Ocean Engineering 66 (2013) 58-68
ELSEVIER
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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
Lyapunov and Hurwitz based controls for input-output linearisation
applied to nonlinear vessel steering
Lokukaluge P. Perera, C. Guedes Soares *
Centre for Marine Technology and Engineering (CENTEC). Instituto Superior Técnico, Technical University of Lisbon, Lisbon, Portugal
CrossMark
A R T I C L E I N F O
Article history: Received 8 May 2012 Accepted 6 April 2013 Available online 8 May 2013 Keywords:
Partial feedback linearisation Input-output linearisation Nonlinear vessel steering Lyapunov based control Hurwitz based control Ship autopilot systems Ship control
A B S T R A C T
T h i s paper focuses on the Lyapunov, H u r w i t z a n d PID based control a p p r o a c h e s for an i n p u t - o u t p u t linearisation a p p l i e d n o n l i n e a r v e s s e l steering s y s t e m . A m a t h e m a t i c a l m o d e l of the v e s s e l steering s y s t e m is d e r i v e d c o n s i d e r i n g a s e c o n d - o r d e r Nomoto m o d e l w i t h n o n l i n e a r m a n o e u v r a b i l i t y features. In general, the control d e s i g n pr oces s of the n o n l i n e a r v e s s e l s t e e r i n g s y s t e m is affected by the nonlinearities that are related to r u d d e r angle, r u d d e r rate a n d v e s s e l h e a d i n g angle. T h e r e f o r e , to s i m p l i f y the n o n l i n e a r v e s s e l s t e e r i n g system, i n p u t - o u t p u t l i n e a r i s a t i o n is p r o p o s e d in this study. D u r i n g this proc e ss ( L e . i n p u t - o u t p u t linearization), the s y s t e m is d i v i d e d in a s y s t e m w i t h linear d y n a m i c s a n d a s y s t e m w i t h i n t e r n a l d y n a m i c s . F u r t h e r m o r e , the stability c o n d i t i o n s of internal d y n a m i c s are a n a l y s e d to observe o v e r a l l stability of the v e s s e l s t e e r i n g s y s t e m . T h e L y a p u n o v , H u r w i t z a n d PID b a s e d controllers are p r o p o s e d for course k e e p i n g and course c h a n g i n g m a n o e u v r e s in vessel steering. F u r t h e r m o r e , t h e overall stability conditions of the p r o p o s e d L y a p u n o v a n d H u r w i t z based controller a r e a n a l y s e d , c o n s i d e r i n g a L y a p u n o v candidate f u n c t i o n a n d the Hui-witz conditions, respectively. Finally, the p r o p o s e d c o n t r o l algorithms are s i m u l a t e d o n a v e s s e l s t e e r i n g s y s t e m and s u c c e s s f u l results w i t h r e s p e c t to c o u r s e k e e p i n g a n d course c h a n g i n g m a n o e u v r e s are p r e s e n t e d in this paper. © 2013 E l s e v i e r L t d . A l l rights reserved.
1. Introduction
One o f t h e m a j o r challenges i n m o d e r n m a r i t i m e n a v i g a t i o n is t o d e s i g n advanced a u t o p i l o t systems t h a t g u a r a n t e e r o b u s t s t a b i l i t y a n d p e r f o r m a n c e s u n d e r v a r i o u s sea c o n d i t i o n s . C o n v e n -t i o n a l m a r i -t i m e n a v i g a -t i o n consis-ts o f a u -t o p i l o -t sys-tems -t h a -t are g u i d e d b y r e m o t e p o s i t i o n i n g systems a n d c o m p l e m e n t e d b y m a r i t i m e s u r v e i l l a n c e sensors. Even t h o u g h , these r e m o t e p o s i -rioning systems a n d m a r i r i m e s u r v e i l l a n c e sensors have a c h i e v e d considerable i m p r o v e m e n t s i n recent years, m a r i t i m e a u t o p i l o t systems are s t i l l u n d e r d e v e l o p e d . As a n e x a m p l e , m o s t m a r i t i m e a u t o p i l o t s y s t e m s are m a i n l y g u i d e d t h r o u g h w a y - p o i n t s a s s o c i a t e d t r a j e c t o r i e s t h a t are b a s e d o n t h e p r o p o r t i o n a l , i n t e g r a l a n d d e r i v a t i v e ( P I D ) t y p e s o f c o n t r o l l e r s ( L i m a n d F o r s y t h e , 1983a, 1 9 8 3 b ) . H o w e v e r , t h e s e g u i d a n c e m e c h a n i s m s a n d c o n t r o l f o r m u l a t i o n s n o t a d e q u a t e f o r m o d e r n m a r i t i m e n a v i g a t i o n r e q u i r e m e n t s e s p e c i a l l y i n c o n f i n e d w a t e r w h e r e r e l a t i v e l y l a r g e a n d p r e c i s i o n c o u r s e c h a n g i n g m a n o e u v r e s are r e q u i r e d . T h e r e f o r e , t o s a t i s f y t h e p r e s e n t m a r i t i m e n a v i g a t i o n r e q u i r e m e n t s v a r i o u s i n t e l l i g e n t g u i d a n c e a n d a d v a n c e d c o n t r o l a l g o r i t h m s h a v e also p r o p o s e d i n t h e r e c e n t l i t e r a t u r e (Possen, 2 0 0 2 ; M o r e i r a et al., 2 0 0 7 ; * Corresponding author.E-mail address: guedess@mar.ist.utl.pt (C. Cuedes Soares). 0029-8018/$-see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.oceaneng.2013.04.002
Perera et a l , 2 0 1 1 , 2012a, 2 0 1 2 b , 2012c, 2 0 1 2 d ; Perera a n d G u e d e s Soares, 2012c.
T h e c o n t r o l f u n c t i o n a l i t i e s i n m a r i t i m e a u t o p i l o t systems can be d i v i d e d i n t o t w o sub-systems: speed c o n t r o l s y s t e m a n d s t e e r i n g c o n t r o l system. The m a i n o b j e c t i v e o f t h e speed c o n t r o l l e r s y s t e m is t o m a i n t a i n vessel's speed, a n d t h e m a i n o b j e c t i v e o f the s t e e r i n g c o n t r o l s y s t e m is t o m a i n t a i n vessel's course. H o w e v e r , t h i s s t u d y is f o c u s e d o n t h e c o n t r o l l e r d e s i g n process o f vessel s t e e r i n g , w h i c h is t h e m o s t i m p o r t a n t aspect o f a m a r i t i m e a u t o p i l o t s y s t e m i n c o n f i n e d w a t e r s . T h e s t u d y deals w i t h t w o p r o b l e m s : t h e i d e n t i f i c a t i o n o f a p r o p e r vessel s t e e r i n g m o d e l a n d t h e d e s i g n o f a n a p p r o p r i a t e c o n t r o l l e r . T h e r e f o r e , t h e d e r i v a t i o n o f a m a t h e m a t i c a l m o d e l f o r vessel s t e e r i n g is a n i m p o r t a n t p a r t o f t h e c o n t r o l l e r design process, w h e r e a l m o s t a l l t h e c o n t r o l strategies i n t h e recent l i t e r a t u r e are based o n s o m e types o f m a t h e m a t i c a l m o d e l s .
T h e r e are several c o n t r o l l e r strategies p r o p o s e d i n t h e recent l i t e r a t u r e w i t h respect to vessel s t e e r i n g a n d t w o m a i n types o f c o n t r o l strategies can be i l l u s t r a t e d : l i n e a r c o n t r o l l e r s a n d n o n -l i n e a r c o n t r o -l -l e r s . The -l i n e a r c o n t r o -l -l e r s are p r o p o s e d c o n s i d e r i n g t h e l i n e a r state or p a r a m e t e r c o n d i t i o n s i n vessel steering. The n o n l i n e a r state or p a r a m e t e r c o n d i t i o n s i n vessel s t e e r i n g are o f t e n linearised a r o u n d specific p o i n t s (i.e. Jacobian l i n e a r i s a t i o n ) i n these l i n e a r c o n t r o l l e r s . Even t h o u g h , t h i s a p p r o a c h c o u l d s a t i s f y t h e r e q u i r e m e n t s i n course k e e p i n g m a n o e u v r e s t h a t m a y
be i n a d e q u a t e f o r course c h a n g i n g m a n o e u v r e s , w h e r e n o n l i n e a r vessel s t e e r i n g c o n d i t i o n s can also be observed.
T h e r e f o r e , a l i n e a r i s e d vessel s t e e r i n g m o d e l is n o t a d e q u a t e f o r c o u r s e c h a n g i n g m a n o e u v r e s i n vessel s t e e r i n g . H o w e v e r , these n o n l i n e a r vessel s t e e r i n g c o n d i t i o n s w o u l d g e n e r a t e stable or u n s t a b l e d y n a m i c s . T h e r e f o r e , t o s a t i s f y t h e n o n l i n e a r state o r p a r a m e t e r c o n d i t i o n s (i.e. stable a n d u n s t a b l e s t e e r i n g c o n d i t i o n s ) i n course k e e p i n g a n d course c h a n g i n g m a n o e u v r e s f e e d -back l i n e a r i z a t i o n is p r o p o s e d i n t h i s s t u d y . F u r t h e r m o r e , t h e n o n l i n e a r c o n t r o l l e r s based o n t h e L y a p u n o v a n d H u r w i t z c o n -d i t i o n s are p r o p o s e -d t o achieve c o u r s e k e e p i n g a n -d c o u r s e c h a n g i n g m a n o e u v r e s i n vessel s t e e r i n g .
