rudder roll stabilisation of shipsof the ship, as indicated in the following considerations....

Rudder Roll Stabilisation of Ships

Jose De Dona

September 2004

Centre of Complex DynamicSystems and Control

Outline

1 Introduction2 Ship Motion Description and Modelling

Motion due to Rudder Control ActionMotion due to the WavesDisturbance Model Parameter EstimationComplete State Space Model

3 Control Problem DefinitionA Challenging Control Problem

4 Control System DesignCapturing All the Objectives in an Optimisation ProblemRecasting the Problem in the RHC FrameworkPosing the Problem as a QPUsing Certainty EquivalenceSummary of the Proposed Control Strategy

5 Constrained Predictive Control of Rudder-Based Stabilisers6 Conclusions


Introduction

We present a case study of control system design forrudder-based stabilisers of ships using RHC.

The rudder’s main function is to correct the heading of a ship;however, depending on the type of ship, the rudder may alsobe used to produce, or correct, roll motion.

Rudder roll stabilisation consists of using rudder-induced rollmotion to reduce the roll motion induced by waves.

An automatic control system is necessary to provide therudder command based on measurements of ship motion.


Introduction

Reduced roll motion is important for it can affect the performanceof the ship, as indicated in the following considerations.

Transverse accelerations that occur due to rolling interrupttasks performed by the crew. This increases the amount oftime required to complete a mission.

Roll accelerations may produce cargo damage, for example,on soft loads such as fruit.

Roll motion increases hull resistance.

Large roll angles limit crew capability to handle equipment onboard, and/or to launch and recover systems.


Introduction

Amongst the different types of stabilisers, rudder-basedstabilisation is a very attractive technique. The reasons for this arethat almost every ship has a rudder (thus no extra equipment maybe necessary), and also this technique can be used in conjunctionwith other stabilisers.

As we shall see, this design problem is far from trivial.


Ship Motion Description and Modelling

To describe the motion of a marine vehicle, two reference framesare considered: an inertial frame and a body-fixed frame.

Oxφ

θy

ψ

z

α

O0

Body-fixed frame

Inertial frame

Rollp Surge

x, u

Pitchq

Swayy, vYaw

r

Heavez,w



The motion of a marine vehicle can be considered in six degrees offreedom.

The table shows the notation used to describe the different motioncomponents.

translation surge sway heave

position x y zlinear rate u v w

rotation roll pitch yaw

angle φ θ ψ

angular rate p q r



For marine vehicles position and orientation are describedrelative to the inertial reference frame, whilst linear andangular velocities are expressed in the body-fixed frame. Thischoice is convenient since some of these magnitudes aremeasured on board, and thus, relative to the body-fixed frame.

A convenient abstraction to obtain a mathematical model thatcaptures the different effects that give rise to ship motion, is toseparate the motion due to control action (rudder motion) fromthe motion due to the waves. This abstraction results in twomodels that can be combined using superposition.


Motion due to Rudder Control Action

The first part of the model can be obtained using Newton’slaws. This approach yields a nonlinear model that describesthe motion components in terms of the forces and momentsacting on the hull. By linearising this model, we can obtain alinear state space model that describes the ship response dueto the rudder (control) action.

A discrete time version of this model can be expressed as

xck+1 = Acxc

k + Bcuk , (1)

where the state xck and control uk , are given by

xck =[vc

k pck rc

k φck ψc

k

]tand uk = αk , (2)

with αk being the current rudder angle.


Motion due to Rudder Control Action

The parameters of the model (1), the values of the entries ofthe matrices Ac and Bc, will vary with the forward speed of thevessel. However, this variation is such that constant valuescan be considered for different speed ranges. Systemidentification techniques and data collected from tests can beused to estimate the parameters for different speed ranges.

The different sets of parameters can then be used in a gainscheduling-like approach to update the model. This helps tominimise model uncertainty.


Motion due to the Waves

The second part of the model incorporates the motioninduced by the waves.

The sea surface elevation can be described in stochasticterms by its power spectral density, or sea spectrum.

