A7l,- AlO 303 COORDIMTED STEERING OF SURF CE S XP(U) VAL 1/1

POSTUOUAOTE SCHOOL MONTEREY CA S S LEE SEP 97

tWUCMASIFIED F/G 13/10S M

sonhhhhhl

1.8.

111111-2

%~..'

NAVAL POSTGRADUATE SCHOOLMonterey, California

00 o

OTtIO

<.NOV 1 9 1987

THESISECOORDINATED STEERINGOF A SURFACE SHIP

by

Sang Sik, Lee

September 1987

Thesis Advisor George J. Thaler

Approved for public release; distribution is unlimited. .

11CUR'?' CL.ASSIFICATION 0; THIS pAt3

REPORT DOCUMENTATION PAGE1& 4 P( SSIA TION Ib AISTICTiVE MARKIN4GS

UNCLA~1?~la WEUNITY CLASSIFICATION AUTHORITY J DISTRIBUTION/ AVAILABILITY OF REPORT

Zb ICLSSIICrIO'OONGROIG SHEDLEApproved for public release;2bOfLASII-AIOIDWNROIG CHDUEdistribution is unlimited

4 PEIRFORMING ORGANIZATION REPORT NUMBER(S) 5 MONITORING ORGANIZATION REPoor NuVSER(SJ.P. -

6a NAME OF PERFORMING ORGANIZATION 613 OFIICE SVMSO. ?a NAME OF MONITORING ORCIANsIA!'ON(if__________________lf_________

Naval Postgraduate Schoo 62Naval Postgraduate School

6( ADDRESS C~ry Star@ a'7P Cad.) 'b ADORE SS (City. State "n ZIP Cooe)

Monterey, California 93943-5000 Monterey, California 93943-5000

8a NAME OF ;UNDINGj SPONSORING Ob OFICE SYMBOL 9 P0OCUREMENT 1NSTRUjMENT DOEN rIC;AlION PjijMgER ~.OZRGANZATION (of aJ0iecablfI

9c AO"RSSC' r Stare and ZIP COO*) 140 SfluRCF OF F(INDING NtIMRERS

PROGRAM PROJECT rAS,( WORK ,NIf

ELEMEFNT NO NO NO ACCESSION NO

( I"'(wde Se","ty) CIaUS0,car,oAl

COORDINATED STEERING OF A SURFACE SHIP

LEE, Sang Sik1"~ "'. CO REPORT 't ME roviqto a )At OF REPORT (Year M.onrA DdyJT77777, hNT

Master's thesis FROM ro 1987 September 846 SLP;'E(%ENTARY NOTArtON

(OSAr, CODES IS Su~iECT TERP.S iConionue on reverieE of necouaty "n dpmrthy by block Aumbof)

I~ oL GROUP SUBsGOLuP Precompensator design to suppress theI undesirable cross-coupling effects A

I -LSr R.AC' (COntinuE on 'evente d nocoua", and 'dontitV by bloc number)

The conventional approach to ship steering is to regard the ship asa single input, single output system without cross-coupling orinterastion between speed, yaw and roll. This approach has foundsuccessful application, particularly in conventional vessels where theamount of cross-coupling is normally slight. But, as a result of tightmaneuvering, the modern warship suffers severe cross-coupling effectsbecause of large control surfaces, high speed and low tonnage.Consequently, the adoption of a multivariable approach to ship steeringwould appear to be more suited for the design of a steering controlsystem.

;0 :) YR,3uTON AVAILASILITy OX ABSTRACT' j21 ABSTRACT SECURITY CLASSiFiC.ATION

(3_CLASSIIEOI.INL'Mltf 0 0 SAME AS RP' 00TIC USERas UNCLASSIFIEDIla 14A'.E Ol' RESPONSIBLE NOIVIDUAL 1115 TELEP"ONE(Indudo Area Code) l2c OF;,( I MBOL

George J. Thaler (408) 646-213 62Tr00 FORM 1473.684 MAR Si APR ed-t-o may be vtod wm't'I euvlaulted SECUAITY CLASSI;iCATION 0$ T'4S PACE

All OtMep od'tont" are obsolete UCASIFIE1NLSSFE

-7 --. 7 ' -- "%

SKCUNITY CLASSIFICATION Of THIS PAGE (When De. "he.r*4

19. ABSTRACT (cont.)

This thesis describes the results of a simulation study ofprecompensator design to suppress the undesirable cross-couplingeffects between speed, yaw and roll.

Simulation studies using DSL and Function Minimization arethe basis for accomplishing the design.

Simulation results presented indicate that the adoption ofa multi-input, multi-output approach would result in a signifi-cant improvement in the combined steering and stabilizationproblem of a warship.

..

-5

S N 0102- LF 014. 6601 UNCLASS I FIED

SOICUMITY C6ASSIICATIOW OF THIS PAGE(W%#n Data angered)

"- . . . ... . . . .. . -,- - -,-. -.-,,..- -.- --,--,- .... .. - - -.-.-.. .-- .-,--- .- - - m """' :" ° '

Winn~ ~ ~ ~ ~ ~~~~~~~~rm. An Pw Poop %, wcwLw N rr~V T7 WV-vwj

Approved for public release; distribution is unlimited.

Coordinated Steering- ___

Of A Surfiice Ship LA F.) r

PTIC TAB

by 1'!: L'u:d ]O

Sang Sik, Lee By-Lieutenant Commander, Rcpublic of Korean Navy Dis~tribut ion/

B.S., Korean Naval Acadcmy, 1978 Availability CodesB.S., Seoul National University, 1984 lAvail and/or

~Dist Special

Submitted in partial fulf illment of the IA,rcquircments for the degree of ______________

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

* from the

NAVAL POSTGRA DUATE SCHOOLSeptember 1987

Author:an I., c

Approved by: -

Gcwge J Taler, esis Advisor

Alex Gerba, Jr., Second RacC

Departmen Electrical and Computer Engineering

Gordon E. Schachcr,Dean of'Science and Engineering

3

Art9e

.yV' A.. * *~ 1.55S

Ir

ABSTRACT

The conventional approach to ship steering is to regard the ship as a single input,

single output system without cross-coupling or interaction between speed, yaw and roll.

This approach has found successful application, particularly in conventional vessels

where the amount of cross-coupling is normally slight. But, as a result of tight

maneuvering, the modem warship suffers severe cross-coupling effects because of large

-" control surfaces, high speed and low tonnage. Consequently, the adoption ota

multivariable approach to ship steering would appear to be more suited for the design

of a steering control system.

This thesis describes the results of a simuiation study of pre-compensator design

to suppress the undesirable cross-coupling effects between speed. yaw and roil.

Simulation studies using DSL and Function Minimization are the basis for

accomplishing the design.

Simulation results presented indicate that the adoption of a multi-input, multi-

output approach would result in a significant improvement in the combined steering

and stabilization problem of a warship. '. . ,

4

• ... . . . . ,:~44~ *2. a . . . . . .-.. ..... .. . -...: ... ......' .*.-.. .-. . -.- .-. .. . . . .- -

"JKArJWN7LAX~-W xwlmvw~l?%~rJr~wnrA 10PT v Ir KXA xfw- x"*A-xwww-w vwvwv vlWrW~vvr~N~v~rjljw 'S

TABLE OF CONTENTS

I. INTRODUCTION....................I

I. BASIC MODEL............................................ 12

II. PRE-CO MPENSATOR. DESIGN.............................. 17

IV. COMPUTER SIMULATION ................................. 27

A. PHILOSOPHY OF FUNCTION MINIMIZATION........... 27B. CHOICE OF COMPENSATORS.......................... 28

C. CHOICE OF DESIRED OUTPUTS.........................9

D. COST FUNCTIONS..................35

E. WEIGHTING FACTORS ............................... 35

F. VARIATION OF NUMBER OF POLES AND ZEROES ........ 54

V. CONCLUSIONS AND RECOMMENDATIONS.................. 64

A. CONCLUSIONS ...................................... 64

B. RECOMMENDATIONS................................. 64

APPENDIX A: SYSTEM BLOCK DIAGRAM FOR SIMULATION ......... 65..d

APPENDIX B: CONSTRAINT PARAM-VETERS OF FUNCTIONMINIMIZATION ................................... 66

APPENDIX C. COMPUTER PROGRAM FOR, UNCOMPENSATEDSYSTEM .......................................... 68

APPENDIX D: COMPUTER PROGRAM FOR ORIGINAL PRE-COMPENSATOR ................................... 69

APPENDIX E: COMPUTER PROGRAM FOR ORIGINALREDUCED PRE-COIMPE'NSATOR ...................... 71

APPENDIX F: COMPUTER PROGRAM FOR AFGENSUBROUTINE ..................................... 73

lp.

S.'

. . . . . . . .. . . . . . . . . . . . .

. . . . .. . . . . . . . . . . . . . . . .

