A two-time scale control law based on singular perturbations used in rudder roll stabilization of ships

(1)

ELSEVIER

Contents lists available at ScienceDirect

Ocean Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g

A two-time scale control law based on singular perturbations

used in rudder roil stabilization of ships

R u - Y i R e n ^ Z a o - J i a n Z o u ^ ' ' ^ ' * , X u e - G a n g W a n g ^

^ School of Naval Architecture, Ocean and Civil Engineenng, Shanghai Jiao Tong University, Shanghai 200240, China ^ State Key Laboratory of Ocean Engineering Shanglial Jlao Tong University, Shanghai 200240, China

CrossMark

A R T I C L E I N F O

Article history:

Received 11 September 2013 Accepted 2 July 2014 Available online 27 July 2014 Keywords:

Rudder roll stabilization Non-minimum phase Time scale decomposition Singular perturbation

A B S T R A C T

A t w o - t i m e scale d e c o m p o s i t i o n m e t h o d is used to a n a l y z e a n d design the r u d d e r roll stabilization (RRS) s y s t e m of ships. In the s u r g e - s w a y - r o l l - y a w s h i p m o t i o n s y s t e m , roll m o t i o n is m u c h faster t h a n the others, the interactions b e t w e e n these fast a n d s l o w d y n a m i c s c a u s e the n o n - m i n i m u m p h a s e behavior in roll d y n a m i c s , w h i c h is regarded as a m a j o r challenge i n RRS control design. A s m a l l p a r a m e t e r e is i n t r o d u c e d to describe the fast roll d y n a m i c s by a s i n g u l a r o r d i n a r y differential equation. By using s i n g u l a r p e r t u r b a t i o n a p p r o a c h e s , t h e s y s t e m is t h e n d e c o m p o s e d into t w o d i f f e r e n t t i m e scale s u b s y s t e m s , a q u a s i - s t e a d y - s t a t e s u b s y s t e m to d e s c r i b e the s l o w d y n a m i c s , a n d a b o u n d a r y layer s u b s y s t e m to describe the fast d y n a m i c s . Separate c o n t r o l strategy is used to stabilize e a c h subsystem and the c o u p l i n g effect b e t w e e n t h e s u b s y s t e m s is also c o n s i d e r e d . A L y a p u n o v f u n c t i o n is constructed for the s l o w s u b s y s t e m a n d robust a n a l y s i s is m a d e to evaluate the u n m o d e l e d d y n a m i c s . Simulation results s h o w the effectiveness a n d robustness of this a p p r o a c h u s e d in RRS s y s t e m .

Due t o t h e r e l a t i v e l y s m a l l m o m e n t o f i n e r t i a c o m p a r e d t o o t h e r degrees o f f r e e d o m (DOFs), t h e r o l l m o t i o n o f a surface ship is easily a f f e c t e d b y the e n v i r o n m e n t a l d i s t u r b a n c e s s u c h as waves a n d w i n d , a n d o f t e n p r o d u c e s t h e largest a c c e l e r a t i o n . Large r o l l m o t i o n is t h e m a i n cause o f seasickness, can g r e a t l y a f f e c t the c o m f o r t o f the passengers, decrease t h e w o r k efficiency o f the crew, d a m a g e t h e cargo, a n d i n some e x t r e m e cases, m a y cause t h e c a p s i z i n g o f t h e s h i p . T h e r e f o r e , ship r o l l r e d u c t i o n has b e c o m e an active research area since 1970s. Criteria o f t h e m a x i m u m r o l l angle f o r d i f f e r e n t w o r k c o n d i t i o n s have b e e n m a d e b y Faltinsen ( 1 9 9 3 ) . I t is suggested t h a t t h e m a x i m u m r o o t m e a n square o f r o l l angle s h o u l d be less t h a n six degrees f o r l i g h t m a n u a l w o r k a n d t h r e e degrees f o r i n t e l l e c t u a l w o r k .

I n t h e p a s t decades, m a n y devices have b e e n d e s i g n e d t o reduce t h e r o l l m o t i o n , b o t h active c o n t r o l a n d passive c o n t r o l devices, such as b i l g e keels, gyroscopic stabilizers, a n t i - r o l l i n g tanks, s t a b i l i z i n g fins a n d m o v i n g w e i g h t s (Trealde e t al„ 2 0 0 0 ; G a w a d e t al., 2 0 0 1 ; Perez a n d Blanke, 2 0 0 2 ; T o w n s e n d e t al., 2 0 0 7 ;

* Corresponding author at: School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China.

Tel./fax: 4-86 21 34204255.

S u r e n d r a n et al., 2 0 0 7 ) . H o w e v e r , all these approaches need extra devices a n d i n s t a l l a t i o n costs, t h u s are u s u a l l y expensive.

A l t h o u g h the original objective o f the r u d d e r is t o steer the ship to a desired course, f o r m o s t surface ships, r u d d e r action can also cause certain roll m o t i o n . So i t is expected t h a t i f the r u d d e r is suitably operated according to t h e r o l l m o t i o n and t h e course deviation, the r o l l angle m a y be reduced to some degree, at the same time the heading is n o t v i o l e n t l y changed. This r u d d e r roll stabilization (RRS) c o n t r o l strategy needs n o extra devices and is relatively cheap, thus has d r a w n m a n y researchers' interests i n the past decades (Van A m e r o n g e n et al., 1990; Blanke and Christensen, 1993; Lauvdal and Possen, 1998; Perez, 2005). M o d e l e x p e r i m e n t s a n d full-scale trails have b e e n made t o evaluate its effectiveness i n practice (Van A m e r o n g e n et al., 1990). In RRS c o n t r o l system, r u d d e r is the only actuator f o r t w o o u t p u t s (roll and heading), thus s u f f i c i e n t band-w i d t h separation o f the t band-w o loops has t o be guaranteed.

T h e r e are also several d r a w b a c k s o f RRS, s u c h as t h e i n e f f i -ciency a t l o w speed a n d severe feedback l i m i t a t i o n s d u e to r u d d e r s a t u r a t i o n a n d rate l i m i t s . Besides, i t is w e l l - k n o w n t h a t ships have n o n - m i n i m u m phase ( N M P ) b e h a v i o r i n t h e r u d d e r - t o - r o l l d y n a m i c s , w h i c h is considered to be one m a j o r c h a l l e n g e f o r RRS ( L a u v d a l a n d Fossen, 1997; Perez, 2 0 0 5 ) . N M P systems have an inverse i n i t i a l response a n d large phase lag. The N M P b e h a v i o r i n r o l l m o t i o n o f t e n causes a f u n d a m e n t a l l i m i t a t i o n i n t h e RRS s y s t e m : d i s t u r b a n c e s a t t e n u a t i o n at some f r e q u e n c i e s w i l l result i n a m p l i f i c a t i o n at o t h e r frequencies. This l i m i t a t i o n t h u s poses a

(2)

R.-Y. Ren et al. / Ocean Engineering 88 (2014) 488-498

t r a d e - o f f b e t w e e n r e d u c i n g t h e r o l l angle at c e r t a i n frequencies and a m p l i f i c a t i o n at o t h e r s (Perez, 2 0 0 5 ) .

The N M P p h e n o m e n o n o f t e n arises f r o m t h e interaction b e t w e e n opposite fast a n d slow d y n a m i c effects i n the system (Perez, 2005). As to t h e ship, t h e N M P behavior i n r o l l m o t i o n is caused b y the fact t h a t t h e roll dynamics is m u c h faster t h a n the other DOFs, Singular p e r t u r b a t i o n approach is such a m e t h o d to analyze and separate the d i f f e r e n t t i m e scale m o t i o n s i n c o n t r o l problems. I n this paper, t h e RRS system f o r ships is decomposed i n t o t w o d i f f e r e n t time scale subsystems, namely the quasi-steady-state ( s l o w ) subsystem and b o u n d a r y layer (fast) subsystem. The c o n t r o l objectives a n d control strategies o f t h e t w o subsystems are treated separately.

Singular p e r t u r b a t i o n approaches have b e e n used i n aerospace i n d u s t r y f o r m a n y years as a time-scale separation technique ( M e h r a , 1979; B e r t r a n d et al., 2 0 1 1 ; Esteban et al., 2013). For example, a t h r e e - t i m e scale c o n t r o l l a w is designed f o r a n o n l i n e a r h e l i c o p t e r m o d e l i n v e r t i c a l f l i g h t (Esteban et a l , 2013). This can be done due to t h r e e d i f f e r e n t t i m e scales o f a l t i t u d e m o t i o n , angular velocity, and t h e associated collective p i t c h angle o f blades. A c o m p r e h e n s i v e l i t e r a t u r e r e v i e w o f singular p e r t u r b a t i o n used i n a i r c r a f t c o n t r o l was m a d e b y N a i d u and Calise (2001). However, despite o f t h e extensive w o r k i n aerospace i n d u s t i y , f e w w o r k o f singular p e r t u r b a t i o n a n d ; time scale separation techniques has been d o n e i n ship c o n t r o l c o m m u n i t y . This is m a i n l y due t o t h e relatively p o o r r u d d e r e f f e c t a n d s i m p l e c o n t r o l objectives f o r a ship conti-ol system. However, w h e n a RRS p r o b l e m is considered, t h e t r a d i t i o n a l 3-DOF m o d e l ( s u r g e - s w a y - y a w ) is c o u p l e d w i t h fast r o l l m o t i o n , and d i f f e r e n t t i m e scale m o t i o n s d o exist i n t h i s system. The concept o f t i m e scale separation based o n singular p e r t u r b a t i o n can be used to analyze such p r o b l e m s i n a n a t u r a l and elegant way. Singular p e r t u r b a t i o n is a m e a n s o f t a k i n g i n t o a c c o u n t t h e o f t e n neglected h i g h - f r e q u e n c y p h e n o m e n a a n d c o n s i d e r i n g t h e m i n a separate f a s t time scale ( K o k o t o v i c et al., 1987). By i n t r o d u c i n g a s m a l l p a r a m e t e r e, t h e f a s t v a r y i n g state variables are described i n t h e f o r m o f s i n g u l a r o r d i n a r y d i f f e r e n t i a l equations (DDEs), t h e e q u a t i o n s b e c o m e singular w h e n e t e n d s t o zero. A s t r e t c h e d t i m e scale is used t o describe the fast d y n a m i c s a n d t h e s l o w state variables are regarded to be c o n s t a n t i n t h i s time scale. A so-called quasi-steady-state e q u i l i b r i u m (QSSE) is used t o pass i n f o r m a t i o n b e t w e e n d i f f e r e n t t i m e scale subsystems.

