The Decoupled Cushion Control in Ride Control Systems for Air Cushion Catamarans

Copyright © IFAC Control Applications in Marine Systems, Fukuoka, Japan, 1998

THE DECOUPLED CUSHION CONTROL IN RIDE CONTROL SYSTEMS FOR AIR CUSHION CATAMARANS

Daniele Bertin*, Sergio Bittanti**, Sergio M. Savaresi** *Calzoni s.p.A., Automation and Control Dept., Via Stendhal, 34, Milano, ITALY. E-Mail: [email protected] **Politecnico di Milano , Dipartimento di Elettronica e Injormazione, Piazza Leonardo da Vinei, 32, 20133, Milano, ITALY. E-Mail: {bittantilsavaresi}@elet.polimi.it.

A well-known limit of air cushion catamarans is the so-called "cobblestone effect", a resonance phenomenon due to the interaction between the ship body and the air cushion dynamics excited by the incoming waves. In this paper the problem of designing a Ride Control System for the attenuation of the wave-induced vertical accelerations of a Surface Effect Ship (SES) is considered. It is shown that the damping of two basic resonant modes can be achieved by two independent control loops . Such a decoupled control scheme exploits the intrinsic orthogonality of the two oscillation phenomena. Copyright @ 1998 1FAC Keywords : Surface Effect Ship; resonance ; decoupling control; identification

I.INTRODUCTION

sea state and the ship speed and it can be shown that the sea state at which the cobblestone effect arises may be quite low. The resonances are within the band related to both the seasickness and fatigue physiological effects on the passengers.

Among the characteristics of the Air Cushion Catamarans (ACC) - also named Surface Effect Ships (SES) (see e.g. Butler, 1985), an important one is that the transmission of the wave effect is made mainly by the air plenum and not directly by the ship hull, possibly increasing the passenger comfort. However, the presence of the cushion is responsible of a resonance phenomenon, called "cobblestone effect", which typically takes place in moderate sea state. In order to reduce the above effect, Ride Control Systems (RCS) for ACC are typically conceived to command a set of vent valve actively controlled in order to stabilize the cushion pressure.

The theory of the acoustic modes has been introduced in S0rensen and Egeland (1995) by means of a modal solution of the linearised boundary-value problem for the spatial pressure variation in the cushion with the seal assumed to be rigid . Such modeling shows a strong similarity with the problem of vibration damping in large flexible space structures. The passivity approach, experienced in that context, has been applied in the design of the Passive Proportional Controller, described in in S0rensen and Egeland (1995); it is related to a Ride Control System that, contrary to previous designs (e.g. Kaplan et aI. , 1981), takes into account also the spatial pressure variation.

The dynamics of the air cushion is characterised by two kind of resonance phenomena: a uniform pressure and a spatially varying pressure term (S0rensen et aI., 1992). The first one ("zero mode") is dominated by the coupling of the cushion lumped volume phenomena with the ship body mass . The second component is related to the pressure propagation along the air cushion (which has generally a prevalent longitudinal dimension) and is termed also "acoustic component". The cobblestone effect depends on both types of resonances. Such resonance modes produce high intensity pressure oscillations when the frequencies of the perturbation (waves) are tuned with the modes themselves. This occurs according to the

The purposes of the paper is to present an alternative scheme for a RCS based on the decoupling of the oscillating modes related to the two main phenomena. The study, developed in the context of the BrireEuram project MAINCOMPSES, considers as target vessel a large and high length-to-beam ACC (Fazzeri er aI., 1997). The control system design is derived by both the theoretical modeling and a black-box identification approach. To this end a sophisticated simulator of a SES (named MACC2), recently

217

developed within the project MAlNCOMPSES (Bertin, 1997), has been used. This simulator has been developed by Riva Calzoni in co-operation with Fincantieri and Ifremer and has been experimentally validated by a scaled model basin tests. Interestingly enough, it can be shown that the control of the two main resonant modes can be achieved by two independent control loops. The controller, preliminarily presented in Bertin et al. (1997b), is reported in the present paper with a more complete treatment including the second acoustic mode and a comparison with the above mentioned Passive Proportional Controller with which it turns out to have many common aspects.

