A Method to Determine the Applicability of Rudder Roll Stabilisation for Ships

Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993

A METHOD TO DETERMINE THE APPLICABILITY OF

RUDDER ROLL STABILISATION FOR SHIPS G.N. Roberts Department o/Control Enginering. Royal Naval Engineering College. MafUllion. Plymouth PL5 3AQ. UK

Abstract. The problem addressed is that of assessing the suitability of Rudder Roll Stabilisation (RRS) for ships. TIris paper proposes a method whereby ship operators can assess the suitability of an RRS System for their ship and allows the operator to establish the benefits of incorporating such a roll stabilisatioo system prior to embarking 00 a costly design exercise. Keywords: Ship Motion Control. Rudder Roll Stabilisation

1. INTRODUCTION

need for a simple, straightforward method to establish the applicability of RRS for a particular ship, and to quantify the benefits to be gained from the installation of such stabilisation system.

A review of conferences or journals dedicated to marine automation which have taken place over the last decade or so will reveal a significant volume of research into the application of the technique of achieving roll stabilisation of ships by means of the rudder. This approach to improving roll damping, known as Rudder Roll Stabilisation, is based on the concept that the inward heel which results from rudder displacement can be utilised to compensate for ship roll motions induced by the sea.

2. THE CONTROL PROBLEM Before discussing the procedure for quantifying the effectiveness of an RRS system for a particular ship it is enlightening to firstly examine the control problem and the solution. The block diagram repre-sentation of the control configuration for RRS is given in Fig 1, where GRR(s) represents the ship dynamics relating roll angle to rudder input, GC1I(S) represents the transfer function of the RRS controller and D(s) represents the sea state disturbance input. It should be noted that this is not a linear process as is depicted by this figure, although, as a first approximation the block diagram adequately describes the control configuration.

The effectiveness of an RRS system is primarily a function of rudder speed, and it has been established, van der Klugt (1987), van Amerongen (1989), that a ruddder speed in the region of 20° .S-l is necessary for an RRS system to be as effective as fm stabilisers. The requirement for high speed rudders does impose significant additional costs penalties for the rudder servomechanism or steering machine as a result of the requirement for additional hydraulic power and a necessity for stronger stock bearings because of the extra torques and increased rudder duty cycles. Consequently, the application of a full RRS system is only a practical solution for new ships where a faster rudder is specified at the design stage, or to ships which undergo a major refit during which the rudder servomechanism is replaced and/or uprated.

2.1 Ship Dynamics The transfer function describing ship dynamics, GRR(s), can be broken down into two main constituent parts; the roll angle to rudder position transfer function, and the non-linear behaviour of the rudder servomechanism. The roll angle to rudder position transfer function will have the standard form :

(1)

It is possible to obtain a measure of roll reduction with rudder speeds slower than the optimum value, Roberts and Braham (1990), and it may be that the roll reduction obtained using the existing rudder system may be adequate. Thus, in order for the ship operator to be able to make this decision there is a

where: ~ is the roll angle (0), b. is the actual rudder angle r), Wo is the undamped natural roll frequency (rad.s-1), 1; is the roll damping ratio, T1 is the lead term

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Sea State Disturbance D(s)

Desired Roll Angle (00)

RRS Controller

RolVRudder Dynamics'

GCR (S)

GRR(S)

J--..-- Stabilised Roll

~(s)

Figure 1. RRS Control Configuration time constant (seconds), T2 is the lag term time constant (seconds) and K' = speed related gain which accounts for variations in rudder force, which increase with ship's forward speed. Generally K' is relatively constant over a reasonable speed range so that gain scheduling can be used to update the controller gain accordingly. Simulated yaw and roll angle time responses of a warship to a step demand applied to the rudder are shown in Figure 1. There are three important ship characteristics, pertinent to RRS, which can be observed in this Figure: (1). The presence of the initial inward heel or non-minimum-phase characteristic, caused by the right hand plane zero (1 - sT1) in equation 1, is established. (2). The magnitude of this initial heel can be observed - this gives an indication of the maximum roll reduction possible, and: (3). The difference between the speed of the roll and yaw dynamics is apparent. This speed difference is a requirement for RRS in order that the larger rudder movement, to control roll motions, has negligible effect on the yaw response.

2.1.1 Rudder Servomechanism The rudder servomechanism contributes the major non-linearity in ship dynamics. The rate at which the actuator can move (slew rate or rudder speed) is a function of the physical size and power of the mechanical system, and the range of actuator movement is restricted by practical considerations. Typically rudder movement is rate limited whenever the error between the actual rudder angle and the demanded rudder angle is greater than 3 ° and for errors of less than 3° the movement is linear, Mort (1982).

