Use of Kalman filtering techniques in dynamicship-positioning systems

M.J. Grimble, B.A., M.Sc, Ph.D., A.F.I.M.A., Sen. Mem. I.E.E.E., C.Eng., M.I.E.E., R.J. Patton, M.Eng.Mem. I.E.E.E., and D.A. Wise, B.Sc, C.Eng., M.I.E.E.

Indexing terms: Control equipment and applications, Optimal control, Simulation, Spectral analysis

Abstract: The position-control systems for dynamically positioned vessels include wave filters to remove thewave motion signals. These ensure that the sytem only responds to low-frequency forces that would cause thevessel to move off-station. Several filters have been proposed and used in this role, and in the followingdiscussion the Kalman filter is considered. The Kalman filter depends upon the model of the vessel, and thedevelopment of such a model is described. Simulation results are given to illustrate the performance of thefilter and the performance of the combined Kalman filter and optimal state-feedfack control system.

1 Introduction

The demand for dynamically positioned vessels for offshoreexploration and production is increasing, and the perform-ance specifications are also becoming tighter.1"4 Theposition-control system must be capable of maintaining areference position and heading under certain specifiedweather conditions. A maximum allowable radial positionerror, which is typically three per cent of water depth, isnormally specified. The control system must also avoidhigh-frequency fluctuations in the thrust demand, aphenomenon known as thruster modulation.

The motions of a vessel5 are often assumed to be thesum of low-frequency motions, due to wind, current andwave drift forces and high-frequency motions, due to thefirst-order wave forces.6 The low-frequency forces (lessthan 0-25 rads"1) would cause the vessel to move off-station and these must therefore be counteracted usingthe vessels' thrusters. The high-frequency cyclic motionscannot be counteracted effectively because the thrustershave a limited thrust capability. The control system mustnot, therefore, respond to the high-frequency motions ofthe vessel. Unnecessary wear and energy consumptionin the thrusters is then avoided.

In a conventional dynamic positioning system usingp.i.d. controllers and notch filters, the wave filters imposea phase lag on the position-error signals. This phase lagrestricts the allowable bandwidth that can be used for thecontroller, while still maintaining the stability marginsrequired for satisfactory controller performance; hence, aninevitable conflict arises between controller bandwidthand filter attenuation. The more effective the wave filterbecomes in reducing thruster oscillations due to the waves,the more restrictions are placed on the controller band-width and hence on the position-holding accuracy. Theseconsiderations have led to the development of a secondgeneration of dynamic positioning systems, designed usingoptimal stochastic control theory and employing Kalmanfilters.

Paper 687D, first received 19th September 1979 and in revised form7th February 1980 (originally presented at Oceanology Inter-national, Brighton, 9th March 1978)

Dr. Grimble and Mr. Patton are with Sheffield City Polytechnic,Pond Street, Sheffield SI 1WB, England and Mr. Wise is with GECElectrical Projects Ltd., Boughton Road, Rugby, Warwickshire,England

IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MAY 1980

The filtering problem is to estimate the low-frequencymotions of the vessel so that control can be applied tominimise the position error. The position measurementscontain noise; however, this is not the source of thisfiltering problem. Even if the position of the vessel could bemeasured exactly, the problem of separating the high- andlow-frequency motions would remain.

The Kalman filter7 involves a model of the systems, andit is therefore particularly appropriate for separating thelow- and high-frequency motions.8 The calculation of theKalman gain matrix is relatively straightforward9 once thesystem has been defined. Most of the following discussionwill therefore be concerned with the system description.

The basic components of the dynamic positioningsystem, that is, the thrusters, position-measuring device andcontrol computer, are first considered. The low- and high-frequency models of the vessel are described, and theseare represented in state-equation form. The Kalman filter isthen defined and the resulting control system is discussed.Finally, typical results are presented from an extensiverange of simulation tests.

2 Dynamic positioning system

The main components in the dynamic positioning systemare the thrusters, the position measurement system and thecontrol computer. The thrusters comprise various com-binations of main engine, tunnel thrusters, steerablethrusters and cycloidal propellers. The first considerationin the design of a dynamic positioning system is to ensurethat the size and dynamic response of the thrust devicesare adequate to meet the operational requirements, underspecified environmental conditions.

