ∈ R propeller diameter. 117, 120

39

Ship

Dr2 > 0 ∈ R2×2 reduced-order damping matrix of the ship. 60Dr3 > 0 ∈ R3×3 reduced-order damping matrix of the ship. 60fni ∈ R Pressure-induced force in the inertial frame on the ith panel.

56, 57GMa ∈ R amplitude of the metacentric height change in waves. 99GMm ∈ R mean metacentric height. 99GMT ∈ R transverse metacentric height. 99k44 > 0 ∈ R linear restoring oecient in roll. 27, 81, 83κ1 ∈ R linear restoring coecient of the 1-DOF roll model. 6971κ2 ∈ R coecient of the change in the amplitude of the linear restoring

moment of the 1-DOF roll model. 70, 73, 74, 76κ3 ∈ R phase of the change of the linear restoring moment of the 1-

DOF roll model. 70, 73, 74, 76κ4 ∈ R cubic restoring coecient of the 1-DOF roll model. 6971κ5 ∈ R amplitude of the external forcing of the 1-DOF roll model. 69,

70, 73κ6 ∈ R phase of the external forcing of the 1-DOF roll model. 70, 71,

73κ ∈ R2×1 hydrostatic coecient vector of the 1-DOF roll model. 70κ ∈ R4×1 hydrodynamic coecient vector of the 1-DOF roll model. 70,

71, 73κd ∈ R derivative heading controller gain. 53κi ∈ R integral heading controller gain. 53κp ∈ R proportional heading controller gain. 53kf ∈ R form factor for viscous correction. 98kg > 0 ∈ R6 gravity-induced forces of the ship. 51ki ∈ R integral speed controller gain. 53kn > 0 ∈ R Nomoto gain. 99kp < 0 ∈ R linear (potential and viscous) damping coecient in roll. 99kp ∈ R proportional speed controller gain. 53kφ > 0 ∈ R restoring torque for the 1-DOF roll model. 70kφ3 > 0 ∈ R cubic restoring coecient in roll. 83, 99kφ ∈ R restoring torque without wave inuence of the 1-DOF roll

model. 70kφ,c > 0 ∈ R restoring torque for the 1-DOF roll model. 62kφ,m > 0 ∈ R restoring torque for the Mathieu model. 62kφt > 0 ∈ R amplitude of the time-varying change in the linear restoring

coecient in roll. 27, 81, 83k|p|p < 0 ∈ R nonlinear damping coecient in roll. 99kp ∈ R6 pressure-induced generalized forces of the ship. 51, 53, 57k ∈ R6 generalized restoring forces/moments of the ship. 51Kr2 ∈ R2×2 reduced-order generalized restoring forces /moments of the

ship, linear part. 60kr3 ∈ R3 reduced-order generalized restoring forces/moments of the ship.

60Lpp > 0 ∈ R Length between perpendiculars. 98

40

Frequency detuning

M > 0 ∈ R6×6 total inertia of the ship. 50, 51, 60m > 0 ∈ R ship's mass. 51, 53m44 > 0 ∈ R total moment of inertia in roll. 27, 69, 81, 83MA > 0 ∈ R6×6 added mass of the ship. 50, 52, 53, 62mni ∈ R Pressure-induced torque in the inertial frame on the ith panel.

57M r2 > 0 ∈ R2×2 reduced-order total inertia of the ship. 60M r3 > 0 ∈ R3×3 reduced-order total inertia of the ship. 60MRB > 0 ∈ R6×6 rigid-body inertia of the ship. 50, 52, 53, 62∇ > 0 ∈ R displaced water volume. 99np > 0 ∈ R propeller revolution per second (rps). 117np,d > 0 ∈ R desired propeller revolution. 117, 118νk > 0 ∈ R kinematic viscosity of the uid. 98ωe ∈ R time-varying encounter frequency. 2729, 5863, 65, 67, 8184,

88, 99101, 103, 107, 113, 133, 143, 148, 149, 154ωe,0 ∈ R initial (nominal) encounter frequency. 103, 110ωe,d ∈ R desired encounter frequency. 106, 110, 111, 124ω∗e,d ∈ R optimal encounter frequency. 106, 110ωφ > 0 ∈ R natural roll frequency. 61, 83, 98, 149rbg ∈ R3 position of the ship's center of gravity in the body frame. 51,

53Rn > 0 ∈ R Reynolds number. 98sc ∈ R1×4 parameter vector for the restoring torque of the 1-DOF model.

62sm ∈ R1×4 parameter vector for the restoring torque of the Mathieu model.

62Sw ≥ 0 ∈ R Wetted (submerged) hull surface. 98Sw,i ≥ 0 ∈ R Wetted (submerged) part of the panel i of the vessel. 51τ c ∈ R6 generalized forces generated by the actuators. 50, 52τ c,r3 ∈ R3 reduced-order generalized forces generated by the actuators. 60τ e ∈ R6 generalized environmental forces. 50, 52τ e,r3 ∈ R3 reduced-order generalized environmental forces. 60tδ > 0 ∈ R time constant of the rudder dynamics. 118, 123tn ∈ R Nomoto time constant. 99Tp > 0 ∈ R propeller thrust. 117tp > 0 ∈ R thrust deduction number. 117, 120Tp,n > 0 ∈ R time constant propeller. 117, 120va > 0 ∈ R advance speed at the propeller. 117wf > 0 ∈ R wake fraction number. 117, 120xu < 0 ∈ R linear (potential and viscous) damping in surge. 98x|u|u < 0 ∈ R nonlinear (quadratic) damping in surge. 98

Frequency detuning

Am,hd < 0 ∈ R2×2 desired closed-loop dynamics for the yaw system in the L1 adap-tive heading controller design. 120

41

Frequency detuning

am,sp < 0 ∈ R desired closed-loop parameter for the surge system in the L1

adaptive speed controller design. 119, 120asp ∈ R known constant for the surge system in the L1 adaptive speed

controller design, asp , xu/ (m+ a11 (0)). 118Be ∈ R2×1 control eectiveness matrix. 111Bhd ∈ R2×1 known constant vector for the yaw system in the L1 adaptive

heading controller design. 120bsp ∈ R known constant for the surge system in the L1 adaptive speed

controller design, bsp , 1/ (m+ a11 (0)). 118Chd ∈ R2×1 known constant vector for the yaw system in the L1 adaptive

heading controller design. 120Chd,lp low-pass lter in the design of the L1 adaptive heading con-

troller, Chd,lp , khdDhd (s) / (1 + khdDhd (s)). 122Csp low-pass lter in the design of the L1 adaptive speed controller,

Csp , kspDsp (s) / (1 + kspDsp (s)). 119Dhd strictly proper transfer function in the design of the L1 adaptive

heading controller, Dhd , (s+ ω0,hd) /(s2 + ωn,hds

). 122

Dsp strictly proper transfer function in the design of the L1 adaptivespeed controller, Dsp , (s+ ω0,sp) /

(s2 + ωn,sps

). 119

Γhd > 0 ∈ R adaptation gain for the L1 adaptive heading controller. 122,123

Γsp > 0 ∈ R adaptation gain for the L1 adaptive speed controller. 119, 120J > 0 ∈ R objective function. 103, 106, 110J1 > 0 ∈ R objective function. 103, 110J2 > 0 ∈ R objective function. 103, 110kg,hd ∈ R feed forward gain of the L1 adaptive heading controller, kg,hd ,

−1/(C>hdA

−1m,hdBhd

). 122

kg,sp ∈ R feed forward gain of the L1 adaptive speed controller, kg,sp ,−1/

(a−1m,spbsp

). 119

khd ∈ R controller gain of the L1 adaptive heading controller. 122, 123kψ,d > 0 ∈ R derivative gain heading controller. 112ksp ∈ R controller gain of the L1 adaptive speed controller. 120kψ,p > 0 ∈ R proportional gain heading controller. 112ku,p > 0 ∈ R proportional gain speed controller. 112ωh > 0 ∈ R cuto frequency of the high-pass lter in the extremum seeking

loop. 106ωhd ∈ R unknown control eectiveness for the yaw system in the L1

adaptive heading controller design. 120ωl > 0 ∈ R cuto frequency of the low-pass lter in the extremum seeking

loop. 107Sf set of feasible solutions for the control allocation problem. 111σhd ∈ R unknown disturbance for the yaw system in the L1 adaptive

heading controller design. 120σsp ∈ R unknown disturbance for the surge system in the L1 adaptive

speed controller design, σsp , τ1e . 118

τωe ∈ R control input, ωe , τωe . 83, 84

42

Frequency estimation

τv,ωe ∈ R virtual control input for the encounter frequency. 111τv,u ∈ R virtual control input for the surge dynamics. 112θhd ∈ R2×1 unknown constant parameter vector for the yaw system in the

L1 adaptive heading controller design. 120θsp ∈ R unknown parameter for the surge system in the L1 adaptive

speed controller design, θsp , x|u|u|u|. 118ζv ∈ R2×1 variation vector. 111

Frequency estimation

F k−1 ∈ R5×5 discrete system matrix. 135Gk−1 ∈ R5×5 discrete system matrix. 135Hk ∈ R5×3 discrete system matrix. 135Kk ∈ R5×5 Kalman gain of the extended Kalman lter. 136ko > 0 ∈ R observer gain in the update law for the frequency estimator of

an unknown sinusoidal signal. 145147, 149, 151, 154ωu,l > 0 ∈ R cuto frequency of the low-pass lter in the second-order speed

reference model. 136P+c,k ∈ R5×5 a posteriori state error covariance matrix of the extended

Kalman lter. 136P−c,k ∈ R5×5 a priori state error covariance matrix of the extended Kalman

lter. 135, 136ϕy ∈ R phase angle the unknown sinusoidal signal. 144Qc ∈ R5×5 Process noise error covariance matrix of the extended Kalman

lter. 135Rc ∈ R3×3 measurement noise error covariance matrix of the extended

Kalman lter. 135σ > 0 ∈ R amplitude of the unknown sinusoidal signal. 144146σr > 0 ∈ R reference value fo the amplitude of the unknown sinusoidal sig-

nal. 146, 151, 154θs ∈ R unknown parameter of the sinusoidal signal, θs , −ω2

e . 144,145, 154

ts > 0 ∈ R sampling time. 135, 149V k ∈ R5 Gaussian white noise process with zero mean and covariance

matrix Rc. 135W k−1 ∈ R5 Gaussian white noise process with zero mean and covariance

matrix Qc. 135x+k ∈ R5×1 a posteriori state estimate of the extended Kalman lter. 136x−k ∈ R5×1 a priori state estimate of the extended Kalman lter. 135ζf ∈ R auxiliary lter state for the estimation of an unknown sinusoidal

signal. 144147ζu > 0 ∈ R damping coecient in the second-order speed reference model.

136zm ∈ R measurement of an unknown sinusoidal signal. 145

43

Part II

Ship Modelling

45

Chapter 3

A Six-Degree-Of-Freedom Ship

Model

In this chapter, an accurate numerical 6-degree-of-freedom (DOF) model of a con-tainer ship in sinusoidal waves is derived. This is motivated by the need to derivea 1-DOF roll model for parametric roll resonance which overcomes the apparentlimitation of the prevalent Mathieu-like model with constant excitation to capturethe underlying physics of parametrically excited roll motion when the encounterfrequency is non-constant. With the possibility of detuning the excitation frequencyand the natural roll frequency in order to control parametric resonance, the needfor a simple analytic 1-DOF roll model, suitable for control synthesis purposes, hasemerged. To that matter, a more complete and complex 6-DOF is derived in thischapter, taking into account the external forces and moments induced on the shipby the hydrostatic and hydrodynamic pressure eld of the surrounding ocean. Therestoring force is computed by integration of the pressure over the instantaneoussubmerged hull. First-order wave-induced eects are included.

It will be shown that the derived 6-DOF model is adequate to handle complexsea states and non-steady ship motions. However, since the 6-DOF ship modelpresented in this chapter is highly accurate and purely numerical by nature, it willbe reduced to a lower-order analytic model in a subsequent chapter. In recent years,several ship models of dierent complexity for parametric roll have been developed[26, 44, 62, 109, 110, 132, 153].

The content of this chapter was published in Breu, Holden, and Fossen [16].

3.1 Equations of Motion

This section introduces the reference frames and presents the equations of motionof the 6-DOF model of the ship.

3.1.1 Reference Frames

Two reference frames are considered; a geographical reference frame xed to theocean surface, and a reference frame xed to the vessel (body frame). Figure 3.1

47

3. A Six-Degree-Of-Freedom Ship Model

depicts the two reference frames in the horizontal plane, that is the z-axis is notshown in Figure 3.1. The following assumption is made:

Assumption 1. The geographical reference frame xed to the ocean surface isinertial.

The ocean surface reference frame is dened by the North-East-Down coordi-nate system in Fossen [41], with the axes accordingly. It is referred to as the inertialframe or the n-frame. The reference frame xed to the vessel is moving with thevessel and it has its origin at a location ob midships along the ship's transversalmidline, so that a portstarboard symmetric ship is mirrored about the body-framexz-plane. The body axes xb, yb and zb coincide with the principal axes of inertia,as dened by Fossen [41]. It is referred to as the body frame or b-frame.

ob

xb

u

vv

β

yb

βw

λ

kw

yn

xn

pn

ψ

χβnw

Figure 3.1: The inertial reference frame and the body frame as seen from the top.

Vectors expressed in the inertial reference frame are written in boldface with asuperscript n (e.g., xn), whereas vectors expressed in the body frame are denotedin boldface with a superscript b (e.g., xb).

The location of the body frame origin relative to the inertial frame, expressedin the inertial frame, is given as pn ∈ R3, as indicated in Figure 3.1. Figure 3.2shows the 6-DOF ship motions with respect to the body frame.

The inertial frame and the body frame coordinate systems can be rotated rela-tive to each other. The rotation matrix R ∈ SO(3) ⊂ R3×3 is associated with thisrotation so that pn = Rpb. R satises R> = R−1, det (R) = 1 [34, 41]. The rota-tion matrix R belongs to the special orthogonal group of order 3, SO (3) ⊂ R3×3,see Egeland and Gravdahl [34] and Fossen [41].

The rotation matrix can be fully parametrized with (no less than) three param-eters. In this work, the so-called rollpitchyaw Euler angles are used. These anglesrepresent simple rotations about the three dierent body axes; roll (φ) about the

48

3.1. Equations of Motion

xb

u (surge)p (roll)

ybv (sway)

q (pitch)

zb

w (heave)

r (yaw)

Figure 3.2: The six-degree-of-freedom ship motions. The body frame coordinatesystem is xed to the origin of the ship. [41]

x-axis, pitch (θ) about the y-axis and yaw (ψ) about the z-axis. See Egeland andGravdahl [34] and Fossen [41] for details.

Combining these three angles into the vector Θ = [φ, θ, ψ]>, we can write R as

R (Θ) =

cθcψ sφsθcψ − cφsψ cφsθcψ + sφsψcθsψ sφsθsψ + cφcψ cφsθsψ − sφcψ−sθ sφcθ cφcθ

(3.1)

where c· = cos (·) and s· = sin (·).

3.1.2 Hydrodynamic Forces and Moments

To derive the hydrodynamic forces and moments on a ship, certain assumptionsare made:

Assumption 2. There is no current.

Assumption 3. The hull can be split into triangular or quadrangular panels, andeach panel can be parametrized as a two-dimensional surface embedded in R3.

Assumption 4. For a maneuvering ship in a seaway, the frequency-dependentparameters of the damping, added mass and Coriolis/centripetal matrices can beapproximated at a constant excitation frequency (see for example Fossen [41]).

Assumption 5. The ocean is innitely deep.

49


Assumption 6. The pressure eld in the ocean is unchanged by the passage ofthe ship (in eect, waves are traveling through the ship's hull.)

The ship's generalized position (position and attitude) vector η ∈ R6 can bedened as

η ,[pn>, Θ>

]> ∈ R6 (3.2)

wherepn ,

[xn, yn, zn

]> ∈ R3 (3.3)

is the ship's position in the inertial frame and

Θ ,[φ, θ, ψ

]> ∈ R3 (3.4)

the vector of Euler angles representing the ship's rotation relative to the inertialreference frame.

The ship's generalized velocity vector ν ∈ R3 can be dened as

ν ,[vb>, ωb>

]> ∈ R6 (3.5)

where

vb ,[u, u, w

]>= R>vn , R>xn = R>

[un, vn, wn

]> ∈ R3 (3.6)

is the ship's linear velocity in the body frame and

ωb ,[p, q, r

]>(3.7)

is the ship's angular velocity relative to the inertial frame, expressed in the bodyframe.

The generalized forces generated by the actuators are τ c ∈ R6 and τ e ∈ R6 arethe environmental disturbances and unmodeled generalized forces.

From Fossen [41], the 6-DOF ship model is

η = J (Θ)ν (3.8)

Mν +D (ν)ν +C (ν)ν + k (η, t) = τ c + τ e. (3.9)

where

J (Θ) =

[R (Θ) 03×3

03×3 G (Θ)

]∈ R6×6 (3.10)

with R as in (3.1) and

G (Θ) =

1 sin (φ) tan (θ) cos (φ) tan (θ)0 cos (φ) − sin (φ)

0 sin(φ)cos(θ)

cos(φ)cos(θ)

, cos (θ) 6= 0 (3.11)

and M , MRB + MA = M> > 0 ∈ R6×6 is the sum of the rigid-body inertiaMRB > 0 ∈ R6×6 and added mass MA > 0 ∈ R6×6, D (ν) ∈ R6×6 the damping

50

3.1. Equations of Motion

matrix, and C (ν) ∈ R6×6 the Coriolis/centripetal matrix; k = kp − kg wherekp is the pressure-induced generalized forces and kg the gravity-induced forces. Dsatises y>D (ν)y ≥ 0 ∀ y,ν ∈ R6.

If

M ,

[M11 M12

M21 M22

]∈ R6×6, M11,M12,M21,M22 ∈ R3×3

and

S (x) ,

0 −x3 x2

x3 0 −x1

−x2 x1 0

∈ SS (3) ⊂ R3×3 ∀ x = [x1, x2, x3]> ∈ R3

then

C (ν) =

[03×3 −S

(M11v

b +M12ωb)

−S(M11v

b +M12ωb)−S

(M21v

b +M22ωb)]

= −C> (ν) ∈ R6×6.

Here, SS (3) is the group of all skew-symmetric matrices of order 3, see Egelandand Gravdahl [34], and Fossen [41].

The gravity-induced forces kg are given by [41]

kg (η) = mg

[R> (Θ) ez(

R> (Θ) ez

)× rbg

](3.12)

where rbg ∈ R3 is the ship's center of gravity in the the body frame, ez = [0, 0, 1]>,m > 0 ∈ R the ship's mass, and g > 0 ∈ R the acceleration of gravity.

By knowing the pressure eld of the surrounding ocean, kp ∈ R6 can be found.At any given point rn in the ocean (in the inertial frame), there will be a localpressure eld Ψ ≈ Ψ (rn, t) ∈ R [122, 155].1 It is assumed (Assumption 3) thateach section of the ship's hull can be parametrized with parameters a ∈ R andb ∈ R, so that the vector rbi (a, b) gives the position of a point on the surface ofpanel i, in the body frame. Dening

Ψi (a, b) , Ψ(Rrbi (a, b) + pn, t

), (3.13)

the generalized pressure-induced force on the ship can be taken as [33, 122, 155]

kp (η, t) =∑

i

∫Sw,i

Ψi (a, b)∂rbi∂a (a, b)× ∂rbi

∂b (a, b) da db∫Sw,i

Ψi (a, b) rbi (a, b)×(∂rbi∂a (a, b)× ∂rbi

∂b (a, b))

da db

(3.14)

where Sw,i ≥ 0 ∈ R is the wetted (submerged) part of panel i and the ship isparametrized so that the normal vector

(∂rbi/∂a

)×(∂rbi/∂b

)2 points out of the

hull. The eects of current and waves can all be accounted for in the force kp [37,122]; however, the eect of current is neglected by Assumption 2.

1Technically, the pressure eld would also be a function of the ship's state, except for As-sumption 6.

2See Edwards and Penney [33] for proof that this is a normal vector.

51


3.2 Computer Implementation of the 6-DOF Model

To implement a computer version of the 6-DOF model (3.8)(3.9), the data from aspecic, 281 m long container ship is used. This is the same ship as used in Galeazziet al. [48], Holden et al. [62], and Holden et al. [61] and its main characteristics aresummarized in Table 3.1 [63].

Table 3.1: Container ship, characteristics [63].

Quantity Value

Length between perpendiculars 281 mBeam amidships 32.3 mDraught amidships 11.8 mDisplacement 76468 m3

Roll radius of gyration 12.23 mTransverse metacentric height 1.84 m

Whereas the model can be used for any sea state in any condition, an imple-mentation suitable for parametric roll in chosen in this work. As such, it is assumedthat the waves are planar and sinusoidal.

3.2.1 Inertia and Damping Forces

The parameters for inertia MRB , added mass MA, and damping D > 0 ∈ R6×6

are computed in ShipX (VERES) [39]. The unmodeled force vector τ e is set tozero and the control force τ c is

τ c = −

kp(u− ubd

)+ ki

∫ tt0

(u (T )− ud (T )) dT

0000

κp (ψ − ψd) + κd

(ψ − ψd

)+ κi

∫ tt0

(ψ (T )− ψd (T )) dT

(3.15)

with ud ∈ R the desired surge speed and ψd ∈ R the desired heading. The rudimen-tary proportional integral derivative (PID) controllers in surge and yaw are thereto keep the ship on course in the presence of the other forces. Without these con-trollers, the simulated ship tends to drift quite heavily o course. The parameters

52

3.2. Computer Implementation of the 6-DOF Model

used are

MRB =

[mI3 −mS

(rbg)

mS(rbg)

Jb

], m > 0 = 7.7358e7

rbg = [−3.7486, 0,−1.120]>

Jb =

1.41e10 0 00 3.70e11 00 0 3.70e11

MA =

0 0 0 0 0 00 7.59e7 0 6.43e7 0 −1.04e90 0 7.80e7 0 −7.8e8 00 6.43e7 0 2.20e9 0 −9.08e90 0 −7.83e8 0 3.39e11 00 −1.04e9 0 −9.08e9 0 4.48e11

D (ν) =

5.66e3 0 0 0 0 00 3.31e7 0 1.50e7 0 1.22e80 0 4.66e7 0 −1.05e9 00 1.50e7 0 2.48e8 0 −2.27e90 0 −4.03e8 0 2.73e11 00 −5.03e8 0 −2.79e9 0 1.35e11

+

[2.83e4|v1| 01×5

05×1 05×5

]

kp = 7.7358e7 , ki = 7.7358e6 , κp = 8.18e11 , κd = 0 , κi = 0

with all values in base SI units (kgms). With these numbers, all forces andmoments of the 6-DOF model (3.8)(3.9) except for kp can be computed.

3.2.2 Pressure-Induced Forces

The nal force that needs to be computed is the pressure-induced spring termkp (η, t). This is computed directly from (3.14) with some further simplications:

Assumption 7. The waves are a simple, planar, standing sinusoid.

Assumption 8. The hydrostatic part of the pressure extends from the instan-taneous ocean surface and down.

Assumption 9. The dynamic part of the pressure extends from the averageocean surface and down.

By Faltinsen and Timokha [36], these approximations and the ones used inderiving the 6-DOF model (3.8) and (3.9) give a rst-order approximation of thewave eects.

53


Pressure Field

From Faltinsen and Timokha [36], it follows that the ocean surface under theseconditions is given by

ζ (t, rn) = ζ0 cos (ω0t− kwrn1 cos (βw)− kwrn2 sin (βw) + αζ) (3.16)

in the inertial frame, with ζ ∈ R giving the wave elevation at a position rn =[rn1 , r

n2 , r

n3 ]>. The instantaneous ocean surface is then at [rn1 , r

n2 , ζ (t, rn)]

>. Theconstant parameters are the wave amplitude ζ0 ∈ R, the frequency of the waves asseen by an observer stationary in the inertial frame ω0 > 0 ∈ R, the wave numberkw > 0 ∈ R, and a phase angle αζ ∈ R. The ocean surface given by (3.16) is forwaves propagating in an arbitrary direction βw ∈ R relative to the inertial xn-axis,as depicted in Figure 3.1. For waves travelling along the xn-axis, (3.16) can besimplied to

ζ (t, rn) = ζ0 cos (ω0t± kwrn1 + αζ) (3.17)

where the plus sign corresponds to waves along the negative xn-axis and the mi-nus sign corresponds to waves along the positive xn-axis. For the derivation of thepressure-induced forces, waves travelling along the negative xn-axis will be con-sidered. The expressions for waves propagating along the positive xn-axis followhowever by a sign change.

