On a Ship Track-Keeping Controller with Roll Damping Capability

IFAC

Copyright © IF AC Control Applications in Marine Systems, Glasgow, Scotland, UK, 2001

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Pu bl ications www.elsevier.comllocate/ifac

ON A SHIP TRACK-KEEPING CONTROLLER WITH ROLL DAMPING CAPABILITY Dorota Lozowicka, I Antonio Tiano,

2.3

and Adam Lozowicki,

I

Technical University ofSzczecin ul. Klemensiewicza 6/1,70-028, Szczecin, Poland e-mail: [email protected]. 2 Department of Information and Systems, University of Pavia, Via Fen-ata 1, 1-27100 Pavia, Italy, e-mail: [email protected]. 31nstitute of Ship Automation, National Council of Researches, Via De Marini 6, 1-16149 Genova, Italy. J

Abstract: In this paper the problem of ship trajectory tracking with £-accuracy is considered. Results presented are based on a three-degrees-of-freedom model with full dynamic interaction between motions in roll, sway and yaw. The aim of control is to minimise the influence of wave disturbances on roll motion and to track with £-accuracy a ship trajectory. For this purpose a linearised and a non-linear ship model are used . The Matlab-Simulink simulations confirm the viability of the proposed method for the high precision control of ship track-keeping. Copyright © 2001 IFAC

Keywords: tracking systems, MIMO, marine systems, disturbance rejection I. INTRODUCTION

to efficiently and safely carry out demanding navigation tasks in a widely varying range of environmental conditions. In principle, the design of such control systems should be based on a multi variable approach, which takes properly into account the couplings between the different motion and determines control systems within the framework of optimal control theory. A number of successful simulation studies and sea trials have been carried out which apparently supported such approach.

The remarkable growth of transport of passenger and cargo at sea as well as the initial exploitation of ocean resources has determined the construction of an increasing number of new surface ships and underwater vehicles. For an efficient operation at sea, it is essential that these craft should be equipped with advanced control systems. Control systems to be installed on board ships are generally designed in such a way to reduce fuel consumption, to minimise disturbing wave induced motion and at the same time to improve navigation accuracy. It has been verified in the recent years that traditional control methods are generally inadequate for designing efficient control systems.

It is worth and surprising enough noting, however, that the adoption of these optimal designs has been until now quite irregular on board new ships, were old fashioned PlO autopilots are often preferred. This apparent discrepancy is partly due to the fact that most of the proposed designs are critically dependent on the availability of accurate mathematical models of ship and environment, which are generally quite complex and difficult to be determined and properly on-line tuned. It may be therefore attractive to explore the applicability of robust control methods, which potentially can reduce the negative effects on control system performance of the uncertain factors affecting the ship dynamic behaviour. Most of these methods can be derived by an H~ optimization approach

Design solutions for marine craft motion control are often rather difficult to find within the context of classical control theory, owing to the intrinsic nonlinear dynamic behaviour of the plant itself and to the disturbances, which act upon it. Many researches and simulation studies have been conducted in the recent years in order to design and put into operation a new generation of ship control systems, such as autopilots, stabilisers and dynamic positioning systems, capable

191

(Francis, 1987) which aims at satisfying the control specifications with a significant rejection of disturbances. It has been shown that this corresponds, in the linear case, to the determination of a stabilising feedback controller that minimises the infinity-norm of a properly weighted system transfer matrix. The weights are chosen in such a way to cope with the main uncertainties affecting the system.

V

l

00256

-0.0122 -4.4802 -0.0300

r -0.0012 -0.2211 -0.0062 -0.0009 0 p = 0.0025 -0.6504 -0.0252 _ -0.0282 0 q; 0 0 o 0 0 0 0 o lit

0.13151 -0.0050 P + -0.0043 b ({J 0 0 !If r

vl

[;l= [~

0

1

0

0

0

0

(1)

r

°l

P

1

({J

In this paper an algorithm for determining a robust control of the ship multivariable track-keeping problem is presented. This algorithm, based on a functional analysis approach, allows to reduce the computational complexity, which generally does not allow an efficient implementation of robust control methods, such as for example the !i-synthesis one (Helton and Merino, 1998), in the case of multi variable systems. Some preliminary simulation results indicating the good achievable performance of the proposed controller, are presented. These results have been obtained during simulation experiments based on a non-linear ship model previously determined from towing tank and sea trials experiments (Blanke and Jensen, 1997).

