OPTIMIZATION OF GUIDANCE AND CONTROL USING … · Vicente Chavez Garcia, Jr. ... 5.8 YAW AND RUDDER vs. TIME (controller C) .. .. 34 5.9 YAW AND RUDDER vs. TINE (state-feedback controller)

RD-A149 956 OPTIMIZATION OF GUIDANCE AND CONTROL USING FUNCTION iMINIMIZATION AND NAVSTAR/GPS(U) NAVAL POSTGRADUATESCHOOL MONTEREY' CA V C GARCIA SEP 84

UNCLAISSIFIED F/G 20/2 N

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111WII~ : '

MICROCOPY RESOLUTION TEST CHART

NATtONAL BUREAU OF STANDARDS- 1963-A

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NAVAL POSTGRADUATE SCHOOLMonterey, California

In0)

DTICo ~ ELETT

FEB 1 1 1985

THESIS -

OPTMIZATION OF GUDACE AD CONTRL USINFUNCTION MINIMIZATION AND NAVSTAR/GPS

by

Vicente Chavez Garcia, Jr.

September 1984

Thesis Advisor: George J. Thaler

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Optimization of Guidance and Control using Master's ThesisFunction Minimization and NXVSTAVGPS September 1984

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Optimization, Performance Criterion, Nmto, Taylor's Series,Function Minimization Subroutine, NAVSTAR/GPS, Space ransportation System(STS), Sea-Land Mclean (ML-7)4~

0. ABSTRACT (Continue on revereo side It necessary usd identify by block mmbon"

A carefully designed controller, tuned to minimize a performance criterionbased on representation of the added drag due to steering, can minimizepropulsion losses. A computer simulation modeling the Sea-Lard Mclean (SL-7)-ontainership was coupled to a function minimization subroutine and a sea-state generator subroutine to accomplish the tuning. Storing these optimalcotroller parameters in a look up table as functions of ship state, sea

*:" state, and encounter angle, this technique can be used as an adaptive .

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UCLASSIFIED. , cumrv CLASaIFCATION oF T1IS PAGE rN.D Aa Eahee .

20. (ontinued

controller. Satellite platform. can give continuus nvrirornental operatingco itions which may be used to select pzoper contro.ler parameters toprovide contimus operation on a mininum of the cost fuction. The SL-7containership on a mininum of the cost function. The SL-7 containershipco.puter model was tested in calm waters and in a seway.

~~/iv C"L. iJ/), --- '> ,,-

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Optisizotion of quidonce aud control usingFunction HIjamization and4 NAVSTIR/GPS

by

Vicente Chavez Garcia, Jr.Lieutenant, United States Navy 8

B.S.E.D N00 eico State aniversity 1978B.S.L, University of Central Florida, 1982

Submitted~in partial fulfillment of the

requirements for the degree of

EASTER OF SCIENCE IN ELECTRICAL ENGINEERING

from the

NAVAL POSTGRADUATE SCHOOL

September 19814

Author: -d =-k

Approved by:_

IB3rrttStIi is 19 n elS

Electrical anA Computer EngineeringiN

Cean of Jneani Engineering

3

ABSTRACT

A carefully designed controller, tuned to minimize a

performance criterion based on representation of the added

drag due to steering, can minimize propulsion losses. A

computer simulation modeling the Sea-Land Mclean (SL-7)

containership was coupled to a function minimization subrou-

tine and a sea-state generator subroutine to accomplish the

tuning. Storing these optimal controller parameters in a

look up table as functions of ship state, sea state, and

encounter angle, this technique can be used as an adaptive

contrcller. Satellite platforms can give continuous environ-

mental operating corditions which may be used to select

proper controller parameters to provide continuous operation

on a minimum of the cost function. The SL-7 containershipcomputer model was tested in calm waters and in a seaway.

4J

* * . .* ..-... , * ** * * .

",, * ..y or-- ..r r --°- ,- .- %- .. o . -- . ., -+-. .-. . .-

TABLE OF CONTENTS

I. INTRODUCTION . . . . . . . . . . . . . . . . . . 11

II. COMPUTER MODELS ................. 13

III. COST FUNCTION . . . . . . . . . . . . . . . . . . 17

IV. CCNTROLLER DESIGN USING 1CSOS . . . . . . . . . . 19

V CCNTROLLER DESIGN USING FORTRAN PROGRAM ..... 26

VI. CCNTROLLER DESIGN IN SEA STATE . . . . . . . . . . 36

VII. AN ADAPTIVE CCNTROLLER .............. 61

VIII. CCNCLUSIONS iND RECOMMENDATIONS FOR FUTURE

STUDY .. . . . . . . . . . .66

APPENDIX A: PROGRAM T0 CALCULATE OPTIMAL GAINS ..... 70

APPENDIX P: EXAMPLE PROBLEM USING ICSOS ....... 86

LIST Of BEFERENCES ................... 92

INITIAL DISTRIBUTION LIST . . . . . . . . 94

Access n" foYr

FT IS

INESPECTEI) Av

-D..st

5 , , , .... . o +o .

A. . . .. -0fl -

LIST OF TABLES

1. THIRD-ORDER NONOTO MODEL FOR THE SL-7 . . . . . . 15

2. HYDRODYNAMIC COEFFICIENTS FOR THE SL-7 ....... 16

3. VEIGHTING FACTCR . . . . . . . . . . . . . . . . . 18

14. REID'S RESULTS ............. .- .20

5. ICSOSRESULTS .................. 20

6. SIMULATION RESULTS STEADY STATE 600 SECS . . . 22

7. SIMULATION RESULTS STEADY STATE 600 SECS . . . 22

8. SIMULATION RESULTS - STEAD! STATE 600 SECS . . . 22

9. SIMULATION RESULTS - STEADY STATE 600 SECS . . . 23


11. SIMULATION RESULTS - STEADY STATE 600 SECS . . 24

12. SIMULATION RESULTS - STEADY STATE 600 SECS . 25

13. SIMULATION RESULTS - STEADY STATE 600 SECS . . ° 2714. SIMULATION RESULTS - STEADY STATE 600 SECS . . . 27



17. SIMULATION RESULTS - STEADY STATE 600 SECS 9 "3018. SIMULATION RESULTS - STEADY STATE 600 SECS . . . 412


20. SIMULATION RESULTS - STEAD! STATE 600 SECS . 4 . 3

21. SIMULATION RESULTS - STEADY STATE 600 SECS . . . 143a 22. SIMULATION RESULTS - STEADY STATE 600 SECS . . . 144




26. SIMULATION RESULTS - STEADY STATE 600 SECS . . . . 46

27. CCHPARISON OF THE MINIMUM COST . . . . . . . . . . 46

28. EFFECTS DUE TO TRANSIENT AND GRADUAL BUILD UP

OF SEA STATE . . . . . . . . . . . ... . . . . . . 47

6

- - - - -°.-

L. * ,

29. SINULATION RESULTS - STEADY STATE 600 SECS . 4 . 8

30. SIMULATION RESULTS- STEADY STATE 600 SECS . • . 49

31. SIlULATION RESULTS - STEADY STATE 600 SECS . . . 49

32. COHPARISON OF IRREGULAR TO REGULAR SEAS

CCNTROLLER GAINS .. ..... . . . . ... . .. .50

33. CCAPARISON OF IRREGULAR TO REGULAR SEAS

CCNTROLLER GAINS . . ............... 50

34. ICSOS OUTPUT . ........ .. ... 90

7.

I.,.

-'-°

. . . .. .-.. *-°°

. "o. °

LIST OF FIGURES

2.1 BLOCK DIAGRAM . . . . . . . . . . . . . . . . . 13

2.2 DETERMINATION OF NOMOTO MODELS .......... 14

4.1 OPTIMIZATION OF CONTROLLER . . . . . . . . . . . 19

4.2 VARIOUS STRUCTURES FOR CONTROLLERS . . . . . . . 21

4.3 OPTIMIZATION OF STATE FEEDBACK CONTROLLER . - . 25

5.1 OPTIMIZATION OF CONTROLLER USING FORTRAN

PROGRAM . ° 26

5.2 TWO STATE SYSTEM . . . . . . . . . . . . . . . . 28

5.3 THREE STATE SYSTEM . . . . . . . . . . . . . . . 29

5.4 YAW vs. TIME (controller A, B; C and

state-feedback) . . . 30

5.5 BUDDER vs. TIME (controller A, B, C and

state-feedback) . . . . . . . . . . . . . . .. 31

5.6 YAW AND RUDDER vs. TIRE (controller A) a . .. 32

5.7 YAW AND RUDDER vs. TIME (controller B) . .. . 33

5.8 YAW AND RUDDER vs. TIME (controller C) . . . . 34

5.9 YAW AND RUDDER vs. TINE (state-feedback

controller) ....... . . . . . . . 35

6.1 OPTIMIZATION OF CONTRCLLER IN SEA STATE .... 38

6.2 ADDED MASS vs. ENCOUNTER FREQUENCY . . . . . . 39

6.3 ADDED INERTIA vs. ENCOUNTER FREQUENCY ..... 40

6.4 ENERGY DENSITY SPECTRUM . . . . . . ...... 41

6.5 YAW vs. TINE 30 DEGREES ........... 51

6.6 RUDDER vs. TIME 30 DEGREES . . . . . . . . 52

6.7 YAW vs. TIME 60 DEGREES .......... 53

6.8 RUDDER vs. TIME 60 DEGREES ...... . 4. .54

6.9 YAW vs. TIME 90 DEGREES .......... .55

6.10 RUDDER vs. T1ME 90 DEGREES ...... . . . . 56

8

6. 11 YAW vs. TIME 120 DEGREES .......... 57

6.12 RUDDER vs. TIME 120 DEGREES ......... 58

6.13 YAW vs. TIBE 15 DEGREES .......... 59

6.14 RUDDER vs. TIME 150 DEGREES . . . . . . . . . 60

7.1 ADAPTIVE CONTROLLER . . . . . . . . . . . . . 63

7.2 DIGITAL BLCCK DIAGRAM .. . . . . .. . .... 648.1 NATCHED FILTER DESIGN . . . . . . . . . . . . 68B. 1 DETAIL BLOCK DIAGRAM .. .. .. .a. .. . .. 87

1.2 IAW AND RUDDER vs. TIME .. . . . . . . . 91

9

7 ~ A*~ . :..:~ b.C ~ ~..i-.K. -"

ACKIOVLEDGERENT

This thesis vas made possible by the continuous support

of my vife, Emulia. I thank J.CDR Jim Cass for installing and

debugging the sea state generator program obtained *from

DTNSRDC. I also give thanks to LT Emmanuel Horianopoulos and

LT Pericles Kyritsis Spyromilios for their assistance in

generating plots and data. Finally, I owe a great debt of

gratitude to Dr. George J. Thaler, for his support, enthu-

siasm, and outstanding example of excellence.

101

I. INTRODUCTION

An overall rise in fuel prices has led to an increasing

interest in the design of autopilots for ships. The purpose

of the automatic steering control is to minimize the propul-

sion losses, which are caused by added drag due to steering

of the ship. Minimizing a performance criterion based on

added drag due to steering can reduce fuel consumption.

Claims by many researchers indicate that a carefullydesigned ccntroller could save from one to two percent of

fuel. For large containerships this could amount to more

than $100,000.00 per year savings.

To study the optimization problem, models of both the

ship and its operating environment are required. What type

of computer model should be used to represent the ship?

Chapter two addresses the developmeunt of several models.

Since the best model was desired it was decided to use the

equations of motion to simulate the ship in our Fortran

program. The basic Nomoto models give an adequate descrip-

tion of ship steering dynamics for design. The Ncmoto

second- and third-order models were developed from the egua-

tions of motion as defined by a series expansion including

all terms (both linear and nonlinear) for which hydrodynamic

coefficients were available. An interactive program that

utilized the Nomoto mcdels to model the ship was also used.

Two independent programs were developed to aid in the design

of the ccntzoller.

What is an adequate cost function which represents the

added drag due to steering? Chapter three addresses the

-.. classical cost functicn used by many researchers.

Since a variety of control algorithms are possible one

* must ask if one algorithm provides a lower minimal cost than

-11

" "-'--.-.',% .2.,', , .' '.'. - ,J.,','.;-......'.-..--.........-......................'."".'.,....-." ..- ..... "." ,..-.," ..- "."

another. Chapters fcur, five and six address the selection

of the ccntroller which provides the ainiaum value of added

drag due to steering.

Ship dynamics change with operating conditions such as

ship steed , sea state , and encounter angle. Therefore an

adaptive controller must be used to provide minimum addeddrag due to steering. Chapter seven development of an

approach to an adaEtive controller utilizing satelliteinfor ma tic n.

Conclusions were drawn from these experiments and arepresented in Chapter eight. This thesis investigated only

course keeping with emphasis on minimizing rudder and yawing

activity to reduce fuel consumption. Presented in this

Chapter are recommendations for future study where the

objective is track fcllowing which would be important for

ships reguired to follow stringent routes. It is also impor-tant fox other systems such as satellites, missiles,aircraft, where the controller minimizes yaw error to keep

the system on track.

12

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

lie. .IRI!ZR KODELS

Th. model which best represents ship-steering dynamics

is a Taylor's series expansion of the force and moment rela-

tionships around a selected steady-state operating point.

The resulting equations are commonly known as the eguations

of notion (Ref. 1]. A computer program was developed using

known available data on the hydrodynamic coefficients for

the SL-7 containership to provide a computer simulation of

the ship. The computer program is shown in Appendix A. -

Z igure 2.1 shows the block diagram. Small yaw command

* angles are used, for example ThgC= 1.0 /57.296 represents a

yaw command change of one degree.

*igur 2.e CONTKOLLERROF

4 t cl udder MOTI3

+ OW.. . OW.e. .m. . .. .~ * %*. .*. .*. .~. . . . . . . . . .

To attain the Nonoto second- and third-order transfer

functions from the equations of notion,. the function mini-mization subroutine was used to obtain the coefficients.

Pigure 2.2 shows the scheme used to obtain the Icuototransfer functions. The computer program is shown inIppendiw A.

-EQUATIONSE

OFEMOTION

FUNCTION +-TEP MINIMIZATION S

Plgur 2.2 MZN UBROIOUTIET EDL

The~~~~~~ ~~~~ 2ooomdl eecekdaantaayi eutfrom~~~~ lierie e#utins

ftoceeding~ totescn-rerNmt qain

Adut oamtr

*(S)/6(S) = K/S*(1*ss) (2.1)

Deriving the second-crder Nomoto transfer function from the

law equation only, the result is2 (S)/6(S) = 0.040893/S*(1 8.539932*S)

and using function mirimization as in Figure 2.2

*(S)/8(S) = 0.0409221/S*(1+8.5520782*S)

and the agreement is obvious. Using function minimization

with both yaw and sway equations with linear terms only, the

results are:

P(S)165)= 0.1072741/S*(1+31.9199524*S)

If the nonlinear terms are included but the perturbaticn is

small

4,I S)/(S} 0.1072082/S*(1+31.8907013*S)

and it is clear that the nonlinear terms contribute little.

Proceeding to the third-order Nomoto equation:

',(S)/a(S) = K*({+T2*S)/S*({ITP1*S)*(I+TP2*S) (2.2)

The parameters were calculated and checked by using functionminimization as in figure 2.2. The results are given in

Table 1. It is clear that the answers obtained by function

Sminimization agree closely with the analytic solutions.

S.