2. Feedback linearisation
2.1. Introduction T h e i m p l e m e n t a t i o n o f a n o n l i n e a r c o n t r o l l e r is o f t e n d e g r a d e d b y t h e c o m p l e x i t y i n t h e d e s i g n processes. The d e s i g n c o m p l e x i t y i n t h e c o n t r o l l e r comes g e n e r a l l y f r o m the n o n l i n e a r state and p a r a m e t e r b e h a v i o u r i n t h e s y s t e m d y n a m i c s , i n w h i c h n o n l i n e a r vessel s t e e r i n g can be observed. T h e r e f o r e , as m e n t i o n e d b e f o r e , t o s i m p l i f y t h e c o n t r o l l e r d e s i g n process, t h e f e e d b a c k l i n e a r i s a t i o n t e c h n i q u e is p r o p o s e d i n t h i s study. The f e e d b a c k l i n e a r i s a t i o n t e c h n i q u e has b e e n used i n t h e l i t e r a t u r e t o a l g e b r a i c a l l y t r a n s -f o r m n o n l i n e a r system d y n a m i c s i n t o -f u l l y o r p a r t i a l l y l i n e a r s y s t e m d y n a m i c s (Slotine a n d Li, 1991) a n d t h a t can also be used t o a v o i d t h e c o m p l e x i t y i n t h e c o n t r o l l e r d e s i g n process.T h e f e e d b a c k l i n e a r i s a t i o n a p p r o a c h s i m p l i f i e s t h e s y s t e m d y n a m i c s a r o u n d a n o p e r a t i n g r e g i o n , w h i c h is t h e m a i n a d v a n -tage i n t h i s a p p r o a c h . T h e c o n v e n t i o n a l l i n e a r i z a t i o n t e c h n i q u e (i.e. Jacobian l i n e a r i z a t i o n ) can s i m p l i f y t h e s y s t e m d y n a m i c s at a s p e c i f i c o p e r a t i n g p o i n t b u t c a n n o t preserve t h e s y s t e m d y n a m i c s a r o u n d o t h e r o p e r a t i n g p o i n t s . T h e r e f o r e , f e e d b a c k l i n e a r i z a t i o n is used t o linearise t h e s y s t e m d y n a m i c s a r o u n d a special r e g i o n . H o w e v e r , feedback l i n e a r i z a t i o n can o n l y be a p p l i e d t o a special class o f n o n l i n e a r systems t h a t are i n a f f i n e f o r m a t (i.e. c o n t r o l -lable c a n o n i c a l f o r m ) . A n o n l i n e a r s y s t e m t h a t is i n a f f i n e f o r m a t can be w r i t t e n as (Guay, 2 0 0 1 ) x = F(x)-hG(x)i(x y , = /7(x) (1) w h e r e x, s y s t e m states; f ( x ) , s y s t e m f u n c t i o n ; C(x), c o n t r o l f u n c -t i o n ; Ux, c o n -t r o l i n p u -t ; y^, o u -t p u -t ; a n d f i ( x ) , o u -t p u -t f u n c -t i o n . H o w e v e r , a n a d d i t i o n a l c o o r d i n a t e t r a n s f o r m a t i o n can be used t o achieve t h e a f f i n e f o r m a t f o r a n o n l i n e a r s y s t e m t h a t has n o t b e e n i n t h e f o r m a t o f ( 1 ) . The n o n l i n e a r vessel s t e e r i n g m o d e l t h a t is i n t r o d u c e d i n t h i s s t u d y is n o t b e i n g p r e s e n t e d i n t h e a f f i n e f o r m a t ; t h e r e f o r e a c o o r d i n a t e t r a n s f o r m a t i o n is b e i n g also i n t r o d u c e d .
I n general, t h e feedback l i n e a r i s a t i o n process can be d i v i d e d i n t o t w o steps ( H e n s o n a n d Seborg, 1997). The flrst, t h e change o f c o o r d i n a t e s o f t h e system states. The second, t h e f o r m u l a t i o n o f t h e c o n t r o l l a w u n d e r f e e d b a c k l i n e a r i s a t i o n . The change o f c o o r d i n a t e s o f t h e system can be c o n s i d e r e d as a t r a n s f o r m a t i o n o f a n o n l i n e a r s y s t e m i n t o a l i n e a r s y s t e m . I n g e n e r a l , t h e f e e d b a c k l i n e a r i s a t i o n can also be f o r m u l a t e d b y i n s p e c t i n g t h e o r i g i n a l n o n l i n e a r s y s t e m a n d selecting a n e w c o n t r o l l a w t h a t can cancel t h e s y s t e m n o n l i n e a r i t i e s (Yang et al., 2 0 0 1 ; R a m i r e z a n d Ibanez, 2 0 0 0 ) . H o w e v e r , c o m p l e x n o n l i n e a r systems can be l i n e a r i s e d c o n s i d e r i n g Lie d e r i v a t i v e based t r a n s f o r m a t i o n s ( H e n s o n a n d Seborg, 1990) as f u r t h e r discussed i n t h i s study.
The change o f coordinates o f the system states can be f u r t h e r categorised into t w o f o r m u l a t i o n s : f u l l state feedback linearisation and partial feedback linearisation. The f t i l l state feedback linearisaflon is
o f t e n refen'ed as input-state linearisaflon t h a t maps t h e linearised inputs i n the enflre state variable space o f the nonlinear system. However, o f t e n nonlinear systems cannot satisfy the f u l l state feedback linearisation condiflons, due to the system i n p u t - o u t p u t relaflonships. I n these situations (i.e. failures i n f u l l state feedback linearisation), t w o soluflons are introduced. The flrst, an arflflcial o u t p u t can be i n t r o -duced to satisfy the f u l l state feedback linearisable condiflons. The second, partial feedback linearisation can be i n t r o d u c e d to s i m p l i f y the nonlinear system. However, the flrst approach is n o t always applicable i n the controller design process, w h e r e the a r f l f l c i a l o u t p u t is d i f f e r e n t f r o m the actual o u t p u t t h a t cannot be observed o r measured by t h e system sensors. Therefore, i n a tracldng s i t u a f l o n the arflflcial o u t p u t w i l l n o t necessarily saflsfy the desired o u t p u t (Henson and Seborg, 1997).
Partial f e e d b a c k l i n e a r i s a t i o n is b e i n g p r o p o s e d i n t h i s s t u d y t o s i m p l i f y t h e n o n l i n e a r vessel s t e e r i n g s y s t e m . P a r t i a l feedback l i n e a r i s a t i o n t h a t is o f t e n r e f e r r e d as i n p u t - o u t p u t l i n e a r i s a t i o n can be used f o r a n o n l i n e a r system, w h e r e t h e s y s t e m is incapable o f f u l l state f e e d b a c k l i n e a r i s a t i o n . F u r t h e r m o r e , i n p u t - o u t p u t l i n e a r i s a t i o n m a p s t h e l i n e a r i s e d i n p u t s w i t h t h e a c t u a l o u t p u t s i n t h e n o n l i n e a r s y s t e m w h i c h is a n advantage i n t h i s a p p r o a c h .
A sector o f s y s t e m d y n a m i c s c a n n o t be d i r e c t l y c o n t r o l l e d or o b s e r v e d i n an u n d e r - a c t u a t e d n o n l i n e a r s y s t e m (i.e. vessel steer-i n g s y s t e m ) . T h e r e f o r e , f u l l state f e e d b a c k l steer-i n e a r steer-i s a t steer-i o n o f t e n fasteer-ils t o s i m p l i f y a n u n d e r a c t u a t e d n o n l i n e a r s y s t e m . H o w e v e r , i n p u t -o u t p u t l i n e a r i s a t i -o n can be used t -o s i m p l i f y an u n d e r - a c t u a t e d n o n l i n e a r s y s t e m , w h e r e t h e c o n t r o l l a b l e o r observable sector o f t h e s y s t e m can o f t e n be l i n e a r i s e d . The r e m a i n i n g sector o f t h e u n d e r - a c t u a t e d n o n l i n e a r system, t h e u n c o n t r o l l a b l e o r unobser-v a b l e sector, is categorised as i n t e r n a l d y n a m i c s t h a t c a n n o t be d i r e c t l y c o n t r o l l e d o r observed b y t h e o u t p u t a n d t h a t c a n n o t be l i n e a r i s e d b y f e e d b a c k l i n e a r i s a t i o n .
The presence o f a n u n c o n t r o l l a b l e o r unobservable sector i n a n o n l i n e a r s y s t e m raises issues o n t h e overall s t a b i l i t y (Terrell, 1999). One s h o u l d note t h a t t h e stable i n t e r n a l d y n a m i c s can generate necessary c o n d i t i o n s f o r the system t o stabilise u n d e r various feedback c o n t r o l laws. Hence, i n p u t - o u t p u t l i n e a r i s a t i o n can also be affected b y stable i n t e r n a l d y n a m i c s , w h e r e t h e s t a b i l i t y c o n d i -t i o n s o f i n -t e r n a l d y n a m i c s s h o u l d be f u r -t h e r analysed u n d e r -t h e b o u n d e d - i n p u t - b o u n d e d - o u t p u t (BIBO) c o n d i t i o n s .
A n o n l i n e a r system w i t h stable i n t e r n a l dynamics is o f t e n referred as a m i n i m u m phase system and a n o n l i n e a r system w i t h unstable i n t e r n a l dynamics is o f t e n referred as a n o n - m i n i m u m phase system. The m i n i m u m phase nonlinear system is o f t e n i d e n t i f l e d b y the zero-dynamics, w h e r e the i n p u t c o n d i f l o n s o f t h e system is observed w i t h respect to t h e zero o u t p u t conditions. The stability c o n d i t i o n s o f t h e zero-dynamics o f a n o n l i n e a r system can be f u r t h e r analysed either b y Lyapunov stability t h e o r e m or H u r w i t z c o n d i t i o n s .
Even t h o u g h , the z e r o - d y n a m i c s can be g l o b a l l y a s y m p t o t i c a l l y stable i n a system, these approaches can o n l y guarantee local stability c o n d i t i o n s , w h i c h is d u e to the applicable r e g i o n o f f e e d -back l i n e a r i s a t i o n . A n o n l i n e a r system w i t h a s y m p t o t i c a l l y stable zero d y n a m i c s is called a s y m p t o t i c a l l y m i n i m u m phase systems. However, a n o n - m i n i m u m phase o r w e a l d y n o n - m i n i m u m phase n o n l i n e a r system can also be stabilized b y selecting a proper/ a r t i f l c i a l o u t p u t or a c o n t r o l i n p u t (Slotine a n d Li, 1991).
T h e r e f o r e , t h e f e e d b a c k l i n e a r i s a t i o n process can be s u m -m a r i s e d as f o l l o w s . The flrst step, t h e n o n l i n e a r s y s t e -m f o r f u l l y o r p a r f l a l l y l i n e a r i z a t i o n s h o u l d be o b s e r v e d . The second step, i f t h e s y s t e m is f u l l state linearisable t h e n i n p u t - s t a t e l i n e a r i s a t i o n s h o u l d be a p p l i e d , o t h e r w i s e i n p u t o u t p u t l i n e a r i s a t i o n (i.e. p a r -tial f e e d b a c k l i n e a r i z a t i o n ) s h o u l d be a p p l i e d . T h e t h i r d step, t h e s t a b i l i t y c o n d i t i o n s o f i n t e r n a l d y n a m i c s s h o u l d be o b s e r v e d , unless t h e s y s t e m is f u l l state linearisable. T h e final step, a p r o p e r c o n t r o l l a w t o stabilise t h e c l o s e d - l o o p c o n t r o l s y s t e m u n d e r f e e d b a c k l i n e a r i s a t i o n s h o u l d be d e r i v e d .
