On the robust design of motion regulator for foil-catamaran in irregular waves

On the Robust Design of Motion Regulator

for Foil-Catamaran

in Irregular Waves

Key-Pyo Rheel, (M) and Sirn-Yong Lee2

ABSTRACT

The motion control of Foil-Catamaran is considered for the sake o,f enhancing the ship mo-tion performance. An advanced design technique oj the ship motion regulator is introduced

and verified bg theoretical calculations and experiments. _{The design technique is}

devel-oped uging the frequency domain analysis based on the spectral analysis of the ship motion in irregular waves, and offers easier and effective methodology to obtain the robustness of the motion regulator. Experimental results show the enhanced seakeeping performance by virtue of the motion regulator and good coincidence with the theoretical calculations.

INTRODUCTION

In recent works, there have been many reserches on the attitude control system for high speed marine ve-hicles, because which is highly demanded for the sea-keeping performance of the future high-speed marine vehicles. In this study, an advanced design technique of motion regulator for foil-catamaran is introduced and verified by theoretical calculations and experiments.

Ship motion may be described by integro-differential equations in time domain using superposition integrals with impulse response functions. In general, the su-perposition integrals are reduced to convolution inte-grals within linear ship motion theory (Kang and Gong, 1995). These time-domain equations of motion make for some complications and difficulties in control system design for ship motion regulation. But these compli-cations and difficulties can be avoided by the frequency-domain analysis. Basic idea is that the spectral anal-ysis method may be useful for estimating the ship mo-tions in irregular waves with regular wave results (Price & Bishop, 1974). And it is also true that a ship motion regulator would preserve robustness, including robust stability and robust performance, in irregular waves, provided that the control system ensures robust-ness with respect to all regular wave components of

the irregular waves. The controlled system dynamics in irregular waves can also be described by superpos-ing controlled ship motions in regular waves, as well as the uncontrolled ship motion dynamics. Under this assumption, named the assumption for ship motion

reg-ulation by authors, the motion regulator can be designd

based on frequency-domain analysis.

In frequency-domain or regular waves, the equations of motion can be reduced to equations of mass-spring-damper systems with frequency dependent coefficients. The frequency dependencies of the hydrodynamic coef-ficients still trouble us in control system design. Nom-inal plant for the ship motion dynamics is prescribed by a linear time-invariant system for the sake of facility in control system design. The frequency dependen-cies are treated as modelling uncertainties, then the de-sign problem, under the assumption for the ship motion regulation, resolves itself into the problem of obtaning the robustness with respect to model uncertainties due to variations of the frequency-dependent system coeff-icients. The nominal plant may be prescribed so that the uncertainties could be as small as possible.

Only heave and pitch motions in head seas are con-sidered. Linear feedback system will be constructed with controller and estimator of constant coefficients.

1Proffeesor, Dept. of Naval Architecture and Ocean Engineering, Seoul National University, Seoul, 151-742, KOREA

‘Research Specialist, Institute of Engineering Science, Seoul National University, Seoul, 151-742, KOREA

The controller is designed by LQ (Linear Qudratic) control theory and the estimator is done by low-pass filter in this study. The control gain and the cut-off frequency of the low-pass filter are determined by per-formance and robustness analysis.

The efficiency of the control system and the valid-ity of the design method are examined by theoretical calculations and verified by experiments.

SYSTEM MODELLING

Figure 1: Inertial reference coordinate system

By virtue ofthe assumption for the ship motion

regula-tion, the frequency-domain analysis comes in useful and

the equations of motion in regular waves are considered here. As we consider heave q3 and pitch qs defined by the coordinate system shown in Figure 1, the coupled-equations of motion in a regular wave can be expressed in the form (m+ ass)ijs + bssqs + Csaqs +assijs + bssqs + c35qJj= F3, (1) a53ij3 + b53fi3 + C53713 +(155 + abb)ijb + bss~s + cshqs = Fci,

where, m and 155 are the mass and the mass moment of inertia of the ship. aij, bzj and cij are added masses, damping coefficients and restoring coefficients respec-tively, in which subscripts i and j indicate the direc-tions of the force and motion respectively. And F3 and F5 are wave exiting forces. The hydrodynamic coefficients and exciting forces are summarized in Ap-pendix A,

Let the state and control input vectors be

x = [7j3 ?j5 ?73q#,

u = [Qc,f %ar,

and let the disturbance or exogenous input be the in-coming regular wave of amplitude <,4 and frequency w d = <,4 eiWt. Then, from the equations of motion, the

system equation is described as

M(w)x = A@)x + B~(w)u + DO(w)d, (2)

where the coefficient matrices are

0 0 1] [ –b33 –b35 –C33 -C35 –b53 –b55 –C53 –C55 1 AO= ~ o 0 0 , 10 1 0 0] [ ~u2$F,fcLa,f ~u2sF,acLa,a –:u2XF,f sF,jcLct,f 1 gu2xF,asF,acLa, a , B(J = o 0

1

[1

The coefficicient matrices are dependent on the wave frequency w or the encounter frequency w., Multiply-ing M-1 to each sides of (2), the system equation is reduced to

x = A(w)x + B(w)u + D(w)d. (3)

The measurement vector z maybe part of the state or manipulated signals. Assume that the measure-ment z is represented by the transference H from the state x as

z= Hx+v, ₍₄₎

where v is the measurement noise. The measurement variables were heave and pitch displacements in experi-ments. The transference H, in this case, is a constant

[ 1

0’ 010’ coefficient matrix, or H =

0001”

Let the system outpu; y be repr~sented by the transference C as

y = Cx. (5)

Since we concern about the heave and pitch motion, let the output variables be heave and pitch displacements, then C = H.

dl Gii A r e

1

&

Figure 2: Colsed-loop configuration of an attitude control system

CONTROL SYSTEM DESIGN

There are two main control objects for the dynamic motions of the marine vehicles or ocean structures. One is the low-frequency motion and the other is the wave-frequency motion (or high-frequency motion) con-trols (Fossen, 1994). Ship manoeuvring, dynamic po-sitioning, etc. are typical ones for the low-frequency motion control. Ensuring tracking performance is the main goal of this control problem. In this paper the wave-frequent y motions , specifically, the heave and pitch motions of foil-catamaran are concerned, and en-suring regulation performance is the main goal of the problem.