Confirmation of adequate thruster size can be obtainedfrom a 'capability study'. The purpose of this study is toestimate the maximum forces and moment to maintainthe position and heading of a vessel in a specified environ-ment of wind, waves and current, and then to ensure thatthe thrust devices can meet this demand. The estimatesinclude allowance for counteracting wind gusts and forthe thrust margin required for adequate control systemresponse to disturbances.

The thrust devises contain both dead-zone and satu-ration. The dead-zone in the thruster control is typically inthe region of 1—2%. Balchen et al.10 suggest figures of

93

0143-7054/80/030093 + 10 $01-50/0

0-3 m and 10 3 rad for the dead-zones in surge, sway andyaw, respectively.

The position of the vessel can be measured by a varietyof methods; a common form of position measurement useseither an acoustic or a taut-wire system or a combinationof both. The optimal method of combining these signalswill not be considered here, although it is related to thefiltering problem. This also applies to the detection oftransducer failure, which is clearly an important problem.

There are several types of acoustic position measuringsystem.4*6 For example, in one configuration, a beacon onthe sea bed emits acoustic pulses, which are detected by aset of four hydrophones mounted beneath the vessel.Signal-processing equipment measures the difference intime of arrival of the pulses at the hydrophones. Fromthis information, knowledge of the system geometry,the water depth and the velocity of sound in water and theposition of the vessel relative to the beacon can becomputed. The system also incorporates a vertical referenceunit so that compensation can be made for the effect ofroll and pitch of the vessel on the measurements. Analternative method of position measurement utilises ataut wire between the vessel and a sinker weight on theseabed.1 The wire is maintained at constant tension bymeans of a constant-tension winch. The angle of the wirerelative to the vertical is used to calculate the position ofthe vessel. The mechanism does not require compensationfor roll and pitch, as in the acoustic system, since it ispendulum-stabilised. The two position-measurement systemshave complementary advantages.1

The position-control system depends on the reliability ofthe control computer and thus, in some cases, dualcomputer systems are used. These have 'bumpless' auto-matic changeover in the event of hardware failure.

The dynamic positioning system is designed to controlthe surge, sway and yaw motions of the vessel i.e. motionsin the horizontal plane. The heave, pitch and roll motionsare not controlled. It is usual to assume that the vesselmotions can be represented by the sum of low- and high-frequency components. The low-frequency motions (in therange 0—0-25 rad s"1) are due to wind, current and wave-drift forces, and the high-frequency motions (in the range0-3—1-6 rad s"1) are due to the first-order wave forces. Thefollowing analysis is based on the above simplifyingassumption.

3 Low-frequency motions of vessel

The low-frequency ship models should(a) enable the wind, current and wave-drift forces to be

estimated(b) describe the vessel dynamics(c) describe the thrust-device dynamics(d) quantify the interaction effects between the thrust

devices, the vessel hull and the current flow.The acquisition of these models involves a comprehensiveseries of tests carried out on models of the vessel, in windtunnels and towing tanks.11 Where such testing is notavailable, reliance must be placed on a combination ofempirical and theoretical results, backed by tank and wind-tunnel test experience, obtained for similar vessels.

The vessel dynamics are represented by nonlinear dif-ferential equations. For control-system design, linearisedequations are required. Unfortunately, the parameters ofthese linearised models are affected by the steady current

flow and the transient force levels applied to the vessel.Selection of the appropriate parameters for the linearisedmodel constitutes part of the art of control-system design.

The nonlinear differential equations relating surge,sway and yaw velocities u, v and r, may be expressed inthe following form:

(M-Xz)u-{m-Yj,)rV = XA + XH(u,v,r)

(M-YJv+iM-XJru = YA + YH(u,v,r)

(/« -Nf.)r = NA+ NH (u, v, r) (1)

where XA, YA represent the applied surge and swaydirection forces due to the thruster action and the environ-ment. Similarly, NA represents the applied turning momenton the vessel. XH, YH and NH represent the hydrodynamicforces and moment due to relative motion between thevessel and the water. The terms X^, Yt and N+ representthe added masses and added inertia which depend on thenature of the body motion and resulting flow pattern.