This paper i n t r o d u c e s t h e s i n g u l a r p e r t u r b a t i o n a p p r o a c h t o analyze t h e ship RRS p r o b l e m . T h e r e are t h r e e m a j o r m e r i t s o f u s i n g this a p p r o a c h i n RRS system.

Firstly, m o r e d e t a i l e d analysis is possible i n time d o m a i n , s u c h as s t a b i l i t y issues a n d t i m e d o m a i n response. U n l i k e t r a d i t i o n a l analysis m e t h o d s , w h o s e e m p h a s i s is o n t h e b a n d w i d t h s e p a r a t i o n i n Bode d i a g r a m c o n s i d e r e d i n f r e q u e n c y d o m a i n , t h i s p a p e r emphasizes t h e separation o f d i f f e r e n t t i m e scale subsystems i n time d o m a i n . The s t a b i l i t y a n d r o b u s t analysis are easily c o n d u c t e d i n t h i s m o d e l , a n d t h e s e n s i t i v i t y analysis t o m o d e l e r r o r s can also be evaluated w i t h i n t h i s f r a m e w o r k , w h i c h are n o t easily c o n -d u c t e -d i n f r e q u e n c y -d o m a i n .

Secondly, t h e time scale d e c o m p o s i t i o n a p p r o a c h a n d separate c o n t r o l strategy s i m p l i f y t h e c o n t r o l l a w d e s i g n f o r RRS system. By u s i n g singular p e r t u r b a t i o n m e t h o d , t h e o r i g i n a l u n d e r a c t u a t e d RRS s y s t e m can be d e c o m p o s e d i n t o t w o single i n p u t single o u t p u t (SISO) subsystems, t h u s is r e l a t i v e l y easier t o o b t a i n t h e a p p r o -p r i a t e c o n t r o l l a w t h a t can stabilize each s u b s y s t e m . T h i r d l y , t h e p r o p o s e d separate c o n t r o l strategy considers t h e i n t e r a c t i o n b e t w e e n d i f f e r e n t time-scale subsystems. T h e c o u p l i n g e f f e c t is i m p o r t a n t i n some cases, a n d s i n g u l a r p e r t u r b a t i o n a p p r o a c h takes t h i s i n t o c o n s i d e r a t i o n t h o u g h QSSE.

I n t h i s paper, a s i m p l i f i e d 3-DOF ( s w a y - r o l l - y a w ) l i n e a r m o d e l is u s e d t o design a n d analyze t h e RRS c o n t r o l l a w . As course k e e p i n g o p e r a t i o n s are c o n s i d e r e d i n m o s t situations, t h e l i n e a r m o d e l has considerable accuracy i n these p r o b l e m s (Perez, 2 0 0 5 ) .

489

Li et al. ( 2 0 0 9 ) used a c o m p r e h e n s i v e 4DOF ( s u r g e s w a y r o l l -y a w ) n o n l i n e a r m o d e l as a v i r t u a l ship f o r s i m u l a t i o n a n d p e r f o r m a n c e e v a l u a t i o n . This n o n l i n e a r m o d e l was o b t a i n e d b y a set o f captive m o d e l tests (Son a n d N o m o t o , 1982). I t is selected as a b e n c h m a r k m o d e l to evaluate t h e p e r f o r m a n c e o f t h e l i n e a r m o d e l i n t h i s paper. The d i f f e r e n t p e r f o r m a n c e s b e t w e e n t h e l i n e a r a n d n o n l i n e a r m o d e l s are e v a l u a t e d .

The s t r u c t u r e o f t h i s p a p e r is as f o l l o w s . Section 2 i n t r o d u c e s t h e n o n l i n e a r a n d l i n e a r i z e d m o d e l s o f m o t i o n o f surface ships, t h e m o d e l o f d i s t u r b a n c e s is also d e s c r i b e d . Section 3 gives a b r i e f i n t r o d u c t i o n t o t h e singular p e r t u r b a t i o n approach, based o n w h i c h t h e RRS c o n t r o l is designed. Robustness analysis o f t h e u n m o d e l e d d y n a m i c s is also m a d e i n t h i s section. Section 4 gives t h e s i m u l a t i o n results. Section 5 is t h e c o n c l u s i o n .

2. Model definition and analysis

I n t h i s section, t h e m o d e l s o f s h i p m o t i o n a n d e n v i r o n m e n t a l d i s t u r b a n c e s are described.

2.1. 4-DOF nonlinear model

A ship i n a seaway moves i n 6-DOFs. T h r e e t r a n s l a t i o n d i s p l a c e m e n t s are used t o d e f i n e t h e l o c a t i o n a n d t h r e e a n g u l a r d i s p l a c e m e n t s are used t o d e f i n e t h e o r i e n t a t i o n . These m o t i o n s are o f t e n described i n t w o fypes o f r e f e r e n c e f r a m e , n a m e l y t h e i n e r t i a l f r a m e a n d b o d y - f i x e d f r a m e .

As s h o w n i n Fig. 1, t h e l o c a t i o n a n d o r i e n t a t i o n o f t h e ship are described i n t h e i n e r t i a l f r a m e , t h e t r a n s l a t i o n d i s p l a c e m e n t s a n d a n g u l a r d i s p l a c e m e n t s are described as [Xo,yo,Zof a n d [4),6,\//f, w h e r e Xo,yo a n d Zo are t h e t h r e e c o o r d i n a t e s o f t h e ship, (p, 9 a n d y/ are r o l l , p i t c h a n d y a w angle, respectively. The c o m p o n e n t s o f t h e f o r c e a n d m o m e n t [ X , Y , Z f , [K,IVI,Nf, t h e c o m p o n e n t s o f t h e t r a n s l a t i o n a l v e l o c i t y a n d the a n g u l a r v e l o c i t y [u,v,wf, [p,q,r]'^, are described i n t h e b o d y - f i x e d f r a m e , w h e r e u, v a n d w are surge, s w a y a n d heave v e l o c i t y , a n d p, q a n d r are r o l l , p i t c h a n d y a w rate, respectively. The r u d d e r angle is expressed as S.

I n t r a d i t i o n a l m a n e u v e r i n g issues, such as c o u r s e - k e e p i n g p r o b l e m , n o r m a l l y only a 3-DOF m o d e l (surge-sway-yaw) is consid-ered. However, w h e n consider the RRS p r o b l e m , a 4-DOF m o d e l i n c l u d i n g the r o l l m o t i o n is needed, I n this paper, a comprehensive 4-DOF nonlinear m o d e l (surge-sway-roll-yaw) is used to describe the RRS system (Fossen, 1994):

(m+mx)ii-(m + my)vr = X (1)

(,m+my)v+(m+mx)ur+myayi- -mylyp = Y (2)

(.Ix+Jx)P-mylyV-mxlxUr+WCM(p = l< (3)

{iz +Jz)r + myayV =N-Yxc (4)

(3)

w h e r e m,Ix and denote the mass and m o m e n t o f inertia o f t h e ship, nix, JTiyJzJx denote the Padded mass and added m o m e n t o f i n e r t i a i n corresponding directions. W is t h e ship displacement. CM is t h e m e t a c e n t i i c height. and ly denote the z-coordinates o f t h e centers o f and iriy respectively, ay denotes the x-coordinate o f t h e center o f niy. Xc is the x coordinate o f the g r a v i t y center. X, Y, K and N are the h y d r o d y n a m i c forces and m o m e n t s i n corresponding direc-tions, w h o s e detailed expressions i n t h e f o n n o f h y d r o d y n a m i c coefficients can refer to Fossen's b o o k (Fossen, 1994).

This n o n l i n e a r m o d e l is r e g a r d e d as o n e o f t h e m o s t c o m p r e -h e n s i v e m o d e l s i n t -h e o p e n l i t e r a t u r e s , i t captures t -h e essential characteristics o f 4-DOF ship m o t i o n . I n t h i s paper, t h i s n o n l i n e a r m o d e l is used f o r s i m u l a t i o n a n d p e r f o r m a n c e e v a l u a t i o n .

2.2. 3-DOF linear model and analysis

A l t h o u g h t h e n o n l i n e a r m o d e l has a h i g h accuracy, its h i g h l y n o n l i n e a r i t y a n d c o m p l e x i t y m a k e i t v e r y d i f f i c u l t t o be used t o analyze a n d design a n RRS c o n t r o l l a w . As t h e course k e e p i n g o p e r a t i o n s are c o n s i d e r e d i n m o s t s i t u a t i o n s , a n d t h e r e are o n l y s m a l l d e v i a t i o n s f r o m t h e steady-state course, t h e l i n e a r m o d e l is e x p e c t e d t o have c o n s i d e r a b l e accuracy (Perez, 2 0 0 5 ) . I n fact, m o s t RRS p r o b l e m s are s t u d i e d i n t h e f r a m e w o r k o f l i n e a r m o d e l s i n t h e o p e n l i t e r a t u r e s ( B l a n k e a n d C h r i s t e n s e n , 1993; Fang a n d Luo, 2 0 0 7 ) . To o u r k n o w l e d g e , t h e o n l y e x c e p t i o n is Laudval a n d Fossen's w o r k ( L a u v d a l a n d Fossen, 1997).