By Faltinsen and Timokha [36], the pressure eld Ψ for waves travelling alongthe negative xn-axis is given by

Ψ (rn, t) = gρζ0ekw max(rn3 ,0) cos (ω0t+ kwr

n1 + αζ) + gρrn3 . (3.18)

The parameters are the acceleration of gravity g and the density of sea water ρ >0 ∈ R. The term gρrn3 is the hydrostatic pressure, and gρζ0e−kw max(rn3 ,0) cos(ω0t−kwr

n1 + αζ) the dynamic pressure.

The Submerged Part of the Ship

The ship's hull, as previously mentioned, is split into panels, each forming a triangleor quadrangle. In the body frame, panel i has corners pbi,j , j ∈ 1, 2, 3, 4 forquadrangles and j ∈ 1, 2, 3 for triangles. In the inertial frame,

pni,j = Rpbi,j + pn . (3.19)

As (3.17) gives the explicit wave surface (in the inertial frame) one can computewhich points, at any given time and physical location, are above or below the wavesurface by solving the equation

e>z pni,j = ζ

(t,pni,j

)

where ez = [0, 0, 1]> is the unit vector in the z-direction. Rather than solving the

equation explicitly, an approximation is used. First, the points are transformedusing

pni,j = pni,j − [0, 0, ζ(t,pni,j

)]> . (3.20)

54

3.2. Computer Implementation of the 6-DOF Model

Any point with a positive z-value is submerged, see Figure 3.4(a)3.4(b).Each panel is individually parametrized with a bilinear interpolation, so that,

for each panel i

pni = k0 + kaa+ kbb+ kabab (3.21)

with a, b ∈ [0, 1) denes all points on panel i.For each partially submerged panel (one with at least one point underwater

and at least one point above water), the parametrization is used to nd where theedges of the panel intersect the water line, and to compute the coordinates of thesepoints. The submerged points and the points in the waterline then make up (one ormore) new panel(s). Note that the panels whose submerged part forms a pentagonare split into three triangular panels, whereas panels whose submerged part formsa triangle or a quadrangle are kept as such, see Figure 3.3.

Water line

Figure 3.3: Cutting partially submerged panels into new fully submerged panels.

The partially submerged panels and the panels wholly above the waterline arediscarded. The fully submerged panels based on the partially submerged panels, andthe original fully submerged panels, are kept. The entire transformationcuttinginverse transformation process can be seen in Figure 3.4.

The approximation to nd the true intersection of the hull and the ocean surfaceis good if the average size of the panels is small relative to the wave length.

Generalized Forces

Before computing the forces and moments, the transformation (3.20) needs to bereversed such that the all points of the panels are expressed in the inertial frame.The panels are

pni,j = pni,j + [0, 0, ζ(t,pni,j

)]> , (3.22)

and these panels are parameterized bilinearly so that

pni = k0 + kaa+ kbb+ kabab (3.23)

55


Water line

Avg. water line

(a) Initial state.

Water line

Avg. water line

(b) Post-transformation.

Water line

Avg. water line

(c) Post-cutting.

Water line

Avg. water line

(d) Reverse transformation.

Figure 3.4: Transforming panels.

with a, b ∈ [0, 1) dene all points on panel i, in the inertial frame. The partialderivatives of pni with respect to a and b can be explicitly found as

∂pni∂a

= ka + kabb ,∂pni∂b

= kb + kaba . (3.24)

It is worth noting that for triangular panels, kab = 0.For each panel, the pressure-induced force in the inertial frame can be computed

as

fni =

∫ 1

0

∫ 1

0Ψ (pni (a, b) , t) [ka + kabb]× [kb + kaba] da db quadrangles∫ 1

0

∫ 1−b0

Ψ (pni (a, b) , t)ka × kb da db triangles

The integration must be done numerically. Although more points could be used,for increased computational speed, only corner points are used in the calculation

56

3.3. Encounter Frequency

of the pressure forces. Thus,

fni ≈

14

∑4j=1 Ψ

(pni(aj , bj

), t) [ka + kabbj

]× [kb + kabaj ] quadrangles

16

∑3j=1 Ψ

(pni(aj , bj

), t)ka × kb triangles

where

a1 = 0, b1 = 0,

a2 = 0, b2 = 1,

a3 = 1, b3 = 0,

a4 = 1, b4 = 1.

The computation for the torque is similar. However, the torque relative to thebody origin rather than the inertial origin is needed. Therefore:

mni =

∫ 1

0

∫ 1

0Ψ (pni (a, b) , t) [pni (a, b)− pn]

quadrs.× ([ka + kabb]× [kb + kabb]) da db∫ 1

0

∫ 1−b0

Ψ (pni (a, b) , t) [pni (a, b)− pn]× (ka × kb) da db triangles

This is approximated as

mni ≈

14

∑4j=1 Ψ

(pni(aj , bj

), t) [pni(aj , bj

)− pn

]quadrs.

×([ka + kabbj

]× [kb + kabaj ]

)16

∑3j=1 Ψ

(pni(aj , bj

), t) [pni(aj , bj

)− pn

]× (ka × kb) triangles

Note that this torque is still in the inertial frame, but relative to the body origin.If the set S consists of all i such that panel i is one of the original, fully

submerged panels or one of the newly created (also fully submerged) panels,

kp (η, t) ≈[R>

∑i∈S f

ni

R>∑i∈Sm

ni

](3.25)

can then be taken to get the total pressure-induced force and moment in the bodyframe.

The system is simulated with a xed time step, and for each time instant, theoutlined procedure for computing kp is performed.

Note that the procedure automatically handles such eects as (rst-order) wave-induced forces and Doppler shift of these. The rst by simple virtue of the pressureeld including the dynamic pressure and the latter by including pn in (3.19).

3.3 Encounter Frequency

To an observer at a xed location on the ocean surface, that is in the inertialreference frame, waves will appear to have a specic frequency of oscillation (or arange of frequencies if the waves are irregular). Ocean waves can be seen as planarwaves [36], and for regular sinusoidal waves, these can be described by (3.17). Toa stationary observer, rn is constant.

57


This is however not true when the observer is moving with the ship at a nonzerospeed. To a moving observer, the waves will appear to behave dierently than tothe stationary observer due to the Doppler eect. The location of the observer inthe inertial frame is taken as rn = pn (i.e., the body origin) and it is assumed,without loss of generality, that pn (t0) = 0. The observer's velocity in the inertialx-direction is then

xn , un (3.26)

so that

ζ (t,pn) = ζ0 cos

(ω0t+ kw

∫ t

t0

un (τ) dτ + αζ

). (3.27)

Since the velocity is given in the body frame in the 6-DOF model (3.8)(3.9),

un = e>xRvb (3.28)

where ex = [1, 0, 0]>.The encounter frequency, the frequency seen by the observer, can then be de-

ned as

ωe ,d

dt

(ω0t+ kw

∫ t

t0

un (τ) dτ

)(3.29)

= ω0 + kwe>xRv

b ≈ ω0 + kwe>xR[vb1, 0, 0]>

= ω0 + kw cos (θ) cos (ψ)u ≈ ω0 + kw cos (ψ)u (3.30)

and (3.27) can be rewritten as

ζ (t,pn) = ζ0 cos

(∫ t

t0

ωe (τ) dτ + αζ

). (3.31)

Note that if un is a constant, then so is ωe ∈ R, and the above simply becomesζ (t,pn) = ζ0 cos (ωet+ αζ).

It is an important fact that whereas one cannot change ω0, it is possible tochange ωe by changing the velocity un or the heading angle ψ ∈ R.

58

Chapter 4

1-DOF Roll Model for Time-Varying

Speed

Based on the 6-DOF ship model of Chapter 3, a 1-DOF roll model is derived in thischapter. The content of this chapter was published in Breu, Holden, and Fossen[16].

The spring term in the full 6-DOF model is analytically unknown. The followingextra assumption is made to derive a 1-DOF roll model:

Assumption 10. The ship is travelling directly into the waves.

Note that the ship is still allowed to change its forward speed. This assumptionwill be relaxed in Chapter 5 when the ship is allowed to change course as well.This assumption implies that the encounter frequency (3.30) simplies further sinceψ = 0:

ωe ≈ ω0 + kwu . (4.1)

Note that, although (3.30) was derived for waves travelling along the negative xn-axis, holds true for all head sea condition, specically for waves propagating alongthe positive xn-axis and the heading angle ψ = π by a sign change in (3.30).

For ships in parametric resonance, it is well-known that the most importantdegrees of freedom are heave, roll, and pitch [62, 110]. Heave and pitch are alreadycoupled, and during parametric resonance these transfer energy to roll.

Disregarding all other degrees of freedom, dene

ηr3 ,[zn, φ, θ

]>, νr3 ,

[w, p, q

]>(4.2)

and note that

ηr3 =

cos (φ) cos (θ) 0 00 1 sin (φ) tan (θ)0 0 cos (φ)

νr3 ≈ νr3 . (4.3)

This 3-DOF model can be written as

M r3 νr3 +Cr3 (νr3)νr3 +Dr3νr3 + kr3(ηr3 , t

)= τ c,r3 + τ e,r3 . (4.4)

59

4. 1-DOF Roll Model for Time-Varying Speed

For simplicity, it is assumed that the velocities in heave and pitch are low, andthat the only coupling between these two degrees of freedom and roll exists in thespring term kr3 ∈ R3. This allows to write

M r3 =

m33 0 m35

0 m44 0m53 0 m55

, Cr3 (νr3) = 03×3 , Dr3 (νr3) =

d33 0 d35

0 d44 0d53 0 d55

where mij and dij are the i, jth element of M > 0 ∈ R6×6 and D (0) from the6-DOF model (3.8)(3.9).

Furthermore, following Holden et al. [62], kr3 can be simplied to

kr3(ηr3 , t

)≈

k33 0 k35

0 k44 0k53 0 k55

ηr3 +

0kzφz

nφ+ kφθφθ + kφ3φ3

0

+ kr3 (t)

with

kr3 (t) = −

azζ0 cos

(∫ tt0ωe (τ) dτ + αz

)

0

aθζ0 cos(∫ t

t0ωe (τ) dτ + αθ

)

where az, αz, aθ, and αθ are constant. Note that kr3 (t) is merely ζ of (3.27)phase-shifted and scaled, eectively sent through a linear lter.

Neither heave nor roll nor pitch are likely to be directly actuated, so τ c,r3 = 0.The unmodeled disturbances τ e,r3 ∈ R3 are also assumed zero. Furthermore, (4.4)is rewritten as

m44φ+ d44φ+ k44φ+ kφ3φ3 = −[kzφ, kφθ]ηr2φ (4.5)

M r2 ηr2 +Dr2 ηr2 +Kr2ηr2 = −kr2 . (4.6)

with

ηr2 = [zn, θ]> , M r2 =

[m33 m35

m53 m55

], Dr2 =

[d33 d35

d53 d55

]

Kr2 =

[k33 k35

k53 k55

], kr2 (t) = −

azζ0 cos


)

aθζ0 cos(∫ t


) .

Note that the ηr2 -subsystem (4.6) is completely decoupled from the roll-subsystem(4.5) and is merely a linear ordinary dierential equation with constant coecientsand a sinusoidal input. If it is assumed that ωe is constant, then the system willhave as a steady-state solution

ηr2 (t) =

azζ0 cos


)

aθζ0 cos(∫ t


) =

[azζ0 cos (ωe (t− t0) + αz)aθζ0 cos (ωe (t− t0) + αθ)

](4.7)

60

where az, αz, aθ, and αθ are dependent on the (constant) forcing frequency, thatis the encounter frequency ωe.

The main purpose of this work is to derive a roll model for time-varying ωe,thus the case is considered when ωe is not constant. Revisiting the equation for ωe,and note

ωe =d

dt

(ω0 + kwe

>xRv

b)

= kwe>xR

(S(ωb)vb + vb

)(4.8)

gives

|ωe| ≤ kw‖ex‖‖R‖(‖S(ωb)vb‖+ ‖vb‖

)= kw

(‖S(ωb)vb‖+ ‖vb‖

)(4.9)

since ‖ex‖ = ‖R‖ = 1.For large ships, neither the acceleration ‖vb‖ nor the term ‖S

(ωb)vb‖ is likely

to be large. To cause parametric resonance, waves have to be approximately of thesame length as the ship, and since kw is inversely proportional to the wave length,kw is likely to be quite low. Thus |ωe| ≈ 0 and a quasi-steady approach can beused. Therefore the solution to (4.6) is taken to be given by the second term onthe right-hand side of (4.7) even when ωe is non-constant.

Inserting the solution (4.7) into the right-hand side of (4.5) it follows

[kzφ, kφθ]ηr2 = kzφazζ0 cos

(∫ t

t0

ωe (τ) dτ + αz

)+ kφθaθζ0 cos

(∫ t

t0

ωe (τ) dτ + αθ

)

= kφt cos

(∫ t

t0

ωe (τ) dτ + αφ

)

where

k2φt = ζ2

0

[k2zφa

2z + k2

φθa2θ + 2kzφkφθazaθ cos (αθ − αz)

]

αφ = arctan

(kzφaz sin (αz) + kφθaθ sin (αθ)

kzφaz cos (αz) + kφθaθ cos (αθ)

).

1-DOF roll model (time-varying speed) Under the stated assumptions, theroll motion can be described by the 1-DOF parametric roll model

m44φ+ d44φ+

[k44 + kφt cos

(∫ t

t0

ωe (τ) dτ + αφ

)]φ+ kφ3φ3 = 0 . (4.10)

1-DOF roll model (constant encounter frequency) If ωe = 0, then the rollmotion is described by the Mathieu equation

m44φ+ d44φ+ [k44 + kφt cos (ωet+ αφ)]φ+ kφ3φ3 = 0 . (4.11)

For both models, the natural roll frequency ωφ > 0 ∈ R is given by ωφ ,√k44/m44.

61


4.1 Model Verication

To verify the 1-DOF simplied roll model (4.10), it is simulated and comparedto simulations of the full 6-DOF model (3.8)(3.9) presented in Section 3.2. Sincethe Mathieu equation (4.11) is commonly used to describe ships sailing with con-stant surge speed experiencing parametric roll resonance, its ability to describe thedynamics of a ship for a non-constant encounter frequency is additionally investi-gated.

In the simulations, the ship described in Galeazzi et al. [48], Holden et al. [61]is used with the same parameters for inertiaMRB , added massMA, and dampingD as in Section 3.2.1. The kinematics are implemented using quaternions, seeEgeland and Gravdahl [34], Fossen [41], instead of the Euler angle representationin the simulations. Those two representations can be used interchangeably and thechoice of representation is to a certain degree arbitrary [41].

As the main dierence between the models is in the spring term, the springterms are compared. Furthermore, the parameters of these are not known a priori,and need to be identied. Denoting the roll angle computed based on the 1-DOFmodel (4.10) and the Mathieu model (4.11) as φc ∈ R and φm ∈ R, respectively,The spring torque for the 1-DOF roll model (4.10) and the Mathieu model (4.11)are dened as

kφ,c (t; s) =

[k44 + kφt cos

(∫ t

t0

ωe (τ) dτ + αφ

)]φc (t) + kφ3φ3

c (t) (4.12)

kφ,m (t; s) = [k44 + kφt cos (ωet+ αφ)]φm (t) + kφ3φ3m (t) . (4.13)

To determine the parameters s = [k44, kφt, αφ, kφ3 ] in (4.12) and (4.13), nonlin-ear least-squares curve tting is used:

sc = arg mins

∑

t

|k4 (η (t) , t)− kφ,c (t; s)|2 (4.14)

sm = arg mins

∑

t

|k4 (η (t) , t)− kφ,m (t; s)|2 (4.15)

where k4 is the fourth element of k in the full 6-DOF model (3.8) and (3.9).The instantaneous encounter frequency in the simplied roll model (4.10) is

calculated from the simulation of the full 6-DOF model by (3.29). However, evenwhen attempting to keep constant speed, the waves cause the ship's speed to os-cillate. This is reected in the 6-DOF model. Using the instantaneous values ofωe, the Mathieu model (4.11) will not oscillate. Thus, a low-pass ltered encounterfrequency is used when simulating the Mathieu model.

4.1.1 Constant Forward Speed

To compare the models when ωe is kept approximately constant, the three modelsare simulated with constant speed (barring small variations due to wave-inducedforces in surge).

In the following, the signals of the 6-DOF model is represented without sub-script, while the subscripts c and m denote the simplied roll equation and the

62

4.1. Model Verication

Mathieu equation, respectively. The simulation parameters and the model param-eters are summarized in Table 4.1.

Table 4.1: Simulation parameters, constant speed.

Quantity Symbol Value

Mean forward speed u 7.90 m/sMean encounter frequency ωe 0.645 rad/sWave amplitude ζ0 2.5 mWave length λ 281 mWave number kw 0.0224 rad/mNatural roll frequency ωφ 0.343 rad/sModal wave frequency ω0 0.4684 rad/s

k44 1.7646e9 kg m2/s2

Model parameters: kφt 7.3224e8 kg m2/s2

Simplied roll equation αφ 0.2295 radkφ3 2.2741e9 kg m2/s2

k44 1.7685e9 kg m2/s2


Mathieu equation αφ 0.2118 radkφ3 2.2691e9 kg m2/s2

Figure 4.1 shows the simulation results for all three models. From Figure 4.1(a)it is evident that the ship is experiencing parametric roll resonance in this scenario.

Figure 4.1(c) compares the spring torque divided by the roll angle of the the full6-DOF model to the ones of the simplied roll equation and the Mathieu equationcomputed by (4.12), i.e., k4/φ versus

kφ,cφc

= k44 + kφt cos

(∫ t

t0

ωe (τ) dτ + αφ

)+ kφ3φ2

c

kφ,mφm

= k44 + kφt cos (ωet+ αφ) + kφ3φ2m .

Once steady-state is reached, there is good agreement between the 6-DOF model(3.8)(3.9) and the two 1-DOF models (4.10) and (4.11).

In this scenario, the 1-DOF model (4.10) and the Mathieu model (4.11) behavealmost identically. This is as expected, since with ωe = 0 the two models areidentical. The slight variations in ωe in this scenario are not enough to cause anysignicant discrepancy.

Up until about 220 s, there is signicant discrepancy between the 6-DOF model(3.8)(3.9) and the two 1-DOF models (4.10) and (4.11). The two 1-DOF models(4.10) and (4.11) go to the maximum roll angle much faster than the 6-DOF model(3.8)(3.9). This is because the two 1-DOF models (4.10) and (4.11) are derivedunder the assumption that heave and pitch are in steady-state. For the rst 200 sor so, that is not the case. Once steady-state heave and pitch are achieved, the6-DOF model (3.8)(3.9) quickly catches up to the two models (4.10) and (4.11).

63


time (s)

rollangle

()

φφc

φm

0 100 200 300 400 500 600 700

-20

0

20

(a) Roll angle.

time (s)

ωe/ωφ(−

)

ωe/ωφ

Filtered ωe/ωφ

0 100 200 300 400 500 600 700

1.87

1.88

1.89

(b) Frequency ratio.

time (s)

torque/roll( kgm

2/s2)

k4/φkφ,c/φc

kφ,m/φm

0 100 200 300 400 500 600 7000

1

2

3 ×109

(c) Spring torque divided by roll angle.

Figure 4.1: Model comparison. φ, φc and φm are roll angles from the 6-DOF andthe 1-DOF models, and the Mathieu equation.

64


4.1.2 Maximum Roll Angle

To compare the models under a wide range of scenarios, the three models are simu-lated for dierent (almost constant) forward speeds and dierent wave amplitudesand compute the maximum roll angle as a function of the encounter frequency andthe wave amplitude. The simulation scenarios and parameters are identical for allthree cases. The spring torque constants for the simplied 1-DOF roll model (4.10)(sc of (4.14)) and the Mathieu model (4.11) (sm of (4.15)) are re-estimated foreach forward speed and each wave amplitude.

It is well-known that parametric resonance occurs at wave encounter frequen-cies approximately twice the natural roll frequency [44]. Therefore the models aresimulated for an initial surge speed of 0.5 to 12.8 m/s, resulting in a frequency ratioωe/ωφ of 1.4 to 2.2. The wave amplitude ζ0 ranges from 0 to 6 meters.

Figure 4.2 depicts the maximum roll angle for the models of dierent complexityas a result of the simulations. The roll amplitude is limited to 90 in the plots for thesimplied roll equation (Figure 4.2(b)) and the Mathieu equation (Figure 4.2(c))for the sake of presentability. Note that qualitatively the simplied roll model(Figure 4.2(b)) is quite close to the 6-DOF model (Figure 4.2(a)), at least for lowwave amplitudes.

The Mathieu equation (Figure 4.2(c)) simulated with the ltered wave en-counter frequency also behaves reasonably well and is almost indistinguishablefrom the 1-DOF model. This is reasonable, as there are only very small variationsin ωe.

4.1.3 Time-Varying Forward Speed

Since the dierence between the simplied roll model (4.10) and the Mathieu model(4.11) only becomes apparent when the speed is non-constant, the system is sim-ulated with non-constant forward speed. The scenario tested is a simple speedchange, so that the desired forward speed ud is given by

ud (t) =

u0 ∀ t ∈ [t0, t1]u0 + l (t− t1) ∀ t ∈ [t1, t2]

u1 ∀ t ∈ [t2,∞)

where l is the desired acceleration and u1 = u0 +l (t2 − t1). This gives an encounterfrequency

ωe (t) ≈

ωe,0 ∀ t ∈ [t0, t1]ωe,0 + kwl (t− t1) ∀ t ∈ [t1, t2]

ωe,1 ∀ t ∈ [t2,∞)

=

ω0 + kwu0 ∀ t ∈ [t0, t1]ω0 + kw [u0 + l (t− t1)] ∀ t ∈ [t1, t2]

ω0 + kwu1 ∀ t ∈ [t2,∞).

Due to small oscillations in surge, ωe does not exactly match the desired value,as seen in Figure 4.3(b). The Mathieu model (4.11) is once again fed the low-passltered values of ωe, while the 1-DOF model (4.10) uses the unltered values. Theparameters used in the simulation are shown in Table 4.2.

65


ζ0 (m)

ωe/ωφ (−)

maxim

um

rollangle

()

0123456

1.41.51.61.71.81.92.02.12.2

0102030405060708090

ζ 0(m

)

ωe/ωφ (−)

20 40 60 80

0

1

2

3

4

5

6

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2

(a) 6-DOF model (3.8)(3.9).

ζ0 (m)

ωe/ωφ (−)

maxim

um

rollangle

()

0123456

1.41.51.61.71.81.92.02.12.2

0102030405060708090

ζ 0(m

)

ωe/ωφ (−)

20 40 60 80

0

1

2

3

4

5

6

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2

(b) Simplied roll model (4.10).

ζ0 (m)

ωe/ωφ (−)

maxim

um

rollangle

()

0123456

1.41.51.61.71.81.92.02.12.2

0102030405060708090

ζ 0(m

)

ωe/ωφ (−)

20 40 60 80

0

1

2

3

4

5

6

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2

(c) Mathieu model (4.11).

Figure 4.2: Maximum roll angle, model comparison.

66


Table 4.2: Simulation parameters, time-varying speed.


Initial mean forward speed u0 7.90 m/sDesired acceleration l 0.005 m/s2

Final mean forward speed u1 9.43 m/sInitial mean encounter frequency ωe,0 0.645 rad/sFinal mean encounter frequency ωe,1 0.680 rad/sSimulation start time t0 0 sAcceleration start time t1 300 sAcceleration stop time t2 607 sWave amplitude ζ0 2.5 mWave length λ 281 mWave number kw 0.0224 rad/mNatural roll frequency ωφ 0.343 rad/sModal wave frequency ω0 0.4684 rad/s

k44 1.7646e9 kg m2/s2


Simplied roll equation αφ 0.2295 radkφ3 2.2741e9 kg m2/s2

k44 1.7676e9 kg m2/s2


Mathieu equation αφ 0.2122 radkφ3 2.2702e9 kg m2/s2

Figure 4.3 depicts the results of the simulation. Again, the ship is in parametricroll resonance, as shown in Figure 4.3(a). The non-constant forward speed resultsin a non-constant encounter frequency and frequency ratio, respectively, see Figure4.3(b). The spring torque divided by the roll angle is compared in Figure 4.3(c) forthe three models. The simplied roll equation is able to estimate the roll motionwell even for non-constant speed, whereas it is apparent that the Mathieu equationis not. It gradually becomes out of phase with the roll motion of the full 6-DOFmodel and never gets back in phase even when steady-state is reached.

It can be concluded that the simulations indicate that the model based on thesimplied roll equation is adequate to describe the ship's dynamics in parametricroll resonance when the wave encounter frequency is non-constant. The Mathieuequation, on the other hand, is not able to capture the dynamics to a sucientextent unless the encounter frequency is very close to constant and only if thelow-pass ltered value of ωe is used.