!If

is deduced. A direct evaluation of system's eigenvalues: 0.0095, -0.2520, -0.0145 + 0.1619i -0.0145 - 0.1619i 0.0225, indicates that this model has an unstable eigenvalue. Equations (1) are a straightforward linearisation of the non-linear model for a ship speed of 12.5 m/s and ship's metacentric height GM = 83 cm. The control system designed for the equations (1) is then compared with a control system for an equivalent parameter tuned model, given by equation : V

-0.0115 -3.2325 -0.1112

I'

-0.0010 -0.2216 -0.0066 -0.0010 0 r

p = 0.0037

2. SHIP MATHEMATICAL MODEL

(p

0

lit

0

-0.06941

01217 1

-0.0050

0.0960 -0.0752 -0.0552 0 p + -0.01038 0

1

o

0

({J

0

0

0

o

!If

0

vl

For the ship motion analysis it is useful to consider the two co-ordinate systems shown Fig. 1. The first one XoYoZ'o is chosen in the earth-fixed point, while the second XYZ in the centre of symmetry of the hull :

I

0

o 0

r

°l P

(2)

I

cp

'I' This model has all stable eigenvalues :-0.0003 , -.2352, -0.0363 + 0.2337i and -0.0363 - 0.2337i. The designed control systems for the above. linearised models will be subsequently used for controlling with £-accuracy of the non-linear model proposed by (Blanke and Jensen ,1997).

If we analyse the ship motion control system, then we must take also into account the mathematical model of the steering machine. The rudder angle and the rudder rate saturations result from the characteristics of the steering machine. Typical constraints are in the ranges:

Fig. 1. Ship co-ordinate system In the paper the pitch and heave motions will be neglected. We will take under consideration the ship motion in three degrees of freedom relating the coupled sway, yaw, and roll response to the rudder at a constant cruising speed.

I ~::; ~ax '" 35 deg;

151 ~ 5 = 2.3 degls or 151 ~ 5 = 4,6 degls. (3) max )

max2

At first the controller for a linear and unstable model of ship motion, given by Blanke and Jansen (1997) in the form:

These constraints will be used for designing the class of admissible input signals (class of given ship trajectories) L2(W) or more generally R(W), described in the next section. If we consider the ship motion mathematical model, we must take into account the influence of the environmental disturbances such as waves, wind and ocean currents. These disturbances

192

can be roughly divided into a high frequency (HF) component and a low frequency (LF) one. For eliminating the influence of the HF disturbances, a conventional filter can be used, as shown in Fig. 2. The problem of eliminating the influence of LF component will be addressed in section 3.

R(w'm)

= {r : r = W(w) for some

WE

R,

IlwllR.$"m <

oo}

(5)

If the constant m is not determined exactly, then the notation R( W) will be used; we will assume in addition that W = P. Then R(P,m) denotes the set of all signals r(t) ER such that the operation P and a constant 0 < m < 00 satisfy the relations

Let C be a set of allowable controllers K that can be used in our control system (4) and let us denote by e~t) the error signal in the control system (4) with the controller KE C, which corresponds to the signal r(t) E R(W,m).

Fig. 2. Filter for eliminating the ship heading

If/HF

Let us denote by e~t) the error signal in the control system (4) with the controller KE C, which corresponds to the signal r(t) E R(W,m).

disturbances on

Definition 1.

3.HlGH PRECISION FEEDBACK CONTROL SYSTEM

The plant P is controlled with £-

accuracy for a signal

r{t) E R{P, m)

by the system

(4) if there exist a controller KC E C such that for We take under consideration the feedback control system given by equations

every

r( t)

E

R..P, ~ the inequality

r

u(t)= K(e(t)) i e(t) = r(t) - p(u(t)) l r(t)= W(w(t))

(7) (4)

is satisfied. Let R be a space L2 (space of signals with bounded energy) or let R be a Marcinkiewicz space M (space with bounded mean power). Laplace transforms of the signals u, w, e, r will be denoted by u(s), w(s), e(s), res) respectively. In the particular case when the plant P is represented by a linear, causal and stationary operation, then the following theorem can be proved.

where r(t), e(t) and u(t) are reference signal, error signal and control signal respectively.

Theorem 1. If for the number k E [k j , k 2 ) where k j , k2 are sufficiently large and for the operation P the inequality

infl.!.. + p{s 1 ~>0

Fig. 3. Feedback control system

res~O k

Let R be a Banach space and let operations P:R~R and K:R~R represent plant and controller regarded as operators between two functional spaces. The operation W(w(t», where w(t) belongs to a Banach spaces R, is the generator of a reference signal r(t). From relations (3) we can conclude that for lel :5: ~ax the controller K has the form

(8)

is fulfilled, then system given by equations (4), controls the plant P with £-accuracy for a signals r(t) E L:!(P) (r(t) E M(P». Theorem I can be generalised in a simple way for a system with n-inputs and n-outputs (MIMO systems). Let the operation P be given by the transfer matrix pes) and let as denote by Aj(S), A2Cs), ... , A-,,(s) the eigenvalues of the transfer matrix pes). The following statement can be proved:

K(e) = ke, kE 9\. It is assumed that r is not a known fixed signal but that it can be modelled as belonging to the class of functions:

193

n

The

closure

of

the

set

Un;,

(see Fig. 4). The error signal has a form e(t) = ['I'o(t)If
where

;=1

n; = {A: ..1.= A;{s)

for res> o} is

equal to

the spectrum of operation P in the space L 2. The first two equations from (4) take the form :

o (11 )

(9)

Theorem 2. If the numbers k; E [k/, k]) , where k/ , k] are sufficiently large and the operation P fulfil the inequality:

1~

min inf l. + A;(s > 0 ; res~O k j

(10)

then the system given by equation (9) controls the plant P with £-accuracy for every signal r(t)EL](P) (r(t)

E

Fig. 4. Track of ship trajectory

M(P)) .