APPENDIX G: COMPUTER PROGRAM FOR FUNCTIONM INIM IZATION ...................................... 74

APPENDIX H: COMPUTER PROGRAM FOR COMPARISONBETWEEN ROBERTS AND F.M ......................... 78

LIST OF REFERENCES ................................................ 81

INITIAL DISTRIBUTION LIST ......................................... 82

6

I! '

LIST OF TABLES

I. ELEMENTS OF MATRIX FOR WARSHIP MODEL [REF. 1 ............ 13 ",p

2. GAIN VARIATION [REF. 1] ........................................ 13

3. MAX VALUES OF INPUT AND OUTPUT ......................... 14

4. ELEMENTS OF DIAGONAL MATRIX R(S) ........................ 19

5. ELEMENTS OF MATRIX A(S) .................................. 19

6. ELEMENTS OF DIAGONALIZING PRE-COMPENSATOR ............. 20

7. FINAL ELEMENTS OF DIAGONALIZING PRE-COMPENSATOR ...... 21

8. PRE-COMPENSATOR GAIN VARIATION ......................... 21

9. FINAL REDUCED PRE-COMPENSATOR ELEMENTS ............... 24

10. OPTIMUM PARAMETER VALUES OF REDUCED PRE-CO M PEN SA TO R .................................................. 30

11. SIMULATION CASES USING WEIGHTING FACTORS(W.F.) .......... 39

12. OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR AT CASE I W.F . .................................. 44

13. OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR AT CASE 2 W.F . ..................................

14. OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR AT CASE 3 W.F . .................................. 46

15. OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR AT CASE 4 W.F . .................................. 47

16. OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR WHEN WE CHANGE K11 ONLY .................... 58

17. OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR WHEN WE CHANGE K33 AGAIN IN TABLE 16 ....... 59

18. THE LIST OF PARAM ETERS ...................................... 67

7

ILIST OF FIGURES

2.1 Multivariable Structure of a Warship Model ........................... 12

2.2 Step Response for Uncompensated Warship Model at RudderD em and ........................................................ 15

2.3 Step Response for Uncompensated Warship Model at Fin Demand ........ 16

3.1 Compensated System Configuration ................................. 173.2 Step Response for Compensated Warship Model at Rudder Demand ....... 223.3 Step Response for Compensated Warship Model at Fin Demand .......... 233.4 Step Response for Compensated Warship model at Rudder Demand

% asing Reduced Order Pre-compensator ............................... 25'.5 Step Response for Compensated Warship model at Fin Demand

using Reduced Order Pre-compensator ............................... 26

4.1 Block Diagram of Function Minimization ............................. 284.2 Output when we use the Optimum values of Reduced Pre-

compensator at Rudder Demand ................................ 314.3 Comparison between Roberts and F.M at Yaw output .................. 32

4.4 Comparison between Roberts and F.M at Roll output .................. 32

4.5 Comparison between Roberts and F.M at Speed output ................ 334.6 AFGEN b.................................................. 364.7 A FG EN 2 ...................................................... 36

4.8 A FG EN 3 ...................................................... 374.9 Stcp Response when = 1, Xr. = . and .=. at Rudder Demand......... 37

4.10 Step Response when kh = 1, r = .6 and k. =. 1 at Rudder Demand ......... 414.11 Step Response when )h= 1, r. = .6 and ks= .4 at Rudder Demand ......... 42

4.12 Step Response when X. = I, .r= I and X= I at Rudder Demand .......... 43

4.13 Comparison between Fig 3.4, Fig 4.2 and Fig 4.9 at Yaw Output .......... 48

4.14 Comparison between Fig 3.4, Fig 4.2 and Fig 4.9 at Roll Output .......... 484.15 Comparison between Fig 3.4, Fig 4.2 and Fig 4.9 at Speed Output ......... 49

4.16 Comparison between Roberts, CASE 4 and CASE I for Fin Outputat 30 Rudder D em and ............................................. 51

8

4.17 Comparison between Roberts, CASE 4 and CASE I for RudderOutput at 30 Rudder Demand ...................................... 51

4.18 Comparison between Roberts, CASE 4 and CASE I for PowerOutput at 30 Rudder Demand ...................................... 52

4.19 Comparison between Roberts, CASE 4 and CASE 1 for Fin Outputat 200 Rudder Dem and ............................................ 55

4.20 Comparison between Roberts, CASE 4 and CASE I for RudderOutput at 20° Rudder Demand ..................................... 55

4.21 Comparison between Roberts, CASE 4 and CASE I for PowerOutput at 200 Rudder Demand ..................................... 56

4.22 Comparison between Table 10 and Table 16 for Roll Output ............. 60

4.23 Comparison between Table 10 and Table 16 for Speed Output ............ 60

4.24 Comparison between Table 10 and Table 17 for Roll Output ............. 61

4.25 Comparison between Table 10 and Table 17 for Speed Output ............ 62

A.I System Block Diagram for Simulation ................................ 65

9

::..

,-.- • , •... ." v ." ." ,.', ", ", -"" " '"-" " " -" - -'"-'""'," "':'"" " ; -"'-..- ...=,, ....- .-....,-..-....:.t'...:.',_. . . .. ,',..-,,, , , .-, , .,. .. ,.'..' -.. .--- -: .. >; '. , " -:

ACKNOWLEDGEMENT

A significant debt of gratitude is owed to Dr. George J. Thaler, for the many

hours of assistance and guidance he has extended, from the author's first course in

control theory through more advanced cources, and specially for his help in the

preparation of this thesis.

Also I wouid like to express my sincere appreciation to Professor Alex Gerba Jr.of the Department of Electrical and Computer Engineering of the Naval Postgraduate

School, my second reader.

To my wife, Yoon Jung and my son, Dong Hoon, for their encouragement and

patience. I am deeply grateful.

Finally, I wish to express my appreciation to the Korean Navy Authority for the

opportunity to study in the Naval Postgraduate School.

10-- * S APi. .

I. INTRODUCTION

Modern warships must be highly maneuverable to satisfy numerous operationalrequirements. However, the steering characteristics of a modern ship are nonlinear, sosevere interaction or cross-coupling exists between the control surface inputs and thecontrolled outputs. Therefore, warship steering is a complex multivariable controlproblem . 'S

Interaction between roll, yaw and speed are pronounced in warships because oftheir length to beam ratio and relatively large control surfaces. However, warships 2should be able to execute high speed maneuvers while maintaining their fightingcapability. This is generally not possible due to severe cross-coupiing.

In this thesis, the development of a pre-compensator to reduce .he cross-couiikngeffects which are present ;n the steering characteristics of a modern ship is introduced.The design method adopted by Roberts [Ref. 11 uses the Direct Nyquist Array (DNA)

frequency response technique as defined by Rosenbrock [Ref. 21 and Fricker [Ref. 31. ,'2This thesis uses the basic ship model and the results which were reported by Robertswhich have the potential to produce improved seakeeping and ship stability. ,rEmphasis in this thesis is placed on optimizing the parameters of a pre-compensatorusing Function Minimization (F.M.) via a digital computer.

Simulation studies employed Function Minimization techniques together with theDynamic Simulation Language (DSL) package.

%%AA4

V

S -'

II I"i

.°. "

- . . . .- . -~- 1

11./-.~~ ~ ~ - - -.

FSVnTN'qN

11. BASIC MODEL

The model and data used in the study is that proposed by Roberts [Ref. 11. The

structure of the ship model is shown in Figure 2.1 and the elements of the transfer

function matrix, G(S), which were derived using curve fitting techniques to measured

step and frequency response data, are given in Table 1.

F IN G~() }ROLLw 23

RUDDER G22(s) YAW

G32(S)

G3 1 (3)

POWER G33 (3) SPEED

Figure 2.1 N1ultiv aria ble Structure of a Warship Model.

The nonlinear nature of the ship dynamics is denionstratcd by the chanige in

steady state gain parameters as shown in Table 2.

12

VVaNM % Kr a7K -. Ju v.r r. . - .- r

TABLE IELEM'VENTS OF MATRIX FOR WARSHIP M'VODEL jRE.F. 11

G11(S) '<4S2 + 0.24S + 1

G12 ( S) 3 ,2 -8. 57S + 1)53. 3S3 + 17. 17S 2 + 9. 52S + 1

K",G,.,(S) I

-- S( 12S' + 32. 25S 2 + ii. 2s + 1)

G Kl1(10os + 1)

G31(S) 582 *3S2*

G32( S)=

G33( S24S + 1

G1 3( S) C;_11 ( S) =G 23( S) 0

TABLE 2.

GAIN VARIATION [RET:. 11

ts)e '<1'1 2 2 '31 K<32 133 -

12 .114 1.18 .01 .058 .096 .1

18 .8 .932 .02 .067 .146 1.0626 .68 39 .021 .068 t65 .053

77,

13

For ease of anal3 sis the systcm inputs and outputs are used as defined below in

Table 3.

TABLE 3MAX VALUES OF INPUT AND OUTPUT

INPUTS MAX VALUE

Fin Angle 1 ±27'

Rudder Angle ±30'

Power +10%

OUTPUTS 1 MAX VALUE

Roll Angle *15

Yaw Angle t120"

Forward Speed 30 Kts

Figure 2.2 and Figure 2.3 show the time responses for the ship model when we

use 10% demands for rtdder and fin at 12 kts.

As can be seen in Figure 2.2, there are pronounced cross-coupling effects in both

roll and speed outputs.

In this thesis, 10% demand for rudder at 12 kts will be always used for

convenience.

14

...". . .,. .q . . . - -. " "I, ,

...

Fir 2

at R "

1 5

155,,'

o.1

-. -., .' ., ' .-.-' ..--., -. -, . - --- .. ..- ., . . .. . .. . . . - .. .- . .. . ., • . .- . .. o . . ., .. , , , -.' ., , , .,, . _ , .,

a; C; C2

IC!I S0

S -7

03ti

Figue 23 Sep Rspose or Ucomensted arsip ode

16o

V- JrWV- VW.A VT 7W-~v 7777VTA.FrPk1rw"(

• A.

III. PRE-COMPENSATOR DESIGN

The aim of including a pre-compensator, K(S), is to decouple the interaction

present in G(S), thus enabling the reduction of interaction or cross-coupling in the

system. The action of the pre-compensator is to propagate the three input demands in

such a way that each input affects its associated output only.

This chapter was extracted From Roberts [Ref. 11.

The system configuration with the pre-compensator included is shown in Figure

3.1.

'A-

+FIN RL 'WANTED ROLL FI-OROLL

PRE-COMANTED COURSE~ p ENSATOP RUDDER SHIP YAW

K(S) G(S)WANTED SPEED POERSPEED

Figure 3.1 Compensated System Configuration.