Nominal Plant

Note that the coefficient matrices of the system equation (3) are dependent on the frequency of mo-tion. This fact still make the control system design be difficult to deal with despite (3) is a linear system.

Uncertainties come out from differences between the true system (the real plant) and the mathematically modelled system(the nominal plant) due to approxima-tion, abbreviaapproxima-tion, reduction etc. for possibility or fa-cility of design and analysis in system modelling. The difference between the nomianl and real plant may be represented in several ways. The simplest forms are the additive representation

Gt=Go+Ga

and the multiplicative representation

G~ = (I+ A)Go

(6)

(7)

where Gt and GO represent the real plant and the nominal plant, and G~ and A are the additive and multiplicative perturbations (Green & Limebeer, 1995). Above two representations are equivalent and any par-ticular selection is purely a matter of convenience.

The size of the uncertainties at any frequency is de-termined by their maximum singular values at that fre-quency. A constant-gain controller and estimator will be used in this study. Accordingly the nominal plant is described as a linear time-invariant system, and the frequency dependency of the system coefficients is con-sidered as a uncertainty. The constant coefficient ma-trices will be selected by fixing the representive or de-sign frequency. The design frequency wd can be se-lected in such a way that the nominal plant make the uncertainty due to the coefficient variations be as small as possible, which can be achieved by selecting &Jdto make the maximum value of the uncertain y’s maxi-mum sigular values, become the smallest in the mean-ingful frequency region, or

~d = m~{@(A(w))}. (8)

Though there are many rooms for uncertainties un-der the linear ship motion theory, but the uncertainty comes out from the frequency dependency is consider-able. And it also estimatable if it is possble to obtain the reasonable information of the ship motions by the-oretical calculations or experiments.

From now, let us confine our analysis to the un-certainty of the frequency dependency, assuming that other uncertainty components are far smaller than the one.

System Transference

Let us regard the system (3) as the real system Gt, and let the coefficients of the nominal plant be those at the design frequency wd A = A(wd) and B = B (~~). The exogeneonous input term has no concern about the controller and estimator design, but will be take into account in performance analysis.

The closed-loop configuration is shown in Figure 2. It can be represented by a unity feedback system for the constant attitude tracking and motion regulation

problem. Using Laplace transform, the system trans-ference of the nominal plant can be described as

Y(S) = Go(S)U(S) + Gd(s)d(s), (9) where

Go = C(S1 – A)-lB and

Gd = C(S1 – A)-lD(w), while the real plant Gt is described as

G, = C(SI – A(w)) -lB(u).

Tracking error e(s), defined by the diffrence between the reference input and the output, is described as

e(s) =r(s) – y(s)

=S(s)[r(s) – Gd(s)d(s)] + T(s)n(s), by means of the sensitivity function

S(s) = [1+ GOKF(S)]-l

and complementary sensitivity (or co-sensitivity) tion T(s) = GoKF(s)[l + GoKF(s)]-l. (lo) (11) func-(12)

For the sake of enhancing the tracking performance and the regulation performance, the magnitude of the sensitivityy function S(s) should be small, because this function indicates the measure of transference from the reference input r(s) and the disturbance d(s) to the tracking error e(s). On the other hand, we can ex-pect good noise suppressing performance by making the co-sensitivity functoin T(s) be small enough.

Stability Analysis of Control System

Nyquist stability criterion is a widely used method to determine the stability of a SISO(Single-Input-Single-Output) system. It can be extended and gener-alized to the MIMO (Multi-Input-Multi-Output) system in terms of GNC (Generalized Nyquist Criterion).

By the GNC, the closed loop system is internally stable if and only if the polar plot of the open-loop transfer function, det(I + GOKF), encircles the origin the same number of poles of the function in open right half plane. This condition is described as

N(O, det(I + GoKF(iu)), OR)

= No. of poles of GO in ORHP. (13) If the nominal plant GO contains the (additive) un-certainty G~ , the GNC can be rewritten as

N(O, det(I + (G. -t Ga)KF(iw)), OR)

= No. of poles of Go+ G. in ORHP. (14)

Provided that the nominal system GO is stable, (14) is satisfied if

?7(G.KF(I + GoKF)-l (iu)) <1 (15)

by corollary in APPENDIX C. Through sequential manipulation with the signlar value inequalities in AP-PENDIX C, the inequality (15) is satified if

where the arbitrary bound y, ~ l/F(Ga).

Fkom the above results, it is found that feedback control can only have a detrimental effect on the sta-bility robustness of the system, if the nominal plant is stable. Specifically this fact is really true for the ship motion regulation problem because ships are nat-urally stable systems to float on the water. The most common reason for introducing feedback control is to enhance the performance in the presence of uncertainty.

Performance Analysis of Control System

Designing a control system is a process to ensure the stability and performance criteria of the closed loop system.

Performance Criteria

Followings are considered as performance criteria of the attitude control problem in general.