60 LP. Perera, C. Guedes Soares / Ocean Engineering 66 (2013) 58-68
The necessary c o n d i t i o n s f o r f u l l y or p a r t i a l l y f e e d b a c k l i n e a r -i s a t -i o n o f a n o n l -i n e a r s y s t e m can o f t e n be o b s e r v e d b y the r e l a t -i v e degree o f t h e system, w h i c h can be d e f i n e d as t h e d i f f e r e n c e b e t w e e n t h e n u m b e r o f f i n i t e poles a n d zeros o f the s y s t e m t r a n s f e r f u n c t i o n . Hence, i f t h e r e l a t i v e degree o f a s y s t e m is less t h a n t h e s y s t e m order, t h e n some zeros t h a t m a y be stable o r u n s t a b l e e x i s t o n t h e s y s t e m . Hovvever, t h e t r a n s f e r f u n c d o n c o n c e p t is n o t a p p l i c a b l e i n n o n l i n e a r systems. Hence, i f t h e r e l a t i v e d e g r e e is less t h a n t h e s y s t e m o r d e r i n a n o n l i n e a r system, t h e n t h e s y s t e m has s o m e i n t e r n a l d y n a m i c s t h a t c a n n o t be o b s e r v e d f r o m t h e o u t p u t . T h e r e f o r e , i n these types o f s i t u a t i o n s the s t a b i l i t y o f i n t e r n a l d y n a m i c s s h o u l d be f u r t h e r analysed t o d e t e r m i n e t h e o v e r a l l s y s t e m s t a b i l i t y . A r e l a t i v e degree o f a n o n l i n e a r s y s t e m can be c a l c u l a t e d by a Lie d e r i v a t i v e s a p p r o a c h . T h e s i m p l i f i e d Lie d e r i v a t i v e c o n d i t i o n s w i t h respect to a r e l a t i v e degree o f a n o n l i n e a r s y s t e m can be w r i t t e n as, several d i f f e r e n t i a t i o n s o f t h e s y s t e m o u t p u t u n t i l t h e i n p u t appears e x p l i c i t l y . T h e f i r s t d i f f e r e n t i a t i o n o f t h e n o n -l i n e a r s y s t e m o u t p u t p r e s e n t e d i n ( 1 ) u s i n g Lie d e r i v a t i v e s can be w r i t t e n as y . = ^ x = ^ ( F ( x ) + C(x)u.) dh{x) Fix) dx dh(x) dx dx = L f / l ( X ) + (Lc/7(X))Ux G(x)Ux (2) A s s u m i n g the c o n d i t i o n t h a t t h e s y s t e m i n p u t c a n n o t be o b s e r v e d b y the first d i f f e r e n t i a t i o n , w h e r e Lcii(x) = 0, t h e n the second d i f f e r e n t i a t i o n o f t h e s y s t e m o u t p u t i n ( 1 ) u s i n g Lie d e r i v a t i v e s h o u l d be p e r f o r m e d a n d t h a t can be w r i t t e n as y x = ^ y x = ^ ( i F / j ( x ) ) = ^ x = ^ ( F ( x ) + G(x)u.) ^ ( ^ ' ' " « > F ( x ) 4 - ^ ^ C ( x ) u . dx dx = Ljh(x) + (LcLlh{x))Ux (3) A s s u m i n g the c o n d i t i o n t h a t t h e s y s t e m i n p u t c a n n o t be o b s e r v e d b y t h e second d i f f e r e n t i a t i o n , w h e r e LcLlh(x) = 0, t h e n t h e t h i r d d i f f e r e n t i a t i o n o f t h e s y s t e m o u t p u t s h o u l d be p e r -f o r m e d . H o w e v e r , i -f L c L -f / j ( x ) * 0 , t h e n i t can be c o n c l u d e d t h a t t h e r e l a t i v e degree o f t h e n o n l i n e a r s y s t e m is 2. O t h e r w i s e , t h i s process s h o u l d be c o n t i n u e d r-l steps, w h e r e the s y s t e m o u t p u t is o b s e r v e d b y t h e i n p u t , I c L p h ( x ) * 0 , w h e r e t h e n o n l i n e a r s y s t e m has t h e r e l a t i v e d e g r e e o f r.
Hence, t h e r e l a t i v e degree c a l c u l a t i o n can be s u m m a r i s e d b y t h e d i f f e r e n t i a t i o n o f t h e n o n l i n e a r s y s t e m o u t p u t i n f o l l o w i n g f o r m a t , w h e r e a n o n l i n e a r s y s t e m t h a t has r e l a t i v e d e g r e e r i n Lie d e r i v a t i v e s o f Levine (2011) LcL'l:ii(x) = 0 f o r k < r - l LcL'f'h(x)T^O (4) T h e r e f o r e , to f i n d t h e r e l a t i v e degree o f a n o n l i n e a r system, t h e Lie d e r i v a t i v e s L c l . f / i ( x ) are d e r i v e d f o r k<r-l s u c h t h a t t h e LcLph(x) is i n v e r t i b l e . One s h o u l d n o t e t h a t i f t h e r e l a t i v e degree o f a n o n l i n e a r s y s t e m is e q u a l t o t h e o r d e r o f t h e s y s t e m , t h e n the s y s t e m is f u l l i n p u t s t a t e linear, o t h e r w i s e t h e s y s t e m is i n p u t -o u t p u t linear. F u r t h e r m -o r e , t h e f e e d b a c k l i n e a r i s a t i -o n a p p r -o a c h n o t o n l y f o r a s i n g l e i n p u t n o n l i n e a r s y s t e m b u t also f o r a m u l t i -i n p u t n o n l -i n e a r s y s t e m c a n also be used ( H o u a n d Pugh, 1997). As t h e f i n a l a n d m o s t i m p o r t a n t step o f f e e d b a c k l i n e a r i s a t i o n , a p r o p e r c o n t r o l l a w s h o u l d be d e r i v e d . T h e c o n t r o l l a w u n d e r
i n p u t - o u t p u t l i n e a r i s a t i o n can be w r i t t e n as
Ux = ^ ( - « ( > ; ) + Vx) (5)
w h e r e Vx is n e w c o n t r o l i n p u t and a{y) a n d /?(y) are c o n t r o l f u n c t i o n s t h a t can be w r i t t e n as
a(x) = L'phix)
/?(x) = LcL^-'h(x)
2.2. Literature review
(6)
T h e r e are m a n y feedback l i n e a r i s a t i o n based c o n t r o l systems p r o p o s e d i n t h e r e c e n t l i t e r a t u r e . The f e e d b a c k l i n e a r i s a t i o n d e s i g n f o r vessel s t e e r i n g w i t h t h e r u d d e r angle a n d r u d d e r r a t e s a t u r a t i o n l i m i t a t i o n s is p r o p o s e d b y T z e n g et al. ( 1 9 9 9 ) . A d a p t i v e f e e d b a c k l i n e a r i s a t i o n a p p l i e d l i n e a r q u a d r a t i c o p t i m a l c o n t r o l f o r ship s t e e r i n g is p r e s e n t e d i n Fossen a n d Paulsen ( 1 9 9 2 ) . H o w e v e r , t h e vessel d y n a m i c m o d e l s i n these studies are l i m i t e d to t h e m o d i f i e d first o r d e r N o m o t o m o d e l , w h e r e t h e n o n l i n e a r vessel s t e e r i n g b e h a v i o u r has n o t b e e n c o n s i d e r e d .
The feedback l i n e a r i s a t i o n can o n l y p r o d u c e a f u l l state or p a r t i a l l i n e a r i s e d system t h a t is associated w i t h a feedback c o n t r o l l a w . H o w e v e r , m o s t o f t h e c o n v e n t i o n a l c o n t r o l strategies can f a i l t o s i m p l i f y the c o n t r o l design process due t o t h e n o n l i n e a r i t i e s a m o n g t r a n s f o r m e d i n p u t s a n d o r i g i n a l o u t p u t s ( H e n s o n a n d Seborg, 1997). T h e r e f o r e , t h e H u r w i t z a n d L y a p u n o v based c o n t r o l strategies are p r o p o s e d i n t h i s s t u d y to be i m p l e m e n t e d o n t h e i n p u t - o u t p u t l i n e a r i s a t i o n a p p l i e d n o n l i n e a r vessel s t e e r i n g system. A l i n e a r s y s t e m w i t h a c h a r a c t e r i s t i c e q u a t i o n t h a t is associated w i t h t h e r o o t s i n t h e l e f t - h a l f plane o f r o o t - l o c u s c a n be cate-g o r i s e d as t h e H u r w i t z c o n d i t i o n s (Clark, 1992). T h e r e f o r e , t h e H u r w i t z c o n d i t i o n s can be used t o d e s i g n a n a p p r o p r i a t e c o n -t r o l l e r f o r -t h e i n p u -t - o u -t p u -t f e e d b a c k l i n e a r i s a -t i o n a p p l i e d vessel s t e e r i n g s y s t e m . L y a p u n o v c a n d i d a t e f u n c t i o n s have b e e n used t o observe t h e s y s t e m s t a b i l i t y c o n d i t i o n s a n d t o d e s i g n c o n t r o l l e r s f o r t h e n o n l i n e a r c o n t r o l systems. A p r e d e f i n e d c a n d i d a t e f u n c t i o n is used t o d e r i v e t h e s y s t e m s t a b i l i t y c o n d i t i o n s a n d t o d e s i g n a n a p p r o p r i a t e c o n t r o l l e r f o r t h e i n p u t - o u t p u t f e e d b a c k l i n e a r i s a t i o n a p p l i e d vessel s t e e r i n g s y s t e m i n t h e sense o f L y a p u n o v ( K h a l i l , 2 0 0 2 ) . T h e r e f o r e , a c o m p a r i s o n a m o n g these t w o c o n t r o l s t r a t e -gies a n d a c o n v e n t i o n a l PID c o n t r o l l e r is f u r t h e r p r e s e n t e d i n t h i s s t u d y .