The ship motion induced by the waves can be interpreted as afiltering process made by the ship’s hull, which has a selectedresponse to certain frequencies and attenuates others. Thefrequency response of the hull due to wave excitation is calledthe ship response operator.



The total effect can be incorporated into our model as a colourednoise output disturbance. The roll motion induced by the waves willthus be modelled with a shaping filter:

xwk+1 = Awxw

k + wk ,

ywk = xw

k + vk ,(3)

where

xw = [pw φw]t,

and wk and vk are sequences of i.i.d. Gaussian vectors withappropriate dimensions.



For a given hull shape, the filtering characteristics of the hulldepend on the forward speed of the ship U, and the heading anglerelative to the waves: the encounter angle χ.

x0

y0

c

λ

Wave profile

U0 deg

90 deg

180 degχFollowing seas

Quartering seas

Beam seas

Bow seas

Head seas



The variations in the characteristics of the motion responsedue to speed and encounter angle are the consequence of aDoppler-like effect. The frequency observed from the ship iscalled the encounter frequency ωe, and is, in general, differentfrom the wave frequency ωw (observed from a fixed-referenceframe):

ωe = ωw − ω2wUg

cos χ. (4)

The encounter effect produces significant variations in themotion response of the ship even for the same sea state,hereby defined by the sea spectrum. Consequently, thevalues of the parameters of the model (3) should be updatedfor changes in different sea states and sailing conditions.


Disturbance Model Parameter Estimation

The model of the disturbance (waves) needs to be updated whenthe sea state or sailing conditions change. Here, we present asimple approach to estimate the parameters based on the controlscheme shown in the figure.

Controlcommand

Motion(measurements)

RudderShipdynamics

Environmentaldisturbances

Kalman filterController / control algorithm

Hydraulicmachinery

Disturbanceparameter estimator

Open loop

Closed loop

Controlaction



If the stabiliser control loop is open—that is, the rudder is kept tozero angle, and only minor corrections are applied to the rudder tokeep the course—we can then use the roll angle and roll ratemeasurements to estimate the parameters of a second ordershaping filter.

Under these conditions, the measurements coincide with the stateof the following shaping filter:

[φw

k+1pw

k+1

]=

[θ11 θ12

θ21 θ22

] [φw

kpw

k

]+

[wφ

kwp

k

].



By defining the vector

θk =[θ11(k ) θ12(k ) θ21(k ) θ22(k )

]t,

we can express the available measurements as

θk+1 = θk + θwk ,

[φw

kpw

k

]=

[φw

k−1 pwk−1 0 0

0 0 φwk−1 pw

k−1

]θk + vk .

(5)

The system (5) is in a form that we can apply a Kalman filter toestimate θk |k from the measurements φw

k and pwk . The variable θwk

represents a small random variable that accounts for unmodelleddynamics.



To assess the proposed method to estimate the parameters ofthe filter and the quality of the prediction using the abovemodel, a series of simulations were performed for differentsea states and sailing conditions.

The parameters to describe the sea state that were used arethe average wave period T and the significant wave heightHs (average of the highest one third of the wave heights).These parameters are used in the International Towing TankConference recommended model for the wave power spectraldensity, commonly termed ITTC spectrum in the marineliterature:

Sζζ(ωw) =172.75H2

s

T4ω5w

exp −691

T4ω4w

(m2sec/rad). (6)



The process for obtaining the ship roll power spectral density isindicated via an example in the figure.

0 1 2 30

1

2

3

Ww [rad/sec]

Sw

ITTC Hs [m]:4, T [sec]:7

0 1 2 30

0.005

0.01

Ww [rad/sec]

Sw

−sl

ope

ITTC Hs [m]:4, T [sec]:7

0 1 2 30

5

10

15

20

Ww [rad/sec]

RO

Mag

Speed [kt]: 15, Enc. Angle [deg]:90

0 1 2 30

0.05

0.1

0.15

0.2

Ww [rad/sec]

ps−

S−

roll

RMS roll:12.0569

0 1 2 30

0.05

0.1

0.15

0.2

We [rad/sec]

S−

roll

RMS roll:12.0569

0 1 2 30

1

2

3

Ww [rad/sec]

We

[rad

/sec

]