Based o n t h e l i n e a r m o d e l , t r a n s f e r f u n c t i o n f r o m r u d d e r t o r o l l a n d r u d d e r t o y a w l o o p s can be easily o b t a i n e d . Some i m p o r -t a n -t concep-ts i n RRS sys-tems, s u c h as n o n - m i n i m u m phase a n d b a n d w i d t h separation, can be clearly i l l u s t r a t e d i n the Bode d i a g r a m .

For s i m p l i c i t y , t h e r u d d e r angle S is r e g a r d e d as t h e o n l y i n p u t i n t h i s paper, t h e surge v e l o c i t y is a s s u m e d t o b e c o n s t a n t w h e n t h e p r o p e l l e r speed keeps u n c h a n g e d ( S k j e t n e a n d Fossen, 2 0 0 1 ) , T h e r e f o r e , t h i s paper assumes u = Uo, w h e r e Uo is a constant. I f w e l i n e a r i z e t h e n o n l i n e a r s y s t e m l o c a l l y a t t h e e q u i l i b r i u m p o i n t [ V o , P o , r o , ( / ' o f = [ 0 , 0 , 0 , 0 f , t h e s i m p l i f i e d 3-DOF l i n e a r m o d e l ( s w a y - r o l l - y a w ) can be o b t a i n e d (Fossen, 1994): M x + Cx = B5 (5) where x = [v,p,r,4>f, B = [ ö i , 0 2 , 0 3 , 0 ] ' '

/ m i l mn 0 0 \

m j i m22 0 0 0 0 m33 0 \ 0 0 0 1 /

/ d ] i

di2 di3 d i 4 \ ^ _ d2l d22 d23 d24 dsi d32 d33 d34 \ 0 1 0 0 / T h e e l e m e n t s i n B, M a n d C are r e l a t e d w i t h t h e p a r a m e t e r s a n d h y d r o d y n a m i c c o e f f i c i e n t s i n Eqs. ( l ) - ( 4 ) , a n d t h e i r d e t a i l e d expressions o f t h e e l e m e n t s i n B, JVf a n d C can be f o u n d i n t h e a p p e n d i x o f Fossen's b o o k (Fossen, 1994). M u l t i p l y i n g b o t h sides o f Eq. ( 5 ) b y iW \ w e o b t a i n x = - M - ' C x + l W - ' B 5 (6) It f o l l o w s : V = auV-i'ai2r+an(/> + auP+YsS (7) r = a2iV-|-a22r-|-a23<;6-i-024P-l-N3(5 (8) <Ï>=P (9)

p = a4iv-t-a42r-f a430+a44P+'Q5 (lO)

w h e r e ay is t h e c o r r e s p o n d i n g e l e m e n t i n t h e m a t r i x - i W ~ ' C ; y^, Ns a n d Kg are t h e c o r r e s p o n d i n g e l e m e n t s i n t h e v e c t o r M ~ ' B . T h e l i n e a r m o d e l equations ( 7 ) - ( 1 0 ) are u s e d f o r t h e d e s i g n o f RRS c o n t r o l l a w i n t h i s paper. Based o n t h i s l i n e a r m o d e l , t h e r u d d e r - t o - r o l l t r a n s f e r f u n c t i o n is o f t h e f o r m (Perez, 2 0 0 5 ) : 4'(s) l<ron(q^-s){q2+s) ^(S) ( P l + S ) ( P 2 + S ) ( s 2 + 2 f ^ f f l ^ S - F t t ) 2 ) >

w h e r e Kmihqi>q2>Pi'P2>^tp a n d are a l l p o s i t i v e constants, w h o s e expressions can be easily d e r i v e d f r o m t h e l i n e a r m o d e l e q u a t i o n s ( 7 ) - ( 1 0 ) . Let N{s) = Krci,(q2+s) D(s) = (Pl +s)(p2 -^s)(s2 + l^^co^s + cop Eq. (11) can be w r i t t e n as tp(s)_N(s)q^ N(s)s 5(s) D(s) D(s) = r , ( s ) - T 2 ( s ) (12) As s h o w n i n Eq. (12), T2(s) has a e x t r a s i n t h e n u m e r a t o r , w h i c h is a c t u a l l y a d i f f e r e n t i a t o r , t h u s r2(s) has a l a r g e r b a n d w i d t h and f a s t e r response t h a n T i ( s ) . I n RRS system, r i ( s ) stands f o r t h e s l o w d y n a m i c s a n d T2(s) stands f o r t h e fast d y n a m i c s o f t h e r u d d e r t o -r o l l s y s t e m .

The m o s t d i s t i n c t i v e time d o m a i n f e a t u r e o f a N M P s y s t e m is t h e i n v e r s e i n i t i a l response, because t h e t e r m s o f T i a n d T2 i n Eq. ( 1 2 ) have t h e o p p o s i t e signs. The p h y s i c a l i n t e r p r e t a t i o n is that, i f a s t e p - l i k e change i n r u d d e r angle is a p p l i e d t o m a k e t h e ship take a t u r n , the r o l l m o t i o n has a m u c h faster response t o the rudder change t h a n other DOFs. However, as l o n g as there is a s m a l l heading deviation, a reaction force induced by h y d r o d y n a m i c effects is m u c h larger t h a n t h a t p r o d u c e d by the rudder, w h i c h is also the m a i n force p r o d u c i n g t h e t u r n . This effect finally makes t h e r o l l angle o f the opposite sign to t h e i n i t i a l response (Perez, 2 0 0 5 ) .

The N M P b e h a v i o r i n r o l l m o t i o n is a c t u a l l y a consequence of t h e i n t e r a c t i o n b e t w e e n fast r o l l d y n a m i c s a n d s l o w y a w d y n a m i c s . This p a p e r w i l l s h o w t h a t t h e s i n g u l a r p e r t u r b a t i o n a p p r o a c h can be used t o separate these d i f f e r e n t t i m e scale m o t i o n s i n a n a t u r a l a n d elegant w a y .

2.3. Disturbance model

The e n v i r o n m e n t a l d i s t u r b a n c e s are v e r y c o m p l i c a t e d , thus i t is p r a c t i c a l t o use o n l y c e r t a i n s i m p l i f i e d m o d e l s t o describe the d i s t u r b a n c e s . U s u a l l y t h e e n v i r o n m e n t a l d i s t u r b a n c e s r e f e r to w i n d forces a n d w a v e forces. W i n d is o f t e n m o d e l e d as a stochastic signal w i t h n o n - z e r o mean, w h i c h w i l l cause a constant r o l l a n g l e a n d s t a t i o n a r y h e a d i n g e r r o r ( V a n A m e r o n g e n et a l , 1990). I n this paper, o n l y w a v e disturbances are c o n s i d e r e d .

W a v e m o d e l s are u s u a l l y d e s c r i b e d b y m e a n s o f f r e q u e n c y s p e c t r u m . I n RRS system, h i g h - f r e q u e n c y r o l l m o t i o n m u s t be r e d u c e d , t h u s I s t - o r d e r waves are c o n s i d e r e d i n t h e s i m u l a t i o n m o d e l . T h i s k i n d o f d i s t u r b a n c e s can be o b t a i n e d u s i n g a 2 n d -o r d e r l i n e a r a p p r -o x i m a t i -o n -o f t h e P i e r s -o n - M -o s k -o w i t z spectral d e n s i t y f u n c t i o n . To find a balance b e t w e e n t h e s i m u l a t i o n v a l i d i t y a n d a u t h e n t i c i t y , m a n y scholars a d o p t e d t h i s m o d e l t o s i m u l a t e t h e w a v e d i s t u r b a n c e s i n RRS s y s t e m ( V a n A m e r o n g e n e t al., 1990; L a u v d a l a n d Fossen, 1998; O'Brien, 2 0 0 9 ) . It is p r e f e r r e d b y ship c o n t r o l e n g i n e e r s , o w i n g t o its s i m p l i c i t y a n d a p p l i c a b i l i t y (Fossen, 1994).

(4)

R.-Y. Ren et al. / Ocean Engineering 88 (2014) 488-498 491

The disturbances o f y a w and r o l l motions Wy, a n d are given by

w^, = h(s)-W](s) (13)

(14)

w h e r e Wi(s) a n d W2(s) are Gaussian w h i t e noises, and t h e shaping filter /i(s) is described as

h(s) =

S 2 + 2 ( * O « O S + (D2 (15)

w h e r e K^, ^o, and Wo denote the d o m i n a t e w a v e s t r e n g t h coefficient, the d a m p i n g c o e f f i c i e n t a n d the encounter w a v e fi-equency, respec-tively. T h e n Wy, and can be regarded as disturbances signals to be added to the s i m u l a t i o n m o d e l .

T h i s m o d e l produces a n a r r o w b a n d t y p e o f disturbances. This n a r r o w b a n d p r o p e r t y is d u e t o t h e c o n c e n t r a t i o n o f w a v e e n e r g y at c e r t a i n f r e q u e n c y , w h i c h is the case i n m o s t o f t e n a d o p t e d w a v e s p e c t r u m m o d e l s , such as t h e P i e r s o n - M o s k o w i t z s p e c t r u m a n d JONSWAP s p e c t r u m (Fossen, 1994). T h e r e f o r e , the g i v e n n a r r o w band d i s t u r b a n c e m o d e l is reasonable.