TABLE 1

TBIRD-ORDIR NOBOTO MODEL FOR T81 SL-7

speed K TZ TP1 TP2knots calc coop alc coop calc coup calc coup

~ ~ ~ ~ 15 ]a 1j:346 1J.946 1j7:58j 197: ~ 3:-:'" ..061 15 67 15 .14 006 75430 J8.B632 .1477 .1477 11.28 11. 8 6.470 6.467 53.793 53.93

15

Analytical equations used to calculate seccnd-crder

Nomotc transfer function coefficients are:

K =N /I =-( -N )IN

6 r z rr

Analytical equations used to calculate third-crder

transfer function coefficients are:

K =(N6 -*Y T/y )/(N -N *(Y -I*)/Y v 6 v r V r v

TP 2 N(- ,)*Y)(Y*UUN*Yr-vT 2((H-I)*(I -I)-Nrl*Y)/(Nv*(Yr-N*U)-! *Nr)r+( r v r r vr

TP1.TP2(?!-YYv*Nr*

The nondimensionalized hydrodynamic coefficients for the

SL-7 containership are shovn in Table 2.

TABLE 2HYDRODYNABIC COEFFICIENTS FOR THE SL-7

axial force lateral force moment z-axis= -0.0001 To~ =-0.00758 Nov = -0.00213

I( = -0.0003 1 = 0.0023 Nr = -0.00105uu r r1 r = 0.0039 1, = 0.00145 N' = -0.0007vr

I vy = -0.0012 T'Vr 0.01 N r = -0.015VVVVr vvrIs = -0.0005 1 ' = -0.008 N' = -0.008vrr vrr

To -0.03 N' = 0.01VVV VVVT r = 0.003 rr = -0.006Err rrr

6 6 = -0.0005 N 6 6 6 = 0.0001

16

*In recent years, zany have studied the proble2 of

(Rsf. 2] (Rtef. 3] [Ref. 4] [Ref. 5] (Ref. 61 (Ref. 7]

B .Ref. 8] (Ref. 93 (Ref-. 10] Ref. 11] [Ref. 12] (Ref. 13]

* [Ref. 14] optimizing an automatic ship-steering controller

*for minimum fuel consumption. It is well known that addi-

tional drag is intrcduced by steering and that both the

rudder motion and the yawing motion contribute to this added

*drag. A measure of the added drag given as a cost function

is

T

j 1/TJ >.44 *2 + )6 dt (3.1)

where =yaw error

6=rudder angle

=weighting factor

while this expression is an approximation, it is conven-

ient for shipboard use because 0.and & are readily measur-

- able. There is no general agreement on numerical values for

* the weighting factor, X and in this study the values used

*were chosed from the work of R.E. Reid (Rief. 7] For the

- 5-.-i.

Ihe weighting factors for the operating range of the

ship are shown in Tate 3.

17

TADLI 3

WEIGHTING FACTOR

ship sPeed weighting factor(knots)

16 16.79623 8.12832 4.2

Reid's work shows the relationship of weighting factor

to the closed-loop natural frejuency, mass, pivot point,

shipspee, Xyr adX1166 hydrodynamic coefficients. It isshown in Equation 3.2. Reid chose a closed-loop natural

frequency of 0.05 tad/sec which experimentally showed at

this frequency, the ueighting factor in the cost function,

provided good representation of the added drag due to

steering.

X 2*a* i1+,x )(CP/L) *w.2 /(P/2) (1*X6 *UZ) (3.2)yr 6

IT. .C.9jUQLLRDESG USING~f ICSOS

The Interactive Control System optimization andSimulation (ICSOS) package finds optimum values f or uzqknown

(free) parameters in a control system design problem and/orperfoxms simulation of the system. An example of usage of

* *ICSOS is shown in Appandix B.

in preliminary studies ICSOS was used with Iomotc modelsto study controller characteristics in calm water. The func-

tion minimization subroutine adjusted the controller parame-

ters to minimize the cost function. Fig are 4I. 1 shows the

scheme used to evaluate the controller. parameters.

uiue .1 OfIIATO O7OT3L3

STEP +

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

Reid [ Ref. 7] uses the second-order Nosoto model of

* equation 2.1 for the Si-? and also uses a contrcller

S descrited by

Gc(S) Ki*(14T1*S) /(14T2*S) (14.1)

His results are given in Table 4.

TABLE 4

REIDI S RESULTS

speed plant weighting controller gainsknots K T factor KI TI T2

16 0.1084 90.36 16.796 0.4556 89.51 10.0623 0.1556 64.67 8.128 0.3769 62.60 8.30832 0.2167 45.45 4.2 0.3188 44.92 7.066

Using this plant and weighting factor values but applying

ICSOS, results were cbtained and shovn on Table 5.

TABLE 5

ICSOS RESULTS

speed plant veighting controller gains costknots K T factor Ri Ti T2 J min

16 .1084 90.36 16.796 .4;4616 90.3459 10.0215 340.86423 .1556 64.67 8.128 .313171 64.6658 8.4640 139.991632 .2167 45.45 4.2 .318645 45.4475 7.0662 60.828

In each case the controller zero (1/Ti) cancel-s the*plant pcle (1/T) . Additional experiments consisting of

inserting arbitrary numbers in the Nomoto equation and

repeating the computer run indicated that this will always

be true. That is, to minimize the cost the plant pole is

*cancelled and a nev ;ole location determined with appropri-

ately adjusted gain.

20

The siaile contrcller of Equation 4.1 is an arbitrarily

chosen structure. To determine the effects of more complex

controllers three additional structures were chosen as shown

-in Figure 4.2. Back of these was used with the Ncuoto

second-order model for the ship at each of the indicated

speeds. The results are shown in Tables 6, 7, and 8.

I..

lKII(I TIS) XI (I +TS)_(I T2S) (I+T2S)(I T3S)

(A) (U)

XIITS)I S k0 KI (I+TlS) +(l+T2S)(lsT4S) (I+T2S)

(0)

Figure 4.2 VIRIOUS STIUCTURIS FOR CONTROLLERS

These results are very interesting. at 16 knots the

coatrcller gain (11), controller zero (1/Ti) and ccntroller

pole (1/12) are essentially the sane for all structures. For

structure B, which includes an additional pole, the function

minimization subroutine tries to drive the additional poleto infinity, and no doubt would have done so if the calcula-

tions had continued. For structure C, which has two Foles

and two zeros, a zero and pole cancel indicating that they

are not needed or wanted. for structure D, the integrator

21

_____.°.

TABLE 6

SIMULATION RESULTS - STEADY STATE 600 SECS

CALM VATER FOR VARIOUS CONTROLLERSFOR FIXED SHIP SPIED ( 16 KNOTS I

NONOTO SECOND-ORDER MODEL (K=. 1084 T-90.36)X= 16.796 OPTIMAL PARAMETER GINS OFVARIOUS CITROLLERS * COST FUNCTION

contr controller gains cost. •Ki Ti T2 T3 T4 Ti J Din

A .4!4616 90.4355 10.0215 - - - 340.864B .444101 90.2950 9.8566 .01 - - 341.046C .454511 90.3685 10.0224 23.085 23.084 - 340.864D .454581 90.3719 10.0222 - - 1E09 340.864

TABLE 7


CALM WATER FOR VARIOUS CONTROLLERSFOR FIXED SHIP SPEED ( 23 KNOTS

NOBOTO SECOND-ORDER MODEL (K=. 1556 T=14.67)X= 8. 128 OPTIMAL PARA METER GLINS OFVARIOUS CtNTROLLERS , COST FUNCTION

contr ccntrcller ains costKi Ti2 T3 T4 J min

A .373171 64.66579 8.463957 - - 139.9916B .340024 79.65872 8.889204 .01 - 140.9338C .373139 64.66855 8.463497 25.9719 25.9738 139.g991

TABLE 8


CALK VA7ER FOR VARIOUS CONTROLLERSFOR FIXED SHIP SPEED ( 32 KNOTS

NOOTO SECOND-ORDER MODEL (K=.2167 T=q5.45)4 *2 C OPTIMAL PARAMETER GLINS OF

VARIOUS TROLLERS , COST FUNCTION

contr controller gains costKI Ti T2 T3 T4 J min

A .318645 45.44747 7.06617 - -60.828]B 18 45.4 5 7.066 *85 60.933

*C :318678 4 551 7.06790 50.1 29 50.04832 60.828

22

....................................... ........ ........... ... ....- °

gain is driven to zero. The saae pattern of results is

* obtained at 23 knots and 32 knots. Note that in all cases

the minimum cost is essentially the same, as would be

expected since all cctrollers are the same.

Using the computer method of Figure 4.1 and the Nomoto

third-order models of Table 1, controllers A, B, C of Figure ":.4.2 were optimized.. The results are shown in Tables 9, 10,

and 11.

TABLE 9SIMULATION RESULTS - STEADY STATE 600 SECS

CALM VA71B FOR VARIOUS CONTROLLERSFOR FIXED SHIP SPEED ( 16 KNOTS )

NONOTO THIRD-ORDER MODEL(K=.073812TZ=22.5673 ,TP1212.9458 TP2 = 107.5853)

X= 16.796 OPTIhAL PARAEETER GAINS OFVARIOUS CfNTROLLERS , COST FUNCTION

contr ccntroller qains costKi Ti T2 T3 T4 J min

A 0.64416104 90.C994 15.27712 - - 370.4023B 0.6441367 84..826 15.78691 .24598 - 374.3808C 0,6151139 107.5782 8.73520 12.9368 24.9676 369.9297

TABLE 10


CALM VATER FOR VARIOUS CONTROLLERSFOR FIXED SHIP SPEED ( 23 KNOTS )

NOICTO THIRD-ORDER MODEL(K. 1067 TZ=15 675 TPI29. 01 IP2=75.13)X 8. 18 OPTIHIL PARANETIR GAINS O1VARIOUS CENTROLLERS , COST FUNCTION

contr controller gains costK1 TI T2 . T3 T4 min

2A 0.5224258 63.13609 12.72212 - - 152.2920B 0.5216467 64.93709 12:63218 .0505174 - 152.5333C C.5001907 75.14852 6.527490 9.039928 18.260 152.2800

Of sajcr interest is the fact that the difference in ..

"cost" between A, B, C is less than one per cent. At each

23

., '.. : ., . • . . .... ..,. . . . . ... . .. ,.., -. . -, .. " . .., ,. , ... , . '.' ., .".., '. ,., ,.'

TABLE 11


CALM WATER FOR VARIOUS CONTROLLERSFOR FIXIE SHIP SPEED j 32 KNOTS )

NOBOTO THIRD-ORD R MODEL(K=.. 1771 TZ=11.2833 TP1=6 4699,TP2=53.7931)o2 1 OPTIMAL PARAETER GAINS OF

VARIOUS UCNTROLLERS , COST FUNCTION

contr controller gaims costK1 Ti T2 T3 T4 J min

A 0.427633 48.66C48 10.744e5 - - 68.09039B 0. 298732 89.40696 15.01033 .0597786 69.32355C 0.417991 53.69S61 4.970016 6.294354 13.85724 68.04735

speed (16,23,32 knots) controller C is "BEST", tut the

difference is slight. Examining the parameter values

obtained for controller C, it is seen that at all three

speeds both poles of the ship are essent'ally cancelled by

zeros of the controller.

These results seen to indicate that the dynamics cf the

plant determines the optimum structure for the controller.

Using a state-feedback controller and Nomoto third-order

models of Table 1, the controller was optimized for various

ship speeds. Figure 4.3 shows the scheme used to evaluate

the state-feedback ccntroller.

Using the scheme of Figure 2.2, with no change in cost

function or weighting, the optimal gains and costs were

determined as shown in Table 12.

*hen comparing the state-feedback controller with

contrcller C, it is seen that at each speed controller C hasa lower cost. Among the controllers consired, controller C

is "BEST" when using the Nomoto third-order model, although

the differences in ccst are not dramatic.

24

.. * -• *. ..... .*

Paramet AjrtPraee

Figure~.3 OPINIIZATION O TT EDAKCNRLE

SIDUIO SROUT TE D!SAE 60SC

Piquie4.3 OTIATIRDA TAECFDAC CONTROLLER

Npeed Desoto third-crder plant weighting controller costkots 9 TZ TPl TP2 factor RI K2 J zin

16 .0738 22.567 12.946 107.583 16.796 4.426 78.004 370.7112J :0715.675 9.014 75.13 8.128 3.103 45.649 152.596

31 .1477 11.283 6.470 53.793 4.2 2.240 27.896 68. 2513

25

V. CONTROLL13 DEIG USING FORTRAPj U2OJA! -

The Fortran program referenced in Chapter two which

*provided a computer simulation of the SL-7 ship was zodi--

* * ied. A function minimization subroutine was coulped to the

simulation and used the subroutine to adjust ccntrcller

parameters to minimize the cost function and to evaluate the

minimum cost. Figure 5. 1 shows the scheme used to evaluate

the cozticile: parameters. This program was used for compar-

ison to ICSOS. The computer program is shown in Appendix A.

TE IT FINIMIaION ATON

SUBROUTINAdj2st

Juf CX*8dt

ligure 5.1 OPTINIZATION 0F CONTROLLER USING FORTRAN PIOGRAN

26

Using the computer method of Figure 5.1 and the nonli-

near equations of notion, controllers A, B, C of Figure 4.2

were optimized. The results are shown in Tables 13, 14, and15.

TABLE 13


CALE WATIR FOR VARIOUS CONTROLLERSFOR FIXED SHIP SPEEDM( 16 KNOTS )

EQUATIONS OF NOTIONX= 16 796 OPTIMAL PARAMETER GAINS OFVARIOUS CdNTROLLERS , COST FUNCTION

contr controller gains cost -K1 T1 T2 T3 T4 J n

A .6 11401 89.81704 15.381699 -- 1.128189B .620050 90.67294 15.542297 0.9201336 - 1.173323C .617326 107.1494 8.597198 13.353928 25.21362 1.126307

TABLE 14


CALE VATIR FOR VARIOUS CONTROLLERSFOR PIXIE SHIP SPEEDES 2.3 KNOTS)

EQUATIONS OF MOTIONA= 8 128 OPTIMAL PARAMETER GAINS OF .VARIOUS CINTROLLERS 8 COST FUNCTION

contr controller gains costKI T1 T2 T3 T4 J min

A .522106 66.33122 12.83327 - 0.4640879B .4S5869 66.151!2 13.01183 0.92783 - 0.4857854C .503967 74.79771 6.65880 9.20533 18.4022064 0.4636095

These results agzee with those obtained by ICSOS andcontroller C provides the minimum cost.

If the assumjtion that the steering dynamics of the ship

is adequately modeled as a second-order system is valid,

then only two states are needed for feedback. For a third-

order system three states are required. The ccntroller

structures are shown cn Figure 5.2 and 5.3. Using the scheme

27. . . . * * * . .:..

p . -",'j' ! U* .&L . Zrit

. . .' - - r .-

TAB-- 15

SIEULATIOI RESULTS - STEADY STATE 600 SECS

CILS VATIR FOR VARIOUS CONTROLLERSFOR FIXED SHIP SPIED ( 32 KNOTS )

EQUATIONS OF NOTIONA a 4.2 OPTIHAL PARABETER GAINS OVARIOUS CCNTROLLERS v COST PONCTION

contr controller gains costKi TI T2 T3 T4 J mn

A .428404 48.65540 10.814426 - 0.2072417B .291732 89.40696 15.010330 0.01 0.218334C .417333 53.09654 5.096548 6.474857 14.0205 0.2071124

of Figure 5.1 with no change in cost function or weighting,

the optimal gains and cost were determined as shown in

Tables 16 and 17. Comparing costs, there is little differ-

ence between the tuc state system and the three state

system. The conlarison between state feedback with

controller C, it is seen that at each speed controller C is

tetter, tut not much ketter.