Once the roll power spectral density is obtained, the rollmotion realisations (time series) are computed as

φ(t) =∑

i

φi sin(ωei t + θi), (7)

where the phases are chosen randomly with a uniformdistribution in [−π, π] and the amplitudes calculated from

φ2i = 2Sφφ(ωei)∆ωi , (8)

where Sφφ(ωe) represents the roll power spectral density(S-roll in the previous figure).The number of regular (sinusoidal) components used tosimulate the time series is normally between 500 and 1000 toavoid pattern repetition depending on the total simulation time.This procedure for simulating time series for a ship responseis a standard practice in naval architecture and marineengineering.



The measurements taken from one of the realisations were used toestimate the parameters. The figure shows the evolution of theestimates of the parameters. A sampling period of 0.25 sec wasadopted based on the value of the roll natural period of the vesselof approx 7 sec.

0 50 100 150 200−1

−0.5

0

0.5

1ITTC Hs [m]:4, T [sec]:7 Speed [kt]: 15, Enc. Angle [deg]:90

Av last 10 est =0.96631

0 50 100 150 200−2

−1.5

−1

−0.5

0

0.5

0 50 100 150 200−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0Av last 10 est =−0.26367

0 50 100 150 2000.88

0.9

0.92

0.94

0.96

0.98

1Av last 10 est =0.95752

Av last 10 est =0.24582

t [sec]t [sec]

t [sec]t [sec]

θ 11

θ 12

θ 21

θ 22



The figure shows the roll angle and rate predictions for two otherdifferent realisations using 5 and 10 step-ahead predictions. Thisconsists of using the measured state as initial condition for model(3) and then running this model forward in time.

0 100 200 300 400−60

−40

−20

0

20

40

60N−step predictor with sampling period Ts=0.25

ActualPredicted N=5

0 100 200 300 400−60

−40

−20

0

20

40

60ITTC Hs [m]:4, T [sec]:7 Speed [kt]: 15, Enc. Angle [deg]:90

ActualPredicted N=5

0 100 200 300 400−50

0

50N−step ahead predictor with sampling period Ts=0.25

ActualPredicted N=10

0 100 200 300 400−50

0

50ActualPredicted N=10

φk, φ

k|k−

Nφ

k, φ

k|k−

N

p k,p

k|k−

Np k

,pk|k−

N

kk

kk



From the above example we can see that the filter convergesrelatively quickly, and the quality of predictions for thebehaviour of the ship can be deemed satisfactory.

We note that, for the chosen sailing conditions, the use of asecond order disturbance model seems to give good results.For other sea conditions, it may be necessary to resort tohigher order models.

This process should be performed before closing the controlloop. The proposed control strategy can be considered aquasiadaptive control strategy. That is, if the sailing condition(heading and speed) or the sea state changes, it may benecessary to open the loop and re-estimate the parameters ofthe disturbance model to avoid significant degradation in theclosed loop performance.


Complete State Space Model

Thus, we have:

A model for the ship motion due to the rudder control action:

xck+1 = Acxc

k + Bcuk ,

where the state xck and control uk , are given by

xck =[vc

k pck rc

k φck ψc

k

]tand uk = αk ,

with αk being the rudder angle.

A model for the roll motion induced by the waves:

xwk+1 = Awxw

k + wk ,

ywk = xw

k + vk ,

wherexw = [pw φw]t,

and wk and vk are sequences of i.i.d. Gaussian vectors.



The complete state space model can then be represented via stateaugmentation as

xk+1 = Axk + Buk + Jwk ,

yk = Cxk + nk ,

where xt = [xck xw

k ].

The following measurements are assumed available to implementthe control:

yk =[pk rk φk ψk

]t+ nk

=[(pc

k + pwk ) r f

k (φck + φ

wk ) ψf

k

]t+ nk ,

(9)

where, nk is noise introduced by the sensors, and rfk and ψf

k arethe filtered yaw rate and yaw angle respectively.