● ship attitude tracking ● ship motion regulation ● noise suppression

● control energy mimization

The ship motion regulation and noise suppression are the main performance criteria in regulator design. The sensitivity and co-sensitivity function are responsi-ble for the regulation and noise suppression respectively as described already in the error dynamics (10). The sensitivity function S(s) defined as eq. ( 11) is the trans-parency from the reference input and disturbance to the tracking error. If the magnitude of the sensitivity function S is small enough, then the tracking or regula-tion performance will be enhanced. Singular values of the tranfer function can be used to determine the sig-nal magnifying characteristic of the transfer function. The maximum (=) and the minimum singular value (g) of arbitrary transfer function G represent, respectively, the maximum and the minimum gain of the transfer function as

. -,

The tracking and the regulation performance are achived by making the magnitude of the sensitivity S(s) be as small as possible, or

Zr(s(iLLJ)) <<1. (18)

The noise suppression performance is achieved by making the magnitude of the co-sensitivity function T(s) be as small as possible, or

ZF(T(iLJ)) >>1. (19)

But it is impossible to make both the functions be small at the same time because

S(s) + T(s) = I

Now we manipulate the above singular value in-equalities for specific performance description. We also let our anlysis confine to the disturbance rejec-tion and the noise suppression performances, because our main considerations are concentrated on ship mo-tion regulamo-tion.

Disturbance Rejection (Regulation)

The exogenous disturbance d, such as wave excit-ing forces, affects the output y, the ship motions in this case, via a transfer function Gd in an undesirable way. The disturbance rejection performance is to re-duce or to eliminate the influence of d on the output Y. Specifically, for a small performance bound ~d, the condition

ensures that

or the sensitivity operator S plays a role in disturbance rejection using feedback.

Through the sigular value operations with the sigu-lar value inequalities , assuming ~d <1 we have

It is found that g(GoKF) >>1, or high-loop gain will enhance the disturbance rejection performance for ar-bitrary disturbance d and those transfer function Gd. It can be found that the tracking performance is also achieved by high-loop gain with the same analysis of the disturbance rejection performance.

Noise suppression

Sensors are never free from noises of mesmrements or else. As the disturbances, the noises are harmful

to the performance of the control system, so their influ-ence on the control system should be reduced as small as possible. Given that noises are inevitable, it is im-portant that their effect be considered in the feedback system analysis and design process.

In (10), the transfer function from the noise n to the output y is the co-sensitivity T. Therefore, for a small performance bound Tn, the condition

B(T(iw)) ~ ~n (iu) (23)

ensures that

II Y(i~) IIS 7n(iw) II n(iw) II . (24) A sufficient condition for (23) in terms of the loop-gain operator is obtained as

F(GoKF(iu)) ~ Yn(iw)

1 + Tn(iu)” (25)

The effects of noises are reduced by low-loop gain, or E(KF) <<0. In the extreme case of open-loop control KF = O, the measurement noise have no effect on the output y. This confirms that the noise suppression conflicts with objectives requiring high-loop gain. It is recognized that, ensuring the noise suppression per-formance should not destroy the disturbance rejection performance, generates an important trade-off in the design of the feedback control system.

It can be found that the energy minimization per-formance is z&o achieved by low-loop gain. It is intu-itively a natural fact because no control pay no energy.

Robust Performance

The actual performance of the real system is re-served by the robust performance. It is not easy to figure out the uncertainty of the nominal plant from the real system quantitatively. Therefore, in general, one may expect that the robust performance will be en-hanced if the robust stability and the nominal perfor-mance in addition to the nominal stability are achieved in design stage of a control system.

Only replacing the nominal plant Go with the real plant Gt, the aforementioned performance anal-ysis techniques are applicable to the robust perfor-mance analysis provided that information about the uncertainties are given. The uncertainty from the frequency-dependency of the hydrodynamic coefficients is a estimatible uncertainty. The frequcy responses of the ship motions in regular waves are useful to verify the robust performance as well as the robust stability.

Control System Design for a Foil-Catamaran Now let us apply the design method considered so far to the 1 meter-long model ship shown in Figure 20. Its principal particulars are described in Table 1.

0.3, I [ 1 a k a 1 4 -1.OO 2 4 6 0Isrlrt(t&.p)

Figure 3: Nondimensional added masses and

0.2s,

-14 ]

n

---+---%’%

,,

:,—

Figure 4: System transferences (U = 1.5 m/see)

Nominal Plant of the Model Ship

Figure 3 shows the heave and pitch added masses and damping coefficients of the model ship without hy-drofoils at zero speed. They are calculated by the strip method (Salvesen et al., 1970), and nondimen-sionalize as a~~ = ass/m, b$3 = b33/(m~fi),

aj5 = a55/(mL~P) and b~5 = b55/(mLPPJm.

Calculated results show sudden changes of the hydrody-namic forces around the nondimensional frequency 2.5. It is known that this sudden changes are due to the in-teractions between hulls of the catamaran. When the ship is advancing, the hydrodynamic forces have simi-lar characteristic. The problem is that, if the nominal plant is selected by fixing a design-frequency, the nom-inal plant will have uncertainty due to the variation of the hydrodynamic forces.

For the case of the model ship with the design speed of the ship of 1.5 m/see, the design frequency was chosen at the nondimensional encounter frequency w,/ ~- = 2.8, followed (8). The singular values of the nominal transference and the real transference are shown in Figure 4 for that case. It is found that the nominal plant GO has noticeable difference with the real plant Gt in the nondimensional frequency range from 2.5 to 4.0. The multiplicative uncertainties, in this case, are shown in Figure 5, and the msximum value of the uncertainty is about 60% at frequency 2.6.

% ‘u I I I I , ,

damping coefficients (U = O m/see)

“w~

‘;:=

A full state feedback controller with constant coeffi-cients is used to regulate the ship motion and a full state estimator is constructed by means of a low-pass filter to estimate all states from noisy measurements for feed-back. The control system constructed with the con-troller and estimater should have the robustness to the aforementioned uncertainty.

Controller Design : LQ regulator

The full state feedback controller is constructed by means of the steady state LQ (Linear Quadratic) con-trol theory. The general features of the LQ control theory are presented in Appendix B.

The control law given from the LQ theory is reduced to a full state feedback with constant control gain, or

u = – Kyc =. R–lBTpx (26)

where R is the input weighting matrix in the LQ per-fromance index and P is the solution of the algebraic Rlccati equation.