67


time (s)

rollangle

()

φφc

φm

0 100 200 300 400 500 600 700 800 900-50

0

50

(a) Roll angle.

time (s)

ωe/ωφ(−

)

ωe/ωφ

Filtered ωe/ωφ

0 100 200 300 400 500 600 700 800 900

1.9

1.95

2


time (s)

torque/roll( kgm

2/s2)

k4/φkφ,c/φc

kφ,m/φm

300 350 400 450 500 550 600

1.5

2

2.5

3 ×109

(c) Spring torque divided by roll angle.

Figure 4.3: Model comparison. φ, φc and φm are roll angles from the 6-DOF and1-DOF models and the Mathieu equation.

68

Chapter 5

1-DOF Roll Model for Time-Varying

Heading and Speed

The complex 6-DOF ship model (3.8) and (3.9) of Chapter 3 is a numerical modelwhich relies on the integration of the hydrodynamic and hydrostatic pressure eldover the instantaneous submerged part of the hull. Though the 6-DOF model ishighly suitable for simulation studies, its usability for control purposes and math-ematical analysis is limited.

In Chapter 4, the three most important degrees of freedom for a ship in para-metric roll resonance were considered, that is, the heave, roll and pitch motions. Byusing a quasi-steady approach an analytical 1-DOF roll model a model with anexplicit functional (rather than a numerical) relationship between time and waveforce was derived which accurately captured the parametric roll motion of a ship.However, the 1-DOF model in Chapter 4 assumed that only the ship's surge speedcould change, while the heading angle was assumed constant.

In this chapter, an analytical 1-DOF roll model is derived where the ship'sspeed and heading angle are allowed to be slowly time-varying. For a large ship,like many ships susceptible to parametric roll [66], this is a reasonable assumption.The content of this chapter was published in Breu, Holden, and Fossen [17].

Motivated by the model introduced in Chapter 4 and following the same deriva-tion, the 1-DOF roll model is tentatively chosen as

m44φ+ d44φ+ κ2 cos

(∫ t

t0

ωe (τ) dτ + κ3

)φ+ κ1φ+ κ4φ

3

= κ5 sin

(∫ t

t0

ωe (τ) dτ + κ6

) (5.1)

where m44 > 0 ∈ R is the sum of the rigid-body moment of inertia and the addedmoment of inertia in roll and d44 > 0 ∈ R is the linear damping coecient inroll. The linear and the cubic restoring coecients in roll are κ1 ∈ R and κ4 ∈ R,respectively. Those are hydrostatic coecients and thus assumed constant. Thecoecients κi = κi (u, ψ), i ∈ 2, 3, 5, 6, may be dependent on the ship's surgespeed and heading angle or both (e.g., κ5 ≈ 0 for head seas).

69

5. 1-DOF Roll Model for Time-Varying Heading and Speed

For simplicity, it is assumed that the waves are traveling along the inertialx-axis, and are sinusoidal.

The model introduced in Chapter 4 and the 1-DOF roll model (5.1) dier inthree major aspects. Whereas the model in Chapter 4 assumes constant coecients,most of the model coecients in (5.1) are explicitly allowed to be dependent on theship's surge speed and heading angle. Furthermore, the model (5.1) incorporatesdirectly excited roll motion by considering the external forcing term on the righthand side of (5.1) (the Froude-Krylov force [122]) which was neglected in Chapter 4due to the ship sailing in head sea conditions. Finally, in Chapter 4, ψ ≡ 0 andu 6≡ 0. Here, ψ 6≡ 0 and u 6≡ 0.

5.1 Model Coecient Identication

In this section, the coecients κi, i ∈ 1, . . . , 6 of the 1-DOF roll model (5.1)will be identied from simulations of the 6-DOF model (3.8) and (3.9).

The hydrostatic coecients κ1 and κ4 are constants, and they can be deter-mined from simulations without waves. κ2 ∈ R and κ3 ∈ R are the coecients ofthe change in the amplitude of the linear restoring moment in roll and its phasedue to the passage of the waves, and account for the coupling of roll to the heaveand pitch motions. The external wave forcing is considered by the right hand sideof (5.1) with its hydrodynamical coecients κ5 ∈ R and κ6 ∈ R.

5.1.1 Hydrostatic Coecients

The constant coecients κ1 and κ4 are found by a free decay test. Dene the springtorque for the 1-DOF roll model (5.1) without wave inuence as

kφ (t; κ) , κ1φ (t) + κ4φ3 (t) . (5.2)

The hydrostatic coecient vector κ , [κ1, κ4]> is identied by a nonlinear least-

square curve tting of (5.2) to the pressure-induced generalized forces in roll ofthe 6-DOF model (3.8) and (3.9) simulated in a calm water scenario. Thus, bydenoting the fourth element of k in (3.9) as k4, κ ∈ R2×1 is given by

κ = arg minκ

∑

t

|k4 (η (t) , t)− kφ (t; κ)|2 . (5.3)

5.1.2 Hydrodynamic Coecients

The identication of the hydrodynamic coecients κ , [κ2, κ3, κ5, κ6]> is done in

the following way.The complex 6-DOF model is simulated for a wide range of constant u, ψ dou-

bles, and the pressure-induced moment in roll (the fourth element of k, that isk4) is saved. The sum of the restoring and external moments in roll is dened as

kφ (t; κ) ,

[κ1 + κ2 cos

(∫ t

t0

ωe (τ) dτ + κ3

)]φ (t)

+ κ4φ3 (t)− κ5 sin

(∫ t

t0

ωe (τ) dτ + κ6

). (5.4)

70

5.1. Model Coecient Identication

The coecients κ ∈ R4×1 for these conditions are then found as

κ = arg minκ

∑

t

|k4 (η (t) , t)− kφ (t;κ)|2 (5.5)

where the values of the hydrostatic coecients κ1 and κ4 are found from the freedecay tests in Section 5.1.1. This procedure results in one set of κs per u, ψ double.The parameters for the simulations of the 6-DOF ship model are equivalent tothose presented in Chapter 4.

Figures 5.15.4 show the results of this procedure for surge speeds u ∈ R rangingfrom 0 m/s to 10 m/s and for heading angles ψ from 0 to 90. Note that headingangles of 0 and 90 correspond to the ship sailing in head sea and beam seaconditions.

The hydrodynamic coecient κ6 represents the phase angle of the externalsinusoidal wave force. Note that, in head sea conditions (ψ = 0) the amplitudeof the external forcing term is zero which can be observed in Figure 5.3. As aconsequence κ6 cannot be estimated in those conditions.

ψ ()

u (m/s)

κ2

0102030405060708090

0

2

4

6

8

10

0.5

1

1.5

2×109

Figure 5.1: Hydrodynamic coecient κ2 and its functional approximation (denotedby +).

5.1.3 Identication of κ

A functional expression for the coecients κ is desired that is valid for a wide rangeof surge speeds and heading angles. This allows the analytical analysis of the roll,giving both easier analysis and better simulation than a numerical model.

The values of κ obtained in Section 5.1.2 are used to derive a functional rela-tionship between ψ and u and κ. From Figures 5.15.4, it appears that κ predom-inantly varies with the heading angle ψ, not the surge speed u. Furthermore, the

71


PSfrag

ψ ()

u (m/s)

κ3

0102030405060708090

0

2

4

6

8

10

-2

-1

0

1

2


ψ ()

u (m/s)

κ5

0102030405060708090

0

2

4

6

8

10

0

1

2

3×108


72

5.1. Model Coecient Identication

ψ ()

u (m/s)

κ6

102030405060708090

0

2

4

6

8

10

0.5

1

1.5

2


relationship appears to be sinusoidal in nature. Therefore κ is expressed as

κ2 = κ2 (ψ) = κ2,1 sin (κ2,2ψ + κ2,3)

+ κ2,4 sin (κ2,5ψ + κ2,6)

+ κ2,7 sin (κ2,8ψ + κ2,9)

(5.6)

κ3 = κ3 (ψ) = κ3,1 sin (κ3,2ψ + κ3,3)

+ κ3,4 sin (κ3,5ψ + κ3,6)(5.7)

κ5 = κ5 (ψ) = κ5,1 sin (κ5,2ψ + κ5,3)

+ κ5,4 sin (κ5,5ψ + κ5,6)(5.8)

κ6 = κ6 (ψ) = κ6,1 sin (κ6,2ψ + κ6,3)

+ κ6,4 sin (κ6,5ψ + κ6,6) .(5.9)

To determine the parameter values for κ, the above functions are curve tted tothe data gathered for κ in the rst step in Section 5.1.2. The results of the curvetting of the functions (5.6)(5.9) are shown in Figures 5.15.4 (denoted by + inthe plots).

The coecients κ5 and κ6, corresponding to the external wave forcing, arevery well approximated by the functional expressions (5.8) and (5.9). On the otherhand, the approximations of κ2 and κ3 which account for the coupling of roll to theheave and pitch motion, are less accurately described by (5.6) and (5.7). Mainlyfor large surge speeds and small heading angles, there is a discrepancy between thehydrodynamic coecients and its functional approximations. In Section 5.2 it willbecome evident that this corresponds to when the ship is experiencing parametric

73


roll resonance. However, it will be shown that the functional expressions for κ2 andκ3 are appropriate to describe the roll dynamics to a satisfactory extent.

5.2 Model Verication

In this section, the 1-DOF roll model with the hydrodynamical coecients givenby the functional expressions presented in Section 5.1.3 is veried. To that matter,the 1-DOF roll model is simulated and it is compared to simulations of the complex6-DOF model of Chapter 3. The simulations are performed using the same modelparameters as in Chapter 4.

5.2.1 Maximum Roll Angle

The 6-DOF ship model (3.8) and (3.9) is simulated for a wide range of surge speedsand heading angles. For each simulation, the surge speed and the heading angleare kept constant (except for small variations due to the limited bandwidth ofthe speed and heading controllers). Then, the 1-DOF roll model (5.1) is simulatedwhere the surge speed and heading angle of the previous simulations of the 6-DOFmodel enter via the encounter frequency (3.30). The maximum steady-state rollangles of the various simulation scenarios for the dierent models are depicted inFigures 5.55.7.

From the simulations of the 6-DOF ship model depicted in Figure 5.5, it isevident that the ship is experiencing parametric roll resonance of up to 30 forlarge surge speeds and small heading angles. This corresponds to an encounterfrequency which is close to twice the natural roll frequency, a well-known criteriafor parametric resonance [100].1

Furthermore, it is noticeable from Figure 5.5 that the ship is suering directlyexcited roll motions at high heading angles, independent of the surge speed. Thosemaximum roll angles are considerably lower than the maximum roll angles causedby parametric resonance and peak in beam sea condition.

Figure 5.6 depicts the simulation results from the 1-DOF roll model where thehydrodynamic coecients are determined numerically from the simulations of the6-DOF model for each simulation scenario. In Figure 5.7 the maximum roll anglesare shown for the 1-DOF model with the hydrodynamical coecients approximatedby the functional expressions of Section 5.1.3.

The plots of the maximum roll angles in Figures 5.6 and 5.7 are very similar.That indicates that the functional approximation of the hydrodynamical modelcoecients of Section 5.1.3 is appropriate.

Comparing the simulations of the 1-DOF roll models in Figures 5.6 and 5.7to the simulations of the 6-DOF model depicted in Figure 5.5, the 1-DOF rollmodels are qualitatively quite close to the 6-DOF model for most of the simulationscenarios. The maximum roll angle is slightly overestimated by the 1-DOF models,however that may be explained by the simplications in the derivation of the 1-DOF models and lack of higher-order restoring coecients.

1The apparent higher roll angle for low, non-zero heading angles might be due to low speedresolution in the border between parametric roll and not parametric roll.

74


PSfrag

ψ ()

u (m/s)

maxim

um

rollangle

()

0102030405060708090

0

2

4

6

8

10

0

10

20

30

40

Figure 5.5: Maximum roll angle, 6-DOF model.

PSfrag

ψ ()

u (m/s)

maxim

um

rollangle

()

0102030405060708090

0

2

4

6

8

10

0

10

20

30

40

Figure 5.6: Maximum roll angle, 1-DOF model.

75


PSfrag

ψ ()

u (m/s)

maxim

um

rollangle

()

0102030405060708090

0

2

4

6

8

10

0

10

20

30

40

Figure 5.7: Maximum roll angle, 1-DOF model, functional expressions.

However, for the 1-DOF roll models, the region of parametric roll resonance atlarge surge speeds extends to higher heading angles than for the 6-DOF model. Thisis slightly less pronounced for the 1-DOF model with the functional expressions forthe coecients.

This eect is probably caused by two factors: the assumption of speed indepen-dence of κ2 and κ3 breaks down when the speed range is very large (see Figs. 5.1and 5.2); and the behavior at the borders of the area of parametric resonance especially the non-smooth borders is dicult to model accurately.

5.2.2 Time-Varying Heading and Speed

Since the 1-DOF roll model (5.1) with functional expressions for the hydrodynamiccoecients was derived to allow time-varying heading angle and surge speed, itis simulated for non-constant heading and speed. The results are compared tosimulations of the full 6-DOF ship model (3.8) and (3.9).

The initial conditions of the simulation scenario are chosen such that the ship isin parametric roll resonance condition, specically ψ (0) = 0, u (0) = 7 m/s. At 200 sthe surge speed is increased gradually to about 8 m/s with a constant acceleration of0.005 m/s2 while the heading angle is increased by 0.1/s to approximately 20. Thesurge speed and the heading angle are depicted in Figures 5.9 and 5.10, respectively.The oscillations in surge and yaw are due to the limited bandwidth of the PIDcontrollers.

Figure 5.8 shows the roll angle of the 6-DOF model φ ∈ R and of the 1-DOFmodel φm. The ship is experiencing large roll amplitudes up to about 20 due toparametric roll resonance. The roll angle of the 1-DOF model is qualitatively veryclose to that of the 6-DOF model, being only slightly underestimated. The sum ofthe restoring and external moments in roll are depicted in Figure 5.11, and theyshow a good match between the 6-DOF model and the 1-DOF model.

76


time (s)

rollangle

()

φφm

0 100 200 300 400 500 600

-20

-10

0

10

20

Figure 5.8: Roll angle.

time (s)

speed(m

/s)

0 100 200 300 400 500 600

7

7.5

8

Figure 5.9: Speed.

time (s)

headingangle

()

0 100 200 300 400 500 600

0

10

20

Figure 5.10: Heading angle.

time (s)

external/restoringmoments

( kgm

2/s2)

k4

kφ

0 100 200 300 400 500 600

-5

0

5

×108

Figure 5.11: Restoring/ external moments in roll.

77


The simulation scenario with time-varying speed and heading angle indicatesthat the 1-DOF model with the hydrodynamic coecients given by heading de-pendent functions is appropriate to describe the ship's dynamics during parametricroll resonance.

78

Part III

Frequency Detuning

79

Chapter 6

Frequency Detuning

The content of this chapter has been published in Holden, Breu, and Fossen [64].Control of parametric roll resonance has attracted considerable research in re-

cent years [15, 4749, 61, 66, 79, 80, 126, 149]. The proposed control methods canroughly be categorized as direct or indirect methods. The direct methods are aimedat directly controlling the roll motion by generating an opposing roll moment, asseen in Galeazzi and Blanke [47], Holden et al. [61], and Umeda et al. [149]. Indi-rect strategies attempt to violate the empirical conditions necessary for the onsetof parametric roll resonance. The potential of violating one of the conditions forthe onset of parametric roll resonance (see [44]) has been eectively shown in Breuand Fossen [15], Galeazzi et al. [54], Jensen, Pedersen, and Vidic-Perunovic [79],Jensen, Vidic-Perunovic, and Pedersen [80], Ribeiro e Silva, Santos, and Soares[126] and may be called frequency detuning. A hybrid approach, doing both at thesame time, is also possible, as seen in Galeazzi et al. [48] and Galeazzi [49].

In this chapter, the indirect approach to control parametrically excited rollmotions is considered. A simple model for parametric roll resonance is the Mathieuequation:

m44φ+ d44φ+ [k44 + kφt cos (ωet+ αφ)]φ = 0

where m44 > 0 ∈ R is the sum of the moment of inertia and the added momentof inertia in roll, d44 > 0 ∈ R the linear hydrodynamic damping coecient, k44 >0 ∈ R the linear restoring moment coecient and kφt > 0 ∈ R the amplitude ofits change, ωe ∈ R the encounter frequency, and αφ ∈ R a phase angle. All theparameters are considered constant.

It is known [100] that such a system parametrically resonates at ωe ≈ 2√k44/m44

(an encounter frequency of twice the natural roll frequency). The encounter fre-quency ωe is the Doppler-shifted frequency of the waves as seen from the ship. Asthe frequency is Doppler-shifted, it can be changed by changing the ship's speed.

The main purpose of this chapter is to show that it is feasible to control para-metric roll resonance by changing the encounter frequency to violate the conditionωe ≈ 2

√k44/m44. In this work, this is called frequency-detuning.

As shown in Chapter 4, the Mathieu-type equations are not valid for non-constant ωe. To design and analyse the control system, the simplied, 1-degree-of-freedom (DOF) Model (4.10) developed in Chapter 4 is therefore used, allowing the

81

6. Frequency Detuning

ship's forward speed to change, but only slowly. For ships susceptible to parametricroll many of which are large [28, 4244] this is not an unreasonable assumption.

Based on the 1-DOF roll model (4.10) in Chapter 4, a simple controller basedon a linear change of the encounter frequency, achieved by variation of the ship'sforward speed, is proposed. It is proven mathematically that the proposed controlleris able to drive the ship out of parametric resonance, driving the roll motion tozero. It is worth noting that the controller is in fact simple enough that a helmsmancan perform the necessary control action, rendering a speed controller unnecessary.

The controller is tested with the simplied 1-DOF Model (4.10) and the full6-DOF Model presented in Chapter 3, and is shown to work as expected in bothcases.

6.1 Roll Model for Non-Constant Speed

As previously mentioned, there is no direct actuation in roll; instead, the encounterfrequency ωe is changed (by changing the forward speed) to detune the encounterfrequency and thus violate a necessary condition for the existence of parametricroll resonance.

The following assumptions are made:

Assumption 11. The ship is traveling in head or stern seas.

Assumption 12. The waves are planar, standing and sinusoidal, with frequencyω0 > 0 ∈ R and wave number kw > 0 ∈ R.

Assumption 13. The ship is changing speed only slowly.

Assumption 14. The ship's sway and heave velocities are small.

Assumption 15. The ship's pitch angle is small.

The ship is traveling with linear and angular velocities

vb = [u, v, w]> ∈ R3 (6.1)

ωb = [p, q, r]> ∈ R3 (6.2)

as seen from the ship. For an observer standing on the mean ocean surface, theship will appear to have linear and angular velocities

vn = Rvb = [un, vn, wn]> ∈ R3 (6.3)

ωn = Rωb = [pn, qn, rn]> ∈ R3 (6.4)

where R is a rotation matrix given by

R (Θ) =

cθcψ sφsθcψ − cφsψ cφsθcψ + sφsψcθsψ sφsθsψ + cφcψ cφsθsψ − sφcψ−sθ sφcθ cφcθ

(6.5)

where c· = cos (·), s· = sin (·) and Θ = [φ, θ, ψ]> are the roll, pitch, and yaw anglesas dened in Fossen [41], see also Chapter 3.

82

6.1. Roll Model for Non-Constant Speed

To analyse the eects of speed changes, a model is needed that is valid for time-varying speed. As discussed in Chapter 4, the commonly used Mathieu equationis not adequate in this case. Thus, the 1-DOF Model (4.10) of Chapter 4 is used,given by

m44φ+ d44φ+

[k44 + kφt cos

(∫ t

t0

ωe (t) dτ + αφ

)]φ+ kφ3φ3 = 0 (6.6)

where φ is the roll angle,m44 the sum of the rigid-body moment of inertia about thex-axis and the added moment of inertia in roll, d44 the linear hydrodynamic damp-ing, k44 the linear restoring moment coecient, kφt the amplitude of its change, andkφ3 > 0 ∈ R the cubic restoring force coecient. These parameters are constant.

Note that the natural frequency of φ is ωφ ,√k44/m44. From Chapter 3, the

encounter frequency ωe is given by

ωe = ω0 ± kwun = ω0 ± kw[1, 0, 0]Rvb . (6.7)

The encounter frequency is the frequency of the waves as seen from the ship. Dueto the Doppler eect, this is not the same as the frequency of the waves seen by astationary observer, ω0. For a ship traveling at constant velocity, ωe is constant andthe Mathieu equation can be used to describe the ship's behavior, see Chapter 4.

If the ship is nonrotating (i.e., ωb ≡ 0), then

ωe = ±kw[1, 0, 0]Rvb ≈ ±kwe>xR[u, 0, 0]>

≈ ±kwu cos (θ) cos (ψ) ≈ ±kwu cos (ψ) (6.8)

by the assumption of small sway velocity, yaw rate and pitch angle. The ship isassumed to be sailing in head or stern seas, that is ωe = kwu (head seas) orωe = −kwu (stern seas).

u can be set directly; this is the forward acceleration and can be changed byincreasing or decreasing throttle. It will, however, be limited, so it is taken to satisfy|u| ≤ umax. Thus, τωe , ωe is taken to be the control input, satisfying

|τωe | = |ωe| ≤ τωe,max = kwumax . (6.9)

Note that the assumption that the forward speed changes only slowly impliesthat umax is quite small. The assumption of slow speed change is a necessity toderive the model (6.6), as detailed in Chapter 4.

As it is apparent from the above equation, τωe,max depends on the size of umax

and kw. For the type of large, slow vessels that are susceptible to parametric roll,umax is likely to have quite a low value. For ships to parametrically resonate, thewave length has to be rather long, or kφt will be too small [62]. A long wave lengthimplies a small kw, since kw = 2π/λ if λ > 0 ∈ R is the wave length. Thus τωe,max

is quite small.

83


6.2 Control Design

The control objective is to design u such that the origin of the roll system (6.6)is (at least) asymptotically stable. Choosing a u so that ωe is equal to the desiredτωe ∈ R is a control allocation (CA) problem.1

6.2.1 Control Principle

The basic control principle is to (slowly) change the encounter frequency from anundesired value ωe,0 to a desired value ωe,1. Tentatively, the controller is chosen as

τωe (t) =

0 ∀ t ∈ [t0, t1)

ε ∀ t ∈ [t1, t2)

0 ∀ t ∈ [t2,∞)

(6.10)

for some small constant ε > 0 ∈ R, with t2 ≥ t1 ≥ t0. The initial time is t0.If ωe (t0) = ωe,0, then

ωe (t) =

∫ t

t0

τωe (τ) dτ + ωe,0 =

ωe,0 ∀ t ∈ [t0, t1)

ωe,0 + ε (t− t1) ∀ t ∈ [t1, t2)

ωe,1 ∀ t ∈ [t2,∞)

(6.11)

where ωe,1 = ωe,0 + ε (t2 − t1). This gives

∫ t

t0

ωe (τ) dτ =

ωe,0 (t− t0) ∀ t ∈ [t0, t1)

ωe,0 (t− t0) + 12ε (t− t1)

2 ∀ t ∈ [t1, t2)

ωe,1 (t− t2) + ωe,0 (t2 − t0) + 12ε (t2 − t1)

2 ∀ t ∈ [t2,∞)

.

If the ship is sailing in head or stern seas and v = w = θ = 0, then ωe (t) =ω0± kwu and the encounter frequency of (6.11) can then be achieved with a surgevelocity of

u = ∓ω0 − ωe (t)

kw= ∓ 1

kw

ω0 − ωe,0 ∀ t ∈ [t0, t1)

ω0 − ωe,0 − ε (t− t1) ∀ t ∈ [t1, t2)

ω0 − ωe,1 ∀ t ∈ [t2,∞)

. (6.12)

Proving that the controller (6.10) works is done in two steps: rst, ensuringthat there exists a (unique nite) solution of (6.6) at t = t2. This step is done inthe Appendix. Secondly, it needs to be proven that if ωe (t) ≡ ωe,1 ∀ t ≥ t2, thenthe solution to the initial value problem

m44φ+ d44φ+ [k44 + kφt cos (ωe,1t+ αφ)]φ+ kφ3φ3 = 0

φ (t2) = φ2, φ (t2) = φ2

(6.13)

1It is also possible to change ωe by changing course (i.e., changing ψ). This will have theunwanted side eect that the ship will be directly excited by waves (i.e., there will be a an externalforce on the right-hand side of (6.6) proportional to the wave amplitude), which may also result inrelatively large roll amplitude in the type of seas that give rise to parametric resonance. Changingψ to change the encounter frequency is investigated in Chapter 7.

84

6.2. Control Design

where

αφ , αφ − ωe,1t2 + ωe,0 (t2 − t0) +1

2ε (t2 − t1)

2

is a constant, goes to zero for all φ2, φ2.