3. Time scale analysis and control design for RRS system

I n t h i s s e c t i o n , a b r i e f i n t r o d u c t i o n o f s i n g u l a r p e r t u r b a t i o n a n d t i m e scale s e p a r a t i o n approaches are g i v e n . T h e s t a n d a r d s i n g u l a r p e r t u r b a t i o n m o d e l f o r 3-DOF ( s w a y - r o l l - y a w ) s h i p c o n t r o l s y s t e m is also d e r i v e d , u n d e r t h i s m o d e l , t h e s l o w y a w s u b s y s t e m a n d f a s t r o l l s u b s y s t e m are separated. C o n t r o l strategies are d e s i g n e d to stabilize each subsystem, a n d t h e f i n a l RRS c o n t r o l l a w is t h e c o m b i n a t i o n o f t h e c o n t r o l laws f o r each s u b s y s t e m .

3.1. Singular perturbation

This part gives a brief i n t r o d u c t i o n o f the m a i n procedure o f singular p e r h t r b a t i o n used i n the control system t o separate d i f f e r e n t time-scale m o t i o n (Kokotovic et al., 1987). Singular p e r t u r b a t i o n and time-scale separation techniques w e r e i n t r o d u c e d t o conti'ol engineer-ing since late 1960s and have been a c o m m o n tool f o r the analysis and design o f control systems (Kokotovic and Sannuti, 1968; Edelbaum and Kelley, 1970; Kokotovic et al., 1987; Esteban e t al., 2013) (Fig. 2).

The s t a n d a r d s i n g u l a r p e r t u r b a t i o n m o d e l is i n the e x p l i c i t statevariable f o r m i n w h i c h t h e derivatives o f s o m e state v a r i -ables are m u l t i p l i e d b y a s m a l l p o s i t i v e scalar e, t h a t is:

x=f(x,z,e,t), x(to)=Xo, XER"

ez=g(x,z,e,t), z(to) = Zo, zeR"

(16)

(17)

w h e r e t h e p a r a m e t e r 0 < e <^ 1 represents a s m a l l constant, x denotes t h e s l o w state variables a n d z denotes t h e f a s t state variables. I t is assumed that t h r o u g h o u t the f o r m u l a t i o n the ftinctions ƒ and g are smooth, and above ODEs have a unique solution. It is also

assumed that the system has an isolated e q u i l i b r i u m at t h e o r i g i n (x = 0 , z = 0 ) .

I n c o n t r o l a n d s y s t e m t h e o r y , i t is o f t e n a c o m m o n e n g i n e e r i n g task t o get a s i m p l i f i e d r e d u c e d - o r d e r m o d e l i n p r a c t i c e . The m o d e l equations (16) a n d (17) are steps t o w a r d r e d u c e d - o r d e r m o d e l i n g . Singular p e r t u r b a t i o n is such an a p p r o a c h to c o n v e r t t h e o r d e r r e d u c t i o n i n t o a p a r a m e t e r p e r t u r b a t i o n p r o b l e m , called singular. I f set e = 0, t h e d i m e n s i o n o f t h e o r i g i n a l s y s t e m equa-tions ( 1 6 ) a n d (17) is r e d u c e d f r o m n - f m t o n, a n d t h e s i n g u l a r d i f f e r e n t i a l e q u a t i o n (17) degenerates i n t o t h e f o l l o w i n g t r a n s c e n -d e n t a l e q u a t i o n :

0 = g ( x , z , 0 , t ) (18)

w h e r e t h e bar is used t o i n d i c a t e t h a t t h e variables b e l o n g to a s y s t e m w i t h e = 0. Due t o t h e a s s u m p t i o n t h a t t h e s y s t e m has an i s o l a t e d e q u i l i b r i u m , t h e n f r o m Eq. (18), z can be d e s c r i b e d as a f u n c t i o n o f x:

z = h(x, t ) (19)

w h e r e z = h(x, t ) is a n associated r o o t o f Eq. (18), i t r e p r e s e n t s t h e quasi-steady-state equilibrium (OSSE) o f t h e fast d y n a m i c s Eq. (17). To o b t a i n t h e r e d u c e d - o r d e r m o d e l , s u b s t i t u t i n g Eq. ( 1 9 ) i n t o Eq. ( 1 6 ) , a n d k e e p i n g t h e same i n i t i a l c o n d i t i o n f o r t h e state v a r i a b l e x{t) as f o r x{t) :

t =f(X, h(X}rOj:l^x(to) = Xo (20)

Eq. ( 2 0 ) can be r e w r i t t e n i n t o a m o r e c o m p a c t f o r m :

x = / ( x , t ) , x(to) = Xo (21)

This m o d e l is called quasi-steady-state subsystem, because z, w h o s e d e r i v a t i v e z =g/e is large w h e n e is s m a l l , m a y r a p i d l y c o n v e r g e t o a r o o t o f Eq, (18), w h i c h is quasi-steady-state f o r m o f Eq. (17). This s u b s y s t e m describes t h e s l o w d y n a m i c s o f t h e s y s t e m a n d also takes t h e fast d y n a m i c s i n t o a c c o u n t b y s u b s t i t u t i n g t h e QSSE i n t o Eq. ( 1 6 ) . S t r e t c h i n g t h e time t o T = x / e , t h e fast s y s t e m becomes

J = g ( x , z ( r ) ) , _ t

~ e (22)

w h i c h is also c a l l e d boundary layer subsystem. It describes t h e fast d y n a m i c s i n a s t r e t c h e d time scale. I n t h i s t i m e scale, x can be t r e a t e d as a c o n s t a n t p a r a m e t e r a n d e d e f i n e s t h e s t r e t c h e d t i m e scale.

As l o n g as the system is d i v i d e d i n t o the slow quasi-steady-state subsystem and the fast b o u n d a i y layer subsystem, the c o n t r o l strategy

Slow part

Quasi-steady-stata model Slow subsystem control law x = f{x,z,e,t) ?=ƒ(?.<) Full model X = / ( X . Z , £ , 0 | — I Q S S E e = -0 2 = h{x,t) Singular ODE EZ = g(x,Z,£,t) Boundary layer equation Fast subsystem control law Full model control law

Singular perturbation control strategy

(5)

can be designed separately i n each subsystem. The c o n t r o l p r o b l e m is thus s i m p l i f i e d and t h e c o n t r o l laws designed t o stabilize each subsystem are relatively easy to o b t a i n . Finally, the c o n t r o l l a w is expressed as

T = t / e , r e s u l t i n g i n the f o l l o w i n g b o u n d a r y layer ( f a s t ) subsystem:

(23)

w h e r e o i and aj- are the c o n t r o l inputs w h i c h stabilize t h e corre-s p o n d i n g corre-s l o w and facorre-st corre-subcorre-sycorre-stemcorre-s.

3.2. Time scale decomposition for RRS system

Despite o f t h e large n u m b e r o f l i t e r a t u r e s p u b l i s h e d i n t h e field o f a v i a t i o n c o n t r o l , f e w w o r k s u s i n g s i n g u l a r p e r t u r b a t i o n m e t h o d have b e e n d o n e i n t h e s h i p c o n t r o l c o m m u n i t y . T h i s is m a i n l y d u e t o t h e r e l a t i v e l y s i m p l e c o n t r o l objectives o f s h i p c o n t r o l such as c o u r s e - k e e p i n g a n d p a t h - f o l l o w i n g , a n d also because o f t h e p o o r r u d d e r e f f e c t . H o w e v e r , f o r t h e RRS c o n t r o l p r o b l e m s , d i f f e r e n t time scale m o t i o n s d o exist i n r o l l m o t i o n a n d o t h e r DOFs. As w i l l be s h o w n , t h e time scale analysis t e c h n i q u e s give a g o o d s o l u t i o n t o these p r o b l e m s . I n s p i r e d b y p r e v i o u s w o r k ( K o k o t o v i c e t al., 1987; N a i d u a n d Calise, 2 0 0 1 ; Esteban et al., 2013), this p a p e r uses s i n g u l a r p e r t u r b a t i o n m e t h o d t o analyze t h e RRS s y s t e m .

In t i m e scale analysis, t h e r e are several ad h o c assessments o f variable's speed, w h i c h is o f t e n d e f i n e d as t h e inverse o f t h e t i m e t h a t a v a r i a b l e takes t o c h a n g e across a s p e c i f i e d range o f values (Esteban a n d Rivas, 2 0 1 2 ) . T h e special n a t u r e o f t h e d y n a m i c s o f ships s h o w s t h a t t h e c o n t r o l signal is a l l o c a t e d i n t o two d i f f e r e n t time scale subsystems, t h a t is, a s l o w s u b s y s t e m a n d a f a s t s u b s y s t e m .