The set of nonlinear dynamic equations for the WimpeySealab model are expressed in normalised form as:11

(1 + 0-044) w' = XA + 0-092 v'2

-0-l38u'U' + l-84rV

(l+0-84)i>' = YA -2-58v'U'- 1-84v'3fU'

+ 0-068 r ' l r ' l -1 .044 r V

(K'Z2

Z+0-0431) f' = NA -0-764u'v'

+ 0-258v'U'-0-162r'\r'\ (2)

where the prime is used to denote the per unit variable, andU' is used to denote the vector sum of the surge and swayvelocities u', v . The above equations describe the low-frequency motions for surge, sway and yaw referring to thevessel's axes.

For the initial tests on the simulation of the full vesselmodel and Kalman filter, a linearised form of the aboveequations has been considered. The use of linear equationsfor the simulation tests results in a straightforward assess-ment of the performance of the Kalman filter, and alsoprovides some insight in to the properties of the system. Anumber of linearised models could be obtained for differentsea currents and Beaufort numbers. The modulus velocitydamping terms indicate, however, that there is no straight-forward method of developing an all-encompassing linearmodel.

The linearised form of the ship equations have, thefollowing state equation form:

(3)

where W/(Y)G/?3 is the control input to the thrusters,tai(t)ER is a white-noise signal representing the randomforces applied to the vessel, and rti(t)£R3 is the wind-force disturbance. Other disturbance forces, such as wavedrift and current forces, cannot be measured, but can beconsidered to produce an unknown mean value on thesignal w/.

94 IEEPROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980

The low-frequency model is illustrated in Fig 1. Let thefollowing matrices be defined as:

y =7i 72

73 74

Vl(4)

where y projects the thruster forces on to the vessel axesand r\ represents the co-ordinate change from earth to vesselaxes. The above matrices can now be combined to form thefollowing system matrix:

A,=

011

T?4

0

r?3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

- 7 7 2

033

7?1

053

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

035

0

055

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

02 71

0

03 73

0

~bi

0

0

0

0

02 72

0

03 74

0

0

-b2

0

0i

0

0

0

0

0

0

0

-b.

(5)

The elements {a^} and {0,} result from the linearisation ofthe nonlinear ship equations. The remaining matrices ineqn. 3 are defined as

D, =

0

0

0

0

0

0

bi

0

0

'01

0

0

0

0

0

0

0

0

0

0

0

0

0

0

b2

0

0

0

02

0

0

0

0

0

0

0

0

0

0

0

0

0

b*

o"0

0

0

03

0

0

0

E, =

01

0

0

0

0

0

0

0

0

0

02

0

0

0

0

0

0

0

0

0

03

0

0

0

(6)

u , —

wind disturbances

controlinputs thrusters

b2

b2

yaw

Fig. 1 Low-frequency model of ship

IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980 95

The low-frequency position of the vessel is given by theoutput equation

yi = QXl (7)

where

0

0

0

T?l

T?3

0

0

0

0

T?2

T?4

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

(8)

The state variables x2, x4, x6 represent the position of thevessel in surge, sway and yaw, relative to the earth axes.The state variables X\, x3, xs are the surge, sway and yawvelocities relative to the vessel axes. The state variables x7,xs and xg represent the thruster forces.

4 High-frequency dynamics

The assumption which is fundamental to the developmentof models for the high-frequency motion of a vessel is thatthe sea state is known and can be described by a spectral-density function. It is assumed that, in the worst case, thehigh-frequency motions of the vessel are not attenuated bythe vessel dynamics. The behaviour of the ship in responseto the high-frequency first-order wave forces may bemodelled by means of separate colouring filters for eachdegree of freedom of motion. It is further assumed that thewind action has a stationary characteristic of white noisewith unit power spectral density12'13, i.e.