T h e r e are m a n y linear and n o n l i n e a r vessel c o n t r o l approaches p r o p o s e d i n the r e c e n t l i t e r a t u r e w i t h respect to t h e vessel n a v i g a -t i o n sys-tems. A s l i d i n g m o d e c o n -t r o l l e r f o r ship course c o n -t r o l has been p r o p o s e d a n d s i m u l a t i o n results are described by Tomera (2010). Yaozhen et a l . (2010) present a f u z z y s l i d i n g m o d e l c o n t r o l l e r f o r a s h i p course c o n t r o l system. The c o m p a r i s o n b e t w e e n the M o d e l Predictive a n d Sliding M o d e Controllers f o r ship h e a d i n g c o n t r o l is presented i n M c G o o k i n et al. ( 2 0 0 8 ) . H o w e v e r these studies are l i m i t e d to linearised vessel steering m o d e l s .
A t e r m i n a l s l i d i n g m o d e f u z z y c o n t r o l a p p r o a c h based o n m u l t i p l e s l i d i n g surfaces f o r a n o n l i n e a r s h i p a u t o p i l o t s y s t e m is p r o p o s e d b y Y u a n a n d W u ( 2 0 1 0 ) . S i m i l a r l y , a p i - t y p e s l i d i n g m o d e c o n t r o l l e r is p r o p o s e d f o r t r a c k i n g c o n t r o l o f a s h i p b y Y u a n d W u ( 2 0 0 4 ) . H o w e v e r , t h e r u d d e r rate e f f e c t s o n vessel s t e e r i n g are i g n o r e d i n these studies, w h e r e a vessel s t e e r i n g s y s t e m is f u r t h e r s i m p l i f i e d . One s h o u l d n o t e t h a t t h e special c o o r d i n a t e t r a n s f o r -m a t i o n a l o n g w i t h f e e d b a c k l i n e a r i z a t i o n is p r o p o s e d i n t h i s s t u d y t o a v o i d t h e r u d d e r rate e f f e c t s f r o m t h e c o n t r o l s y s t e m . Vessel m a n o e u v r i n g p e r f o r m a n c e a l o n g a d e s i r e d p a t h u s i n g s l i d i n g m o d e c o n t r o l is p r o p o s e d i n S k j e t n e ( 2 0 0 5 ) , w h e r e t h e s i m u l a t i o n results are l i m i t e d t o c i r c u l a r m a n o e u v r e s . S i m i l a r l y , a s l i d i n g m o d e c o n t r o l l e r f o r h a r b o u r m a n o e u v r e s o f a large s h i p is
p r o p o s e d b y Le et a l . ( 2 0 0 3 ) . H o w e v e r , the s h i p m a n o e u v r e s are s u p p o r t e d b y e x t e r n a l forces (i.e. t u g b o a t s ) i n t h i s study, w h e r e t h e ship as a f u l l y a c t u a t e d s y s t e m is assumed.
F u r t h e r m o r e , a s l i d i n g m o d e c o n t r o l l e r f o r t r a j e c t o r y t r a c k i n g o f a m a r i n e vessel is p r o p o s e d i n Cheng et al. ( 2 0 0 7 ) . S i m i l a r l y , a vessel as a f u l l y a c t u a t e d s y s t e m is assumed i n t h i s study, w h e r e t h e surge force, s w a y f o r c e a n d y a w m o m e n t are available. It is w e l l k n o w n t h a t t h e vessel s t e e r i n g s y s t e m is associated w i t h u n d e r - a c t u a t e d c o n d i t i o n s , w h e r e the s w a y m o t i o n c a n n o t be d i r e c t l y actuated. Even t h o u g h these studies have p r o p o s e d t h r u s t e r s f o r t h e s w a y a c t u a t i o n , a f u l l y a c t u a t i o n s i t u a t i o n u n d e r h i g h speed a n d r o u g h sea c o n d i t i o n c a n n o t be a c h i e v e d b y t h e vessel.
T h e t h e o r e t i c a l a n d e x p e r i m e n t a l results f o r p a t h c o n t r o l o f a surface s h i p i n r e s t r i c t e d w a t e r s are p r e s e n t e d by Z h a n g et al. ( 2 0 0 0 ) . T h e c o n t r o l a p p r o a c h is based o n an i n p u t o u t p u t l i n e a r -i s a t -i o n based s l -i d -i n g m o d e c o n t r o l l e r . T h e d y n a m -i c a d a p t -i v e s l i d i n g m o d e c o n t r o l l e r f o r t r a c k i n g o f s h i p s t e e r i n g is p r o p o s e d i n C h e n g et al. ( 2 0 0 6 ) . H o w e v e r , these studies are l i m i t e d t o s m a l l h e a d i n g angle v a r i a t i o n s . I t is o b s e r v e d t h a t f o r large h e a d i n g angle changes, large r u d d e r angle v a r i a t i o n s are r e q u i r e d . H o w -ever, t h e r u d d e r angle a n d r u d d e r rate are associated w i t h some a c t u a t i o n l i m i t a t i o n s , t h e r e f o r e t h e h i g h e r c o n t r o l gains r e q u i r e d b y t h e p r o p o s e d m e t h o d s c a n n o t be achieved, w h e r e t h e s t a b i l i t y i n t h e c o n t r o l l e r c a n n o t be p r e s e r v e d .
O p t i m i s a t i o n b a s e d c o n t r o l strategies b y t w o types o f p e r f o r -m a n c e i n d i c e s f o r course k e e p i n g (i.e. to -m i n i -m i s e h e a d i n g error, p r o p u l s i o n losses a n d r u d d e r a c t i v i t i e s ) a n d course c h a n g i n g (i.e. t o m i n i m i s e h e a d i n g e r r o r ) m a n o e u v r e s are p r o p o s e d b y L i m a n d Forsythe (1983a, 1 9 8 3 b ) . S i m i l a r l y , the s i m u l a t i o n s are l i m i t e d t o s l o w a n d s m a l l changes i n h e a d i n g angle c o n d i t i o n s , w h e r e t h e adequate p e r f o r m a n c e i n t h e p r o p o s e d c o n t r o l approaches c a n n o t be e v a l u a t e d .
T h e r e are m a n y o t h e r a p p l i c a t i o n s w i t h respect t o f e e d b a c k l i n e a r i s a t i o n has b e e n p r o p o s e d i n the f o l l o w i n g studies o f Guay ( 2 0 0 1 ) , Y a n g et al. ( 2 0 0 1 ) , R a m i r e z a n d Ibanez ( 2 0 0 0 ) , H e n s o n a n d Seborg ( 1 9 9 0 ) , S p o n g (1994), a n d L i u e t al. ( 2 0 0 4 ) .
It is o b s e r v e d t h a t t o achieve a large h e a d i n g angle change w i t h i n a s h o r t t i m e p e r i o d , a large r u d d e r angle v a r i a t i o n w i t h i n a s h o r t time p e r i o d is r e q u i r e d , i n w h i c h preserves t h e c o n t r o l l e r p e r f o r m a n c e . H o w e v e r , t h e r u d d e r a c t u a t i o n is a f f e c t e d b y t h e r u d d e r angle a n d r u d d e r rate l i m i t a t i o n s , w h e r e m a n y vessel s t e e r i n g s i m u l a t i o n s are l i m i t e d t o s m a l l h e a d i n g angle v a r i a t i o n s .
A large h e a d i n g angle change w i t h i n a s h o r t time p e r i o d can be a v o i d e d b y d e s i g n i n g a p r e - f i l t e r t h a t can reduce t h e s u d d e n h e a d i n g angle v a r i a t i o n s . T h e r e f o r e , a p r e - f i l t e r can reduce t h e large e r r o r v a r i a t i o n s , w h i c h e v e n t u a l l y reduce t h e large r u d d e r angle a n d r u d d e r rate r e q u i r e m e n t s t o m a i n t a i n t h e s t a b i l i t y i n t h e c o n t r o l l e r .
I t is w e l l k n o w n t h a t t h e vessel s t e e r i n g s y s t e m is a s l o w response system. Hence, t h e s y s t e m does n o t r e s p o n d t o s u d d e n v a r i a t i o n s o f t h e r e f e r e n c e a n d c o n t r o l i n p u t s . T h e r e f o r e , t h e p r o p o s e d p r e - f i l t e r c a n s l o w d o w n t h e vessel reference h e a d i n g a n d c o n t r o l i n p u t s t h a t are s l i g h t l y faster t h a n t h e vessel response a n d t h a t c a n i n t r o d u c e constrains o n t h e c o n t r o l l e r b e h a v i o u r . Hence, t h e large v a r i a t i o n s i n t h e vessel h e a d i n g r e f e r e n c e are r e d u c e d t h a t e v e n t u a l l y reduce large v a r i a t i o n s i n t h e c o n t r o l i n p u t s (i.e. t h e large r u d d e r angle and rate v a r i a t i o n s ) , w h e r e t h e p r o p o s e d p r e - f i l t e r c a n preserve t h e s t a b i l i t y o f t h e c o n t r o l l e r .
The o r g a n i s a t i o n o f this paper is as f o l l o w s . Section 2 c o n t a i n s a m a t h e m a t i c a l m o d e l o f vessel steering, w h i c h is c o n s i d e r e d f o r t h e c o n t r o l l e r design. T h e f e e d b a c k l i n e a r i s a t i o n a p p l i e d n o n l i n e a r vessel s t e e r i n g s y s t e m is discussed i n Section 3. The c o n t r o l design is p r e s e n t e d i n Section 4. Section 5 contains a d e t a i l e d d e s c r i p t i o n o f t h e c o m p u t a t i o n a l s i m u l a t i o n s . Finally, t h e c o n c l u s i o n is p r e -sented i n Section 6.
3. A mathematical model of vessel steering
3.3. Sway-yaw sub-systemsT w o c o o r d i n a t e systems f o r vessel s t e e r i n g are p r e s e n t e d i n Fig. 1. The c o o r d i n a t e systems can be d e f i n e d as X„Y„Zn, t h e e a r t h fixed c o o r d i n a t e s y s t e m a n d XbV/^Zi,, t h e vessel b o d y fixed c o o r d i -n a t e s y s t e m . The vessel states are d e f i -n e d as u, surge l i -n e a r v e l o c i t y ; v, s w a y l i n e a r v e l o c i t y ; w , heave l i n e a r v e l o c i t y ; p , r o l l a n g u l a r v e l o c i t y ; q, p i t c h a n g u l a r v e l o c i t y ; r, y a w a n g u l a r v e l o c i t y ; X, surge f o r c e ; V, s w a y f o r c e ; Z, heave f o r c e ; K, r o l l m o m e n t ; M , p i t c h m o m e n t ; a n d N , y a w m o m e n t .
