Some aspects of the stability of automatic course control of ships

•5

A R C H I E F

Lab. V . Scheepsbouwkunde

Technische Hogeschool

Delft

Hydrodynamics

Section Report No. H y - 1 6 . April 1973

L A Brief Description of the

HyA Large Amplitude PMM System

hy M.

S. C H I S L E T T and L. W A G N E R S M I T T

2. Course Stability While Stopping

by L. W A G N E R SMITT and M. S. C H I S L E T T

3. Some Aspects of the Stability of

Automatic Course Control of Ships

by M O G E N S I. B E C H

ocQ e®ra/ao@8o©CQn

AUTOMATIC COURSE CONTROL OF SHiPS

,M. I . Becbf'

A short discussion ofthe ship's transfer fiinction is given, based on the second-order differential equation.-The influence ofthe steering gear is pointed out.' Based on Bode diagram representatibni thé principal.require-' ments to the controller transfer.function are deducted in conjunction with a short discussion of the total

closed-loop block diagram. The target for control and a possible optimization criterion is suggested.

1 I N T R O D U C T I O N

W I T H THE INCREASING SIZE of ships' DOW operating on relatively long and uninterrupted voyages, there is a natural demand to diminish propulsion expenses as much as possible. Likewise, with the waterways becoming gradu-ally more crowded, the requirements for accurate control of thé ships in straight sailing and course alterations are . increasing. A better general appreciation of the inter-action between ship, steering gear and autorpilot is there-fore necessary (i)t.

This paper describes some aspects ofthe ship's transfer fimction as deduced from the second-order non-linear equation, and underlines the importance of a smooth, steering characteristic for proper hahdlihg. •

Limitations on handling are sometimes' imposed by steering gear parameters,-which very often are (iictate.d only by classification Tequirements.

. The results of improper steering gear parameters are mentioned as observed from the Bode diagrams.

Referring , mainly to Nomoto's work on propulsion losses resulting from steering activity, the effeas of varia-tions in the closed-loop transfer function are discussed, resulting in requirements to a preferred transfer function.

1.1 Notation . .

The commonly accepted notation recommended by the Manoeuvrabiiity Committee of the 10th I . T . T . C . has been adopted. A résurné is given below.

\X)Y,N Forces and turning monient referred to

amidships.

Velocities and rate of turn. Rudder angle.

Distance from amidships to centre of gravity. Mass of ship. • '

Mass moment of inertia with respect to a vertical 2r-axis through amidships. •

Partial derivatives are wrinen by subscripts to the dif-ferentiated function, e.g'.

» = iV„ s -tr'- .etc. u,v,r 8 m L = 0 8v

Differentiation-with respect to time is written as dS

d^ S =

dt

^19^2^ "ƒ this paper was accepted for pttblicacion on 11th January

j* Decca-Arkas AjS, Copenhagen. j- Refere?ices are given in Appendix

17.L-{Journal Mechanical Engineering Science .

2 N O N - L I N E A R M O D E L E Q U A T I O N A N D I D E N T I F I C A T I O N O F T H E N O N - L I N E A R I T I E S

The Unearized equations of motion of a ship moving in the horizontal plane, in deep unrestricted water and at eon-stant speed, can be solved with respect to ifi to the well-known second-order differential equation:

'•iT2l?+(^i + T2);A-|-^i = ^(rg8-K8--8^) (17.1)

where h,. ;is the rudder angle corresponding to i/ü = = ^ ==.'S = 0.

Ships, and especially...unstable ones, can only be accurately; described by equation (17.1) in a very small range'of and 8.

For any significant range of manoeuvring, the values of '"i^aj ( n + '^a) k Will depend predominandy upon the

• value of ii.

Rewriting equation (17.1) we obtain (2):'

in which {Yi-mXNt-Q-{Y:.-mx^XNi-mxc,) _ iNi-mXo)Y,-iY--m)N, • _ - .N,Y,-Y,N, . J _ _ Y^(N,-mxoUo)-N,{Y,-muo) -. • riTz (Yi-mXN^-I,)-(Yf-mx^XNi-mXo) (17.3) (17.4) (17.5) ' + 5 Fig. 17.1. Notation

{

\ M. I . B E C H and (Yi-mXNr-moUo) + i N r - m U l r J iYi-mXNf-L)-iYf-mx^)iNi-mx^) . . . (17.6) An examination of the coeflBcients in equation (17.2) shows that ( I / T J + I/TZ), A/TITZ and T3 all remam fairly constant during manoeuvres for a given ship at constant speed, whereas I/TITZ is far from heing constant.

The substitution

Ta/ T1T2 renders

. (17.7).