5. CONTROL SYSTEM FOR THE LINEARISED SHIP MODEL

4. SHIP TRAJECTORY TRACKING Let us take under consideration the control system given by equation (4). Let the control signal u(t) consist of the rudder angle ~t) as a function of time.

Let us take under consideration the ship model given by equation (I). For stabilising this system and for minimising the influence of the waving disturbance PH, the additional feedback loop as in Fig. 5 is used. An analogous system can be used for minimising the influence of disturbances, for the stable model given by equation (2). For the feedback system shown in Fig. 5 and associated to the stable model, the following polynomials have been found:

We will assume that the plant P consists of the ship motion model, the steering machine and a filter of HF disturbances (see also Fig. 2). We assume additionally that we do not know exactly the ship speed on the reference ship trajectory. In this case the following procedure can be used, through which the influence of a (LF) disturbances can be effectively eliminated.

p(5) 8(5)

As a reference signal we take the vector r(t)=[ 'l'o(t), Po(t)( The coefficient p oet) is connected with roll angle (j"I,t) and it will be assumed that Po(t) = O. The first coefficient 'l'o(t) , is connected with ship course If
=

-0.01035 4 -0.00245 3 +0.00015 2 +0.00005 55 +0.30835 4 +0.07315 3 +0.01325 2 +0.0005 (12)

The relation between 0 and 'V is given by:

'1'(5) 8(5)

0.05 4

-

0.00505 3 - 0.00055 2 4

-

0.00035

55 + 0.30835 + 0.07315 + 0.01325 2 + 0.0005 (13 ) 3

The element given by the transfer function :

is used as controller. In a previous paper (Lozowicki and Tiano, 2000), a PD controller was used. But the results of minimisation the influence of disturbances are much better for the case with controller (14).

and centre in the actual ship position {x(t),y(t)}. The first coefficient 'l'o(t) of the reference signal r(t) is defined as the angle between the line passing through the two points {x(t),y(t)} , {xo(t),yo(t)} and the Xo-axis

194

w

It should be noticed that controller (14) can be used both for stable plant (2) and for unstable plant (1).

w

I--=----.<

E

A reference signal generator

The control system for minimising the course disturbance I{IH is shown in Fig. 6, where the relation between l{Iand 8 has the form (13). Controller Kc2 (s) has the form

Based on Theorem I and Theorem 2, we can conclude that the system shown in Fig. 7 controls the considered ship model (1) or (2) with £-accuracy with respect to any reference signal belonging to the class R(W,m) = L 2(P,m). As already stated, to the class ofa reference signals L2(P,m), belong all signals from L2 space, with bounded (in the sense of the norm of L2 space) first five derivatives.

PH

Fig. 7. A high precision feedback control system with disturbances I{IH and PH Some simulations have been carried out by using the non linear complete ship mathematical model and periodic sea waves disturbances on both yaw, sway and roll motion, in the MA TLAB-Simulink programming environment. In Fig. 8 a result of disturbance PH and damped roll signal is illustrated. In Figures from 9 to 12, the desired ship coursechanging l{Io , the realised ship course If/, the error signal e and roll signal P are shown. The results of this simulation confirm the above theoretical considerations as well as the high precision control of the ship motion achievable with the proposed method.

P ~s)/O(s)

8

~ +

k1e.sTI

k2

Fig.5. Control system for minimising the influence of roll disturbance PH

'If

'lfH

0

-0.1

ffJ(s)//l(s)

-0.15 -0.2 "----"----'-''------'''---'-------'''-----'''----'''-----' o 50 100 150 200

~ +

k~3e ·,r]

Fig. 8. Time histories of roll disturbance damped roll rate p k.

Fig.6.Control system for minimising the influence of course disturbances I{IH

195

PH

and of

0 .5

2 1.8

0

1.6 1.4

-0.5

1.2 -1 0.8

-1 .5

0.6 -2

0.4 0.2

-2 .5 0

0

50

100

150

0

50

100

150

200

200

Fig. 12. Time history of roll rate signal p Fig. 9. Time history of the ship desired course lfIo REFERENCES Blanke M. And Jensen A.G. 1997. Dynamic properties of container vessel with low metacentric height. Transactions of The Institute of Measurement and Control. 19 (2), pp.79-83. Francis A. 1987. A course in H ~ control theory, Springer-Verlag, Heidelberg. Helton J. and Merino O. 1998 Classical Control using H~ methods. SIAM, Boston. Lozowicki A. and Tiano A 2000. On the design of a high precision track keeping system for a 5th IFAC Conference on containership. Manoeuvring and Control of Marine Crafts MCMC2000, Aalborg, Denmark, pp.123-128.

2r---------------~--------------__,

1.8 1 .6 1 .4

1.2

0.8 0 .6 0.4

0.2

°0~~-----50--------1~00--------15-0------~ 200

Fig. 11. Time history of the realised ship trajectory IfI

Fig. 12 Time history of the error signal e

196