Note that we have only considered the wave disturbance affectingthe roll angle and the roll rate but not the yaw. The reason for thisis that, in conventional autopilot design, the yaw is filtered and onlylow frequency yaw motion is corrected. This is done to avoid theautopilot making corrections to account for the sinusoidalwave-induced yaw motion. Therefore, in the problem we areassuming that the yaw is measured after the yaw wave filter.


A Challenging Control Problem

The design of a control strategy that uses the rudder forsimultaneous course keeping and roll reduction is not a simple taskand must be performed so as to best deal with the following issues:

Underactuated System. One control action (rudder force)achieves two control objectives: roll reduction and lowheading (yaw) interference.

A key fact to understand, for the successful application of thistechnique, is that the dynamics associated with therudder-induced roll motion are faster than the dynamicsassociated with the rudder-induced yaw motion. Thisphenomenon depends on the shape of the hull and thelocation of the rudder and the centre of gravity of the ship.



Uncertainty. There are three sources of uncertainty:

First, there is incomplete state information available (thenecessary sensors can be very expensive).

Second, there are disturbances from the environment(wave-induced motion) that, in principle, cannot be known apriori.

Third, there is uncertainty associated with the accuracy of themodel.

Nonminimum Phase System. The response of roll due torudder action presents nonminimum phase dynamics. Thisimposes fundamental design limitations and trade-offs.



Input constraints. The rudder action demanded by thecontroller should satisfy rate and magnitude constraints.

Rate constraints are associated with safety and reliability. Byimposing rate constraints on the rudder command, we ensurean adequate lifespan of the hydraulic actuators.

Magnitude constraints are associated with performance andeconomy. Large rudder angles induce flow separation (loss ofactuation and poor performance), and a significant increase indrag (resistance). Also, it is desirable to reduce the maximumrudder action at higher speeds to reduce the mechanicalloads on the rudder and the steering machinery.



Output constraints. Since the rudder affects the shipheading, it may be necessary to include constraints on themaximum heading deviations allowed when the rudder is usedto reduce roll.

Unstable plant. The response of yaw to rudder action ismarginally unstable: there is an integrator.

Based on the above considerations, it is evident that the problemof rudder roll stabilisation of ships is a challenging one and, assuch, the chosen control strategy plays an important role inachieving high performance.


Output Feedback Control

The basic control objectives for the particular motion controlproblem being addressed here are as follows:

(i) minimise the roll motion, which includes roll angle andaccelerations;

(ii) produce low interference with yaw;

(iii) satisfy input constraints.



In a discrete time framework, all the above objectives are capturedin the following optimisation problem.

Definition (Output Feedback Control with Input Constraints)

Find the feedback control command uk = K(yk ) that minimises theobjective function

V = limN→∞

1N

E N∑

k=0

ytk Qyk + (yk+1 − yk )tS(yk+1 − yk ) + utk Ruk

(10)

subject to the system equations

xk+1 = Axk + Buk + Jwk ,

yk = Cxk + nk ,

and the input constraints |uk | ≤ umax and |uk+1 − uk | ≤ δumax.



Choosing the matrices Q and S as:

Q = diagQp , 0,Qφ,Qψ,S = diagSp , 0, 0, 0,

the objective function (10) becomes (assuming no sensor noise)

V = limN→∞

1N

E N∑

k=0

[Qpp2

k + Qφφ2k + Sp(pk+1 − pk )2

]+ Qψψ

2k + Rα2

k

.

(11)

The objective function (11) incorporates a term that weights the rollaccelerations via the difference pk+1 − pk . The reason is that bothroll angle and roll acceleration affect the performance of the ship;and are directly related to the criteria often used to evaluate shipperformance in the marine environment.


Finite Horizon Optimal Control Problem

The problem defined above is not easy to solve due to thepresence of constraints. We will approximate its solution using thecertainty equivalent solution of an associated finite horizonproblem, together with a receding horizon implementation.

Given the initial condition x0, we seek the sequence of controlmoves

uopt0 (x0), . . . , uoptN−1(x0) (12)

that minimises the objective function

VN 12

xtNPxN +

N−1∑j=0

12

(xtj Q xj + utj R uj + utj T xj + xtj T tuj), (13)

subject to

xj+1 = Axj + Buj ,

yj = Cxj ,(14)


Finite Horizon Optimal Control Problem

and the constraints

|uj | ≤ umax and |uj+1 − uj | ≤ δ|umax.