The approach used in this study is to develop a pre-compensator, K(S), which

totally diagonalizes the pair G(S)K(S) ( - Q(S) ). This is in effect non-interacting

control and can result in the elements of K(S) having high order which may prove a-4NI

problem when it comes to implementation, particularly if this is to be achieved using

analog techniques. If necessary the complexity of the elements of K(S) can be reduced

using standard reduction routines while maintaining diagonal dominance in G(S) K(S).I

17

.%%

The diagonalizing pre-compensator was produced using the method suggested byFricker [Ref. 3]. This technique produces an initial "ideal" pre-compensator K(S) which

diagonalizes Q(S). The elements of K(S), which can be high order, are then reduced toa simpler form while ensuring that diagonal dominance in Q(S) is maintained. Thisinvolves expressing G(S) as:

(eon 3. 1G(S) = R(S) A(S)

where R(S) is a diagonal matrix formed by extracting common row elements from G(S)so that A(S) contains numerator polynomial elements only.

The decoupling pre-compensator is therefore given by:

K(S) = A(S) 1 (eqn 3.2'

However, this method is only possible if IA(s)j = 0 has all stable factors, i.e., allroots of the charactoristic equation are in the left half plane. If this is not the case.

* then K(S) is formed from:

Adjoint A(S) (eqn 3.3)K(S) = A(S)

Realization Factor

where the realization factor will contain the stable factors of IA(S) together with othersuitable lag elements to make K(S) physically realizable.

Tie elements of the diagonal matrix R(S) and the elements of A(S) so formed aregiven in Table 4 and Table 5 respectively. As JA(S) contains all stable factors, theideal pre-compensator can be formed directly from Equation 3.2. The results of thisoperation are given in Table 6.

It is necessary to scale the columns of K(S) to ensure that the steady-state gainsof the diagonal elements of Q(S) remain the same as those of G(S). The elements ofthis final pre-compensator are given in Table 7. Table 8 gives the speed related gain

variation.

Figure 3.2 and Figure 3.3 show that the addition of the pre-compensator has

resulted in an improvement in outputs at 12 kts.

18

%-P

TABLE 4

ELEMENTS OF DIAGO'NAL MATRIX R(S)

IR11(S)(S- + .06S +.25)(S- + .02S + .154)(8.2S + 1)

1R,(S)

-- S(.434S + 1)(6.62S - 1)(4.18S

(S(S +(3 .199S .99)(10S +1)(24S 1)

TABLE 5

ELEMENTS OF MAATRIX A(S)

A1 (S) = .25K<11(S' + .2S +.15)(8.2S + 1)

A1 () .1K(S .06S + .25)(-8. 57S + 1)12(S) = K<,2 (S

A.2( S) = K',2 ( .2'A~ 3 S K,('O*S 1.)(10.05S

31(S) = A.,() + ,(S

IA(S)! =,2( S C .2S 2 + 2 .:)S .2S)

,3( S) = K38.2S +) 10.) 0.0 S 1 )

This "ideal pre-compensator can 'ie reduced to ,he individual ciements ,C Kts

using step response data.

After the reduction process is completed, the elements of the reduced pre-

compensator so formed are given in Table 9. The terms KC1. Kc2 ' and K13 given inI

Tabie 9 are the speed related compensator gain changes necessary for the pre-

19

76

......................

TABLE 6ELEMENTS OF DIAGONALIZING PRE-COMPENSATOR

00649

IKIs 53. 19S 3 + 17. 136S2 + 9.5S + 1I-00454( -34. 143S' 2S2 - 8.29S +1)!K~() 3. 19S3 + 17. 136S + 9.5S+ 1

K,,(S) =.00285

.00376(10S 1)=534S5 278S4 +182. 6Sj 46. ISS- 11. SS 1

.00013( 80050S5 15650S 4 20665S3 +2255S 2 3S+5 352S' - 326S5 21113S4 646S3 162S- 21. JS 1

00285:< 3(s =100S3 4-30S- +12S -1

K, 3( S) = K2 ( S) =1 I( 3 S) = 0

compensator to cope with the non-linearity of the warship model and these are the

same as in Table 8.Figure 3.4 and Figure 3.5 show that interactions between inputs and outputs

have not oeen eliminated in the system's time response when we use the reduced order

pre-compensator also.

20

117Tpm iw~wiit.1 nr W.,VV'

TABLE 7FINAL ELEMENTS OF DIAGONALIZING PRE-COMPENSATOR

[KJJS Z~ 53.19S3 + 17.136S2 + 9.5S + 1

Kc(-34. 143S3 + 2s' - 8. 29S + 1)K~2 S) 53.19S' + 17.-136S' + 9.5S + 1

Kc.,(10S + 1)i3() 534S5 278S4 + 182. 6S3 + 46. 15S 2 11.5SS + 1

K- (8050S3 + 15650S4 + 20665S3 +2255S 2 -,3S + 1)K()=5352S + 3326S~ 2113S~ + 646S3 162S 21. 6S +

KS33) loos +30S2 +12S +1

I13( S) = K.I1(S) = '-3( S) =0

TABLES8

PRE-COMPENSATOR GAIN VARIATION

Smed Kc Kc2

12 -1.6 .579 .0456

is-5.2 11..11 -3.0C'2

26 -5.6 1.274 I-4.08

21

TV w -wNr 7 -to

J, LbLh0

c;* 4

......a.a. .. al tj

Figue 3. Stp Repons fo ConpcnstedWarsip o

at Rude Dead

- a a a 22

V w~w~wvIwxwiiwv

L

6 thCiL sC

NN

.I. .... .. a liFigure 3.3 StpRsos o4opnae asi oe

Na Fi Demand.

23.

TABLE 9

FINAL REDUCED PRE-COMPENSATOR ELEMENTS

1RK1 1(S )

8. GS 1

K,(-34.14S3 * 2S 2 - 8.3S + 1)53. 2S 3 17. l3S2 9. 5S + I

RK22(S) I

RK31( S )=

14.8S2 + 1. 3S + 1)

=K KcI(2QOOS 2 - 2S + 1)

RK32 (S) 130S' + 25S' + 12.6S I 11

RK3 3( S) =

RK13 (S) RK 1 (S) RK33(S) 0

24

V7 -. -. -Y -y - .

C4'

4 C;

15o

IO

- --. -T ,-

I'p

6 UN-.'

4 C; C

............. --

Figure 3.5 Step Response for Compensated Warship niodelat Fin Dernand using Reduced Order Pre-compensator,

26

N, .. . ..- , . . . ., . ., . .

F .

IV. COMPUTER SIMULATION".

From the results discussed so far (all of which are from Roberts but wererepeated at N.P.S. as part of this thesis), it can be seen that the pre-compensator

substantially reduces the interaction intensity of the compensated system and results in

improved ship stability and minimal loss of speed while executing normal maneuvers.

Also. it has been shown that the outputs of ship's rol angle, heading angle and speedcan be controlled by a low-order pre-compensator. However. it can also be seen -hat

'arge cross-coupling effects exist which cause long settling times. Therefore, a method

must still be devised which controls the interactions between inputs and outputs and

shortens settling "ime. The outputs of ship's roll angle. heading angle and speed can be

changed by a Jew ,-actors.

First of ail. this thesis will introduce the philosophy of Function Minimization

(F.M.). which is the main theme of this paper and introduce a few factors, which have

important effects on outputs.

Next, using above factors, this thesis will design other pre-compensators and

simulate them.

A. PHILOSOPHY OF FUNCTION MINIMIZATION

Classical control theory has historically been applied to the design problem with

,he assistance of graphical presentations and trial and error methods. Such methods

have been quite successful in the development of good control systems, but do not

answer the question "Is this the best system possible?".

If a given function has a minimum within the range of permitted parameter

variation, there exist numerical methods which can be used to find the minimum.

These numerical methods have been programmed and most computer libraries contain

one or more subroutines which can be used for Function Minimization. This thesis

uses the HOOKE [Ref. 41 subroutine of DSL.

Since one has the freedom to select the cost function to be used, certainly any of

those used within optimal control theory can be chosen. We can therefore design an -.

"optimal controller" without using the conventional theoretical approach.

We can also use any of the well known performance indices as a cost function,

i.e., JE2 dt, J!EJ dt, flElt dt are easily evaluated and minimized.

27

S::.

A

In addition, one can select the cost function to suit the particular specification of

the problem.

Figure 4.1 shows the block diagram for simulation using Function Minimization.

R FIN i.

YOLL COM PRE-COMPEN RUDDERP I

YAW COM SATOH POWER ---- r,

DE I RED'.'DESIRED _FUNCTION K

POLLYAW,SPEED 4-

VARIATION -,MINIM ZATION 4

COSTFUNCTION

Figure 4.1 Block Diagram of Function Minimization.

B. CHOICE OF COMPENSATORS

Use of Funcion Minimization in the computer is relatively expensive, so some

preliminary analysis and design is desirable to avoid excessive computer time. For a

simple system ( one output ) a BODE design or a ROOT LOCUS design might bc a U'

good starting point. The compensator thus found is then simulated and improved

through use of Function Minimization. For the ship problem in this thesis, the

preliminary work was done by Roberts as can be seen in Chapters 11 and Ill.

28

This thesis uses Roberts' compensators and outputs as a starting point. The

desired output curves were obtained by modifying the known outputs. The questions

to be answered are:

* Can better performance be obtained using the original compensator butrequiring a better output?

* Can the compensator be reduced and the resulting output be as good or betterthan the original compensators?

* Can a better output be obtained by increasing the order of the compensator?

Vvhen the Function Minimization subroutine minimizes the function, it movesthe poles and zeroes of the compensator to the best locations in the S-plane. If the

compensator has too many poles and zeroes, the program tends to set Z = P for the

unneeded poles and zeroes, thus the compensator order may be reduced.

In many cases, even though the poles and zeroes may not exactly cancel, they are

placed so close to each other that they contribute very little to the result and so wemay etlFciently cancel them. Thus one can start the design by simply choosing each K

to have numerous zeroes and poles as can be seen in Table 7 and Figure 3.2, which isthe original pre-compensator. However, this thesis will show us that we don't need

that many poles and zeroes for the choice of compensators.

