International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 1

Number 3

September 2007

313

The Use of Backstepping Method to Ship

Course Controller

A. Witkowska

University of Technology, Gdansk, Poland

R. Smierzchalski

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: The article systematises and perform approaches the new concept of the ship autopilot in which

control rules are derived for nonlinear controllers designed with the aid of the backstepping method and used

for controlling the ship’s motion on its course. The objectives, approaches and problems were described. The

design is very interesting has goals to create closed-loop systems with desirable stability properties in the

regulation and tracking problems with a uniform asymptotic stability, rather than analyze the properties of a

given system. The symulation were performed on the tanker model and were comparised in the system with

PD controller.

1 INTRODUCTION

In marine navigation, it is required the skills

of determination of ship position, appointment of

proper ship course as well as the keeping on

appointed course. Numerous investigations

performed in the past were oriented on designing an

integrated ship control system. Despite significant

improvement in automation, the course control is

still an active field of research, especially in low

speed regimes. The navigation at this speed is

difficult due to manoeuvring problems connected

with a relatively big mass of the ship and limited

dimension of the rudder, which must be significantly

deflected to obtain the required change of ship’s

course. This effect is especially noticeable on

tankers. Reduced controllability of those ships can

be compensated by the use of automatic control

systems, which change the course of the ship in a

desired way by proper movements of the rudder.

Nowadays, autopilots installed on ships usually

use the algorithm of PID controller. The measured

ship course is compared with the required (set) value

and the calculated difference makes the input signal

passed to the controller. The control signal, obtained

at the output of the controller, is then transmitted to

the servo-mechanism of the steering gear and

provokes a required change in the rudder deflection

angle. The automatic ship course control system

(autopilot) is expected to execute two tasks. The first

task consists in course changing when the ship

moves along the desired trajectory, and in this case

the manoeuvre should be performed fast and

precisely. This is of especial importance when the

maneuveres are performed in high-traffic water

regions, or in restricted waters. The second task

consists in keeping the ship on the desired constant

course – in this case the rudder activity and the so

called “ship yawing effect” should be minimised to

reduce fuel consumption. The article systematises

and perform an approach of the new concept of the

ship autopilot in which control rules are derived for

nonlinear controllers designed with the aid of the

backstepping method and used for controlling the

ship’s motion on its course. The design sets is very

interesting has goals to create closed-loop systems

with desirable stability properties in the regulation

and tracking problems with a uniform asymptotic

stability, rather than analyze the properties of a given

314

system, because systems that posses it can deal

better with perturbations and disturbances.

Therefore the method backstepping matter first of

all in the ship automation , in which were required

the stability of work of arrangement as well as safety

of ship quidance of on appointed course aside from

of influence the disturbances and perturbances. It

deal tanker ships, container as well as passenger

ship.

2 BACKSTEPPING METHOD

2.1 Historical outline

The difficulties observed in ship control mainly

result from neglecting nonlinear dynamic

characteristics and changes in ship motion

parameters. Numerous attempts, published in the

literature, to overcome these difficulties make use of

methods that linearise the system for certain

operation points, like the feedback linearisation

method, for instance. These methods, however,

return solutions which are not fully satisfying and

the linearized systems do not reflect the true

proprieties of real object.

In recent ten to twenty years a number of new

methods were developed for designing controllers to

control nonlinear dynamic systems. These are mainly

recursive methods, such as backstepping, forwarding,

and various combinations of them. A common

concept of the abovenamed basic recursive methods

is the design of a globally stable control system,

having a cascade structure, for a class of nonlinear

dynamic systems. In particular, the backstepping

method is based on the Lyapunov function theory

(La Salle 1966) but its origin can be found in some

theories of linear control, such as the feedback

linearisation method or the LQR method.

The beginning of development of the

backstepping method in application to nonlinear

control system designs can be dated on the turn of

Eighties and Nineties of the last century. A list and

discussion of publications issued in that time can be

found in an overview by Kokotović and Arcak

(Kokotovic 2001), as well as in Fossen (Fossen

2002).

The backstepping method is based directly on

the mathematical model of the examined system,

introducing to it new variables in the form depending

on the state variables, controlling parameters, and

stabilising functions. The task of a stabilising

function is to compensate non-linearities recorded

in the system and affecting the stability of its

operation. The linearisation methods used in the

feedback-based systems usually aim at eliminating

non-linearities existing in the system. The use of the

backstepping method makes it possible to create, in

an arbitrary way, additional nonlinearities and

introduce them to the control process to eliminate

undesirable nonlinearities from the system (Fossen

1998). This is of high importance in case of ship

control systems in which removing all nonlinearities

would require the information on accurate models

of all existing non-linearities, hardly available

in practice. The backstepping method allows to

obtain global stability in cases when the feedback

linearisation method only secures local stability.