Since the states of the system are motion displace-ments and veocities, the components of the control gain matrix are considered as proportional and deriva-tive gains of a PD(proportional-derivaderiva-tive) controller. The control gains can be determined by the tuning the weighting matrices Q and R in LQ control.

Let us consider the diagonal weighting matrix for

the state Q= qll o 0 0 0 qzz o 0 0 0 qss o ₁ “ 10 0 0 q,,j

Since the state is defined as x = [7j3 Tj5 113 V5]T, the components ql 1to q44are weigntings for suppressing the heave velocity, pitch velocity, heave displacement and pitch displacement in turn. The higher weighting is, the more the corresponding state will be suppressed. In general, the weighting matrix is determined by con-sidering the balancing of each channels. So to speak the reciprocal value of the square of the state’s ampli-tude or rms (root-mean square) value could be used for the weighting. But since this method does not con-sider the precise situation and dynamic property of the system, the designer should make efforts to find out better weighting by trial and error.

let also the weighting matrix R related to control input be a diagonal one

Since the control is defined as u = [CZ.,f aC,a]T, the components rl I and rzz are weigntings for minimizing the control efforts of fore and after hydrofoils respec-tively.

The components of the weighting matrix R can be determined so that the two control outputs will have little difference, or will be well balanced. The mag-nitude of available lift force and the relative position from center of rotational motion (assumably center of gravity) determine the weighings. For the case of the model ship in this study, the fore and apt hydrofoils are identical, but their longitudinal locations are different. The fore hydrofoil is located at 0.32 meter and the apt one is -0.43 meter from center of gravity. The ratio

rll /r22 = 1.0/0.8 gave a good balancing for this case.

A method to find the weighings Q and R, is rec-ommended as to find parameters aa, i = 1, . . . , 5 in the following equations, so as to achieve the satisfactory per-formance. 911 = ~1933 q22 = Q2q44 (911 + W933) = Q’3(q22+ a2944) rll = ~4~zz qll = ct’5rll

al indicates the ratio between heave damping and restoring forces supplied by the controller and CYZdoes

the ratio between pitch damping and restoring forces. a3 is a suppressing balance between heave and pitch mode. a4 is for balancing of the control outputs. ”

a5 is a very important parameter and should be determined through the robustness and performance analysis. This parameter indicates the compromising index between the regulating performance and control energy, The lager a5 be, the more can be enhanced the regulating performance if the control energy are suf-ficiently available without saturation. But, as we’ve already examined, the high-loop gain have decremen-tal effect on the noise suppression performance as well as the robust stability. Therefore the parameter a5 should be determined through the performance and robustness analysis in connection with the estimator. Other parameters may also request slight variation dur-ing the tunndur-ing operation of a5.

The weighting matrices determined by aforemen-tioned method as follows for the case of the model ship

r50 o 0 07

And, in this case, the control gain become

[

~ = 4.7535 –1.6825 3.0649 –3.1602 1 3.5900 2.4160 4.3914 3.6632 “

Estimator Design : Low-pass Filter

The full state estimator is constructed by use of low-pass filter. General techniques are introduced in Ap-pendix B. It is well known that the derivative control, or motion damper plays an important role to obtain the regulating performance. if the motion velocities cannot be measured directly, which is the situation of experiments in this study, they should be estimated by an estimator.

In general, Kalman filter is used to estimate the low-frequency motion of ship manoeuvring or drifting of ocean structures (Sub, 1991; Fossen, 1994). But, the estimation of the wave-frequency motion makes for some complications due to the frequency-denpendency of the hydrodynamic coefficients. The hydrodynamic coefficients of catamaran, in particular, have more com-plicated frequency-dependencies than mono-hull ship due to the interaction between twin hulls. Special care must be devoted to constructing a shaping fil-ter. As it is well known, if the system dynamics as-sumed in the Kalman fiter are far unacceptable, the time-propagation process in Kalman filtering will be no longer valid and moreover contaminate the estimation, It is beyond the scope of the present work and left to advanced works.

61

4.0W3SI , 1 1 , , 1 I i g -,ow* w“ -1.1y4 _{16 18 20} -2.60C-02 . _2~ 1! =’ -3.= w -3me-q; I 16 18 20 time (see)

Figure 6: Measurements in a calm water test

~ 3+OAB3 ‘g ‘-mJ- Me-OS O.oe+ooo _{10 20 30 40 50}

!.:m

~-

---}

---

Figure 7: Power Spectral densities of the sig-nals ,,:~~~ 2.0A5 ---+--- -- :

~o,m

:R

*H:DXL3

---:

----

--

----:

Figure 8: Motion Spectra (U = 1.5 m/see, llli3=0.032m, T0 =0.9 see)

The characteristic of a low-pass fiter is determined by the cut-off frequency, which eventually determine the filter gain. Figure 6 shows the measured motion of the model ship in calm water. born the power spec-tral densities of the measured signals shown in Figure 7, it is found that the measurement noises have wide-banded spectra and can be modelled a white Gaussian noises. The motion spectra of the model ship in irreg-ular waves - the sea condition will be presented in the section of EXPERIMENTS - have most of energy in the frequency range 5N15 rad/sec (=0.7958N2.3873 Hz) as shown in Figure 8. So, in experiments, the cut-off fre-quency was chosen to be 10 Hz, and the characteristics of the low-pass filter are shown in Figure 9.

If the ship motion frequency is 15 rad/see, esti-mated motion will have 10 %-reduced amplitude and 200-shifted phase compared to the real ship motion. The amplitude-reduction and phase-shift of the estima-tion should also be overcome by the robustness of the control system.

