Sensitivity reduction in ship-manoeuvringperformance via nonlinear compensation

M.J. Ashworth, B.Sc, M.Phil., Ph.D., C.Eng., M.I.E.R.E., andProf. D.R. Towill, D.Sc, C.Eng., M.I.Prod.E., F.I.E.R.E., F.I.P.C.

Indexing terms: Feedback, Nonlinear systems, Sensitivity

Abstract: During a manoeuvre it is frequently found that a ship's yaw dynamics vary widely. To obtain aconsistent response to the helm, it is therefore advisable to provide an autopilot to cope with plant uncer-tainties. Using the method of Horowitz, a high-loop-gain linear system may be designed which achieves anacceptable spread of transient responses. Unfortunately, there is a penalty to be paid in the form of excessiverudder activity when course keeping. It is therefore desirable to find a way of reducing the loop-gain require-ment by attempting to remove some of the plant ignorance. The paper therefore proposes to reduce theplant uncertainty by using an inverse nonlinear compensator derived from a simple linear time-invariantmodel of the ship. It is then possible to control the resultant modified (and of reduced ignorance) plant witha much smaller loop gain. The paper shows that the design results in consistent manoeuvring performancecoupled with much reduced rudder activity during course keeping. The complete compensator is readilyrealisable in DDC.

1 Introduction

Methods of frequency-domain synthesis for linear time-invariant feedback systems, with the objective of achievingbounded sensitivity of response despite significant parametricvariation of the plant, have been available for many years[1—3]. Synthesis begins with the definition of the sensitivityrequirement and its formulation in the frequency domain andproceeds to the derivation of a minimum-magnitude loopfunction which is dependent upon the degree of plant uncer-tainty. When, however, the plant is inherently nonlinear andthe variability of performance arising from this source can beattributed instead to the uncertainty of a postulated equivalentlinear system, the hypothetical range of plant parametricvariation may have to assume exessive values. The designer isthen impelled to implement an extensive loop bandwidthwhich, with the existence of measurement noise or disturbancein the feedback path, can lead to either the saturation of theplant drive or inordinate control activity [4].

In such a situation, the solution may simply be to effect areduction of the noise level by the use of improved or higher-quality transducers or a rearrangement of the physical layout.With certain systems, however, the noise may arise as anunavoidable consequence of the environment, and the designermay be faced with no option other than to limit the allowableregime of plant variation or to accept an inferior sensitivityperformance from his system. Such cases have been reported[5—7] where, rather than accept the latter negative alternative,a reduction of the extent of the plant variational regime isbrought about the expedient of inverse nonlinear precom-pensation. Cancellation of major nonlinear behaviour need notbe exact since any remnant uncertainty of the modified plantwill have its effect curtailed by an outer linear minor feedbackloop, which will now be of a much reduced bandwidth.

This paper discusses the application of the technique ofinverse nonlinear precompensation to the synthesis of acontroller for a marine vessel with inherent nonlinear hydro-dynamic feedback on manoeuvring and which experiencesconsiderable yaw disturbances stemming from proximate

Paper 2171D, first received 12th May and in revised form 25th August1982Dr. Ashworth is with the Department of Electrical, Electronic &Control Engineering, Edinburgh Building, Sunderland Polytechnic,Chester Road, Sunderland SRI 3SD, England, and Prof. Towill is withthe Department of Mechanical Engineering & Engineering Production,University of Wales Institute of Science & Technology, King EdwardVII Avenue, Cardiff CFl 3NU, Wales

obstacles (vessels, canal walls, harbour obstructions, sea flooretc.) and the ubiquitous sea motion.

2 Loop-function synthesis of linear systems

The incremental sensitivity function, as defined by Bode [8]and Truxal [9], which relates variation of the system functionT{s) to that of the plant P(s), is applicable only for infinitesi-mal changes and therefore implies some lack of confidencewhenever the plant undergoes even moderate variations. Toovercome this difficulty, Horowitz [1] introduced a large-scale sensitivity function SJ for the situation when the nominalplant P0(s) underwent a large-scale change to some variantP(s), causing the designed nominal closed-loop system functionTo (s) to be modified to T(s). Thus

1

AP(s)/P(s) l+L0(s) 0)

where AT=T- To, AP = P — P0 and L0(s) is the nominalloop function. Eqn. 1 is readily manipulated to give Horowitz'sfundamental relationship:

y(/w*)(2)

Application of eqn. 2 defines a boundary Bh at co = cofe uponwhich —Z,o(/cOfe) ensures satisfaction of the specified closed-loop tolerances. Location of — Z,0(/cjfe) within the region yk

(see Fig. 1) beyond Bk implies overdesign in that the magnitudeof the loop function is unnecessarily large, resulting in adiminished set {To/T} and increased transducer plant noiseamplification.