A s s u m i n g , t h e c o n s t a n t f o r w a r d speed, Uo, o f ocean vessel n a v i g a t i o n a n d t h e s w a y - y a w s u b - s y s t e m can be c o u p l e d , t h e vessel l i n e a r s t e e r i n g s y s t e m can be w r i t t e n as
m(v + Uor + Xci-) = Y^v + Yrr - l - YsSn -F Y^v - I - Yf-r
hr -F mxdv -F Uor) = N„v + Nrr -F NSSK + Ni,v + Nj-r ( 7 ) ' w h e r e t h e s y s t e m p a r a m e t e r s can be w r i t t e n as m , mass o f t h e
vessel; UQ, c o n s t a n t surge l i n e a r v e l o c i t y ; XQ. d i s t a n c e t o t h e c e n t r e o f g r a v i t y ; SR, r u d d e r angle; and k, i n e r t i a o f t h e vessel a l o n g t h e z-axis. F u r t h e r m o r e , the respective h y d r o d y n a m i c c o e f f i c i e n t s c a n be w r i t t e n as Yv, Vr, Ys, Vy a n d Yf, h y d r o d y n a m i c c o e f f i c i e n t s o f t h e s w a y m o t i o n a n d Nv, Nr, Ns, Ny a n d Nr, h y d r o d y n a m i c c o e f f i c i e n t s o f t h e y a w m o t i o n . The state space m o d e l d e s c r i b i n g t h e vessel l i n e a r s t e e r i n g system, i n t r o d u c e d i n ( 7 ) can be w r i t t e n as MRi>vr + NRiUo)l^vr = BRSR (8) w h e r e t h e s y s t e m states c a n be d e f i n e d as, Pvr=lv r f , a n d t h e m a t r i c e s o f MR, NR{UO), a n d BR can be f u r t h e r r e p r e s e n t e d as MR: NR{UO)--m-Yy mxc-Yr m x c - N v h-Nr -Yv mUo-Yr -Nv mxcUo-Nr (9) A s s u m i n g t h a t MR is p o s i t i v e d e f i n i t e t h e vessel l i n e a r s t e e r i n g s y s t e m i n ( 8 ) can be w r i t t e n as -MR-^NR{Uo)L'yr+MR ^BRSR (10) A = The s y s t e m m a t r i c e s A a n d B i n ( 1 0 ) can be r e p r e s e n t e d as Qii au g _ 021 022 ' ~ b2 (11)
62 LP. Perera, C. Cuedes Soares / Ocean Engineering 66 (2013) 58-68 w h e r e t h e respective c o e f f i c i e n t s o f m a t r i c e s / I a n d B i n (11) can be w r i t t e n as a i l = a i 2 = 021 = 022 = 01 = 02 = (Iz-Nr)Yy + {Yr-mxc)N, (m-Yi,)(Iz-Nr)-(mxc-Yi,)(mxc-Nr) {Iz-Nr)(Yr-mUo) + (Yi-mxc)(Nr-mxGUa) (m-Yv)(Iz-Nr)-{mxG-Yy)(mxc-Nr) (m-Yy)Nv + (Nv-mxc)Yv (m~Yi,)(Iz-Nr)-(mXc^ (m-Yv)(Nr-mxcUo) + -Yi,)(mxc-Nf) (Ni-mXc)(Yr-muo) (m-Yy)(Iz-Nf)-(mxc-Yv)(mXc-Nr) (Iz-Nr)Ys + (Yt-mxc)Ns (m-Yv)iIz-Nr)-(mXc-Yi)(mxc-Nr) {m-Yi,)Ns + (Nv-mXc)Ys (m-Yi,)(Iz-Nr)-(mxc-Yi,)(mxc-Nr) (12)
3.2. The modified Nomoto model
T h e s e c o n d - o r d e r l i n e a r N o m o t o e t al. ( 1 9 5 7 ) m o d e l c a n be d e r i v e d f r o m ( 1 2 ) b y e l i m i n a t i n g t h e s w a y v e l o c i t y , v, w h i c h is r e s u h i n g i n rir2i^ + ( r i -\-T2)r-\-r = l(R(T35R-\-SK) ( 1 3 ) w h e r e t h e r e s p e c t i v e vessel s t e e r i n g p a r a m e t e r s are c o n s i d e r e d as Tu T2, T3 a n d KR a n d can be w r i t t e n as r i T 2 + 1 2 = 011022-012021 O n -I-O22 0 1 2 0 2 1 - 0 1 1 0 2 2 Ö2 KR- 0 2 1 ^ 1 - 0 1 1 ^ 2 0 1 1 0 2 2 - 0 1 2 0 2 1 (14) A d e t a i l e d d e s c r i p t i o n o f t h e s t a b i l i t y o f vessel s t e e r i n g w i t h r e s p e c t to t h e above p a r a m e t e r s is p r e s e n t e d i n Perera a n d Guedes Soares (2012a, 2 0 1 2 b ) . The s e c o n d - o r d e r l i n e a r N o m o t o m o d e l , i n (14), can be r e w r i t t e n c o n s i d e r i n g the vessel h e a d i n g angle, 1//
5R) (15)
The second-order N o m o t o m o d e l i n (15) is adequate f o r course keeping manoeuvres, b u t this m o d e l may n o t be s u f f i c i e n t f o r course changing manoeuvres, w h e r e the nonlinear conditions i n vessel steering cannot be incorporated. Furthermore, the nonlinear steering conditions can generate stable and unstable steering conditions i n vessel navigation. Furthermore, the same vessel can demonstrate unstable steering conditions due to the sea or cargo conditions. Therefore, the vessel steering system presented i n (15) s h o u l d be m o d i f i e d to include n o n l i n e a r steering conditions as proposed i n A m e r o n g e n a n d Cate (1975), w h e r e the vessel heading rate, ij/, is replaced by a n o n l i n e a r f u n c t i o n KRH(4/). Hence, (15) can be w r i t t e n as
T2 hT2 KR hT2 (16) C o n s i d e r i n g t h e n o n l i n e a r f u n c t i o n t h a t c a p t u r e s vessel n o n -l i n e a r s t e e r i n g c o n d i t i o n s , w h e r e H(ij/) = n^tj/+ n2ii'^, i n w h i c h is c o m m o n l y r e f e r r e d as t h e s t e e r i n g characteristics f u n c t i o n . T h e r e -f o r e , ( 1 6 ) can be w r i t t e n as
1//^^' = -d\ii,-d2ia2>r' + aw) + diidiÖR -1- 5R) (17)
w h e r e t h e n o n l i n e a r s t e e r i n g p a r a m e t e r s are d e f i n e d as n , a n d 02, and t h e i n i t i a l p a r a m e t e r s o f vessel s t e e r i n g s y s t e m are d e f i n e d as (^1 = 7 7 + 7T' ^2 = 7 ^ a n d da = T3. Hence, (17) can be r e w r i t t e n as
V^^^^a^ijf^+a2ij/ + a3W + f)\SR-i-/32SR ( 1 8 )
w h e r e the final p a r a m e t e r s o f vessel s t e e r i n g can be d e f i n e d as
aj=-n2d2, a2=-n,d2, a 3 = - d ; , pi=d2, a n d P2=d2d3.
4. Feedback linearisation
4.1. Coordinate transformation One s h o u l d n o t e t h a t t h e n o n l i n e a r vessel s t e e r i n g s y s t e m d e r i v e d i n ( 1 8 ) consists o f r u d d e r a n g l e a n d r u d d e r rate e f f e c t s t h a t c a n n o t be c o n t r o l l e d s i m u l t a n e o u s l y by a s i m p l e c o n t r o l s y s t e m . F u r t h e r m o r e , t h e vessel s t e e r i n g s y s t e m i n ( 1 8 ) is n o t i n t h e a f f i n e f o r m as discussed p r e v i o u s l y . T h e r e f o r e , t h e n o n l i n e a r vessel s t e e r i n g m o d e l i n ( 1 8 ) c a n n o t be used f o r f e e d b a c k l i n e a r i s a t i o n . Hence, t o c o n v e r t t h e vessel s t e e r i n g s y s t e m i n t o a n a f f i n e f o r m a t (i.e. t o r e m o v e t h e r u d d e r rate effects f r o m vessel s t e e r i n g ) , a special algebraic c o o r d i n a t e t r a n s f o r m a t i o n is p r o p o s e d as X, = v / X 2 = V / X3=v-nSR ( 1 9 ) Hence, a p p l y i n g t h e c o o r d i n a t e t r a n s f o r m a t i o n i n ( 1 9 ) t o t h e n o n l i n e a r vessel s t e e r i n g m o d e l i n ( 1 8 ) can be w r i t t e n as Xl = v ^ = X2 X 2 = V> = X 3 -I- J'J Ö R X 3 = v ' ^ ' - ) ' l 5 f i = a i X 2 ^ + «2X2 -FaaXs + / ? 2 5 R - ) ' , 5 R (20) A s s u m i n g t h e c o n d i t i o n s o f = / ? 2 , t h e n o n l i n e a r vessel s t e e r i n g m o d e l i n ( 2 0 ) can be w r i t t e n as X , = X 2 X 2 = X 3 + / ? 2 5 R X3 = « 1 X 2 ^ - ^ 0 2 ^ 2 + « 3 X 3 - H / ? l 5 j ! ( 2 1 ) One s h o u l d n o t e t h a t t h e m a i n o b j e c t i v e o f t h e p r o p o s e d t r a n s f o r m a t i o n is t o r e m o v e r u d d e r rate e f f e c t s f r o m t h e n o n l i n e a r vessel s t e e r i n g s y s t e m a n d t o t r a n s f o r m t h e s y s t e m i n t o a n a f f i n e f o r m a t . T h e r e f o r e , t h e r u d d e r a n g l e as t h e m a i n c o n t r o l i n p u t is i l l u s t r a t e d i n the s y s t e m i n ( 2 1 ) . Hence, t h e n o n l i n e a r vessel s t e e r i n g s y s t e m p r e s e n t e d i n ( 2 1 ) c a n be w r i t t e n i n a f f i n e f o r m a t x = F(x) + C(x)Ux (22) w h e r e t h e respective s y s t e m m a t r i c e s can be w r i t t e n as T X = [Xi X2 X 3 j - , U X-2 m = X3 «1X2^ + « 2 ^ 2 0 = SR «3X3 C(x) = (23)The o u t p u t o f vessel steering, t h e h e a d i n g angle o f t h e vessel, c a n be w r i t t e n as yx = h(x) = w (24) Hence, t o s i m p l i f y t h e n o n l i n e a r vessel s t e e r i n g s y s t e m i n ( 2 2 ) , t h e feedback l i n e a r i s a t i o n t e c h n i q u e is c o n s i d e r e d i n f o l l o w i n g s e c t i o n . 4.2. Relative degree As t h e flrst step o f f e e d b a c k l i n e a r i s a t i o n , t h e r e l a t i v e degree o f t h e s y s t e m s h o u l d be i n v e s t i g a t e d t o v e r i f y w h e t h e r the n o n l i n e a r vessels s t e e r i n g s y s t e m is i n p u t - s t a t e o r i n p u t - o u t p u t l i n e a r i s a b l e .