STEP - 'e+ 8 EQUATIONS

FN OMOTION

A d2+ 82 1dt Adjust Parameter

,-."Figure 5.2 TWO STATE SISTE-D

28

. ° * * ..... * o* *S -'o

STgEP .3 TESAEQSUSTINS

;-LT I RE...TS...ST.... SCA-.A ! MOTIINSSHP TPED

'"SUBOUINAdjust- K3:,'

':'" Parameter ::

FUNCTION DEE -T.MINIMIZATI-ON K2SUB OUAdjus Prmeter

2GAINS 2O2

I0 J 0 d Adjust Parameter

Fi 1gure 5.3 TR STM ST

".. -'.'.~.26'

"' TAB.LE 16 '"

SIBU:IITIONi R:ISULTS - ST:EAD: STATE 600 SECS "-

• ~CALB ATlIR FOR VARIOUS SHIP SPENDS"'',"."~~~ BUATIONS OF NfOTION --

!-" OPTIBif - PARABZTER GAINS FOB"'"~Tin STATE SYSTEN

.speed gains weighting cost.-Rnots K1 x2 f actor J min"-:

:i: 13 4l.4033689 775016 16 796 1 128771.-

3.0889006 45 26377M 8.128 .4646050H 2.2342062 2 .6808014 4.2 .2075207

i/

lote that for the Nomoto model studies yaw error and

rudder angles were measured in degrees; when the equations

of notion were simulated yaw error and rudder were in

radians. Thus the numerical values of the cost# J, are

different. /

Transient response plots for controllers A, B, C, and

three state-feedback at ship speed 32 knots are shown'in

ligures 5.4 - 5.9.

29

°,i. . .-. . . o.

TABZ 17


CALD EATER FOR VARIOUS SHIP SPEEDSEQUATIONS OF NOTION

OPIUMrNJ PARA ETER GAINS FORTHRE1 STATE SYSTE'

seed gains wei hting costors 1 2 13 fa tor J sin

4 8617249 87.7073364 99. 802704 16.796 1.1282891740983 56. 82 88. 913391 128 .4643548S6150 3 1144 186035 4.2 .2074225

/ .32 KNOT5

TRIA VS TIE-rcEC8 KK MDO AB,,C COMP

FEC- Aq

3io .0 4.0 mi.o 40 16.0 o . 0~ li .o 4. 0~ d,.-

Figure S.4 TAN vs. 1IBZ (controller A. B* C and state-feedback)

30

M u s 1r o .. s .. g zao? -s. s.. is

: .-,-. .?. ... ?.; . . ... .. , ,..? , , ... ,? . , .. ., .].. ..,.. l. _.,._.,.,,,.. .. _. ,_- ..-... . . .. -,,.,. ... .. ,;

.3%

32 KNOTS

RLIOE VS TINC-FTECBACK AM IB COWt

.3C

'31

32 KNOTS

viNok~kx VS TI1CCOIC A

6.050 0.071. 40.6 15.0 110.0 371.0 ado.* ai1.0 210.0 A7.6__ -__TIME

Pigure S5.6 TIN AND RUDDER vs. TIE (controller A)

32

32 KNOTS

Ytei-RACW VS TIM-COlEN5ATOR 8

RUCI

S.J S.0 10.0 A1.5 UI.0 16.0.im0lma10.0 204.0 211.8 210.0 10TIMC

Figure 5.7 TAV IND BUDDED vs. TIRE (controller B)

33

32 KNOT

I IYF1W-UOE VS TIME-COMPrN5ATOR C

II

RUOC

0.0 2.0 S6.0 75.0 1i.0 liS.0 350.0 275.0 20.0 m2.0 2SO.O'dS.o 30

Pigure 5.8 TAN IND BUDDED vs. TIME (controller C)

34'

m a

Tim

*Figure 5.9 TAN AND BUDDER vs. TINE. (state-feedback ccatroller)

35

VI..CON7IOLLER DESIGN IN SEA STATE

A sea state generator was coupled to the Fortran

program, so that the function minimization subroutine could

be used in the presence of the sea state. The sea state

generator was an elaborate program obtained from DTNSBDC.

This program generates added mass and inertia values as

functions of encounter frequency and also calculates the

forces and moments. The forces and moments are generated

and stored in a look up table which was coupled to the equa-

tions of motion. Figure 6. 1 shows the scheme used to eval-

uate the controller parameters. The computer prograz is

shown in Appendix A.

The cptimal gains obtained by the calm water study of

Chapter five were use as the initial guess in evaluating the

optimal ccntroller parameters in the presence of a seaway.

For ccmparison, studies of the value of the cost function

using calm water gains in sea state were obtained; then the

function minimizaticn subroutine was allowed to adjust

controller parameters in the presence of several sea states

and encounter angles. The entire study was done at a ship

speed of 32 knots. The added mass and inertia change withrespect to encounter frequency as shown in Figures 6.2 and

6.3. Figure 6.3 is ncndimensionalized by dividing the added

inertia by the mass of the displaced water and the square of

the length between pexpendiculars. To convert back to dimen-

sionalized units of lb-ft-sec 2, multiply the graph points by

2.581E12. Since the sea state is represented by irregularvaves, the waves impinging on the ship hull contain the

total energy density spectrum composed of many frequencies

and the ship responds to an average value of added zass andinertia. The values used for this study was obtained at

36

36 ' '

p . :". .",'-''.i,"' :" ." .", ." .- - .' ,. ,;.'- -" .". ., " .,., - J•."- "-"- ' ' '''"- . . .

encounter frequency cf 0.75 rad/sec from our sea stategenerator. This frequency gave us values for added mass and

2 inertia representative of an average value. The energy

density spectra for various sea states are shown in Figure

6.4. The added mass for sway was changed from 2.6457E06

lb-secz/ft for calm water to 2.3043E06 lb-sec2/ft for a

seaway. The added moment of inertia for yaw was ch'anged

from 1.42E11 lb-ft-Sec2 for calm water to 1.5096E11

lb-ft-sec2 for a seaway. All other hydrodynamic cceffi-

cients were kept constant at calm water values. The results

are shown in Tables 18 - 25. In certain sea states and

encounter angles the calm water optimal gains performed well

as shcwn by calm water cost value when compared to sea state

cost value. In most cases the function minimization subrou-

tine found new gains with lower cost function values in

seaway as compared to using calm water gains. In the calm

water evaluation, the system was perturbed with a one degreecourse change, but the course change was not included in the

seaway tests. The difference in cost values is attrituted to

the difference in operating conditions.Using the Proportional, Integral and Derivative (PID)

contrcller Equation 6.1 with no change in cost function , -

the function minimization subroutine was used to adjust

contrcller parameters to minimize the cost function andevaluate the minimum cost. The results are shown in Table

26. When comparing the PID with controller A, it is seen

that at each encounter angle, ccntroller A is better. Theseresults agree that in a seaway controller A provides the

minimum cost.

Table 27 shows ccmparison of the minimum cost functionfor controller A, controller C, and PID. The study was done

at ship speed of 32 knots and at sea state 4. Controller A

provides the minimum cost.

37

o . . . . . . . . . . .• . • ." -~~~~~.. .. . . ... '%- ' ° , , % , % % %. .. ., .-. -.- ,- -** .% -%****• °*.*'.'%.' s.'.. . '..., . .,•' . . . .%° , "°" " % %-

°....... . .

SEPr''- , EQUATIONS

x CONTROLLER EUONS

| ~Adjust

Parameters

FUNCTIONMINIMIZATIONSUBROUTINETJ'f (X*2+ 82 )dr _ _

Pigure 6.1 OPTIMIZATION OP CONTROLLER IN SEA STATE

The optimal gains obtained in the presence of sea state

was done over using a simulating time of 600 seconds. The

sea state program is designed to provide gradual increase in

the forces and moments during an initial time interval..

This is done to minimize initial condition transients in the

ship dynamics. There will-unavoidably be some transient

effects, however, and these could affect the value of the

cost, J, determined during the 600 seconds of simulation. To

determine whether such initial transients had. any signifi-

cant contribution to the value of i, additional simulation

runs were made with the controller parameters fixed at their

optimal values. However, evaluation of the cost, J, was -

started only after 300 seconds of simulation had elapsed.

38

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

ADDED MASS COEFFICIENTSSWAY WRT SWAY ACCELERATION ____

. .............. ..... LEG EN D ......... ...........

16 KNOTS

C4 . ................. ................. .......

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

0.0 0.5 1.01 .5iENCOUNTER FREQUENCY (RAD/SEC)

Figure 6.2 ADDED BASS vs. ENCOUNTER FREQUENCY

39

z. .. . . . . . . . . . . . . . . . . . . . . . .. .. .. . . . . . . . . . . . . . . . . . . . . . .

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

z.... ............. .... .... .................. L G N

r., . .. . .. .. ... ......... .. . . ......... .... .......2 3.KN O T S

rT-r

0*

......... ....... ................... ......... 2 N T

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

SEASTATE 4,5,&6

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

• -" .., _.._ _. _. .. .. .. ... ....._ .. .. ... ........ _ ... .......... ..... .... ..... .... _.. .. ._, : ..- -.... ..

. . . . . .... . . ... ................. I

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

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

.. ..... .. . ......... ............. ....... ... ... .... .... ..~ } . .- i .i .......... ....i...... ....... ... .. ......V ......- LEGEND, .... ......

k ! i ii~i i'l i ! BEAUFORT 4 I, , , .... .... ! . I. .. .... ... . .... ... .. .. ..... B E A U F O R T 5 i i°~ ~~~~~ ..." . . .i .. --....- - IT ..

:- ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~.. .... ..... i.. ..i-.i....i..i.. .. -...! ....... I.. ....

4.. . . . . . . . -

, .......... r .. ....... .F , ~ ... ..r ...... ........................... .. ....... ......... ........ i "i ..... ......... ! ..... i......... ... .... ......K ! .. .. ......... ....... ......... ... .... .

"~~ ~ ...........iiii ...... ..... ....... r. i....•~~~~~ ~. ..........i,.... ..... ......... .. . .............

0.0 0.4 0.8 ;.a 16CIRCULAR FREQUENCY (RAD/SEC)

Figure 6.4 ElilG DENSITY SPECTRUM

41

TABLE 18


FIXED SHIP SPEED (32 KNOTS) IN A SEAWAY

SHIP MODEL: E UATIOES OF NOTION- TANvC=O. 0

SEA STATE 1

CONTROLLER A

encounter controller gains sea state cost withangle cost calm waterdegree KI 1 T2 J min J

0 .14284037 18.6554395 10.814426 .61745E-34 .61745E-3430 1.1561117 29.3693695 1.4592390 .2870198 .512840260 1.4033298 10.6530075 1.1086683 .1342071 .215472690 .2969198 58.2413940 1.8758221 .1300669 .1565958120 .1761794 299.999512 30.7967834 .05741726 .0727727150 2.e430557 5.2826872 .8887696 .0219070 .0939400180 1.6211386 14.0782928 2.0712433 .0051925 .0095694

TABLE 19

" . SIMULATION RESULTS - STEADY STATE 600 SECS

FIXED SHIP SPEED (32 KNOTS) IN A SEAWAYSHIP MODEL: EQUATIONS F NOTION

SEA STATE 2

CONTROLLER A

encounter controller gains sea state cost withangle cost calm waterdegrEe K1 1 T2 J min J

0 .42840370 48.6554395 10.814426 .61745E-34 .61745E-3430 .27997030 249.9305 19 857742 .84774852 .08862256 .575100 24.38 62 2.5079853 .04104504 .0535 87§90 1.3577642 9.49564080 1.1068363 .02650556 .0483197

• • 120 1.1208973 25.4498596 4.0224676 .04928402 .0717524150 2.S777727 16.2154541 .56274800 7.5751530 28.1294403180 .61420630 .482041200 6.2521963 .000124338 .0002445

42

~. ........................................

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

TABLE 20


PIKED SHIP SPEED (32 KNOTS) IN A SEAWAYSHIP MODEL: EQUATIONS OF NOTION

T NC=0.

SEA STATE 4

CONTROLLER A

encounter controller gains sea state cost withangle cost calm waterdejree Ki Ti T2 J min J

0 .284037 48 o'c40 10.814426 .620598E-34 .62C598E-3430 .9815440 5o7.-336 .6999879 .02854677 .039589260 .6201209 40.C556 19.606873 .09375697 .103269690 1.809746 36.01225 6.324708 1.5171340 4.1623011120 5.195190 18.92513 .6999907 9.991730 48.970703150 1.446776 16.89375 .5265408 16.67052 24.822098180 .1000000 1.000000 20.149999 .30739631 .0076657

TABLE 21


PIKED SHIP SPEED (32 KNOTS) I A SEAUA!SHIP NODEL: EQUATIONS OF MOTION

SEA STATE 6

CONTROLLER A

encounter controller gains sea state cost withangle cost calm waterdegree K1 1 T2 J min J

0 4j;4037 48.6cc3955 10 8144264 .50899E-32 .50899e-3230 2.715786 10.41721832 .5342450 1.4287940 4.7472401060 1.7228041 8.4014740 .5141125 1.5827220 3.4274492090 1.e584366 37.1672655 .5792384 4.5505371 13.2757149

120 3.3122489 106.722259 .9260592 22.108002 94.5497589150 .2854474 157.483887 119.981018 .81100580 1.50448510180 .8053379 .75733550 6.04484460 .07365978 .142564400

43

TABLE 22

SINULATION RESULTS - STEADY STATE 600 SACS

FIXED SHIP SPEED (32 KNOTS) IN A SEABA!SHIP ECDEL: EQUATIONS OF NOTION

dA C=0.0

SEA STATE I

CONTSOLLER C

encounter controller gains sea calmangle cost costdegree 11 Ti T2 T3 T4 J min J

0 1.61345 16.8755 "5.0481 47.3405 1.73759 .14E-33 4cE-3330 .957558 12.6178 .32113 43.3531 8.15752 .324739 :35773360 .781984 17.6475 9.22485 13.9438 16.7663 .159710 .20313790 .417332 53.0965 5.09655 6.47857 14.0205 .148588 .148588120 .417332 53.0965 5.09655 6.47857 14.0205 .077918 .077918150 2.13735 18.8265 17.5778 25.1516 21.1481 .031496 .081612180 .957558 12.6178 7.32113 43.3531 8.15752 .006566 .008172

TABLE 23

SINULATION RESULTS - STEAD! STATE 600 SECS

FIXED SHIP SPEED (32 KNOTS) IN A SEAVA!SHIP NOBEL: E ATIONS OF NOTION

!aNC=0.0

SEA STATE 2

CONTROLLER C

encounter controller gains sed calmangle cost costdegree K1 T1 T2 T3 T4 J Din J

0 .4173J3 53.0965 5.09655 6.47486 14.0205 .18E-33 .18E-3338 .849594 19.9913 !:34138 20. 3578 13.7487 .05438 .061184

.417333 53.0965 .09655 6.4 486 14.0205 .044 56 .04453690 .781984 17.6475 9.22485 13.9438 16.9438 .033518 .048467120 .880395 21.4597 10.9255 9.24547 11.0667 .055292 .056288158 .899999 15.7103 1.11632 41.3275 2. 9012 10763' 23 8522188 .440916 .093671 17.7305 25.2103 5.04178 .800125 .06163

44

. .- -

TABLE 24


PIXED SHIP SPEED (32 KNOTS) IN A SEAVAYSHIP MODEL: EQUATIONS OF MOTION

YA[C=0""

SEA STATE 4

CONTROLLER C

encounter controller gains sea calmangle cost costdegree El Ti T2 T3 T4 J amin J

0 1.22424 70.3578 61.3016 10.5467 61.8215 .62E-34 .14E-3330 .6S0573 20.1488 20.3214 5.10369 19.7841 .034033 .071S7860 .782547 12.6178 13.7713 21.5637 21.5637 .098914 .244369 -

90 2.22895 51.7744 54.3190 17.3522 6.07814 1.57368 2.98305120 3.72749 85.4697 40.6999 8.52234 1.35207 10.3530 37.8988150 .417333 53.0965 5.09655 6.47486 14.0205 20.3956 20.3956180 .059166 .286208 19.5103 46.4767 22.3286 .007397 .099413

TABLE 25


FIXED SHIP SPEED (32 KNOTS) IN A SEAWAYSHIP MODEL: EQUATIONS OF NOTION

SEA STATE 6

CONTROLLER C

encounter ccntzoller gains sea calmangle cost costdegree X1 Ti T2 T3 T4 J min J

0 2.33178 52.0881 95.7892 31.6564 11.6959 .49E-32 .19E-3130 2.08709 73.1270 76.7193 12.1726 16.4711 2.00375 3.e337960 2.00128 71.6612 77.6750 13.1170 17.0912 2.15428 3.4179490 .957558 12.6178 13.7713 7240670 15.4611 5.763SS 8.80971

120 3.10589 81.8044 38.2439 91.2237 9.14683 24.9099 72.6716158 11250 70.3578 61.3016 35.5894 61.8215 2.50971 7.5002218 1:2875 1.87828 413.4698 49.9147 11.2599 .078894 .930885

45*. . . . . .. -- -.

