log approach for the high-precision track control of ships

7/25/2019 LOG approach for the high-precision track control of ships

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MARINE CONTROL

LOG approach for the high precision track control of

ships

T.

Holzhuter

Indexing terms: Track control, LQG-control, Kalman ilter

Abstract: Design ideas and experiences gained

during the development of an adaptive high

precision track controller for ships are

summarised. The controller is installed

on

a

number of different ships, using a variety of

position measurement systems. The design follows

the spirit of the LQG paradigm combined with a

model following feedforward strategy. The ship is

kept to a manoeuvring trajectory through a

combination of feedforward and LQG feedback

control. The variances and weighting coefficients

for the LQG controller are chosen systematically.

The large number of design parameters is reduced

by appropriate model scaling. In addition, a

decomposition structure of the Kalman filter is

exploited to reveal the important tuning

parameters.

1 Introduction

The development of a commercial track controller has

been refined over several years.

A

track controller for

special applications such as mine hunting and dredging

has been developed, using specialised position measure-

ment systems [l]. This is based on an adaptive course

controller

[2]. A

standard track controller for commer-

cial ships is now installed on a number of different

ships. Operating experiences with this controller have

been very satisfying, and are presented in detail in

[3].

The classical track control approach, which viewed

the track control loop as an additional feedback loop

around the course-keeping loop of the standard autopi-

lot, was felt to be inadequate when dealing with high

precision track control. The state space approach pro-

vides a more direct and lucid solution.

With the use of GPS as a standard position reference

system, a problem appeared, which seems to be typical

for the shipbuilding sector. Traditionally, GPS receiv-

ers and autopilots are manufactured by different firms

and thus only a narrow interface between the two

The filtering process is thus intermingled with the auto-

pilot, as the filtering should be based on an adequate

model of ship motion. This is not really surprising from

a theoretical point of view, although it is not reflected

by the industrial division of labour. The two communi-

ties should work more closely, and one aim of the

present work is to aid this communication process.

The proposed general controller design combines the

spirit of the LQG paradigm

[4]

with a model following

feedforward strategy. For track changes, a model based

manoeuvring trajectory is constructed,

on

which the

ship is retained by the track-keeping controller. This

feature allows smooth track changing manoeuvres and

can also be used to impose reliable predicted manoeu-

vring tracks on the radar screen.

The LQG feedback part of the track controller

involves a considerable number of tuning parameters.

This number may be reduced by the systematic use of

scaling and a simplification of the Kalman filter, which

also allows a more intuitive understanding of the

design choices involved.

c

Y

Fig.

1

Coordinate system for the description

o

ship motion

The angle 7p is the difference between actual heading and trac k course , U is the

forward velocity measured by the log,

v is

the cross velocity to starboard,

x

is

the position along the track, y is the cross-track-error

2 Ship

model

devices exists. Filtering of GPS data is crucial, how-

ever, when high accuracy is to be maintained during

track changing manoeuvres. Simple low pass filtering

of the individual position components is disastrous.

For a commercial

track

controller

the

amount of instal-

lation work has to be quite low,

so

the model must

have a small number of parameters, identifiable from

on-board measurements. The linearisation of the

EE,

1997

hydrodynamic equations of motion of a ship leads to a

IEE

Proceedings online no. 19971032

second-order model in the state variables: rate of turn

r

Pape r first received 3rd A pril an d in revised form 3rd October

1996

and sway velocity v See Fig.

1

and [ 5 ] . This model is

Th e a u t h o r i s wth Ha mb u rg Po l y t e c h n i c , Sa n d k a mp 13a, D-24259

often ill-conditioned with respect to identification when

Westensee , German y using on-board measurements [6]. The well-known

IEE

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Theory Appl., Vol. 144, No. 2, March 1997

121


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simplified model after Nomoto

[7]

seems preferable.

For the sway velocity, v = ar

is

assumed. The parame-

ters

K,

T of the simple linear model in eqn. 1 can be

roughly derived from general ship data.

2)

L

U

T=To -

K=K o -

U

L

The remaining states can be deduced from kinematic

relations according to Fig.

1.