The f i r s t Lie d e r i v a t i v e c o n d i t i o n d e r i v e d i n ( 4 ) c a n be i m p l e m e n -t e d f o r -t h e n o n l i n e a r vessel s -t e e r i n g s y s -t e m i n ( 2 2 ) a n d c a n be w r i t t e n as Ö / ! ( X ) , LcL°h(x) = Lgh(x): dx -G(x) dhix) dX2 mm' dX3 = [ 1 0 0 ] = 0 (25) Since, t h e f i r s t Lie d e r i v a t i v e c o n d i t i o n , L c f . ° / ! = 0 , t h e n t h e second Lie d e r i v a t i v e c o n d i t i o n s h o u l d be c a l c u l a t e d a n d t h a t can be w r i t t e n as L C 4 / I ( X ) = L G dh(x) dy 'dim dXt Fix) dim dX2 dhix)' dX3 [ 1 0 0 ] = [ 0 1 0 ] X2 a,X2^ -I-«2X2 ^3 « 1 X 2 ^ - I - « 2 ^ 2 + « 3 X 3 = ^27^0 03X3 = Lc[X2] (26)
Since, t h e second Lie d e r i v a t i v e c o n d i t i o n , LcLlhix)*0, i t can be c o n c l u d e d t h a t t h e r e l a t i v e degree o f t h e s y s t e m is i n t h e o r d e r o f 2 a n d t h e n o n l i n e a r vessel s t e e r i n g s y s t e m is i n t h e o r d e r o f 3, w h e r e t h e r e l a t i v e degree is less t h a n t h e system order. T h e r e f o r e , t h e n o n l i n e a r vessel s t e e r i n g system is n o t i n p u t - s t a t e l i n e a r i s a b l e b u t i n p u t - o u t p u t l i n e a r i s a b l e w i t h r e s p e c t t o t h e s y s t e m o u t p u t (i.e. t h e vessel h e a d i n g ) . 4.3. Input-output linearisation T h e i n p u t - o u t p u t l i n e a r i s a t i o n a p p r o a c h f o r t h e n o n l i n e a r vessel s t e e r i n g s y s t e m i n ( 2 2 ) can be w r i t t e n as z, = L ° / i ( x ) = h(x) = x , Z 2 = 4 h ( X ) = dh(x) dx
_ dim dim) dhix)
— ÖX, dX2 dx. !<2 X3 = [ 1 0 0 ] « 1 ^ 2 -I- « 2 ^ 2 + "3X3 X2 X3 a i X 2 ^ + « 2 X 2 + « 3 ^ 3 = X2 (27)
w h e r e Z7 a n d Z2 are n e w l y d e f i n e d s y s t e m states. Hence, t h e c o n t r o l f u n c t i o n s o f a(x) a n d /?(x) i n ( 6 ) can be w r i t t e n as a(x) = Ljh(x) = LFLlh(x) [0 1 0 ] X2 X3 « 1 X 2 " + « 2 ^ 2 + « 3 X 3 = X3 /?(x) = L c 4 / ! ( x ) = L c t r / i ( x ) 0 = [ 0 1 0 ] / ? 2 | (28) Hence, t h e c o n t r o l l a w u n d e r i n p u t - o u t p u t l i n e a r i s a t i o n (i.e. f e e d b a c k l i n e a r i z a t i o n ) i n ( 5 ) can be w r i t t e n as Ux= -^i-Xi-l-Vx) P2 (29) a n d the i n p u t - o u t p u t l i n e a r i s a t i o n (i.e. f e e d b a c k l i n e a r i z a t i o n ) a p p l i e d vessel s t e e r i n g s y s t e m can be s u m m a r i s e d as Z , = Z 2 Z 2 = V x (30) I n t e r n a l d y n a m i c s o f t h e i n p u t - o u t p u t l i n e a r i s a t i o n a p p l i e d n o n l i n e a r vessel s t e e r i n g s y s t e m can be w r i t t e n as X 3 - « 3 - | ) x 3 - « , Z 2 ^ - a 2 Z 2 = | v x (31) One s h o u l d n o t e t h a t t h e n o n l i n e a r vessel s t e e r i n g s y s t e m p r e s e n t e d i n ( 1 8 ) has b e e n s e p a r a t e d i n t o t w o sectors o f a s y s t e m o f l i n e a r i s e d d y n a m i c s i n ( 3 0 ) a n d a system o f n o n l i n e a r i n t e r n a l d y n a m i c s i n ( 3 1 ) b y i n p u t - o u t p u t l i n e a r i s a t i o n . T h e r e f o r e , t h e s t r u c t u r e o f n o n l i n e a r i n t e r n a l d y n a m i c s w o u l d n o t p l a y a n i m p o r t a n t p a r t o f t h e c o n t r o l l e r design process as t h a t c o u l d be separated f r o m l i n e a r d y n a m i c s . H o w e v e r , t h e c o n t r o l l e r d e s i g n process d e p e n d s o n t h e s t a b i l i t y c o n d i t i o n s o f i n t e r n a l d y n a m i c s . Hence, t h e s t a b i l i t y c o n d i t i o n s o f i n t e r n a l d y n a m i c s i n ( 3 1 ) s h o u l d be f u r t h e r i n v e s t i g a t e d t o observe t h e o v e r a l l s y s t e m s t a b i l i t y . 4.4. Internal dynamics
The stability conditions i n internal dynamics o f the i n p u t - o u t p u t linearisation applied nonlinear vessel steering system can be observed b y its zero d y n a m i c conditions as discussed previously. The zero d y n a m i c conditions o f internal dynamics o f the i n p u t - o u t p u t linear-isation applied nonlinear vessel steering system can be achieved t h e under the conditions o f
X, = z , = 0 X, = Z 2 = 0 Z2 = = 0 ( 3 2 ) A p p l y i n g t h e zero d y n a m i c c o n d i t i o n s i n ( 3 2 ) i n t o i n t e r n a l d y n a m i c s o f t h e i n p u t - o u t p u t l i n e a r i s a t i o n a p p l i e d n o n l i n e a r vessel s t e e r i n g s y s t e m i n ( 3 1 ) c a n be v y r i t t e n as X 3 + ^ - « 3 X 3 = 0 "2 (33) One s h o u l d o b s e r v e t h a t t h e p a r a m e t e r c o n d i t i o n s i n ( 3 1 ) c a n be w r i t t e n as a, <0, 02 < 0, 0 3 < 0, /?, > 0 a n d /?2 > 0 f o r t h e n o n l i n e a r stable vessel s t e e r i n g s y s t e m . Hence, t h e c o n d i t i o n s o f
^ £ 1 - 0 3 ^ = J - - l - i + J - > 0 are a l w a y s s t r a t i f i e d . T h e r e f o r e , ( 3 3 ) is
H u r w i t z , w h e r e t h e zero d y n a m i c c o n d i t i o n s are a s y m p t o t i c a l l y stable, w h e r e t h e r o o t s o f i n t e r n a l d y n a m i c s are o n t h e l e f t - h a l f p l a n e o f t h e r o o t locus (Ogata, 1997). Hence, t h e stable d y n a m i c c o n d i t i o n s f o r the i n p u t - o u t p u t l i n e a r i s a t i o n a p p l i e d n o n l i n e a r vessel s t e e r i n g s y s t e m can be c o n c l u d e d .
5. Controller design
T h e p r o p o s e d c l o s e d - l o o p c o n t r o l s y s t e m i n vessel s t e e r i n g is p r e s e n t e d i n Fig. 2. T h e s y s t e m consists o f t h r e e m a i n u n i t s p r e -f i l t e r , vessel s t e e r i n g system, a n d c o n t r o l s y s t e m . T h e p r e - -f i l t e r derives t h e r e f e r e n c e h e a d i n g angle, a n d t h e h e a d i n g rate w i t h respect to t h e r e q u i r e d h e a d i n g angle.The s y s t e m c o m p a r e s t h e r e f e r e n c e h e a d i n g w i t h t h e a c t u a l h e a d i n g i n t h e vessel t o calculate t h e h e a d i n g error. T h e n t h e a n g l e d i f f e r e n c e (i.e. h e a d i n g e r r o r ) w i l l f e e d b a c k i n t o t h e c o n t r o l system, w h e r e t h e c o n t r o l s y s t e m w i l l generate a n a p p r o p r i a t e
64 LP. Perera, C. Cuedes Soares / Ocean Engineering 66 (2013) 58-68
Pre-filler Conlroller system
Vessel steering system
Fig. 2. The closed-loop control system.
c o n t r o l i n p u t t o t h e r u d d e r a c t u a t i o n system. T h e r u d d e r angle v a r i a t i o n s w i l l generate t h e r e q u i r e d s t e e r i n g c o n d i t i o n s t o achieve t h e d e s i r e d m a n o e u v r e s , w h e r e t h e a p p r o p r i a t e r u d d e r angle a n d r u d d e r rate c o n d i t i o n s generate t h e d e s i r e d h e a d i n g angle c o n d i t i o n s t h r o u g h t h e vessel s t e e r i n g system.
The vessel s t e e r i n g s y s t e m can be f o r m u l a t e d b y a n o n l i n e a r m a t h e m a t i c a l m o d e l t h a t has p r e v i o u s l y f o r m u l a t e d i n (18), w h e r e t h e c o n t r o l s y s t e m consists o f the p r o p o s e d c o n t r o l a l g o r i t h m s . T h r e e c o n t r o l a l g o r i t h m s c o n s i d e r e d i n t h i s s t u d y are t h e H u r w i t z , Lyapunov, a n d PID based c o n t r o l l e r s . The PID c o n t r o l l e r is p r o posed as a n a d d i t i o n a l s i m u l a t i o n to c o m p a r e t h e s y s t e m p e r f o r -mances u n d e r t h e L y a p u n o v a n d H u r w i t z based c o n t r o l l e r s i n t h i s s t u d y .