Output (controlled) variables:

Pu: uniform pressure (it IS assumed to be the average along the longitudinal axis of the vessel). p AF:= p A - P F: differential pressure at the aft and the fore of the air cushion. This secondary controlled variable is relevant since it might be responsible of high-frequency (I Hz or more) pitch oscillations.

-P

AIR CUSHION

\I~ U~

The paper outline is as follows. In Section 2 the mode ling of the ACC dynamics from first-principle and from identification is presented and discussed. On the basis of such a model, a decoupled control scheme is designed as reported in Section 3. Its performance, together with the comparison with alternatives controllers, is illustrated in Section 4.

I ~,

rp

-111)011141. -----'.

PAF Fig.l . Input-Output variables Throughout the paper the assumption of collocated pressure sensor (actuator located in correspondence of a sensor) is assumed . The modeling of the process under study has been approached in the two different way that are presented in the following sections. The two approaches allow to perform a cross validation of the models and play a complementary role in the synthesis and analysis of the control system.

2.MODELING THE ACC DYNAMICS The work here described is referred to a specific configuration of the ship, which is representative of a significant set of ACC, especially of large size, and characterized by two actuator groups located at about the cushion ends. The control law is be applied to air flow modulators constituted by conventional vent valves or by the more innovative combination of vent valve with active fan inlet guide vanes. The approach followed in the design of the servo for the coordination of both the type of actuators has been described in Bertin et at. (1997a).

2.1 The theoretical model The theoretical model here considered is based on the one developed in S0rensen and Egeland (1995). In its complete formulation this model takes into account the infinite modes of acoustic oscillations. The encountered wave excitation can involve only a limited range of frequencies . Moreover, the validity of the model for higher modes may become questionable since for high frequencies a 3D modeling may be advisable. For these reasons we will focus, in the sequel, on a finite dimensional modelisation. Precisely, only the first two acoustic modes are considered. Indeed the analysis based on the first two acoustic modes is considered to be fair enough for the present target vessel. A corresponding state-space model can be directly derived with the standard structure:

The target vessel is a large ACC with a mass of about 5000 t and a cushion length Lc=145 m. For that ship, it has been shown that the acoustic modes could not be negligible (Bertin et al., 1997c). The I/O variables we consider (sketched in Fig. l) are the following: Input (control) variables: q A' q F: net air flows delivered by the fan-valve

systems at the aft (A) and at the fore (F) of the air cushion, respectively. They reflect the optimal actuator configuration, in order to compensate the two principal resonant modes. Notice that the real control variable are in fact the blades angles of the IGVs and valves; however, such primary control inputs are governed and co-ordinated by a servo system (see Bertin et al., 1997a) whose main role is to reduce the power absorption required at the fan shaft. Moreover, notice that (due to technological reasons) the aft fan and the associated vent valve are not exactly located at the rear extreme of the cushion, but at a distance XA about 1/5 of its length. As it will be shown, this fact has a non-negligible effect onto the RCS control law.

X= A {

x + B u + EX

(1)

y=Cx

with state vector:

x(t) = [173 175 iJ3 iJ5 Pu PI P2 /11 /12 the

input

u(t) = [:;

and

1

y(t)=

output

[~uF

1

vectors

r'

respectively

the encountered wave disturbance input X. The state variables 113,115 are respectively the heave position,

218

the pitch angle, ~I' ~2 are respectively the mode I

for the modeling of the air cushion dynamics part and is characterized by a finite-volume representation of the longitudinal cushion dynamics . Precisely the representation of the cushion pressure dynamics is able to reproduce the propagation by means of an arbitrary number of discretized volumes . In the present case 5 volumes have been considered. A detailed modeling of the seals (Bertin et at., 1998) is obtained from scale model experiments data.

and 2 time varying pressures. -The expressions of matrices A, B, C, E (reported in Bertin and Savaresi, 1997), cannot be included due to the limited space available. This model presents the following differences with respect to the model presented in S0rensen et at. (1992). - the control signal u is in term of air flow instead of vent valve area and consequently the uniform pressure damping term in A doesn't include the component due to the actuators;

The following 2-inputsl2-outputs dynamical system has been considered as reference:

p,,(s)= F;!(S)qA(S)+ F!c(S)qF(S) { PAF (s) = F2! (s )q A (s) + Fc2 (s )q F(s)

- the uniform pressure is considered as accessible variable and therefore included in y. Note that this imply that the matrix C cannot be factorised as C= BT P with P diagonal, a representation which was most useful in the stability analysis in S0rensen et al. (1992).