It has been shown [10, 11] that an optimum Lo exists andis unique (with only two minor restrictions for nonminimum-phase systems) and that it lies on every Bk over the completefrequency range. The optimal design is defined as that which,while satisfying the system specifications, minimises A" of eqn.3:

is

lim L0(s) = —s*°° s

(3)

where e = the pole-zero excess.Graphical techniques of loop-function synthesis have been

presented [3] which facilitate the determination of the loop

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magnitude boundaries. Alternatively, the computational prob-lem posed, involving as it does the statistical variation of anumber of plant parameters, is one which is eminently suitedto computer-aided-design procedures [4].

3 Correlation of time/frequency-domain specifications

It has been seen that the sets which define the variationalregions of plant and system function are given in terms offrequency responses and that it is therefore necessary tohave means by which a translation may be effected from thetime domain when specifications have been set in terms ofthe transient response. Now, whereas the definition of thefrequency-domain sets is rendered straightforward by MonteCarlo methods [4], it is necessary to invoke additional pro-cesses when variation is known only from boundaries ofacceptable time responses, since the rigorous translation ofsuch bounds has not yet been achieved.

region yk foracceptable-Lo(ju;k)

Fig. 1 Loop-function magnitude boundary B^ upon which — Lo (ju>h)ensures the specified restriction of {T0/T}k for a given variation {P0/p)k

Procedures have been reported in the literature [7, 12, 13]of which that based on the application of the transient responsesensitivity function of eqn. 4 is quite transparent:

bat/at(4)

where u{t) is the transient response and at is the ith variableparameter of the plant. It is readily shown [14] that, if theindividual parameters at vary in a random manner, the bound-aries of the transient response will be given by

3au3i/(0

dat

2\ 1/2(5)

where a refers to the relevant standard deviation and it isassumed that all perturbations are incremental and do notresult in system instability.

Examination of the set of curves of S%. leads to the assign-ment of allowable variations Aa{ to satisfy the responsetolerance and hence to the specification of frequency-domainvariations.

4 Estimation of rapidly changing dynamics

The dilemma which confronts the designer is that reduction ofloop bandwidth to minimise the degrading effects of noisewill give rise to a worsening of the ability of the closed-loopsystem to counteract the effects of changing dynamics. Amuch lower bandwidth could be tolerated if the varyingdynamics could be tracked via estimation procedures through-out the transient periods. This would eliminate the necessityto resort to the alternative of nonlinear compensation.

Investigations have been carried out [7, 15] on the abilityof estimation algorithms to handle the changing dynamics ofa particularly nonlinear system using data logged during trials,

which consisted of tape recordings of system responses toPRBS, sinusoidal and step change stimuli. Preliminary identifi-cation of the model type and order was carried out along thelines of Box and Jenkins [16] to provide a basis for sub-sequent spectral analysis, employing maximum entropy anddiscrete Fourier transform methods, and recursive least-squaresestimation.

However, the results of the estimation were disappointingin that any nonlinear dynamical variation was found to besubstantially complete, over the range of transients con-sidered, before the variance of the estimates had fallen tovalues which were better than that known a priori from thepredicted tolerances of the linearised model. It should ofcourse be stated that the method of recursive least squares isintended for linear stationary processes, whereas the systemconsidered was nonlinear and nonstationary and was subjectedto disturbances which were highly coloured. It would beoptimistic to expect good results from such an unpromisingbasis. It became obvious therefore that the tracking of thevariable parameters of the plant over the period of a transientwas out of the question.

5 Inverse nonlinear compensation

Following Horowitz [6], let the input x and output y of theplant be related as follows:

y = Q (6)

where the set of plants Q may be nonlinear and time varying.Further, let the set of Y of acceptable responses y for all Q inresponse to system demands r £ R be defined by

Y(s) & T(s)r(s) (7)

Eqn. 7 ensures that the definition of Y for the inverse match-ing covers those responses encountered under actual operationalconditions. Now define a nominal response y0 such that

y0 k h* u = v Q = Qo (8)

where h (f) =£f l H(s) and H(s) is a desired linear time-invariant model of plant behaviour. A cancellation functionA is now constructed:

x =

Hence

q = qo

pep

(9)

(10)

where p(s) =H(s) for q—q^. In essence, a model-followingsystem has been defined with y(t) following v(t) exactlywhenever the plant assumes its nominal form, with the non-linearities of q being compensated for by those of A. Themodified plant is therefore as shown in Fig. 2. This modificationthus shows variations from H(s) only if the parameters of thenonlinear time-varying plant change, and not because of thenonlinearities themselves.