6.2.2 The System in the Time Interval t ∈ [t2,∞)

From the results in the Appendix, it is known that there exists a nite solutionto (6.6), valid at t = t2. From t ≥ t2, the trajectories of the system will be thesolution to the initial value problem (6.13).

From Nayfeh and Mook [100], it is known that there are parameter values of ωe,1which ensure that the trajectories of the system (6.13) go to zero. If it is assumedthat ωe,0 ≈ 2ωφ (where parametric resonance of (6.6) is known to occur), it ispossible to nd theoretical values for the regions of stability from the approximatemethods of Nayfeh and Mook [100]; specically Theorem 6.1 is established followingthe material in Nayfeh and Mook [100, Section 5.7.3], however for a time-varyingexcitation frequency.

Theorem 6.1 (Frequency detuning). The behavior of (6.13) can be categorizedinto three dierent categories, depending on the value of ωe,1:

If 0 ≤ ωe,1 ≤ ωe, then the origin of (6.13) is globally attractive.

If ωe < ωe,1 ≤ ωe, then the origin of (6.13) is unstable, and there exists ahigh-amplitude, stable limit cycle. All trajectories of (6.13) converge to thislimit cycle, with the exception of those starting in the origin.

If ωe,1 > ωe, then the origin of (6.13) is locally stable, there exists a high-amplitude, stable limit cycle and a slightly lower-amplitude, unstable limitcycle.

ωe and ωe are the solutions to the equations√

1− d244ω

2e

k2φt

− m44ω2e

kφt

(2

√k44

m44ωe− 1

)= 0 (6.14)

√1− d2

44ω2e

k2φt

+m44ω

2e

kφt

(2

√k44

m44ωe− 1

)= 0 . (6.15)

In this theorem, asymptotic stability of limit cycles follows the denition of Khalil[84, Denition 8.1].

Proof. To simplify the analysis, the alternative dimensionless time scale is dened:

T ,1

2ωe,1t+ αφ (6.16)

giving

d

dt=

1

2ωe,1

d

dTd2

dt2=

1

4ω2e,1

d

dT 2.

85


Using primes to indicate derivatives with respect to T , the system (6.13) isrewritten as

φ′′ + 2ιγφ′ + [κ+ 2ι cos (2T )]φ+ αιφ3 = 0 (6.17)

where

ι =2kφt

m44ω2e,1

γ =d44ωe,1

2kφt

κ =4k44

m44ωe,1

α =2kφ3

kφt

are all positive dimensionless parameters. It is assumed that ι is small.Equation (6.17) is known to parametrically resonate if κ ≈ 1 (i.e., ωe,1 ≈ 2ωφ;

the encounter frequency is twice the natural roll frequency).Using an O (ι) (big O notation) approximation to the solution of (6.17), Nayfeh

and Mook [100] derives a solution using the method of multiple scales (see Nayfehand Mook [100]) given by

φ = a cos (T − β/2) +O (ι) (6.18)

where a and β are slowly time-varying.Dening the alternative (also dimensionless) time scale

t = ιT (6.19)

(which is slowly varying) and letting

√κ = 1− ισ (6.20)

(with σ representing the nearness of κ to unity, and thus the system to paramet-ric resonance), a and β satisfy the (nonlinear homogenous ordinary) dierentialequations

da

dt= − a

2√κ

sin (β)− γa (6.21)

adβ

dt= 2σa− a√

κcos (β)− 3α

4√κa3 . (6.22)

The aβ system has equilibrium points (corresponding to a steady-state peri-odic motion of φ, i.e., a limit cycle) given by

a = 0, β arbitrary (6.23)

86

6.2. Control Design

(the trivial solution) and

sin (β) = −2√κγ, cos (β) = 2σ

√κ− 3α

4a2 . (6.24)

Since√κ = 1− ισ, the non-trivial steady-state solution of φ has the amplitude

a2 =8σ

3α± 4

3α

√1− 4γ2 (6.25)

where only the positive root is relevant.If 2γ > 1, then (6.25) has no real roots and only the trivial steady-state solution

exists. As this is equivalent to high damping, if 2γ > 1, parametric resonance willnot occur. (The origin of (6.17) is then globally attractive for all ωe,1).

If 2γ ≤ 1, then there is one real root of (6.25) if 2|σ| <√

1− 4γ2, and two if2|σ| >

√1− 4γ2. The condition 2σ = −

√1− 4γ2 corresponds to (6.14) (giving

ωe) and 2σ =√

1− 4γ2 to (6.15) (giving ωe).Figure 6.1 illustrates the stability properties of (6.17) for the dierent cases.

Dashed lines represent unstable equilibrium values of a for dierent values of σ,and solid lines stable equilibrium values.2

σ

a

RI RII RIII

Figure 6.1: Stability regions of (6.17), theoretical.

In Region RI , there is only the trivial solution. From Nayfeh and Mook [100],this is globally attractive.

In Region RII (where parametric resonance occurs), the trivial solution is un-stable, and there exists a large-amplitude steady-state solution, a limit cycle. Apartfrom the case where φ (t2) = φ (t2) = 0, this limit cycle is globally attractive [100].

2It is worth noting that Figure 6.1 bears strong similarity to a cross-section with the waveheight kept constant of the simulation of the full 6-DOF Model (3.8)(3.9) of Chapter 3, exceptthat in that chapter there is no evidence of the high-amplitude solutions of (6.17) in Region RIII .The stability regions indicated from simulating the 6-DOF Model are shown in Figure 6.2.

87


σ

a

RI RII RIII

Figure 6.2: Stability regions of the 6-DOF Model (3.8)(3.9) of Chapter 3, simula-tion.

Region RIII has three equilibrium values, and is somewhat more complicated.The value a = 0 (equivalent to φ = 0) is (locally) asymptotically stable. However,there exist two limit cycles, one high-amplitude and one low-amplitude. The high-amplitude one is (locally) asymptotically stable, whereas the low-amplitude one isunstable.

Based on the proof of Theorem 6.1, it can be concluded that it is possible that, ifone increases ωe so that ωe,1 2ωφ (i.e., σ 0), φ does not go to zero but insteadto the high-amplitude limit cycle. If one instead decreases ωe so that ωe,1 2ωφ(i.e., σ 0), φ will go to zero no matter how large φ (t2) is. This is illustratedFigure 6.3.

This suggests that reducing the encounter frequency is the most sensible choice,and, in fact, the only option that can be guaranteed to work. It is, however, worthnoting that the analysis is based on a simplication of the ship dynamics. Thehigh-amplitude limit cycle has not been observed in the simulations with the morephysically accurate 6-DOF ship Model (3.8) and (3.9) of Chapter 3. Galeazzi et al.[54] and Ribeiro e Silva, Santos, and Soares [126] came to the opposite conclusionregarding speeding up versus slowing down. But bear in mind that in Galeazzi etal. [54], the conclusion was largely predicated on the need to have sucient speedfor ns (which were used in addition to speed change) to be eective.

None the less, decreasing the encounter frequency has another benet: If itis assumed that σ starts at zero and slowly increases, trajectories will tend togo to a higher-amplitude limit cycle as the steady-state value of a increases withincreasing σ in Region RII . However, if instead σ is decreased, the trajectories willtend to go to a lower-amplitude limit cycle even if still in parametric resonance.This phenomenon has been observed in the simulations with the most accurate

88

6.3. Simulation Results

σ

a

RI RII RIII

t0×t1×

[t2,∞)×

[t2,∞)×

Figure 6.3: Control of parametric roll resonance: Increasing vs decreasing the en-counter frequency.

6-DOF Model (3.8) and (3.9) of Chapter 3, so there is reason to suspect that thisholds true for real-world cases.

6.3 Simulation Results

To test the validity of the controller (6.10), the closed-loop system is simulatedusing both the simplied model (6.6) and the full 6-DOF Model (3.8) and (3.9)of Chapter 3, in three dierent simulation scenarios. In all scenarios, the initialconditions are chosen such that the ship was experiencing parametric roll resonance.

In accordance with the open-loop simulations in Chapter 4, a reduction of thefrequency ratio to ωe,1/ωφ < 1.7 will lead the ship out of the region where the shipis susceptible to parametric roll resonance.

Three dierent scenarios are simulated:

1. Slow deceleration. The controller is turned on after parametric roll has al-ready fully developed.

2. Slow deceleration. The controller is turned on before parametric roll has fullydeveloped.

3. Fast deceleration. The controller is turned on before parametric roll has fullydeveloped.

The simulation parameters (the same as those used in Chapter 4 are listedin Table 6.1. The control parameters are found in Tables 6.26.4. The simulationresults are summarized in Table 6.5, and can be seen in Figures 6.46.6.

Figure 6.4 shows the simulation results for the controlled system in comparisonwith the uncontrolled system for the rst scenario. It is obvious from Figure 6.4(a)

89


Table 6.1: Simulation parameters.


Wave amplitude ζ0 2.5 mWave length λ 281 mWave number kw 0.0224 Natural roll frequency ωφ 0.343 rad/sModal wave frequency ω0 0.4684 rad/sSimulation start time t0 0 s

k44 1.7533× 109 kg m2/s2

Model parameters kφt 7.1373× 108 kg m2/s2

(simplied roll equation) αφ 0.2741 radkφ3 2.2627× 109 kg m2/s2

Table 6.2: Control parameters, Scenario #1.


Control action ε −1.7889× 10−4 rad/s2

Maximum deceleration v1,max 0.008 m/s2

Initial forward speed v1 (t0) 7.44 m/sInitial encounter frequency ωe,0 0.6346 rad/sFinal encounter frequency ωe,1 0.5831 rad/sFinal forward speed v1 (t2) 5.14 m/sController turned on t1 300 sController turned o t2 588 s






90







Table 6.5: Simulation results, maximum roll angles.

Scen.Simplied 1-DOF model Full 6-DOF model

Uncontrolled Controlled Red. Uncontrolled Controlled Red.

#1 25.34 25.34 0% 23.34 23.34 0%#2 25.34 22.40 11.6% 23.34 20.33 9.0%#3 23.57 13.71 41.8% 17.99 4.83 73.2%

that the ship is experiencing large roll amplitudes caused by parametric resonance.The frequency ratio is gradually decreased after 300 s (Figure 6.4(c)), which causesthe expected gradual reduction of the roll motion to zero.

The simulation results with the full 6-DOF Model (3.8)(3.9) of Chapter 3 areshown in Figure 6.4(b). The controller works equally well with the more complexmodel.

Of course, since the controller is turned on only after parametric roll has fullydeveloped, the maximum roll angle in Scenario #1 is the same for the controlledand uncontrolled cases. The steady-state roll angle is zero as predicted.

The simulation results of the second scenario are shown in Figure 6.5. In thisscenario, the encounter frequency is reduced when the roll angle is much lowerthan in the rst scenario, early enough that parametric rolling has not yet fullydeveloped (specically, when the roll angle is about 5). Figure 6.5 shows that boththe simplied 1-DOF model and the full 6-DOF model behave similarly.

However, despite the controller being turned on when roll is only at 5, themaximum roll angle is not greatly reduced compared to the uncontrolled case.This is simply because the ship is moving very slowly out of resonant condition.The steady-state roll angle is none the less zero, as predicted.

To get the ship to move out of resonant condition before the roll angle hasreached dangerous levels requires, as it turned out, signicantly faster decelerationthan in Scenarios #1 and #2, even if the controller was turned on at a lower rollangle.

To this eect, Scenario #3 was simulated. The controller is turned on early, ata time when the roll angle is about 2. The ship is decelerating at twice the rate

91


time (s)

φ()

ControlledUncontrolled

0 100 200 300 400 500 600 700 800

-20

0

20

(a) 1-DOF model.

time (s)

φ()


0 100 200 300 400 500 600 700 800

-20

0

20

(b) 6-DOF model.

time (s)

ωe/ωφ(−

)

Controlled (actual)Controlled (desired)Uncontrolled

0 100 200 300 400 500 600 700 800

1.65

1.7

1.75

1.8

1.85

(c) Frequency ratio.

Figure 6.4: Simulation results, Scenario #1.

92


time (s)

φ()


0 100 200 300 400 500 600 700 800

-20

0

20

(a) 1-DOF model.

time (s)

φ()


0 100 200 300 400 500 600 700 800

-20

0

20

(b) 6-DOF model.

time (s)

ωe/ωφ(−

)


0 100 200 300 400 500 600 700 800

1.65

1.7

1.75

1.8

1.85



93


of Scenarios #1 and #2. Also, both the initial and nal encounter frequencies arelower in Scenario #3 than in the two others. The results are plotted in Figure 6.6.

time (s)

φ()


0 100 200 300 400 500 600 700 800

-20

0

20

(a) 1-DOF model.

time (s)

φ()


0 100 200 300 400 500 600 700 800

-20

0

20

(b) 6-DOF model.

time (s)

ωe/ωφ(−

)


0 100 200 300 400 500 600 700 800

1.65

1.7

1.75

1.8

1.85



From Figure 6.6, it is visible that the controller is capable of reducing theroll angle suciently fast such that the maximum roll angle is only 1/2 (1-DOFmodel) to 1/4 (6-DOF model) of that of the uncontrolled case. Interestingly, fromFigure 6.6(b) it can be seen that the controller works signicantly better for the

94


full 6-DOF model than the simplied 1-DOF model. In steady-state, the roll angleis zero, as expected.

From the simulation results, it is seen that the controller is capable of bringingthe ship out of parametric resonance and assuming sucient deceleration capa-bility reduce the maximum roll angle signicantly. It is also vital to turn on thecontroller as early as possible. The simulations conrm the theoretical derivationspresented in Section 6.2.

How practical the controller is in a real-world scenario depends almost entirelyon the ability of the captain (or the automated systems) to detect parametricresonance, and the ability of the ship to rapidly decelerate. If these capabilitiesare present, then the controller could prove useful. In the absence of one or bothof these abilities, the practicality of the controller is limited, at least on its own.However, it might be possible to pair it with another control scheme such as nsas done in Galeazzi et al. [48], Galeazzi [49], u-tanks [63], gyro stabilizers or otheractive controllers.

95

Chapter 7

Frequency Detuning by Optimal

Speed and Heading Changes

Frequency detuning control approaches are designed to change the frequency of theparametric excitation for instance via the Doppler-shift of the encounter frequency that is, the frequency of the waves as seen from the ship. The Doppler-shift canbe achieved by variations of the ship's speed and heading angle.

Whereas the eectiveness of frequency detuning control to stabilize paramet-rically excited roll oscillations in ships has been reported, little research on howto change the encounter frequency with respect to optimality has been conducted.Since changes of the ship's speed and heading angle result in a shift of the en-counter frequency, optimal control methodologies can be used to determine theoptimal encounter frequency and the optimal set-points for the ship's speed andheading angle.

In this chapter, three optimal control methods for the stabilization of para-metric roll resonance are proposed. As a rst approach, the application of a modelpredictive control (MPC) method to ships experiencing parametric roll resonance isproposed. Constraints on inputs and states as well as an objective function aimingto violate one of the empirical conditions for the onset of parametric roll resonanceare formulated in the MPC framework. It is illustrated in simulations that theproposed MPC approach is apt to be used for the stabilization of parametricallyexcited roll oscillations.

Secondly, the extremum seeking (ES) methodology is adapted to iterativelydetermine the optimal set-point of the encounter frequency. The mapping of theencounter frequency to the two controllable states, the ship's forward speed andheading angle, is a constrained optimization problem which can be posed in atwo-step sequential least-squares formulation. By dening an appropriate objectivefunction and designing globally exponential stable speed and heading controllers,it is shown that the proposed ES controller is able to stabilize the roll oscillationscaused by parametric excitation eectively.

Finally, an extension to the ES control approach is presented, aiming at ro-bustness. To that matter, two modications are suggested. Firstly, the speed andheading controllers are formulated in the framework of L1 adaptive control which

97

7. Frequency Detuning by Optimal Speed and Heading Changes

guarantees robustness while still having fast adaptation. By doing so, robustnessis increased with respect to model uncertainty, lack of knowledge, and boundedexternal disturbances. Secondly, the ES loop is modied towards limit cycle mini-mization, which replaces the objective function of the previously presented controlapproach, thus relaxing the assumption that the frequency range for roll parametricresonance is a priori known.

The material presented in this chapter has been published in Breu and Fossen[15],Breu and Fossen [18], and Breu, Feng, and Fossen [14].

7.1 3-DOF Ship Model

A simplied 3-DOF model of the ship describing the coupled motions in surge, rolland yaw is used to represent the ship dynamics. The roll dynamics are structurallyrepresented by the 1-DOF roll model of Chapter 4. In addition to the linear damp-ing in (4.10), quadratic damping is also considered and the linear and nonlinearrestoring moment is expressed in terms of the mean metacentric height and theamplitude of its change in waves, respectively. Similarly, the model for the surgespeed is enhanced by more realistic quadratic damping. The dynamics of the yawmotions is described by a rst-order Nomoto model. The following assumption ismade:

Assumption 16. For a maneuvering ship in a seaway, the surge and yaw motionsare approximated by the zero-frequency potential coecients while added massand damping in roll is approximated at the natural roll frequency ωφ > 0 ∈ R.Furthermore, the uid memory eect are neglected.

With the stated assumptions, the surge dynamics may be expressed as, seeFossen [41],

(m− a11 (0)) u−(xu + x|u|u|u|

)u = τ1e + τ1c (7.1)

where, neglecting the sway dynamics, u is the ship's forward speed andm−a11 (0) >0 the total and added mass in surge; xu < 0 ∈ R and x|u|u < 0 ∈ R are the linear(potential and viscous) and the nonlinear damping coecients in surge, respec-tively, where x|u|u is modelled using the ITTC surge resistance curve [41]

x|u|u = −1

2ρSw (1 + kf )

0.075

(log10Rn − 2)2 , Rn =

uLppνk

.

The water density is denoted by ρ > 0 ∈ R, the wetted surface of the hull bySw ≥ 0 ∈ R, and the form factor yielding a viscous correction by kf ∈ R. TheReynolds number Rn > 0 ∈ R depends on the length between perpendicularsLpp > 0 ∈ R and the kinematic viscosity of the uid νk > 0 ∈ R.

Motivated by the results presented in Galeazzi et al. [54], Neves and Rodríguez[110], Shin et al. [132] and based on the 1-DOF roll model of Chapter 4, the linearand nonlinear restoring moment for the surface vessel are approximated by

k1,4,6(η1,4,6

)≈

0ρg∇GMTφ+ kφ3φ3

0

(7.2)

98

7.2. Encounter Frequency Revisited

where g > 0 ∈ R is the acceleration of gravity, ∇ > 0 ∈ R the displaced watervolume, and GMT ∈ R the transverse metacentric height given by

GMT = GMm +GMa cos

(∫ t

0

ωe (τ) dτ

). (7.3)

Here, GMm ∈ R is the mean metacentric height, GMa ∈ R the amplitude of themetacentric height change in waves and ωe the encounter frequency. This modeltakes into account velocity changes since ωe is allowed to vary with time.

The roll dynamics is then

(Ix − a44 (ωφ)) φ− kpφ− k|p|p|φ|φ+ ρg∇GMmφ

+ ρg∇GMa cos

(∫ t

0

ωe (τ) dτ

)φ+ kφ3φ3 = 0 (7.4)

where φ is the roll angle, Ix − a44 (ωφ) > 0 the total moment of inertia and addedinertia in roll, kp < 0 ∈ R the linear damping, k|p|p < 0 ∈ R the nonlinear dampingcoecient in roll, and kφ3 > 0 ∈ R is the cubic coecient of the restoring momentin roll.

For simplicity, it is assumed that the ship is controlled by a single rudder suchthat

τ6c = −nδδr (7.5)

where δr ∈ R denotes the rudder angle. Then, the yaw subsystem can be approx-imated by a rst-order Nomoto model with time and gain constants tn and kn,respectively, see Nomoto et al. [114] or Fossen [41]:

tnr + r = knδr . (7.6)

The Nomoto constants tn ∈ R and kn > 0 ∈ R can be related to the hydrodynamicship coecients such as the acceleration derivatives and the velocity derivatives.These coecients may be approximated by considering the geometrical dimensionsof the ship, that is, the length between the perpendiculars and the draft of theship, as stated by Clarke, Gedling, and Hine [30].

The control inputs to the 3-DOF ship model (7.1), (7.4), and (7.6) are thethrust force in surge τ1c , as well as the rudder deection δr. Note, that the rollequation (7.4) is not directly actuated. The regulation of the roll motion has tobe achieved indirectly by exploiting the coupling of the roll to the surge and swaymotions.

7.2 Encounter Frequency Revisited

Here it is considered that the waves are not necessarily travelling along the x-axis.Consequently, the waves can be described by

ζ (t,pn) = ζ0 cos(ω0t− kw>pn + αζ

)(7.7)

where ζ (t,pn) is the sea surface elevation at a location pn at a time t. The vector pn

is expressed in the inertial frame. The amplitude of the sinusoidal wave is ζ0 ∈ R,

99


the modal wave frequency ω0, and the initial phase shift αζ ∈ R. The wave vectorkw ∈ R3 implicitly denes the wave number kw:

kw = kwew

where ew ∈ R3 is the propagation vector, satisfying ‖ew‖ = 1. The wave length fora planar wave is

λ =2π

‖kw‖=

2π

kw(7.8)

and the phase velocity is

cw =ω0

‖k‖ =λ

Tw(7.9)

with tw > 0 ∈ R the period. It is assumed that ζ0, ω0 and kw are constants forsimplicity.

To an observer at a xed location in the inertial reference frame, the frequencyat which the waves encounter the ship equals the modal wave frequency. Whenmoving with the ship at a nonzero velocity, however, this is not true. A moving shipcauses a shift in the peak frequency of the wave spectrum which can be accountedfor by introducing the encounter frequency.

From Figure 3.1 it is evident that the encounter angle expressed in the inertialframe is given by

βnw = βw + ψ .

Without loss of generality it is assumed that βnw ∈ R is constant; that is, the wavesare always coming from the same compass direction. The horizontal velocity ofthe ship ν1,2 is expressed in the body frame and can readily be expressed in theinertial frame:

ν1,2n = R1,2 (ψ)ν1,2 (7.10)

where R1,2 (ψ) ∈ SO (2) is the rotation matrix about ψ.The peak frequency shift of the wave spectrum is due to the Doppler shift. The

projection of the ship velocity ν1,2n on the wave vector kw is

vp = vpew = ‖ν1,2‖ cos (βnw − χ) ew (7.11)

where the course angle χ ∈ R is the sum of the heading angle ψ and the sideslipangle β ∈ R. The encounter frequency, that is the frequency of oscillation of thewaves as it appears to an observer on the ship, can then be calculated by consideringthe Doppler shift and by combining (7.10) and (7.11):

ωe = ω0 (1− vp/c)

= ω0

(1− kw

ω0‖R1,2 (ψ)ν1,2‖ cos (βnw − χ)

)

= ω0 − kw√u2 + v2 cos (βnw − ψ − β) . (7.12)

Under the assumption of deep water (hw ≥ λ/2), where hw > 0 ∈ R is the waterdepth, the dispersion relationship holds:

kw =ω2

0

g. (7.13)

100

7.3. Model Predictive Control

To decouple the surge model from the sway-yaw subsystem, it is assumed that theforward speed of the vessel is slowly time-varying only, which implies:

‖ν1,2‖ =√u2 + v2 ≈ u (7.14)

and that there is no ocean current present. From Figure 3.1 it is apparent that thesideslip angle β is

β = arcsin

(v

‖ν1,2‖

)≈ v

‖ν1,2‖ (7.15)

when β is small. Since the sway component of the ship velocity is neglected, thesideslip angle is disregarded as well.

Hence, the encounter frequency can be expressed by

ωe (u, ψ, ω0, βnw) = ω0 −

ω20

gu cos (βnw − ψ) . (7.16)

7.3 Model Predictive Control

MPC is a rather recent control methodology which is characterized by the usageof an explicit plant model to predict the output of the process. This prediction isconsequently used to nd an optimal control signal which minimizes a speciedobjective function. The MPC formulation allows to address the constraints of thestates and the input explicitly. MPC has been successfully applied to a wide varietyof control problems and the increasing availability of computing power has onlyadded to its popularity in both academia and industry, see, for example Camachoand Bordons [27]. In ship control, MPC formulation has been applied among othersto autopilot control design, roll stabilization, fault-tolerant control of a propulsionsystem, tracking, and control of ship n stabilizers, see Kerrigan and Maciejowski[83], Naeem, Sutton, and Ahmad [99], Perez [122] and Perez and Goodwin [121].The material in this section was published in Breu, Feng, and Fossen [14].

7.3.1 Model Predictive Control Applied to Ships in Parametric

Roll Resonance

The basic structure of an MPC setup is depicted in Figure 7.1. An MPC algorithmconsists generally of the following elements, see Camacho and Bordons [27]:

Prediction model

Objective function

Optimizer to obtain the control law

The strategy of MPCs can be summarized in a three step loop which is performedat each time instant, see for example Camacho and Bordons [27]:

1. The future process outputs are predicted for a prediction horizon N , depend-ing on the past inputs and outputs and on the future control signals.