TABLE 26

SIMULATION EISULTS - STEADY STATE 600 SECS

FIXED SHIP SPEED (32 KNOTS) IN A SEAWAYSHIP MCDEL: E CATIONS OF NOTION

SEA STATE 4

PID CONTROLLER

encounter ccntroller gains sea stateangle Ki Ti T2 J min

30 .95263100 4.20720860 .69368610 .0298561960 .68631890 12.5794449 8.2121658 .0973051290 2.5809155 12.14247589 .77810380 1.5915950120 4.9198265 12.5986176 .67592390 10.708980150 1.3970e23 15.7682953 .51991180 17.427200

8(S)/ ,f(S) = K1 lI*TI*S / (1+T2*S)**2 (6.1)

TABLE 27

COMPARISON OF TEE 8MINUM COST

SHIP SPEED (32 KNOTS)SEA STATE 4

enccunter controller controller controllerangle A C PIDdegree J Din J Din J sin

30 .02854677 .034033 .0298561960 .093756S7 .098914 .0973051290 1.5171340 1.57368 1.5915950120 9.9917300 10.3530 10.708980

150 16.670520 20.3956 17.427200

The value obtained was then doubled and compared with the

result of evaluating J over the full 600 seconds. Comfarison

of Table 28 with cost values in Tables 18, 19, 20, 21 shows

only small differences.

To ottain insight into the stochastic process of irreg-

ular seas, a deterministic process was studied. The Fortran

46

% •

".; --" " "' - ." ." "'' "," "-'"- -' - ."'"-.1."" ".'"- -"""""''""""".-°.'"--.-.' "-''-'-" " "-"" "- -"--

TABLE 28

EFFECTS DUE TO TRANSIENT AND GRADUAL BUILD UP OF SEA STATE

INTEGRATION OF COST FUNCTION ( 300 TO 600 SECS)FIXED SHIP SPEED (32 KNOTS) IN A SEAWAY

sea Encounter ccntroller gains cost coststate angle Ki Ti T2 J min 2*J min

1 60 1 4033298 10.650075 1. 1086683 .0641122 .12,822442 60 .955751C0 24.3813f3 2.3079853 .0199731 .03994624 60 .62012090 40.805560 19.606873 .0515974 .10319486 60 1.7228041 8.4014740 .51411250 .7906179 1.581236

program was modified to minimize the cost function in the

presence of a regular sea. To allow comparison with

previous work the encounter frequency of 0.75 rad/sec was

used and scaled the amplitude of the regular sea to its

Frospective sea state. The entire study was done at a ship

speed of 32 knots. The results are shown in Tables 29, 30,and 21.

Table 29 shows that for regular seas the ccntrcller

parameters do not change significantly. for different sea

states; but as sea state increases, the cost value increases

due to the increase in yaw moment and sway force on the

ship. Tables 30 and -1 also show that the controller parame-

ters do not change significantly from sea state to seastate. However, an encounter angle of 90 degrees shows a

relatively high cost compared to costs calculated for 60 and

120 degrees at a given sea state. To account for this

anomaly, the following is suggested. In the regular sea,

the added mass and inertia were known for a given encounter

frequency, while in the irregular sea a gepresentive average

value was used. The 5ethod used to obtain the average might

not represent the actual average. Also, it seems reasonable

to suppose, that the assumptions of the function weighting

factor are satisfied for all encounter angles; that is, the

weighting function (Eq. 3.2), which appears in the cost

47S. . . . . .. . . . . .- . '

function (Eq. 3.1), does totally represent the added drag

for all encounter angles. Future study is needed tc ansver

these questions.

The sea state in the deterministic model is represented

by regular waves. On this description, the waves impinging

on the ship hull correspond to only one frequency in the

energy density spectrum. In the case of irregular seas,

however, the spectral components change for different

states, as shown in Figure 6.4. Thus comparison of the

contrcller parameters obtained for regular seas with results

for irregular seas is not justified. The function minimiza-

tion subroutine adjuEted controller parameters to minimize

the cost function for either case (irregular or regular

seas) as shown in tables 32 and 33.

TABLE 29

SIMULATION RESULTS - STEADY STATE 600 SXCS

FIXED SHIP SPEED (32 KNOTS) IN A REGULAR SEAWAYSHIP aCtEL: EQUATIONS OF NOTION

YAC=O.0ENCOUNTEB FREQUENCY = 0.75 RAD/SEC

ENCOUNTER ANGLE = 60 DEGREES

CONTROLLER Asea ccntroller gains cost

state KI TI T2 J min

1 .1449795 141.383179 32.9405670 .0007645822 .1534657 129.987473 31.4042358 .0030564344 .1514665 135.798737 32.9749756 .0093454796 .1533340 135.488495 33.5585632 .022174600

" .

Note that in bcth the deterministic and stochastic

models, among the ccntrollers considered, controller A is"BEST" in a seaway disturbance, although the differences in

cost are not dramatic.

Finally, the observed dependence of optimal ccntroller

gains on sea state and encounter angle suggests that an

48

•-..;...'..:...... . ........................ .... .. ........... . -. . . .,... . . .*..-. -. ,. *., -., ,-..,.,,

TABLE 30


FIXED SHIP SPEED (32 KNOTS) IN A REGULAR SEAVA-SHIP BCDEL: E UATIONS OF NOTIONYfgC=O.0

ENCOUNTES FREQUENCY = 0.75 BAD/SECSEA STATE 4

CONTROLLER A

encounter ccntroller gains costangle Ki Ti T2 J amn

30 .2351043 102.021973 28.3396912 .002S8530060 .1514665 135.798737 32.9749756 .00934547990 .4964442 66.546493 49.7598267 .048143C90120 .1327230 149.540543 33.6013489 .038937880150 .4536914 70.566528 31.5839539 .062534153

TABLE 31


FIXED SHIP SPEED (32 KNOTS) IN A REGULAR SEIAYSHIP MCDEL: EQUATIONS OF MOTION

YAIC=O. 0ENCOUNTEE FREQUENCY = 0.75 HAD/SEC

SEA STATE 6

CONTROLLER A

encounter ccntroller gains cost--angle Ki T1 T2 J min

30 .2370022 100.122940 28.0581207 .00709287860 .1533340 135.488495 33.5585632 .02217460090 .5210407 62.153702 49.9858093 .112772680120 .1414837 142.695160 35.3171234 .091541650150 .4587426 71.451385 33.4568024 .14461582S

adaptive controller zust be used to provide a continuous

minimum on the cost function.

After obtaining tle optimal gains for controller A, to

observe the behavior of the rudder and yaw motion of the

ship, transient respcnse plots were obtained for controller

A at ship speed of 32 knots and sea state 4 for various

encounter angles as shown in Figures 6.5 - 6.14. Note the

49

.. ::Q -:- :-: :Kk:&4K.§>I.

TABLE 32

COAPABISCN OF IRREGULAR TO REGULAR SEAS CONTROLLER GAINS

SEA STATE 4

CONTROLLER A

encounter controller gainsangle Ki Ti T2

30 (irregular) .9815440 5.733036 .699987930 (regular) .2351043 102.021973 28.3396912

60 (irregular) .6201209 40.805560 19.606873060" (regular) .1514665 135.798737 32.9749756

90 (irregular) 1.809746 36.012250 6.324708090 (regular) .4964442 66.546493 49.7598267

120 (irregular) 5.195190 18.925130 .6999907120 (regular) .1327230 149.540543 33.6013489

150 (irregular) 1.446776 16.893750 .52654C8C0150 (regular) .4536914 70.566528 31.5839539

TABLE 33

COMPARISON OF IRREGULAR TO REGULAR SEAS CONTROLLER GAINS

SEA STATE 6

CONTROLLER A

encounter controller gainsangle K1 Ti T2

30 (irregular) 2.9715786 10.4721832 .534245030 (regular) .23700220 100.122940 28.05812C7




(irregular) .2854474 157.483887 119.9810181 0 (regular) .4587426 71.451385 33.4568024

increase in both rudder and yaw amplitude as the encounter

angle increased. This is due to the increase in yaw moment

and sway force on the ship.

50

-,

--.-. !-.-. .-.-. o-o-., " -.... , . ... -. -.. . _ ... _ .T - - . . .... . . . . . .. . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . .. . .. . . .. . -- ..... .

32 KNOTS-SER STATE 4

ENCOUNTER ANGLE 30 DEGREES

6,

i

S

i,°

. ., .. . . .

2 -.

0.0 50.0 100.0 ,s0.0 200.0 "2,,.U 300.0 *Sb.0 ICo.n 150.0 500.C 5s'.0 60 i:"- 'TI ME

~igure 6.5 IIIg vs., flHE 30 DEGREES•

I-I

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

- - -1. -.-.- '.- -,A ..? .. -

352 KNO.TS-SER STATE 4


Ia.

80.0 56.0 100.0 350.0 2.0 250.0 36 0 l . 6. .0 050.0 6

Pigure 6.6 RUDDER vs. TIRE 30 DEGREES

52

32 KNUTS-SEA STATE 4

ENCOUNTER ANGLE 60 OLGRELS

6 i

La

r !c'

'l ~ IB

'd. ' . 2C o 0 r 0 ' 3 o. '5. C. 45. 5 M '. '5.0

.... ..... ... ... ... T IV j ._

aJ

i Figure 6.7 TAV vs. TIRE 60 DEGREES

* - -. 53

uric,o -I

•.Figure..7 I . ......E 60 .D..GRE.ES..

32 KNOTS-SEAi STATE 4

ENCOUNTER HNGLE 60 DEGREES

31n

raa

TIM

Fig/4 7!8 fUDRv.TME 6 ERE

54\~

U... 7

32 KNOTS-SE STTE4-

ENCOUNTER RNGLE 90 UE.REE5

C, .

0 • .

I-

TIM

Figure~~~~.' 6.9 YW1s".N 0DERE

o -N

S.

.. ur .. .AI .s .ZH .0 .)3 R E ..

p.. ,a=

• :ii2.a

Fiur .9T° s.TN.90DGRE

.i." -'55

.32 KTS-SEA STATE 4

ENCOUNTER ANGLE 90 ULGREES

TIE-

" r

q

" V

Figure 6.10 RUDDZR vs. TIBE 90 DEGREES

56

32 K.NUT-SER STATE 4


9.

19~

9t

G.0 W0a60 11. ':. i. 3 . 5. 6. 03 " 0 S00 6

,1TIM

Fiur 6. 11TNv.TM 2 ERE

d5

32 iNOTS-SEA STATE 4


9

9

t .t

TIME

' ,

I'I

..- 0.0 50.0 100.0 150.0 200.0 250.0 3030.0 75,00 400.0 450.0O 500.0 550.0 150

_____TIME -

Pigure 6.12 RUDDER vs. TIBE 120 DEGREES

58

32 KNOTS-SEA STATE 4

ENCOUNTER ANGLE 150 DEGRLES

A 4k

C-0 56- Id. 1i. 26 0lGQ 3 . . 6 0 4.

Iv"4'

.Pi •

C.9

Jigure 6.13 711i vs. 21.81 150 DEGREES .

il

59 "'lg

:o . . ...

, % o .. *: . § * . . .* , = , . • ... . * * 1, h h .. ;. . . -•.. • . % . = . % o , . . . . - .. . . , . % % , % % .,.==e~oo %'-," ,,...................................................................................................................................- o*.o oo" o".o

32 KtNi'TS-S T~T

ENCOUNTER flNGLE 150O DEGREES

9

TIM

Figure 6. 14 BUDDER vs. TINE 150 DEGREES

60

VII. 13 ADAPTIVE CONTROLLER

In a seaway, the controller gains changed dramatically

for changes in sea state and encounter angle. An adaptive

controller must be used to provide continuous operation on a

minimum of the cost function. This Chapter addresses a

theoretical design of an adaptive controller.

In the future, there will be better measurement of navi-

gation than can be provided by conventional equipment on

board a ship. Presently the Navy is involved in a program

that will provide precision navigation data. The

NAVSTIP/GLCBAL POSITION SYSTEM (GPS) [Ref. 15] [Ref. 16]

aRef. 17] will provide extremely accurate three-dimensional

positicn and velocity information to users anywhere in the

world. The position determinations are based on the measure-

ment of the transit time of BF signals from four satellites

of a total constellation of eighteen. This system is sched-

uled to be fully operational in 1988. At present (1984)

there are four NAVSIAR/GPS satellites in operation which

allows three to four hours per day of navigation time.

Already the Texas Instrument Company markets a receiver for

this system where GPS can be used.

Tbe Navy Remote Ocean Sensing System (NROSS) [Ref. 18]

will te able to detezmine wind velocities over the world's

oceans with an accuracy sufficient to determine ocean

surface waves. It's objective will be to acquire global

ocean data for operation and research use by both the mili-

tary and civil sectors. This system is scheduled to launch

its first satellite in June 1989.

The scheme for an adaptive controller is shown in Figure

7.1. Having stored the optimal controller parameters in a

look up table as functions of ship speed, sea state, and

61

* ** - - *-*.-..

encounter angle, the ship operating condition must he known

so that the table is useful. NAVSTAR/GPS would identify

ship speed and NROSS would identify sea state and encounter

angle. The optimal Farameters can then be looked up and

inserted into the controller. This should place system oper-

ation near the minimuu J. To ensure fine tuning, a micro-

processor programmed, with the function minimization or-line

in machine language, with inputs of yaw error and ruddermoticn of the ship would accomplish the fine tuning rapidly.Since tle subroutine is written in Fortran (as used for this

study) this would be inappropriate for on-line use.

The adaptive controller can be performed with digitalcircuits rather than analog components. Garcia [Ref. 19]demonstrates the process for converting an analog ccntrcller

into a digital controller. Figure 7.2 illustrates theprocessing of the majcr components in a digital controller.

An analog component circuit can be replaced by an analc to

digital converter, a digital processor, and a digital toanalcg ccnverter. Some of the benefits which can be realized

by doing this are:

1. A high-speed processor could actually prccess anumber of multiplexed signals, performing processing func-

tions on a number of independent channels.

2. The processing function is permanent in software,unless deliberately changed, and will not drift with age.

3. The processing function can be changed without

changing components, merely by changing software.