The effective, or modified, plant set is therefore definedby

(11)

The desirable model of plant behaviour H(s) therefore exhibitsuncertainty arising from mismatch between the plant and theinverse compensator. The set {p} can be determined by com-puting v for all y G Y and hence carrying out an identificationor by employing the sensitivity method of Section 3.

228 IEEPROC, Vol. 129, Pt. D, No. 6, NOVEMBER 1982

The advantage of the approach adopted is that [6], for alarge class of nonlinear systems, the cost of feedback can bereduced greatly since, in the absence of uncertainty, there isno need for the provision of feedback.

6 Approximate nonlinear ship model

Using standard nomenclature [17], the equation of motion ofa ship under the influence of an external yawing moment TV ofhydrodynamic origin is given by

Izr + mxG (v + ru) = N

where

./V = g(u, v, r, it, v, r,8,8)

= N^v + N;.r + mxGru

(12)

uH(s)

linearmodel

V

irP

A(s)

werse-nrecomper

X

Dnlinearsator

q ( )

plant

modified plant

Fig. 2 Inverse nonlinear compensation to limit the variability of theeffective plant prior to design

and u, v, r and 5 are instantaneous values of ship speed, swayrate, yaw rate and rudder angle. Hence on substitution for TV ineqn. 12, we have

= f (13)

Thus, in the steady state, the right-hand side of eqn. 13 can beequated to zero and will express the steady-state relation-ship between yaw and sway rate and rudder angle whichdefines the Dieudonne spiral curve. The latter is normallydetermined during ship trials and expressed in terms of onlyrudder angle 5 and yaw rate r. A common approximationemployed to represent the spiral curve is that of a cubic; hence

f ^ N8(8 -a0 -axr- (14)

There can be no theoretical justification for eqn. 14. It isassumed for simplicity and is tolerated only in so far as itsuse yields results of practical value.

Using data for the USS Compass Island [18], it is readilyshown [15] that the reduced eqns. 13 and 14 lead to thefollowing ship model (see Fig. 3):

A{U)r+B{U}r+r = -K(d + T{U}8)-Cr (15)

0.015

0.010

- 0005

80 160t.s

Fig. 3 Yaw-rate responses of the USS Compass Island and the modelgiven in eqn. 15 for a rudder demand of—0.4 rad

•ship

where U is the ship's speed prior to the initiation of the ma-noeuvre. The elimination of sway rate from the equations ofmotion and the gross simplification of the introduction of thesteady-state spiral curve into a dynamic situation markedlyalter the prediction of ship behaviour. It is pursued, however,to simplify the setting-up of the controller without referenceto hydrodynamic data.

fm rA

rB \ . * r

j+

CM3

i

-1/K1*Ts

Fig. 4 Inverse nonlinear ship compensator to predict the actualrudder requirement

The inverse nonlinear compensator A can now be con-structed as shown in Fig. 4, where rm is the yaw-rate require-ment of the 'preferred' linear model H(s).

7 Simulation with a nonlinear compensator

Becuase the modified plant will operate ultimately in closedloop, the poor prediction achieved in the preceding Sectionshould be tolerable. Initially, it is necessary to decide upon thepreceding linear model H(s) of Fig. 2. Consideration of thecombined behaviour and plant suggested the generalisedsecond-order model of eqn. 16, which has been consideredcomprehensively in Reference 19:

\+bs + cs2 (16)

The function of eqn. 16 cannot reproduce faithfully the largeovershoot and extensive 'tail' of the ship yaw response to steprudder demands, but can provide a compromise predictionfor both large and small rudder demands. Thus, for {K, a, b,c}= {—0.035, 5, 10, 100}, the step change rudder-demandresponses of the 'modified ship' system are illustrated in Fig. 5in comparison with those for the uncontrolled ship. It is seen

0.06

0.04

0.02

1- model

Fig. 5 Normalised yaw-rate responses of the modified ship system andthe uncompensated ship to given step rudder-change demands

a Ship response (6^ = — 0.1 rad)b Compensated ship response (5^ = — 0.1 rad)c Ship response (5^ = — 0.4 rad)d Compensated ship response (6^ = — 0.4 rad)

IEEPROC, Vol. 129, Pt. D, No. 6, NOVEMBER 1982 229

that there is considerable improvement in invariance ofresponse with changing rudder demand. This is achieved eventhough the linear model H(s) is of simple form, with noncriticalparameter settings and with an inverse compensator derived ina simplistic manner.