G e n e r a l l y speaking, a s h i p is a s l e n d e r body, t h u s Qx+Jx)< (h+Jz) h o l d s f o r m o s t surface ships. I n fact, (Iz+Jz) is o f t e n 4 0 times l a r g e r t h a n (Ix+Jx) (Fossen, 1994). Due to t h e r e l a t i v e l y s m a l l m o m e n t o f i n e r t i a a n d l a r g e r e s t o r i n g f o r c e i n r o l l m o t i o n , t h e r o l l m o t i o n has a m u c h f a s t e r response speed c o m p a r e d t o t h e y a w m o t i o n . N a t u r a l l y , w e c a n choose t h e s m a l l p a r a m e t e r e as £ = (.h+Jx)/Qz+Jz)<'i. T h e n , t h e r o l l m o t i o n is c o n s i d e r e d t o be g o v e r n e d b y s i n g u l a r ODEs r e p r e s e n t i n g a f a s t subsystem, j u s t b y m u l t i p l y i n g t h e r o l l d y n a m i c s e q u a t i o n s b y e. T h e rest o f t h e m o t i o n s are c o n s i d e r e d as s l o w subsystem. T h a t is.

w r erp

( f,(y,r,(P,p,S) \

r f2{v,r,(l),p,S) ep (24) t h e l i n e a r i z e d s i n g u l a r p e r t u r b a t i o n s h i p c o n t r o l s y s t e m can be d e s c r i b e d as t h e s l o w p a r t : v = auV+anr+ai3</>+ai4p + ys5 \j/ = r r = a2]V + a22r+a23(p+a24P+Ngö a n d t h e fast p a r t : = a34P ep = d4xV+a42r+a43(j)+d4i,p+Ni;5 (25) (26) (27) (28) (29) w h e r e «34 = e = ( / ; < + J J / ( / z - t - ; j , 041 = £ 0 4 , , 0 4 2 = 6 0 4 2 , 043 = 6043, 044 = £ 0 4 4 . Ns= eNs- T h e i n i t i a l c o n d i t i o n s are

[v(fo), v / ( f o ) , r(to), (pitolpito)]^ = [Vo, V/Q, TQ, 9^0. Pol'' (30)

The t i m e - s c a l e d e c o m p o s i t i o n is a c h i e v e d b y s t r e t c h i n g t h e fast subsystem's t i m e scale. The s t r e t c h e d time scale is g i v e n b y

deb - ^ = g^(v,i//, r,(p,p) = a34P dp -^ = gpiv,W<r,(P,p) = 041 v-i-a42r-Fa43<^-l-a44p+JVa5 (31) (32)

Let [(/>,p]'' = h(y,\fr,r,5)eR^ r e p r e s e n t t h e QSSE o f t h e b o u n d a r y layer subsystem w h e n s e t t i n g e = 0, t h a t is:

g4,(y>WJ,<P,P) = 0

gp(v,y/J,4),p) = 0

(33)

(34)

T h e bar here d e m o n s t r a t e s t h a t t h e variables b e l o n g to a system w i t h 6 = 0. T h e n s o l v i n g t h e Eqs. ( 3 3 ) a n d ( 3 4 ) , results i n Ö4iV-HÖ42r-HNg(5 - Ö 4 3 (35) (36) p = 0 Set t h e values o f (p a n d p i n t h e s l o w s u b s y s t e m t o ^ a n d p , by s u b s t i t u t i n g Eqs. ( 3 5 ) a n d ( 3 6 ) i n t o Eqs. ( 2 5 ) - ( 2 7 ) , and k e e p i n g the same i n i t i a l c o n d i t i o n s as Eq. ( 3 0 ) , t h e s l o w quasi-steady-state s u b s y s t e m can be o b t a i n e d :

7 = auV+aur+Ys5 (37)

W=f (38)

f = a2:V + a2zF+NsS (39)

w i t h t h e i n i t i a l c o n d i t i o n :

[v(to), W(tol r(to)f = [Vo, y/o, rof (40)

w h e r e v,\//,r d e n o t e t h e q u a s i s t e a d y s t a t e variables, On = 0 n ^13041/043,012 = «12 013042/043, Ys = V^J^ 013^^/043, 021 = 021 -023 0 4 1 / 0 4 3 , 0 2 2 = 0 2 2 - -023042/043, a n d = - O23 iV5/a43. T h i s p r o c e d u r e is a c t u a l l y c o n s i d e r i n g t h e c o u p l i n g e f f e c t of t h e r o l l m o t i o n o n t h e y a w d y n a m i c s b y s u b s t i t u t i n g t h e QSSE o f t h e f a s t b o u n d a r y l a y e r s u b s y s t e m i n t o t h e s l o w p a r t o f the s y s t e m Eqs. ( 2 5 ) - ( 2 7 ) .

3.3. Control design for RRS system

The m a i n goal f o r s h i p c o n t r o l s y s t e m is to k e e p t h e h e a d i n g at a d e s i r e d course. For RRS systems, i t is also r e q u i r e d t o reduce the r o l l angle as m u c h as possible, at t h e c o n s t r a i n t o f rudder s a t u r a t i o n a n d speed l i m i t .

I n s i n g u l a r p e r t u r b a t i o n a p p r o a c h , t h e separate c o n t r o l strategy is used t o d e s i g n t h e c o n t r o l l a w : t h e q u a s i - s t e a d y - s t a t e subsystem is used t o d e s i g n t h e y a w d y n a m i c s , a n d t h e b o u n d a r y layer s u b s y s t e m is used t o c o n t r o l t h e r o l l m o t i o n . Fang a n d Luo (2007) also used t h e c o n c e p t o f separate c o n t r o l i n RRS p r o b l e m , however, t h e y d i d n o t c o n s i d e r t h e c o u p l i n g e f f e c t i n t h e i r separate control strategy. T h e i r w o r k s h o w s t h a t t h e separate c o n t r o l has better h e a d i n g p e r f o r m a n c e b u t w o r s e r o l l r e d u c t i o n p e r f o r m a n c e c o m -p a r e d to the c o m -p a c t c o n t r o l s t r a t e g y w h i c h considers the c o u p l i n g e f f e c t . I n t h e p r e s e n t paper, t h e s i n g u l a r p e r t u r b a t i o n m e t h o d takes i n t o a c c o u n t t h e c o u p l i n g e f f e c t b e t w e e n r o l l and h e a d i n g b y s u b s t i t u t i n g t h e QSSE i n t o t h e s l o w subsystem.

T h e use o f s e q u e n t i a l time scale d e c o m p o s i t i o n p e r m i t s to d e s i g n c o n t r o l strategies f o r S w h i c h is t h e s u m o f t w o c o m p o -nents, 5 = S:i,+S^, w h e r e 0^ = ry,(v,if/,r) is used t o stabilize the s l o w h e a d i n g s u b s y s t e m a n d 8^=r^{v,\i/,r,(p,p) is used t o reduce t h e fast r o l l m o t i o n .

(6)

R.-Y. Ren et al. j Ocean Engineering 88 (2014) 488-498 493

3.3.1. Control law for slow subsystem

T h e c o n t r o l o b j e c t i v e f o r the s l o w quasi-steady-state s u b s y s t e m is t o k e e p t h e h e a d i n g at a d e s i r e d course. W i t h o u t loss o f generality, t h i s paper sets the d e s i r e d y a w angle t o be ifr^ = 0 ° .

A L y a p u n o v f u n c t i o n is c o n s t r u c t e d f o r t h e s l o w quasi-steady-state s u b s y s t e m e q u a t i o n s ( 3 7 ) - ( 3 9 ) , w h i c h w i l l g u a r a n t e e t h e s t a b i l i t y o f ( / / a n d r. As l o n g as t h e c o n t r o l l a w is o b t a i n e d , i t c a n be s h o w n t h a t t h e s w a y v e l o c i t y v is also g u a r a n t e e d t o c o n v e r g e to zero. This also coincides w i t h t h e f a c t t h a t a h e l m s m a n u s u a l l y o n l y uses t h e h e a d i n g angle a n d h e a d i n g rate t o g u i d e his s t e e r i n g action, t h e s w a y v e l o c i t y is l e f t to be d a m p e d o u t b y itself. For s i m p l i c i t y , t h e bar over t h e quasi-steady-state variables is n e g l e c t e d f r o m here o n .

Select t h e L y a p u n o v f u n c t i o n F(t) > 0 as

Fit): lV^ + lk2W^+^k3r^>0 (41)

w h e r e k^,k2,k3 are n o n - n e g a t i v e constants; take t h e d e r i v a t i v e o f F(f) w i t h respect t o time: F(t) = ;<i VV + k2y/y/+/<3 r r s u b s t i t u t e Eqs. ( 3 7 ) - ( 3 9 ) i n t o Eq. ( 4 2 ) , i t f o l l o w s : F ( f ) = /<i v(a „ v + a u r + Y s S y , ) + k 2 y / r + k3r(a2^v+a22r+NsSyr) (43) by u s i n g f u l l state f e e d b a c k o f t h e s l o w s u b s y s t e m : 5y, = av+br+aj/ (44) w h e r e a, b a n d c are t h e c o r r e s p o n d i n g f e e d b a c k gains, Eq. ( 4 3 ) c a n

be r e w r i t t e n as

F(t) = v\krau+k^Ysa} + r^(k3a22+k3N5b)

+vr(kxan + hYsb-i-k3a2^ -H/CaN^a) +vi;/(kiYsc)+ry/(k2 + k3N5C} c h o o s i n g -e-/<3a22 Ns k3Ns w h e r e e is a p o s i t i v e constant, t h e n F ( t ) = - e r ^ < 0 k3Ns (45) (46) (47)

Since w e set k-[ = 0, above process o n l y proves t h a t y/ a n d r a s y m p t o t i c a l l y c o n v e r g e t o zero, w h i l e s w a y v e l o c i t y v is n o t g u a r a n t e e d to converge t o zero. H o w e v e r , f o r RRS a n d course k e e p i n g p r o b l e m s , s w a y m o t i o n is n o t a k e y issue, besides, t h e s w a y v e l o c i t y w i l l d a m p o u t v e r y f a s t i f a c o n s t a n t h e a d i n g angle is guaranteed, d u e t o t h e f a c t t h a t t h e l a t e r a l d a m p i n g f o r c e is u s u a l l y v e r y large f o r a surface ship.

By trial and error over d i f f e r e n t values o f /<2, ks and e, appropriate values can be d e t e r m i n e d according to Eq. (46), these parameters should consider b o t h t h e r u d d e r l i m i t a t i o n and t h e response characteristics o f the y a w m o t i o n .