One of the earliest books on backstepping control

methods was published by Krstić, Kanellakopoulos

and Kokotović (Krstic 1995). In there, especial

attention was paid to adaptive and nonlinear control

of SISO-type systems, with some extension to

MIMO-type systems. Another concept how to apply

the backstepping method in control system design

was proposed by (Sepulchre 1997). The method

developed by him took into account acceleration

increment inertia for cascade control systems.

(Krstic 1998) extended the topic, focusing on

the stabilisation problem in stochastic nonlinear

systems.

2.2 The backstepping approaches

The backstepping method was used in numerous

engineering applications, among other cases for

designing a system that controls the flight trajectory

(Harkegard 2003), in the spaceship observation

process (Krstic 1999), in the designs of industrial

systems, electric machines and nonlinear systems of

wind turbine-based power production, as well as in

robotics for controlling a robot moving along a

desired trajectory. In particular, the backstepping

method can an be an effective tool in adaptive

control designs for estimating parameters,

(Fang 2004, Jiang 2002) and solving various optimal

control problems. Moreover, the control algorithms

based on the backstepping method make it possible

to design a robust, nonlinear controller that limits the

effect of disturbances acting both in deterministic

and stochastic manner (Do 2004, Skjetne 2005). As

a result, a control process is obtained which is

globally stable in the entire area of its operation.

In the marine technology, the presented

backstepping method was used in the systems that

steer the ship on its course (Do 2004, Pettersen

2004), to secure course stabilisation. In 1999, Fossen

published a work (Fossen 1999), which focused on

practical use of the backstepping method in

mechanical systems and its application to ship

steering.

315

2.3 The optimization of the system designed by

backstepping

However, attempts to apply backstepping method

this method in real marine systems revealed

numerous problems which needed solving. One of

them is the structure and selection of the stabilisation

functions and identification of their parameters. In

order to obtain optimal quality of control for the

designed nonlinear course controller, its parameters

need tuning. The choice of the parameters of

backstepping ship course controller with regard to

compound ship models is not an easy task to do

taking into consideration the nonlinear working

system and the complicated control unit structure.

The impediment is the change of the system

dynamics depending on the working point and stem

parameters time variability which was caused by the

course modification, speed, loading state or the

influence of the environment disturbance. The

analysis of the regulation system structure taking

into consideration parameters variability could lead

to more precise control over the vessel movement in

various system working conditions. The design

systems, presented in the literature, that make use of

the backstepping method are optimised using

classical methods, usually based on the H

∞

method

and solution of Hamiltonian Jacobi Bellman

Equation and the Riccati equation (Ezal 2000; Krstic

1999).

In the symulations performed in this article the

parameters of the nonlinear control structure were

tuned to optimise the operation of the control

system. The optimisation was performed using

genetic algorithms.

3 STRUCTURE OF CONTROL SYSTEM

For convenience, backstepping have been introduced

using a system consisting of a nonlinear subsystems

and a integrator chain. However, these procedures

are applicable to larger classes of systems. In

backstepping method arduous and time-consuming

calculations were introduced therefore in this article

was limited to performance the symulations results

for this method only (Krstic 1995).

In present work backstepping method was applied

in system showed on Figure 1. In the window „Ship”

the equations of the ship dynamics characteristics

were modelled. In the present investigations, the

mathematical model of the dynamical characteristics

of the ship was taken from a model tanker described

by Astrom and Wittenmark in „Adaptive Control”

(Astrom 1989) and modelled by a nonlinear third-

order differential equation, referred to as the Bech

and Wenger’s model. The model was complemented

by the dynamics of the steering gear, shown in

Figure 2.

Trajectory

controllerr

Course

controller

Steering

gear

Ship

Dynamic

Ship Model

(

)

(t)

(

)

(

)

Fig. 1. The block scheme of arrangement of steering

the movement of ship

The input signal passed to the steering gear comes

from the autopilot and has the form of the set rudder

angle,

)(t

, while the output signal is the current

rudder angle,

(t). For the majority of ships

the rudder angle and speed of its change are kept

within certain limits (Amerongen1982) where

max

35 [deg], 2.3

≤≤

max



7 [deg/s].

max

s+1

Steering machine

Fig. 2. The block scheme of steering gear

It is usually required for the steering blade to

move from one limiting position to the other in time

shorter than 30 [s]. In this article it was assumed

that the rate of rudder motion is approximately

limited to

max



[deg/s] until

3≤−

δδ

[deg],

when the rudder operates in the linear region of the

characteristic. The maximum rudder angle is

max

35 [deg] .