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Assuming that some means is employed to keep the rudder demand below the maximum excursion limits the effect of the non-linearity is to introduce a phaselag whenever the rudder demand is rate limited. Sharif (1992) has shown that by using frequency domain techniques a linear approximation to the servomechanism may be obtained, the process leading to:

3..(s) 3 j.s)

1 (1 + as)(l + bs)

(2)

where bd is the desired rudder angle (0) Sharif found that the coefficients, a and b, varied with rudder speed and also the magnitude of rudder demand.

2.3 ControUer Design The aim of the roll stabilisation system is to ensure that the roll moment generated by the control system opposes the roll moment generated by the waves. Two distinct control philosophies have dominated the design of active roll stabilisation systems, with classical frequency-domain sensitivity analysis being the basis of the early fin/roll controller designs Connoly (1968), Tinn (1970), whilst more recently the Linear Quadratic Gaussian (LQG) technique has been used successfully, Byrne et al. (1986), Katebi et al. (1987) and others. These design approaches are equally applicable to controllers designed for rudder/roll stabilisation, although the majority of designs reported in the open literature have employed LQG or derivatives of LQG. As the aim of the paper is to present a methodology which can be easily used to establish the feasibility of an RRS system, the type of control is incidental, the overriding requirement is that controller design should be relatively straightforward. It is therefore proposed that controller design is to be achieved via frequency response methods using a controller structure G~(s), Lloyd (1988), where:

Gc.CjwG ••Cio'> lOCUS (3)

where Ks is the speed dependent gain, K is the roll angle sensitivity, KR is the roll rate sensitivity and KA is the roll acceleration sensitivity. Here the controller has to compensate for the phaselag introduced by combined effect of the rudder to roll dynamics (equation 1) and the rudder servomechanism (equation 2). The coefficients, Av A2 and A3 of the denominator of equation 3 are selected so that the phase-lag introduced at the ship's undamped natural roll frequency is negligible. In this way the required phase lead becomes only a function of the numerator coefficients.

INCREASlNGW

Figure 2. Typical RRS Nyquist Locus

If ~. is defined as the unstabilised roll, ~. as the stabilised roll and ~r as the roll reduction then:

2.3 Predicted RoD Reduction In this section it is shown that by adopting a frequency response representation it is possible to predict roll reduction. Assuming the linear representation of the rudder servomechanism, the rudder roll control loop is defined by the transfer function : ~s)

D(s)

1

(5)

ie:

(4)

then:

where D(s) represents the sea state disturbance input.

and

~,

=

1

(6)

~

=

1 _ _ _ _1=--_ _ 1 + GcijfJ».GmlifJ»

(7)

, as

for

any

particular

frequency

I 1 + Gcijro ).G/lJf(jro) Ican be measured directly from the Nyquist plot, an estimate of roll reduction may be achieved using equation 7. What is required therefore to be able to give an estimate of the roll reduction which would be obtained with a RRS system is:

Using classical frequency-domain sensitivity analysis it is seen that roll reduction will occur providing that I 1 + Gcijro).G/lJfVro) Iis greater than unity over the frequency range of interest. The performance of the roll stabilisation system can therefore be assessed by consideration of the Nyquist locus of Gc.(jro).GRR(jm). Figure 2 gives a typical roll control loop Nyquist At a particular frequency on the locus locus.

(1). details of the roll to rudder transfer function (equation 1).

I 1 + Gcijro).G/lJfVro) I represents the distance from the (-1,0) point to the G~(jm).GRR(jm) locus, and hence roll reduction occurs for all those frequencies for which the Nyquist locus lies outside the unit circle centred at (-1,0). Conversely, amplification of roll motion will occur at frequencies for which the Nyquist locus lies inside the unit circle.

(2). The rudder speed in order to obtain the linearised transfer function (equation 2). (3). Details of the coefficients of the controller G~(s), (equation 3). The key to this estimation process is therefore a knowledge of the coefficients of equation 1, ie. K', T1,T1, 1;; and m., as without these it is not possible to establish the coefficients for equation 4. In an earlier paper, Roberts (1993), a method was given whereby the coefficienrnts of equation 1 could be obtained from simple ship trials. In this paper controller synthesis based on such data will be presented.

Uoyd (1988) has shown that satisfactory roll reduction over the frequency range of interest can be achieved if the phase angle of G~(jm).GRR(jm) is zero at the ship's undamped natural roll frequency m•. Consequently roll controller design involves selecting the controller coefficients in order to introduce a phase-advance equal to the phase-lag resulting from the warship's roll dynamics and the rudder servomechanism, whilst at the same time ensuring that the frequency at which the G~(jm).GRR(jm) locus enters the unit circle is kept as high as possible.