In this thesis, as shown in Table 9 and Figure 3.4, the reduced order pre-compensator was used.

The optimum parameter values of this reduced order pre-compensator, which

reduces the corresponding cross-coupling, can be determined by Function

Minimization and Table 10 shows the optimum parameter values of the reduced order

pre-compensator in Table 9 by Function Minimization.

Figure 4.2 shows the positive damping effect on the system's time response ,.%hen

optimum parameter values are used in the reduced pre-compensator.

Figures 4.3 through Figure 4.5 show the difference between Roberts' method andthe Function Minimization method, where the continuous line is the output of Roberts -_4and the dotted line is the output of Function Minimization.

As shown in Figure 4.4 and Figure 4.5, cross-coupling is decreased dramatically

in the case of Function Minimization.

C. CHOICE OF DESIRED OUTPUTSOne can always choose the ideal output as the desired output for the Function Js

Minimization subroutine. However, this may not be a good choice because that

29

,..-,.-...-.-,, ,: .,..,...,. ,... ... ,,..,,-[,.] .,j, .,,. . ... ,,".....".".,......"...."...............................lai~amal

TABLE 10OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR

I RX11(S) = 8.6S + 1

K.2-72. 14S3 1.98925S 2 - 2.3S 1)RK,,( S) = 1(31-23

S76. 2S + 28. 63S + 11. 5S + 1

RK,2(S) = 1

.63525R -131 (S ) =32.05S2 + 2. 975S + 1

K3( 1741. 25S 2 - 1. 94375S + 1)RK32 (S) = 9. 25S + ii. 25S' + 47. IS + I* I 1RK3 3 (S) = 11.4S + .

P K13(S) = RK2 1 (S) = RK33(S) = 0

particular output may be impossible to achieve. Although the minimization process

will determine a closest fit, if the cost function is a "least squares" function the

solution may not be acceptable. For example, with the ship problem of this thesis, the

ideal output might be

* Turn radius of two ship lengths

* Zero speed change

, Zero roll angle

None of these characteristics are possible. The results of a FunctionMinimization design will not satisfy any of them and the design achieved may not beacceptabie. The desired output must therefore be chosen realistically, i.e., within the

physical capabilities of the system. For the ship control problem, outputs were chosenbased upon the results obtained by Roberts. In order to obtain better performance the

desired outputs were chosen to be similar.

30

.... ... . .. . . ... A

Laia

C2.

6 I9 6 *Ph

7 To 'H-

69 W*I

C;N

AW-A

6 6 4

-- --- --Fiur 4. Oupu whe we us ha-iu a f eue r-o pnao

ato Rude Dm-d

63

ywp. V%4.N

jlr.T3

II

qU

6 kw 6ht

vp 4; c; ,

JMO

Figure 4.3 Comparison between Roberts and F..M at Yaw output.

32

KNT~ Kyim-irrig mR -7 -- v 1%,-%; %r-%r-a-,w- - w -%rw- - w k:

C'S

JP

V.

Ad

* e.

333

3SS29 alil

34-

TWOWPVIVNV MV PC-~ MR .'%FP- -

This thesis chose 3 desired system outputs , each consisting of a specification for

heading, roll and speed. These desired output curves (AFGENI, 2 and 3) were

obtained by modifying the known outputs (Figure 3.3) using Arbitrary Function

Generators (AFGEN) provided by DSL.

Figure 4.6, Figure 4.7 and Figure .4.8 show AFGEN1. AFGEN2 and AFGEN3

.espectively.

We can know :hat the A F GEN 3 is better than the other two cases by the trial

and error method. Therefore this *thesis will use the AFGEN3 for desired outputs "or

another simuiarion.

D. COST Fu.NCTIONS

When usingl Function \Iinimiuzation as a design too[, a cost function must be

chosen. This cost !Eunct-on ,s asuailv an intezrai.

IOne - osiib~e -rccedure s -n oose -.he . esired performance as a rc',crence ina

'eilec, . Ost -urictwn VM.r'c :S -fle .nreirai c i ,le quare )t' the .1iaTerence ',-etrween

aesirzu )utnut and. actuai -.utnut. When the svstem has several outputs, the cost

:aunction must ;onsi"Jer iil of nem. usuailv as a weighted sum.

E. WEIGHTING FACTORS

For zhe -iree outputs sxvstem, the cost :unction is of the torm

E jEI dt kE2dt - XJE 32 jt.

There are no fixed rules for choosing the values factors k-,X. and X.One

approach is as follows:

" Select -,he Ouzzput which is considered most important. say n 1. 1Use this as ari!ferenc e and select k, =Lo.

* Base the vajue assigneJ to -.he ; econd weiahting factor. ;~ n -his case. n -he.mporzzance ), output %itn respect rio output J . utput i s equaiivimportant ais output 1-0hnt. =l. IfA -. s haif as mportant. -hen k,

* The third weighting factor is criosen in like manner. If the third ourput :s lessimportant than the First. then k. ..k I t a s also !ess :woortant than hne

*econd output. -hen ~ .

The actual numbers chosen. :"Or c:xamvle 1, ) are simply estimate-, I ased o-n

experience. The designer may decide to change them af-ter stud':ing simulation results

of a First design.

In the case of ship control, the primary output is heading. Therefore, assign kh a

value of 1.0. Some change in speed is unavoidabie and the desired response should

35

.7I

C

ITS

.9; C;

IW

Figure 4.6 AFGEN 1.

36

373

%a

IsJ

CD 43

AN.

6.4b

c; C;*C;

------ -----

Figue 4. A FEN 3

384

: ~ ~ ~ ~ - -7ri'r.;,; '' - . . . .. -Y- .- . :,. ., . ,7- _ - .:

show this. However, if the desired response cannot be achieved perhaps more a",

fluctuation in speed would be acceptable. For example, a value of approximately 0.1

could be assigned to If the desired roll output is twice as important as the speed

output, then %r would be assigned a value of 0.2.

Table I I shows each case for simulation using weighting factors.

TABLE I 1

SIMULATION CASES USING WEIGHTING FACTORS(W.F.)

CASE Xh A I OUTPUTS

1 2 . Figure 4.9

S6 .i Figure 4.0-

3 1 .6 .4 Figure 4.11

4 1 1 1 Figure 4.12

CASE4 is from Table 10, which uses Xh = 1, Xr = 1 and k. 1 and this wash. r

repeated in Table 11 for comparison when we consider the weighting factors.

Table 12 through Table 15 show the elements of the resulting compensators for

each of the above when we used the F.M. approach.

When F.M. was used for all coefficients in compensators, RK1 :, RK31 and RK 2

were :e narameters that ,he F.M. subroutine adjusted.

.

O'r

39

..

'CV

AUA

6I

\ ..

\ %

\~

\ . ,.

d qi p

Fiur 49 \c Repos whn =I .2ad =.

at\ Rude Dend

IJ "

7400

.........................-

at Rudder Demand~*-.. .'*-*.*s. %

-p ~ 0

* II,

9;0

41

6~

C2 C

6

C C; 0

11~~~ ~ 1O1 X 6 n

at6dc ead

- - C CC -42

do

.,.,

\! *"°

w a'a

6 I1

\ -, -.-

6m Ihc

Figure 4.12 Step Response when Ih !, Xr 1 and X.at Rudder Demand.

43

.. ;....., ...,., ...,,, .,......, -..,v->......... .... ,.-.-....... . ..--. ,-,.. -- , o".--

TABLE 12

OPTIMUM PARAM-ETER VALUES OF REDUCED PRE-- COMPENSATOR AT CASE I W.:.

1RKI(S) = 8.6S 4 1

K,(-47.64S3 + 2.0C0625S2 - 2.3S 1RKI2(S) = 367.7S' + 27. 38S + 11. 375S + I

RK22(S) = I

.577125RK3( S) =20. 425S 2 + 1. 8625S + 1

KCI(1915.625S 2 - 2.001875S + 1)K32 (S) = 90. 625S3 + 53. 125S2 + 23.85S + 1

1RK3 3 (S) = 11.4S + 1

RK 13(S) = RK21IS) = RK 33(S) 0

44

444

. . .. . . - - - , - , -. -,- .- - - - -' .' .'- -' - - : - .x .. " - - ':- ' - . ' - - - - - ' > " " : " " "

TABLE 13

OPTIMIUM PARAMIE ER VALUES OF REDUCED P'RE-COMIPENSATOR AT CASE 2 W.F.

RKII(S)=8.6 6S 1

= K-,(-66. 14S' + 1. 99525S - 3.3S 1)

45.2S3 + 31. 13S2 s 05

RK,,( S) =1

603999R 31 (S) = 58S2 424

RK3(S= KI( 1835S' - 1. 975S +1)RK32( S) 53S 3 + 80S2 + 34. 6S + 1

1RK 33( S)

11. 4S + 1

R13(S KS ) ) K3 3( S) =0

45

77777"'417

TABLE 14

OPTIMUMI PARAM'i itL VALUES OF REDUCED) PRE-ICOMNPENSAT: OR AT CASE 3 W.F.

RKIIs) =8.6S + 1

K P. 1 () j(-56.14S 3 +1. 996S2 - 2. 3S +1)

71. 2S- 27. 13S 2 +1J..25S 1

RK22 ( S) = 1

RK 31 (S) 25 S + 2. 35S + 1

RK32 (S) 1842. 5S' - 1. 9775S + 1)RK 3( S 56.5 5 3 +77. 5S 2 +33. 6S + 1

RK33 (S)11.4S + 1

RK13 ( S) =RK 2 1(S) =RK 33 (S) =0

46

P P .

p

p

TABLE 15

OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR AT CASE 4 W.F.