(rgS+S) (17.8) The main non-Unearities of equation (17.1) have there-by been lumped in the steering characteristic H(4i) = S,

which defines the rudder angle 8 required to outbalance the forces and moments acting upon the hull in a steady turn with rate 0.

Hiiji). can be determined, in general, by the reversed

spiral test, or in cases of dynamic stability by the Dieu-donne spiral test.

^\;^ndling bad t Ordinary unstable Marginally stable^. Stable r. Good o r ~ ~ ^

fair \ •'Super stable ^

Infirittely stable*^ Very bad ~~— /'Reversed loop'^ 1 V

Fig.' 17.2.. Principal characteristics of the r-S curve Journal Mechanical Enginaering Science

3 T R A N S F E R . F U N C T I O N '

Linearizing equation (17.8) in a small range of fi around a given value 0o we obtain; . .

. = —(r38-l-/18+8o) With So =,H(0o).

The Laplace transform of equation (17.9) is

(17.9)

^2} T1T2

d ^ -d f

(17.10).

which is consequendy the transfer function of the ship. The gain coeflScient, kr^jr^T^, is rather constant, being the product oikfr^Tz and T3, and so is the zero — llr^.

-The poles arCj in addition to one in origp, the roots of the characteristic equation

, H ( - ! - + i W ^ - . ^ = 0 (17.11). \ T l Tz/ TiTa dip or as '+{-+~)x+— = 0 . (17.12) d<ji 1 (17.13)

dH{>i>)ld4i, being the slope of the steering characteristic,

will vary as a funcdon of 0. This causes the poles — I / T I and — 1/T2 to vary as follows:

di/j >0i > 0 < 0 dil> = 0; - 1 = 0 - i < 0 ship unstable ship marginally stable Xr, + T,y^dH(,l>) dip <0i ^1 Ta

ship stable, real roots < ~4^7jr~' complex roots; ship oscillatory stable Consequently, as seen from the typical steering characteristics (Fig. 17.2), a normal ship will become in-creasingly stable as the rate of turn increases.

The possibihty exists, at least in theory, that a ship may be dynamically unstable in straight saüiiig, but when going through mcreasing degrees of stabihty as the rate of turn mcreases, it may finally become oscillatory stable with complex poles in the ttuning condition.

Abnormahties in the steering characteristics may bé reflected iii abrupt and seemingly random variations in steering performance.

The available input point for conttol ofthe ship's heading is, in general, thé conttol point of the steering gear.

If, for the sake of simplicity,, the common hydrauhc steering gear with conttolled deUvery pumps is considered, the accessible point will be the input point óf the floating lever.

Any necessary equipment for Operating the floating lever can normaUy be made economi<ally with suflBcient bandwidth to. justiiy that only the steering géar ttansfér function is considered in an approximated linear analysis, provided reasonably large signal amphtudes are con-sidered.

In a small signal analysis, the non-linear switching nature of the normal ^input controller becomes pre-dominant. The block diapa.m of such a hydrauhc steering gear is shown in Fig. 17.3.

then: A s I then 1 1_ (17.14) (17.15) (17.16) (1,7.17) (17.18) A common value of ^Smax is 6°.

^^mftx will depend mainly upon the number of pumps in operation at any time. ' .

With both pumps in operation, the steering gear must be within classification requirements, and a common value is 37s.

SAs) < g y A ó ( s ) - ^

Fig. 17.3. Block diagram of steering gear

Wé then obtain:

0° ^

I f only one! pump is operating, the value = 4 s is more likely. . •

For the small conttol amphtudes of interest in a. sttaight course stabihty analysis we will always have:

]ASit)\<A8^^ . . (17.19)

and we may therefore apply the ttansfer function [equa-. tion (17.16)] without further resttictions as far as the main steering gear is concerned.

5 S H I P + S T É E R I N G G E A R T R A N S F E R F U N C T I O N : B O D E D I A G R A M

If we now assume that the effective rudder angle S appear^ ing in equations (17.1)-(17.10) is Unearly related to the. actual physical rudder angle (2) (3), wé may write the combined ship-steering gear ttansfer function as seen from, floating lever input point to the Chip's heading as foUows: .

^ ^ ( s +

-8,(i) f 2 ^ I \ (17.20)

Vl Ta/ Tj^T2 dill i \ - T R /

This describes the. conttol object, which normaUy has to be taken as it is found, and it is therefore reasonable to

= - 0 000035s"5 I <imrji)\ d^A

- ( ^ - ^ ) = 0 0 8 5 - ' = 004s-'

L- Lji—i—I I I 1 111 \—I—I I l l l l l 1—u—I I I i - ' i I r - I r<>i., I I - , •, I

0001 001 . ' O ' l 1 10

rqd/s

Fig. 17.4. Ship+steering gear—principal transfer fimction: Bode diagram with asymptotes

0001 •. OOI Q.1

Fig. 17-.5. Ship+steering gear: Bode diagram varying value of H{ifi)

discuss shortly what requirements it poses on a successful controller.'