In order to estimate the model (4), the system (namely the MACC2 simulator) has been excited on q A and

q F (around a suitable equilibrium condition) using two independent, zero-mean, band-limited white noises, and the corresponding output signals has been recorded. The parametric identification has been done using a standard Least Square technique (ARX models) . The Bode diagrams of the estimated transfer functions are depicted in Fig.2. Precisely in Fig 2a (modulus and phase) FII and FI2 are given, whereas in Fig 2b (modulus and phase) F22 and F21 are given. The order of the estimated transfer functions is 2, which revealed to be enough for an accurate fitting of the I/O signals. The inspection of the Bode plots reveals the following peculiar features of the dynamical

For the controller design a simplified model is considered. Some minor effects are neglected to put well in evidence the main structure of the process to be controlled . These include the effects of the hydrostatic restoring forces (much weaker with respect to the cushion lift) and the effect of the pitch velocity on the acoustic modes. In the end one obtains the following Laplace transform representation.

p" (s) = F;Js XqA (s )+qAs)) {PAF(s) = F;(s Xa,qA (s )-qAs ))- F;(s X~qA (s )+qF(S))

(4)

(2)

relationships between

{q A' qF}

and

{Pu, PAF } . First

of all the first resonant mode (zero-mode) at about 0.7 Hz (mainly affects the mean pressure Pu) and the acoustic modes I at 1.2 Hz and 2 at about 2.4 Hz. Secondly it can be seen that FII is approximately coincident with FI2 in the frequency band of interest for the mode 0; therefore one can simply write

Pu(s)= FO(S)[qA(S)+qF(S)] with

e

-

I

RL

(X

1

The analysis F21 (s) '" -0.63·

PeOVeO qfo . 2pco

= ~~ aIr leakage conductance

l1~n = m +

c

'" FI2 (s).

This is a confirmation of

of

Fn(s)

F22 and F21 shows that is in the frequency band of

interest for the mode I; therefore we can formulate equation (4) as follows:

Wwo

n = cos -L x A''

FII (s)

the structure of the equation (2).

where (see Bertin and Savaresi, 1997 for a complete list of symbols):

C =

Fo(s) =

(X,

• -

PAAs) = ~(s)[0.637A(S)-qAs)]- ft;(s lO.637A(s)-qF(s)]

2n = cos-x A

where

Le

PI(S)= F22 (s) and P2 (s)

(5)

takes into account

the residual effects relevant around the mode 2. It is interesting to observe the structural correspondence of the above equations (5) with the first-principle model (2) where parameter (x, should be selected as (X 1 = 0.63 . This value of (x, corresponds to a

A"

2.2 The Input-Output identified model In order to validate model (3) a black-box identification procedure applied to data collected by the above mentioned MACC2 simulator has been used. The simulation model is completely non-linear

position of the actuator XA = 0.283 Le = 40.5 m that matches quite weII to the position in the simulator.

219

a)

1(1'

10·'

10'

,0'

conside red as the deeoup ler at the input variable s of the system. It is worth noting that the effect of the second term in the right hand side of (3.2b) can be reduced by a suitable position ing of the aft flow qA in the RCS . The value of XA that nullify the coeffici ent a in (3.3), so that only input Uj affects (7), turn out to be XA =LJ3. The validity of the adopted approxi mation will be tested in section 4.

Frequency (faO-sec)

400

200

O' --'-~~ ,O-., -'--~i.i.U."'-"f'~-'--~CU,'" -"----'---'--'.i..i..Cii .2~)OL.,---'-.....i..i~.i.U,OFreQuency (rad'sec)

On the basis of the estimat ed model (7) , it is now easy to design a simple but effectiv e control scheme for the attenuation and dampin g of the first two resonan t modes of an ACe. In accorda nce with the approac h we have conside red a simple proport ional controll er as shown in Fig.3, where po", is the nomina l air cushion overpre ssure. The overall control law, named Decoup led Cushion Control (DCC) , is represe nted by:

b)

u=

HC[P~I - Pu]

(9)

PAF

with

H=rl: , a,K"~ Fl eQuency(ra~sec )

I+a, (Kern and

1r~

I KAF = I+a, I+a, -KAF

I+a,

l+a,

1

I diaJK l+a,

2

1(10)

K

&."