It is found that the control system constructed by the designed controller and estimator ensure

stabil-bEEEEEl

Frequency (rsd)

Figure 9: Characteristics of the low-pass fil-ter

it y from the results shown in Figure 10. The nomi-nal and robust performances are verified by Figure 11. The disturbance rejection performance is examined by the sensitivity function in low-frequency range, and the noise suppression performance is examined by the co-sensitivity function in high frequency range. But the results indicate only information about the maxi-mum and minimaxi-mum discrepancies of the performances. Once the nominal performance are achieved, the robust regulating performance or the robust disturbance rejec-tion performance can be verified by the frequency re-sponses of the ship motion in regular waves, from which the performances of the concerning channels can reex-amined.

In this study, an LQ controller and a low-pass fil-ter were used to construct the ship motion regulator. Other techniques or theories may also be available such as pole-placement, eigen-value and eigen-structure as-signment, adaptive control theories, etc. Whichever method is used, the robustness and performance are checked, and the method of this study give a way to do that, H~ control theory are can be more easier tool

0.s0- 1 1

I

Figure 10: Robust Stability Analysis

than the method of this study, but Hm filter problem may give troubles to designer due to the things as we’ve discussed about using the Kalman Filter.

EXPERIMENTS

Model tests were performed in SNUTT(Seoul Na-tional University Towing Tank). A l-meter model of catamaran was made of wood as shown in Figure 20. Its principal particulars are given in Table 1. The model ship was equipped with a couple of hydrofoils made of duralumine and servo-motor system to actu-ate the hydrofoils.

The hydrofoils are rectangular-shaped of NACA 0012 section with the chord length of 0.03 m and the span length of 0.15 m, They are located at Station 2.0 and Station 9.5. Fore-foil is not located at foremost part of the ship to avoid exposing to the air by excessive pitching of the ship. All of them are submerged 0.03m below the bottom of the model to insure that they do not break the water surface and pull an air cavity when the ship is heaving and pitching in waves. The sizes of the hydrofoils are determined to give enough control energy, but those sizes are not adequate to exert enough forces to lift up the ship.

Test Settings

Figure 21 shows the test settings. A counter mass towed the model ship to be freely surging, heaving and pitching. Letting surge motion to be free is impor-tant otherwise the heave and pitch motions can be dis-torted. Heave and pitch were messured by means of 6-DOF motion-measurement device which is equipped with six potentiometers and mounted on the hull deck. Measured heave and pitch displacements are used for the feedback signal of the control system. Surge dis-placement was also measured to figure out the ship re-sistance. Local vertical accelerations were measured at FP (Forward Perpendicular), midship, and AP (Af-ter Perpendicular) using three strain-type

accelerome-~:m!=

---

+--

---

---

---1 1

10 20 100 200

Frequency( rmlhec )

Figure 11: Performance Analysis

ters mounted on the hull deck. Encountered wave el-evation was measured by a condenser-type wire probe mounted on the towing carriage. The wave probe was located far in front of the model ship so as not to be affected by ship-generated waves.

Two AC servo-motors were mounted on the deck in order to actuate the hydrofoils. A circular disk was attached at the pole of them. The disk and the hy-drofoil were connected with a linear rod to transfer the command input, where the rot ation center and radius of the hydrofoils were to be them of the disks identically.

Test Program

Experiments consisted of tests in regular waves and long-crested irregular waves. The regular-wave tests consisted of two different speeds of 1,0 and 1.5 mlsec in head sea. Test conditions for regular-wave tests are summarized in Table 2. The conditions are selected so aa to cover up the significant frequency range.

The irregular-wave experiments were performed at speeds of 1.0 and 1.5 mlsec in head seas at four differ-ent sea conditions. The sea conditions are tabulized in Table 3. Sea condition numbers are selected to be the sea states with respect to the 100-m prtotype ship. The long-crested irregular waves are generated by the wave maker according to the ITTC sea spectrum (Lloyd 1989).

Test Procedure

The experiments were conducted in three diffrent configurations: the hull as originally built; the hull with fixed hydrofoils; and with control actions of the hydro-foils. The second and third programs were performed in a same trial, while the first was performed separately. Figure 22 shows the test scheme. For each condi-tion in regular-wave tests, data collected for 20 seconds with sampling rate of 100 Hz. The hydrofoils were fixed in first 10 seconds, then control actions were exe-cuted for the last 10 seconds. In irregular tests,

to-63

0.02 2 -0.01 a ‘I?J -0.00 @ =*4.01 -1Cm I , -o.02~ ₅I _{10 15 JO-’} 1s a so 8 -lsO _{5 10 15 20} 0.02 0.01 go.w ‘+.01 #20 _{5 10 15 20} 0.2 ~ 0.: -0.1 ‘O 5 10 15 20

time (see) time (see)

Figure 12: Time histories in regular waves (U = 1.5 m/see, w. = 9.0 rad/see)

0.01 0 -0.00 e

:X

time (see) time (see)

Figure 13: Time history in irregular waves (U = 1.5 m/see, H1/~ = 0.032 misec, To = 0.9 see)

tal measuring time for each conditions was 30 seconds. Control actions began at 15 seconds.

When the control actions started, a pre-filter (win-dow fuction) made a smooth starting for the 1 second so that the starting control actions were not to be abrupt or impulsive ones.

RESULTS AND DISCUSSION

Time Histories of the States and Controls

Figure 12 shows the time histories of the ship mo-tions, controls, measured wave and local vertical accel-erations at FP (Forward Perpendicular), midship, and AP (After Perpendicular) in regular wave test. Figure 13 shows the time histories of them in irregular wave test.

It is ascertained that ship motions are fairly re-duced when the control actions sxe executed. The measurement noises are amplified during the control action, and they give harmful effects on the feed-back sigmd. Measured accelerations show the situa-tion clearly. The sources of the noises were devices’

actuations. It is thought that careful noise solution may be necessary to enhance the performance of the control system.