• Fig. 17.4 shows a typical Bode, diagram, corresponding to equation (17.20), illustrating the mutual, situation of cross-over frequencies and asymptotes in the case- of a dynamically stable ship.

Fig. 17.5 shows the corresponding family of curves for a hyppthetical case of a fixed set of constant coefiSciems but varied" degree of stability from very stable tb Very unstable. . . - .

It- will be seen that, particularly in the case of a very unstable ship, the only frequency range where an accept-able phase margin may be achieved is restricted to a region around l/rg to ( l / f j - l - l / T a ) . It is fmther seen tiiat tiie presence of I/TR drives the phase lag in excess of 180°

for higher &equéncie"s. ^ Consequendy, if the steering gear becomes too slow and

thereby I / ^ R approaches too closely to (l/Tj-t-l/T-a), then •the stable control of the vessel will become more diflScult

if not impossible. '. Auto p i l o t ' Telem'otor 'posiLion servo Hydraulic . s l c e r i n g gear Ship A-R(l>5rpHl(I^S<-CR^CRl W'-'"'CR"'**'rD'^ I 05-3 Ii P • 4 TpH. 120 s D = 0 5 ° 24Ó s = 0 5 ° •' 480 s V 00 s 2" -kcR' 1-8 . • . ' . ' 4° 3-5-28 s ,' TD Ö'l-l-25-2-5-3-75 s Somai - 5°-35'' . . • ". S 3°/s TH': 5Qm s ,|J.8 _{'a I max} TR = Ta =; 2 s

Fig. .17.6. Auto-pilot j ship block diagram

m,

^1

2h

"2m" '^'^^^

? i ^ = ^ - l ^ ( ' ^ - c o n s t . )

dip ' stability index

. 2 0 s ' < l ^ ) < + 2 0 s

i 15 < a < 1-9

.TCR' 3-5-30 S . • 1Z0.Z40.-'180,«.S

Fig. 11.7: Auto-pilot controller transfer function: Bode diagram, asymptotes only

It is immediately apparent from the Bode diagram that any controller that shall render satisfactorily stable steer-ing, at least of an unstable ship, wiU have to apply phase lead—i.e. differentiation—over a certain range of frequen-ciesj and this differentiation may. most profitably be apphed in the above-mentioned range from l/rg to (1/TI-)-1/T2). Even in the stable case it is an advantage to apply differentiation in this range. The amount of dif-ferentiation needed will mainly depend upon the value of

dH(ip)lé<ji and the proximity of TR.

At frequencies in excess of the steering gear cut-off. frequency any loop gain will proyide Uttie to control and merely add noise, i.e.'steering gear wear.

6 O V E R A L L B L O C K D I A G R A M A N D C R I T E R I O N F O R O P T I M U M A D J U S T M E N T

The óvéraU block diagram óf ship, steeriiig gear and

auto-pilot is iUustiated in Fig. 17.6. The compass transfer •function has been neglected in the block diagram under

the assumption that if is ideal. .

The parameters that are avaUable for adjustment are the time constants (automatic permanent helm), (counter rudder time constant) and TJ, (damping), and the coefBdents (iudder gam) and (counter rudder) in. the controUer, as well as the deadbandwidth D and thé teleinotof speed Sj in thé telemotor position servo. ' The objea of adjustmeiit will be to achieve a closed-loop performance which best satisfies the requirements of a safe and'economic voyage from A to B.

Safe operation aiid' handUng .demands that yawing motions due to sea disturbances and course .alterations commanded via the auto-püot should be settied in an aperiodic manner and normaUy as quickly as compatible with hull and steering gear dynamics [equation (17.20)]..

=-20s

= - 0 000036 5-3

-270

Fig. 17.8. Optimization óf auto-pilot on stable ship: Bode diagram

f. M . I . B E C H

Fig. 17.9. Optimizationofauto-pjlot on stable ship: Nichols diagram.

This corresponds to a flat closed-loop response with the widest bandwidth compatible with the transfer fiinciion (17.20). ' . .

It wiU later .be seen that this requirement happens to coincide With demands for optimum propulsion economy. Propulsion losses due to rudder and hull motions, have been studied by, amongst others, Nompto and Motoyama (4) and Nomoto (5).