O t h e r m e t h o d s such as t h e classical p o l e - p l a c e m e n t m e t h o d and slide m o d e c o n t r o l l e r (Fang a n d Luo, 2 0 0 7 ) can also be used t o stabilize such s u b s y s t e m , h o w e v e r , c o n s t r u c t i n g such a L y a p u n o v f u n c t i o n is c o n v e n i e n t t o evaluate t h e n o n l i n e a r e f f e c t o n t h e system's s t a b i l i t y , w h i c h w i l l be s h o w n l a t e r i n Section 3.4.

3.3.2. Control law for fast subsystem

T h e f a s t b o u n d a r y layer s u b s y s t e m Eqs. ( 2 1 ) a n d ( 2 2 ) can be s t a b i l i z e d b y s e l e c t i n g t h e c o n t r o l signal S^. To c o n s i d e r t h e s l o w subsystem's s l o w v a r y i n g r u d d e r e f f e c t o n t h e f a s t s u b s y s t e m , s u b s t i t u t i n g t h e 5^ i n t o t h e f a s t subsystem, t h e fast s u b s y s t e m c a n be d e s c r i b e d as dd) ^ = 041V-I-042 r-F 043 «6 + a 44P-F iVa(5v'+15^) = a43'P+a44P+NsS4, + C{v, y/, r) (48) w h e r e G(v,y/,r) = ia4i+aNg)v+cNsy/+(a42 + bNs)r, t h e v a r i a b l e s v,y/ a n d r are r e g a r d e d as c o n s t a n t i n t h i s s t r e t c h e d t i m e scale. This s u b s y s t e m can be w r i t t e n as a m a s s - s p r i n g - d a m p i n g s y s t e m : w h e r e 2 CO - 034^43 - 0 3 4 0 4 4 2a43%/-a34a43 (42) G = a 3 4 G ( v , v ^ , r ) (49) (50) (51) (52) (53)

(On is the n a t u r a l f r e q u e n c y o f t h e r o l l system, f is t h e d a m p i n g c o e f f i c i e n t o f t h e s y s t e m . G(v, y/, r) can be r e g a r d e d as a c o n s t a n t d i s t u r b a n c e i n t h e f a s t time scale, w h i c h w i l l cause a steady r o l l response. T h i s d i s t u r b a n c e d e m o n s t r a t e s t h e s l o w subsystem's e f f e c t o n t h e fast s u b s y s t e m . F r o m Eq. ( 4 8 ) , t h e e q u i l i b r i u m p o i n t o f t h e fast s u b s y s t e m is G(v, yr, r ) 0 0 = -- 0 4 3 (54) Po = 0 (55) (f)Q is r e g a r d e d as a c o n s t a n t e q u i l i b r i u m p o i n t i n t h e s t r e t c h e d time scale. I n t h i s s i t u a t i o n , t h e c o n t r o l l a w is t o s t a b i l i z e t h e r o l l angle (p t o its e q u i l i b r i u m p o i n t cpa, r a t h e r t h a n z e r o . This is q u i t e i m p o r t a n t especially i n t h e case w h e r e v,y/ a n d r have r e l a t i v e l y large values, f o r e x a m p l e , i n a t u r n i n g o p e r a t i o n o r s u d d e n l y c h a n g i n g course c o n t r o l , t h e s h i p w i l l have a large r o l l angle e q u i l i b r i u m p o i n t . I n these cases, i f t h e p r o p o r t i o n a l c o n t r o l l e r ( P - c o n t r o l l e r ) o f t h e r o l l angle is used, t h e f e e d b a c k l a w s s h o u l d be t h e f o r m o f kp(ip-(pQ) r a t h e r t h a n kpip.'Vor s i m p h c i t y , t h i s p a p e r o n l y uses a d e r i v a t i v e c o n t r o l l e r ( D - c o n t r o l l e r ) o f t h e r o l l angle. T h e i n t e n t i o n is t o increase t h e d a m p i n g r a t i o o f t h e s y s t e m , w h i c h means t h e f e e d b a c k is t a k e n as 2^sCo„ dtp (56) w h e r e ^g is a p o s i t i v e constant. S u b s t i t u t i n g t h e f a s t c o n t r o l l a w Eq. ( 5 6 ) i n t o ( 4 9 ) , t h e t o t a l d a m p i n g r a t i o o f t h e r o l l s y s t e m becomes = (^+^s) > w h i c h means the system w i l l have a higher d a m p i n g ratio u n d e r the c o n t r o l law. Thus a faster d a m p i n g speed i n roll m o t i o n is expected.

T h e r e f o r e , b y t r e a t i n g t h e fast a n d s l o w s u b s y s t e m s separately, t h e final c o n t r o l l a w is

(57)

w h e r e the expressions o f 5^ and 5^ are given b y i n Eqs. (44) and (56). H o w e v e r , i t is n o t c o m p l e t e l y e q u i v a l e n t b e t w e e n t h e q u a s i -steady-state s u b s y s t e m e q u a t i o n s ( 3 7 ) - ( 4 0 ) a n d t h e f u l l s y s t e m e q u a t i o n ( 2 4 ) , t h e d i s c r e p a n c y b e t w e e n t h e t w o m o d e l s is t h e f a s t t r a n s i e n t . T h e separate c o n t r o l d e s i g n is t o m a k e t h e associated s u b s y s t e m stable a n d w i t h a p r e s c r i b e d d e s i r e d d y n a m i c s . H o w -ever, t h i s does n o t g u a r a n t e e t h e a s y m p t o t i c s t a b i l i t y o f t h e f u l l s y s t e m . F o r t i j n a t e l y , u n d e r several a s s u m p t i o n s , t h e f u l l m o d e l can also be g u a r a n t e e d t o be stable i f e i's s u f f i c i e n t l y s m a l l ( K o k o t o v i c

(7)

e t al., 1 9 8 7 ) . I n s i n g u l a r p e r t u r b a t i o n a p p r o a c h , i t is an i m p o r t a n t issue t o d e f i n e t h e b o u n d s o f t h e s i n g u l a r l y p e r t u r b e d p a r a m e t e r e. As t o the RRS s y s t e m , e is o f t e n less t h a n 0.025. M o r e details a b o u t t h e s t a b i l i t y issues c a n r e f e r to K o k o t o v i c et al. ( 1 9 8 7 ) .

3.4. Robust analysis in yaw motion

To c o n t r o l t h e h e a d i n g at a desired v a l u e is o f p r i m a r y i m p o r t a n c e i n RRS c o n t r o l system. T h e quasisteadystate s u b -s y -s t e m i-s u-sed t o de-scribe t h e h e a d i n g c o n t r o l -s y -s t e m , a n d t h e s u b s y s t e m Eqs. ( 3 7 ) - ( 3 9 ) is p r o v e d t o be stable b y c o n s t r u c t i n g a L y a p u n u v f u n c t i o n . H o w e v e r , n o n l i n e a r i t i e s are n e g l e c t e d i n t h i s l i n e a r i z e d m o d e l . Besides, t h e r e d u c e d - o r d e r s l o w s u b s y s t e m is o b t a i n e d b y s u b s t i t u t i n g t h e QSSE o f t h e b o u n d a r y layer subsys-t e m i n subsys-t o q u a s i - s subsys-t e a d y - s subsys-t a subsys-t e subsyssubsys-tem, subsys-t h i s p r o c e d u r e does n o subsys-t c o n s i d e r t h e t r a n s i e n t i n t e r a c t i o n e f f e c t b e t w e e n r o l l a n d y a w m o t i o n s . To evaluate t h e i m p a c t s o f these factors o n t h e s l o w subsystem's s t a b i l i t y , especially t h e y a w m o t i o n , t h e f o l l o w i n g m o d e l is u s e d i n t h e robustness analysis: = '• (58) i-= a2iV-\-a22r-{-Ns8-i-A (59) w h e r e A captures t h e m o d e l u n c e r t a i n t i e s w h e n d e r i v i n g a r e d u c e d - o r d e r l i n e a r e q u a t i o n . The s w a y d y n a m i c s is n e g l e c t e d h e r e because t h a t t h e s w a y v e l o c i t y is o f t e n v e i y s m a l l a n d i t can d a m p o u t b y i t s e l f To analyze t h i s u n m o d e l e d d y n a m i c s e f f e c t o n t h e s t a b i l i t y o f t h e system, t h e s i m i l a r r o b u s t analysis a p p r o a c h is used to evaluate t h e h e a d i n g c o n t r o l system's s t a b i l i t y as L i e t a l . ( 2 0 0 9 ) , a n d t h e s a m e a s s u m p t i o n is m a d e as i n t h e i r w o r k : Assumption 1 : A satisfies \A\<rQ + Yjv\+y,\r\ Assumption 2: v satisfies \v\^ro+Y,.\r\ (60) (61)

w h e r e Yo,yv>Yr>7o<7r are a l l p o s i t i v e constants.