- 4. Accuracy can be made very high and can be changed*- merely by changing scftware.

5. Processing, which previously required large compo-cents such as inductors in low-frequency controllers, can

now be performed by very small digital circuits.

62

- ~ K2 2::.~2 ::%:.:: K.-:.:.Q.Q. .§4.:..~ T.*:"

NROSSReceiver

U SEA STATEENCOUNTER ANGLE

*owA, DIGITALI

MIOCRSOROCSO

FUNCTIONMINIMIZATIONSUBROUTINE

T

Figure 7.1 ADAPTIVE CONTROLLER,

63

X(t) A/D DGTLDA YtInput CONVERTER PROCESSOR CONVERTER output

Figure 7.2 DIGITAL BLOCK DIAGRAM

-In converting an analog controller to a digital

*controller, the process can be broken down into the

following steps:

* .1. Determine the desired analog transfer functioE.

* . 2. Set the sampling frequency.3. Ipply the bilinear z-transformation.

5 4. Match one point in the s domain to the z domain.

5. Obtain the optimum constant coefficients.

- .6. Obtain the digital transfer function.

7. Ottain the simulation diagram.

Che optimal ccutroller parameters can be stored in

memory. Intel company markets a II megabit non-volatile

* -read/vrite bubble memory. It is supported by a TSLI

64

contrcller which provides a black box interface. It is easy

to use and can be used with any 8- or 16-bit microproces-

sors. Tke bubble meucry advantage is:

1. Fast access time compared with disk or tape.

2. Wca-volatile.

3. Vide temperature range of operation.

h 4. Vcrking storage.5. Pcrtable operation

6. Low power.

7. High reliability.

65

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

VIII. CONCLUSIONS IND RECOMMENDATIONS FOR FUTURE STUDY

A. CCNCLUSIONS

In designing the controller, three different ship acdels

were used. Using the second-order Nomoto model Equation 2.1

allowed comparison of results with Reid's [Ref. 7] [Ref. 10]

work. It is clear that the answers obtained by function

minimization agree closely with Reid's results as shown in

Tables 4 and 5. A better description of the ship is the

third-order Nomoto model which involves both the sway andyaw equations. This zodel includes the two dominating poles

of the ship. The best model to describe the dynamics of theship is a Taylor's series expansion. This allowes both

linear ard nonlinear terms in the equations of moticn to

affect tl.e design of the contrcller.

To determine which controller structure would prcvide

the minimum cost due to steering, various structures werestudied. It was found that the dynamics of the plant deter-

mines the optimum structure for the controller. In calm

water study, when using a second-order Nomoto model, thebest structure was ccntroller A. When the third-order Ncmoto

model Equation 2.2 was used the best structure wascontrcller C, but the difference is slight. Observe that in

each case the contrcller zeros cancel the plant poles. When

the equations of motion were used for the plant, the best

structure was controller C. When the equations cf motion

were coupled to a sea state generator and the cost function

was sinisized in the presence of a seaway, the best struc-

ture was controller A. This study concludes that ccntrcller

A should be used.A function minimization subroutine is an engineer's tool

which can be used in zany engineering problems. Previously a

66

.-. *..:-* .

matched filter was designed for the Naval Postgraduate

School research project on the Space Transportation System

(STS) for the Get Away Special Program. It was matched to

the signature of the auxillary power unit (APU) on board the

space shuttle. The goal was to turn on the solid state

recording system before lift off, to record the acoustic

power generated inside the shuttle bay. Basically the

matched filter is a finite Impulse Response (FIR) filter

with the w,'ights calculated to obtain the least squared

error of the desired output when the input is the signature

of the APU. Figure 8.1 shows the scheme used to evaluate

the FIR weights.

B. RECCEMENDATIONS FCR FUTURE STUDY

In the future most ships both military and commercial

will have GPS receivers as part of their navigation equip-

ment. Using extremely accurate three-dimensional position

and velocity infcrmation from satellite platforms will allow

ships to navigate accurately in and out of ports. The func-

tion minimization subroutine is a powerful tool for

designing the contrcller. This routine simply takes the

inputs that require minimizaticn and adjusts the parameters

to accomplish this task. The cost function for the added

drag due to steering is a function of yaw error and rulder

motion. The use of function minimization and NAVSTAB/GPS

provides the means for optimization for guidance and cont-

roll. There are several areas that need future study and

work.

1. Should the objective change to track following then

it is necessary to minimize the yaw error only. This would

be very important both militarily and commercially should a

port be mined. If the ship could follow a stringent route,

knowledge of mine locations would allow access.

67

Finite Impulse Response Filter

INPUT

X(n) DELAY X ') DELAY X(n- DELAY X(n-3) DELAY X(n-N)

2 N

yln), + di(n)

Adjust Act ual Desiredotroler$ fOrot e(niaraser s Output e Output

FUNCTION MINIMIZATION

COST FUNCTION: (en)dt

Figure 8.1 HATCEED FILTER DESIGN

2. 1 controller for orbit keeping for satellites withBigh-Energy Laser weaEons would be very important. The small

far-field spot size cf a focused laser beam can be selec-tively focused on the most vulnerable component on the

target, Facilitating precision energy deposition, andgreatly increasing the probability of a kill.

3. In adaptive controller to minimize track error on

board a cruise missile could be programmed for selective

targets.

68

• . . ."

4. Military and commercial aircraft can tinefit just as

do ships ty reducing drag to minimize fuel consumption.

69

~L

APPENDIX A

PROGRAM 70 CALCULATE OPTIMAL GAINS

The program is set up to calculate the optimal gains for

contrcller A. It is referenced in Chapter five and six. It

can easily be modified to obtain optimal gains for the rest

of the ccntrollers. After obtaining the optimal gains the

program most be modified to do a simulation. The prograz has

sufficient comments for appropriate changes. It is refer-

enced in Chapter two.

Ibhis program cac be modified to obtain the Ncmaoto

models. It is referenced in Chapter two. The following need

to be changed.

C GAIN COEFFICIENTS TO BE OPTIMIZED

=XX (1)

TP1=XX (2)

zIX (3)

IP2=XX (4)

C EERCE SIGNAL TO £RIVE RUDDER (YAW ACTUAL - YAW CCMMANL)

c FOR EQUATIONS OF EOTION. -

D=YAW - YAWCC ERROR SIGNAL TO rEIVE RUDDER (YAW COMMAND - YAW ACTUAL)

C FCB INCHOTO 3RD OBLER MODEL.

D2=YAVC-YAW2

11= (D2-I2)/TP1

13=K*(TZ*XI+12)

.4= (X3-X5)/TE2C INTEGRATION

12=X2 XI*DELI

15=X5+X4*DELI

YAW2=YAW2 X5 *ELT

C CCST FUNCTION

IDIFF=-TDIFF+ (YAW-YAW2)**2

70

* . . . .-.. . .. . ..- *% *. * ...- -. . . ...-. ,%. , " . -%, %, °

% %% .. - ,-o,.,-.-.

* ,...-_. "

PROGRAM TO CALCULATE OPTIMAL GAINS FOR CONTROLLER

//GAECIA JOB (2220,0356),IRESEARCIII CLASS=J// EXEC FRTIXCLGPt IM'S=DPREGION=102K//FOEI.SYSIN DDCC THIS PROGRAM WILL OBTAIN TEE CONTROLLER OPTIMAL GAINS.C I IS REFERENCED IN CHAPTER 5.CC IN ORDER TO PERFORM SIMULATION ONLY WHEN GAINS HAVE BEENC OBTAINED CHANGE XS(* TO 11*) AND DELETE XU(*)oAND XL(*).

DINENSION XS (),XU ),XL(3)IS (1)=.427IS 2 =48.66IS 3) =10.7

C ISIJ IS THE STARTING GUESSC XL I IS THE LOVER LIMIT FOR THE I'TH VARIABLEC ZU I IS THE UPPIE LIMIT FOR THE I1TH VARIABLE

XL 1) =.1U (1)=10.

XL (2) =1.XU (2) =200.XL (3) =.10XU 3 =100.-

C A DESCRIITION OF IRE FOLLOWING PARAMETERSC IS DISCUSSED IN ECXPLI

R=9./13.NTA=10 00NPR=100NAV=ONV=3IP=O

C TEE FOLLOWING STATEMENT MUST BE CHANGED TOC CAlL PLANT{X)-C If ONLY SIHULATICN IS WANTED

CALL BOXPLXI(V,NAV,NPR,NTA,R,XS,IP,XU,XL,YMN,IER)NRITE 6 25)

25 FORMAT l,' OPTIMAL GAINS,/)DO 30 1=1.3

30 WRITE{6, 1)I,IS(I-40 FORMAT (IX,' X(',I2,')=',F1&.7)

STOPE NDSUBROUTINE PLANT{XX

C SUBRCUTINE PLANT (XX) SIMULATES THE SHIPCOMMON TDIFFRZAL*8 L, L2 L3 14 L5,L6REAL*8 X'XD6T YbfYOT.U UDOT,V VDOT YAN R,RDOTREAL*8 TIME IRIE XU6OT XUU XVB XVf,XDbDEAL*8 YV,Y,!,YVVRgYVAR,YVVfRRR,YDDDYVDOTHEAL*8 NV NRH E NVVR NVER NVVV NRRRNDDDNRDOTREAL*8 RH6 . I Ft! .Z,XP MASSDELTREA1*8 DYAWE,fAVWIYA CEsi ISR LAMDA,DREAL*8 K1,T1,2,D,X2,Di2,Ci(11',S"DIMENSION XX (3)

CC CLOSE LOOP ANALYSIS WITH FILTERCC INITIAL CONDITIONS FOR INTEGRATIONC SIMULATION END TIME IN SECONDS

ETIME=600.TIME=O. 0ICCUNT= 1

C INITIALIZE THE CCST FUNCTICNISE=0.0ISR=0.0TDIFF=0.0LAMDA=4.2

71

S.. * ~~~v ............................................... '

C GAIN COEFFICIENTS TO BE OPTIMIZEDi'i: K1=XX (1) -71=XX (2)T2-XX 3)

C Z*IDCI4,YDYOT ARE FIXED COORDINATES ON EARTHX=0.0

XDOT=0. 0YDOT=0.0

C UoUDCT,VVDOT ARE FIXED COORDINATES ON SHIPv=o.oUDOT=O. 0VDOT=O. 0YAN=O.0R=O.0RcOT=0. 0

C CEDEBED SPEED IN FEET/SECC 54.01 FT/SEC=32 KNOTS

UC=54.01C AT STEADY STATE ACTUAL SPEED (U) = COMMAND SPEED (UC)

U=aCC D = BUDDER ANGLE

D=0.0L=880.5L2=L**2L3=L*L*LL4=L*L3LS=L*L4L6=L*L5

C SEA DISTURBANCEC FCBCES IN X I DIRECTION COMPUTED IN POUND FORCEC MCNENTS IN

F1=0.F7=0.I Z=O.

C ISEA IS A SWITCH ; ISEA=O (CALM WATER) ISEA=1 (SEA STATE)ISEA1

C HYDRODYNAMIC COEFFICIENTS ARE INSERTED HERE AS PARAMETERSDBH=1.98761ASS= (.00 44 * 1.5*RHO*L3)I2=10.00028 * .5*RHO*L5)YAW =0.0X2= 0.0DX2=0.0

200 CCNTINUES=DSQRT U**2+V**2)

C INPUT YAW OMMANDIAC=0.0IF (TIME.GE.O.0) YAWC=0.0

C EERBO SIGNAL TO DRIVE RUDDER(YAW ACTUAL - YAW ORDERED)C ( CCMPENSATOR FILTER )

YAWE=YAW - YAWCD12= (YAE-X2)/42D=K1*(T 1*DX2 +2)

C AXIAL FORCE HYDRCDYNAK.IC COEFFICIENTS SURGE)C XUDOI IS THE ADDEZ MASS TERM WHICH MUST BE CHANGED FORC DIFFERENT ENCOUNTER ANGLES , SPEED , ENCOUNTER FREQUENCYC

XUDOT=(-.0001)*(. 5*RHO*L3)XU= (-0.0253) *(.5*R O*L2*S)

UU= (-0.0003) * (.5*RHO*L2)XR = (0.0039) * (.5*RHO*L3)*- -IVV (-.0012 *(.5*RHO*L2 -IDD= (-0.0005) (.5*RHO*L2*S**2"

C LATERAL FORCE H Y ODYNAMIC COEFFICIENTS (SWAY)IV= (-0. 00758) * 5*HHO*L2*S)

B= (0.00231* .(.RHO*L3 SYE= 0.00145 . HO* L2***2)!VVR (0. 0 1) *,(.B*RHO*L3/S)

72

IVR= -0.008)*1.5*RHO*L4/S)YVVV= (0.03 .5*RHO*L2/S)YRRR= (0.03 *(.5*RHO*L5/S)YDDD = -0.0005) *(5* R H O * L21S**2)

C YUDOl IS THE ADD E MASS TERM WHICh MUST BE CHANGED FORC DIFFERENT ENCOUNTER ANGLES, SPEED , ENCOUNTER FREQUENCYCC lVDOT=(-0.0039)* (.5*2110*L3)=5C SPIED=32 KNOTS ENCC , TER ANGLE ENCOUNTER FEQ =.75

I VDOT=-2. 3643EC6 G 5U"C RCREWT ABOUT Z-AXIS HYDRODYNAMIC COEFFICIENTS (YAW)

: (-0.00213 *".5*RHO*L3*S"NR= (-0.00 105 .5*RHO*L4*S)ND=-. 00071 (.5*RHO*L3*s**2)NVV =-0.015) * .5* HO*L4/S}NVRE (-0.008) * .5*RHO*L5/S}NVVV= (0. 016* (..*RHO*L3/S)NRBE= -. 06) *.5*RHO*L6/S)NDDD=( .0001 * ( 5*RHO*L3*S**2)

C NRDOT IS THE ADDED INERTIA TERM WHICH MUST BE CHANGED FORC DIfFEEENT ENCOUNTER ANGLE , SPEED , ENCOUNTER FREQUENCYCC NRDOT=(-0.00027)*(.5*RHO*L5)C SEEED=32 KNOTS ENC UTER ANGLE =150 ,ENCOUNTER FREQ =.75

NBDOT=-1.5096E+1C SeITS SEA STATE TO ZERO

iEfISEA. EQ. 1) GO TO 30FX=.FY=O.MZ= 0.GC TO 35

C TABLE LOOK UP OF SEA DISTURBANCEC UNIT 12 HAS THE SEA STATE rATA MIMED CHC IT MUST BE SYNCHECNIZED BY APPROPRIATELYC CALLING CH IN THE PROPER TIME IN THE LOOP.C TEE SEA DATA WAS CREATED FOR 600 SECONDSC WITH AN INCREMENTAL INTERVAL OF 1 SECOND.30 READ (12) CH

FX=CH (3)F=CH (4)"Z=CH 8)

35 CCNTINUEC U ICIUAL SPEEDC UC CCMMANDED SPEEDC IE = PROPELLER THEUST

XP=-XUU*UC**2C EQUAIICNS OF MOTICNc FOR CONSTANT SPEEr COMMENT THE NEXT TWO INSTRUCTIONS

UDOT=( (XVR + PASS)*V*R + XUU*U**2 + XVV*V**21 + XDD*D*D + YX + XP )/(MASS-XUDOT)

DOT=(YV*V + (YR-,ASS *U *R + YD*D + YVVR*V**2*R1 + YVER*V*R**2 + YVVV*V**32 + YRRR*R**3 + YDDD*D**3 + FY )/(MASS-YVDOT)RrOT=l NV*V + NR*R + ND*D NVV R*V**2*R1 + NY R*V*R**I2 + NVVV*V**3

2 + NRRR*R**3 + NDDD*D**3 + MZ )/(IZ-NRDOT)C VEEN TO PRINTOUT

IF 4 ICOUNT.EQ.11) GO TO 50GC TO 300C CCNVEET RADIANS TC DEGREES

50 YAiDEG= YAW*57.296RDEG=R*57. 296RDDEG=RDOT*57 .296DDEG=D*57. 296

YAWC=YAVC*57.296C WRITE 6 100) 1IE IP X IDOT YYDOT D.C ,1I T 6DOT V IDO ItYAWEG RDEG RDDEG,DDEG

100 F CfMA1'61X 'TIMi=',F8. 3.' SEC Xf=' F1o0.2, LBF X='1,F8.2, FT xICT=',F8.I,' FT/SEC T=',F8.2,

73

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

........................................................ ....0. . . . . . . . .

