Linear optimal feedback control design of a surface piercing hydrofoil craft in longitudinal waves

(1)

ThCI-fl4OLOGY REPORTS OF THE OSAKA UÑ][VRSITY Vol. 42, No 2122 (October, 1992) pp.399-410

(Received April 11, 1992)

Linear Optimal Feedback Control Design of

A Surface Piercing Hydrofoil Craft

in Longitudinal Wavés

Masami HAMAMOTO and Hiroshi YAMAKAWA (Department ofNaval Architecture and Ocean Engineering)

Abstract

This paper is concerned with the dynamics and control of a surface piercing hydrofoil craft supported by surface piercing fore fOil and fully submerged aft foil and eqúipped flaps to each of fôils. A hydrofoil craft designed this way is expected to keep self-stability and have an efficiency of reducing oscillatory motions in longitudinal waves. Optimal feedback control method has been applied to this type of hydrofoil craft and linear systems analysis has been carried out. Simulation results have indicated the fèasibility of gxeatly decreasing the longitudinal motions within the limit of flap angles to be required.

1. Introduction

Hydrofoil crafts, in generai, are divided into two types by means of shape of the foil. One is surface piercing t'pe, and the other is fully submerged type. Surface piercing type has an

advantage of self-stability at the running in waves, while fully submerged type must be

equipped so with much costiy controller to stabilize. However, oscillatory motions of sur-face piercing hydrofoil crafts due to waves are relatively higher than that of fully submerged

hydrofoil crafts because of damping and restitutive forces generated by V-shaped foil.

Accordingly, surface piercing hydrofoil crafts are restricted to use in a calm water such as a lake.

Applying flaps to each foil of the craft supported by surface piercing fpre foil and fully submerged aft foil can be considered thàt the craft system can still keepself-stability and

controllability in waves. It is difficult, however, to design the foil and the cdntrol properties at the running condition because the range of possible vertical motions is extremely narrow around free: surface.

There are three important parts to successfully design the controlled syste.m of this hydro-foil craft. They are as fôllows:

designing non-controlled hydrofoil craft itself, determining the feedback contrOl gain,

evaluating the maximum angles of both fore and aft flaps. The flow chart of this procedure is shown in Fig. 1.

To keep the system controllable the hydrofoil craft model investigated in this papér has

been so designed that the center of mass is moved forward and the project area of aft foil is 399

//

TECHNISCHE

Laborum voar

6thmmetha

Archief Mekelweg 2,2628 CD DeNt

(2)

400 M. HAMAMOTO and H. YAMAKAW

(Staticbalance) (Dynamicbalance)

Equationsofmotion

vectrdêörFptiôn

Riccati'seqiiition)4-<tfwlCtiOri

Feedback control gain

AnaJys of fre4úency response

(Maximum angles of flaps)

N

Yes

p)

Fig. 1. Procedure for design of controilled flaps.

made smâll. Thus, the damping force cán be decreased and the hydrofoil craft can take full advantage of contrcl efforts. The general arrangements of this model are shown in Fig.2.

To determine the optimal feedback control gain, the mathematical model of this hydro-foil craft in longitudipal waves has been: clarified. The gain has been defived with use of

optimal feedback control method by solving Riccati's equation.

l'o evaluate the maximum angles of flaps linear system analysis has been carried out.

o,

I-Principal Dimensionsof Modcl

Fig. 2-. General arrangements of the model of hydrofoil craft.

Slii lèñgth - i .5in

l)is(ance Irorn fore foil to aftföil

10m Breadth 0.4i1i

Dcpth .. 0.2ii

l)iiplaccíacnt I l5kf

(3)

2. Equations of Motion

The space fixed axis and the body fixed axis are defined as shown in Fig.3. The equations of motion concerned with heave and pitch öf the hydrofoil-craft in longitudinal waves under the assumptions of small pitching motion and constant forward Velocity, U, are described as

UÜ )= AZ,

I =AM,,

(1)

where m is mass of the craft and I, is the moment of inertia. U

AZ

Öptimal Feedback C nttol of a Hydrofoil Craft in Lòñgitúdinal Waves 401

-

P U Caa A1 a

mZf(lf)

pn±a(ì+la)

+mj

±mza i

AM =

jP U2 CLaf(AfAaf + AAfaf)If

P U Ciao Aa A aa l

+1fl lf(WlfO) mzaia(iì±1a)

- mzf If wf ± mza la twa,

Fig. 3. Coordinate systems.

u2 CLaf(AfAGf.± AAfaf)

>

A.Z apd A M consist Of force or momentactng fore fOil añd aft foil, acting to added mass of each foil, and generated by wave particks arOund fore and aft

fils, so that

(4)

402 M. HAMAMOTO and H. YAMAKAWA

Table 1. List of Terms.