Test in Regular Waves

Figure 14 and 15 show the frequency responses of the ship motion in regular waves. The results show poor agreement between the theoretical calculations and experimental results in the vicinity of the reso-nant frequency, but show good coindence at other fre-quency region. The discrepancies may be come out from using strip method and the potential-based calcu-lation. Advanced researches are demended for them. It is found that the controlled ship motions are dras-tically reduced and the ship motion regualtor achieves the robust regulation performance over all frequency range.

In ship attitude control problem, desired constant attitude tracking is also the matter of concern as well as ship motion regulation. To achieve the tracking per-formance, high-loop gain should be ensured at lower frequency than motion frequency and at zero frequency.

2,0 _I - U/aIOw(C4L) _{I II} l---1 # 1 ‘me/ sqrt(g&) -2.0 1 - WIkw&CO.lnll(cd.) 1.5---+

--0 Wffoul&CmlrolOIX$l.) u \ 1.0-F“ 045.--- ~--- -l---1 1 0.OO _{2 4} 6

Figure 14: Frequency Response of Motion (U = 1.0 m/see)

2.0 I H whWII(GL) M WIrolls(cd.) - WIrdll&mnlrd(CA) 1.s --, 0 *O**) ❑ W/folll(Exp.) I 0 \ 1.0 -- c-0.s---1 1 I I 1 0.OO 2 4 6

II

11

‘[11

m-a “1am (Cal.) - w/folk& CC..(IQI(COL)

1.s ---- ---L--- ---- 0 W/ofolk mm.) 11 d--~4---ll---l---l

Figure 15: Frequency Response of Motion (U = 1.5 m/see)

It can be easily achieved by adding integral control and low-frequency balancing (Lewis, 1992). It is found that the calculated pitch motion are amplified at low frequency region, but the problem can also be overcome by an integral control.

Another considerations required to enhance the reg-ulation performance of the low-frequency ship motions or attitude tracking. To achieve those performance, the control energy should be large enough to do that, in other words the large hydrofoils should be adopted. The problem is closely related to the ship resistance problem, because the large hydrofoils will increase the ship resistance.

Test in Irregular Waves

Figure 16 and 17 show the significant ship motions in irregular waves. It seems that the results show good agreement between theoretical calculations and experimental results in spite of the poor estimation of frequency responses in the vicinity of the resonant fre-quency by the strip method. It is found that the cal-culated ship motions under estimate the ship motions at sea condition 3. The ship motions are very small

compaxed to the measurement noises at that condition. So It is thoght that the experimental results may be unreliable for that condition. At sea condition 6, it is found that the regulation performance do not reach the calculated performance.

Those results may be caused from the saturation of the control input. From the time histories, we’ve no-ticed that the saturations of the control were already met sometimes at sea condition 5.

Figure 18 and 19 show the measured vertical accel-erations at FP (Forward Perpendicular), midship and AP (After Perpendicular). It is found that the ship motion regulator reduces the vertical accelerations as desired.

CONCLUSION

In this study, an advanced design technique of ship motion regulator for foil-catamaran is introduced and verified by theoretical calculations and experiments.

Concluding Remarks

1. A ship motion regulator designed based on the assumption for ship motion regulation showed good

SeaConditions

Figure 16: Significant ship motions

;W

---Sea Conditions

Figure 17: Significant ship motions

ulating performance in irregular waves as well as in regular waves. Accordingly it can be said that the design technique ensures the robustness of the motion regulator, including the robust stability and the robust performance.

2. A ship motion regulator should be designed so as to have the robustness with respect to the variation of the frequency-dependent hydrodynamic coefficients.

3. The robust regulation performance or the ro-bust disturbance rejection performance can be verified by frequency responses of the ship motions in regu-lar waves, while the frequency response of the nominal plant can be used for analyzing the nominal regulation performance.

4. Ship motions of catamaran in regular waves are badly estimated by the strip method or the potential-based analysis in the vicinity of the resonant frequency. However the strip method gives not so bad estimation of the ship motions in irregular waves.

REFERENCE

{1] ~strom, K. J. an Wittenmark, B., Adaptive con-trol, Addison Wesley Publishing Company, 1995.

i/

I

5!!!2E!J..Z!I

Sea Conditions

in irregular waves (U = 1.0 m/see)

,

E2Ei H

-

[2] ~strom, K. J. an Wittenmark, B,, Cornputer-Controlled Systems, Prentice Hall, 1997.

[3] Bryson Jr., A. E., and Ho, Y. C., Applied Optimal Control, John Wiley & Sons, 1978.

[4] Chen, C. T., Linear System Theory and Design, CBS College Publishing, 1970.

[5] Fossen, T. I., Guidance and Control of Ocean Ve-hicles, John Wiley & Sons, 1994.

[6] Green, M. and Limebeer, D. J. N., Linear Robust Control, Prentice Hall,1995.

[7] Hadler, J. B., Lee, C. M., Birmingham, J. T. and Jones, H. D., “Ocean Catamaran Seakeeping De-sign, Based on the Experience of USNS Hayes”, SNAME ‘I’r., pp. 126-161, 1974.

[8] Hamming, R. W., Digital Filters, Prentice Hall, 1989.

[9] Kashiwagi, M. “Heave and Pitch Motions of a Catamaran Advancing in Waves”, Proc. of FAST’93, pp. 523-534, 1993.

[10] Kang, C.-G. and Gong I.-Y., “Time Domain Sim-ulation of the Motion of a High Speed Twinhull with Control Planes in Waves”, Proc. of FAST’95, pp. 1031-1041, 1995.

[11] Lewis, F. L., Optimal Estimation, John Wiley&

“IE?!!$!

~

i!