It FT YDOT=I,F8.4,9 FT/SECI./2x,1 UC=1 F8. ,SFT/SEC U=9,F8°4 1 FT/SE6 UHOT='FO6 H0.9

4 ' FT/SEC**2 V=1 P8 4, FT/SEC VD6T=',F O.6,5' FT/SEC**2, / 2X IYAiC'o F8.4,I DEG YAW =H'15.7,6' DEG YAW fiAE=1 F15.7 1 DEG/SEC YAW AC6El='

715.7,' DEG/SEC$*2',/,2 ,$RUDDER =1,F15.7,1 DEG ',/-ICCINT= 1

C TEST IF WANT TO STOP300 IF (TIME.GE.E7IME) GO TO 400

C INTEGRATION STEP SIZE DE-TDELT= 1.0

C INTEGRATIONU=U+UDOT*DELTV=V+VDOT*DELTR=R+BDOT*DELTYAW=YAW+E*DETX2=X2+DX2*DEL-

C CCNVEET SHIP TO fIXED COORDINATES ON EARTHXDOT=U*DCOS (YA%)-V*DSIN (YAW)YDOT=U*DSIN (YAW) +V*DCOS (YAW)X=X+XDOT* DELTY=Y+YDOT*DELTTIME=TIP1E+DE17ICCUNT-ICOUNT41ISE=ISE + LAMtA*YAVE**2ISB=ISR + D**2GO TO 200

C J= TDIFF = COST FUNCTION400 IDIF=ISE+ISR

WRITE(6,500) ISE ISR,TDIFF,KIT1 T2500 FORMAT(' , I F15.7 ' TOTAL='1F157,2xK=',F15.,2X I,F15.7,2X :2=' ,F15.7)

AEWIND 12EETU RNEND

C DELETE ALL THE FOLLOWING SUBROUTINE IF SIMULATICN ONLYC AND NOT OPTIMIZATION IS WANTEDCC SCEROOTINE BOFLX (CATEGORY HO)CC PURPOSECC BOXPLX IS A SUBROUTINE USED TO SOLVE THE PROBLEM CFC locacting A MINIMUE (OR MAXIMUM) OF AN ARBITRARY OBJECT-C ive function SUBJECT TO ARBITRARY EXPLICIT AND/CRC implicit constraints by tHE COMPLEX METHOD OF N.J. BOX.C explicit constraints are dEFINED AS UPPER AND LOWERC bounds on.the independent variables IMPLICIT constraintsC may be arbitrary function of the varIABLES. TWO FUN-C ction subprogram to evaluate the objective FUNCTION ANDC imElicit conSTRAINtS RESPECTIVELY must be SUPPLIEDC b; the user (see EXAAPLE BELOW). B6PLX ALSO HAS tHEC o'ticn to perform integer programming, where the valuesc 01 the independent variables are res ricted to integers.CC USAGECC CALL BOXPLX (NV,NAV,NPR,NTA,RXS,IP,XU,XL,YMN,IER)CCC DESCRIPTION OF PARAMETERSCC NV AN INTEGER INPUT DEFINING THE NUMBER OF INDEEENrENTC VARIABLES OF THE OBJECTIVE FUNCTION TO BE MINIMIZED.C NOTE: MAXIMUM NV + NAV IS PRESENTLY 50. MAXIMI NV ISC 25. IF THESE LIMITS MUST BE EXCEEDED PUNCH A SOURCEC DECK IN THE USUAL BANNER, AND CHANE THE DIMENSIONC STATEMENTS.

74

_-_--.. *-.;..'V.

CC NAV AN INTEGER INPUT DEFINING THE NUMBER OF AUXILIARY varC iaELES THE USER WISHES TO DEFINE FOR HIS OWN CONVENIENCE.C TYFICALLY HE MAY WISH TO DEFINE THE VALUE OF EACH IMPLICIC CONSTRAINT FUNCTION AS AN AUXILIARY VARIABLE. IF THISC IS DCNE, THE OPTICIAL OUTPUT FEATURE OF BOXPLI CAN BEC USED TO OBSERVE THE VALUES OF THOSE CONSTRAINTS AS THEC SOIUTICN PROGRESSES. AUXILIARY VARIABLES IF USED,C SHOULD BE EVALUATEE IN FUNCTION KE (DEFINED BELOW).C NAY MA BE ZERO.CC NPI INPUT INTEGER CONTROLLING THE FREQUENCY OF OUTP.UTc desired for dia gMnctic purioses.C IF NPR .LE. 0 NO UTPU W LL BEC PROEUCED BY B6XPLX. OTHERWISE THE CURRENT COMPLEX CFC K= 2*NV VERTICES AbD THEIR CENTROID WILL BE OUTPUT AFTERC EACH NPR PERMISSIBLE TRIALS. THE NUMBER OF TOTAL TRIALSC NUMBER OF FEASIBLE TRIALS NUMBER OF FUNCTION EVALUAIICNSC AND NUMBER OF IMPLICIT CONSTRAINT EVALUATIONS ARE IN-C CIUDED IN THE OUTPUT.C ADDITIONALLY (WHEN NPR .GT. 0) THE SAME INFORMATIONC WILL BE CUTP6T:CC 1) IF THE INITIAL EOINT IS NOT FEASIBLEC 2) AFTER THE FIRST COMPLETE COMPLEX IS &ENERATEDC 3) IF A FEASIBLE VERTEX CANNOT BE FOUND AT SOME TRIAL,C 4) IF THE OBJECTIVE VALUE OF A VERTEX CANNOT BE MAIEC NO-LCNGER-WORS7.C 51 IF THE LIMIT ON TRIALS (NTA) IS REACHED ANDC 6) WHEN THE OBJECTIVE FUNCTION HAS BEEN UNCHAN6ED FORc 2*BY TRIALS, INDICATING A LOCAL MINIMUM HAS BEENC FOUND.CC IF THE USER WISHES TO TRACE THE PROGRESS OF A SOLUTION,C A CHOICE OF NPR = 25, 50 OR 100 IS RECOMMENDED.CC NTA INTEGER INPUT OF LIMIT ON THE NUMBER OF TRIALSC allowed in the calculation.C IF THE USER INPUTS NTA .LE. 0 A defaultC VALUE OF 2000 IS USED. WHEN THIS LIMIT IS REACHEDc CONTROL RETURNS TO THE CALLING PROGRAM WITH THE BESTC ATTAINED OBJECTIVE FUNCTION VALUE IN YMN, AND THE BESTC ATTAINED SOLUTION POINT IN IS.CC R A REAL NUMBER INPUT TO DEFINE THE FIRST RANDC NUMBERC USED IN DEVELOPING THE INITIAL COMPLEX OF 2*NV VERTICIES. --C GT. R .LT. 1 IF R IS NOT WITHIN THESE BOUNDS,C TWILL BE REPLACt BY 1./3.CC XS INPUT REAL ARRAY DIMENSIONED AT LEAST NV NAV.c the first nv must contain aC FEASIBLE ORIGIN FCB STARTING THE CAL-C CULATI7CN. THE LAST NAV NEED NOT BE INITIALIZED. UPCNC RETURN FROM BOXPLX THE FIRST NV ELEMENTS OF THE ARRAYC CONTAIN THE COORDINATES OF THE MINIMUM OBJECTIVEC function AND THE REMAINING NAY (NAY .GE. O) CONTAIN THeC values of THE CORRESPONDING AUXILIARY VARIABLES.CC TP INTEGER INPUT FOR OPTIONAL INTEGER PROGRAMMING.C if ip=l, THE VALUES OF THE INDEPENDENT VARIABLES WILLC be rplaced WITH INTEGER VALUES (STILL STORED AS BEAL*).CC zU A REAL ARRAY LIMENSIONED AT LEAST NV INPUTTING THEC U er BOUND ON EACH INDEPENDENT VARIABLE (EACH EXPLICITC cHSTRAINT). INPUT VALUES ARE SLIGHTLY ALTERED BY BOXPLI.CC XL A REAL ARRAY EIMENSIONED AT LEAST NV INPUTTING THEc lover bound on each independentC VARIABLE, (EACH EXPLICIT CONstraint).

75

. . . . . .. . . .

C NOTE: FOR BOTH XU AND XL CHOOSE REASONABLEC VALUES IF NONE ARE GIVEN NCT VALUES WHICH AREC magnitudes ABOVE CR BELOO THE EX7ECTED SOLUTION.C input values are SLIGHTLY ALTERED BY BOXPLX.CC YMI THIS OUTPUT IS THE VALUE (REAL*4) OF THE OBJECTIVEC funcTIcN,CORRESPONEING TO THE SOLUTICN POINT OUTPUT IN XSCC IER INTEGER ERROR RETURN. TO BE INTERROGATED UPCNC return FECM BCXPLI. IER WILL BE ONE OF THE FOLLOWING:CC =-I CANNOT FIND 1EASIBLE VERTEX OR FEASIBLE CENTROID --C AT THE START OR A RESTART (SEE 'METHOD$ BELOW).C =0 FUNCTION VALUE UNCHANGED FOR IN' TRIALS. (WHEREC N=6*NV+1O) THIS IS THE NORMAL RETURN PARAMETER.C =1 CANNOT DEVELOP FEASIBLE VERTEX.C =2 CANNOT DEVELOP A NO-LCNGER-WORST VERTEX.C =3 LIMIT ON TRIALS REACHED. (NTA EXCEEDED)C NOTE: VALID RESUIIS MAY BE RETURNED IN ANY OF THEC ABOVE CASEE.CC EXAMPLE OF USAGE

C THIS EXAMPLE MINIMIZES THE OBJECTIVE FUNCTION SHOWN INC the EXTERNAL FUNCTION FE(X). THERE ARE TWO INDEPENDENTC varIABLES X11) & X(2) AND TWO IMPLICIT CONSTRAINTC function X(3) 8 X(4) WHICH ARE EVALUATED AS AUXILIARYC varia-les (see EXTERNAL FUNCTION KECX) ).CC DIMENSION XS(4),XU(2),XL(2)CcCC S7AETING GUESSC it1) = 1.0C XS(2) = 0.5CC UPPER LIMITSC XU(ll = 6.0c I U(2 = 6.0CC lOWEr LIMITSC XL( = 0.0

cC C T= 9./13.C NTA = 5000C NR = 50C NAV = 2C NV = 2C IP = 0CC .-C CALL BOXPLX (NfV,NAV NPH NTA R IS IP XU,XL,YMN,IER)C WRITE(6 1) (XS(I I=I 4)YANIER)C 1FCEMAT$1 W//6, ISN LOCATED AT (XS(I)=).c 2 4l(e137 54 )C AND VALUE IS ,,E13.7,, IER =,15)

C S7OPC ENDCCCCC FUNCTION KE (X)C EVALUATE CONSTRAINIS. SET KEO IF NO IMPLICIT CONSTRAINTC is viCLATED, OR S1T KE=1 IF ANY IMPLICITc ccnstraint is violated.C DIMENSION X(4)C X2 = 2C KE =

C 1(3) = X1 + 1.732051*X2C I (X(3) .LT. 0. .OR. X(3) .GT. 6.) GO TO 1C X(4) = 1/1.732051 -12

76

• .. ... .. ... %.. ,• -.. . .. ", ..- ,. o. o....-.-. -.... ... . . ,. . . . . . . .-. - ..

-*.....-.- . ..... .......- ... . .:.i; - ._-...._..Z '

K. C IF IX(4) .GE. C.) RETURNc 1KE= 1C RETURNC ENDCC-" cc ::C FUNCTION FE(l)C DIMENSION X(4)ccCC TEIS IS THE OBJECTIVE FUNCTION.C FE= -X(2)**3 (9.-(X (1)-3.) **2)/(4 6.76538))C RETURNC ENDCC METHODCC THE CCMPLEX METHOD IS AN EXTENSION AND ADAPTION OFc the sinple method cf linea I rogramming.C STARTING WITH ANY CNE feasibe point in n-dimensionC A "CCMPLEX" OF 2*N vertices is constructed byC SELECTING RANDOM ECINTS WITHIN THE feasibleC REGICN. FOR THIS PURPOSE N COORDINATES ARE FIRSTC RANDCI.LY CHOSEN WITHIN THE SPACE BOUNDED BY EXPLICIT CCN-C STRAINTS. THIS DEFINES A TRIAL INITIAL VERTEX.c it is then checked for possible violationC OF IMPLICIT CONSTRAINTS. IF one or more are violated,C THE TRIAL INITIAL VERTEX IS DISPLACED half of itsC DISTANCE FROM THE CENTROID OF PREVIOUSLY SELECTED initialC VERTICES. IF NECESSARY THIS DISPLACEMENT PROCESS ISC REPEATED UNTIL THE VERTEX HAS BECOME FEASIBLE. IF THISc FAIL TO ha pen after 5*n+10 displacementsC THE SOLUTION IS AEANDONED. AFTER EACH VERTEX IS ADDEDC TO THE COMPLEX THE CURRENT centroid is checked fcrC FEASIBILITY. IF IT IS INFEASIBLE, the last trailC VERTEX IS ABANDONED AND AN EFFORT TO GENERATE an alter-C ATIVE TRIAL VERTEX IS MADE. IF 5*N+10 VERTICES AREC ABANDCNED CONSECUTIVELY, THE SOLUTION IS TERMINATED.C

. C IF AN INITIAL COMPLEX IS ESTABLISHED, THE BASICc co putation loop iE initiated.C TH SE INSTRUCTIONS FIND THE CURRENT WORST vertex thatC IS THE VERTEX WITH THE LARGEST CORRESPONDING value forC THI OBJECTIVE FUNCTION AND REPLACE THAT VERTEX BYC ITS OVER-REFLECTICN THROUGH THE CENTROID OF ALL CIHEE"c vertices. (if the vertex to beC REPLACED IS CONSIDERED AS A VECTOR IN n-s aceNC ITS OVER-REFLECTICb IS OPPOSITE IN DIRECTION IN-C CREASED IN LENGTH EY THE FACTOR 1.3, AND COLLINEAR WITHC THE REPLACED VERTEX AND CENTROID OF ALL OTHER VERTICES.)CC WHEN AN OVER-REFLeCTION IS NOT FEASIBLE OR REMAINSc WORST IT IS CONSIDERED NOT-PERMISSIBLEC AND I DISPLACED EALFWAY TOWARD THE CENTROID.C AFTER FOUR SUCH ATTEMPTS ARE MADE UNSUCCESSFULLYC EVERY FIFTH ATTEMPT IS MADE BY REFLECTING THE OFFENDINGc VERTEX THROUGH THE PRESENT BEST