E, space-fixed coordinate axis X, Z body-fixed coordinate axis

heave of hydrofoil craft

O pitch of hydrofoil craft

heaving deviation from free surf äce «oil bone)

0o. trim angle (föil boñe)

u, w : translational velocity along the X and Z axis respectively U : ship speed

L : ship length

m : displacement

mzf : added mass of fore. foil

m0 added mass of aft foil moment of inertia

distance from center of mass to fofe foil along X axis 'a distance from center of mass to aftfoil along X axis

h1 : average, strut length of fore foil ha : strut length of aft foil

T : dihedral angle offôre foil C : chord of fore: and aft foil

f

: average of submerged depth of fore and aft foil (foil bone)

fj : equivalent submerged depth öf fóre foil

Aj project area of fore foil (foil bone) prOject area of aft foil

attack angle offore foil (foil bone) attack angleof aft foil (fóil bone)

aOf : zero lift angle of fore fOil

zero lift angle of aft foil

a : down wash of aft foil

CLCJ : lift coefficient slope of fore foil

CL.a : lift coefficient slope of aft foil ôj : flap angle of fore foil

flap angle of aft foil coefficient of fòre flap coefficient of aft flap wave height

velocity potential of wave amplitude of wave phase velocity of wave wave number wave length wave frequency encOunter frequency gravitational acceleration time a c k : : À : 'w : g : t :

(5)

Optimal Feedback Control of a Hydrofoil Craft in Longitidinal Waves 403

b

where Af, Xf, and a are written as

2(

eG e wf)

tanr

afO + Wif 8 wj

Vf +Kof ôf,

e'

±Kaôa5

u

ax

(4)

and each of coefficients is shown in Table 1.

The relatiçnship of velocity in the directiOn of Z With velocity in the direction of is

wG+U8.

Substituting Eqs.(2), (3) and (4) toEq.(1) using Eq.(5) makes

a

c +a2G +a3G ±a48

+a58 +a68 =f(t)+K1(t),

+b28 +b30 +b4G +b5G +bG =g(t)+Km(t),

Where

a1 =m+mç

2 Ciaf4f + CLú0Aa'

a2=--PU

_LI i 2Ca1

a3---PU2CLaftF

a4 = (mzflfmzala),

a5

(mzf+mza)U_!PU2

CLAfPCLaaAaIQ

a6 P u2 (cLÎA! + CL0aAa CL Caf If)

_tant'

b1 = (ii +mzf If +mza

1a2),

b2 = ('ñzflfPP!zala) u

!p

u2

CifAfli_CLaaAaI

2 U.

b3 =PU2 (CLafAfif_CLaaAaia

CLa

Cafi

tant' J '

b4

mzf if m i,

u2 CLafAf if CLaaAa la

(6)

i

2Cal

Laf

tanr

f(t) and g(t) represent external force and moment as follows:

f(t) =

a7 +a8 1+a9 wa +aiô +a11 +a12 e +a13e

g(t) =

b7 ±b8 +b9 +b10 +b11 +b12 -k +b13e , (8) where i 2Ca a7 -i-P U2 CLQJ

tant'

1 2 CLaIAf

a8=--PU

1 2 CLaA0

a9=-PU

ù

alo =mzf, all m47,

a,2 U2 ÇLjAf, : b12 = U2 CLfAf If,

a13

=

U2Cjaa Aa, b13

=

U2 CL.aoAa 1a

Kj(t) afld Km (r) denote force and moment generated by both flaps as follows: Kf(t)= Cf16'f+ C12à'a, Km(t)= C1fl1&f+ Cm,a, (10) w he re

i

c1 = - -j P U2 Cf A1K,

i

C12 jPU2CLaaAaK&

=

-

u2 CL4f I K,

i

Cm, P U2 CLaa A la K,

(il)

bl(j =mzf If,

b7

=

jP U2 CLO! 2Caf if.

tant'

bs=pU2

CLf4fif

2 U b9

=

-1--pU2

aAala

2 U b11 mzala, (7)

and ôj and 8a are angles of each flap. Also, the values of K81. and K8 are shown in Táble L

(7)

Usng these vectors, the feedback contr011ed system of the hydrofoil cra.ft can be derived from the equations of motion for heave and pitch obtained in the previous section as fOl-lowS:

xAx +Bu,

y=Cx,

where

/

Optimal Feedback Control of a Hydrofoil Craft in Longitudinal Waves 405

3. Linear Optimal Feedback Control

To apply linear optimal feedback control the system has to be describedwith vectors. The

state space vector, x, the output vector, y, and the control input vector, u, have been takçn

as

X-=

¡1= 8f j F (12)

and the mätr x, F, denotes feedback control gain matrix.