-

---L---4---3

,

1*

:

Sea Conditions Sea Conditions

Figure 18: Measured Accelerations in Irregular Waves (.U = 1.0 m/see)

-1

L,

-.-,-,

-,-,

1

.,r**-~qjl

‘-~

,

-*am

I

“’~sliiir--

““

‘“’-””

‘

‘

‘

---‘:e~:u’:m

---

~---

-,

---

---

q

---Sea Conditiom Sea Conditions Sea Conditions

Figure 19: Measured Accelerations in Irregular Waves (U = 1.5 m/see)

Sons, 1986.

[12]Lewis, F. L., Applied Optimal Control and Esti-mation, Prentice Hall, 1992.

[13] Lloyd, A. R, J. M., SEAKEEPING : Ship Be-haviour in Rough Weather, Ellis Horwood Limited, 1989.

[14] Maciejowski, J. M., Multivariable Feedback De-sign, Addison Wesley Publishing Company, 1989. [15] Newman, J. N., Marine Hgdrodynarnics,

Cam-bridge, Mass., MIT Press, 1977.

[16] Newman, J, N., “The Theory of Ship Motions”, Advanced in Applied Mechanics, vol. 18 [ Ed. by Chia-Shun Yih ], pp. 221-283, Academic Press, 1978.

[17] Philips, C. L. and Nagle, H. T., Digitai Control System Analysis and Design, Prentice Hall, 1995. [18] Price, W. G. and Bishop, R. E. D., Probabilistic Theory of Ship Dynamics, Chapman and Hall, 1974. [19] Rhee, K. P., Lee, G. J. and Lee, S. Y., “Attitude

Control of Foil-Catamaran”, Proc. of the 10th Ko-rea Automatic Control Conference-Int’1 Program,

Oct. 1995.

[20] Rhee, K. P. and Lee, S. Y., “Regulation of Mo-tion Responses of Foil-Catamaran by Experiments”, China-Korea Marine Hydrodynamics Meeting, Aug. 1997.

[21] Salvesen, N., Tuck, E. 0. and Faltisen, O., “Ship

Motions and Sea Loads”, SNAME ~., vol. 78, pp. 250-287, 1970,

[22] Sub, S.-H., “Improved Methods for Processing Po-sition Information for Dynamic Posit ioning”, Ph. D Thesis, The university of Michigan, 1991.

APPENDIX A : EQUATIONS OF MOTION

If the ship has a pair of hydrofoils located at fore and apt part of the ship to control the longitudinal motion, hydrodynamic coefficients and wave exciting forces in (1) are described by superposing hydrodynamic forces acting on the hull (denoted by subscrip II) and those due to hydrofoils (denoted by subscript F) as

ass =aH,ss + aF,f + aF,a,

U35 =aH,35 — xF,faF,33,f — XF,aaF,33,~,

U53 =aH,53 — XF,faF,33,f — XF, aaF,33,~7

a55 =aH,55 + X~,faF,33,f + xi, aaF,33,a,

b33 =bH,33 + ~U(SF,fCLa,j + SF,. Calm,.) +&( SF,f@f + SF, OCV,a), ‘U(xF,f SF, fcLa,f b35 =bH,35 – ~ + XF,aSF,aCLa,a) 8 – ~p(xF,fSF,f Cv,f + ZF,aSF,aCv,O), 67

b53 b55 C33 C35 C53 C55 F3 =bH,53 – ~U(xF,fSF,fCLa,f + xF,aSF,acLa,a) 8 – ~p(XF,f SF, f Cy f + XF,aSF,aCV,G), ‘bH,55 + ~U(x$,fSF,fCLff,f + x?’,aSF,aCL~,a) + &p(x\, f SF, f CV, f + x%,aSF,aCV,a)~ =cH,33,

=cH,35 + ‘U2(SF,f CLa3f + SF,aCLa,a),

2 =cH,53 ,

P2

=CH,55 + –U (XF,f SF, fCLa,f + XF,aSF,aCLa,a),

2

‘fH,3 (A eiwt + :U2(SF,fCL,~,f~C,f

In the above equations, pand U are the density of water and the constant advancing speed of ship. Subscripts j and a means that the terms are related to the fore and apt hydrofoils respectively. SF,. are the planar area of the hydrofoils, or the value (chord x span) for rectangular hydrofoil. CLa,* = dCL,*/d~ are the lift-curve slopes and a,-,. are the commanded angle of attacks. CV,* are the viscous drag coefficients for pla-nar vertical motoin of the hydrofoils. The lift-curve slope of a hydrofoil should be determined by consid-ering the effects assciated with the aspect ratio, the reduced frequency, the catamaran hulls, and the free surface. In this work, Lawrence and Gerber’s method is used to obtain the lift-curve slopes, which are intro-duced in (Hadler et .al 1974). The free surface effect is ignored since the hydrofoils, for the case of the model ship, are submerged enough from the free surface and the effects are negligible.

APPENDIX B : LQ CONTROLLER AND

LOW-PASS FILTER

LQ controller

If the control system is controllable then the steady state LQ(Linear Quadratic) control can be used, which is a optimal control rule minimizing the following infi-nite horizon performance index

J= E{/W xT(t)Qx(t) + uT(t)Ru(t)dt} , (B.1) o

where Q and R are weighting matrices, and E indi-cates the expectation of the sequence. We can also

achieve the 2-norm bound of the transparency from “the disturbance to the output RYd with the above optimal control.

The solution of the above optimal control problem with the constraint of plant dynamics can be obtained by the solution of algebraic Riccati equation P as

U(t) = –R–lBTPx(t), (B.3) and O = ATP + PA – PBR-lBTP + Q. (B.4) Low-Pass Filter An estimator can be filter as F(s) =

constructed using a low-pass

[1 so 10s — 710’ (B.5)

lJ

where the filter time constant T is determined by the filter cut-off frequency wC(rad/see) or fC(Hz) as

1 1

‘=G==

The estimator can be realized and implemented as the following discretized system

&k+l ‘@F&k + rF%

(B.6) kh =& + DF~h ‘

where the coefficient matrices are

(-e-y _{00 01}

@F=

1

1

_-)

—e–u

r)

(l-e-+)

(1–%)

.