. C VERTEX INSTEAD OF THROUGH THE CENTROID. IF 5*n+10C DISPLA5EMENTS AND CVER-REFLECTIONS OCCUR WITHOUT AC SUCCESSFUL (PERMISSIBLE) RESULT THE CURRENT BESTC VERTEX IS TAKEN AS AN INITIAL FEASIBLE POINT FOR Ac RESTART RUN OF THE COMPLETE PROCESS.C RESTARTING IS ALSO UNDERTAKEN WHEN 6*nv 10 CONSECUTIVEC TRIALS HAVE BEEN MIDE WITH NO SIGNIFICANT CHANGE IN THEC VALUE OF THE OBJECTIVE FUNCTION. IN ALL CASESC RESTARTING IS INHIEITED IF THE LAST RESTART DIb NOTC PRODUCE A SIGNIFICANT IMPROVEMENT IN THE MINIMUMc ATTAINED.C

77

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

C IT IS RECOMMENDED THAT THE USER READ THE REFERENCE FORC FURTHER USEFUL INFCRMATION. IT SHOULD BE NOTED THAT THEC ALGORITHM DEFINED THERE HAS BEEN ALTERED TO FIND THEC CONSTRAINED MINIMUM, RATHER THAN THE MAXIMUM.CCCC REMARKSCC THE INTEGER PROGRAMMING OPTION WAS ADDED TO THIS EROGRAMC AS SUGGESTED IN REFERENCE (2). A MIXEDc inte q er/continuouE variable version of boxtlx * --C WOULD BE EASY TO CBEATE BY DEclaring "ip" o be an arrayC OF NV CONTROL VARIABLES WHERE IP (ii=1 would indicateC THAT THE I-TH VARIABLE IS TC BE CONFINED to integerC VALUES. EACH STATEMENT OF THE FORM 'IF (IP .EQ.)' etc.C WOULD THEN NEED TC BE ALTERED TO 'IF (IP(I) .EQ. 1)' etc.C WHERE THE SUBSCRIPT IS APPROPRIATELY CHOSEN. NC MALLY,C kU AND XL VALUES ABE ALTERED TO BE AN EPSILON #WITHIN'c actual valuesC DECLARED BY THE USER. THIS ADJUSTMENT IS NOT MADEC WHEN IP=1.CC NOTE: NO NON-LINEAR PROGRAMMING ALGORITHM CAN GUARANTEEc that the answer found is the globalC MINIMUM. RATHER THAN JUST A local minimum. howeverC ACCOBDING TO REF. 2 THE COMPLEX method has an advantageC IN THAT IT TENDS 16 FIND THE GLOBAL minimum moreC FRfQUENTLY THAN MANY OTHER NCN-LINEAR PROGRAM-C MING ALGORITHMS.CC IT SHCULD BE NOTEE THAT THE AUXILIARY VARIABLE FEATUREc CAN ALSO BE USED IC DEAL WITHC PRCELEMS CONTAINING EQUALITY CONstraints. any equalityC CCNSTRAINT IMPLIES THAT A GIVEN VARiable is not trulyC INDEPENDENT. THEREFORE, IN GENERAL, ONE variableC INVCLVED IN AN EQUALITY CONSTRAINT CAN BE RENUMBERED fromC THE SET OF NV INDEPENDENT VARIABLES AND ADDED TO 1HE SETC OF 1AV AUXILIARY VARIABLES. THIS USUALLY INVCLVESC renumbering THE INLEPENDENT VARIABLES OF THE GIVENC prcblemC SUBRCUTINES AND FUhCTICNS REQUIREDiCC SUBROUTINE 'BOUT' AND FUNCTION IFBV I ARE INTEGRALC parts of THE BOXPIX PACKAGE.I!CC TWO FUNCTIONS MUST BE SUPPLIED BY THE USER. THE FIRST,c ke(x) is used to evaluate the implicitC CONSTAAINTS. SET KE=O AT THE beginning of the functionC TEEN EVALUATE THE IMPLICIT CONstraints. in the exampleC AEOVE, THE FIRST CCNSTRAINT, X(3), must be within theC RANGE (0. °LE° X(3) °LE. 6jh THE SECOND constraint x(4)C MUST BE .GE° 0. . IF EIT ER CONSTRAINT IS not withinC THESE BCUNDS CONTFOL IS TRANSFERRED TO STATEMENT 1C AND KE IS SET TO "1" AND CONTROL IS RETURNED TO BOXfIX.CC THE SECOND FUNCTICB THE USER MUST PROVIDE EVALUATES TIEc objective function. it isC CAILED FE(X) AS SHCWN IN THE EXAMple above, and feC MOST BE SET TO THE VALUE OF THE OBJECTIVE functionC CCERESPCNDING TO CURRENT VALUES OF THE NV INDEPENDENTC VARIABLES IN ARRAY 1X9.CC REFERENCESCC BOX, M. J. "A NEW METHOD OF CONSTRAINED OPTIMIZATICNC and a CCMPARISON WITH OTHER METHODS",C computer journal, 8 apr. '65, PP. 45-52.C

78

° /, . .

C BEVERIDGE G., AND SCHECHTER R., "OPTIMIZATION: THEORY ANDC PRACTICE", ;CGRAW-EILL, 1970.CC EEOGRAMMERCC R.R. HILLEAFY 1/1966.C REVISED FOR SYSTEM 360 4/1967C CORRECTED 1/1969C REVISED/EXTENDED BY L.NOLAN/R.HILLEARY 2/1975"C CORRECTED 8/1976CCC ........... ......... ~......@ *.

CC

SUBROUTINE BOXPLX INV,IAV,NPR,NTZ,RZ,XS,IP,BU,BLYMN,IER)DIMENSIN V450,50), FUN(50), SUXJ(25), CEN(25), XS(NV)

C 1,:u(nv),bl(NV)KV = 5EP = 1.E-6NIA = 2000IF (NTZ.GT.0) biA = NTZE = HZIF (R.LE.0O..OF.R.GE.1.) R=1./3.NIT = NV+NAV

CC TOTAL VABS, EXPLICIT PLUS IMPLICITNT = 0C CURRENT RIAL NO.

NET = 0C CURRENT NO. OF PERMISSIBLE TRIALS

NIFS =0

C CURRENT NO. OF TIMES F HAS BEEN ALMOST UNCHANGEEC -

C CHECK FEASIBILITY OF START POINTC

DO 4 I=I NV

II (BLjI)!LE.%1) GO TO 1II = -I= BL (I)

GC TO 21 IF (BU(I).GE.Nl) GO TO 3II = I

V7 = BU(I I)2 IF (NPR.GT.0) WRITE (6,49) 13 V(I,1) V

CEN(I) = VTIFLIP. EQ.) GC TO 4BL() =BII)AIIAX1 (EP EP*ABS (BL (I)))

B I BU I)-AMAX1 EPEP*ABS(BU I4 SU V

CC

NCE 1C NUMBER CF CONSTRAINT EVALUATIONS

I=1IF (KE(V(1,1)) ).EQ 0) GO TO 5IF (NPR . GO TO 12WEIE 16,50)GO TO 2 -

5 HEE= 1CC AU1MEEE OF VERTICES (K) = 2 TIMES NC. OF VARIABLES.

K = 2*NVCC NUMBER OF DISPLACEMENTS ALLOWED.

79

* - .•!"&.-]

NLIA 5*NV.10CC DUMBER OF CONSECUTIVE TRIALS WITH UNCHANGED FE TCc terainate.

NCT = NLIE +NVALPHA = 1.3FS K

FKM FK-1.B77A = ALPHA+l.

CC INSURE SEED OF RANDOM NUMBER GENERATOR IS ODD.

ICR = R*1.E7

C IF (MOD(IQR,2).EQ.0) ICli=IQRe1Q1

C SET UP INITIAL VERTICESFUN 1(1) = FE (V (1, 1))YIIN =FUN (1)11F 1.

C UNOLD = FUN~i)

DC 15 I=2,KFI FI+1.LIMT = 0

7 LIMT = LIIIT+lC

*C IND CALCULATION IF FEASIBLE CENTROID, CANNOT BE FOUND.11 (LIMT. GE. Nul)~ GO TO 11

CDC 8 J=l,NV

CC EANLCII NUlBER GINERATOR (EANDU)

IQR = IQR*65539IF (IQR.LT.O) IQB = IQR+21L47L83647+1E)X,=IQRR X = RQX*.4656613E-9

V iBLpJ 4RQX* (EU J) -BL (3' )I CC IE .1 (J,I) =AN T(V (J, 145-~ C

DO 10 L=1,NLIMSCE = NCE+lIF (KE(V(1,I)).EQ.O) GO TO 13

CDO 9 J=1,NV

V7=.5 *(V J,I)+CENJ)IF (IP. EQI 1AlNPV+

V JTC

*10 CCNTINUEC

11 IF (NPR LE 0) GO TO 12

12 L BU1 (NT,NPT,NFENCE,NV,NVT,V,I,FUN,CEN,I)

GO To 48C

13 DO 14l J=1 NVSUMN) = HUN ) V J,I)

143 CEll (J = S U3 J)IFlCC IRY TO ASSURE FEASIBLE CENTROID FOR STARTING.

NCE = NCE+1IF (KE(CEN).EC.0) GO To 60SU (3) =SUMJ -V(p, I)GO 0O 7

60 NEE = NFE+1FUN4(I)= FE(V(1,I))

15 CCI INUE

80

CC END CF LOOP SETTING OF INITIAL COMPLEX.

IF (NPR.LE.O) GO TO 17CALL BOUT N NPT,NFE,NCE,NV,NVT,V,K,FUN,CEN,O)

C FIND THE WORST VIRTEX, TEE 'JITH.J= 1

CDO 16 I=2 KIF (FUN(J).GE.TUN(I)) GC TO 16

16 CCNTINUECC EASIC LOOP. ELIMINATE EACH WORST VERTEX IN TURN.C it must become NC LONGER WORST NOT MERELY IMPROVED.C find next-to-vertex, THE 'JN'TH ONE.

17 JN = 1IF (J.EQ.1) JN = 2

CDC 18 I=1 KIF (I.EQ.oJ GC TO 18IF (FUN(JN) .GE.FUN(I)) GO TO 18JN = I

18 CCNTINUECC LIMT = NUMBER CF MOVES DURING THIS TRIAL TOWARD THEC centroid DUE TO FUNCTION VALUE.

LIMT = 1CC CCMEUTE CENTROII AND OVER REFLECT WORST VERTEX.C

DO 19 1=1 NVEN= VIJ

SCM I (1) = Ml( -VTCEN(I) = SUM I) /FKMVT = BETA*CEN (I)-ALPHA*VT

C IF (IP.EQ.1) 11 = AINT(VT..5)~CC INSURE THE EXPLICIT CONSTRAINTS ARE OBSERVED.

19 V(I,J) = AAAX1(AMIN1(VT,BU(I)),BL(I))C

N7 = NT+1CC CHECK FOR IMPLICIT CONSTRAINT VIOLATION.C

20 DC 25 N=1,NLIMNCE = NCE 1IF (KE(V(1,J)).EQ.O) GO TO 26

CC EVERY 'KV'TH TIME OVER-REFLECT THE OFFENDING VERTEXC thrcuqh the BEST *ERTEX.

IF (MOD N,KV). E.O) GO TO 22CAIL FBV (K,F i)

CDO 21 I=I NVVT = BETA4V (I ,)-ALPHA*V (I,J)IF (IP.EQ.1) 1 = AINTVT .5)

21 V(I,J) = AMAX1(AMIN1(V ,BU(I)),BL(I))C

GC TO 24CC CCNSTRAINT VIOLATION: MOVE NEW POINT TOWARD CENTROID.C

22 DC 23 I=1oNVVT = .5*(N(IV(I J)IF (IP.EQ 1) = AiNT(VT+.5)V(I,J) =VT

23 CCNTINUEC

81

24 NT = NT+125 CCNTINUE

CIIR = 1

cC CANNOT GET FEASIELE VERTEX BY MOVING TOWARD CENTROID,C CR EY OVER-REFLICTING THRU THE BEST VERTEX.

IF (NPR.LE. 0) GO TO 42WRITE (652 )NTJ"CALL BOUT (NT,NPT,NFE,NCE,NV,NVT,V,K,FUN,CEN,J)GC TO 42

K C FEASIBLE VERTEX FOUND, EVALUATE THE OBJECTIVE FUNCTICN.26 NTE = NFE+1

FUNTRY = FE(V(1,J))CC TEST TO SEE IF FUNCTION VALUE HAS NOT CHANGED.

AfO = ABS(FUNTEY-FUNOLD)AMX = ANAX1 (AES(EP*FUNOID) ,EP)

CC ACTIVATE THE FOLlOWING TWO STATEMENTS FOR DIAGNCSTICc pi-rroses on l.C WITE (6,94) J,AFO,A!X,FUNTRY,FUNOLD,FUN(J),FUN(JN)

1 NIFS.NC 99 CRMAT (1X,13 ,6E15.7,2I5)

IF (AFDO.GT.AX) GO TO 27N7FS = NTFS+]If (NTFS.LT.NC) GO TO 28IER = 0IF (NP.3.LF.O) GO TO 42WRITE 6,3 KCALL BOUT (NT,NPT,NFE,NCE,NV,NVT,V,K,FUN,CEN,O)GO TO 42

27 NIFS = 0CC IS TEE NEW VERTEX NO LCNGEB WORST?

28 IF (FUNThY.LT.FUN(JN)) GO TO 34CC TRIAL VERTEX IS STILL WORSTi ADJUST TOWARD CENTbOID.C EVERY 'KV'TH TIVE OVER-REFLECT THE OFFENDING VERTEXC tbrcuph the BEST fERTEX.

LIN LIMT+ 1IF (OD(LIT KV) NE. 0) GO TO 30

C CAL FB (K,fUb, 1-)

DC 29 I=I NVV7 = BETA4 V(I ,)-ALPHA*V I,J)IF (IP.EQ. 1) l = AINT VT+.5)

29 V4I,J) = AMAX1|AMIN1(VT,BU(I)),BL(I))GO TO 32

C

30 Do 31 I=1 NVVT = *5* (E I 1 V(,fIF (IP. EQ.1) V AINT(VT+.5)

I CVT31 CNIJUE

C32 IF (LIMT.LT.NLIM) GO TO 33

CC CANNOT MAKE THE 'J'TH VERTEX NO LONGER WORST BYc disilacin towardC THE C NTROID OR BY OVER-REFLECTING THRU THE BEST VERTEX.

IER = 2IF (NPR .LE. 0) GO TO 42WITE 652 IT JCALL BOUT (NTNPINFE,NCE,NV,NVT,V,K,FUN,CEN,J)GC TO 42

33 NT = NT+-

82

. . . . . .

GO TO 20CC SUCCESS: WE HAVE A REPLACEMENT FOR VERTEX J.

34 FUN(J) = FUNTBYFUNOLD = FUNTEYNPT = NPT+.

CC EVERY I00'TH PERMISSIBLE TRIAL RECOMPUTE CENTROIDC summation to AVCID CREEPING ERAOR.

IF (MODNPT,1O).NE.O) GO To 37C

DO 36 I=I NVsuC(I) = 6.DC 35 N-. K

35 SUM(I) = §M(I)+V(I,N)C

CEN4I)= SUM (I)/FK36 CONTINEC

LC = 0GO TO 39

C37 DO 38 I=1,NV38 SUM(I) = SU(-I)+V(I,J)C

LC = JC

39 IF (NPR. LE.O) GO TO 40If (MOD(NPT,NPR).NE.0) GO TO 40

CCALL BOUT (NTNPT,NFE,NCE,NV,NVTV,K,FUN,CEN,IC)

CC BAS THE MAX. NUMEER OF TRIALS BEEN REACHED WITHCUTC ccnvergence? iF NOT, GO TO NEW TRIAL.