As p r e s e n t e d i n Fig. 2, t h e closed-loop c o n t r o l s y s t e m consists o f a p r e - f i l t e r , w h o s e m a i n o b j e c t i v e is to derive s m o o t h c o n t i n u o u s f u n c t i o n s f o r the vessel reference heading, y/^, and t h e h e a d i n g rate, i j / r , c o n s i d e r i n g t h e r e q u i r e d h e a d i n g angle, iff'-^. T h e r e f o r e , the p r e -f i l t e r -f a c i l i t a t e s s m o o t h t r a n s i t i o n o -f t h e re-ference h e a d i n g angle and h e a d i n g rate c o n d i t i o n f o r the c o n t r o l l e r . The p r o p o s e d p r e -f i l t e r can be w r i t t e n i n t h e -f o l l o w i n g -f o r m u l a r i o n
(34) w h e r e f r a n d cor are the p r e - f i l t e r parameters.
The p r o p o s e d c o n t r o l l e r s f o r the i n p u t - o u t p u t l i n e a r i s a r i o n a p p l i e d n o n l i n e a r vessel s t e e r i n g s y s t e m are discussed i n t h e f o l l o w i n g secrions. T h e c o n t r o l m e t h o d o l o g i e s are d e r i v e d c o n -s i d e r i n g t h e i n p u t - o u t p u t l i n e a r i -s a t i o n a p p l i e d n o n l i n e a r ve-s-sel s t e e r i n g s y s t e m i n ( 3 0 ) a n d (31).
5.1. Hurwitz based control
The respective h e a d i n g e r r o r c o m p o n e n t s i n vessel s t e e r i n g can be w r i t t e n as e2=Z2-y'r (35) w h e r e e,, h e a d i n g e r r o r ; 62, h e a d i n g e r r o r rate;i//r , r e f e r e n c e h e a d i n g a n d 1//^, r e f e r e n c e h e a d i n g rate. The d i f f e r e n r i a r i o n o f t h e respective h e a d i n g e r r o r c o m p o n e n t s i n ( 3 5 ) can be w r i t t e n as e i = Z l -Vr (36) I t is a s s u m e d t h a t t h e second d i f f e r e n r i a r i o n o f t h e r e f e r e n c e h e a d i n g i n t h e p r e - f i l t e r is n e g l i g i b l e , w h e r e i//r = 0. This c o n d i r i o n has b e e n a s s u m e d t o r e d u c e the c o n t r o l g a i n r e q u i r e m e n t t h a t w o u l d i n f l u e n c e t h e r u d d e r a c t u a t i o n system. One s h o u l d n o t e t h a t t h e r u d d e r a c t u a t i o n s y s t e m is associated w i t h r u d d e r angle and r u d d e r rate l i m i t a r i o n s . T h e r e f o r e , t h i s c o n d i t i o n c o u l d p r e -serve t h e c o n t r o l l e r s t a b i l i t y a n d ( 3 6 ) can be w r i t t e n as e, = z i -6 2 = 2 2 V r (37) The e r r o r d y n a m i c c o n d i r i o n s o f t h e vessel s t e e r i n g s y s t e m can be w r i t t e n as
w h e r e iu a n d l<2 are respective e r r o r d y n a m i c constants. T h e r e f o r e , t h e respective e r r o r d y n a m i c constants, l<, a n d k2, s h o u l d be p r o p e r l y selected t o s a t i s f y t h e H u r w i t z c o n d i t i o n s . W h e n , t h e respective e r r o r d y n a m i c constants, l<, a n d k2 are H u r w i t z , t h e n ( 3 8 ) converges a s y m p t o r i c a l l y t o zero, w h e r e t h e closed l o o p s y s t e m is e x p o n e n r i a l l y stable. A p p l y i n g t h e e r r o r c o n d i r i o n s i n ( 3 7 ) i n t o ( 3 8 ) results i n
62 + /<2e2 + /<iei = 0
Hence, ( 3 9 ) can also be w r i t t e n as
èi = 6 2 é 2 = - ' < i e i - / c 2 e 2 (39) (40) T h e r e f o r e , t h e n e w c o n t r o l i n p u t u n d e r t h e H u r w i t z c o n d i t i o n s c a n be w r i t t e n as Vx = -l<\e^-i<2e2 (41) Hence, c o n s i d e r i n g ( 4 1 ) t h e m o d i f i e d f e e d b a c k c o n t r o l l e r l a w d e r i v e d i n ( 2 9 ) can be w r i t t e n as iix=-^(-X3-/<iei -/<:2e2 ) ( 4 2 ) P2
5.2. Lyapunov based control
L y a p u n o v s t a b i l i t y t h e o r e m is one o f t h e n o n l i n e a r m e t h o d s t h a t guarantees t h e s t a b i l i t y o f a closed l o o p c o n t r o l s y s t e m . T h e r e f o r e , t h e s t a b i l i t y c o n d i t i o n o f a n o n l i n e a r c o n t r o l s y s t e m is o f t e n analysed b y L y a p u n o v s t a b i l i t y t h e o r e m . To analyse t h e s t a b i l i t y c o n d i t i o n s o f t h e vessel s t e e r i n g s y s t e m , a f o l l o w i n g L y a p u n o v c a n d i d a t e f u n c t i o n is p r o p o s e d V = l e j + i e l (43) The d i f f e r e n t i a t i o n o f t h e L y a p u n o v c a n d i d a t e f u n c t i o n i n ( 4 3 ) c a n be w r i t t e n as V' = e i è , -F 6262 (44) A p p l y i n g t h e e r r o r c o n d i t i o n s p r e s e n t e d i n ( 3 6 ) i n t o ( 4 4 ) c a n be w r i t t e n as V = (Z, -y/r)(Z2-Vr) + (Z2-V'r)Vz = (Z2-V/r)(Zl " V r + ^x) (45) C o n s i d e r i n g t h e c o n t r o l i n p u t as Vx = - Z l -F V r - Z 2 + (46) A p p l y i n g t h e c o n t r o l i n p u t ( 4 6 ) i n t o ( 3 9 ) can be w r i t t e n as V = - ( Z 2 - v . r ) ' < 0 (47) Hence, i t is o b s e r v e d t h a t ( 4 7 ) a l w a y s makes a n e g a t i v e s e m i -d e f i n i t e c o n -d i t i o n ( A i c a r -d i et al., 1995). C o n s i -d e r i n g LaSelle's t h e o r e m ( K h a l i l , 2 0 0 2 ) , t h e c o n d i t i o n o f V = 0 f o r z, = Z 2 = 0 is r e q u i r e d t o p r o v e t h a t the c o n t r o l l e r is g l o b a l l y a s y m p t o t i c a l l y stable. H o w e v e r , o n l y t h e c o n d i t i o n o f V = 0 f o r Z2 = 0 c a n be p r o v e n i n t h i s s i t u a t i o n , a n d t h e r e f o r e t h e c o n t r o l l e r c a n achieve s e m i g l o b a l s t a b i l i s a t i o n . The e s t i m a t e d r e g i o n o f a t t r a c t i o n , i2, t h a t preserves the c o n t r o l l e r s t a b i l i t y is g i v e n hy Q = Z2eR^ : V<c. The c o n s t a n t , c, c a n be chosen large e n o u g h , w h e r e V is r a d i a l l y u n b o u n d e d . The r a d i a l u n b o u n d e d n e s s i n V g u a r a n t e e s t h e b o u n d e d n e s s o f t h e e r r o r t h a t c o r r e s p o n d s t o any b o u n d e d i n i t i a l c o n d i t i o n s . Hence t h e largest i n v a r i a n t set, E, t h a t c o r r e s p o n d s t o a n y b o u n d e d i n i t i a l c o n d i t i o n s can be w r i t t e n as
Hence, c o n s i d e r i n g ( 4 6 ) , t i i e m o d i f i e d feedbacl< c o n t r o l l e r l a w d e r i v e d i n ( 2 9 ) can be w r i t t e n as
Ux = ^ ( - X 3 - Z , + Wr-22 + V'r) (49)
5.3. PID control
I t is obsei-ved t h a t m o r e t h a n h a l f o f t h e i n d u s t r i a l c o n t r o l l e r s are based o n PID o r m o d i f i e d PID t y p e a p p l i c a t i o n s (Ogata, 1997). I n PID c o n t r o l , an accurate s y s t e m m o d e l is n o t r e q u i r e d f o r t h e i m p l e m e n t a t i o n , w h i c h is an advantage over o t h e r c o n t r o l l e r s . The PID c o n t r o l l e r can be w r i t t e n as
Vx = -kpe^-ki e^(T)dT-koe2 ( 5 0 )
w h e r e hp, kj and ko are p r o p o r t i o n a l , i n t e g r a l a n d d e r i v a t i v e c o n t r o l gains, respectively. Hence, c o n s i d e r i n g ( 5 0 ) , t h e m o d i f i e d f e e d b a c k c o n t r o l l e r l a w d e r i v e d i n ( 2 9 ) can be w r i t t e n as
Ux = ^ ^ - X 3 - / < p e , - k , e, ( T ) d T - / < D e 2 ^ ( 5 1 )
6. Computational simulations
6.3. Summarised system dynamicsA s u m m a r y o f system dynamics i n n o n l i n e a r vessel steering t h a t has been used f o r t h e c o m p u t a t i o n a l simulations can be w r i t t e n as
X l = = X2 X2 = l/> = X3 + Ux = Vx X3 = « 1 X 2 ^ + «2X2 + ( " 3 - ^ ) ^ 3 + ^ V x (52) w h e r e "x = 5R = J - (-X3 + Vx) and n = ^ 2 6.2. Controller parameters T h e c o m p u t a t i o n a l s i m u l a t i o n s o f t h r e e c o u r s e c h a n g i n g m a n o e u v r e s o f a n o n l i n e a r vessel s t e e r i n g s y s t e m are p r e s e n t e d i n Figs. 3 a n d 4 . T h e t o p p l o t o f F i g . 3 consists o f t h e vessel r e s p o n s e o f r e f e r e n c e h e a d i n g , a c t u a l h e a d i n g u n d e r L y a p u n o v b a s e d c o n t r o l w i t h o u t r e f e r e n c e h e a d i n g r a t e (LBC w / o R H R ) , L y a p u n o v based c o n t r o l w i t h r e f e r e n c e h e a d i n g r a t e (LBC w / RHR), H u r w i t z based c o n t r o l ( H B C ) a n d PID c o n t r o l ( P I D ) , r e s p e c t i v e l y . O n e s h o u l d n o t e t h a t t h e H u r w i t z b a s e d c o n t r o l l e r has n o t b e e n u s i n g t h e f e e d b a c k o f t h e r e f e r e n c e h e a d i n g r a t e . T h e m a i n r e a s o n b e h i n d t h i s s e l e c t i o n is t o r e d u c e t h e l a r g e r c o n t r o l g a i n t h a t c o u l d be d e m a n d e d b y t h e c o n t r o l l e r f r o m r u d d e r a u c t i o n s . As t h e r u d d e r a c t u a t i o n s y s t e m is l i m i t e d b y t h e r u d d e r r a t e a n d r u d d e r a n g l e l i m i t a t i o n s , t h i s a p p r o a c h m a k e s r e a l i s t i c c o n d i t i o n s i n vessel s t e e r i n g . T h e b o t t o m p l o t o f Fig. 3 c o n s i s t s o f t h e v e s s e l r e f e r e n c e h e a d i n g e r r o r u n d e r t h e c o n t r o l l e r s o f LBC w / o RHR, LBC w / R H R , HBC a n d PID, r e s p e c t i v e l y .