2 =ucos$++sin$+d, 3 )

l =

usin$ + v

cos$ +

d,

(4)

The forward velocity of the ship is measured by the

speed log, and can thus be viewed as a known parame-

ter. The linearisation of this model is straightforward

and the fully linearised model is given in eqn. 5 .

f + \

0

0 0

0

0 0 0 0

+

T 0 1- 0 0 0 0 0 0 r

c

0

0

0 0

U?

0 0 0 0

5

= 0 0

0 -2Dh-wh

0 0 0 0

f

Y U

a 0 0 1 0 0 y

b 0 0 0 0

0

0 0 0 0

b

d?J 0 0

0 0 0

0 0 0 0 d ,

X 0

0

0 0 0

0 0 0 1

X

dz

0

0 0 0

0

0 0 0 0 d ,

0 0

0

0 0 0 0

0 0 0

0 0 0

0

o u

0 0 0

0 0 0 1

(5)

The model includes a coloured noise to model peri-

odic disturbances of the heading signal, due to coupled

roll and pitch motion in rough sea. The dominating

frequency of this disturbance model with states

(


3/7

From the typical accuracies of position and heading

measurements, it is clear that the position measurement

does not contribute to an increase in the estimation

accuracy of the heading and turnrate. Estimation

of

the

states xk

k = 1,

..., 5 of the heading submodel should

therefore be independent of the position measurement

y,. This introduces further structural zeros into the

Kalman gain matrix, namely L , = 0. The result can

easily be checked numerically by letting R2 become

large compared to

R1,

while the ratios Q41R2and QslR2

remain constant. Taking L, = 0 leads to a suboptimal

filter, which in principle has

a

higher covariance

P

given by the Lyapunov equation

P = A(I LC)P(I LC)TAT+ ALRLTAT

+

BQBT

(9)

where A ,

B ,

C

are the state space matrices of the discre-

tised system. The increase of P over the optimal Po

provides a measure for the performance deterioration

due to the suboptimality. The actual increase in estima-

tion variances

of

less than 1% is negligible, as may be

seen from the Appendix by comparison of cases 1

and 2.

The decomposition of the heading and the track sub-

models can be driven even further. Setting

Lyw= 0

removes the direct influence of the heading innovation

on the position estimate. The gyro information still

finds its way to the position estimate in the next time

update, and so the deterioration of performance should

be small. The Appendix shows that the increase in esti-

mation variances due to this simplification is less than

10%. The increase is even less if the sampling time for

the gyro signal is shorter than that for the measure-

ment.

The above simplification treats the heading and turn-

rate estimates as a deterministic input to the track

model. The cross track channel, with the exception of

the deterministic input

Q

and i

=

/a,

orresponds to

the submodel of the forward channel, which is dis-

cussed in Section

4.

With these simplifications, the

three submodels are totally decoupled and each meas-

urement only affects the states of its respective sub-

model. The disturbances for the two position

measurements are assumed to be equal. For symmetry

reasons, the filter response and the gains for the two

track channels should also be equal. The suboptimal

Kalman gain matrix is

L =

The full submodel decomposition also proves advanta-

geous under limitations on software or cpu-time, as the

Kalman gains for the two track channels may be calcu-

lated analytically, as in Section 4.

The proposed approach reduces the number of

parameters to a set of five tuning knobs, which roughly

correspond to the following controller characteristics.

Typical values are also given, and justified later:

IEE Proc-Control Theory Appl. , Vol. 144,

No.

2, March

1997

Q 1 * l R ,

= lo4 Low pass for course filtering

Q2*lRW

lo3 Estimation time for the rudder bias

Q3/R, =

1

Intensity of wave filtering

QplRp

-

Low pass for position filtering

QdlRp

- Estimation time for the drift

Note that due to scaling, Q = Q I F and

Q;

=

Q2P. For the track-related variances, no scaling is

used because the dynamic changes with environmental

conditions and position fixing instrumentation, rather

than with the ship parameters. The choice of variances

for the two track submodels is discussed in Section 4.

The other variances determine the Kalman filter for the

heading submodel. As a consequence of submodel

decomposition, only the ratios QklR1k = 1, ..., 3 are

relevant. Selected frequency responses of the Kalman

filter for the heading submodel are shown in Fig. 2a.

This Kalman filter has two inputs, the measured head-

ing + and the rudder 6. The transfer function @/+, is

simply a low pass and the transfer function ?I+, is a

differentiator with low pass filtering.