8 Tolerancing of the modified ship system

Trivial manipulation of the model function of eqn. 16 leadsto the transient response sensitivity function which defines,as a function of time, the dependence of the magnitude of theresponse upon the parameters of the model. Thus the variationof the yaw rate, rm(t), with respect to the parameter b isobtained from

„ , . , . , *m -Kb(l+as)0 db/b 1 + 2bs + (2c + b2)s2 +2bcs3 + c2s*

(17)

Examples of these functions are shown in Fig. 6, of which themost influential is that with respect to the gain K.

0.03

-0.01

Fig. 6 Yaw-rate response sensitivity function of the model H(s) forK = -0.03,a = 5,b

Salt)-o-o— Sb It)

Sc(t)

Assuming parameter independence, eqn. 5 is employed todefine the fractional tolerance F of the yaw-rate response:

F =1

.(0 p

2 \V2(18)

where the summation is over all model parameters/?. Examin-ation of F reveals that a variability of K of AK/K = 0.367 isadequate to account alone for the total variation of the plant.It is therefore assumed that a, b and c are invariant, but thatK has a tolerance of = 37.6%.

9 Loop design for the modified ship system

Referring now to the linear-system design procedure of Sec-tion 1, we have that the plant-variant boundary PO/P(J'OJ) issimply a straight line along the real axis:

(19)

Table 1: Amplitude-ratio tolerance range of the third-order coefficient-plane model to achieve step response variation of Fig. 7

CJ Amplitude-ratio tolerancerad/s %

0.020.030.040.050.08

1.332.543.674.616.23

+ 0.91+ 1.79+ 2.68+ 3.44+ 4.80

1.0-

80t s

160

Fig. 7 Acceptable variation of the course-change response of thecompensated and closed-loop ship

There remains the decision as to the extent of the set {To/T}and as to the nominal system function To to be achieved. Afruitful approach for guidance in the selection of the latter isto refer to the behaviour of standard model functions. Forthe system in hand, it was considered that an appropriatemodel would be a third-order coefficient-plane model [19]with no numerator dynamics. In normalised form, the systemmodel function is

Tm(S) =1

1 + 2.5S + 2.0S2 +S3 (20)

where s = 5/10.73. Monte Carlo examination of the variationof Tm(S) as a function of gain provided the acceptable ampli-tude-ratio tolerance range of Table 1. The loop magnitudeboundaries Bk corresponding to the closed-loop tolerancesof Table 1 and the given Po/P are shown in Fig. 8. Theseboundaries exhibit the sharp V-form characteristic of plantswhich have major gain variation. Operation of the CAD

Fig. 8 Loop magnitude boundaries corresponding to the third-ordercoefficient-plane model of closed-loop ship behaviour of Fig. 7

a cjfe = 0.02 rad/sb u>k - 0.04 rad/sc cjfc = 0.05 rad/sd cjfe = 0.08 rad/s

230 IEEPROC, Vol. 129, Pt. D, No. 6, NOVEMBER 1982

compensation procedure referred to in Section 2 providedthe loop function also illustrated in Fig. 8, comprising:

forward path gain = 3.75

forward path compensation =

feedback path function = (1 + 10.5s + 56.25s2)

(21)

It will be noted that a double pole has been included at s =— 0.5 to provide adequate attenuation of the loop function athigher frequencies. On reduction to third order by absorptionof the 'far-off poles, we have

Tm(s) =1

1 + 2.195+ 2.00S2 + 53 (22)

where s = 5/8.86. Reference to standard forms [19, 20]reveals that the transient response has been modified slightlyas a result of the change incurred in Tm(s), such that a lowlevel of overshoot (< 10%) now exists.

Fig. 9 Complete closed-loop configuration for course-change controlof the ship, including compensation terms (eqn. 21), linear modelH(s) and inverse nonlinear compensator A

Fig. 9 illustrates the final feedback design combining thesensitivity-based linear compensation functions of eqn. 21,the approximate model H(s) of eqn. 16 and the inverse com-pensation of Fig. 4.