In the window „Course controller” the ship

course controller was placed.. It was accepted in

system constant speed equal 5 [m/s]. The tanker

model is led along the course defined by the turn

points, which was used to computation the angle of

ride sets among present position of tanker and the

closest point of turn. The defined heading angle is

determinated trigonometrically on the base of

straight line between the present tanker location and

the position at the turning point. On Figure 1 the (x,

y) they are the current co-ordinates of position of

tanker got from GPS however the (x

, y

) they are

the co-ordinates of point of turn. The controller of

trajectory makes possible manoeuvring the ship in

reference to position. The procedure backstepping

used to design of nonlinear functions describe the

structures of applied controllers was exactly

performed in article (Witkowska 2007) and in this

article was developed on trajectory controller.

316

4 SIMULATION RESULTS

The investigations consisted in comparing the results

of the tuned nonlinear controllers having four

parameters with the conventional PD controller. To

compare results PD controler was tuned by the same

genetic algorithm in the same algorithm working

conditions.

Figure 3 presents the results of the simulation

tests performed with two controllers: the

conventional linear PD controller the results of

which are marked with dashed line, and the

nonlinear controller, marked with continuous line.

All controllers were tuned for the ship dynamic

characteristic equations corresponding to the

ballasting state, but in this part of analysis in the first

1000 [s] of the tests, the mathematical model of the

ship made use of the parameters corresponding to

the ballasting state, while during the remaining time

the full load parameters were applied. For the

sytuation shown on a Figure 3 the exact values of the

time quality coefficients, determined from the step

response of two controllers for two load states, are

collected in Table 1-2, where the used symbols are

the following: t

– the rise time, calculated as the

time interval during which the output signal has

changed from 10% to 90% of the set value, yust, Mp

– maximum over-regulation, expressed in percents

and calculated as Mp = 100% (ymax – yust)/ yust, t

– the time of control, calculated as the time interval

from zero to the instant at which the controlled

(output) signal reaches steadily the 1% accuracy

zone of the set value, J

– the quality integral

coefficient described by equation (11),

Table 1. Estimated values of time quality coefficients for

balasting state

Ballasting state

[s]

[%]

[s]

[-]

170.68

0.81

308.15

268.6663

Backstepping

131.71

0.18

261.77

233.9747

Table 2. Estimated values of time quality coefficients for full

load state

Full load state

[s]

[%]

[s]

[-]

148.34

3.29

508.38

156.2852

Backstepping

115.09

18.00

439.77

154.307

Figure 4 presents an example ship trajectory with

the beginning at point (0,0) and the initial ship

course ψ

= 0 [deg]. Figure 4 compare trajectories

for two systems: with PD course controller (dashed

line) and course controller designed by backstepping

method (solid line). In this case the tanker has the

parameter set for ballasting state. The tanker model

is led along the course defined by the following turn

points. The successive turning points are marked in

the table by circuit.

On Figure 5, there are the temporary graphs of

variables occurrent in process steerings on trajectory

from Figure 4 {they were noted two ride set - for

arrangement with PD controller (dashed - dot line);

for arrangement with backstepping controller

(dashed line) ,as well as the answering them real

rides of ship: PD (dotted line), backstepping (solid

line).

Fig. 3. Comparing results of simulation with controllers: PD

(dashed line), nonlinear backstepping controller (solid line)

Fig. 4. Ship position along the set trajectory (circuits) -

comparing results of simulation with tuned controllers: PD

(dashed line), nonlinear backstepping controller (solid line)

Fig. 5. Ship courses and ship rudder angles for trajectory from

Figure 4

317

5 CONCLUSION

Moreover, in order to obtain the reference data for

comparison, a conventional PD controller was

examined, which was also tuned with the aid of

genetic algorithms for the same conditions as in the

case of the nonlinear controllers.

The quality of operation of the examined

controllers was evaluated from the tests checking the

effect of ship parameter changes. Two states of ship

load were analysed, which were the ballasting and

the full load. Step responses were examined to the

set ship course change by 40 [deg]. As shown in

Table 2, the tests have revealed that the obtained

results are comparable for controllers when the ship

was in the ballasting state, slightly better results

were obtained for the backstepping method. When

the ship was in

the full load state better results were

produced by the PD controller than by the nonlinear

controller designed using the backstepping method.

The reason of this regularity lies in the fact that the

parameters of the controllers were only tuned for the

ballasting state and then were used unaltered for the

full load state, which was the source of some error. It

turned out that the backstepping method is more

sensitive to changes of parameters than the PD

controller, which seems to be more robust.

On the ground the simulating investigations it is

possible to affirm with proposed arrangement

automatic the steerings the ship to possibly

efektywnie practical to manoeuvring with oiler in

operations of change of ride and the tailing of

trajectory. The conducted investigations proved, that

the arrangements of automatic steering the

movement of ship from used the backstepping

method are effective and with success very they can

replace manual tanker control.

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