S£(co) = 10 (9)

For example, the high-frequency surge motions can begenerated by feeding an appropriate transfer function withwhite noise. This transfer function is chosen to minimisethe error between the true motion spectrum and theapproximate spectrum being generated. The most suitableapproximation depends upon the assumed sea spectrum,significant wave height and the order of the model. Aninternationally accepted wave-energy standard is thePierson-Moskowitz power spectral-density function givenby the following nonrational expression:14

CO(10)

where co is the angular frequency in rads l, A =4-894,B = 3 • 1094/(7*1/3 ) 2 . The term hV3(m) is defined as thesignificant wave height. By finding the stationary pointfor Sn(to), the resonant frequency of the spectrum canbe found as

co4 = 4B/5 or ton = (45/5)1/4 rads (IDThe sea spectrum described by the Pierson-Moskowitzexpression has been approximated by a rational propertransfer-function representation using the followingidentity:

S (co) — \G(jco)\2 S(co) (12)

The form of the spectral density So (co) indicates that thetransfer function G(s) should consist of at least twocascaded second-order sections with numerator s3. A singlesection of the transfer function may be expressed as15

(13)

and the resulting gain G (co) may be written as

|G(co)| = co H/ = 1 1 L 1

J(14)

where n = 2, bt are gain constants, com- are the zth sectionresonant frequencies and the Q factors Qi are definedby

Qi = coni/2$i (15)

and f,- is the z'th section damping coefficient.In order to determine the parameters of the resulting

fourth-order transfer function it is necessary to performthe following minimisation:

AS = min / j v[Sn(co) - |G(co)|2]2 dco (16)

where Sn(co) is defined in the frequency interval (0, coi)and the spectrum is typically represented by 250 points.

The above minimisation may be effectively performedby the Gauss-Newton algorithm or nonlinear least squares,together with a logarithmic or Bode form of the gainfunction.15 The operation of taking logSi0 results in alinearisation of the gain expression to obviate the difficultieswhich would otherwise arise from attempting the mini-misation with squared terms present. A typical approxi-mation result is shown in Fig. 2.

The high-frequency model of the vessel, for one degreeof freedom, therefore has the form

G(s) =Ks2

s4 + c^ s3 + a2 s2 + a3 s + a*

(17)

The high-frequency vessel motions can be generated usingthe following system in state-equation form:

x=Ahx+Dh<»h (xh(t)<ERn, 08)

g-5

<K 06 08 K) V2 k \ i : 6 18log10 UJ \

-10

Fig. 2 Least-squares fit to Pierson-Moskowitz spectrum

Significance wave height = 9-2 m

S2 + 2s$i + CO] a.

= 3-68= 1-40= 0-38= 0-56

96 IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MAY 1980

where

Ah =

Aiu

0

0

0

A%

0

0

0 Dh=

Df? 0

0 D%

(19)

and the submatrices for surge, sway and yaw each have thefollowing form:

A%u =

Dsh

u =

0

0

0

~Q\

0

0

0

dsu

1

0

0

0

1

0

asu

0

0

1

—a

(20)

The high-frequency component of the vessel position isgiven by the output equation

yh = chxh (21)

where

Ch =

0

and each submatrix has a similar form:

Cshu= [0 0 0 Ksu]

(22)

(23)

As an alternative to the above model, the high-frequencymotions can be generated using harmonic oscillators.10

Since the dominant frequency of the wave motion is timevarying and unknown, the frequency must be estimated inreal time. This gives rise to a nonlinear estimation problemwhich requires the use of an extended Kalman filter. Thereare advantages of using this approach; however, the majordisadvantage is that, when the estimator reaches steadystate and is tracking the dominant wave frequency, themodelled spectrum is not a good representation of the sea-wave energy spectrum.

An extended Kalman-filtering scheme which is beingdeveloped does not suffer from these problems and involvesa more realistic sea-wave model. This is similar to thescheme presented here, except that the wave-modelparameters are estimated recursively.16 A full descriptionand analysis of the extended Kalman filter for thisapplication will be published in the near future.

It has also been shown that an alternative extendedKalman filter is possible for dynamic vessel-positioningsystems.17 This filter will adapt to changing weatherconditions by means of the estimation of only one high-frequency model parameter.