2. An optimization problem is solved to determine the set of future controlsignals, minimizing the objective function.

101


Model +

Optimizer

Past inputsand outputs Predicted

outputs

−

Referencetrajectory

Future errors

Costfunction

Constraints

Futureinputs

Figure 7.1: Basic structure of a model predictive controller (Camacho and Bordons[27]).

3. The rst sample of the control signal is sent to the process and the steps 13are repeated at the next time instant.

In the context of the control of parametric roll resonance of ships, an approachfeaturing the MPC formulation to control the ship's forward speed and headingangle simultaneously in order to damp the roll motion is used. Furthermore, itis assumed that no control input aects the roll dynamics, that is τφ = 0. Theship's surge speed and heading angle can however be changed. This results in atime-varying encounter frequency and the transients due to heading and speedchanges must be taken into account. By changing the speed and heading activelyit is possible to violate the frequency condition for parametric roll resonance. Tothat matter, the MPC formulation is adapted to nd the optimal surge speed andheading angle to achieve a regulation of the roll motion while taking into accountconstraints on the inputs as well as on the states.

7.3.2 State-Space Model

The 3-DOF ship model (7.1), (7.4), and (7.6) is considered. The encounter fre-quency (7.16) couples the roll dynamics to the surge and yaw dynamics, respec-tively. For convenience in the notation the system dynamics are expressed in statespace form with the matrices as in Chapter 3.

x = f (x, τ ) (7.17)

y = g (x) (7.18)

102

7.3. Model Predictive Control

where x =[η1,4,6,ν1,4,6

]>and

f (x, τ ) =

[J1,4,6

(η1,4,6

)ν1,4,6

M−1[τ −C

(ν1,4,6

)ν1,4,6 −D

(ν1,4,6

)ν1,4,6 − k

(η1,4,6

)]]

g (x) = x

τ = τ e + τ c .

7.3.3 Objective Function

The objective function is crucial in the statement of the MPC problem with respectto the performance of the proposed control methodology. The main condition forparametric resonance to occur is the tuning of the parametric excitation frequencyto the natural frequency of the roll motion [100]. Specically, when the roll systemis parametrically excited at double the natural roll frequency, primary parametricresonance might develop:

ωe ≈ 2ωφ . (7.19)

Following that reasoning, the objective function for the MPC scheme is constructedin a way to avoid that critical frequency ratio. Additionally, a cost functional whichaccounts for the deviation of the initial encounter frequency, is included. Thiscorresponds to the idea of penalizing the deviation from the initial surge speedand from the initial heading angle, representing a very crude attempt to include amission specic objective. It should be noted that the chosen objective function,especially the mission related part, could of course be enhanced. However, in theabsence of true mission goals (since they dier from situation to situation), thechosen objective function may be seen more as a proof of concept than an advancedsetup. To that matter, the following objective function for the MPC statement isproposed:

J = w1J1 + w2J2 (7.20)

where wi, i ∈ 1, 2 are weights. The objective function J > 0 ∈ R consists of twocost functionals

J1 = c1e−c2(ωe−2ωφ)2 (7.21)

J2 = c3 (ωe − ωe,0)2. (7.22)

Here, ci > 0, i ∈ 1, 2, 3 are constants. Eq. (7.21) represents the penalty of theship not violating the frequency condition (7.19), and (7.22) penalizes the deviationof the ship from its nominal cruise condition given by ωe,0 ∈ R the encounterfrequency (7.16) with the nominal set points for the ship's surge speed u0 ∈ R andheading angle ψ0 ∈ R.

7.3.4 Obtaining the Control Law

To obtain the control signal, the objective function (7.20) has to be minimized ateach time instant. The minimization of (7.20) is subject to equality constraints

103


which, for a state space model as presented in Section 7.3.2 are the model con-straints given by (see Camacho and Bordons [27])

f (x, τ ) = 0 (7.23)

y − g (x) = 0 . (7.24)

Furthermore, the minimization of (7.20) is as well subject to inequality con-straints expressed as

y ≤ y (t+ j) ≤ y, ∀j = 1, N (7.25)

τ ≤ τ (t+ j) ≤ τ , ∀j = 1,M − 1 (7.26)

∆τ ≤ ∆τ (t+ j) ≤ ∆τ , ∀j = 1,M − 1 (7.27)

where N and M are the prediction horizon and the control horizon, respectively.The solution of the problem to minimize the objective function (7.20) with themodel constraints (7.23)(7.24) and the inequality constraints (7.25)(7.27) is nota trivial one. It generally involves solving a nonconvex, nonlinear problem.

The nonlinear model predictive control (NMPC) problem in the form of a gen-

eral nonlinear programming problem with w =[τ>,x>,y>

]>is, see Camacho and

Bordons [27],

minw

J (w) (7.28)

subject to: c (w) = 0, h (w) ≤ 0 .

Here, c corresponds to the equality constraints (7.23)(7.24) and h to the inequalityconstraints (7.25)(7.27).

The optimization (7.28) is performed by using the TOMLAB Optimization En-vironment (TOMLAB/NPSOL)1, see Holmström, Göran, and Edvall [68].

7.3.5 Simulation Results

The ship is simulated by applying the nonlinear MPC scheme to the system. Theinitial values for the simulations are chosen such that the ship is experiencingparametrically excited rolling. The nominal cruise condition is chosen as u0 =7.5 m/s and ψ0 = 0. It is assumed that the ship is initially in head sea condition,that is βnw = π. Table 7.1 lists the model parameters. In the simulation results, thecontrolled variables are denoted by the subscript c.

The MPC control, initially turned o, is activated when the roll amplitudeexceeds φ = 3 for the rst time. In Figure 7.2, the roll angle and the frequencyratio is shown for the controlled and the uncontrolled scenario. Figure 7.2(a) depictsthat the ship is experiencing large roll angles due to parametric roll resonance inthe uncontrolled scenario. However, the reduction of the frequency ratio reducesthe roll angle quickly when the MPC is active.

The ship's forward speed and the thrust in surge is shown in Figure 7.3 and theship's heading angle and the rudder deection are depicted in Figure 7.4. Again,the controlled and the uncontrolled scenario is shown.

104

7.4. Extremum Seeking Control

Table 7.1: Model parameters, adopted from Holden et al. [62].


Moment of inertia, roll Ix 1.4014× 1010 kg m2

Added moment of inertia, roll a44 −2.17× 109 kg m2

Nonlinear damping, roll k|p|p −2.99× 108 kg m2

Linear damping, roll kp −3.20× 108 kg m2/sWater density ρ 1025 kg/m3

Gravitational acceleration g 9.81 m/s2

Water displacement ∇ 76468 m3

Mean meta-centric height GMm 1.91 mAmplitude of meta-centric height change GMa 0.84 mRestoring coecient kφ3 2.9740× 109 kg m2/s2

Mass m 7.6654× 107 kgAdded mass, surge a11 −7.746× 106 kgLinear damping, surge xu 0 kg/sWetted surface S 11800 m2

Form factor kf 0.1 −Ship length Lpp 281 mKinematic viscosity νk 1.519× 10−6 m2/s

Nomoto time constant tn −160.15 sNomoto gain constant kn −0.1986 1/s

Natural roll frequency ωφ 0.3012 rad/sModal wave frequency ω0 0.4353 rad/s

Figure 7.5 shows the cost dened by the proposed objective function. Finally,in Figure 7.6, the roll angles is shown, when the MPC is activated at dierent timeinstants, corresponding to roll angles of 3, 5 and 10, respectively.

7.4 Extremum Seeking Control

ES control is a real-time optimization method, popular in both research and in-dustry. It is characterized by the online tuning of the a priori unknown set pointof a system to achieve an optimal output, with respect to an objective functionalfor example. ES is not model basedit is applicable also when the model is notperfectly known. In the ES methodology, a perturbation signal is added to thesystem to nd an estimate of the gradient of the objective signal. In Ariyur andKrsti¢ [2], a thorough introduction to ES control, including many applications invarious research areas, is presented. A more recent treatment of the ES control canbe found in Zhang and Ordóñez [157]. The material in section was published inBreu, Feng, and Fossen [14].

1See http://www.tomopt.com for information about the TOMLAB Optimization Environ-ment.

105

http://www.tomopt.com


time (s)

rollangle

()

φφc

0 200 400 600 800 1000-20

0

20

(a) Roll angle.

time (s)

frequency

ratio(−

) ωe/ωφ

ωe,c/ωφ

0 200 400 600 800 1000

1.8

1.85

1.9


Figure 7.2: MPC Roll angle 7.2(a), frequency ratio 7.2(b).

7.4.1 Extremum Seeking Applied to Ships in Parametric Roll

Resonance

In this section, the ES method is adapted to the regulation of parametrically excitedroll motions in ship as depicted in Figure 7.7. The proposed, overall scheme consistsof the shipthe surge and the yaw motions are coupled to the roll motion by theencounter frequency (7.16)the control block and a dynamic feedback loop (ESloop).

The output J of the objective function has an extremum, that is a minimum ora maximum, at ω∗e,d ∈ R. The ES loop adds a slow perturbation to the best currentestimate of ωe,d ∈ R in order to iteratively and online tune the parameter ωe,d toits optimal value ω∗e,d.

By assuming the perturbation signal to be suciently slow compared to theopen-loop dynamics, the system can be viewed as a static map and its dynamicscan be neglected for the ES loop. The high-pass lter s/ (s+ ωh) with the cutofrequency ωh > 0 ∈ R serves to eliminate the oset of the cost signal J andthe second perturbation creates a sinusoidal response of J . Adding a sinusoidalperturbation signal to the best estimate of ωe,d causes the two sinusoids to be inphase or out of phase depending on whether the best estimate is smaller or biggerthan its optimal value ω∗e,d. Whereas the low-pass lter ωl/ (s+ ωl) with the cuto

106


time (s)

forw

ard

speed(m

/s) u

uc

0 200 400 600 800 1000

5

6

7

(a) Forward speed.

time (s)

thrust

(N)

τ1

τ1c

0 200 400 600 800 10000

2

4

6 ×105

(b) Thrust.

Figure 7.3: MPC Surge speed 7.3(a), thrust τ1c 7.3(b).

frequency ωl > 0 ∈ R is used to extract the oset caused by multiplying the twosinusoids, the integrator in the ES loop gives the approximate gradient update law.

The proposed ES control schemes requires three time scales in the overall sys-tem. Since it is assumed that the map from the reference to the output of theobjective function is a static map, the time constant of the plant needs to be thefastest. Furthermore the perturbation signal must be suciently slow compared tothe plant, or it would not be fed through the plant properly. The lters give an es-timate of the gradient update law, implying that their time constants are requiredto be slower than those of both the plant and the perturbation signal.

The encounter frequency ωe (7.16) depends among others on the ship's forwardspeed and heading angle which are controllable. The best estimate of the optimalencounter frequency is therefore mapped to the desired surge speed ud and thedesired heading angle ψd by a nonlinear CA strategy as depicted by the controlblock in Figure 7.7. Speed and heading controllers are then used to compute therequired thrust in surge τ1c and the rudder deection δr.

It is noteworthy that the optimal set point of the encounter frequencyand asa matter of fact, the ship's speed and heading angleis a priori not known. Therelies the power of the ES control which iteratively tunes the encounter frequency, asthe parameter of the feedback loop, to its optimal value which minimizes a dened

107


time (s)

headingangle

() ψ

ψc

0 200 400 600 800 1000

0

10

20

30

(a) Heading angle.

time (s)

rudder

angle

() δr

δr,c

0 200 400 600 800 1000-10

-5

0

5

10

(b) Rudder deection.

Figure 7.4: MPC Heading angle 7.4(a), rudder deection 7.4(b).

time (s)

cost

(−)

0 200 400 600 800 10000

1

2

Figure 7.5: MPC Cost.

108


time (s)

rollangle

()

φ3

φ5

φ10

0 200 400 600 800 1000-20

0

20

Figure 7.6: MPC Roll angle: Comparison when the controller is activated at 3,5, and 10, respectively.

Extremum seeking loop

Ship

ControlSpeed

Controller

SurgeSystem

ControlAllocation

FrequencyCoupling

RollSystem

HeadingController

YawSystem

+s

s+ωh

ks

ωl

s+ωl×

Objective

Function

ud τ1c u

ψd δrψ

a sin (ωt)a sin (ωt)

ωe,d

ωe

J

φ

Figure 7.7: Extremum seeking control applied to ships in parametric roll resonance.

109


objective functional. Hence, the vital role of the choice of the objective function isapparent in the performance of the ES control applied to parametric roll resonance.

Treating the encounter frequency as the sole parameter in the proposed ES con-trol scheme seems advantageous compared to the formulation as a multiparameterES method. In particular, CA allows to take into account restrictions on the ship'sspeed and heading angle as well as on their variation rate.

7.4.2 Objective Function

The objective function is one of the key factors with respect to the performance ofthe proposed ES control method applied to ships in parametric roll resonance, asdepicted in Figure 7.7. Its choice determines the ability to regulate the roll motionas well as it accounts for mission dependent restrictions.

It is well known that certain ships are prone to experience parametric rollresonance when the encounter frequency is close to double the natural roll frequencyof the ship.

The objection function is constructed as the weighted sum of two cost func-tionals, accounting for the frequency condition (7.19) and the derivation from thenominal cruise conditions, respectively:

J = w1J1 + w2J2 (7.29)

where w1 and w2 are the weights. The two cost functionals are expressed by

J1 = c1e−c2(ωe−2ωφ)2 (7.30)

J2 = c3 (ωe − ωe,0)2 (7.31)

where ci > 0, i ∈ 1, 2, 3 are constants. Equation (7.30) represents the penaltyof the ship not violating the frequency condition (7.19). Equation (7.31), on theother hand, penalizes the deviation of the ship from its nominal cruise conditionexpressed by the nominal encounter frequency ωe,0, that is the encounter frequency(7.16) with the nominal set points for the ship's surge speed u0 and heading angleψ0. By the choice of the constant parameters ci, i ∈ 1, 2, 3 in (7.30)(7.31), theshape of the cost functionals can be adjusted.

It is apparent from the denition of the objective function (7.29) that in orderto avoid parametric roll resonance the objective has to be minimized. Thus, theES loop is designed such that its parameter ωe,d is iteratively tuned to the optimalvalue ω∗e,d, resulting in a minimum of the objective function.

7.4.3 Control Allocation

The CA block depicted in Figure 7.7 maps the parameter of the ES loop ωe,d the desired encounter frequency to the desired trajectory of the control variables,that is the ship's desired surge speed ud and heading angle ψd. Revisiting (7.16),the desired encounter frequency is approximated by a rst-order Taylor expansion,

110


taking into account small variations of the ship's forward speed and heading angle:

ωe,d (u+ ∆u, ψ + ∆ψ, ·) = ω0 −ω2

0

gcos (βnw − ψ)u− ω2

0

gcos (βnw − ψ) ∆u

− ω20

gsin (βnw − ψ)u∆ψ . (7.32)

Here, it is assumed that the desired encounter frequency can be achieved by adeviation of ∆u ∈ R and ∆ψ ∈ R from the ship's forward speed and heading angle,respectively. Equation (7.32) suggest that the virtual control input τv,ωe ∈ R canbe chosen as

τv,ωe = − g

ω20

[ωe,d (u+ ∆u, ψ + ∆ψ, ·)−

(ω0 −

ω20

gcos (βnw − ψ)u

)]

= − g

ω20

[ωe,d (u+ ∆u, ψ + ∆ψ, ·)− ωe (u, ψ, ·)] . (7.33)

The relation between the virtual controller (7.33) and the variations in surgespeed and heading angle can be expressed by the constrained linear mapping

τv,ωe = B>e (u, ψ, βnw) ζv (7.34)

ζv,min ≤ ζv ≤ ζv,max (7.35)

where the control eectiveness matrix Be (u, ψ, βnw) and the variation vector ζv aregiven by

Be (u, ψ, βnw) =

[cos (βnw − ψ)

sin (βnw − ψ)u

], ζv =

[∆u∆ψ

]. (7.36)

The constraints are expressed in (7.35) where ζv,min and ζv,max denote the lowerand upper bounds on ζv ∈ R2×1, respectively. The desired ship's surge speed andheading angle are then merely

ud = u+ ∆u (7.37)

ψd = ψ + ∆ψ . (7.38)

According to Härkegård [57], Wang, Yi, and Fan [154], the CA problem (7.34)(7.35) can be split up into a two-step sequential least-squares problem to nd thevariation of the ship's forward speed and heading angle:

Sf = arg minζv,min≤ζv≤ζv,max

‖W τv

(B> (u, ψ, βnw) ζv − τv,ωe

)‖ (7.39)

ζv,opt = arg minζv∈Sf

‖W ζ

(ζv − ζv,d

)‖ (7.40)

where W τv and W ζ are weight matrices. First, the set of feasible solutions Sfthat minimize B> (u, ψ, βnw) ζv − τv is computed. Then, the best solution thesolution which minimizes W ζ

(ζv − ζv,d

) is determined. ζv,d is the vector of

desired variations in ship's surge speed and heading angle and is presumably null.The sequential least-squares problem (7.39)(7.40) is solved in Matlab using

the Quadratic Programming Control Allocation Toolbox (QCAT) (see Härkegård[58]).

111


7.4.4 Speed and Heading Controllers

The speed and heading controllers determine the appropriate thrust in surge andthe rudder deection from the desired surge speed and heading angle, respectively,see Figure 7.7.

By assumption, the inner control loop, consisting of the ship and the controlblock in Figure 7.7, is designed such that its dynamics can be neglected for theES loop. Thus, the inner control loop needs to be considerably faster than theoverall closed-loop system, yielding that the controllers are required to be fast incomparison to the perturbation signal and the lters of the ES loop.

Speed Controller

The surge dynamics are given by (7.1). Assuming, that the mass and the dampingterms are perfectly known, the speed controller can be designed by using feedbacklinearization (FL):

τ1c = (m+ a11 (0)) τv,u +(−xu − x|u|u|u|

)u . (7.41)

By taking the virtual control input τv,u ∈ R as an ordinary proportional controller,the closed-loop surge dynamics become

u = τv,u = −ku,p (u− ud) , ku,p > 0 (7.42)

where ku,p > 0 ∈ R is the controller gain, chosen such that the error dynamics areglobally exponentially stable (GES); see Fossen [41] or Khalil [84].

Heading Controller

The yaw dynamics is represented by the rst-order Nomoto model (7.6). To designthe heading controller, is is assumed that ψ ≈ r and that the rudder deection δris the control input:

δr = −kψ,p (ψ − ψd)− kψ,d(ψ − ψd

), kψ,p, kψ,d > 0 . (7.43)

The desired yaw rate ψd is generated by using a third-order reference model. Theproportional and derivative gains, kψ,p > 0 ∈ R and kψ,d > 0 ∈ R, in (7.43) aredetermined such that the closed-loop dynamics

tnψ + (1 + knkψ,d) ψ + knkψ,pψ = knkψ,pψd + knkψ,dψd (7.44)

are GES (Fossen [41] or Khalil [84]).

7.4.5 Stability Considerations

It can be proven that the ES parameter converges to a neighborhood of its optimalvalue and that the ES algorithm is exponentially stable (see Krsti¢ and Wang

112


[87] and Breu and Fossen [15]). Consider the single-input, single-output nonlinearsystem:

x = f (x, u) (7.45)

y = h (x) (7.46)

where x ∈ Rn is the state vector, u ∈ R the input, y ∈ R the output; f : Rn×R→Rn and h : Rn → R are smooth. The control law u = α (x, θ) is parametrized by θ,and assumed to be smooth. The closed-loop system corresponding to (7.45)(7.46)then becomes

x = f (x, α (x, θ)) (7.47)

and it has equilibria parametrized by θ. For the stability analysis, the followingassumptions are made (see Krsti¢ and Wang [87]):

Assumption 17. There exists a smooth function l : R→ Rn such that:

f (x, α (x, θ)) = 0 if and only if x = l (θ) . (7.48)

Assumption 18. The equilibrium x = l (θ) of (7.47) is locally exponentially stable(LES) with decay and overshoot constants uniform in θ for each θ ∈ R.

Assumption 19. There exists θ∗ ∈ R such that:

(h l)′ (θ∗) = 0 (7.49)

(h l)′′ (θ∗) > 0 . (7.50)

Assumptions 1718 guarantee the robustness of the control law with respect toθ, i.e. any equilibria produced by θ can be stabilized by the control law. Assump-tion 19 implies that the output equilibrium map has a minimum when θ = θ∗.

It was proven by averaging for a static system and by the singular perturbationmethod for a dynamic system that (7.47) converges to a unique, exponentiallystable, periodic solution in a neighborhood of the origin (Krsti¢ and Wang [87]).The perturbation signal and the lters in the ES loop determine the size of thisneighborhood.

Due to the three dierent time scales in the proposed ES control (see Sec-tion 7.4.1) the plant the surge and yaw subsystems can be viewed as a staticmap. The ES parameter ωe, determined from the ship's forward speed and headingangle, parametrizes the equilibria of the plant. The speed and heading controllersensure local exponential stability of the equilibria which may be produced by the ESparameter ωe, see Section 7.4.4, and the objective function dened in Section 7.4.2fullls locally Assumption 19. Thus, the parameter ωe converges to a neighborhoodof its optimal value ω∗e .


The ship is simulated by applying the ES control methodology to the system.The initial values for the simulations are chosen such that the ship is experiencingparametrically excited rolling. The nominal cruise condition is chosen as u0 =

113


time (s)

rollangle

()

φφc

0 200 400 600 800 1000 1200-20

-10

0

10

20

(a) Roll angle.

time (s)

frequency

ratio(−

) ωe/ωφ

ωe,c/ωφ

ωe,d/ωφ

0 200 400 600 800 1000 1200

1.75

1.8

1.85

1.9


Figure 7.8: Extremum seeking Roll angle 7.8(a), frequency ratio 7.8(b).

7.5 m/s and ψ0 = 0. It is assumed that the ship is initially in head sea condition,that is βnw = π. Table 7.1 lists the model parameters. In the simulation results, thecontrolled variables are denoted by the subscript c.

The ES control is initially deactivated but is turned on the time instant whenthe roll amplitude exceeds φ = 3 for the rst time. The roll angle and frequencyratio with and without the ES control is shown in Figure 7.8. Note that, in the un-controlled scenario, the ship is experiencing parametric roll resonance with high rollamplitudes; see Figure 7.8(a). It is furthermore apparent that, when the ES controlis activated, the ship is driven out of the frequency ratio relevant for parametricrolling and consequently the roll motion is reduced signicantly.

The frequency ratios ωe,d/ωφ and ωe,c/ωφ in Figure 7.8(b) denote the desiredfrequency ratio as output of the ES feedback loop and the actual, controlled, fre-quency ratio, thus indicating the ability of the controllers to track the desiredencounter frequency.

Figure 7.9 shows the ship's surge speed and the thrust in surge, whereas Fig-ure 7.10 depicts the ship's heading angle and the rudder deection for both theuncontrolled and the controlled scenario.

Both the ship's surge speed and heading angle follow the reference trajectory,determined by the CA block. Due to the perturbation signal in the ES control, the

114

7.5. L1 Adaptive and Extremum Seeking Control

time (s)

forw

ard

speed(m

/s) u

uc

ud

0 200 400 600 800 1000 12005

6

7

8

(a) Surge speed.

time (s)

thrust

(N)

τ1

τ1c

0 200 400 600 800 1000 12000

0.5

1

1.5

2 ×106

(b) Thrust.

Figure 7.9: Extremum seeking Surge speed 7.9(a), thrust τ1c 7.9(b).

ship's surge speed and heading angle show an expected oscillatory behavior.The cost as dened in the objective function is shown in Figure 7.11 and Fig-

ure 7.12 shows a comparison of the roll angle, when the ES control is activated atdierent roll angles, that is at 3, 5 and 10, respectively.

7.5 L1 Adaptive and Extremum Seeking Control

In Section 7.4, the proposed control approach does not address robustness con-siderations. In the objective function of the ES feedback loop it is assumed thatthe encounter frequency and its range where parametric roll is happening, are apriori known. Although the encounter frequency may be observed, the design ofa (possibly nonlinear) observer is not a trivial task. Furthermore, the speed andheading controllers in Section 7.4 rely on the knowledge of the surge and yaw modelcoecients.