In which At is the sampling period.

APPENDIX C : SINGULAR VALUES

Some of the properties of sigular values are exam-ined without precise descriptions or proofs. More de-tails about them are found in Green and Limebeer 1995. And the criterions in the robust anlysis and the per-formance analysis are also derived using the properties of the singular values.

General Properties

Singular values can be obtained by singular value decomposition. Singular values of an arbitrary trans-ference G are regarded as a gain factor for the trans-ference G.

The maximum singular value T(Q) and the mini-mum singular value Q(Q) of a complex matrix Q play a particular important role in our amdysis and are given by the identites

5(Q) = ,rn~l [IQuII,

and

g(Q) = ,lmn~l IIQuII,

where the vector norm is the Euclidean norm.

When Q is square, c(Q) > 0 if and only if Q is nonsingular. In this case,

~(Q–l) = ~. _(cl)

Q(Q)

Inequalities

Using the fact that the maximum sigular value de-fines an induced norm, we have the following inequali-ties for the compolex matrices Q and R.

I@(Q) - F(R) I S F(Q + R) < T(Q) + i7(R) (C.2) z(Q)F(R) ~ ?T(QR) ~ i7(Q)~(R) (C.3)

dQ)dR)

S IZ(QR)< ZF(QMW

dQ + R) (c ~,

~

Corollary : Closeness to Singularity

Let Q and R be p xp complex matrices and suppose also that Q is nonsingular. Then

m(R) < c(Q) ~ (Q + R) is nonsigular (C.6)

and

min

Rdet(Q+R)=O F(R) = Q(Q). (C.7)

Derivation of Robust Stability Criterion

If the nominal plant Go contains the (additive) un-certainty G. , the GNC can be written as

N(O, det(I + (G. + Ga)KF(jw)), ~R)

= No. of poles of Go+ G. in ORHP.

Assume that the nominal plant is stable. The Nyquist diagram should across the origin of the com-plex plane when the number of encirclement of the di-agram is changed as the uncertainty Ga become larger from O. The change of the number of encirclements means that the closed-loop system become unstable due to the uncertainty. Therefore, the closed loop system remain stable if det(I + (G. + Ga)KF(jw) # O for ar-bitrary uncertainty G..

If there exists (I+ GoKF)-l,

det(I + (G. + G.)KF)

= det(I + G.KF(I + GOKF)-l) det (I + GOKF).

Because the nominal plant is assumed to be stable, or the second term in the right hand side will not vanish, the first term should not be vanish to ensure the robust stablity, or

det(I + G.KF(I + GOKF)-l) #O. (C.8)

By the above corollary, (C.8) is satisfied, if

F(G.KF(I + GOKF)-l) <1,

and using the sigular value inequality it is also satisfied by

1

F(KF(I + GOKF) (iw)–l ) < ~ 7s1 &(G.(iw)) –

where ~~ is the reciprocal bound of the uncertainty for a specific system,

Using the singular value inequalities, above condi-tions are reduced as follows.

77(KF(I + GOKF)-l) s y. 1 * ?7(KF)

Q((I + GOKF)) < 7S

From this results, it is found that excessive high-loop gain can instabilize the closed-loop system if there exist uncecertaint y.

Derivation of Performance Criteria

Using the general properties and inequalities, the performance criteria used for performance analysis are derived. The system trasferency G in follows may be replaced by Go for the nominal performance analysis, while Gt for the robust performance analysis.

Disturbance Rejection

The bound criterion for disturbance rejection

is reduced ti(sGd(j(d)) < ~d M follows. F((I + GKF)–l Gd) ~ yd ~ 5((1 + GKF)–l)5(Gd) ~ ~d 1 * c((I + GKF))6(Gd) < ‘d Noise Suppression

The bound criterion for the noise suppression per-formance F(T(@)) s ~n

II

_I

Figure 20: Body plane of the model ship

g

h-li-@-#t

Comcwter _BEE

P. AMP P. AMP P. AMP c. AMP

surge Haaw Piloh

— Wmw

—-1

1.

_{_yl}

‘““

““-T=zl17i

Probe

Figure 21: Body plane of the model ship

s

for Data Maaauremant for Initial State

Timar Start

Measurement

Cutting Outllers

Stata Eatlmation

To:Control SterfTime Ym Calculating Contr018 * Control Out -

‘f’Ftne’Time

r-zxu-l

25

Figure 22: Test procedure

Table 1: Principal Particulars of the Model Ship

Model Ship Particulars

Length(LPP), m 1.000

Beam each hull (B),m 0.100

Draft(T), m 0.070

Half distance beteen Hull, m 0.080

Displaced Volume, m3 0.010

Vertical Center of Gravity, m 0.098 Longitudinal Center of Gravity, m 0.010

Hydrofoil Particulars I 1

I

~

* All lengths are measured realtive to the midship and free surface.

Table 2: Test Conditions for Regular-Wave Tests

we, radisec 4.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 15.0 * 1.2771 1.9157 2.2349 2.5542 2.8735 3.1928 3.5120 3.8313 4.1506 4.7891 Vm = 1.0 m/see (F. = ( ~ 3.0511 4.2010 4.7246 5.2212 5.6945 6.1476 6.5828 7.0021 7.4071 8.1797 i19) w 6.6213 3.4926 2,7613 2.2611 1.9008 1.6310 1.4224 1.2572 1.1234 0.9212 Vm = 1.5 m/see (F. = 0,479) w, radlsec 2.8007 3.7963 4.2448 4.6681 5.0698 5.4531 5.8203 6.1731 6.5133 7.1604

Table 3: Test Conditions for Irregular-Wave Tests

Sea Conditions H113, m To, sec

3 0.014 0.60

G

71