5 Kalman-filtering problem

As there is a large amount of literature available on stateestimation and the Kalman filter7'18"20 it suffices here to

outline the basic features of the method while, at the sametime, giving a full description of the filter model. TheKalman filter provides unbiased low-frequency stateestimates for state-feedback control.

The state equations of the vessel can be written in theform which is normally used for specifying the Kalmanfiltering problem, i.e.

x =

z =

where

A =

D =

Cx + v

Ui 0

LO Ah_

[0 Dh\

[o

F \El

(24)

(25)

and

C = [C, Ch)

(26)

(27)

The state, control and disturbance vectors, together withthe process noise vector, are, respectively,

x = n =

The noise signal wz (process noise) is used to model theeffect of random wind fluctuations and unmodelledphenomena (for example, errors in the low-frequencymodel dynamics). It is assumed that to i is a white-noisesignal with zero mean and a Gaussian distribution. Anestimate of the variance of « j due to wind gusts can bebased on an estimate of the variance of the wind velocityabout a given mean level obtained from the Davenport windgust spectrum.29 The noise signals v which are present inthe position-measuring system are assumed to be additive,of zero mean ard Gaussiar white. Once the covariancematrices of the above noise processes and the covariance ofthe initial state vector have been specified, the Kalman-filtering problem is completely determined. The differentialequation for the Kalman estimator can be expressed as

x = Ax-K(Cx-z)+ Bu +Enl (29)where the Kalman gain matrix can be partitioned into thelow- and high- frequency gain matrices as follows:

K = (30)

The Kalman-filter gain matrix K can be calculated bysolving a matrix Riccati equation or, alternatively, thesteady-state values of A" may be calculated using s-domainmethods.21

The vessel and the resulting filter are as shown in Fig. 3.

6 Control-system design

The way in which the Kalman filter is used in the overallcontrol system is shown in Fig. 4. Recall that the controlsystem must respond to the low-frequency vessel motionsbut not to the high-frequency motions. The state-feedbacksignal is therefore taken from the low-frequency sectionof the Kalman filter. If the filter is working efficiently,

IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y 1980 97

the position-error signal will only contain low-frequencyposition errors and, thus, thruster modulation will beminimised. The controller also has a wind feed-forwardsignal input, via the controller Gw(i), and this loop canbe designed in a similar manner to that used for existingcontrol systems.

It is standard practice in both optimal and classicaldesign procedures to include an integral term in thecontroller, to allow for unknown disturbances.22 In theship model, these disturbances are due to the wave drift andcurrent forces. The controller can be designed usingoptimal control theory; however, the final design must alsosatisfy certain classical performance criteria.23 It is alsonecessary to simulate the complete nonlinear system sincethe design is based upon a linearised model.

The optimal-control strategy for the stochastic systememploying a Kalman filter can be split into two distinctprocedures:

(a) Find the conditional mean estimate of the currentstate vector using the Kalman filter.

(b) Find the optimal feedback on the assumption thatthe conditional mean estimate of the current state vectoris the true system state.The above strategy is often referred to as the certaintyequivalence principle,24 which emphasises the fact that theoptimal feedback will treat the conditional mean stateestimate as the true state. This is also referred to in theliterature as the separation theorem,25 which indicatesthat the stochastic-control problem is solved via thecombination of two separate problems: optimal estimation,together with deterministic optimal control.

The performance index to be minimised by thecontinuous-time controller is given as

and

= Urn E(if [(x(t)-ri)TQ(x(t)-ri)

u(t)TRu{t)} dt (31)

The matrices Q and R are the m-square and /-square positive-definite constant-weighting matrices chosen for a particulardeterministic system performance.