1 "RI(s) = .8.6S 1

RK( K,(-72. 14S3 + 1.98925S2 - 2.3S I)12(S = 76.2S 3 + 28. 63S 2 + 11. 5S + 1

RK 2(S) = 1

•63525RK 31 (S) = 32. 05S2 + 2.975S + 1

K_ 3 (1741.25S 2 - 1.94375S + 1)32(S) 9.25S1 + 111.25S 2 + 47.IS 1

1-RK33(S) 11.4S + 1

RK 13(S) = RK21 (S) = RK33(S) = 0

From simulation, it appears that the-most desirable output can be obtained at -h I k = .2 andX = X 1.-' -r s •'

Figure 4.13 through Figure 4.15 show the comparison between the Figure 3.4 for

Roberts, Figure 4.2 for CASE 4 and Figure 4.9 for CASE 1. Where the output of

Roberts is from Table 9, which gives the final reduced pre-compensator elements, the

output of CASE 4 is from Table 10, which gives the optimum parameter values of

reduced order pre-compensator by Function Minmization. The output of CASE I is

the best case when we consider the weighting parameters for CASE 1, 2 and 3 in Table

11.

The continuous line is used for Roberts, the dotted line for CASE 4 and the

dashed line for CASE I outputs.

Figure 4.16 through Figure 4.18 show the comparison between the above three

cases for Fin, Rudder and Power respectively.

o47 .

*. ,

MAd

6S

CiS

-IRa

Figure 4.13 Comparison bctweccn Fig 3.4, Fig 4.2 and Fig 4.9at Yaw Output,

48

WrW IIWNu- -6"'qJ1 V~ wjl *vlv**I~ -rI-~~~r~P *** U** -- .

fAS

-4D

c; 4

Figue 414 Cmpaisonbeteen igat Rll Otput .4,Fig .2 ad Fi 4.

49S

v- v -vv,

TV".- V-%

AD

c6 C3

6 c9

C; C;U,

6 J C afm C;

Pat Spc Output.-

65

-rjgU J .A L.PFJFIW . -j-.ju. .-- ' .

.00.

kA'

I w 'S..

Figue 416 Cmpaiso beteenRobetsCAS 4 ad CSE 0

fur in utpu at30 udde Deand

519

%.d

68

-I

L•J,

.I-

133MO3) 183808

6 4

Figure ,4.17 Comparison between Roberts, CASE 4 and CASE I

for Rudder Output at 3' Rudder Demand.

52

-- + --- ---'- '- .- --- -

-" -'-"-- ". -'.,.' . ..--: ':,+'; i . ... + a-. ,k ' i iiia Uk

i" i' " i i.i'+--1. " i.-+ -. +. .," .- -. + . .. .. -- '- -. . -, - - . -. -. - -. .- __ _ _

CD-

vX.

PI.cmd

.............

C2

AMU

6 lb

------ -----

6 %b

Figue 418 Cmpaiso beteenRobetsCAS 4 ad CSE

for owerOutut a 3' uddr Deand

530

The simulation obtained so far was performed using three degrees for rudder

demand and we can know that there is no problem about maximum values of input

and output shown by Table 3 for each output.

As can be seen in Figure 4.16 through Figure 4.18, CASE 1 is better than the

other two cases. But we have to consider carefully the physical realization in Figure

4.1S because the power curve varies very fast. Therefore when we design the pre-

compensator using weighting factors or other factors, we have to always consider

physical realization limitations.

Figure 4.19 through Figure 4.21 show the comparison between the above three

cases for Fin, Rudder and Power when we use twenty degrees for rudder demand.

From the above outputs, we can see that CASE 4 is better than the other two

cases when we use twenty degrees for rudder demand.

F. VARIATION OF NUMBER OF POLES AND ZEROES

As shown by Table 7, the original pre-compensator by Roberts requires complex

mathematical equations for a complete and detailed description. For many problems

in the analysis and design of many modern dynamic systems, a simplified description,

i.e., a low order model, is adequate and desirable. This thesis is concerned with the

development of such low order models for the pre-compensator.

Two types of situations are commonly encountered in practice:

" A system exists and can be tested, but its equations are not well known or notclearly defined.

* A high order complex model of the system is known and can be used, butit is undesirable for the problem to be studied.

In either case the response of the system to a chosen signal can be obtained, and

a low order model developed which has essentially identical outputs for identical inputs

as the higher model.

By carefully planned studies it should be possible to see how much reduction in

order can be achieved, how closely the behavior of the low order compares to that of

the system and perhaps a best or optimum order can be found for the reduced order

models.

In addition, by careful selection and classification of the po!e-zero geometry of

the high order system, it is hoped that a correlation may be found between such

geometry and that of the best low order model. In any event, it is anticipated that

some rules may be established for the choice of the number of poles and zeroes in the

54

% ", "~~~...'....".'..'...:.. ............. .. ... . . . . . . . .

J -w

7 o.

(338930 WY) I

TIM lo N

Figure 4.19 Comparison between Robcrts, CASE 4 and CASE Ifor Fin Output at 200 Ru~dder Dcmand.

*7-'.7 .7 7,e

.-

40

LjLLA,

.

C6k

Figure 4.20 Comparison between Robcrts, CASE 4 and CASE 1for Rudder Output at 20" Rudder Demand.

56

LLI

6~C

.. 7

'V N N (V

Figure 4.21 Comparison bctwcen Rohcrts, CASE 4 and CASE Ifor P'owcr Output at 200 Rudder Dcmand.

57

low order model, such that evaluation of the pole, zero and gain will require only a few

computer runs.

As shown in Figure A.1 of System Block Diagram for Simulation, K11 and K.2

affect the Fin output of the system and K31, K32 and K33 affect the Power output of

the system. Therefore when we change the order of these elements, outputs, i.e., roll

and speed will be changed by them also.

Unfortunately there is no known mathematical basis for choosing the "best"

order for a low order model for each element of the pre-compensator. For

convenience, this thesis has changed the order of K and K33 for two cases oniy.

Table 16 and Table 17 show the values of element for :wo cases and Figure 2

through Figure 25 show the outputs for comparison between original reduced order

pre-compensator by Table 10 and the above two cases respectively. Other elements are

the same as in Table 10.

TABLE 16

OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR

WHEN WE CHANGE K11 ONLY

97. 18759S + 3. 94875K(11 (S) = 7.53125S2 + 21.075S + 12. 025

-K 2 (-72. 14S3 + 1.98925S2 - 2. 3S 1)K12 ( S) = 76. 2S 3 + 28. 63S2 + II. 5S + 1

K,,(S) = I

.63525K 31( S)= 32. 05S 2 + 2.975S + 1

K 1741. 25S 2 - 1. 94375S + 1)K(32 ( S) C3 +.5s 111.25S2 47S 19.25S 3 + I.2S 47.1IS + I

1K 3 3 (S) = 11.4S +

K 13 (S) = K21 (S) = K3 3 (S) = 0

58

.',.

.

PL - - -- - " •- -.. . . . .".. ."-... ." '.. .

TABLE 17

OPTIMUM PARAMETER VALUES OF REDUCED PRE-COMPENSATOR

WHEN WE CHANGE K33 AGAIN IN TABLE 16

97. 18759s - 3.94875

7.53125S2 + 21.075S .12.025K 2 S ) = Kc2(-72. 14S-' + I. 98925SL 2. 3S )"'

.6.2S 3 28. 63S+ 1. 8

K22(S) = 1.63525

( 31 (S) 32. 05S2 + 2. 975s 1

)(1741. 25S ' - 1.94175S9.25S3 + 111.25S2 + 47. IS + 1

98. 1875S + 2.0562533n(S) = 6. 6562532 + 28. 05625S . 4. 43625

KI13 (S) = K, 1 (S) = K3 3 (S) = 0

,5,

-5

By the same algorithm, we can approach the appropriate order of pre-

compensator and the values of each element for better design.

As can be seen in Figure 4.22 through Figure 4.25, when we put the additional

7ero and Pole to the eiement of' 'he pre-compensator. :he cross-coupling of the roil

output Is decreased dramatically.

59

* In

Li

~e1

%ol

III i2

C; C; C.

66

C,

0%

NA

6 jW Lb -

fj Co ca 0

C; C;

for Spe Output.

61J.S.

ca

"p

-

-',

.ar

6 ( oIgu

61..

Figure 4, opaio btcclb, 1, an ,6bl 17,,,,; ,: ,.For Rol Ouput.

. ,'3r62/

L 1.1 11 .a

• Id/ L: 1,< w6 0I, lO/!la'

...

ca60

UgU

--- --- --- --- --

Figue 4.5 C mparsonbet"cen rabc 10and-rabe 1

for Sccd Otput

63~

V. CONCLUSIONS AND RECOMMENDATIONS

A. CONCLUSIONS 4

In this thesis, the development of a pre-compensator to reduce the undesired and

highly nonlinear crcss-coupiing effects in the maneuvering characteristics of a modern

warship has been presented. Simulation results have shown that Function

Minirruzaion nrocedures tbr coordinated steering of a surface hhip would ,iiicantlv

improve ship tabiiitv. :ninirmze .oss -of s-eed and reduce :he ,nteraction ntensitv n'

tLhe compensated system.

Simulation results have also shown that the reduced pre-compensator can be

Jeter-mnied bv Function .\linirmization directly rrom an ideal pre-compensator based onavlert -tecutcattons.

B. RECOMMENDATIONS

Computer simulation for Function Minimization leads to the followingS,

recommendations:

* This thesis uses the "HOOKE" .;ubroutine of DSL for Function Minimization.It has certain constraint parameters, i.e., ITMAX, CFT. Simulation outputsare changed by changing initial values and the step size for FunctionMinimization. Therefore, for given specifications initial values and step size forFunction Minimization by constraint parameters should be determined byexperience.

" In this thesis, it has been shown that Function Minimization can be used forvarious cases as shown in Table 11. A particular case for needed specificationshould be determined and simulated by trial and error method 'or dIementoptimum vaiues which have undesired cross-couoiing effects.