40 IF (IT.GE.NTA) GO TO 41

C NEXT-TO-WORST VERTEX NOW BECOMES WORST.J = JNGO TO 17

41 IER = 3I (NPR.GT.0) iRITE (6,54)

CC COlIECTOR POINT FOR ALL ENDINGS.C 1) CANNOT DEVELCE FEASIBLE VERTEX. IEiR = 1C 2) CANNOT DEVELOP A NO-LCNGER-WORST VERTEX. IER = 2C3 FUNCTION VALUE UNCHANGED FOR K TRIALS. IEE = 0C 4 LIMIT ON TRIALS REACHED. IER = 3C 5 CANNOT FIND FEASIBLE VERTEX AT START. IER = -1

4 CCNTINUECC FIND BEST VERTEX.

CALL FBVIF (IER.GE.3)GO TO 44

CC RESTART IF THIS SOLUTION IS SIGNIFICANTLY BETTER THANC the previous, OR IF THIS IS THE FIRST TRY.

IF NPR.LE.0) GO TO 43WRITE 6 55)(P YMN FUN M

43 I7F (UN(!1).GE.IAYN 0 T04IF (ABS FUN (M)-YMN).LE. AMAX1(EP,EP*YMN)) GO TO 47

CC GIVE IT ANOTHER TRY UNLESS LIMIT ON TRIALS REACHED.

44 YMN = FUN4M(M)FUN(.) U (H)* C

DO 45 I=I NVCEN II = V I,M)SUM I = V I'M

83

45 V Ii,1) = V(1,E.)C

DC 46 I=1,NVT46 XS(I) = V(I, M)C

IF (IER.LT.3) GO TO 647 IF (NPR.LE0) GO TO 48

CALL BOUT (NT NPT NFE,NCE,NV,NVT,V,K,FUN,V(1,M),-1)WRITE (6,56) fUN(A)

48 RETURNC

49 FORMAT (50HOINDEX AND DIRECTION OF OUTLYINGivariable at starti5)

50 FORMAT (50 HOIMPLICIT CONSTRAINT VIOLATED AT1start. dead end.

51 FCRMAT ('OCANNCT FIND FEASIBLE',I4,'TH VERTEX ORIcentroid at start.')

52 FC3MAT (!OHOAT TRIAL 14,54H CANNOT FIND FEASIBLEivertex which is no21CNGER WORST 14 15X 'RESTART FROM BEST VERTEX.')

53 FORMAT (40HOHUNTIO& HAS BEEN ALMOST UNCHANGEL1for i5,7h trails)

54 FCRMAT (27HOLIeIT ON TRIALS EXCEEDED.)55 FORMAT(' OBEST VERTEX IS NO.1,13,'OLD MIN WAS1,E15.7,

1 ' NEW MIH IS 'oE15.7)56 FORMAT ('OMIN CB ECTIV1 FUNCTION IS 1,E15.7)

ENDSUBROUTINE FBV (K,FUN,M)DIMENSION FUN (50)M 1

CDO 1 I=2 KIF (FU(A ).LE.TUN(I)) GO TO 1

1 CCNTINUEC

RETURNENDSUBROUTINE BOUT (NT,NPT NFE,NCE NV,NVT,V,K,FN,C,IK)DIMENSION V(50 ,50), FN( O, C(2B)WRITE (6,4) N7,NPT,NFE,NCE'

CDC 1 I=1,KWRITE (65) FN(I),V p I),J=1,NV)IF (NV.,.NV) GoO 1NVP = NV+1WRITE (6,6) (VIJ,I),J=NVE,NVT)

1 CGNTINUEC

IF (IK.NE.0) GC TO 2C

WRITE (6,7) (C(I),I=1,NV)RETURN

2 IF (IK.GE.0) GC TO 3WRITE (6,8) (C(I),I=1,NV)RETURN

3 WRITE (6,9) IK,(CtI),I=1,NV)RETURN

C4 FCRMAT ('ONO. TOTAL TRIALS = 1,15,4X,1'no. feasible trails = ' i5,4x2'NO. FUNCTION EVALUATIONS = ',I5,4X,31nc. constraint evaluations = , i5/400 FUNCTICN VALUE' 6K 1 NDEINDENT VARIABLES/5deRpendENT OR IMPLICIT tCNSTRAINTS" E

5 FC MAT (11H ,E18.72X,7E14.7/(21X,714.7))6 FORMAT (21X 7E1.7)7 FCRMAT (1OHOCENTOID 11X 7E14 .7/(21X 7E14 7))8 FCfiMAT ('0 BEST VERTEX1,7X,7E14.7/(21X,7E11.7))

84

- .. .

9 FGRMAT ('OCENTROID LESS fX#,12,2X,7E14.7/(21x,17e114.7))ENDFUNCTION FE XDIMENSION X3?CCHACN TDIFCAlL rP.LJAI,

. .FE=TDIFFRETURNENDFUNCTION XE(IDIMIENSION IM3

BETURNEND

//GO.SYSIN DD*

//GC.PI12FOO1 DD DIS.E=SHR,DSN-=lSS.S2160.A341

85

EXAMPLE PROBLEMl USING ICSOS

The purpose of this example is to demonstrate the

* performance of the prcgram. Consider the control system of

Figure 4.1 with contioller C. Figure B.1 shows the block

* diagram to evaluate the contrcller parameters.

86

r .

00

0 too Cj

D IS IES

I-A

I0

0

Ilit

2 H 1 87

I +S

The differential equations describing the system and its

desired Ferformance are:

i2=DX2

I-=DX4i'=DX5

R=DR

Defining the following cost function:

J= fLAMDA*YAVI**2+D**2) dt

Defining the special functions:

YAVE=YAWC-YAW

D12= (YAWE-X2) /12

X3=K1* (T1*DX2 +12)

D14= (X3-X4) /T4

d= (t3*dx4+x4)

dx5= (d-x5)/tpl

xe=k*(tz*dx5+x5)

dr= (x8-r)/tp2

Defining the constants:

!1WC=1. 0

K=. 14771

T2=11.2833

TP1=6.4699

TE2=53. 7931

LAMDA=4. 2and using YAWC=1.0 the optimal solution found by

the Frogram is:

K1=.4179916

71=53.69932

72=4.970023

T3=6.294369

14=13.85919COST=68.04735

88

, - -

Table 34 shows the specifications of this problem with

the free parameter cptimum values found. Figure B.2 shows

the actual yaw and rudder response.

89

J. .

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

* . . . . . . . . . . . . . _ _-_. . . . . . . . .

TABLE 34

ICSOS OUTPUT

---- SPECIFICATICNS-----

VARIABLES CINITIAL CCNCITIONS:

X4 z .0X5 = .0R = .0YAW =.0TIME = 00

CON~cIAN7S:YAWd = 1.0O0COO0K = *4771'0000C12 11*283300C0TP1 = 6e4699-10CO3TP2 = 53o7931O0COLAMOA = 4s2OCOCC30OFREE PARAMETERS:Kl : CV= .4179S16 LL= oCCY) LL= 1.ooYVY0TI : OVa 53.69S28 LL= 1*~0OO00-O LL= 1C.'0000)T2 : OV= 4*970C29 LL-. 1.OOCOflO UL= !CeOrV)0QT3 : OVa 6o294369 LL= l*(0Co00 UL= 11%,11V0nT4 : OV= 13.85517 L1= 1.COCOOO LL= 2C*3000C

SPECIAL FUNCTICNS:YAWE YAimC-YAh

* DXZ (YAWE-X2)/T2X3 aK1*(T1*0)X24X2)DX4 IX3-X4)/74

* 0 = (T3*CX4+X4)0X5 =(C-X5)/TF1X8 -K*(TZ*DX5+)t5)DR I X8-RI/TP2

DERIVATIVES:D(X2 /D(TIME --

DX 2O(X4 /O(TIME ==

0X401X5 /D(TIME --

DX 5D(R /O(TIME) =

ORD(YAW /fl(TIME ==

R* D(J /O(TIME ==

LAICA*YA WE**2+D**2

OT L: ACTUAL YAW ANC RUDDER RESFCNSETABULATE: TIPE 0 R YAW

AT INTERVAL 2.CCOCfOOOPLOT: 0 YAW

AGAINST: 7IME AT INTERVAL 2.00O')1COO

* END' CALCULATION WHEN TIN'E .GE. 600*CCO

90

............ ~rrr.m~......................rrrr W

*P. £ - P. .E 0

A _____ _________________

~ I *-~~* 4

4. 44

* . I -.

* . a* 9

3 0 4a *a *

4 I* 0* I

* t. .

I 4 *

3 4

I 0 j E

* I..1 4

e~a * .

ml 4 . Em

I 4 . . I

* .4 4 . 4.

* 4. * I I 0

* 4 . I

I -. * U)

* ** 3 4 0

* I 4

a 4 . U1 * .

.~.

0

* I~ 400* 4 MI

1~* 0*

** 0 0

*00

4* .4 a,4. 0 *ii 4 S I 1. 1.4

* I4* 9 .

44.4

* I.44

~; . . .4 * .*44* .4 ~ 4 9 9. -

P. * P.

4 4 0

* -9. 0 . 4 .4 . .4 * 0 0 0

91 ~.

LIST OF REFERENCES

1. Ccmstock, J.P., Principles of Naval Architecture, pp463-489, The Sotiey-oT al -c icEngineers, 1967.

2. Ncmoto, K. and Motoyama, T "Loss of Propulsive PowerCaused by Yawing with aaricular Reference to auto-matic Steering", Japan Shipbuilng ad MarineEngS4neering, Vcl. 120.-,796.

3. Motora, S. and Koyama, T., "Some Aspects of AutomaticSteering of Ships', Japan Shipbuilding and MarineZngineering, Vcl. 3, No.-4-Jull-196.

4. 13th ITTC Repozt of Performance Committee, On theAdded Resistarce due to Steerinq on a Straiqht C6urse,

5. Bech, M.I., "Some Aspects of the Stability ofAutomatic Course Control of Ships" J of 8ech.gngineering Science, Vol. 14, No. 7 1972.-

6. van Amerongen, J. and van Nauta lemke, H.r., OltimumSteering of Ships with an Adaptive Auto _5ti,PMoceezingg- of"-5-5 SI CUtroI ysTem Sympcsium,Annapolis, Md 1S78.

7. Reid R.E. , Improvement of Ship Steering Contrcl forMerchant Shi Es __Phas_l NaEionaI -ariti me -s a rcCeT -Kiins-tcitff,-NZ-ork, October 1978.

8. Kallstrom, C°G. "Et Al Adaptive Autopilots forTankers", Automatxca, Vol. 15, 1979.

9. Astom1 K.J., Desn of Fixed Gain and Adak.iveAutcpilots based on -4-4 5moT-HodI- PFZee1igs-oT3 oi off "I"M--St--n -- u-om-Coutrol, Genoav,

Ia y, pp 225-23 June 1980.

10. Reid, R.E. and Moore J.W., Optimal Steerinj Contrcl ofH igh Speed Con tainershi, Ynai,* .IIontIl 'uI gnPioc 1980 ACC. San Francisco,Ca, pp WA1O-BI, August 1980.

11. van Amerongen, V. and van Nauta Lemke, H.A., Criteriafor ptiamum Steering of Ships, Proceedings or3-posumon naShi tee ring AUtomatic Control, Genova,Imal , June 19eu

12. Kallstrom, C.G. and Norrbin, N.H.q PerformanceCEterja for Ship Autopilots: An Ana jysis oT- _-NI -. o

92

....,,~. ... . .. . . ............ . . , ... .... ,..,.,°... ,.... . . .. . . : . . . . . - . ,,. • ., . , .i:,

Exjeriments, Prcceedings of Symposium on Ship Steering1X51om a-C-ontrcl, Genova, Italy, pp 23-41, 1980.

13. Clarke D., Development of a Cost Function forAuto. iot Optiziza-i-n, -- Proeeain;3f .7ymjoT um onBEIP Se-ring-IMtoiaic Control, Genova, Italy 1980.

14. Garcia V.C. and Thaler, G.J., Comparison of steerinqControl Arithms for Utiaiz-if0tP1oFs-, paper'9UEsentea ar_.SZVentrF£np ConaTro± 54UYss ympojsium,ath,tunited Kingdom, 27 September 198 .

15. Milliken, R.J. and Zoller, C.J., "PrinciplE ofOperation of NAVSTAR and System Characteristics",Navigation: Jcurnal of The Institute of Naviqajtijn,Vol 25, pp 95-T,--N.-2,-3min-TW7B.

16. Nan Dierendonck, A.,J., Russell, S.S., Kopitzke, E.R.and Birnbaum, M., "The GPS Navigation Message",Navi ation: Jcurnal of The Institute of Navilation,Vol 25, pp 147:637o,---7o.-27-umF-T97.- - --

17. Jorgensen, P.S., "NAVSTAR/Globai Positionirg System18-Satellite Ccnstellaticns", Naviqation: Journal ofThe Institute cf Navigation, Vol 27, pp 89-1U07I-r-. 27

18. NASA Technical Memorandum 86248, NASA OceanicProcesses Proqram, by Esaias, W.E., Greaves,--3- .P'U7W7-,-,-- -omas, R.H., Townsend, W.F., kilsonU.S., pp II-11 and 11-18, May 1984.

19. Garcia, V.C., A Real Time Microprocessor Based Di italLead-Lag gZ- .asTer's Resea rc

BE5 t-7niversy -o£ tU-Ural Florida, Orlando, 1982.

93

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

INITIAL DISTRIBUTION LIST

No. Copies

1. likrary, Code 0142 2Naval Postgraduate SchoolMonterey, Califcrnia 93943

2. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22314

3. Professor Georqe J. Thaler, Code 62Tr 5Lepartment of Electrical andComputer Engineering, oNaval Postgraduate SchoolMcnterey, Califcrnia 93943

4. Professor Robert D. Strum, Code 62StDepartment of Electrical andComputer Engineeringonaval Postgraduate School

Saonterey, Califcrnia 93943

5. CDR. Bob Hiem, DENNavy DeptCP-43EWashington D.C. 20350

6. Department Chairnan, Code 62RrDepartment of Electrical andComputer Engineering,Naval Postgraduate SchoolMonterey, Califcrnia 93943

7. 11. Vicente C. Garcia, USN 2P.O. Box 985Artbony, New Mexico 88021

8. Cfficer in ChargeAttn: F. C. GarciaFleet Analysis CenterCode 8234Corcna, Califorria 91720

9. Er. C.H. Wash, Ccde 63 WXDepartment of [eteorolog1Naval Postgraduate SchoolMonterey, Califcrnia 93943

1 10. Professor Hung-Mcu Lee, Code 62LhDepartment of Electrical andComputer Engineerinq,

* naval Postgraduate 2choolMonterey, Califcrnia 93943

11. Attn: Mr. George CurryG.S. Henschell9 Holt DriveP.O. Box 30Newburyport, Massachusetts 01950

94

12. Attn: Mr. Charles WagnerG.S. Henschell9 Hclt DriveP.O. Box 30Deuturyport, Massachusetts 01950

13. iT. Kritsis-Spyzomilios Pericles.:. SMC 1 54, NPSMonterey, Califcznia 93943

14l. IT. Horiano oulcs EmannuelSriC 1863, M PMonterey, Califcxnia 93943

+a

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Documents

OPTIMIZATION OF GUIDANCE AND CONTROL USING … · Vicente Chavez Garcia, Jr. ... 5.8 YAW AND RUDDER vs. TIME (controller C) .. .. 34 5.9 YAW AND RUDDER vs. TINE (state-feedback controller)