Here a robustication of the ES control approach of Section 7.4 is considered.To that matter, the ES is adapted for limit cycle minimization, see Ariyur andKrsti¢ [2]. The objective function from the previous section is replaced by a limitcycle detection yielding an ES scheme which is independent of the knowledge of the

115


time (s)

headingangle

() ψ

ψc

ψd

0 200 400 600 800 1000 1200

-30

-20

-10

0

(a) Heading angle.

time (s)

rudder

angle

() δr

δr,c

0 200 400 600 800 1000 1200

-20

-10

0

10

20


Figure 7.10: Extremum seeking Heading angle 7.10(a), rudder deection 7.10(b).

time (s)

cost

(−)

0 200 400 600 800 1000 12000

0.2

0.4

0.6

Figure 7.11: Extremum seeking Cost.

116


time (s)

rollangle

()

φ3

φ5

φ10

0 200 400 600 800 1000 1200-20

0

20

Figure 7.12: Extremum seeking Roll angle: Comparison when the controller isactivated at 3, 5, and 10, respectively.

encounter frequency. Furthermore, the speed and heading controllers are designedby the recently introduced L1 adaptive control methodology, see Hovakimyan andCao [70]. The proposed ES control for roll parametric resonance is robust withrespect to model uncertainties, model mismatch, and disturbances.

In this section, the 3-DOF ship model consisting of the dynamics in surge (7.1),roll (7.4), and yaw (7.6) are considered. Additionally, the actuator dynamics isaccounted for. The surge and yaw systems are coupled to the roll system by theencounter frequency, see (7.16).

Consider the surge dynamics given as in Section 7.1, that is by (7.1). Theeective thrust is τ1c = (1− tp)Tp, where Tp > 0 ∈ R is the propeller thrust andtp > 0 ∈ R the thrust deduction number. The propeller thrust is dependent on thepropeller shaft speed np > 0 ∈ R, measured in revolution per second (rps), andthe advance speed at the propeller va = (1− wf )u with the wake fraction numberwf > 0 ∈ R, see Blanke, Lindegaard, and Fossen [11], and thus the eective thrustmay be expressed as

τ1c := (1− tp)Tp = (1− tp)(Tp|np|np |np|np + Tp|np|va |np|va

). (7.51)

The coecients in (7.51) can be approximated by linearizing the nondimensionalpropeller thrust coecient [11], resulting in

Tp|np|np := ρD4pα1 > 0 Tp|np|va := ρD3

pα2 < 0 . (7.52)

Here, ρ is the water density, Dp > 0 ∈ R the propeller diameter, and α1,2 areconstants. The propeller dynamics is modelled as a rst-order system with thetime constant Tp,n > 0 ∈ R and the desired propeller shaft speed np,d > 0 ∈ R np,das input:

Tp,nnp + np = np,d . (7.53)

Roll System

The roll dynamics is given by (7.4) in Section 7.1. Note that there is no directactuation of the roll dynamics.

117


Yaw System

Consider the yaw dynamics given by the rst-order Nomoto model (7.6). The rud-der dynamics is accounted for by the rst-order system with the time constanttδ > 0 ∈ R and the desired rudder deection δr,d ∈ R:

tδ δr + δr = δr,d . (7.54)

Encounter Frequency

Neglecting the sway dynamics, the encounter frequency can be written as

ωe (u, ψ, ω0, βnw) := ω0 −

ω20

gu cos (βnw − ψ) (7.55)

with the heading angle ψ, the modal wave frequency ω0, and the encounter angleβnw expressed in the inertial reference frame, see also the previous chapters.

7.5.1 Adaptive Control Design

The speed and heading controllers are designed in the framework of the L1 adap-tive control theory which guarantees robustness in the presence of fast adaptationby decoupling the adaptation and the robustness, see Hovakimyan and Cao [70].To that matter, the low-pass ltered parametric estimate is used in the control law.Furthermore, L1 adaptive control yields guaranteed, uniform, and decoupled per-formance bounds on the model states and on the control signals. For the controllerdesign, the following assumptions are made [70]:

1. The unknown parameters are uniformly bounded.

2. The variation rate of the unknown parameters is uniformly bounded.

3. The uncertain system input gain is partially known.

4. The unmodelled actuator dynamics are partially known.

7.5.2 L1 Adaptive Speed Controller

The L1 adaptive speed controller generates the desired eective thrust in surgeτ1c,d and, assuming both positive forward speed and propeller shaft speed, thecorresponding desired propeller shaft speed np,d can be computed from (7.51).This is then the input to the actuator dynamics (7.53) to obtain the actual eectivethrust which enters (7.1).

The surge dynamics (7.1) can readily be formulated in the L1 adaptive con-trol framework [70, Sections 2.42.5] by dening the two known constants asp ,xu/ (m− a11 (0)) and bsp , 1/ (m− a11 (0)) and the unknown parameter θsp ,x|u|u|u| with the partial derivatives of θspu being semiglobally uniformly bounded;the unknown, uniformly bounded, and semiglobally uniformly rate bounded distur-bance is dened as σsp , τ

1e . Additionally, consider the control input τ1c as the

sum of an adaptive component τ1c,a and a component τ1c,m , − (asp − am,sp) /bspu

118


yielding the desired closed-loop dynamics specied by am,sp < 0 ∈ R which isHurwitz. The resulting model is then

u = am,spu+ bsp

(τ1c,a + θspu+ σsp

), u (0) = u0 (7.56)

ysp = u . (7.57)

State Predictor

The state predictor for (7.56) and (7.57) is, see Hovakimyan and Cao [70],

˙u = am,spu+ bsp

(τ1c,a + θsp‖u‖∞ + σsp

)(7.58)

ysp = u , u (0) = u0 . (7.59)

Adaptation Laws

The adaptation laws for the unknown adaptive parameters in (7.56) and (7.57) are,see Hovakimyan and Cao [70],

˙θsp

= Γsp

Proj(θsp,−uP

spbsp‖u‖∞

), θ

sp(0) = θ

sp,0(7.60)

˙σsp = ΓspProj(σsp ,−uPspbsp

), σsp (0) = σsp,0 (7.61)

where u := u−u is the estimation error, Γsp > 0 ∈ R the adaptation gain, and Psp =P>sp > 0 solves the algebraic Lyapunov equation a>m,spPsp + Pspam,sp = −Q, forQ = Q> > 0 arbitrary. The projection operator Proj (·) ensures the boundednessof the adaptive parameters.

Control Law

The control law for the L1 adaptive speed controller is then, see Hovakimyan andCao [70],

τ1c,a (s) = −kspDsp (s) (ηsp (s)− kg,spud (s)) (7.62)

where ud (s) and ηsp (s) are the Laplace transforms of the reference signal and

ηsp := τ1c,a + θsp‖u‖∞ + σsp, respectively; kg,sp , −1/

(a−1m,spbsp

)ensures tracking

of the reference and ksp is the feedback gain. The strictly proper transfer functionDsp (s) , (s+ ω0,sp) /

(s2 + ωn,sps

)is chosen to account for the actuator dynamics

and it leads to the low-pass lter

Csp (s) ,kspDsp (s)

1 + kspDsp (s), Csp (0) = 1 . (7.63)

Design Considerations

The design parameters of the L1 adaptive speed controller are the frequencies ofthe transfer function Dsp leading to the desired characteristics of the low-passlter (7.63), the desired closed-loop dynamics specied by am,sp in (7.56), andthe adaptation rate Γsp in (7.60) and (7.61). The following values were chosen:

119


am,sp = −0.05, ω0,sp = 0.0167 s−1, ωn,sp = 0.05 s−1, Γsp = 108. To guarantee theperformance and the robustness of the controller, ksp ∈ R in (7.62) is chosen suchthat the L1-norm condition holds, see Hovakimyan and Cao [70]:

‖Gsp (s)‖L1<ρr − ‖ξsp (s)‖L1‖ud‖L∞ − ρin

Lρrρr +B(7.64)

where

ξsp (s) , Hsp (s)Csp (s) kg,sp

Hsp (s) , (sI− am,sp)−1bsp

Gsp (s) , Hsp (s) (1− Csp (s))

and the other variables as in Hovakimyan and Cao [70]. Numerical evaluation of(7.64) suggested ksp = 3. The model parameters are as in Section 7.3 and addi-tionally: tp = 0.1, wf = 0.1, Dp = 8 m, α1 = 0.4243, α2 = −0.9435, Tp,n = 20 s.

Numerical Performance Analysis

Figure 7.13 shows the performance of the L1 adaptive speed controller. The speedreference ud is the superposition of a sinusoid with amplitude 0.25 m/s, frequency0.02 s−1, and oset 7 m/s and a ramp starting at 0 m/s at 472 s with a slope of0.01 m/s, saturated at 2 m/s. The unknown parameters are initially zero. The sam-pling frequency is 20 Hz. The initial speed decrease is due to the reference osetand the unknown parameters.

Note that although the nonlinear damping is unknown, the L1 adaptive speedcontroller is able to track the reference signal well.

7.5.3 L1 Adaptive Heading ControllerThe L1 adaptive heading controller generates the desired rudder deection δr,d. Theactual rudder deection which enters the yaw dynamics is obtained by consideringthe actuator dynamics (7.54). Assuming that ψ ≈ r and by dening the statesx := [ψ, r]

>, the yaw dynamics can be rewritten as

x = Am,hdx+Bhd

(ωhdδr + θ>hdx+ σhd

)(7.65)

yhd = C>hdx , x (0) = x0 . (7.66)

The matrix Am,hd is Hurwitz all its eigenvalues have strictly negative real parts and it represents the desired closed-loop dynamics; Bhd ∈ R2×1 and Chd ∈ R2×1

are known, constant vectors, ωhd ∈ R is the unknown control eectiveness withknown bounds, θhd ∈ R2×1 and σhd ∈ R are the unknown, uniformly bounded andrate bounded constant parameter vector and disturbance, respectively.

Am,hd :=

[0 1am1

am2

], Bhd :=

[01

], ωhd :=

kntn

θhd :=

[−am1

−am2− 1/tn

], Chd :=

[10

].

120


time (s)

forw

ard

speed(m

/s)

uuud

0 200 400 600 800 10000

5

10

(a) Forward speed.

time (s)

effectivethrust

(N)

τ1c

τ1c,d

0 200 400 600 800 10000

1

2

3 ×106

(b) Eective thrust.

time (s)

propellerspeed(rps)

np

np,d

0 200 400 600 800 10001.5

2

2.5

3

(c) Propeller shaft speed.

Figure 7.13: Tracking performance, speed controller.

121


The design of the L1 adaptive heading controller follows the material presented inHovakimyan and Cao [70, Section 2.2].

State Predictor

The state predictor of (7.65) and (7.66) is given by, see Hovakimyan and Cao [70],

˙x = Am,hdx+Bhd

(ωhdδr + θ

>hdx+ σhd

)(7.67)

yhd = C>hdx , x (0) = x0 . (7.68)

Adaptation Laws

The adaptation laws for the unknown adaptive parameters in (7.65) and (7.66) are,see Hovakimyan and Cao [70],

˙θ

hd= Γ

hdProj

(θ

hd,−x>P

hdB

hdx), θ

hd(0) = θ

hd,0(7.69)

˙σhd

= Γhd

Proj(σ

hd,−x>P

hdB

hd

), σ

hd(0) = σ

hd,0(7.70)

˙ωhd

= Γhd

Proj(ω

hd,−x>P

hdB

hdδr

), ω

hd(0) = ω

hd,0(7.71)

where the estimation error is x := x− x, Γhd > 0 ∈ R is the adaptation rate andP hd = P>hd > 0 solves the algebraic Lyapunov equation A>m,hdP hd +P hdAm,hd =

−Q, for an arbitrary Q = Q> > 0. The boundedness of the adaptive parametersis ensured by the projection operator Proj (·).

Control Law

The control law for the L1 adaptive heading controller is, see Hovakimyan and Cao[70],

δr (s) = −khdDhd (s) (ηhd (s)− kg,hdψd (s)) . (7.72)

Here, ψd (s) is the Laplace transform of the reference signal, ηhd the Laplace trans-

form of ηhd := ωhdδr + θ>hdx+ σhd; kg,hd ∈ R ensures tracking of the reference and

is kg,hd , −1/(C>hdA

−1m,hdBhd

), and khd ∈ R is the feedback gain. The strictly

proper transfer function Dhd (s) is chosen with respect to the actuator dynamics,that is Dhd (s) := (s+ ω0,hd) /

(s2 + ωn,hds

), which leads to the low-pass lter

Chd,lp (s) :=ωhdkhdDhd (s)

1 + ωhdkhdDhd (s), Chd,lp (0) = 1 . (7.73)

Design Considerations

The design parameters are chosen as am1= −0.01 and am2

= −0.14, yielding thedesired closed-loop performance. The L1-norm condition is, see Hovakimyan andCao [70]

‖Ghd (s)‖L1Lhd < 1 (7.74)

122


where

Lhd := maxθhd∈Θhd

‖θhd‖1

Hhd (s) := (sI−Am,hd)−1Bhd

Ghd (s) := Hhd (s) (1− Chd,lp (s)) .

By evaluating (7.74) numerically, khd = 2500 was chosen. Furthermore, Γhd = 100,ω0,hd = 0.0667 s−1, ωn,hd = 0.2 s−1. The model parameters are as in the previouschapter, additionally: tδ = 5 s.

Numerical Performance Analysis

In Figure 7.14, the heading angle ψ, its estimate ψ, and the rudder deection areshown when the desired heading angle ψd is the superposition of a sinusoidal signalwith amplitude 5 and frequency 0.05 s−1 and a ramp function. The ramp functionis zero before 200 s and has a slope of 0.1 s−1 and saturation at 10, afterwards.The unknown parameters ω and θ are initially zero, and the yaw system is exposedto a sinusoidal disturbance σ (t) with amplitude 0.0005 and frequency 0.1 s−1. Thesampling frequency for the simulation is 100 Hz.

time (s)

headingangle

()

ψ

ψψd

0 100 200 300 400 500-10

0

10

20

(a) Heading angle.

time (s)

rudder

deflection()

δrδr,d

0 100 200 300 400 500-40

-20

0

20

40


Figure 7.14: Tracking performance, heading controller.

123


Note that the L1 adaptive heading controller is able to track the reference signalwell, even in presence of the sinusoidal disturbance. This, although the controleectiveness and some model parameters are unknown.

7.5.4 Extremum Seeking Control with Limit Cycle

Minimization

ES control as a non-model-based, adaptive control method has been shown toeectively stabilize the roll motion of ships in roll parametric resonance, as seen inSection 7.4. To that matter, the encounter frequency is online and iteratively tunedto its optimal value yielding a violation of one of the empirical conditions for theonset of roll parametric resonance. Here, a modication of the control scheme as inSection 7.4 is considered in order to take into account robustness considerations.

The roll system described by the model in Section 7.1 has both stable and unsta-ble limit cycles, dependent on the value of the encounter frequency, see Chapter 6.This suggests the use of the ES control methodology with limit cycle minimization,as presented in Ariyur and Krsti¢ [2], to reduce the size of the limit cycle to a min-imum. Thus, the ES scheme as in the previous section is modied to incorporatethe ability for minimization of the limit cycle.

Figure 7.15 depicts the suggested ES control scheme. The overall system consistsof an inner control loop the ship dynamics and the speed and heading controlsystem and an outer control loop with the limit cycle amplitude detector and theconventional ES loop consisting of the perturbation signal and the lters to obtainthe approximate gradient update law. Refer to Ariyur and Krsti¢ [2] for a detailedintroduction to ES control.

Extremum seeking loop

+ ks

ωl

s+ωl× −s

s+ωh

Control&

Ship

Limit CycleAmplitudeDetector

a sin (ωt)a sin (ωt)

ωe,d φ A

Figure 7.15: ES with limit cycle minimization, adapted from Ariyur and Krsti¢ [2].

The inner control loop is shown in Figures 7.16 and 7.17. The ship dynamics isdescribed by the 3-DOF model surge, roll, and yaw as presented in Section 6.1.The roll system is coupled to the surge and the yaw system by the frequencycoupling.

The CA from the desired encounter frequency ωe,d, the output of the ES loop,to the desired forward speed ud and heading angle ψd is formulated as a sequential

124


Ship

Surge

SystemControl

System

Frequency

Coupling

Roll

SystemYaw

System

np,d u

δr,dψ

ωe φωe,d

Figure 7.16: Control system and the ship.

least-squares problem, as in Section 7.4. The L1 adaptive speed and heading con-trollers from Sections 7.5.2 and 7.5.3 track the desired references for forward speedand heading angle, see Figure 7.17.

Speed

ControllerControl

AllocationHeading

Controller

ud

ψd

ωe,d

np,d

δr,d

Figure 7.17: Control system: control allocation and controllers.

Figure 7.18 shows the limit cycle amplitude detector block which is added tothe overall scheme as depicted in Figure 7.15. The serial connection of a high-passlter, a squaring function, a low-pass lter, and a square root function extracts theamplitude of the roll angle φ, see Ariyur and Krsti¢ [2].

ss+Ωh (·)2 Ωl

s+Ωl

√2 (·)

φ A

Figure 7.18: Detector of the limit cycle amplitude, see [2].

The ES loop, shown in Figure 7.15, then tunes iteratively the amplitude of theroll angle to a minimum, yielding a trajectory for the desired encounter frequencywhich is tracked by variations of the ship's forward speed and heading angle. Notethat the encounter frequency as the parameter of the ES loop does not have to beperfectly known.

The parameters of the ES control loop, namely the lter coecients and thetime constant of the perturbation signal in the outer control loop, determine to-gether with the dynamics of the inner control loop the stability and the performanceof the ES control with limit cycle minimization. By requiring dierent time scales

125


for the plant with the controllers, the perturbation signal, and the lters in the ESloop, it can be proven by averaging and singular perturbation, that the proposedcontrol scheme minimizes the limit cycle of the roll motion; refer to Ariyur andKrsti¢ [2] for a thorough mathematical analysis.


The 3-DOF ship model of Section 6.1 is simulated with the proposed ES control loopwith limit cycle minimization of Section 7.5.4 and the speed and heading controllersas in Section 7.5.1. The model parameters are the same as used in Sections 7.3and 7.4. The initial simulation parameters are such that the uncontrolled shipis experiencing heavy roll oscillations due to roll parametric resonance; that is,initially, the forward speed is u0 = 7 m/s, the heading angle ψ0 = 0, and the shipis in head sea condition, thus βnw = 180. The ES control is activated at the timeinstant when the roll amplitude exceeds 3 for the rst time. The simulation resultsare shown in Figures 7.197.21. The uncontrolled, controlled, and desired signalsare denoted without subscript and with subscript c and d, respectively.

Figure 7.19 depicts the roll angle and the encounter frequency for both theuncontrolled and controlled case. It is noteworthy that the roll motion is reducedto a neighbourhood of zero when controlled; the speed of it largely dependent onhow quick the ship's forward speed and heading angle can be changed. Then, thefrequency ratio ωe/ωφ is reduced, resulting in a violation of the frequency conditionωe/ωφ ≈ 2. Compared to the ES scheme suggested in Section 7.4, this is achievedwithout the need to formulate the frequency condition explicitly in the objectivefunction of the ES loop, yielding robustness with respect to uncertainties in theknowledge of the encounter frequency.

Figure 7.20 shows the forward speed, the eective thrust, and the propeller shaftspeed. When uncontrolled, the forward speed is constant, whereas it is reducedwhen controlled. The L1 adaptive speed controller is able to track the desiredspeed reference with some lag, mainly due to the unknown adaptive parameters,namely the nonlinear damping.

The heading angle and the rudder deection are depicted in Figure 7.21. Theuncontrolled heading angle is almost constant, whereas when controlled it is grad-ually increased to its maximum, ψmax = 25, specied in the CA formulation.Note that, although the forward speed changes immediately after the ES controlis activated, the heading angle remains zero for quite a while. This is due to theformulation of the CA and its constraints. Note also that the L1 adaptive headingcontroller handles the sinusoidal disturbance, as specied in Section 7.5.3.

126


time (s)

rollangle

()

φφc

0 200 400 600 800 1000-20

-10

0

10

20

(a) Roll angle.

time (s)

frequency

ratio(−

) ωe/ωφ

ωe,c/ωφ

ωe,d/ωφ

0 200 400 600 800 10001.7

1.8

1.9

(b) Encounter frequency.

Figure 7.19: Roll angle and encounter frequency.

127


time (s)

forw

ard

speed(m

/s) u

uc

ud

0 200 400 600 800 10004

5

6

7

(a) Forward speed.

time (s)

effectivethrust

(N)

τ1

τ1c

τ1c,d

0 200 400 600 800 10000

5

10 ×105

(b) Eective thrust.

time (s)

propellerspeed(rps) np

np,c

np,d

0 200 400 600 800 1000

1.2

1.4

1.6

1.8

2

(c) Propeller shaft speed.

Figure 7.20: Speed, eective thrust, and propeller shaft speed.

128


time (s)

headingangle

() ψ

ψc

ψd

0 200 400 600 800 1000

0

10

20

(a) Heading angle.

time (s)

rudder

angle

() δr

δr,c

0 200 400 600 800 1000

-20

0

20

40


Figure 7.21: Heading angle and rudder deection.

129

Part IV

Wave Frequency Estimation

131

Chapter 8

Model-Based Frequency Estimator

Some of the active control methods from Chapter 6 and 7 do to a certain extent relyon the knowledge of the modal wave frequency ω0 > 0 ∈ R and the wave encounterfrequency ωe ∈ R. In a practical situation, it is however not trivial to obtain goodestimates of both of them. In this chapter and the following one, two approaches toestimate the two crucial frequencies online, are investigated. The material presentedin this chapter was published in Belleter et al. [9] and can also be found in Belleter[8]; the work leading to the material of this chapter was initiated and carried outduring a traineeship of Dennis Belleter in Trondheim when I was co-supervisinghim. My contribution was introduction to parametric roll resonance, provision ofmathematical models, general tutoring and regular inputs. My contribution withrespect to writing the article was mainly proofreading and advisory.

Assuming constant heading, the encounter frequency ωe (3.30) is dependenton the ship's surge speed. Here, results of a model-based frequency estimator arepresented, namely an extended Kalman lter (EKF). An introduction to the EKFgenerally can be found in Simon [133] and Stengel [139]; in the context of marineapplications, Fossen [41] might be of interest.

The 2-degree-of-freedom (DOF) ship model under investigation is a state-spacemodel with ve states. These states are the roll angle φ, the roll rate φ, the surgespeed u, the modal wave frequency ω0, and the wave encounter frequency ωe.Collecting the states in a state vector for convenience of notation yields

x =[φ, φ, u, ω0, ωe

]>. (8.1)

For the study at hand it is assumed that the rst three states, that is the rollangle φ, the roll rate φ, and the surge velocity u are measured. The remaining twostates, that is the modal wave frequency ω0 and the wave encounter frequency ωeare to be estimated by the EKF.

A simplied 2-DOF version of the ship model of Section 7.1 is considered withthe roll dynamics given by

m44φ+ d44φ+ ρg∇GMmφ+ ρg∇GMa cos (ωet)φ = 0 . (8.2)

Note that (8.2) is a Mathieu-type roll model which is valid for constant encounterfrequency ωe only, as discussed in Chapter 4. This chapter investigates the appli-cability of the EKF to estimate an (almost) constant encounter frequency, barring

133

8. Model-Based Frequency Estimator

wave disturbance and measurement noise, and thus the simplied model (8.2) isappropriate. The case of a time-varying encounter frequency due to heading an-gle and speed changes will be covered in Chapter 9. For the sake of readability,γ1 = GMa/GMm and γ2 = ρg∇GMm are used.

The surge dynamics is modelled as

m11u (t) + d11u (t) = τ1c +∑

i

bi cos (ωit) (8.3)

where τ1c is the control input in surge and∑i bi cos (ωit) is a disturbance due

to waves. The modal wave frequency ω0 is assumed to be constant and the waveencounter frequency is given by, see Chapter 7,

ωe (u (t) , ω0) = |ω0 +ω2

0

gu (t)| (8.4)

where it is assumed that the sway velocity is negligible and the vessel is sailing inhead sea condition.

The nonlinear model for the system consequently becomes

x = f (x, τ c, t) =

x2−(d44x2+γ2(1+γ1 cos(x5t))x1)

m44

τ1c

m11− d11x3

m11+∑iBi cos (ωit)

0

x24

(τ1cm11− d11x3m11

+∑i Bi cos(ωit)

)g

(8.5)

y = h (x) =[x1, x2, x3

]>. (8.6)

8.1 Design of the Extended Kalman Filter

In this section an algorithm for a discrete-time EKF obtained from Simon [133] ispresented. However, to implement the EKF algorithm, the nonlinear model (8.5)and (8.6) needs to be linearized and discretized rst. The linearization and dis-cretization results in

xk = F k−1xk−1 +Gk−1τ c,k−1 +W k−1 (8.7)

yk = Hkxk + V k (8.8)

where

F k−1 = exp

(∂f

∂x

∣∣∣x+k−1

ts

)(8.9)

Gk−1 =

∫ ts

0

exp

(∂f

∂x

∣∣∣x+k−1

τ

)dτ

∂f

∂τ c(8.10)

Hk =∂yk∂x

∣∣∣x−k

. (8.11)

134

8.1. Design of the Extended Kalman Filter

Here ts > 0 ∈ R is the sampling time chosen as ts = 0.1 s seconds andW k−1 ∈ R5

and V k ∈ R5 are Gaussian white noise processes with zero mean and covariancematrices Qc ∈ R5×5 and Rc ∈ R3×3, respectively. The linearized and discretizedstate-space matrices are F k−1 ∈ R5×5, Gk−1 ∈ R5×5, and Hk ∈ R5×3. Notethat the subscript k denotes the value of the variable at time step k, whereas thesubscript k − 1 is the value at the previous time step k − 1.