The feedback control gain to be obtained is the solution of Riccati's equation and it can be derived from the condition that the cost function shown below is taken the minimum

value. The cost

functIon, J, is . -(.)Q,,(t) + uT (r)Ru (t) Jdt, (15) q1 o b 0 a1 o b4 O

.0

O i 0 O O

i

0, O

Ï

0 a4 o. b1 O a4 O bi O O Ö O o O

i

O O O i O

i

u=

/

'

/

-- Fx,

1

1 /

a2

.1

b5

i

I Cf O Cm1

\

O

a3 a5

0 0

b6 b2

O

i

O Cm2 O

a6

0

b3

O

/

(13) (14)

(8)

r

where Q and R are veight matriceS and have to be taken as symmetrie matrices as follows:

Each element of matrices may take arbitrary vàlueS, bût determines the contr011er prop-rty and/or system propeprop-rty. Thus, the appropriate values are given by trial and error.

Riccati's equatiOn is given as

PA +ATPPBRBTP+cTQC=0.

(17)

Then, the feedback control gain matrix is Obtained using the solution, P, as follows:

F=KIBTP.

₍₁₈₎

4. Linear Systems Analysis

In the previOus section, the optiÌîiàl feedback control gain has been obtained. TO analyze this system in frequency domain, the transfer functions of the system need to be obtained. the configuration Of block diagram f this feedback controlled System is shown iñ Fig.4. Gs

(we),

G, and

K ( We )are all frequency transfer functions. Each of these transfer functions

represents the Fourier transfer of the ecjüauöns Of mOtion for the hydrofoil craft, for the

controller, and for the feedback control gain, respectively..

Fig. 4. Block diagram configuratiOn of feedback controlled system.

Each of the output signals is described as

¿I (W)= Gs (We)(We),

X(We)=c(we)+d (We),

C(We)=GFAU(we),

(19)

where A notation means Fourier transfer, and c represents force and inment, from flaps

and disdisturbances.

The overall transfers function for y as output and d as input, and u as output and d as QW O O Qe R O O R (16)

(9)

Optimal Feedback Control of a Hydrofoil Craft in Lontudinal Waves 407

input are as fôllows:

n

n -1

y(cue)= [I+GS(We)GFGFK(We)] G5(We)d (We),

(We) = - [I+ K (We)G5(We)_{GF J} _{K (}we) G5 (0.'e)d (°e). (20)

5. Simulation Results

Simulation tests of frequency response in regular longitudinal waves have been carried

out. Three cases of controlled system and non-controlled System are testified. Each value of weight matrices for each case is shdwn in Table .2. Frequency responses of heave and pitch in heading and following sea are shown in Figs.5 to 10, and oscillatory motions have been greatly reduced, especially, In low encounter frequencies. Also, the response of both flaps

and disturbances are shown in Figs.11 to 12. From the response of flaps, the. maximum

angles are evaluated 15 degrees and meet the restriction of arising stall ät 17.5 degrees. Tablè 2. The diagonal components of weight matnces.

-

Contr.oUed Cane j Controlled Case 2 Controlled Case 3 ° ° ° Experimental data (Non-controlled) 00 2.0 4.0 6.0 8.0 10.0 A/L

Heave response l.a heading sea

X X

00 2.0 4.0 6.0 8.0 10.0

AIL Pitch respônse in heading sea

/ à 3.0 Controlled Case j Controlled Case 2 - Controlled Case 3 X X X K Experimental data (Noncontrollrd) AIL Heave response in following sa

-X

00 10 2.0 3.0 4.0 3.0

AIL Pitch response in following sea Fig. 5. Heave and pitch response in heading sea. Fig. 6. Heave and pitch response in following seai

COntrolled case i Q Qe = = 1.00 1.00 R1 = 0.01 R0 = 0.ÓÓi Controlled case 2 _{QG =}

Q =

1.00 1.00 R R0 = 0.005 0.001 Controlled case.3 _QcG

Qe =

1.00 1.00 R0 =

k1ò.óoi.

0.001

8/k a

1.0 Controlled Case I 3.0

-

Controlled Cftae 1

-Controlled Cale 2 Controlled Case 2

0.8 Controlled Case 3 2.4

-

Controlled Case 3

xx" Experimental data x o o o Experimental data

(Noncontrolled) (Non-controlled) 0.6 1.8 X 0.4 1.2 _X X 0.2 X 0.6 2.4 1.8 1.2 0.6

(10)

408 M. HAMA OTÓ and H. YAMAKAWA 5f(deg) &(deg) 5(deg) io.ó - Controlled Cao i Controlled Cace 2 - Controlled Cace 3 a - 0.025 a elk g 0.05 - Controlled Cane I 0.04 --.-- Controlled Cace 2 - C6ntiollrd Cane 3 a Q.025a 0.03 0.02 0.00 2.0 - 4.0 6.0 8.0 10.0