The weighting matrices for the optimal state-feedbackperformance index were chosen as

R =50 x 103

0

0

20 x 103(32)

Kalman filter

Fig. 3 Kalman filter applied to ship-position measurements

98

Q =

300

0

0

0

0

0

0

0

0

104

0

0

0

0

0

0

0

0

300

0

0

0

0

0

0

0

0

104

0

0

0

0

0

0

0

0

103

0

0

0

0

0

0

0

0

103

10

0

0

0

0

0

0

0

r10050

0

0

0

0

0

0

50

100

(33)

7 Discussion of simulation results

The computer plots shown in Figs. 5 to 8 illustrate theresponse of the linear Kalman filter for a simulatedBeaufort 8 sea state (wind speed approximately 19 ms"1).The vessel is uncontrolled and is assumed to be subjected toa steady zero mean, Gaussian wind disturbance having thefollowing sway force and turning moment variances:

Sway = 4 x 10"6 per unit (12 x 103 (kN)2)

Yaw = 9 x 10"8 per unit (250 x 106 (kNm)2)

Fig. 5 demonstrates that the Kalman filter provides un-biased estimates of the heading-angle state variable. Notethat this signal is not corrupted by the sea-wave signal, asrequired. Fig. 6 shows the filter response for yaw rate(state 3 in the low-frequency model). Fig. 7 shows the timeresponse of selected elements of the Kalman gain matrix.Note that the transients die away rapidly so that constant(optimal) steady-state filter gains can be used in practicefor the linear (or linearised) system. Fig. 8 illustrates thehigh-frequency model position estimate (dotted curves)together with the corresponding plant value (continuouscurve). The high-frequency motions are generated bymeans of the fourth-order rational approximation to thePierson-Moskowitz sea-energy spectrum.

Figs. 9 to 11 correspond to the simulation of the 2-input,2-output sway and yaw optimal control system, whichincludes the Kalman filter in the feedback loop. Fig. 8shows the controlled low-frequency position state and itsestimate. The closed-loop system is a state regulator withthe heading-angle reference set at approximately 2-8° andall other reference inputs at zero. These responses illustrate

wind measurement

low-frequencystate modelof vesselhigh-frequency | +

state modelof vessel

controller filter

Fig. 4 Position control system and Kalman filter

IEEPROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980

0

-5

X

S-15

0 20 time, s60 80 100

.2-20

o°--25o>%

-30

-35

Beaufort 8

Fig. 5 Yaw low-frequency position and position estimate (state 4)

-5-

Fig. 6 Yaw low-frequency velocity and velocity estimate (state 3)

20time, s

Fig. 7 Low- and high-frequency Kalman gains for yaw position

60 80 100

• low frequencyhigh frequency. (Filter and dynamics — Beaufort 8)

Fig. 8 Yaw high-frequency position and position estimate (state 13)

IEEPROCEEDINGS, Vol. 127, Pt. D, No. 3, MAY 1980 99

005

O0 04g

a003

0

a-0 02

^001

nnn

- /

• /

x4

Beaufort 8

>

1

0 20 40time, s

Fig. 9 Yaw low-frequency position and position estimate (state 4)

60 80 100

,X3

Fig. 10 Yaw low-frequency velocity and velocity estimate (state 3)

Fig. 11 Sway and yaw positions and position estimates

100 IEEPROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y1980

clearly the result of applying high feedback gains. Thesehigh gains have resulted in good wind disturbance rejection,low thruster modulation levels and low interaction (about2%) between sway and yaw.

Fig. 10 shows the low-frequency velocity state and itsestimate. Fig. 11 shows the combined high-frequency andlow-frequency signals for both sway and yaw positionunder closed-loop control. By comparison of Fig. 9 withFig. 11, one can clearly see that the Kalman filter produceslow-frequency position signals with low-residual high-frequency oscillation (thruster modulation).

The results of digital-simulation tests on the deterministiccontrol-system model (incorporating the low-frequencydynamics only) are shown in Fig. 12. These results alsoshow direct comparison between the nonlinear and linearlow-frequency models for sway and yaw motions only.The surge motion is decoupled from the sway and yawmotions in the linearised model, and this motion can bemodelled separately. The comparison indicates that thelinear quadratic optimal-control design can be applieddirectly to the nonlinear vessel system.