" In this thesis. it has been assumed that the stabilizer fin to yaw cross-couplingterm, G21 (S), is a nuli entry for the class of warship considered. However, it isrecommended that cross-coupling between these parameters be considered infurther studies on general surface ships.

* urther research shouid invesugate ship characteristic constraints.

' Further research should investigate the etfects of various sea state conditions,speeds and maneuvering.

64

IV~~ -C - -J Z- -7.

APPENDIX ASYSTEM BLOCK DIAGRAM FOR SIMULATION

+ Gi

.

SIN K33(5) G3 3(3) *SE

Figure A.l System Block Diagram for Simulaticn.

65

APPENDIX BCONSTRAINT PARAMETERS OF FUNCTION MINIMIZATION

To use HOOKE, the user must initialize the following arguments and, in the

main program.

CALL HOOKE (X, STEP, N, ITMAX, CFTOL, ALPHA, BETA, CF, Q, QQ,W, IPRINT, MINMAX)

All of these arguments must be initialized in MAIN, except for X, CF, Q, QQ,and W. Recommended values for ALPHA and BETA are ALPHA = 2 BETA = 0).5.

All of the arrays, i.e., X, STEP, Q, QQ and W, must be declared and dimensioned in

MAIN.

6

I

66"

.. . . . . . . . . . ... ... . ... ... . . . . . . . . .- ... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ." hi'-

TABLE 18

THE LIST OF PARAMETERS

ARGUMENT MEANING

It he array ofNaaee aus h user mustprogram or in MAIN.

STEP .an array of dimension N containing the initialstensiZes to be usedl In the searc~.

the number of loarameters (a positive integer,N !at most 15).

the maximum number of function zalls --o be .

_:JAX perf ormed.

t-he error in the criterion function to beOFTOL reached before the program terminates

(difference between the current value and theprevious stage value).

the factor of (Y - X) which :.s added to Y to get,ALPHA XN dEW; a number greater than or equal to 1.

the ste~size reduction factor; a number betweenBETA 0.a.id 1

CF the -iilue of the criterion function.

arrays of dimension N, to be used as work space.Q2,QQ, W T~iey must be deci ared and diJ.mensioned in thb J

Y1A:n. program.

'an integer flag:= 0 for no intermediate printoutIPRINT I 2.1 for intermediate printouti

of X, CF, the number of function evaluation andnoification of step-reductio-n.

MINT~,an integer :flaq: = -1, searches for a minimumML ~ ~ 1NAX+ searches :'or a maximum

67 >

p. APPENDIX C

COMPUTER PROGRAM FOR UNCOMPENSATED SYSTEM

TITLE SIMULATION OF UNCOMPENSATED SYSTEMARRAY A 1( ),B 1(3,),A2-(2),B2(,4),A3(lI),B3(5),A4( 2).B4(4),....

A5(I;",B5(3'),A6(1 ),B6(2)TA B LE A 1(1) = 1. B1- 3) = 4_24.LA 2(1- 2)=-. 57 ,1. B 2(1-) f 3. 3,17..

9.52. 1,A3( l) = 1B3(1-5) = 1'2.32.25.1 1.2. I.0.A4(I -2)= 10. 1....

B6 I-')=2-4.1CONST K '. 4K2QI.2=.I:2 'L5.3=.9.3=.,.

RL =0.,YW -= (.,SP =0.DERIVATIVE

FIN= 0.0*STEP(0)RLDDER= S. "STEP(0)

?OWER=.).>7STEP(0'iROL 1 TNR..LAIB. *IROLL: =TR'NFR I.3,RLA2.B-.K12'-RLDDER)ROLL= ROLLI4 ROLL2YAW= TRNFR(O,,4,YW.A3,B3,K22* RUDDER)SPEED 1 T-RN'FR( 1,3,SP,A4,B4,K31*FIN)SPEED2 =T RNFR(0,2,SP.A5,B5, 2RUDDER)SPEEDS TRNFR(0, I ,SP,A6,B6. K3 3*POWER)SPEED = -SPEED 1-SPEED2 + SPEED3

CONTROL FINTIM = SO.:'RINT I.,ROLL.YAW,SPEEDSAVE (Sh O.I.ROLL,YAW,SPEEDGR.-PF-IfG['SI.DE= TEK6IS,PO=0.5) TIME(LE= S..L'N= SEC) ROLL(LO= -1,LI I,...

SC- .25.NI= S).YAW(LO=-l,LI= 3'.SC= 25 .NI= S..SPEED(-O= -1.LI = .. SC = .25.N I=

L,-BEL ILSTEP RESPONSE FOR, UNCOM'VPEN\SATED WARSHIP MIODELA:T 12 KTSLABELGI 10 RUDDER DEMANDENDSTOP

68

APPENDIX DCOMPUTER PROGRAM FOR ORIGINAL PRE-COMPENSATOR

TITLE SIMULATION OF ORIGINAL PRE-COMPENSATORARRAY A1( I),B 1(3),A2(2),B2(4),A3( I),B3(5),A4(2),B4(4),..

A5(1 ),B5(3)jA6( I ,B6(2)ARRAY C 1( ),D 1(4).C2(4),D2(4),C3(2),D3(6),C4(6).D4(7),C5(l1),DS(4)TABLE Al(l)= 1,BI(1-3)= 4,.24,1,A2(1-2)= -8.57.1,B2(1-4)= 53.3,17.17,...

9.52.1.A3(1)= 1,B3(1-5-)= 12,32.25,11.2,1,0,A4(1-2)= 10.1....B4( 1-4) = 240,58,26, 1,A5( 1) =I.B5(1-3) = 240,34, 1,A6(1) I 1....B6(1-2)= 2-4.1

TABLE C1(l)= I, D1(1 -4) =5 3.19,17.13 6,9.5, 1,...

D2(1-4)= 53'.19.17.13,9.5,1,C3J(1-2)= 10,1....D 3( 1-6) 534,278, 182.6,46. 15,11.5,1....C4(1-6) = 80050.156 5020665.2' .55,3,..D4( 1- 7)= 53 52,3 326,2113,646, 162,2 1.6, 1,...C5(1)= 1,D5(1-4)= 100,30,12,1

CONST K1 1=0.1 14,K12=0.18,K22= .01,K31 =0.058,K32=0.096,K33=0.1...RL=0.,YW=0.,SP=0.

CONST GO 0 ,GC)O = 0,GCJ = 0,GC4 = 0,GC5 = 0DERIVATIVE

RIN= 0.0*STEP(0)YIN= 3.*STEP(0)SIN = 0.*STEP(0)ElI = RIN-ROLLE2 = YIN-YAWE3 = SIN-SPEEDFINI = TRNFR(0,3,GCIO,C1 ,D I.E 1)FI-N2 = TRNFR(3,3,GC2O,C2,D2,.1.6*E2)FIN = FINI -~ FIN2ROLL I = TRNFR(0,2RL,A1,BI1,K Il*FIN)RUDDER= YIN-YAWROLL2= TRNFR(1.3,RL,A2,B2,K 12* RUDDER)ROLL = ROLL I + ROLL2YAW = TRNFR(0,4,YW.A3,B3,K22* RUDDER)SPI = TRNFR(1,5,GC30.C3,D3,0.579'E 1)SP2 =TRNFR(5,6,GC4O,C4,D4,0.0456*E2')SP3 = TRNFR(0,3,GC5O,C.5,D5,E3)SPEEDI = TRNFR(1,3,SP,A4,B4,K31I*FIN)SPEED2 = TRNFR(0,2,SP,A5,B5,K32*RUDDER)POWER= SP3-SP2-SPlSPEED3 = TRN FR(0, 1,SP,A6,B6,K33* POWER)SPEED= SPEED3-SPEED2-SPEED I

69

.- 77-77- -

CONTROL FINTI M =80.

*RINT 1.,ROLL,YAW,SPEEDr SAVE (SI) 0.1,ROLL,YAW,SPEED

GRAPH(G1'SI,DE=TEK61 8.PO = ,.5) TIME(LE =8.,UN =SEC) ROLL(LO =-2,LI = I,SC =.5,NI =8),YAW( LO =-2,LI = 3,SC = .5,NI = 8),....SPEED(LO =-2,LI =4,SC = .5,-Nl 8)

LABEL(GI)STEP RESPONSE FOR COMPENSATED WARSHIP MODEL AT 12 KTSLABEL(GIJ 10 RUDDER DEMANDEN.\DSTOP

70

APPENDIX ECOMPUTER PROGRAM FOR ORIGINAL REDUCED PRE-

COMPENSATOR

TITLE SIMULATION FOR ORIGINAL REDUCED ORDER PRE-CO)/PENSATORARRAY A 1(1 ),B 1(3),A2(2),B-2(4),A3(lI),B3(5),A4(2),B,4(4),...

A;( I).BS;(3 ),A6( 1),B6(2)

TABLE.AI(I)= .II3 1...A(-=-.7.B -Y=3.1."..0.2iA ( 1.B3(1-5')= 12.32.25.1 1.2.l.0.A (1-2)= 10.1....

36(1-2)= 24,1TABLE Cl(1)= 1,DIl(l1-2) = S.6.lI.C2YL-4) = -34. 14;,2.-S. 3.1....

DY 14)=53.2,1. 13,9.S.1.C3( 1l=1LD3' 1-3)= 14.3.1.3.I....

C_51. 1.D 5(-2)= 1.CONST K 014K20i.2=.IK1=)0SK2'.9.3=...

R L=0YW =0.. S P0.CONST KC1 =-I.6,KC>0=(.59',KC3=).0456....