AIL

Pitching acceleration in heading sea 0.0 2.0 4.0 6.0 8.0 10.0

AIL

Vertial acceleration in heading sea

00 1.0 3.0 3.0 4.0 5.0

AIL

CoStrolled fare flap angle in foilciing Sea

&(deg) 20.0 16.0 12.0 80 4.0 0.0 10 2.0 3.0 4O 5.0 AIL Venial acceleration in following sea.

elk g - Controlled Cane 1 Controlled Cane 2 - Contlled Cane 3 a 0.025. - Controlled Cane 1 Controlled Cane 3 - Controlled Cane 3 a-0.025m 00 0.0 2.0 3.0 4.f 5.0 AYL

Pitching accéleration In following sea

Fig. 9. Vertical and pitch acceleratjon in heading, Fig. 10. Vertical and pitch acceleration in following

sea. sea. 30.0

-

Controlled Cane I Controlled Cane 2 Cane 16.0 Controlled 3 a 0.025 a 12.0 8.0 4.0 00 2.0 4.0 6.0 8.0 10.0 00 10 2.0 3.0 4.0 5.0

AIL

AIL.

Còntrôlléd aft flap éngle. In heading sea CoStiolled 5f t flap angle In follawing sea

Fig. 7. Controlled fore flap and aft frap in heading Fig. 8. Controlled fore flap and aft frap in

follow-sea. ing sea.

0.0 2.0 4.0 6.0 8.0 10.0

A IL

Controlléd fore flap angle in heading sSa

/ g ó.5 ¡

-Controlled Cace 1 0.4 Controlled Cane 2 - Controlled Cane 3 a 0.025a 0.3 0.2 0.1 10.0 8.0 6.0

k

4.0 2.0 1.0 0.8 0.6 0.4 0.2 0.10. - Controlled Cane L 0.08 Controlled Cane 2 - Controlled Ckne 3 a - 0.025 a 0.06 0.04 0.02

(11)

Optimal Feedback Control ofâ Hydrofoil Craft in Longitudinal Waves 409

A IL Amplitude of wave force anA moment

0.0 2.0 4.0 0 8.0 10.0

Phase angle of wave fbroe and moment

(kg br kg-m) 20.0 16.0 1.2.0 8.0 4.0 00 LO 2.0 3.0 4.0 5.0 AIL Amplitudè of wave fOrce and moment

(deg) 200.0 100.0 a 0.025. zw M. zw M AIL 3.0 4.0 2.0 6. Concluding Remarks

The dynamics of the hydrofoil craft supported by surface piercing fore foil and fully sub-merged aft foil in longitudinal waves have been clarified and the control of the craft using

linear optimal feedback control method has been successfully applied. It implies that the hydrofoil craft system designed this way can be controlled successfully in longitudinal

waves.

-References

M. Hamamoto, T. Enomoto, and H. Manabe, "Dynamics and Control of Surface Piercing Hydrofoil Craft in Longitudinal Waves proceeding for The Kansai Society of Naval Architects, Nov 1991 Sperry Piedmont Company Division of Sperry Rand Corporation, Virginia, "Final Raport, Autopilot System Study for Hydrofoil Craft," Aug., 1962.

M. Oka et al., "Active Control Systems of Hydrofoils," Control of Marine Vehides, Sixth Manne Dynarrics Symposium, The Society of Naval Architects of Japan, Dec. 1989.

B V Davis and G L Oates, Hydrofoil Motions in a Random Seaway" Fifth Symposium Office of

Naval Research Department of the Navy, p611, Sep., 1964.

J A Keunmg A Calculation Method for the Heave and Pitch Motions of a Hydrofoil Boat in Waves I. S. P., 26(302), p217, Oct., 1979.

G F Franklin et al Feedback Control of Dynamic Systems Addison Wesley 1986 G. F. Franklin et al.,. "Digital Control of Dynamic Systems," Addison Wesley, 1990. R. N. Bracewell, "The Fôurier Transform and Its Applications," McGraw-Hil!, 1965

D.McRúeret al., "Aircraft Dynamics and Automatic Control," Princeton University P-ress, 1973. S. Lefschetz, "Stability of Nonlinear Control Systems," Academic Press, Inc., 1965.

li) (1 W. Housner and D. E. Hudson, "Applied Mechanics Dynamics," D. Van Nostrand Company, Inc., 1959.

12) C. L. Liu and J. S. W. Liu, "Linear Systems Analysis," McGraw-Hill, 1975.

Fig.l1: Disturbance in heading sea. Fig. 12. Disturbance in following sea.

a 0.025.

-100.0

-200.0