I n Li et al.'s ( 2 0 0 9 ) w o r k , t h e y e x p l a i n e d t h a t i n A s s u m p t i o n 1 is used t o c a p t u r e t h e effects o f surge speed a n d o t h e r u n c e r t a i n t i e s o n y/ a n d r d y n a m i c s , YO is i n t r o d u c e d t o d e m o n -strate b o u n d e d h i g h e r o r d e r n o n l i n e a r t e r m s i n t h e c o n t r o l i n p u t s , a n d u n c e r t a i n t i e s i n r t e r m are c a p t u r e d b y Y,-- I n t h e p r e s e n t s t u d y , i t is also a s s u m e d t h a t t h e t r a n s i e n t i n t e r a c t i o n e f f e c t s b e t w e e n r o l l a n d y a w m o t i o n s are c a p t u r e d b y YO,YV a n d Yr- A s s u m p t i o n 2 i n t e n d s t o e v a l u a t e t h e b o u n d a r y o f s w a y v e l o c i t y v, w h e r e f o captures t h e phase l a g b e t w e e n the response v a n d r, a n d f r f o r t h e p r o p o r t i o n a l r e l a t i o n s h i p b e t w e e n v a n d r. S u b s t i t u t i n g Eqs. ( 5 8 ) a n d ( 5 9 ) a n d t h e f e e d b a c k l a w Eq. ( 4 4 ) i n t o t h e d e r i v a t i v e o f t h e L y a p u n o v f u n c t i o n Eq. ( 4 3 ) , i t f o l l o w s : F(t) = k2y/y/-i-k3ri' = r2V/r+/<3r(a2i v + a 2 2 r - f - Ï V a 5 + z l ) =-er'^-\rk3rA ( 6 2 ) A c c o r d i n g t o t h e t w o a s s u m p t i o n s :

F(t)<~er'^ + k3\r\(Ya+Y, -YAn)

<-er^ + k3\r\{YQ+Y^(yo+Yr\n) + YAy\) = -dor^-t-Zoirl w h e r e do^e-k^Yr-kiYvYr (63) (64) io = hYo+k3YvYr (65) A c c o r d i n g to Eq. (63), i t is o b v i o u s t h a t F{t) < 0 i n t h e r e g i o n

l'1<-

(66) I t s h o w s t h a t as l o n g as t h e y a w rate r is r e s t r i c t e d i n t h e region V, t h e h e a d i n g c o n t r o l s y s t e m c a n be r o b u s t l y stable e v e n i f there exists u n m o d e l e d d y n a m i c s . I f t h e m o d e l u n c e r t a i n t y is n o t s i g n i f i c a n t , t h e v a l u e o f v, r a n d (p can be m a d e r e l a t i v e l y small, b y p r o p e r l y selecting t h e l o w s u b s y s t e m c o n t r o l l e r gains a, Ö, c and f a s t r o l l m o t i o n g a i n t h u s i t can be g u a r a n t e e d t h a t r e D . T h e r e f o r e , t h e r o b u s t s t a b i l i t y can be g u a r a n t e e d .

4. Simulation results

4.J. Simulation model description

A 4 - D O F ( s u r g e - s w a y - y a w - r o l l ) n o n l i n e a r m o d e l o f a S175 c o n t a i n e r s h i p is used t o evaluate t h e p e r f o r m a n c e o f the d e r i v e d RRS c o n t r o l l a w . This n o n l i n e a r m o d e l w a s o b t a i n e d b y a set o f captive m o d e l tests (Son and N o m o t o , 1982). I t is c o m p r e h e n s i v e a n d accurate, t h u s has o f t e n b e e n u s e d b y m a n y scholars to s i m u l a t e t h e 4 D O F s h i p m o t i o n . T h i s p a p e r takes i t as a b e n c h -m a r k -m o d e l t o evaluate t h e p e r f o r r n a n c e o f t h e l i n e a r -m o d e l . Both t h e m o d e l s are a d d e d w i t h t h e same w a v e d i s t u r b a n c e s and r u d d e r c o n t r o l l a w . T h e m a i n d a t a o f t h e s h i p are d e s c r i b e d i n Table 1. The d e t a i l e d i n f o r m a t i o n a b o u t t h e n o n l i n e a r m o d e l can be f o u n d i n Son a n d N o m o t o ( 1 9 8 2 ) a n d Fossen ( 1 9 9 4 ) .

The t i m e d o m a i n s i m u l a t i o n o f t h e s h i p m o t i o n is c o n d u c t e d by u s i n g t h e f o u r t h - o r d e r R u n g e - K u t t a m e t h o d w i t h a t i m e i n t e r v a l o f 0.1s. The r u d d e r s a t u r a t i o n a n d r a t e l i m i t s ( | 5 | < 2 0 ' ' and |<5|<5''/s) are c o n s i d e r e d i n t h e f e e d b a c k d e s i g n a n d s i m u l a t i o n . The t o t a l s i m u l a t i o n time is Ttotai = 1 2 0 0 s, a n d t h e i n i t i a l c o n d i -tions are chosen as VQ = 0, i/Zg = 0, ro = 0,tpo = 0, a n d Po = 0- The s h i p speed is a r o u n d 7.2 m/s. I n o r d e r t o t e s t i f y t h e c o n t r o l laws' e f f e c t i v e n e s s i n s t e e r i n g o p e r a t i o n , t h e d e s i r e d h e a d i n g angle is set t o be 0 d u r i n g t h e first stage, a n d changes t o 1 0 ° a t 3 0 0 t h second, t h e n t u r n s back t o 0 ° again at 6 0 0 t h second. The c o n t r o l feedback gains are chosen as a = 0 . 0 3 , b = 1.35, c = - 2 , a n d (*a = 0.077. These c o n t r o l p a r a m e t e r s are selected b y t a k i n g t h e r u d d e r l i m i t s a n d t h e ranges o f t h e state v a r i a b l e s i n t o c o n s i d e r a t i o n . The w a v e d i s t u r b a n c e t o r o l l m o t i o n is a d d e d d i r e c t l y i n t o t h e r i g h t side o f t h e s t a n d a r d e q u a t i o n : 4^=f{v,y/,r,(p,p,S)-i-w^ (67) T h e w a v e s h a p i n g p a r a m e t e r s are selected as /Cw = 8 x l 0 ^ ^ ^0 = 0.075, a n d CL>O = 0 . 2 1 . W I is t h e gaussian w h i t e n o i s e w i t h v a r i a n c e o f c7\ = 0.5 a n d a zero m e a n .

Roll d i s t u r b a n c e s w i t h a d o m i n a t e f r e q u e n c y near t h e ship's n a t u r a l f r e q u e n c y ( « „ 0,22 rad/s) are u s e d t o create a r e l a t i v e l y large r o l l angle t o evaluate t h e RRS p e r f o r m a n c e , o t h e r frequencies are also tested. T h e r o l l m o t i o n d i s t u r b a n c e s are s h o w n i n Fig. 3.

Table 1

Principal particulars of S175 container ship.

Item Symbol Value

Length I 175 m

Breadth B 25.4 m

Mean draft d 8,5 m

Displacement volume V 21,222 m^

Keel to transverse metacenter KM 10,39 m

Keel to buoyancy center KB 4,62 m

Block coefficient _CB 0,559

(8)

R.-Y. Ren et al. / Ocean Engineering 88 (2014) 488-498 495 200 300 400 600 600 700 Ti5r!3(second) 800 900 1000 , x 1 0 440 450 460 470 480 Time(second) 490 60O

Fig. 3 . Wave disturbances of the rol! motion, the shaping function parameters are selected as K„ = S x 10~'',fo = 0.075, and = 0.21.

Bode Diagram

10"' 10" 10' Frequency (rad/s) ^

Fig. 4. Bode diagram of yaw spectrum and roll spectrum.

Fig. 4 d e m o n s t r a t e s t i i e o p e n l o o p Bode d i a g r a m o f t h e r u d d e r -t o - y a w if/(s)/S(s) a n d r u d d e r - -t o - r o h (p(s)/S(s) f r e q u e n c y responses for t h e s h i p . As s h o w n i n t h e u p p e r m a g n i t u d e d i a g r a m , t h e r e is e n o u g h b a n d w i d t h s e p a r a t i o n b e t w e e n t h e r u d d e r - t o - y a w a n d r u d d e r - t o - r o l l loops. The c u t - o f f f r e q u e n c y o f y a w s p e c t r u m is a r o u n d 0.063 rad/s, w h i c h is m u c h s m a l l e r t h a n t h e n a t u r a l r o l l f r e q u e n c y . The o p e n - l o o p g a i n o f y a w response is less t h a n - 20 dB near t h e n a t u r a l f r e q u e n c y , w h i c h m e a n s t h a t t h e r u d d e r m o v i n g at such a f r e q u e n c y has v e r y l i t t l e i m p a c t o n y a w m o t i o n . W h i l e at such f r e q u e n c y , t h e o p e n - l o o p g a i n o f r o l l response is a r o u n d 4 d B . This b a n d w i d t h s e p a r a t i o n makes i t possible t o design t h e RRS s y s t e m f o r t h i s ship. The N M P p h e n o m e n o n i n r o l l m o t i o n can be f o u n d i n t h e phase d i a g r a m i n Fig. 4, w h i c h d e m o n s t r a t e s a large phase l a g a n d a large range o f phase angle.

4.2. RRS performances in nonlinear model

The p e r f o r m a n c e s w i t h a n d w i t h o u t t h e RRS c o n t r o l p a r t i n t h e n o n l i n e a r m o d e l are d e m o n s t r a t e d i n Fig. 5.