The augmented state x in (14) is given by

x =[vc pc rc φc ψc φw pw

]t.

The matrices in the objective function are

Q = (A − I)t(CtSC)(A − I) + CtQC ,

R = Bt(CtSC)B + R , T = Bt(CtSC)(A − I).

The matrices Q, S and R are the parameters defining the objectivefunction (10), and the matrices A, B describing the augmentedsystem are

A =

[Ac 00 Aw

], B =

[Bc

0

], (15)

The matrix P in (13) is taken as the solution of the correspondingdiscrete time algebraic Riccati equation.


Posing the Problem as a QP

The QP solution of the above problem is given by

uopt(x0) = arg minLu≤W

12

ut(H1 + H2)u + ut(F1 + F2)x0, (16)

where

H1 = ΓtQΓ + R, H2 = TΓ + ΓtTt,

F1 = ΓtQΩ, F2 = TΩ,

Q = diagQ , . . . , Q , P,R = diagR , . . . , R,T = diagT , . . . , T ,


Converting the Problem to a QP

Ω =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

AA2

...

AN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, Ω =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

IA...

AN−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

Γ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

B 0 · · · 0AB B · · · 0...

.... . .

...

AN−1B AN−2B · · · B

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Γ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 · · · 0B 0 · · · 0...

.... . .

...

AN−2B AN−3B · · · 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.


Converting the Problem to a QP

The matrices L and W that define the constraint set in (16) aregiven by

L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

IE−I−E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ; W =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Mmag

Mrate

Mmag

Mrate

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦where I is the N × N identity matrix and

E =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 · · · 0−1 1 0...

. . .. . .

...

0 · · · −1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, Mmag =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣umax

...

umax

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , Mrate =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

u−1 + δumax

δumax

...

δumax

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The above problem is be solved on line, and the implicit recedinghorizon feedback control law is implemented, that is,

uk = KN(x0(yk )) = uopt0 (x0(yk )),

the first element of the optimal sequence (12).Centre of Complex DynamicSystems and Control

Summary of the Proposed Control Strategy

To summarise the proposed control strategy, the following stepsare envisaged at each sampling instant:

(i) Take measurements, that is, obtain yk (see (9)) and theprevious control action uk−1.

(ii) Update the state prediction (17) and estimate the state xk |kusing (18) and the measured output.

(iii) Using uk−1 and the initial condition x0 = xk |k solve the QP (16)to obtain the sequence of controls (12).

(iv) Update the control command uck = uopt0 (x0).


Constrained Predictive Control of Rudder-BasedStabilisers

Simulation studies were carried out to assess the performance of arudder-based stabiliser designed according to the proposedstrategy.

A speed of 15 kts for the simulations. This is the nominal speed ofthe vessel chosen and also the speed at which this vesselperforms its missions most of the time.

The performance was assessed for the following scenarios:

Case A: Beam seas (χ = 90 deg), Hs = 2.5 m, T = 7.5 sec;

Case B: Quartering seas (χ = 45 deg), Hs = 2.5 mT = 7.5 sec;

Case C: Bow seas (χ = 135 deg), Hs = 4 m, T = 9.5 sec.



The wave heights (Hs) were chosen to represent moderate andrough conditions under which a vessel the size of this naval vesselcan be expected to perform. The particular wave average periodsT are the most probable periods for the chosen wave heights inocean areas around Australia.

The control action was updated with a sampling rate of 0.25 sec.The maximum rudder angle was limited to 25 deg, and maximumrudder rate was limited to 20 deg/sec.



The performance was assessed via

(i) percentage of reduction in roll angle variance and RMS value;

(ii) yaw angle RMS value induced by the rudder;

(iii) percentage of reduction of motion induced interruptions [MII].

MII is an index that depends on the roll angle and roll accelerationand yields the number of interruptions per minute that a worker canexpect due to tipping and loss of balance.