GC 10 0,GC2O= .GC30 0,GC4O0,GCS = 0DERIVATIVE

RIN = 0.0*ST EP(0)YIN= 3.*STEP(O)SIN = 0.*STEP(0)ElI = RIN-ROLLE2 = YIN-YAWE3 =SIN-SPEEDFINI = TRNFR(0,1,GCl,CI,D LE 1)FIN2 = TRNFR(3,2-,GC2O,C2,D2,KC1 *E2)FIN= FIN I - FIN2ROLL I = TRNFR(0.2.RLAI1.31,K I I1*FIN)RUDDER= YIN-YAWROLL2 = TRNFR( 1,3,RL,A2,B2,K 12*RUDDER)ROLL= ROLL I + ROLL2YAW = TRNFR(0,4.YW,A3,B3,K22* RUDDER)SP I = TRNFR(O.2.GCS-O.C3.D3,KC:*E 1)SP2 = TRNFR(2.3,GC'0,C-4.D4,KC3 4 E2)SP3 = TRNFR(O,1 ,GCSO,CS.D5,E3)SPEEDI = TRNFR(l,3.)SP,A4.B4,K3 IFIN)SPEED2 =TRNFR(0,2C,SP,A5,B5,K32* RUDDER)POWER= SP3-SP2-SPISPEED3 =TRNFR(QJ 1,SP,A6,B6,K33 'POWER)SPEED = SPEED3-SPEED2-SPEEDI

CONTROL FINTIM=Yl80.

71

F d- . . .

*RINT 1.,ROLL,YAW,SPEED*RINT l.,FIN,RUDDER,POWERSAVE (SI) .1,ROLL,YAW,SPEEDGRAPH(G1,'S I,DE =TEK618,PO =O,.5) TIME(LE =8.,UN= SEC) ROLL(LO=-2,LI = I....

SC =.5,NI = 8),YAW( LO=-2,LI =3,SC =.5,-NI = 8),...SPEED(LO = -2,LI = 4,SC = .5,NI = 8)

LABEL(GI)STEP RESPONSE FOR COMPENSA -IED WARSHIP MODEL AT 12 KTSLABEL(GI)FIGURE 3.4ENDSTOP

7 2

APPENDIX F-A*COMPUTER PROGRAM FOR AFGEN SUBROUTINE

TITLE SIMULATION FOR desired outputs using afgen subroutineAEGEN 'RL=0,0,1,-.03,4,-.l,0,0,16,0,23,0,29,0,36,0....

42,0,48,0,54.0,61,0,67,0,80,0AFGEN 'YW =0.0,1 ,.00006,4,.006,8,.04, l0,.07,20,.2 9,30O,55,40,.S,80, 1.61AFOE'N SD=0,0,1,-.004.0,-.01,21,0,31,0,41,0,51,0,70,0,80,0DERIVATIVE

X =TIMEROLL = NLFGEN(RL.X)YAW = NLFGEN'(YW,X)SPEED= NLFGEN\(SD,X)

CONTROL FINTI M = 80*RINT 1..ROLLYAW,SPEEDSAVE (SI) 0.1.ROLL.YAW.SPEEDGRAPWH(GI S1,DE=TEK6I8,PO=0,.5) TIME(LE= 8.,LN\= SEC',ROLL(LO=-d.LlI1,.

SC =.5,NI =8).,YAW(LO= -2,LI =3,SC= .5,NI =8)....

SPEED( LO -2.LI= 4,SC =.5,NI = )LABEL(Gl) AFON I FOR FIGURE 4.6ENDSTOP

73 *

APPENDIX G

COMPUTER PROGRAM FOR FUNCTION MINIMIZATION

D COM.NION'HA'NDJ,'FLAG,ER,K1,K21,K3,K4,K5,K6,K7,K8,K9,KO 1,1(I 11D COMMON'HANDJK3,K14TITLE SIMULATION FOR FUNCTION."MINIMIZATION SUBROUTINEARRAY A 1( 1),B I(3),A2(2),B2(4),A-)(1I),B3(5),A4(2),B4(4),A5(lI),B5(3),....

A6(l1),B6(2)STORAG C 1(1).D 1(2),C2(4),D2(4),C3( I ),D3)(3),C4(3),D4(4),C5( 1),DS(2)TABLE Al(1)= l.Bl1-3)=4,.24,1,A2(1-2)= -8.57,1,B2(1-4)= 53.3,17-1 7 ....

9.52.1.A3(1)= I.B3(1-5)= 1,2511.10A4-)=10.1,...B4( 1-4) = 240,58,26, 1,A5( 1) = I1,B5( 1- 3) = 240,34, 1,.A6(I) =1,1601-2) =24,1

CONST KO I = -34.14. K20 =2,K(30 = -8.3,1K40 =53.2.K50 = 17.13, K60 = 9.5, K7 0= .579....K SO= t 4.S. K90 = t. 3.K 100 = 2000, KI 10= -K,120 = 130, K130-= 25....~K140= 12.6

CONST C11= . 114C12=.IS,C22=.03 =.058,C32=.096,C3-3=.I,IC=0,...KCl =-1.6.KC3=O.0456

PARA.M KLIMIN= -50,KPlMAX= 200,K2MIN =.0I,K2MAX = 200,K3MIN = -10,K3MAX= 200..K4.MIN = .0,K4.MAX = 200,K5MIN = .0,K5MAX= 200,K6MIN = .I,K6MAX =200..

K7MIN = .01,K7MVAX =200,KSlN'\= .01,KS.MAX= 200,K9MIN\= .01,K9M.VAX = 200,.KIOMIN= .01,KIOMAX= -300,KllMIN=-I0,KI.MAX= 200,KIl2MIN= .01....KI2MAX- 200,KI3MIN=.01,KI3MNAX=200,Kl4MIN=.1,K14MAX= 200

AFOEN RL =0.0,1,-.03,4,-. 1,10,0,16.0,23,0,29,0,36,0,...-42,0,48,0,54,0,61,0,67,0,80,0

AFGEN 'YW =0,0,1 ,.00006,4,.006,8,.04, 10,.07,20,.29,30,.55,40,.8,80,1 .61AFGEN SD=0,0,1.-.004,10,-.01,21,0,31,0,41,0,51,0,70,0,80,0INITIAL SEGMENT

I F(FLAG. LT.0.) K I = KO01IFfFLAG.LT.0.) K(2=1K20I FiFLAG.L T.0.) K3-= K30I F(F LAG -LT.0.) K4 = K40I F(FLAG. LT.0.) K(5 =K50IF(FLAG.LT.0.) K6= K60IF(FLAG.LT.0.) K7= K70IF(FLAG.LT.0.) K8= KSOIF(FLAG.LT.0.) K(9= K90IF(FLAG.LT.0.) K(10= K100I F(FLAG. LT.0.) K1I1I= KilO0I F(FLAG. LT.0.) K(12 =1K120I F(F LAG. LT 0.) K(13 = K130I F(FLAG .LT.0.) K 14 =K 140FLAG= FLAG+ IIF((KI.LE.KIMIN).OR.(K 1.GE.KIMAX)) THEN

74

KI =KOl

ENDIFIF((K2.LE.K2MIN).OR.(K2.GE. K2MAX)) THEN

K2= K20ENDIF

.

IF((K3.LE.K3MIN,).OR.(K3.GE.K3MAX)) THENK3 = K30

ENDI FIF((K4.LE.K4MIN).ORP.(K4.GE.K4M',AX)) THEN

K4 = K-40ENDIFIF((K5.,LE.K5MIN).OR. K5.GE.K5SMAX)) THEN

K5~= K OEN, DIEFIF((K6.LE.K6MIN-).OR. K6.GE.K6MAX)) THEN

K6 = K60ENDIFF6,.EKI R.K EKA) THEN

K7= K70EN D IFIF(( KS.LE.KSMI'N '.OR.( KS.GE.KS.MAX)) THEN

KS= KSOENDIFI F4iK9. LE. K9M IN').OR.( K9.GE. K9 MAX)) THEN

K9= K90END IFIF((K l0.LE.KIOMI-N).OR.(K1O.GE.KIOMAX)) THEN

K1O= K100ENDIFJF((K II.LE.K I IMIN).OR.(K I IGE.KIMAX)) THEN.4

ENDIFIF((K12.LE.KI2MIN\).Ok (KI2.GE.K'12MAX)) THEN

K,2 = K 120

IF((K13.LE.Kl3MIN).OR. ,K13.GE.Ki3MAX)) THENK13= K130

ENDIFIF((K 14.LE.KI4MIN.OR.A KI1.GE.Ki4M'vAX',) THEN

K14= K1,40END I FCH(I)= 1DIl(l) = S.6DI(2)= IC2(1)=K IC2() K 2C2(3)= K3

75

. ~ ~ Y

C2(4)= 1D2(1) = K4D2(2) = K5D2(3) = K6D2(4) = 1C3(l) = K7D3(1)= KSD"(2)= K9D3(3)= 1C4(fl)= K 10C-42)= K I

D411)= K121)4(21 = K 113D43)= K 14D4(4) =

D5I= 1).

DERIVATIVE=

RIN\ (O.o*STEP(O)YIN~ ='.STEP(o)SIN = O).*STEP(O')ElI RIN-ROLLE 2=Y N-YA WE3 = SIN'-SPEEDFINI = TRNFR(0,1I,IC,C1,DI,E 1)FIN2 =TR-NFR(3).3,IC,C2,D2.KC I *E2)FIN= FINi - FI"N2ROLLI = TRNFR(O,2,IC,A1,B1,CI *FIN\)RUDDER= YIN-YAWROLL2-' TRNFR( 1.3.IC,A2,.,CI*RU:DDER)ROLL = ROLL I -~ ROLL'-YA.-W =TRNFR(O.-.IC.A3.B.3.C22* RUDDER)SP I = TRNFR(O,Z,IC,C3.D3).ELI)SP2 =TRNFR(2,3,IC,C4,D4.KC3*E2)

SP3 =TRNFR(O.1,IC,C5,D5,E3')

SPEEDI = TRNFR(12_.IC-.A4,B4.C3 I1*FIN)SPEED= TRNFR(O.2.IC.A5.B5.C-j R UDDER)POWER= SP3-SP2-SPISPEED3 =TR-NFR(O, I,IC,A6,B6,C--3POWER)SPEED= SPEED3-SPEED2-SPEEDIX=TIMEYNLI = NLFGEN(RL,X)YNL2= NLFGEN\(YW,X)YN L3 = NLFGEN(SD,X)EE I ((ROLL-YN L1)"2)

76

- -~ *.. . . . . . . . . . . . . .