AI'

l+a,

KeAF are the tuning knobs of the controll er). A

Ffequercv(radlsec )

F

Fig.2. Bode plots of F iJDECOUP LER - Eq . (31 )

3.THE CONTR OL SCHEM E The speeial struetur e of the obtaine d model (see (2) or, equival ently , (5» suggest s to define the following two new control inputs

Fig.3. High-fr equency Decoup led Control ler. This control scheme is appeali ng due its simplic ity. In addition it is transpa rent and it makes possible to handle separate ly the two class of resonan t modes: one for the mean cushion pressur e and one for the spatially distribu ted one. Precisel y , since the cobbles tone effect is to mainly due to the uniform pressur e Pu, a high proport ional gains Kern is advisab le to compen sate for the associat ed heave oscillati ons. The role of the second loop is that of slightly damp the first acoustic resonan ce, leaving , in case, to the fins-bas ed control loop the task of reducin g the pitch oscillat ions. The gains Kern and KeAF can be selected by a trial and error method based on simulat ion tests or, alternat ively, by a root locus approac h on the base of the theoreti cal model.

(6)

ll () (t).::LJ F(I)+qA (r) { 1I (T)·-a LJ,,(t)- q F(t) I

1

so that the model takes the form (7)

where 112

= -LJ F -

a 2q A = a· LlO + b· Uj

•

(8)

Note that the cross-te rm F2 turns out to be compar atively small. It is obvious that, if such a crossterm would be neglecte d , then (7) would be constitu ted by two non-inte racting subsyst ems and therefor e there is the possibil ity to simplify the control design task. In this sense, equatio n (6) can be

220

4.SIMULA TIONS AND COMPARISONS

A first analysis can be referred to the theoretical model (I) . Both the controller are able to select the damping of the first two modes (zero and acoustic one) on the base of the proper selection of the two gains. A characteristics of the DCC is that the damping of the two modes can be made easily the acting separately on the gains. In fig . 5 the root locus obtained by augmenting Ku and with KAF=OA is shown. The open-loop poles of the modes from 0 to 2 are close to the Imaginary axis at increasmg frequency . Only the closed loop related to mode 0 moves to the negative real axis .

The effect of the spatial pressure control loop of the DCC is put well in evidence in the following figure representing two simulations in the same sea states with controller parameters: a) Ku = 10, KAF = 5; b) Ku = 10, KAF = 0 (without acoustic mode control loop). The controller b) consists basically of a traditional RCS law and doesn't take into account the spatial pressure aspect. a)

Oecoupled Control

·5

·1 0

." -30

I (Hz]

b)

Fig. 5.

-25

-20

-15

-\0

·5

Root locus with KAF=OA, xA=O. 1 Lc. (The open loop poles are indicated with 'x ' ) .c. _ ' /t'

f-------...:' 1·; ,:.. ~'" "

",

..

r " .

Fig. 6. I [Hz]

FigA. Vertical accelerations on the ship with DCC (, vs. a uniform-pressure controller (b).

:.":"-.: " ;, , '

. .... ( . '"

,

Root locus for increasing gains for xA=0.28 Lc.a) PPC, b) DCC.

j

'" >, . f·

From the plots of Fig. 4 it is evident the effect of t~ uncompensated acoustic mode 1 on the vertic; accelerations.

0'---_............. ·.·· 1

In order to discuss some of the features of t~ proposed control law a comparison with the simpi robust proportional controller proposed in S0rense and Egeland (1995) and termed Passive Proportion; Control (PPC) hereafter. The similarity between the two control laws is very high being based both on a simple proportional feedback of cushion pressures. Contrary to the DCC the PPC control scheme is based on a set of ·individual loops that control the local pressure. In the present case the order of the controller is two according with the number of independent controlled air flows; consequently it has two gains to be selected.

Fig. 7.

' ...

~ ~.~.