C10

ISI

~o~

1ci2

1CT21/=oo

lo2

d2

loo lo2

f

requencywT, rad

a

b

t i me t l T

frequency w, radls

d

Fig.

2

ulmun filter f o r heading submodel

(a) Transfer functions @ &,',and I^*/I/I~

(b) Variation

of

transfer function

IQ,

with pi

E

{

lo6, O4 IO }

( c )

Speed of bias estimation

for Q, E {lo4,10 , O2}

(d) Intensity of wave filtering

Q, E (0,

0.2, I}, T

=

20

and

K

=

0.2

The variance ratio Ql/Rvcontrols the low pass filter-

ing of the heading signal. Fig. 2b shows the filtering for

different values of this parameter. Practical considera-

tions indicate that the time constant should roughly be

a factor of ten smaller than T, leading to a typical

choice of

Ql*lRv*

=

lo4.

The variance Q2 characterises the sensitivity to

changes of the rudder bias b; in traditional PID-con-

trol, this is chosen via the integral part. The frequency

response for this state is rather involved, as this esti-

mate depends

on

both inputs tp and 6. A choice of Q2

is made simply by looking at the time response of the

estimated rudder bias to a step in the true rudder bias.

A time constant of 3

5

T seems reasonable for this

estimation process

so

from Fig. 2c, a typical choice is

If Q 3 > 0, additional band pass filtering is intro-

duced. The necessary intensity of wave filtering changes

strongly with weather conditions. It is therefore advisa-

ble to estimate this variance online

[SI.

Fig.

2d

shows

the wave filtering effect for different values of Q3. The

e2 103.

123


4/7

transfer function is given for an unscaled situation to

give a realistic impression of the involved frequency

range.

The full decomposition of the heading and track

model using the Kalman gains in eqn. 10 only has

an

obvious advantage if the online solution of the Riccati

equation poses a problem with respect to processor

time or software limitations. However, the decoupling

with L, (case 2 in the Appendix) is strongly advisable,

as

there is a serious performance deterioration problem

when the full Kalman gain matrix is used. In the latter

case, the measurement of the cross position component

regularly leads to

a

correction of the heading, rate of

turn and rudder bias. This influence

is

negligible if the

relative sizes of the noise variances R nd R2 are prop-

erly chosen and well behaved white noise disturbances

occur. If the position signal is corrupted with jumps,

then the corresponding large innovations lead to unac-

ceptable steps in the heading model states. This severe

performance problem is effectively solved by the gain

matrix structure with L, = 0.

4 Position filter

The estimation process

of

the three submodels for

heading, forward and cross position can be decom-

posed without significant error. The stochastic models

for the two track channels thus become identical. The

model for the forward channel y,

d,

is

Ym= ( 1 0 )

(y

(11)

The deterministic input, which is different for the two

channels, is omitted because it is not relevant to the

calculation of the gains. The Riccati equation for the

continuous system in eqn.

11

can be solved analytically,

and with Qp = Q4 = Q6 and Qd = Q5 = Q7, leads to

Kalman gains

Idy = I d s = l / Q d / R p

13)

It should be noted that the gains depend only on the

ratios QplRpand QdlR,. The transfer function for the

estimation of the drift IS

(14)

d , S

y m

-

+

( l y / l d y ) ~

+

(1 / l d y )S2

For d[Qd/Rp] 2 Q IRp the damping is Dp 1. In this

case, the sum of t t e two time constants serves

as

an

equivalent time constant

A

simple choice is

Qp = 0,

which assumes that all

deviations from the true position must be attributed to

the drift. However, this is not realistic because in the

cross track motion, for example, the errors in the

parameter

a

would then be interpreted

as a

drift. Some

choice Qp

>

0 has to be made.

If the ship

is

exposed to

a

step in the drift velocity,

the measured position ym shows a ramp increase. The

I24

drift is estimated from eqn. 14 by differentiation and

low pass filtering. See Fig.

3a.

The response of the

position estimate may be understood more clearly by

considering the transfer function

The response for a ramp increase is shown in Fig. 36.