ElI- INTGRL(IC,EEI):iEE2 =((YAW.YNL2)**2)

-%

E22- INTGRL(IC,EE2)EE3 =((SPEED-YNL3)**-2)

E33= I-NTGRL(IC,EE3)E= Eli +E22+E33

CONTROL FINTIM=50,DELT=.o1TERMIINAL

ER= EE NDSTOPEQ RT RN

IMIPLICIT REAL*8fA-FiO-Z)DI-MENSION Xi 4,T~ 4Q 4Qi)W'4STEP 1)=4.STEP2-)= .001STEP(3=.STEP(4) = 4.S TE P(5)= 2STEP(6)= 1.STEP(U) = 0.005STEP(S)= I.STEP(9) = 0. 1STEP(I0)= 15.STEP(I 1) =A.f05S TEP(1 2) = 7.STEP(l 3) = 5.STEP(14) =2.N= 14ITMAX = 200CFTOL= .0000001ALPHA =2BETA = .5I PR INT = 0M INMAX=-ICALL HOOKE(X,STEP,N,ITMAX,CFTOLALPHABETA,

*CF.QQQWIPRI-NTM INMAX)STOPEND

77

7:A7

* APPENDIX H

COMPUTER PROGRAM FOR COMIPARISON BETWEEN ROBERTSAND F.M.

T ITLE S.IMULATION FOR COMPARISON BETWEEN ROBERTS AND F.M.***** ROBERTS OU'TPUT ***ARRAY A 1l(1 ),B 1(3),A2(2).B2(4),AJ( I ),B3(5),A4(2),B4(4),...

A5( 1).B5(3)),A6(I1),B6('2)APRAY C I( 1),D 1(2).C2( 4)D2(4),CS'( 1),D3(3),C4(3),D4(4),C5(lI),D5(2'1TAB\1LiE.Al 1,)= 1Bl 1-)4.41A(-2 85,,2 -4=5.,.I..

9 .5,1.A'I> .B3(1-5i-= 12, 32-25.112.0A(-)10,.B-4 1-4) = 240,.;S,26, 1,A --fI) = 1.B35(1..3)=240,34. 1,A6(l)= 1,...B 6kI-:)= 2.4,1

TAB LE CIl(l)=E 1(1 -2) =S.6. 1,C2(1-4)=31.281D2 1-45='.S...I . 13,9.3. I.CS'() I 1,D3(13=1......

C4 -2=20"-21DJ -4,=130.25,1 2.6,1 ......C5' 1 l.D5 ( 12)-= 11.4.1

CONST Ki I 1 0. 1 1-4.K12 = 0. 18.K22= .01.K31 = 0.058,K32= 0.096,K33 -. 1,....R L =O.AYW = .. SP = 1.

CON ST KC = -1.6,KC2 = 0.57/9,KC3 =0.0-456,....

GC 10 = 0,GC2'O = 0,GC3O 0,GC40 0,GC5O 0DERIVATIVE

RIN = 0.0*STEP(0)YIN\= 3.*STEP0)S IN= 0. *STE P(0)El RIN-ROLLE 2 Y IN-YA WE3=SIN-SPEEDFIN I = TRN FR(0.I.,GClO,CI.D 1,E 1)F 1N 2 = T RN F R3.3.G C 20,C2. D 2,K CI ~E 2F I= F IN'I - FI \2ROLL I = TR.NFR(0,2.RL,A 1.B3I.K I I1*FIN)RLDDER= YIN-YAWRO L L2 = T RN FR( I ,3, RL,A2, B 2.K 12 2*RUD D ER)ROLL = ROLL I - ROL L2YAW= TRNFR(0,4.YW.A3.B3,K22* RUDDER)SPI = TFRNFR(0.2,GC20O,C3-,D3.KC2*E1,S P2 = T RN FR('',3,G C4.C4. D4. K C3 * E2 )S P3 = T RN FR(0, I ,GC 50, C 5.D5, E3)SPEED I = TRN FR(I,3.SP,A-4,B4, K31 *FIN)SPEED2 = TRNFR(0,2,SP,A5,B5,K32'RL DDER.)POWER= SP3-SP2-SPlSPEED3 = T RN FR(0, 1,SP,A6,B6, K33 'POWER)SPEED= SPEED3-SPEED2-SPEED I

78

FUNCTION MINIMIZATION OUTPUT *

ARRAY AAl( I),BB 1(3),AA2(2),BB2(4),AA3-( I),BB3(5),AA4(2).BB4(4),....AA5(l1),BB5(3),AA6(l ).BB6(2)

ARRAY CCI(I),DDI(2),CC2(4),DD2(4),CC3( 1),DD3(3)ARRAY CC4(3 ),DD4(4),CC5( I),DD5, 2)TABLE AA1( 1)= 1,BBI(1-3)= 4,.2 4.I,AA2(1-2)= -8.57,1,...

3BB2(1--)= 5'.S.17.17,9.5 -.1...

BB.4(1I-4}= 240;,5S,26,1.AA5( 1 = I.BB5( I-3,)= 240.3-;.1,AA6( 1= 1....

TA -,1 = .DD5 = 1.4,1,CC 1 52

~R~\ ~.,TE Pf0

[[2 - =Y'iN-YYAW[EE' = SSIN-SSPEED

FIN I = IRN\FRi0.i.GGC':0.CEDD I.EE IFFIN2 =TR\FR(3.3,GGC20,CC2,DD2.KKCI*EE2'FFIN = FFIN I - FFIN2?RROLL I = TRNFRo0.2,RRL.AA1.BBI.KKI 1 *FFIN)PRL D DE = YYI N- NY. VW.,R0OL L -= TR F RiI. 1.,DRL., \A2, B B2. KK 12 R R L D D ETRROLL = RROLLI - RolL2'

N'= TNR .YW\ABBK2RLDDE,SSPI RF'.GQ.C.D.E

S S PE EI = T RNF R( 1,3, S SP.AA%,-\B B4. KK3 I *F FI\)S S PEE 2 = T RN FR(0. 2.S SP...A.B B;, K K3 2*R RL DD EP PO0WE R SS P SSP-S.SP I

SSPEED SE3-S3=~sPL

CO\TROL FINTIM = 81.*RINT I..FFIN,RRLDDER.RROLLYYA,- 'SSPEEDSAVE (SI) 1) 1, ROL L, RROLL.YAW.Y YAV.SPEED,SSPEE DGRA-'PH(GI SI.DE= TEK6IS,PO='),.5) TIMvE(LE= S.,LN= SEC) YAW(LO= -2.LI= I..

SC-=.5.NI= SLYY AW(LO= -2.LI = 4.SC =.5,'NI =8)LABE LG I )CO.MPA RISON B ETWEE N ROBERT AND F.MLABEL(GI)FOR YAW OUTPUT0

7 9

AL0

GRAPH(G2:,SI,DE = TEK618,PO= 0,.5) TI ME(LE = 8.,U.\= SEC) ROLL(L0 -2.LI = 1,...SC = .5,NI = 8),RROLL(LO =-2,Ll 4,SC = .5,NI = 8)

LABEL(G2)COMPARISON BETWEEN ROBERT AND F.MLABEL(G2)FOR ROLL OUTPUTGRAPH(G3,!S1,DE= TEK6IS,PO=O,.5) TIME(LE= 8.,UN = SEC) SPEED(LO= -2,LI 1,..

SC = .5,NIl = 8),SSPEED( LO = -211l =4,SC = .5,NI* = 8)LABE L(G3)COM PARI SON' BETWEEN ROBERT AND F.MLABEVkG3)FOR SPEED OUTPUTEN DSTOP

del

S0S

LIST OF REFERENCES

1. Roberts, G. N. Integrated Control of Warship Vfaneuvering. .N.S. (Journai ofNaval Science). Vol. 12, no. 4, 1986.

2. Rasenbrock, H. H. Computer Aided Control System Design. Academic Pr-ess,

S. F icker. A. j. A Direct Method f'r Designing De-coupling Pre-compensator jorM1uiuivariable Systems. Measurement and Control, Vol. 6, No. 4, pp. 225 -232,

1984.

4. 1.3.M. DYnamnic Simulation Language. 1987.

.. -------------------------------.- "--..'.' -'. .-. . . . .

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Technical Information Center 2Cameron StationAlexandria, VA 22304-6145

2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, CA 93943-5002

3. Department Chairman, Code 62 1Dept. of Electrical and Computer EngineeringNaval Postgraduate SchoolMonterey, CA 939,43-5000

Prof / George J. Thaler, Code 62Tr 7Dept. of Electrical and Computer EngineeringNaval Postgraduate SchoolMonterey. CA 93943-5000

5. Prof. Alex Gerba, Jr., Code 62Gz IDept. of Electrical and Computer EngineeringNaval Postgraduate SchoolMonterey, CA 93943-5060

6. G. N. Roberts IControl Engineering DepartmentRoyal Naval Eagineering CollegePlymouth, United Kingdom

7. LCDR Lee, Sang Sik 8Dept. of Electrical EngineeringNaval Academy, Jinhae City, Gyungnam 602-00Republic of Korea

3. Naval Academy Library 2Jinhae City, Gyungnam 602-00Republic of Korea

9. CDR Chil K. BackSMC 1646, Naval Postgraduate School .Monterey, CA 93943-5000

10. Ismail UnluSMC 2075, Naval Postgraduate SchoolMonterey, CA 93943-5000

82

.............

I1. Prof. D. R.Towill1University of Wales Institute of Science and TechnologyCardiff, United Kingdom

12. Mr T. Weather Ford190 Gardenia AvenueICamarillo, CA 93010-1908

83

1% %

%~)