10 Simulation results

The success of the sensitivity reduction procedure outlined inthis paper to achieve its objectives is not in question (see thelimited variational range of response of Fig. 7). Rather, it isthe problem of noise amplification observed in Reference 21which has resulted in the need to investigate nonlinear com-pensation. Further, having decided on this manner of com-pensation, it is necessary to verify that the disturbance rejectionproperties of the outer feedback have not been degraded.

20

? 10

•I og1

!" 1 0

-20

0J01

Fig. 10 Loop frequency responses with alternative controllers inrelation to a typical sea-state spectrum

a P(J'OJ), linearised shipb L (}UJ) for linear controllerc L(J<JJ) for equivalent linear-nonlinear controllerd Pierson-Moskowitz sea-state spectrum

10.1 Response of the nonlinear compensated system withadded sea-state disturbance

A sea-state disturbance signal was generated by first defininga 20 kn Pierson-Moskowitz spectrum and then employing theIDFT technique to produce the discrete sequence. Fig. 10shows this spectrum in relation to the frequency response ofthe ship alone and those of the ship with linear [21] and'inverse nonlinear compensation. It is observed that the ratioL(s)lP(s), which defines the high-frequency noise transmissionto the plant input, has been greatly reduced. However, in-terpretation of these responses is not straightforward sincethey are in reality only linearised equivalents of decidedlynonlinear functions.

The disturbance sequence was scaled to the level shown inFig. l ie before injection into the feedback path, whichresulted in the rudder activity shown in Figs, lib and c.It is seen that in comparison with the results obtained withpurely linear feedback controls [21], where the full range ofallowable rudder motion was exited, the achieved rudderactivity is slight.

10.2 System response to a disturbing proximate shipThe disturbance considered [21] is that arising from thepassage of an overtaking vessel moving at 18kn when thecontrolled ship is moving at 15kn with an initial transverseseparation of 132 ft (the perturbing vessel is assumed topossess infinite inertia). Fig. 12 shows the resulting motion of

yawdisturbance, rad

actual rudderactivity, rad

-Q02-

actual rudder 02 •activity, rad

0

200

-Q2t .s

200

Fig. 11 Assumed yaw disturbance having the spectral properties of a20 kn Pierson-Moskowitz wind-generated sea and its effect on rudderactivity, with and without a superimposed course-change demand

a Yaw disurbanceb Actual rudder in response to ac Actual rudder in response to a 0.2 rad course change with sea noise

8000 distance,?t

time.s60 / J200

motion of theperturbing shi

200

Fig. 12 Comparative motion of the controlled ship and a perturbingproximate ship of infinite inertia on initially parallel courses, theformer ship moving at 15 kn and the latter at 18 kn

IEEPROC, Vol. 129, Pt. D, No. 6, NOVEMBER 1982 231

the perturbed vessel which, as a consequence of the lowerequivalent bandwidth of the nonlinear design, is slower andresults in a 22 ft displacement towards the perturbing shipbefore counteractive rudder motion takes place. This responseis only slightly worse than that experienced previously [21],but one which is still acceptable and which calls for only 2° ofrudder at most.

11 Conclusions

The attitude adopted in this paper is that a marine vessel canbe considered to have linear dynamics under manoeuvringconditions only if it is further assumed that the parametersof the transfer function of the ship vary over a wide range.Such an assumption leads to difficulties, however, when linearfeedback is invoked to limit the extent of the closed-loopmanoeuvring variability, in that the extensive loop bandwidthwhich is found necessary leads to excessive rudder activityarising from the feedback of sea state [21].

The vehicle adopted for validation of the design procedurewas the USS Compass Island using hydrodynamic parametersfrom captive model tests [18]. The complete digital simulationof the surge, yaw and sway equations of motion, involvingcrossproduct variables of up to third degree, was found tocompare well with actual manoeuvring data [22]. In practice,without the benefit of preliminary hydrodynamic modelling,the parameters of eqn. 15 would be obtained from sea trialsvia RLS estimation while on a straight course, with K and Cdetermined from the results of spiral tests [17]. Again,accuracy is not of prime importance since the linear modelH(s) is noncritical.

It has been shown that the use of even an approximateinverse nonlinear controller can reduce significantly the rangeof variation attributed to the hypothetically linear ship. Thatvariation which is still evident (see Fig. 5) exists as a con-sequence of remnant nonlinearity and imperfect modelling,although now of a much reduced range. This permits anequivalent reduction of the bandwidth of the nominal loopfunction designed on the classical basis of Horowitz [2], wherethe modified plant can now be assumed to exhibit linearperturbations [23]. The compensation arrived at for the USSCompass Island is given by eqn. 21. It is seen that the closed-loop sensitivity specifications have been satisfied without anygross problems of sea-induced rudder activity.