The covariance matrix of the measurement noise Rc is chosen as a constantdiagonal matrix with entries based on the variances of the sensors, that is,

Rc = diag(σ2φ, σ

2φ, σ2u

). (8.12)

The estimation of the process noise covariance matrix Qc is more complicated. Forthe measured states it should represent the uncertainty in the model equations.Note that for equations that are modelled exactly, like the rst equation of (8.5),process noise can be omitted or given a small positive value for numerical purposes.For the states to be estimated it should represent the covariance of the noise drivingthe state estimates. These driving noise terms for the wave frequency and waveencounter frequency should not be chosen too large or the lter might become tooaggressive in its adjustments. Moreover the state estimates might converge to thewrong value ifQc is too large. Especially the entry for the wave frequency should bechosen very small since estimation of the wave frequency is a parameter estimationproblem. Making Qc a constant diagonal matrix:

Qc = diag(qφ, qφ, qu, qω0 , qωe

). (8.13)

Now the EKF can be initialized by:

x+0 = E (x0)

P+c,0 = E

[(x0 − x+

0

) (x0 − x+

0

)T ] (8.14)

where x+0 denotes the original estimate for the state and P+

c,0 is the initial stateerror covariance.

Then with each discrete-time step of the Kalman lter the following has to becalculated. First the time update is done:

P−c,k = F k−1P+c,k−1F

Tk−1 +Qc (8.15)

x−k = fk−1

(x+k−1, τ

1c,k−1

)(8.16)

with P−c,k ∈ R5×5 the error covariance matrix of the error x+k−1 − x−k . In the time

update the model is applied to the system to update the state estimate. The stateerror covariance is updated using the linearized system matrices (8.9)(8.11), thea posteriori state error covariance from the previous time step and the covarianceof the process noise.

135


The measurement update can then be done as in Simon [133],

Kk = P−kHTk

(HkP

−kH

Tk +Rc

)−1

(8.17)

x+k = x−k +Kk

(yk − hk

(x−k))

(8.18)

P+c,k = (I −KkHk)P−k (I −KkHk)

T+KkRcK

Tk (8.19)

whereKk ∈ R5×5 is the Kalman gain, x+k ∈ R5×1 is the a posteriori state estimate

after the k-th time step and P+c,k ∈ R5×5 is the error covariance matrix of the

error x+k − x−k . The Kalman gain is calculated using the linearized measurements

of (8.11), the a priori state error covariance and the covariance of the measurementnoise. The estimate of the states is adapted by multiplying the error in the measuredvariables with the Kalman gain. Finally the a posteriori state error covariance iscalculated using the Kalman gain and the covariance of the measurement noise.

As mentioned in Simon [133] alternate formulations for Kk and P+c,k may be

chosen. However, the formulations chosen here are to guarantee that P+c,k is sym-

metric and positive denite as long as P−c,k is positive denite.

8.2 Frequency Estimation

The EKF of Section 8.1 will now be applied to the ship model (8.5) and (8.6) tocreate a EKF for the ship model. The performance of the EKF will be analysed ina simulation study. However, to perform the measurement update, the EKF needsa measurement of the roll angle, roll rate, and the surge velocity. Hence, a model togenerate a ctive measurement is created rst. This model includes a second-orderwave model and velocity controller. The wave model is based on the second-orderwave model found in Fossen [41] and takes the form:

h (s) =2ζwω0σws

s2 + 2ζwω0s+ ω20

(8.20)

with ζw > 0 ∈ R as a damping coecient and σw > 0 ∈ R is a constant describingthe wave intensity. Both of these parameters are determined by a linearization ofthe wave spectrum as in Fossen [41]. The velocity controller is designed such thatit keeps the velocity at a specied reference value, ur ∈ R. The desired velocity udis generated by a second-order reference model:

ud + 2ζuωu,lud + ω2u,lud = ω2

u,lur (8.21)

with ωu,l = 0, 1 rad/s and relative damping ζu = 1.Combining (8.21) with the surge model (8.3) results in

u = − d11

m11u+

1

m11τ1c (8.22)

τ1c = m11

(ud +

d11

m11ud

). (8.23)

136

8.2. Frequency Estimation

The following acceleration prole is obtained:

u = ud −d11

m11(u− ud) . (8.24)

The wave disturbance and the measurement noise are added to the velocityafter numerical integration to create a disturbed velocity prole. In addition to thewave model and the velocity controller, measurements of the roll angle and the rollrate are generated with the roll model (8.2). The EKF tracks these proles.

For the unmeasured variables a condence interval for the estimates is alsogiven. This condence interval is calculated using the diagonal elements of thestate error covariance matrix resulting in:

x99% = ˆx± 3√Pc,ii . (8.25)

A simulation study is now preformed in which the EKF needs to estimate awave frequency of 0.4684 rad/s. In the simulation study the ship accelerates to aconstant velocity of 5.66 m/s. This velocity is chosen such that the wave encounterfrequency is about twice the ship's natural roll frequency given the specied wavefrequency. As a consequence, the ship is experiencing parametric rolling which isapparent from the roll angle and roll rate plots in Figures 8.1(a) and 8.1(b). Theinitial conditions for the simulation study are chosen as:

[φ φ u ω0 ωe

]>=[

2π90

π90 5 1

2 0.62]>

. (8.26)

The simulation results for the measured variables can be seen in Figure 8.1.It can clearly be seen from Figure 8.1 that estimation of the measured variablesroll angle, roll rate, and surge velocity is trivial and they are estimated nearlyperfect. For the velocity the EKF is tuned such that the lter does not follow allthe disturbances on the velocity. This improves the estimate of the wave encounterfrequency, which strongly dependents on the velocity. Note that the maximum rollangle that is achieved in the simulation is about 0.5 rad which corresponds to about29 and that the ship rolls with a angular speed of 0.15 rad/s which corresponds toabout 9 deg/s. This is quite a heavy resonance. The resonance is lost after about2400 seconds. The eect of this on the estimation of the unknown frequencies willbe demonstrated later.

The results of the estimation of the wave frequency can be seen in Figure 8.2.Figure 8.2 has a plot of the wave frequency estimate including the condenceinterval and a plot with a detail of the mean of the wave frequency estimate.

From Figure 8.2 it can be seen that the mean is estimated quite well untilthe resonance is lost after this the estimate retains a bias. This is due to thefact that the estimation of the wave frequency is a parameter estimation process.This means that the estimate for the wave frequency is not adjusted in the timeupdate (8.15) and (8.16) since ω0 = 0. Hence, the state estimate is only adjustedby the measurement update (8.17)(8.19). This makes the adjustment of the stateestimate dependent on the size of the Kalman gain and the error in the measuredvariables. When the resonance is lost at about 2400 seconds, the error in the stateestimate of the roll angle also reduces signicantly and the wave frequency stateestimate is hardly adjusted.

137


Time (s)

Rollangle

() Measured roll angle

Estimated roll angle

0 500 1000 1500 2000 2500 3000 3500 4000-40

-20

0

20

40

(a) Measured and estimated roll angle.

Time (s)

Rollrate

(/s)

Measured roll rateEstimated roll rate

0 500 1000 1500 2000 2500 3000 3500 4000-10

-5

0

5

10

(b) Measured and estimated roll rate.

Time (s)

Surgevelocity

(m/s)

Measured surge velocityEstimated surge velocity

0 500 1000 1500 2000 2500 3000 3500 40004

4.5

5

5.5

6

(c) Measured and estimated surge speed.

Figure 8.1: EKF frequency estimation of the measured states for the ship in reso-nance.

138


Time (s)

Wavefrequency

(rad/s)

True valueKalman filter estimateConfidence interval

0 500 1000 1500 2000 2500 3000 3500 4000

0.2

0.4

0.6

0.8

(a) True and estimated wave frequency with condence interval.

Time (s)

Wavefrequency

(rad/s)

True valueKalman filter estimate

0 500 1000 1500 2000 2500 3000 3500 40000.460

0.465

0.470

0.475

0.480

(b) True and estimated wave frequency detailed view.

Figure 8.2: EKF frequency estimation of the wave frequency with detail of themean estimate.

From Figure 8.2 it can also be seen that the condence interval for the wavefrequency estimate only converges very slowly. This can also be attributed to thefact that the estimation of the wave frequency is a parameter estimation process.Since the adaptation of the state error covariance of the wave frequency in (8.13) isgoverned by the small process noise term driving the estimate of the wave frequency.

Figure 8.3 shows the results of the estimation of the wave encounter frequency.The top plot in Figure 8.3 shows the wave encounter frequency estimate and itscondence interval. The bottom plot in Figure 8.3 shows a detail of the estimateof the wave encounter frequency and its condence interval.

From Figure 8.3 it can be seen that the estimate for the wave encounter fre-quency converges to the correct value quite fast. Moreover the condence intervalalso converges to within acceptable bounds quite fast. Note here that just likewith the estimation of the velocity the lter is tuned such that the estimate tracksthe mean of the wave encounter frequency and does not try to track all the distur-bances. This would cause a lter that is too aggressive and is more likely to diverge.A close inspection of the wave encounter frequency estimate shows that when theresonance is lost, the estimate of the wave encounter frequency is adjusted less andthe condence interval increases a little. This is because the noise terms becomemore dominant for small values of the states. When the noise is larger with respect

139


Time (s)

Encounterfrequency

(rad/s)

Generated measurementsKalman filter estimateConfidence interval

0 500 1000 1500 2000 2500 3000 3500 40000.55

0.6

0.65

(a) True and estimated wave encounter frequency with condence interval.

Time (s)

Encounterfrequency

(rad/s)


0 500 1000 1500 2000 2500 3000 3500 40000.593

0.593

0.594

0.595

0.595

(b) True and estimated wave encounter frequency detailed view.

Figure 8.3: EKF frequency estimation of the wave encounter frequency with detailof the mean estimate.

to the states the uncertainty in the state estimates becomes larger. Hence, the stateerror covariance increases.

Good initial performance of the lter is obtained by a suitable choice of theinitial state error covariance. For the measured states this is not an issue, howeverfor the unmeasured states it is. If the initial error covariance is chosen too small thelter will be sluggish and it will take longer for the estimate to converge or it mightnot converge at all. If the initial state error covariance is chosen too large the lterwill react very aggressive in the beginning which can cause the state estimate toovershoot the true value and oscillate around it for a while. Both under tuning andover tuning of the initial state error covariance can cause the wave frequency todiverge. Hence, it is important to have a good initial estimate of the wave frequencywhen initializing the lter.

Figure 8.4 shows the wave frequency and wave encounter frequency for a lterwith large initial error covariance to illustrate the mentioned problems.

For convergence the choice of the process noise covariance is also important.Large values for the process noise covariance will allow the lter to track the noisysignals of the velocity in Figure 8.1 and the wave encounter frequency in Figure 8.3more closely. However, large values for the process noise covariance will also cause

140


Time (s)

Wavefrequency

(rad/s)

True valueKalman filter estimate

0 500 1000 1500 2000 2500 3000 3500 4000

0.2

0.4

0.6

0.8

(a) True and estimated wave frequency.

Time (s)

Encounterfrequency

(rad/s)


0 500 1000 1500 2000 2500 3000 3500 40000.593

0.593

0.594

0.595

0.595

(b) True and estimated wave encounter frequency with condence interval detailedview.

Figure 8.4: EKF frequency estimation of the wave frequency and the wave encounterfrequency with large initial error covariance.

the lter to be more sensitive to e.g. a loss of the resonance which can cause thestate estimate of the wave frequency and wave encounter frequency to divergeand their condence interval to increase. However the lter will always need someprocess noise to adjust the estimates.

141

Chapter 9

Signal-Based Frequency Estimator

In the marine community it is common practice to obtain the wave frequency usingfast Fourier transform (FFT) frequency spectra [35]. The obvious disadvantage ofthis approach is that creating a FFT frequency spectrum takes time, resultingin back-dated information for the wave frequency estimate. Moreover due to themoving window and long acquisition times, the FFT is not able to estimate a time-varying wave frequency. Hence, since the FFT is generated with a moving windowit will not detect a time-varying wave encounter frequency until the data is withinthe window.

For those reasons, the design of a frequency estimator that estimates the waveencounter frequency online from measurements is investigated. To that matter, anonlinear frequency estimator which is globally K exponentially stable and has noinitial conditions that need to be tuned for convergence is presented. Furthermorethe frequency estimator is signal based and hence not inuenced by modeling errorsin the equations or in the model parameters. An additional advantage of this is thatthe estimator is not ship-specic and one does not have to spend a lot of time andmoney to model the ship and estimate its parameters. It just needs a measurementof the roll angle from which it can estimate the wave encounter frequency.

In steady-state the roll angle is a sinusoidal signal. The problem of estimatingthe frequency of a sinusoidal function online is a well studied problem in systemtheory. Global frequency estimators are discussed by Marino and Tomei [91] andXia [156]. Moreover, in both Marino and Tomei [91] and Xia [156] an extensionto the estimation of multiple unknown frequencies is presented. Another approachusing adaptive notch lters is taken in Hsu, Ortega, and Damm [71] and is based onthe continuous-time version presented in Bodson and Douglas [13] of the discrete-time Regalia's adaptive notch lter [123]. The lter applied in this chapter is basedon Aranovskii et al. [1], but extended with a switching gain here, to handle changesin the measured signal during the estimation.

The roll angle is a suitable signal to be used in the estimator, since it is, asmentioned before, sinusoidal in steady-state but also directly dependent on thewave encounter frequency exciting the ship. The ship will be in parametric rollresonance when the ship is excited with a wave encounter frequency that is abouttwice as large as its natural roll frequency, ωe ≈ 2ωφ. The ship in parametric roll is

143

9. Signal-Based Frequency Estimator

excited indirectly (through the nonlinear coupling to the heave/pitch motion) andwill oscillate with half the encounter frequency. Furthermore, if the ship is not inhead sea condition, the roll motion will be directly excited by the waves with thewave encounter frequency.

The material of this chapter has been published in Belleter et al. [10] and mayalso be found in Belleter [8]. The work leading to the material of this chapterwas initiated during a traineeship of Dennis Belleter in Trondheim and mostlycarried out after his return to Eindhoven. I was co-supervising him during hisstay in Trondheim and actively contributed with regular feedback on the workat hand. I was involved in the amplitude estimator, the mathematical modellingand the provision and handling of the experimental data. The article was mainlywritten by the rst author and I contributed with proofreading and helping withthe mathematical stability proof.

9.1 Frequency Estimator Design

In this section, the frequency estimator from Aranovskii et al. [1] is extended witha switching gain. The frequency estimator is designed to estimate the frequency ofa measured signal of the form:

y (t) = σ sin (ωet+ ϕy) (9.1)

with σ > 0 ∈ R the unknown amplitude, ωe the unknown frequency, and ϕy ∈ Ris an unknown phase. A sinusoidal signal in this form can be represented by thedierential equation

y = −ω2ey = θsy (9.2)

with θs = −ω2e . For the estimation of the frequency of the signal (9.1) an auxiliary

lter is used. This lter is given by:ζf,1 = ζf,2

ζf,2 = −2ζf,2 − ζf,1 + y,(9.3)

with ζf := ζf,1. Using the lter (9.3), the model (9.2) can be represented in theform

y = 2ζf + ζf + θsζf . (9.4)

This can be shown using the Laplace transform of (9.3) with zero initial con-ditions:

Y (s) = s2Zf (s) + 2sZf (s) + Zf (s) (9.5)

with Y (s) the Laplace transform of y (t) and Zf (s) the Laplace transform of ζf (t).Using the fact that

Y (s) =s2 + 2s+ 1

(s+ 1)2 Y (s) ; Zf (s) =

1

(s+ 1)2Y (s) (9.6)

and that the Laplace transform of (9.2) is

s2Y (s) = θsY (s) , (9.7)

144

9.1. Frequency Estimator Design

it is straight-forward to see that

Y (s) =2s+ 1 + θs

(s+ 1)2 Y (s) = 2sZf (s) + Zf (s) + θsZf (s) . (9.8)

It is then possible to transform (9.8) to the time domain:

y (t) = 2ζf (t) + ζf (t) + θsζf (t) (9.9)

which is equal to (9.4).Using (9.3) and (9.4) the update law for the parameter θs ∈ R can be formu-

lated. For this purpose a new measurement is introduced of the form:

zm = ζf = y − 2ζf − ζf . (9.10)

Substituting (9.4) in (9.10) results in

zm = θsζf . (9.11)

For this signal an observer signal may be dened as

zm := θsζf (9.12)

with zm an estimate of zm ∈ R, and θs (t) the estimate of θs which is adapted bythe observer. The update law can then be chosen as

˙θs = koζf (zm − zm) (9.13)

with ko > 0 ∈ R the observer gain. The resulting frequency estimator algorithmbecomes

ζf,1 = ζf,2

ζf,2 = −2ζf,2 − ζf,1 + y˙θs = koζf

(ζf,2 − θsζf,1

).

(9.14)

This algorithm and its stability proof in Aranovskii et al. [1] are based on a constantobserver gain ko. In the following, the frequency estimator is extended with aswitching gain.

9.1.1 Switching Mechanism

The switching mechanism needs to be related to the amplitude σ of the signal y.Hence, the amplitude of y relates to the amplitude of ζf ∈ R, which determines

the size of zm. In turn zm determines the size of ˙θs. With the same value for ko it

can be veried that | ˙θs| is larger when the amplitude of ζf is larger. This impliesthat for the same ko the frequency estimate can be more readily adjusted for largervalues of the amplitude of ζf and hence a larger amplitude of y. To be able tohave the same readiness to adjust the frequency estimate for small y, resulting in

145


a small amplitude of ζf , it is proposed to have a large gain for small values of theamplitude of y. To avoid too aggressive adaptation when y is large, it is proposesto have a small gain when the amplitude of y is large. Moreover it is desirableto have a large gain when initializing the estimator. Hence, y will be small then,but the initial error can be large and fast convergence is required. The switchingmechanism for the gain is chosen as:

ko (|σ (t)|) =

ko,1 if t < tinit

ko,2 if t ≥ tinit ∧ |σ| > σr

ko,3 if t ≥ tinit ∧ |σ| ≤ σr(9.15)

with ko,1, ko,3 ≥ ko,2.Hence, the gain should switch to a higher value if the amplitude of y is smaller

than σr > 0 ∈ R. The reference value σr is chosen such that it is reasonable toassume that the system is no longer in parametric resonance when the amplitudeis smaller than σr i.e. small roll angles caused by direct forcing or irregular seaconditions.

9.1.2 Amplitude Estimator

To implement the switching mechanism (9.15) online the amplitude of the measuredsignal y as to be known. Since the amplitude σ of y is not measured, it has to beestimated. This may be done by using the squared signal of (9.1):

y2 (t) =σ2

2(1− cos (2ωet+ 2ϕu)) . (9.16)

The squared signal (9.16) can be passed through a low-pass lter to obtain theamplitude of the squared signal σ2/2. The amplitude σ of y can then by obtainedby simply multiplying this by two and taking the square root, see also Section 7.5The amplitude estimation can be done online by taking parts of the recent historyof the roll angle measurement and using the amplitude estimate at the current timestep to compare against the reference σr. Do note that if tlp seconds of data areused there is a small delay in the amplitude estimate, however since the amplitudeestimator only plays a supporting roll in the estimator the delay does not harmthe operation or degrades the performance of the frequency estimator.

9.1.3 Stability Proof

In this subsection it is shown that the frequency estimator isK-exponentially stable.In Sørdalen and Egeland [134] K-exponential stability is dened in the followingmanner.

Denition 9.1. Consider the nonlinear, time-varying system

x = f (x, t) x ∈ D ⊂ Rn , t ≥ t0 . (9.17)

System (9.17) is K-exponentially stable about the origin if there exist a neighbour-hood Ω ⊂ D about the origin, a positive constant λ, and a function h (·) of class

146

9.1. Frequency Estimator Design

K i.e. strictly increasing such that all solutions of x (t) of (9.17) satisfy

‖x (t)‖ ≤ h (‖x (t0)‖) e−λ(t−t0) ∀x (t0) ∈ Ω ∀t ≥ t0 (9.18)

where the constant λ and the neighborhood Ω are independent of t0 and ‖·‖ denotesa norm in Rn. If (9.18) is satised for Ω = D, then (9.17) is globally K-exponentiallystable around the origin.

According to Denition 9.1, if system (9.17) is globally K-exponentially stablearound the origin, then it is uniformly globally asymptotically stable as denedby e.g., Theorem 4.3 in Khalil [84], and in addition it has an exponential rate ofconvergence.

The update law of the Aranovskii frequency estimator is given by

˙θs = koζf

(θsζf − θsζf

)(9.19)

with ζf = σf sin (ωf t) and the estimation error is dened as

θs := θs − θs . (9.20)

Assuming a constant frequency θs this results in the error dynamics

˙θs = θs − ˙

θs = 0− koζf(θsζf − θsζf

)= −koζ2

f θs . (9.21)

It should be noted that ko is a positive valued adaptation gain and ζf is a stateof the lter (9.3) and is a sinusoidal signal if the lter is excited by a signal of theform (9.1), as a consequence ζ2

f = σ2f sin2 (ωf t) ≥ 0.

It is desired that the error dynamics (9.21) is K-exponentially stable about zero.The solutions of (9.21) satisfy

‖θs‖ = ‖θs (t0)‖e−ko∫ tt0ζ2f (τ) dτ

= ‖θs (t0)‖e−ko∫ tt0σ2f sin2(ωfτ) dτ (9.22)

with∫ t

t0

σ2f sin2 (ωfτ) dτ =

σ2f

2(t− t0)−

σ2f

4ωf[sin (2ωf t)− sin (2ωf t0)] . (9.23)

Substituting (9.23) in (9.22) and rewriting results in

‖θs‖ = ‖θs (t0)‖ekoσ

2f

4ωf[sin(2ωf t)−sin(2ωf t0)]

e−koσ

2f

2 (t−t0) . (9.24)

From (9.24) it can be seen that

‖θs‖ ≤ ‖θs (t0)‖e2koσ

2f

4ωf e−koσ

2f

2 (t−t0) . (9.25)

If (9.25) is compared to (9.18) it can be seen that there is a suitable candidate for

λ, which is koσ2f/2, and there is a function h

(‖θs (t0)‖

)of class K with respect to

‖θs (t0)‖:

h(‖θs (t0)‖

)= ‖θs (t0)‖e

2k0σ2f

4ωf . (9.26)

147


Hence, there is a positive constant λ, and a function h (·) of class K such that theconditions of Denition 9.1 are satised. Consequently, (9.21) is K-exponentiallystable. Since, (9.25) satises (9.18) for Ω = D, (9.21) is globally K-exponentiallystable around the origin.

9.2 Case Study: Wave Frequency Estimation

In the case study the roll angle φ serves as input y (t) for the frequency estimator.The roll angle is a suitable input since it is sinusoidal in steady-state, measurable,and directly related to the wave encounter frequency.

To generate the measurements for the roll angle φ, a second-order model de-scribing the roll dynamics is used. The operation of the K-exponentially stablefrequency estimator is veried for various cases. In Chapter 8 a model for the rolldynamics of a ship with a constant wave encounter frequency obtained from Fossen[41] was used. Here, the 1-DOF model for a time-varying wave encounter frequencybased on the 1-DOF roll model in Chapter 5 is considered. For readability, the rollmodel is restated. The roll dynamics for the ship in Table 9.1 can be representedin the form

m44φ+ d44φ+

[k44 + kφt cos

(∫ t

t0

ωe (τ) dτ + αφ

)]φ

+ kφ3φ3 = kwφ sin

(∫ t

t0

ωe (τ) dτ + αwφ

) (9.27)

with

ωe =d

dt

(ω0t+ kw

∫ t

t0

un (τ) dτ

)

=d

dt

(ω0t+ kw

∫ t

t0

u (τ) cos (ψ) dτ

)(9.28)

where un is the velocity with respect to an inertial frame xed to the ocean surfaceand u is the ship's velocity in surge expressed in a body xed frame.

For constant forward speed u, this model can be simplied to

m44φ+ d44φ+ [k44 + kφt cos (ωet+ αφ)]φ

+ kφ3φ3 = kwφ sin (ωet+ αwφ) .(9.29)

The model parameters and their values for the ship under investigation can befound in Table 9.1.