",

"

'.' : ;",.~ ". -

Root locus for increasing gams for xA=O.1 Lc.a) PPC, b) DCC.

One interesting aspect related to the above simple controllers is the behavior of the non controlled modes (in this case we restrain to the second mode) . As the gains increases (assumed the same value) the mode zero and one poles approach the real axis while the

221

second mode poles return toward the Imaginary axis (tending to the open-loop zeros). The results obtained with the two controllers, shown in figures 6 and 7, are anyway different. In particular the resonance frequency of mode 2 becomes significantly lower in the case of the PPC, approaching the mode 1 frequency if the controlled flow is located more afterwards (see fig.7) .

groups located close to the cushion ends. Using two independent modelling procedures (a black-box identification and a first-principle-based approach) it has been shown that a SES is mainly characterised by two resonant modes, which are linked to the two control variables via a simple relationship. By exploiting such a relationship, it has been shown that the two main resonant modes can be treated independently, using a simple algebraic decoupling law. Simulations with alternative controllers shows interesting properties when applied to the present target vessel.

This fact is reported to explain, at least partially, the results of simulation shown in figure 8. They refers to the mean pressure PSD (acting directly on the heave acceleration) with different values of controller gains . What is obtained is a decreasing pressure spectrum with the DCC as the gains increase. On the contrary the PPC shows an optimal value of the gain above which a resonance peak appear and augment with the gains. The frequency of the resonance, that corresponds to the one derived from the theoretical model (see Fig.6) , is low enough to be excited by the encountered waves .

REFERENCES D.Bertin, S.Bittanti, S.Savaresi . "Non-Linear Design of an IGV-Fan Servo System for Surface Effect Ships". NAV&HSMV'97 1nl. Conf., Naples 18-21 March 1997. D.Bertin, S.Bittanti, S.Savaresi . "Control of the Wave Induced Vibrations in Air Cushion Catamarans". CCA ' 97, 5-7 OcLI997, Hartford (USA) .

Mean Pf9 ssur e Oecouphng conI/cl sl1ategy

4000

A

3500

,

\

11

3000

2000

!

D.Bertin. "The SMACC2 Simulation Model of the Ship Dynamics for the Ride Control" . MAlNCOMPSES Report TEC1411D90510I132. Riva Calzoni, 1997.

i :

l

\:

i

\1

I

D.Bertin, S.Savaresi. "Study on the RCS Control MAINCOMPSES Report Algorithms". TEC1431D911l01lI8. Riva Calzoni, 1997.

I I

i

/

1000

I

: I

\

I

1500

!

I

I

I

2500

i

h i 11 ~~\ ~

S.Bittanti, S.Savaresi, D.Bertin. "A Neural-Network Simulator for the 3-Lobes Stern Seal of a S.E.S.". Intelligent Automation and Soft Computing (1998).

;\=5

5()()

0

05

-"--.---- i

1.5

E.A.Butler. "The Surface Effect Engineers Journal , 1985 .

25

Frequency (Kz) 3

Naval

' eanp ra ssure · PPC M

<10

5

K='/

I

Fazzeri R., Massa L. , Saione S.. "Outline Specification of MAINCOMPSES Target Vessel" . MAlNCOMPSES Report TECllllAOOllOII06. Fincantieri, 1997.

i I

2

5

ii

'I

!

K",5

A.J.S0rensen, O.Egeland. "Design of Ride Control System for Surface Effect Ships using Dissipative Control". Automatica, Vol.31 No.2, pp. 183-199, 1995 .

i

/\

5

A.J .S0rensen, S.Steen, O.M .Faltinsen. "Cobblestone Effect on SES". Intersociety High Performance Marine Vehicle Conference-HPMV'92, American Society of Naval Engineers, Washington D.e , 1992.

i i

/11

1

1f t r~W f;-\

~

0

Ship".

0.5

15

25

Freque1'lCy(Hz)

Kaplan P., Bentson 1., Davis S.. Dynamics and Hydrodynamics of Surface-Effect Ships. SNAME Transactions, Vo1.89 , 1981,211-247.

Fig. 8. Controllers with different gains vs . mean pressure.

5.CONCLUSIONS The problem of designing a Ride Control System for a large air cushion catamaran has been considered. The configuration of the actuators is characterised by two

222