The tuning was performed in beam seas, and then the parametersof the controller were fixed for the rest of the simulation scenarios.The only part of the controller that changed with each sailingcondition was the model for the disturbance, which was estimatedprior to closing the loop.

Different prediction horizons were tested. As expected, for shortprediction horizons (N = 1 and N = 2), the performance waspoorer than for longer horizons (N = 5 and N = 10). For a horizonof 10 samples periods, an improvement of 10% in roll reductionwas achieved with respect to that obtained with N = 1. Forhorizons longer than 10 sample periods there was no significantimprovement. Therefore, this is the horizon that was adopted so asto limit the size of the QP problem.


Simulation Results. Case A: Beam seas

0 50 100 150 200 250 300−20

0

20ITTC spectrum, (Hs:2.5[m], T :7.5 [sec]), Speed: 15 [kt], Enc. Angle:90[deg]

Rol

lOpen loopClosed

0 50 100 150 200 250 300−20

0

20

Rol

l Rat

e

50 100 150 200 250 300

−20

0

20

Rol

l Acc

0 50 100 150 200 250 300−10

0

10

Yaw

50 100 150 200 250 300

−20

0

20

time [sec]

Rud

der


Simulation Results. Case A: Beam seas

This case is close to the worst condition that the ship canexperience in regard to roll motion for the assumed sea state:beam seas. The wave period in this case is close to thenatural roll period, which is approximately 7 sec. Therefore,the roll excitation due to the waves is close to resonance.Notwithstanding this, we can still observe good performance.

Results obtained from over 20 different realisations indicateroll reductions on the order of 55–60% for roll RMS values. Asignificant improvement is achieved, however, in regard to MII:80–90% reduction.

From the rudder action depicted in the figure, we can see thatthe controller generates a command that satisfies themagnitude constraints.


Simulation Results. Case B: Quartering seas

50 100 150 200 250 300

−10

0

10

ITTC spectrum, (Hs:2.5[m], T :7.5 [sec]), Speed: 15 [kt], Enc. Angle:45[deg]

Rol

lOpen loopClosed

0 50 100 150 200 250 300−10

0

10

Rol

l Rat

e

50 100 150 200 250 300−10

−505

Rol

l Acc

0 50 100 150 200 250 300−20

0

20

Yaw

0 50 100 150 200 250 300−50

0

50

time [sec]

Rud

der


Simulation Results. Case B: Quartering seas

The performance in quartering seas decreases significantly.This is expected due to the low encounter frequency of thedisturbance.

As depicted in the figure, due to the high interference with yawfor sailing conditions having low encounter frequency, it maybe necessary to incorporate an output constraint in order tolimit the maximum heading deviation. The low encounterfrequency, here appearing in quartering seas, may also bepresent in other sailing conditions if the sea state is given byvery large period waves produced in severe storms.


Simulation Results. Case C: Bow seas

50 100 150 200 250 300

−10

0

10

ITTC spectrum, (Hs:4.5[m], T :9.5 [sec]), Speed: 15 [kt], Enc. Angle:135[deg]

Rol

lOpen loopClosed

0 50 100 150 200 250 300−20

0

20

Rol

l Rat

e

0 50 100 150 200 250 300−20

0

20

Rol

l Acc

0 50 100 150 200 250 300−10

0

10

Yaw

0 50 100 150 200 250 300−50

0

50

time [sec]

Rud

der


Simulation Results. Case C: Bow seas

This case presents the best performance despite the moresevere sea state: 4 m waves. If we compare the RMS of roll inopen loop with that of case B, we can see that these aresimilar. However, due to the higher encounter frequency of thedisturbances in case C, the roll reduction is significantly better.


Conclusions

We have presented a case study and control system designfor a problem of significant practical importance.

The simplifying assumptions have been kept to a minimum.Hence, almost all aspects of the design process have beenaddressed to some degree.

The RHC formulation offers a unified framework to addressmany of the difficulties associated with the control systemdesign for this particular problem: multivariable nature,constraints, uncertainty, stochastic disturbance rejection.

The simulations presented illustrate the performance of RHCand suggest that the methodology should be successful inpractical applications.


rudder roll stabilisation of shipsof the ship, as indicated in the following considerations....

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