Fig. 5(a) s h o w s v e r y s i m i l a r response p e r f o r m a n c e s o f t h e y a w m o t i o n s u n d e r these t w o o p e r a t i o n s . The r e s u l t i n d i c a t e s t h a t t h e d e s i g n e d h i g h f r e q u e n c y r u d d e r o p e r a t i o n 8^ is f a r b e y o n d t h e -without RRS -with RRS 400 600 800 Time(second) 400 600 BOO Time(second)

c

30 20 2, 01 10 CD c

ra

0 V -10 •a -10 Z3 -20 a: -20 -30 400 600 800 Time(second) 1200 Fig. 5. (b) roll

Nonlinear model simulation results with and without RRS: (a) yaw angle, angle and (c) rudder angle.

b a n d w i d t h o f y a w m o t i o n , a n d t h u s has l i t t l e i m p a c t o n t h e y a w m o t i o n . The h e a d i n g angle can be r e s t r i c t e d at t h e d e s i r e d course w i t h considerable accuracy u n d e r the h i g h frequency r u d d e r i n p u t

The r o l l p e r f o r m a n c e s are s h o w n i n Fig. 5 ( b ) . The r o l l angle can reach + 1 5 ° u n d e r t h e w a v e d i s t u r b a n c e s w i t h o u t RRS, w h i l e i t is l i m i t e d w i t h i n + 5 ° w h e n t h e d e s i g n e d RRS c o n t r o l l a w is o n . A t m o s t t i m e , t h e r o l l a n g l e is r e s t r i c t e d w i t h i n + 3 ° . The p e r f o r m a n c e m e e t s t h e s t a n d a r d a n d c r i t e r i o n m a d e b y Faltinsen f o r m a n u a l a n d i n t e l l e c t u a l w o r k ( F a l t i n s e n , 1993). Fig. 5(c) d e m o n s t r a t e s t h e r u d d e r i n p u t s . I t s h o w s t h a t t h e r o l l r e d u c t i o n is at t h e expenses o f h i g h f r e q t t e n c y r u d d e r o p e r a t i o n s . T h e r u d d e r m o v e s at a f r e q u e n c y s i m i l a r t o t h e r o l l m o t i o n ' s n a t u r e f r e q u e n c y . I n t h i s case, d u e t o t h e r e l a t i v e l y l o n g r o l l p e r i o d o f t h e ship, m o s t o f t h e r u d d e r o p e r a t i o n s are b e l o w t h e r u d d e r s a t u r a t i o n a n d rate l i m i t s , a n d m a k e t h e d e s i g n e d RRS c o n t r o l l a w s to have a s a t i s f y i n g p e r f o r m a n c e .

4.3. Comparison between linear and nonlinear models

I n t h i s paper, t h e RRS c o n t r o l l a w is d e r i v e d f r o m t h e r e d u c e d -o r d e r l i n e a r m -o d e l , s-o t h e accuracy -o f t h i s l i n e a r m -o d e l is -o f i m p o r t a n c e . I t is t h u s necessary to e v a l u a t e t h e accuracy o f t h e l i n e a r m o d e l . For t h i s p u r p o s e , t h e 4 - D O F n o n l i n e a r m o d e l is u s e d as a v i r t u a l s h i p f o r s i m u l a t i o n a n d p e r f o r m a n c e e v a l u a t i o n ( L i et al., 2 0 0 9 ) . B o t h t h e l i n e a r a n d n o n l i n e a r m o d e l s are u n d e r t h e RRS c o n t r o l l a w a n d w a v e d i s t u r b a n c e s .

The s i m u l a t i o n results are s h o w n i n Figs. 6 - 8 . Fig. 6 i l l u s t r a t e s t h e y a w m o t i o n p e r f o r m a n c e s o f t h e t w o m o d e l s . I t s h o w s t h a t t h e d i f f e r e n c e i n the y a w angles b e t w e e n l i n e a r a n d n o n l i n e a r m o d e l s is i n d i s t i n g u i s h a b l e f o r m o s t o f t h e time, e x c e p t f o r s o m e peak a n d t r o u g h values, at w h i c h t h e n o n h n e a r d y n a m i c s a n d c o u p l i n g e f f e c t are r e l a t i v e l y larger, t h u s s o m e d e v i a t i o n s a p p e a r b e t w e e n l i n e a r a n d n o n l i n e a r m o d e l s . Fig. 7 s h o w s t h e r o l l m o t i o n p e r f q r m a n c e o f t h e t w o m o d e l s . Despite o f t h e s i m i l a r i t y , t h e r o l l m o t i o n o f t h e n o n l i n e a r m o d e l is

(9)

- - nonlinear model — linear model

O 200 400 600 800 1000 1200 Tlme(second)

Fig. 6. Yaw angle performances of the linear and nonlinear models with RRS control strategy.

400 420 440 460 480 500 520 Tlme(second)

Fig. 7. Roll angle performances of the linear and nonlinear models with RRS control strategy. 1200 Fig. 8. control 680 700 720 Tlme(seoond)

Rudder angle performances of the linear and nonlinear models with RRS strategy.

a l i t t l e s m a l l e r t h a n t h a t o f t h e l i n e a r m o d e l f o r m o s t o f t h e t i m e . This is m a i n l y d u e to t h a t t h e n o n l i n e a r i t i e s o f t e n o f f e r t h e system a n o n l i n e a r d a m p i n g e f f e c t w h i c h t e n d s t o m a k e t h e s y s t e m m o r e stable. Thus t h e designed c o n t r o l l a w based o n t h e l i n e a r m o d e l

tends to give a m o r e conservative c o n t r o l s t r a t e g y a n d m a k e the real s y s t e m safer.

Fig. 8 s h o w s t h e d i f f e r e n c e s o f t h e r u d d e r o p e r a t i o n s . Similar w i t h t h e r o l l p e r f o r m a n c e , t h e r u d d e r o p e r a t i o n i n nonlinear m o d e l is also a l i t t i e s m a l l e r t h a n t h a t i n l i n e a r m o d e l . T h e rudder o p e r a t i o n m e e t s t h e r u d d e r s a t u r a t i o n a n d rate l i m i t at a r o u n d the 6 5 0 t h second w i t h a f i x e d slope. T h i s r u d d e r s a t u r a t i o n is t o some degree i n e v i t a b l e i n RRS c o n t r o l strategy. A b i g c h a l l e n g e i n RRs c o n t r o l d e s i g n is t o m a k e a t r a d e - o f f b e t w e e n t h e RRS perfor-m a n c e a n d t h e r u d d e r o p e r a t i o n l i perfor-m i t s . I n t h e p r e s e n t case, the d e s i g n e d c o n t r o l l a w is w e l l w i t h i n t h e r u d d e r l i m i t a t i o n f o r the m o s t t i m e , t h u s a g o o d p e r f o r m a n c e is expected.

4.4. Track keeping performances

T r a c k k e e p i n g p e r f o r m a n c e s are v e r y i m p o r t a n t i n s h i p m o t i o n c o n t r o l . A l t h o u g h m o s t RRS designs are o n l y c o n s i d e r e d i n course k e e p i n g o p e r a t i o n s , also t h e t r a c k k e e p i n g p e r f o r m a n c e s h o u l d be c o n s i d e r e d , w h e n d e s i g n i n g a RRS s y s t e m f o r r o l l r e d u c t i o n . I n fact, t h e t w o c o n t r o l objectives have a l o t i n c o m m o n , t h e ship's t r a c k k e e p i n g s y s t e m can be d e s i g n e d f r o m t h e course keeping s y s t e m b y i n c l u d i n g a n a d d i t i o n a l p o s i t i o n f e e d b a c k (Velagic et al., 2 0 0 3 ) .

I n t h i s paper, a s i m u l a t i o n is c o n d u c t e d t o evaluate t h e validity o f t h e d e r i v e d RRS c o n t r o l l a w i n t r a c k k e e p i n g p r o b l e m s . The yaw disti)j;bance is also c o n s i d e r e d i n t h i s s i m u l a t i o n , w h e r e the gaussian w h i t e noise w i t h a variance o f (72 = 0.5 a n d a zero mean is a d o p t e d , i t is f i l t e r e d b y t h e s h a p i n g f i l t e r / ( s ) to create the y a w d i s t u r b a n c e . T h e t r a c k k e e p i n g c o n t r o l l a w is selected as ST = S-i-Ccid. w h e r e S is t h e p r e d e f i n e d RRS c o n t r o l i n p u t ; d is the d i s t a n c e f r o m t h e s h i p to t h e p a t h ; C d = 0 . 0 0 2 , w h i c h is t h e gain of t h e p o s i t i o n f e e d b a c k ; t h e d e s i r e d p a t h is s i m p l y selected as x - y = 0, w h e r e x a n d y are t h e p o s i t i o n c o o r d i n a t e s . T h e initial p o s i t i o n o f t h e s h i p is selected as ( 0 , - 8 0 0 ) . A h e a d i n g c o n t r o l is n e e d e d t o t r a c k t h e p a t h , i n w h i c h case, t h e c o u p l i n g e f f e c t o f the y a w m o t i o n , s w a y m o t i o n a n d t h e r o l l m o t i o n m a y be a n issue. All t h e o t h e r p a r a m e t e r s are k e p t t h e same as i n t h e previous s i m u l a t i o n case. T h e s i m u l a t i o n results are s h o w n i n Figs. 9 and 10.

Fig. 9 i l l u s t r a t e s t h e t r a c k k e e p i n g p e r f o r m a n c e w i t h and w i t h o u t t h e RRS c o n t r o l l a w . The p e r f o r m a n c e is s a t i s f y i n g even w i t h a large i n i t i a l p o s i t i o n d e v i a t i o n . As s h o w n i n t h i s figure, the track p e r f o r m a n c e s are veiy close i n b o t h cases, w h i c h d e m o n -strates t h a t the t r a c k k e e p i n g p e r f o r m a n c e o f t h e s h i p is n o t highly a f f e c t e d b y t h e h i g h f r e q u e n c y p a r t o f t h e RRS c o n t r o l l a w . Fig. 10 gives t h e r o l l p e r f o r m a n c e s w i t h a n d w i t h o u t RRS c o n t r o l strategy. I t s h o w s t h a t t h e r o l l angle can be e f f e c t i v e l y r e d u c e d w h e n the RRS is o n . I n fact, f o r m o s t o f t h e t i m e , t h e r o l l angle can be r e s t r i c t e d w i t h i n + 5°,

H o w e v e r , d u e t o t h e r u d d e r l i m i t s , a t r a d e o f f b e t w e e n t h e track k e e p i n g p e r f o r m a n c e a n d t h e r o l l r e d u c t i o n p e r f o r m a n c e is always n e e d e d . I f a f a s t e r t r a c k k e e p i n g p e r f o r m a n c e is r e q u i r e d , w h i c h can b e a c h i e v e d b y i n c r e a s i n g the p o s i t i o n f e e d b a c k g a i n Cd. then

3500 4000