The maximum estimation error decreases with

Zd y

Increasing

Qp

while leaving

Tdom

rom eqn. 15 constant,

leads to

a

decrease in

ldy

=

d[QdIRp].

hus the maxi-

mum estimation error is decreased. See Table 1 for

a

systematic variation. Considering typical rates of

change for the drift,

a

sensible choice is Tdo,= 100s.

F

.8

b

t i m e t , s

C

Fig. 3 Response of

position

Kalman j l t e r to

a

step in the drift

d,

f o r

dif

ferent

values

of Q

(a)

Estimated driftPJy,

(b) Error 2 of the estimated cross position

c) Response

of

position estimate

y^

to

a

step in the measured position

Table

1: Combinations

of

Qp

and

Q, for t h e e a s e s in

Fig.

3

Case Q Qd D a m p i n g

1 0

4 . 10-8 D, =

id2

2 8 .

O

1 .6 .

10-7

Dp=

1

3

1 10-2 1.2 . IO

D,

> 1

4

1 IO-' 1.1 .

10-5

D,>

1

Note tha t the var iance 0 i s a d ju s t e d t o g i ve r o u g h l y e q u a l

t im e r e sp o n ses

o f

the d r i f t es t im at ion see F ig .

3a)

If

Q,

is increased, the estimation of the position

becomes faster than the estimation of the drift. See Fig.

3b. This is desirable because the control loop also has a

drift compensation capability, which relies on the rec-

ognition of position errors. Fig.

3c

shows the estima-

tion error, which in turn results in

a

reduced track

error. However, the increase of

Qp

also increases the

sensitivity to the measurement noise of the position, so

that

a

medium value must be chosen,

as

in case 3.

The Kalman gains are calculated for the linearised

model. It is not advisable to use the linearised equa-

tions in the track extrapolation step (time update) of

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the online Kalman filter. For low speed and high cur-

rent, a considerable angle may be present between the

track course and the heading of the ship, leading to

noticeable estimation errors. A nonlinear extrapolation

step is therefore strongly recommended:

X:+l

=

f(kt)+BS

(17)

Xt+ = + L q -

CX,)

(18)

Here, 2* denotes the extrapolated state estimate,

while 2 is the state estimate after the measurement

update. The geometric nonlinearity of the system is

preserved in eqn. 17 of the Kalman filter by using a

discretised version of eqns. 3 and

4

in the nonlinear

function f x ) , instead of the linearised equations

according to eqn. 5 . This is just a simple case of the

extended Kalman filter technique.

5

LQG

contro l ler

The state space approach allows a straightforward

switching of control modes without distortions. The

Kalman filter can estimate the full state irrespective of

whether all the states are used for control. It should be

noted that the controllable part of the linearised model

is only of order

3

(+(d i j ) + i ) S 19)

The LQG controller minimises the stochastic per-

formance criterion, J

=

E xTWx

+ 8)

with x

= r ,

qj

y)*. The weighting of the rudder is chosen as unity,

without loss of generality since the optimisation only

depends on relative weightings. With diagonal weight-

ing matrices in the absence of specific information, the

criterion reduces to

J

=

E

(X,y2

+

+

X,T

+

S 2 )

(20)

The simplest choice would be A, = Ar =

0

for track

control, and Ay = kr =

0

for course control. This would

leave only one tuning parameter, which controls the

bandwidth of

the system. A slight modification is pro-

posed later.

If the weighting coefficients k are taken as fixed,

there is an undesirable variation of the closed loop

dynamics with the type of ship and its speed. This was

reported to produce difficulties in other applications of

LQG

control to ship steering [lo]. As discussed in Sec-

tion 2, the dependence on the individual ship may be

considerably reduced by scaling the model. The scaled

version of this model is given in eqn. 7. Rewriting the

performance criterion of eqn. 20 in scaled variables and

dividing the criterion by Kg2,eads to the scaled

weightings

A = X+ KT)2 (22)

A

= X,K 23)

Note that the scaling factors for Ay and kq are only

mildly speed-dependent, since according to eqn.

2,

U T

TJ

ToLand KT -

KoTo.

After scaling, the natural choice

is Ay*

=

1 for track control and AV

=

1 for course con-

trol.