The parameters in Table 9.1 are the same as in Chapter 4. The measurementsgenerated using model (9.27) or (9.29) can be used as input to the frequencyestimator (9.14). As a rst attempt a constant wave encounter frequency (9.29) isestimated with a constant estimator (9.13) gain.

The result of the measurement generation can be seen in the top plot of Fig-ure 9.1. The measurements in Figure 9.1(a) are generated using the initial condi-tions: [

φ0 φ0 u0

]>=[0 0 9

]>(9.30)

148

9.2. Case Study: Wave Frequency Estimation

Table 9.1: Model parameters.


The total moment of inertia in roll m44 1.41 · 1010 kg m2

The linear damping coecient in roll d44 2.48 · 108 kg m2/sThe linear restoring coecient in roll k44 1.7533 · 109 kg m2/s2

Amplitude of the varying linear restoringcoecient in roll

kφt 7.1373 · 108 kg m2/s2

Phase of the varying linear restoring co-ecient in roll

αφ 0.2741 rad

The cubic restoring coecient in roll kφ3 2.2627 · 109 kg m2/s2

The wave number kw 0.0224 rad/sThe wave frequency ω0 0.4684 rad/sThe surge velocity u variable m/sThe heading angle ψ 10·π

180 radAmplitude of the direct induced roll kwφ 8.627 · 107 kg m2/s2

The phase of the direct induced roll αwφ 0.2741 rad

where u0 = 9 m/s is chosen such that ωe = 0.6697 rad/s. This fulls the criterionfor the onset of parametric roll ωe ≈ 2ωφ = 2

√k44/m44, with ωφ > 0 ∈ R the

natural roll frequency of the ship.The simulations are made with a xed time-step solver running at 10 Hz. Hence,

the sampling time ts = 0.1 s.In Figure 9.1(a) it is shown that the generated measurement behaves as ex-

pected. The resonance builds up after initialization and then stabilizes to a limitcycle with constant amplitude.

The frequency estimate and the frequency estimation error can be seen in thebottom plots of Figure 9.1. The initial conditions for the frequency estimator arechosen as:

[ζf,1,0 ζf,2,0 θs,0

]>=[0 0 −0.25

]>. (9.31)

Note the dierence between θs,0 and the initial estimate of the frequency in Fig-ure 9.1, which is proportional to the square of the initial wave encounter frequency

ωe,0 dened by ωe,0 = 2√|θs,0|.

Figure 9.1 depicts the problem under consideration in this chapter. If the rollangles are small and the gain is small, the frequency estimate will not converge untilthe resonance has developed. When the roll amplitude increases from t = 100 sto t = 200 s it is evident that the convergence rate of the frequency estimateincrease. Also, it is noteworthy from Figure 9.1(b) that if the value of ko is larger,convergence will start for smaller roll angle amplitudes. Hence, the lter convergesfaster. After convergence the estimates shows small oscillations around the correctfrequency instead of converging to zero steady-state error. This can be explainedby considering the frequency estimate update law (9.13) in which the dierencebetween two sinusoidal signals is taken.

149


Time (s)

Rollangle

()

0 200 400 600 800 1000

-20

-10

0

10

20

(a) Roll angle.

Time (s)

Encounterfrequency

(rad/s)

True encounter frequencyko = 0.5ko = 1ko = 5

0 200 400 600 800 1000

0.7

0.8

0.9

1

(b) True and estimated encounter frequency.

Time (s)

Estim

ationerror(rad/s)

ko = 0.5ko = 1ko = 5

0 200 400 600 800 1000-0.4

-0.2

0

0.2

0.4

(c) Estimation error.

Figure 9.1: Roll angle measurement, frequency estimate, and estimation error forthe ship in parametric roll resonance.

150


The sinusoids are in practise never exactly identical causing a periodic estima-tion error. The error after convergence is analysed using the mean-squared-error(MSE) of the estimate from t = 400 s, when the resonance is fully developed, untilt = 1000 s. The MSE is dened as:

MSE =1

n

n∑

i=1

(ωe,i − ωe,i)2 (9.32)

with i = 1, 2, . . . , n where n is the number of time steps in the simulation. Theresults of the MSE calculations can be seen in Table 9.2.

Table 9.2: Mean Squared Error compari-son, constant encounter frequency.

Estimator gain MSE

ko = 0.5 5.8779 · 10−6

ko = 1 1.5892 · 10−5

ko = 5 4.1546 · 10−4

From Table 9.2 it is evident that the MSE is lower for small gains and higher forlarge gains. Considering this combined with the convergence analysis in Figure 9.1it becomes clear that the gain ko is a compromise between convergence rate andsteady-state error. The switching gain will allow us to use the benets of both alarge and a small gain by using the large gain for convergence and then switchingto the small gain for a small steady-state error.

In the next step, it is veried if the frequency estimator can be applied to caseswith a time-varying velocity and hence a time-varying wave encounter frequency.To do this the ship will initially be at a constant speed of 9 m/s again and thenfrom t = 600 s to t = 800 s the forward speed is linearly increased to 12 m/s.

The initial conditions for the lter and the 1-DOF ship model are chosen as(9.31) and (9.30), respectively. The simulation is performed for dierent gains ko,that is ko = 0.5, 1, 5. The resulting roll angle φ can be seen in the top plot ofFigure 9.2.

In Figure 9.2(a) it is shown that when the surge speed increases, the roll an-gle amplitude increases. The frequency estimate and the estimation error for thismeasurement can be seen in the bottom plots of Figure 9.2.

From Figures 9.2(b) and 9.2(c) it is furthermore noticeable that the lter canstill estimate the wave encounter frequency when it is time-varying. The errorincreases only slightly when the frequency rst starts to rise, however the estimatequickly converges again.

The MSE calculations for the dierent gains ko for the time-varying encounterfrequency are summarized in Table 9.3. Similarly to the case of a constant encounterfrequency, the choice of the gain ko is a trade-o between fast convergence andsteady-state error.

Here, the switching mechanism (9.15) is tested with the gain values ko,1 = 10,ko,2 = 0.5, and ko,3 = 100. The reference for the switching gain is chosen as σr = 8,the time for the initial gain tinit = 100 s, and the time for each low-pass amplitude

151


Time (s)

Rollangle

()

0 200 400 600 800 1000-40

-20

0

20

40

(a) Roll angle.

Time (s)

Encounterfrequency

(rad/s)

True encounter frequencyko = 0.5ko = 1ko = 5

0 200 400 600 800 1000

0.7

0.8

0.9

1


Time (s)

Estim

ationerror(rad/s)

ko = 0.5ko = 1ko = 5

0 200 400 600 800 1000-0.4

-0.2

0

0.2

0.4


Figure 9.2: Roll angle measurement, frequency estimate, and estimation error forthe ship, simulated using the roll dynamics of (9.27).

Table 9.3: Mean Squared Error compari-son, time-varying encounter frequency.

Estimator gain MSE

ko = 0.5 3.2498 · 10−5

ko = 1 3.5536 · 10−5

ko = 5 4.5757 · 10−4

152


estimate is chosen as tlp = 40 s. Initial conditions for the lter and the model areagain chosen as in (9.31) and (9.30), respectively.

The velocity prole of the ship is chosen such that she is cruising at a constantforward speed of 9 m/s for the rst 1000 s. After this the speed is lowered to5 m/s in 200 s. Because of this the parametric roll resonance dies out and afterthe transient period you are left with the direct wave induced roll. The generatedmeasurement for this case can be seen in the top plot of Figure 9.3. In additionto the roll angle, Figure 9.3(a) depicts also the estimate for the amplitude at eachtime step.

Time (s)

Rollangle

()

Roll angleEstimated amplitude

0 500 1000 1500 2000

-20

-10

0

10

20

(a) Roll angle and estimated roll amplitude.

Time (s)

Encounterfrequency

(rad/s)

True encounter frequencySwitching koFixed ko

0 500 1000 1500 2000

0

0.5

1


Time (s)

Estim

ationerror(rad/s)

Switching koFixed ko

0 500 1000 1500 2000-0.5

0

0.5


Figure 9.3: Simulation results where the ship leaves parametric resonance afterabout t = 1100 s.

From Figure 9.3(a) it is noteworthy that the ship behaves as expected. The

153


rst 1000 seconds the ship is in parametric resonance, after that the resonancedamps out and just direct wave induced roll remains. Moreover it is shown thatthe switching reference, σr, is crossed after about 1100 seconds. The frequencyestimate and the estimation error for this measurement and the settings for theswitching mechanism can be seen in the bottom plots of Figure 9.3. Figure 9.3(b)also shows a measurement that does not have the switching mechanism, but justan initialization gain of ko = 10 and then after 100 seconds switches to a constantgain of ko = 0.5.

Another thing that is noticeable in Figure 9.3(b) is that the estimates for thefrequency make a jump at the moment the estimates cross the value of our referenceσr. This is because as explained earlier when the ship is in parametric resonanceit oscillates with a frequency ωe/2 and when the ship only experiences direct waveinduced roll it oscillates with a frequency ωe. This means that at the moment thegains are switched the way ωe is calculated from the lter state θs switches as

well. When in parametric resonance ωe = 2

√|θ| when not in parametric resonance

ωe =

√|θ|. From Figure 9.3 it is visible that as long as the ship is still in para-

metric roll resonance the estimators perform equally well. However as soon as theparametric roll resonance is lost the lter responses start to dier. The gain of thexed-gain estimator is too low to obtain convergence in this simulation time. Theswitching gain frequency estimator, however, does converge quite swiftly after thetransient period in the roll is over. Moreover it can be seen that when the estimatehas converged again, the high gain estimator provides a good steady state erroragain. Hence, when the resonance is damped out and just direct wave inducedroll remains, the switching gain frequency estimator is superior to a xed gainfrequency estimator.

9.3 Verication in Irregular Waves

In this section, a very brief verication of the frequency estimator and gain switch-ing mechanism on experimental data gathered in a towing tank test at the MAR-INTEK Marine Technology Center at the NTNU in Trondheim, is given. The datais gathered in irregular waves. The data and the amplitude estimate can be seen inthe top plot of Figure 9.4. The frequency estimate can be seen in the bottom plotof Figure 9.4. The parameters of the gain switching mechanism (9.15) are chosenas: ko,1 = 10, ko,2 = 0.5, ko,3 = 100, σr = 5, and tinit = 40 s.

It is visible in Figure 9.4 that the estimator is successful in estimating thefrequency in irregular waves. However, it can be seen that the errors are slightlyhigher than for regular waves, especially when gain switches. The increased errorafter switching is expected since the roll angle is still close to the switching thresholdand the gains are selected such that the estimator should function for much smallerroll angles.

154

9.3. Verication in Irregular Waves

Time (s)

Rollangle

()

Roll angleEstimated amplitude

0 200 400 600 800 1000-20

-10

0

10

20

(a) Roll angle and estimated roll amplitude.

Time (s)

Encounterfrequency

(rad/s)

True encounter frequencyEstimated encounter frequency

0 200 400 600 800 10000.4

0.6

0.8

1


Figure 9.4: Frequency estimation in irregular waves.

155

Part V

Closing Remarks

157

Chapter 10

Conclusions

This thesis had the ultimate goal to investigate on the possibility of reducing theeect of parametric roll resonance or possibly prevent the onset of parametricallyexcited roll motions. To achieve the former, control methods have been analysed inorder to detune the parametric excitation frequency and the natural roll frequency.In the course of the PhD work, an appreciable amount of eort has been spent onderiving suitable control models which were simple enough to use for analysis andcontroller development when the encounter frequency is changing. In the following,the contributions will be reviewed separately.

6-DOF Ship Model

A 6-DOF ship model for parametric roll resonance that, unlike most models inliterature, is valid for both constant and non-constant ship velocity was developed.The resulting model is a complex and accurate 6-DOF model which considers theinduced external forces and moments due to the hydrostatic and hydrodynamicpressure eld of the surrounding ocean, including the eect of waves. Gravity andviscous damping are also accounted for.

The 6-DOF model assumes that the pressure eld is unchanged by the passageof the ship, and wave induced eects are in practice limited to rst-order approxi-mations. The model is not analytical, and is therefore only suitable for simulations.The model was implemented in Matlab/Simulink using data from a specic, 281 mcontainer ship.

1-DOF Roll Model For Time-Varying Speed

Assuming that the forward speed and thus the wave encounter frequency is slowlytime-varying only and by using a quasi-steady approach to derive explicit time-domain solutions to the heave and pitch motions (which are the modes most tightlycoupled to roll), a simplied 1-DOF (roll) model which is very suitable for controland analysis purposes, was derived. As it turns out, for constant wave encounterfrequency, the 1-DOF model is identical to a Mathieu-type equation, which is

159

10. Conclusions

commonly used to describe ships in parametric roll resonance. However, unlike theMathieu equation, the model is suitable also for non-constant velocity.

The verication of the proposed 1-DOF model against the complex 6-DOFmodel was done in simulations with constant and non-constant encounter frequen-cies, and it was shown that it is able to qualitatively match the results of thefull 6-DOF model in a wide range of conditions. Furthermore, the Mathieu-typeequations was shown to not be capable of capturing the roll dynamics when theencounter frequency is time-varying.

1-DOF Roll Model for Time-Varying Heading and Speed

The 1-DOF roll model was extended to incorporate both slowly time-varying head-ing angle and surge speed. The proposed 1-DOF roll model is based on the explicittime-solutions of the heave and pitch motions by a quasi-steady approach.

The hydrodynamic coecients of the 1-DOF roll model were identied froma complex and accurate 6-DOF model of a container ship for certain xed surgespeeds and heading angles. The 6-DOF model accounts for rst-order generalizedpressure forces by integrating the instantaneous pressure eld of the ocean numer-ically over the instantaneous submerged part of the ship hull.

Based on these parameters, functional expressions for the hydrodynamic modelcoecients dependent only on the heading angle were found. The parameters ofthose functions were found by curve tting the hydrodynamic coecients to thecoecients obtained by the 6-DOF ship model for a wide range of ship speeds andheading angles.

It was shown in simulations that the proposed 1-DOF roll model qualitativelydescribes the results of the 6-DOF model well for various speeds and heading anglesand also captures the roll dynamics closely when the speed and the heading angleare time-varying.

Frequency Detuning

A necessary, but not sucient, condition for parametric resonance is that the fre-quency of the parametric excitation lies within a certain narrow band. For ships,this frequency can be changed (due to the Doppler eect) by changing the velocitysetpoints. A controller for parametric roll resonance in ships that takes advantageof this and that can drive the roll motion to zero, was derived.

Based on the derived 1-DOF roll model it was mathematically proven that asimple controller incorporating a linear change of the wave encounter frequency,accomplished by changing the forward speed of the ship, can regulate the rollmotion. This was demonstrated in simulations using both the simplied modeland the 6-DOF model. The controller is so simple that it can be implemented bya helmsman, even without a speed controller on board.

However, while the controller drives the roll angle to zero, the transient behaviorcan be problematic. Even if the controller is turned on at a very early stage, theship will have to be capable of a fairly rapid speed change to prevent high transient

160

roll angles. Frequency detuning does have the advantage that it can easily be pairedwith direct actuation, such as the use of u-tanks, ns or gyro stabilizers.

Frequency Detuning by Optimal Speed and Heading

Changes

The simple speed controller was improved by considering the possibility of changingthe heading angle. Three active control approaches for the stabilization of paramet-ric roll oscillations by frequency detuning through the simultaneous variations ofthe ship's forward speed and heading angle thus controlling the frequency of en-counter were proposed. Those control strategies feature optimality considerationswith respect to the optimal speed and heading changes to stabilize parametricallyexcited roll motion.

A model predictive control (MPC) was considered as a rst approach for the sta-bilization of parametric roll resonance. By explicitly formulating both constraintson the input and the states as well as an objective function which accounts forthe parametric roll resonance condition, a controller was presented that eectivelydrives the ship out of parametric resonance and reduces the roll motion signi-cantly.

Furthermore, the methodology of extremum seeking (ES) control was appliedto control ships exhibiting parametric roll resonance. The encounter frequency istuned in real time to its optimal set-point by dening an appropriate objectivefunction. The encounter frequency commands are mapped to the ship's forwardspeed and heading angle by formulating the control allocation (CA) problem inthe sequential least-squares framework, taking into account constraints on the ac-tuators. The speed and heading controllers guarantee exponentially stable originsof the tracking error dynamics.

Both the ES and the MPC were successfully veried in computer simulationsand it was shown that the combined variation of the ship's forward speed andheading angle in both control approaches is ecient to stabilize the roll motion ofa ship experiencing parametric roll resonance.

The ES control to the stabilization of roll parametric resonance by a combinedvariation of the ship's forward speed and heading angle were extended to take intoaccount robustness considerations. To that matter, speed and heading controllerswere designed in the framework of L1 adaptive control theory, a breakthrough inadaptive control where robustness is guaranteed in the presence of fast adaptation.Both the speed and heading controllers were shown to be able to track sinusoidalramp reference signals well, despite model uncertainties, unknown initial condi-tions, and unknown time-varying disturbances, resulting in a major improvementof the robustness.

Furthermore, a modication of the ES control was proposed. Instead of con-structing an objective function dependent on the knowledge of the encounter fre-quency range susceptible to parametric resonance, a control strategy featuring limitcycle minimization was suggested.

The suggested control approach was simulated for a ship in roll parametricresonance. The simulation results show the capability of the proposed ES control

161

10. Conclusions

scheme with limit cycle minimization combined with the L1 adaptive speed andheading controllers to stabilize parametrically excited roll motions eectively, evenin the presence of uncertainties, and disturbances.

Wave Frequency Estimation

For most active control methods attempting to detune the frequency of the para-metric excitation and the natural roll frequency, the knowledge of those frequenciesis crucial. In an attempt to contribute to the development of robust and stable fre-quency estimators for ships in parametric roll, two frequency estimation schemeswere proposed.

An EKF used as a state observer was designed to estimate the wave frequencyand wave encounter frequency of a ship experiencing parametric roll. This observeris designed on the basis of a ship model. The estimation of the wave frequencyproved to be quite dicult due to the fact that the estimation of the wave fre-quency is a parameter estimation process. It was shown that this parameter esti-mation process needs the parametric resonance to readily adjust the state estimate.Moreover the condence interval only converges very slowly since it is only adaptedby the small process noise term driving the lter. The estimate for the mean ofthe wave encounter frequency is good and converges fast. The condence intervalfor the wave encounter frequency also converges to an acceptable value quite fast.It has to be noted though that estimation of the wave encounter frequency alsobecomes more dicult when the resonance is lost. The lter is sensitive to tuningof the initial state error covariance matrix and to the tuning of the process noisecovariance matrix. This can cause a long time for convergence of especially thewave frequency. Hence, for good performance of the lter and convergence it isimportant to make a suitable choice for the process noise error covariance and theinitial state error covariance.

Furthermore, the frequency estimator for measured sinusoidal signals rst pre-sented in Aranovskii et al. [1] was extended with a switching gain. The motivationfor this is to estimate the wave encounter frequency for a ship at forward speedin a seaway. The switching gain is implemented to estimate the wave encounterfrequency when the ship is in parametric resonance and when the ship is not inparametric resonance. The switching is dependent on the amplitude of the mea-sured signal, which is also estimated online.

The frequency estimator was successfully applied in a simulation for constantspeed. The observer gain was analysed and it is a compromise between steady-stateerror and convergence rate. The frequency estimator was also successfully appliedto estimate the wave encounter frequency for time-varying speed.

162

Appendices

163

Appendix A

Appendix

In this appendix, the existence and uniqueness properties of (6.6) is proven.

From Nayfeh and Mook [100] follows the behaviour of the system when ωe is aconstant, but not when it is changing. A unique nite solution of (6.6), valid alsofor time-varying ωe, needs to be guaranteed.

To prove the existence (and uniqueness) of the solution to (6.6), the followingtheorem and lemma will be used; they are repeated here for convenience:

Theorem A.1 (Khalil [84, Theorem 3.3]). Let f (t, x) be piecewise continuous int and locally Lipschitz in x for all t ≥ t0 and all x in a domain D ⊂ Rn. Let W bea compact subset of D, x0 ∈W , and suppose it is known that every solution of

x = f (t, x) , x (t0) = x0

lies entirely in W . Then there is a unique solution that is dened for all t ≥ t0.

Lemma A.2 (Khalil [84, Lemma 3.2]). If f (t, x) and ∂f∂x (t, x) are continuous on

[a, b]×D, for some domain D ⊂ Rn, then f is locally Lipschitz in x on [a, b]×D.

Taking x =[φ, φ

]>, (6.6) can be rewritten as

x =

[x2

− d44m44

x2 − 1m44

[k44 + kφt cos

(∫ tt0ωe(τ) dτ + αφ

)]x1 −

kφ3

m44x3

1

]

=

[0 1

− k44m44

− d44m44

]x+

[0

− kφtm44

cos(∫ t

t0ωe(τ) dτ + αφ

)x1 −

kφ3

m44x3

1

]

= Ax+ g (t, x1) (A.1)

165

A. Appendix

with

f (t, x) ,

[x2

− d44m44

x2 − 1m44

[k44 + kφt cos

(∫ tt0ωe(τ) dτ + αφ

)]x1 −

kφ3

m44x3

1

]

A ,

[0 1

− k44m44

− d44m44

]

g (t, x1) ,

[0

− kφtm44

cos(∫ t


)x1 −

kφ3

m44x3

1

].

Lemma A.3. There is a unique solution of (A.1) (and thus (6.6)) dened forall t ≥ t0.

Proof. It is clear that f (t, x) of (A.1) is continuous in x for all x ∈ R2. It is alsocontinuous in t for all t ≥ t0, as long as ωe (t) is piecewise continuous. The choiceof ωe satises this.

The partial derivative of f with respect to x is given by

∂f

∂x(t, x) = A−

[0 0

kφtm44

cos(∫ t


)+ 3

kφ3

m44x2

1 0

](A.2)

which, by the same argument, is continuous in x for all x ∈ R2 and t ≥ t0. ByKhalil [84, Lemma 3.2], f is therefore locally Lipschitz in x for all t ≥ t0 and allx ∈ R2. The rst part of Khalil [84, Theorem 3.3] is then satised.

To prove that the trajectories of the system are bounded, the following Lya-punov function candidate is used:

V =1

2x>Px+

1

4

(1 +

m44

d44

)kφ3x4

1 (A.3)

with

P =

d44 + k44

(1 + m44

d44

)m44

m44 m44

(1 + m44

d44

) = P > 0 . (A.4)

The time derivative of V along the trajectories of the system (A.1) is given by

V = x>P (Ax+ g(t, x)) +

(1 +

m44

d44

)kφ3x3

1x2

= −(k44 + kφt cos

(∫ t

t0

ωe(τ) dτ + αφ

))x2

1 − d44x22 − kφ3x4

1

− kφt cos

(∫ t

t0

ωe(τ) dτ + αφ

)(1 +

m44

d44

)x1x2

≤ − (k44 − kφt)x21 − d44x

22 − kφ3x4

1 + kφt

(1 +

m44

d44

)|x1||x2| . (A.5)

166

While k44 > kφt, V is only negative denite for suciently small values of kφt.If kφt is suciently small, then the origin of the system (A.1) would be globallyuniformly exponentially stable, by Khalil [84, Theorem 4.10]. A priori it is knownthat this is not the case; in parametric resonance, the origin is, in fact, unstable.

However, V can be used to prove that the trajectories of (A.1) are bounded.For |x1| ≥ µ > 0⇒ ‖x‖ ≥ µ it holds that

V ≤ − (k44 − kφt)x21 − d44x

22 − kφ3x4

1 + kφt

(1 +

m44

d44

)|x1||x2|

≤ −d44x22 − kφ3µ2x2

1 + kφt

(1 +

m44

d44

)|x1||x2|

= −(1− δ)d44x22 − (1− δ)kφ3µ2x2

1

+ kφt

(1 +

m44

d44

)|x1||x2| − δd44x

22 − δkφ3µ2x2

1 (A.6)

for some δ ∈ (0, 1). Furthermore, the term

kφt

(1 +

m44

d44

)|x1||x2| − δd44x

22 − δkφ3µ2x2

1

is negative semidenite if

k2φt

(1 +

m44

d44

)2

≤ 4d44δ2kφ3µ2 ⇒ µ ≥ 1

2δ√d44kφ3

kφt

(1 +

m44

d44

). (A.7)

Therefore, for µ satisfying the above inequality,

V ≤ − (1− δ) d44x22 − (1− δ) kφ3µ2x2

1 (A.8)

which is negative denite. By Khalil [84, Theorem 4.18] the trajectories of (A.1)are bounded for any initial condition x(t0).

Therefore, the second condition of Khalil [84, Theorem 3.3] is satised, and thereexists a unique solution of (A.1) (and thus (6.6)) that is dened for all t ≥ t0.

167

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