Furthermore, it has been demonstrated that the capabilityof the autopilot to reject external disturbances due to ships inclose proximity has not been seriously impaired, despite thereduction in bandwidth. Improvement in this aspect of perfor-mance is a question of trade-off between it and an acceptablelevel of rudder excitation. Finally, it is implicit within theprocedure that the behaviour of the loop function is con-strained at high frequencies to maintain system stability giventhe range of variation of its amplitude ratio. This latter pointhas been discussed elsewhere [4].

12 References

1 HOROWITZ, I.: 'Fundamental theory of automatic linear feedbackcontrol systems', Trans. IRE, 1959, AC-3, pp. 5-19T

2 HOROWITZ, I.: 'Synthesis of feedback systems' (Academic Press,1963)

3 HOROWITZ, I.M., and SIDI, M.: 'Synthesis of feedback systemswith large plant ignorance for prescribed time-domain tolerances',IEEE Trans., 1972, AC-20, pp. 84-97

4 ASHWORTH, M.J., and TOWILL, D.R.: 'Realization of plantsensitivity specifications via multi-loop feedback compensation'.IFAC symposium on computer-aided design of control systems,Zurich, 1979

5 HOROWITZ, I.M., GOLUBEV, B., and KOPELMAN, T.: 'Flightcontrol design based on nonlinear model with uncertainparameters', J.AIAA. 1980, 3, pp. 113-118

6 HOROWITZ, I.: 'Improvement in quantitative nonlinear feedbackdesign by cancellation', Int. J. Control, 1981, 34, pp. 547-560

7 ASHWORTH, M.J., and TOWILL, D.R.: 'An approach to auto-pilot design based on nonlinear ship models with uncertain para-meters', Sixth ship control systems symposium, Ottawa, 1981

8 BODE, H.W.: 'Network analysis and feedback amplifier design'(Van Nostrand, 1945)

9 TRUXAL, J.G.: 'Automatic feedback control system synthesis'(McGraw-Hill, 1955), p. 120

10 HOROWITZ, I., and SIDI, M.: 'Optimum synthesis of non-minimum-phase feedback systems with plant uncertainty', Int. J. Control,1978, 27, pp. 361-386

11 GERA, A., and HOROWITZ, I.: 'Optimisation of the loop transferfunction', ibid., 1980, 31, pp. 389-398

12 SIDI, M.: 'Synthesis of feedback systems with large plant ignorancefor prescribed time-domain tolerances'. Ph.D. thesis, WeizmanInstitute of Science, Israel, 1973

13 KRISHNAN, K.R., and CRUIKSHANKS, A.: 'Frequency domaindesign of feedback systems for specified insensitivity of time-domain response to parameter variation', Int. J. Control, 1977, 25,pp. 609-620

14 ASHWORTH, M.J.: 'Feedback design of systems with significantuncertainty'. (Research Studies Press, 1982), pp. 75-77

15 ASHWORTH, M.J.: 'Computer-aided design of ship steering sys-tems'. Ph.D. thesis, UWIST, Cardiff, Dec. 1981

16 BOX, G.E.P., and JENKINS, G.M.: 'Time series analysis: fore-casting and control' (Holden-Day, 1976)

17 COMSTOCK, J.P.: 'Principles of naval architecture' (SNAME,1967)

18 STROM-TEJSEN, J.: 'A digital computer technique for predictionof standard manoeuvres of surface ships'. Report 2130, DTMB,Washington, USA, 1965

19 TOWILL, D.R.: 'Transfer function techniques for control engineers'(Iliffe, 1970)

20 TOWILL, D.R.: 'Coefficient plane models for control system analy-sis and design' (Research Studies Press, 1981)

21 ASHWORTH, M.J., and TOWILL, D.R.: 'An assessment of thepotential of sensitivity design in the optimization of control strat-egies for manoeuvring vessels'. ILN symposium on ship steering andautomatic control, Genoa, Italy, June 1980

22 MORSE, R.V., and PRICE, D.: 'Manoeuvring characteristics of themariner type ship (USS Compass Island) in calm seas'. SperryGyroscope Company report GJ-2233-1019, 1961

23 HOROWITZ, I.M.: 'Synthesis of feedback systems with nonlineartime-varying uncertain plants to satisfy quantitive performancespecifications' Proc. IEEE, 1976, 64, pp. 123-130

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