A

mild variation in the interval

ky*

E

[0.1,

lo],

depending on the seastate, seems advisable. From the

intention of track keeping, i.e. minimising the variance

IEE Proc -Control T he ory A pp l , Vol 144, No 2, March 1997

of track deviations, the obvious choice is kq*

=

AT*=

0 .

However, the damping is improved by an additional

weighting of the heading; from Fig. 4a sensible choice

is kv* = loay*.

I

1

I I

0 100 200 300

time t s

Fig.

4 Response to unit step in the reference sign alfor dijj erent eightings

_ _ _ A;

= 0.

______ A y = 10h

*

In both cases A , =

1.

The track control

loop uses state feedback. I) track &ror, (ii) heading

In addition to the feedback gains, a disturbance com-

pensation using the estimated rudder bias and cross

drift must be applied. The compensating rudder for a

bias b is 6 = b /K. The stationary value of the rudder to

compensate a drift

dy

is zero. However, there has to be

an offset AQ in the set value of the heading

(24)

A+ = - arcsin-

U

For small relative drift values this can be linearised,

leading to a gain k dU in the drift, x7 = dy . The full

state feedback gain K is then

K =

( k q ,

k ,

O

0

k , O

0 )

( 2 5 )

To illustrate the control performance, Fig.

5

demon-

strates the transient behaviour for a step in the drift.

The maximum track deviation under this severe distur-

bance is less than 55m.

-8

i v)

0

100

200 300 LOO 500

ti me

t, s

Fig. 5

Controlperformance or a unit step in the drift

(i) Drift estimate

(0.1

i s ) ,

(ii) true position (IOm), (iii) position estimate (IOm),

(iv) heading

(0.1"),

(v) rudder (O.Io). Ship parameters:

K

= 0.2P1,T = 20s,

U

10m/s, Q = -50mirad

The sensitivity and complementary sensitivity func-

tions for the track control loop are shown in Fig. 6 for

different weighting coefficients Ay, which determine the

bandwidth of the controller. Although the plant has

two outputs, these transfer functions are scalar as there

is only one plant input. The norm 171wincreases with

bandwidth,

as

expected, indicating that the robustness

with respect to errors in the model parameters, e.g. the

rudder gain K, decreases. The norm of the minimal

destabilising multiplicative perturbation

lAlm

=

1/1

qW

decreases from

lAlm

= 0.72 for Ay =

0.1,

to

/AIm

= 0.62

for

Ay

= 1 0 .

125


6/7

Fi .6 ensitivity function s S s) and complementary sensitivity functions

T(?j

of the LQG truck conrroL erfor d@ren t values

of

the weighting E

{ O . I , I , O}

LQG controllers do not provide guaranteed stability

margins, although the actual margins of the proposed

LQG controller are reasonable. The phase margin for

Ay = 1 is 6 = 56, and the gain margin is = 10. The

stability margins decrease with the degree of wave fil-

tering

as

this is a strongly model-dependent operation.

With Q3 = 1, the phase margin goes down to 6

=

40

and the gain margin to

E

=

2.9.

Some investigations

into the robustness properties with respect to errors in

the ship parameters have been presented in [111.

100

80

2 60

$ 40

.-

7J

2 o v

201

E 10

2

k o

L

V

-10

-201 , ,

,

, , , , , ,

b

L I

I

0

1000

2000 3000

4000

5

t ime , s

C

Fig. 7 Heading, track error and rudder angle during a typical voyage

a

Heading,

b

Track error, c Rudder angle

6 Operational experience

A

track controller designed according to the ideas

described in the previous Sections has been developed

and improved in recent years. The first application was

126

in the high-precision track control of mine hunters [l].

The intention was to apply the controller to commer-

cial ships as soon as a demand for track control arose

in this field. The controller was subsequently adapted

for use as a piece of standard equipment

on

commer-

cial ships. The controller is currently installed on a

number of different vessels, such as large ferries, differ-

ent types of cargo ships and special vessels with Voith-

Schneider propulsion. Several different position meas-

urement systems have been used, including differential

GPS,

which was used for the present results.