The two major contributions tp propulsion losses are inertial hull drag

: .. (17.21) and thé rudder drag due to rudder aetionj which consists inainly of the longitudinal component of the hydro-dynamic force acting nearly perpendicularly on to the rudder blade due: to rudder angle. Hull rhotions are set up partly by steering action and partly by wave action.

An analysis shows that in a periodic yaw set up by steering activity, /8 and ifi will be periodic functions' of time with only a little phase difference. '

The resulting time average of equation (17.21) repre-sents a considerable propulsion loss which is roughly proportional to the square of the yaw amplitude.

The ampUtudès of both jS and ip are proportional to the amplitude of S, and over the significant range of frequenr. cies ÜJ > |1/TI| are inversely proportional to frequency: Consequendy, this part of the propulsion loss will, be proportional to the square of the rudder amphtude and inversely proportional to the square of the steering fre-quency. The rudder drag is roughly proportional to the square ofthe effective rudder angle, and consequendy the time average of rudder drag becomes proportional to the square of the rudder deflection amplitude, but indepen-dent of steering frequency, .

If hull motions are set up as a result of wave disturbance in predominandy obhque following seas, Eda and Crane (6) have found that ^ and ifi in equation (17.21) will vary in hear quadrature phase relationship to each other. Consequendy, in this case the time average of equation (17.21) will be very nearly zero.

: Considering now the optimum closed-loop response from a propulsion economy point Of view, it becomes apparent that any system resonance wiM tend to filter and amphfy a considerable amount of steering activity from the spectrum of sea disflirbances. This will worsen as the relative damping conesponding to the closed-loop system root pairs decreases.

The bandwidth of the closed-loop response should be as wide as practicable, in order that the necessary steerr ing activityj even when aperiodic, may occur in the highest possible frequency range, because hull losses are inversely proportional to steering frequency with given rudder amphtude.

One item, which so far has-been completely neglected ih this discussion, is the telemotor position servo in the overall block diagram, Fig. 17.6. As mentioned above, this part ofthe system contains two parameters available for adjustment, i.e. telemotor speed S, and deadband-width . D. The telemotor speed Sj should merely be ~ adjusted sUghtiy in excess of maximum rudder velocity 1^01 max in order to preserve available bandwidth. The deadbandwidth Z) is inherendy associated with the relay switching systems, which, purely for reasons of com-ponent economy, forms part of almost all existing interface equipment between electrical command and actual steering gear. In many cases a switching type of control is the only, access tó the steering gear itself.

The deadband is a non-hnearity, which becomes

-Z70

d//(.iA)

= ZOs =-0-000036

Fig. 17.i0. Optimization of auto-pilot on unstable ship: Bode diagram

M . I . B E C H

80" -70" -60" -50" -40" -JO" -sé -i(f 0

Fig. 17.11. Optimization of auto-pUot on unstable ship; Nichols diagram

Journal Mechanical Engineering Science

no signal

let^ÏS^Ïr^

loop, i.e. without stewing ^ ' " ' ^ ' ^ ^ ^ J ' °Pen

oft^TÏt^^t^^e'^-dcJm

s i n Ï Ï e r d d ~ j ; r ^

unavoidable result! ^ oscillation is die' This may often also be the case f n r , ^ •

switching controls it dircSl, ' " •"'"-starboard

(m .b^isi^isistr™^'"'^^^^

muItipUed by 5,) then constant TH • osciUations. • '"•"^""P ' ™ ^ i ° t o sustained

40 ^ 7 " ? ^"'^^ minimum value of D is 2° rudder « n „ i . 4 - 5 bemg more common rudder angle, a d o ° S V ° t h r A S s t r ? ' ^ ^ ° " J " ' ^ ' ^ - ( P - - t e d ) '

, P A R A M E T E R S

•Fig- 17 4 The m,mloi I '"''H and r^ in

above the critical freonenrv ^° ^^'"^ or s t r S i n l ^ f n i n T r " " approKimatioh are

iUu-ü^^'ulTl^: o ^ f ° t t — only

thereforrbe'eSeae^t'rr';.,''^^^ modifications may scale, and sma^sSle '''¥P''^' "deluding

large-optimu«.anl:^tJSr-^^^^

A P P E N D I X 1 7 . 1 REFERENCES ( l ) ABKOWTTZ, M A 'T

Steering Lid m a n t u v J b ^ " s h i p ^ h y d r o d y n a m i c s , manoeuvres based on ftS'-.Iie

l™!^^-^.?"

( 4 ) . NOMOTOJ K. and MoTnviMa -r <T '

-caused by yawW J ' , P'opulsion power

(6) EDA, H . and CRANE, C . L . , Tr 'Re,^,r^j, - u¬

,. Hty, Part I I I ,

Davidson 1:^. R^^r.Twey^'