The t o p p l o t o f Fig. 4 consists o f t h e vessel r u d d e r angle u n d e r t h e c o n t r o l l e r s o f LBC w / o RHR, LBC w / R H R , HBC a n d PID, respectively. T h e b o t t o m p l o t o f Fig. 4 consists o f t h e b o u n d e d d i s t u r b a n c e s t h a t w e r e i n t r o d u c e d i n t o i n t e r n a l d y n a m i c s o f t h e vessel s t e e r i n g s y s t e m t o observe t h e s y s t e m p e r f o r m a n c e u n d e r t h e above c o n t r o l l e r s . The b o u n d e d i n t e r n a l d i s t u r b a n c e s t h a t m o d e l l e d as w h i t e Gaussian noise d i s t n b u t i o n s are assumed i n t h i s study. T h e p r o p o s e d c o n t r o l l e r s are i m p l e m e n t e d o n t h e IVIATLAB s o f t w a r e p l a t f o r m .
T h e parameters f o r t h e n o n l i n e a r vessel s t e e r i n g s y s t e m are considered as a i = -0.3710 ( l / r a d ^ ) , a 2 = - 0 . 4 3 4 0 ( l / s ^ ) , a 3 = - 3 . 4 0 0 0 (1/s), /?! = 0.3500 ( l / s ^ ) , a n d / J 2 = 0 . 1 2 2 5 (1/s^). S o m e o f these p a r a m e t e r s are extracted f r o m the s t u d y ( A m e r o n g e n a n d Cate, 1975), a n d o t h e r parameters are generated b y t r i a l a n d e r r o r calculations c o n s i d e r i n g t h e vessel response u n d e r stable s t e e r i n g c o n d i r i o n s . I t is assumed t h a t t h e r u d d e r angle is associated w i t h t h e m a x i m u m l i m i t o f ;r/4 ( r a d ) as p r e s e n t e d i n t h e t o p p l o t o f Fig. 4, Vessel heading Reference heeijing L 3 C w / e R H R LBCw/BHR
-/'-
HBC PIDReference heading error O.B , 0.5 0.-1 0.3 -0.2 - • 01 0 -0.1 -0.2 •0.3 -0.4 - LBC w/o RHR L B C v / f f l H R NSC 40 60 BO Time (sl
66
t h e r e f o r e t h e r u d d e r angle v a r i a t i o n s w i t h i n this r e g i o n are expected, w h e r e the c o n t r o l l e r can preserve its stability. Further-m o r e , the paraFurther-meters f o r the p r e - f i l t e r are considered as f r = 1-0 a n d ( O r = 1 . 0 .
T h e p a r a m e t e r s f o r t h e H u r w i t z c o n d i t i o n s are c o n s i d e r e d as k j = 0 . 1 ( l / s ^ ) a n d /<2=0.8(l/s). One s h o u l d note t h a t t h e L y a p u n o v based c o n t r o l l e r has n o t b e e n i n v o l v e d w i t h any p a r a m e t e r c o n d i t i o n a n d the c o n t r o l l e r is p u r e l y based o n t h e f e e d b a c k o f t h e vessel states o f vessel r e f e r e n c e h e a d i n g , h e a d i n g rate, vessel actual h e a d i n g a n d h e a d i n g rate. T h e PID c o n t r o l p a r a m e t e r s c o n s i d e r e d f o r the s i m u l a t i o n s c a n be w r i t t e n as kp^l.O (1/s^), k,=OA ( l / s 3 ) a n d k D = 1 . 0 ( l / s ) .
7. Discussion
One s h o u l d note t h a t a n o n l i n e a r vessel steering system c o n sidered i n this study is l i m i t e d to stable n o n l i n e a r steering c o n d i -tions. H o w e v e r , a n o n l i n e a r vessel steering system w i t h unstable n o n l i n e a r steering c o n d i t i o n s w i l l be considered as a f u t u r e s t u d y o f this w o r k .
A b r i e f o v e r v i e w o f t h e c o m p u t a t i o n a l s i m u l a t i o n results are f u r t h e r discussed i n t h i s section. The r e f e r e n c e h e a d i n g t h a t is g e n e r a t e d u n d e r t h e p r e f i l t e r to c o n s t r a i n t h e c o n t r o l p e r f o r -m a n c e is p r e s e n t e d i n t h e t o p p l o t o f Fig. 3. The vessel actual response u n d e r the L y a p u n o v based c o n t r o l l e r w i t h t h e reference h e a d i n g rate (LBC w / R H R ) generates t h e fastest p e r f o r m a n c e , w h e n i t converges to the r e f e r e n c e h e a d i n g . T h i s is m a i n l y due to t h e a d d i t i o n a l f e e d b a c k i n p u t o f t h e vessel reference h e a d i n g rate t h a t is used i n t h e c l o s e d - l o o p c o n t r o l l e r . H o w e v e r , t h e vessel response is associated w i t h an a d d i t i o n a l s l i g h t o v e r s h o t i n c o m p a r i s o n to the o t h e r c o n t r o l l e r s . F u r t h e r m o r e , t h e LBC w / RHR f e e d b a c k n o t o n l y generates t h e f a s t e r vessel response b u t also r e q u i r e s a larger r u d d e r angle t h a t is b e y o n d r u d d e r angle l i m i t a t i o n s . Some o v e r s h o o t c o n d i t i o n s can also be o b s e r v e d f r o m t h e r e f e r e n c e h e a d i n g e r r o r i n t h e b o t t o m p l o t o f Fig. 3 u n d e r t h e
same c o n t r o l l e r . As p r e s e n t e d i n t h e b o t t o m p l o t o f Fig. 4, t h e r u d d e r response o f LBC w / R H R goes b e y o n d t h e r u d d e r angle l i m i t a t i o n s , w h i c h is a d r a w b a c k i n t h i s c o n t r o l l e r .
T h e r e f o r e , t h e L y a p u n o v based c o n t r o l l e r w i t h o u t r e f e r e n c e h e a d i n g rate (LBC w / o RHR) is c o n s i d e r e d . T h e vessel response u n d e r t h e LBC w / o RHR generates t h e second f a s t e s t response w h e n i t is c o n v e r g i n g to t h e r e f e r e n c e h e a d i n g . H o w e v e r , t h e c o n t r o l l e r does n o t generate a s l i g h t o v e r s h o o t as f o r t h e p r e v i o u s case. T h e r e f o r e , t h e r u d d e r angle r e q u i r e m e n t o f t h e LBC w / o RHR is s m a l l e r c o m p a r e d t o t h e r u d e r angle r e q u i r e m e n t o f t h e LBC w / R H R . H o w e v e r , t h e c o n t r o l l e r LBC w / R H R has a f a s t e r response w h e n c o m p a r e d t o t h e c o n t r o l l e r LBC w / o RHR.
T h e vessel response u n d e r t h e H u r w i t z based c o n t r o l l e r (HBC) generates t h e s l o w e s t response w h e n i t is c o n v e r g i n g t o t h e r e f e r e n c e h e a d i n g . H o w e v e r , t h e c o n t r o l l e r does n o t generate a n o v e r s h o o t as f o r t h e f i r s t s i m u l a t i o n . The r u d d e r a n g l e v a r i a t i o n o f t h e HBC is s m a l l e r w h e n c o m p a r e d t o t h e r u d e r a n g l e r e q u i r e -m e n t s o f a l l o t h e r c o n t r o l l e r s . H o w e v e r , t h e vessel r e s p o n s e u n d e r t h e HBC is also s l o w e r w h e n c o m p a r e d t o t h e o t h e r c o n t r o l l e r s . T h i s is m a i n l y d u e t o t h e l o c a t i o n o f t h e r o o t s i n t h e c o n t r o l l e r u n d e r t h e H u r w i t z c o n d i t i o n s . One can argue t h a t t h e r e s p o n s e c a n be f a s t e r b y f o r m u l a t i o n o f t h e r o o t s l o c a t i o n i n t h e c o n -t r o l l e r . H o w e v e r , -t h e o v e r s h o o -t a n d o s c i l l a -t i o n c o n d i -t i o n s i n -t h e vessel r e s p o n s e w i l l g e n e r a t e w i t h t h e f a s t e r v e s s e l r e s p o n s e u n d e r t h e HBC.
T h e vessel response u n d e r t h e PID c o n t r o l g e n e r a t e s a m o d e r -ate response w h e n i t is c o n v e r g i n g t o t h e r e f e r e n c e h e a d i n g . H o w e v e r , t h e c o n t r o l l e r generates o v e r s h o o t as w e l l as o s c i l l a t i o n c o n d i t i o n s . One c o u l d argue t h a t the v a r i a t i o n s i n PID c o n t r o l gains c o u l d r e s u l t i n b e t t e r vessel s t e e r i n g s y s t e m responses, w h e r e t h e s e t t l i n g t i m e c o u l d also be c h a n g e d . I f t h e PID c o n t r o l l e r has a l a r g e r s e t t l i n g t i m e the response o v e r s h o o t w i l l be r e d u c e d a n d i f t h e PID c o n t r o l l e r has a s h o r t e r s e t t l i n g t i m e t h e n t h e response o v e r s h o o t w i l l be increased. T h e r e f o r e , t h i s is a g e n e r a l b e h a v i o u r o f PID c o n t r o l a n d an a p p r o p r i a t e s e t t l i n g t i m e w i t h r e s p e c t t o t h e o t h e r c o n t r o l l e r s are 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 s , w h e r e t h e