A general impression of the track-keeping quality

of

the system may be gained from Fig. 7.These data were

recorded on a voyage of the Norwegian ferry ship

Kong Harald. The track error during track-keeping

phases is below 5m, and the typical standard deviation

is 3m. During track changes, the track error is meas-

ured against the internally precomputed manoeuvring

trajectory. The error during these phases is typically

below 1Om. According to the navigation officers,

90%

of the voyages on the Kong Harald are performed in

track-keeping mode. The remaining 10% use course

control and hand steering mode. A detailed presenta-

tion

of

performance results can be found in

[3].

7 Conclusions

A systematic way to choose the considerable number of

design parameters for an LQG track controller has

been discussed here. Commercial track controllers

designed along these lines have been successfully

installed on a number of ships. Detailed performance

results have been presented in a recent paper [3]. Typi-

cally, the track errors achieved are well below 10m.

The approach presented uses a disturbance model

suitable for Kalman filter design. However, due to the

simplified model assumptions, the actual disturbance

variances are difficult to obtain. As an alternative, the

variances are used here as advanced tuning knobs

which reflect the specific disturbance structure of the

plant. The actual variances can thus be selected from a

general knowledge of the desirable observer response.

The described position filtering technique was origi-

nally designed for special navigation systems such as

Syledis or Doppler Sonar, and was later successfully

used with

GPS

receivers.

As

discussed in Section 1, the

position filtering and control process requires further

integration and this problem is addressed by the

present study.

If track errors are to be further decreased, a future

research problem is the robust estimation

of

sway

velocity during manoeuvres. In particular, an improved

capability to distinguish a stationary drift motion from

the transient, manoeuvre-induced sway motion would

be desirable.

References

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lt h IFAC World Congress, Tallin, Estonia, 1990,

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2

HOLZHUTER, T., and STRAUCH, H.: An adaptive autopilot

for

ships: design and operational experience. Proceedings

of

10th

IFAC World Congress, Munich, Germany, 1987,

pp.

226-231

HOLZHUTER, T., and SCHULTZE,

R.:

Operating experience

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M.J.,

and KATEBI, M.R.: LQG design of ship

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J.:

Adaptive steering of ships model refer-

ence approach, Automatica, 1984,

20

(l ), pp. 3-14

11 HOLZHUTER, T.: On the robustness of course keeping autopi-

lots. Workshop on Artifical intelligence and a dvanced technology

in marine automation

CAMS 92, Genova, 1992, pp. 235-345

9 Appendix: Optimal and suboptimal Kalman

gains and their respective increases of variance

Ship parameters are K

=

0.2s-,

T

= 20s, U

=

10m/s, a

=

-50drad.

The relative increase is given by PkklPkko)

1.

Case 1:

Optimal Kalman filter with variance

Po

2.85

l o p 1

-2.35. lop5

9.58.

lo-

-1.36. l o p 5

-7.70.10 5.50. lo-

2.04. lo p2 4.85. lop4

1.22.10-6

2.65.

5.56.10-5

5.85.

/ O \

Case 2 : Suboptimal filter with

L, = 0

2.85

*

lo-

9.58. lo-

L =

( .33.lOW2

7.70. 10 5.50.

lo-

2.04. lo- 4.85. lop4

1.22 10-6

2.65.10-7

.67.10-3.56.10-5

1.60.10-3

1.31.10-4

2.57.10-4

diag(P)

=

P k k

= 6.32.

5.85.

lo-

pkkp l k ) =

Case 3 : Suboptimal filter with L, =

Lyw

= 0

[ 4.85.10-4.50 10-)

2.85

*

1 O - I 0

9.58

L = 3.33.10-

1.22 10-6

6.32.

l o p 7

2.65.10-7

.94.10-5

2.67.10-3

1.60.10-3

pkkp l k )

=

8.00.10-5

diag(P) = Pk,+=

6.09.

lo-

4.06 * lop2

.81 *

lo-

Case 4:

As

case 3, but with

Lyy

aken from the continu-

ous system approximation Ld = L, At according to Sec-

tion

4

L =

( )

2.8 5. 10-1 0

9.58.

lop2

5.48

*

5.00.10-4

1.22.10-6

diag(P) =

P k k

=

6.09.

5.95.10-5

2.67.10-3

1.60.10-3

p k k / p l k )

=

4.09. lop2

7.04. lo-

IEE Proc.-Control Theory Appl. , Vol. 144 No. 2, March 1997

127

log approach for the high-precision track control of ships

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