3.3 ControDer Synthesis Consider the loop transfer function G~(jm).GRR(jm), as previously stated the requirement is for a phase shift of

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4. CONCLUSIONS

zero degrees at the ship's undamped natural frequency . Once the parameters for the ship dynamics (equation 1) and the rudder servomechanism (equation 2) are established the phase lag at wn can be computed. All that remains is to select the numerator coefficients for equation 3 such that the overall phase shift is zero ie. for s jWn' the values of K, KR and KA which render zero. the imaginary part of G~(jW)GRR(jW) Alternatively the values of K, KR and KA can be determined by equating the resulting phase-lead (8J, at the frequency wn' to the phase-Iag from equation 1 (8 1), equation 2 (8 2) and the denominator of equation 3. (8 3), ie:

In this paper a method of predicting the feasibility of an RRS system for ships has been described. The method described provides ship operators with a means of establishing at an early stage the suitability of incorporating an RRS stabilisation system. These procedures have been followed in a successful design evaluation, Dathan (1992), where a twelve per cent roll reduction was predicted and up to twenty per cent roll reduction was achieved during full scale sea trials.

=

6I°

(8)

82.°

(9)

6° 3

(10)

5. REFERENCES van Amerongen, l , (1976)., 'Adaptive steering of ships - A model reference approach to economical course-keeping', PhD Thesis, Delft University of Technology . van Amerongen, l , (1989)., 'Rudder roll damping experience in the Netherlands', IFAC Workshop, Expert Systems and Signal Processing in Marine Automation, Lyngby, Denmark. Byrne, l , Katebi, M.R. and Grimble, MJ. (1987)., 'LOG autopilot and rudder roll stabilisation control system design', 6th. Ship Control System Symposium, The Hague Connolly, lE., (1968)., 'Rolling and its stabilisation by active fms', Transactions Royal Institute of Naval Architects Vol.111 , pp21-48. Dathan, T.J., (1992)., 'Rudder roll stabilisation Peacock Class', MSc Dissertation, RNEC Report SP-92003. RESTRICTED Katebi, M.R., Wong, D.K.K. and Grimble, MJ., (1987). 'LOG autopilot and rudder roll stabilisation control system design', 8th. Ship Control System Symposium, The Hague . van der Klugt, P.G.M., (1987)., 'Rudder roll stabilisation', PhD Thesis, Delft University/van Reitschoten and Houwens BV, Rotterdam. Uoyd, A.RJ.M., (1988). 'Seakeeping - ship behaviour in rough weather', Ellis Horwood, London. Lewis E.V. in Comstock, J.P., (1966). 'Principles of Nayal Architecture', Society of Naval Architects and Marine Engineers, New York. Mort, N., (1982)., 'Autopilot design for surface ship steering using self-tuning controller algorithms', PhD Thesis, University of Sheffield. Roberts, G.N. and Braham, S.W ., (1990)., 'Warship roll stabilisation using integrated control of rudder and fins', 9th. Ship Control Systems Symposium, Bethesda, USA. Roberts, G.N., (1993). 'A note on the applicability of rudder roll stabilisation for ships', Proc. American Control Conference, San Francisco. Sharif M.T., (1993)., 'Frequency response of a rudder servomechanism', RNEC Research report RR92028. Tinn, S., (1970)., 'Control system for active fm roll stabilisation, Naval Engineers Journal, pp78-85.

and the total phase-Iag at wn is given by: (11) Now if the parameters of equations 1 and 2 are known, then 8. can be calculated from equation 11, and the required phase-lead, 8e , which the controller has to introduce is also defined, ie:

6° c = 6,° where:

= tan-I 6° c

(12)

KRw" 2.

(13)

K - KAw.

However, it is clear that with one equation and three unknowns, this condition can be achieved by a large number of combinations of controller sensitivities with each combination providing a degree of roll reduction. In selecting the controller sensitivities the aim is to produce roll reduction over the largest bandwidth for which the G~(jW) . GRR(jW) locus lies outside the unit circle whilst at the same time minimising the effect on yaw. Katebi et al (1987) has shown that the latter constraint is achieved by ensuring that the roll angle sensitivity is small compared to the roll rate and roll acceleration sensitivities. Therefore by pre-selecting K to be unity a relationship between KR and KA can be established using:

(14)

Finally Ks is selected, by considering the data in Sharif (1993) so that the maximum rudder movement will be at an appropriate level.

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