The 12 differential equations comprising the nonlinear

,^nonlinear model

30 35 40 45 50

28

<-, 2e*220

E16

I 1 2i/i 8o£ 4

nonlinear model

linear model

0 5 10 15 20 25 30 35 40 45 50h time.s

1 4

1 2"nonlinear model

5\ 10 /' 15 20 25 30 35 40 45 50

o€ 10

//Minear model

-•, nonlinear model

\

linear model

25 30 35 40 45 50time.s

Fig. 12 Closed-loop response

a State 1b State 2c State 3d State 4

and linear low-frequency models were integrated simul-taneously, for comparison, using the proportional state-feedback control law with gains given by Kp (from eqn. 38,above). The numerical-integration algorithm used for thispurpose is a Runge-Kutta-Merson routine with step-sizecontrol, which is based on error bounds on the solution.

8 Conclusions

The Kalman filter has the advantage that it contains moreinformation about the plant than, say, a frequency-domainnotch filter. The theoretical performance of the Kalmanfilter should therefore be better than those notch filters.However, the Kalman filter normally also has the dis-advantages of greater complexity and of requiring moreextensive computing facilities.

It has been shown by simulation that the constant-gainor Wiener filter can be used for this application. Further-more, it has also been shown that the high-gain optimal-control design is relatively insensitive to changes in thelow-frequency model parameters.

Care must be taken to ensure that divergence of theKalman filter cannot occur from a build-up of modellingerrors.17'18 There are methods of safeguarding againstdivergence, but the most practicable approach is to usea bounded-filter design to bound the variance of theestimation error. For example, the Kalman filter can bedesigned for a worst-case sea spectrum, say Beaufort 9, andcan then be used for a range of sea states including that fora relatively calm sea.

It is interesting that the order of the optimal stochasticcontroller is the same as the equivalent p.i.d. controllerwith the notch filter scheme.

9 Acknowledgments

We are grateful for the support of the UK Science ResearchCouncil. We should also like to thank Dr. F. Boland of theUniversity of Sheffield for his advice during the project.

10 References

1 BALL, A.E., and BLUMBERG, J.M.: 'Development of a dynamicship-positioning system', GEC J. Set & Technol, 1975, 42,pp. 29-36

2 GRAHAM, J.R., JONES, K.M., KNORR, G.D., and DIXON,T.F.: 'Design and construction of the dynamically positionedglomer challenger', Mar. Technol. 1970, pp. 159-179

3 'Havdrill', Holland Shipbuilding, 1973, pp. 740-7454 SJOUKE, J., and LAGERS, G.: 'Development of dynamic

positioning for IHC drill ship'. Presented at the OffshoreTechnology Conference, paper OTC 1498, 1971

5 ENGLISH, J.W., and WISE, D.A.: 'Hydrodynamic aspects ofdynamic positioning', Trans. North East Coast Inst. Eng. Ship-build. 92, pp. 53-72

6 BRINK, A.W., VAN DEN BRUG, J.B., TON, C, WAHAB, R.,and VAN WIJK, W.R.: 'Automatic position and heading controlof a drilling vessel', Inst. TNO Mech. Constr. (Netherlands), 1972

7 KALMAN, R.E.: 'A new approach to linear filtering andprediction problems', Trans. ASME, 1960, pp. 35—45

8 GRIMBLE, M.J.: The application of Kalman filters to dynamicship positioning control'. GEC Engineering Memorandum EM188, Feb. 1976

9 JAZWINSKI, A.H.: 'Stochastic processes and filtering theory'(Academic Press, 1970)

10 BALCHEN, J.G., JENSSEN, N.A., and SAELID, S.: 'Dynamicpositioning using Kalman filtering and optimal control theory',Automation in Offshore Oil Field Operation, 1976, pp. 183 —188

11 WISE, D.A., and ENGLISH, J.W.: Tank and wind tunnel testsfor a drill ship with dynamic position control'. Presented at theOffshore Technology Conference, Dallas, paper OTC 2345,1975

IEEPROCEEDINGS, Vol. 127, Pt. D, No. 3, MA Y 1980 101

12 PHILLIPS, O.M.: 'On the generation of waves by turbulentwind',/. FluidMech., 1957, 2, pp. 414-445

13 MILES, J.W.: 'On the generation of surface waves by turbulentshear flows', ibid., I960, 7, pp. 469-478

14 PIERSON, W.J., and MOSKOWITZ, L.: 'A proposed spectralform for fully developed wind seas based on similarity theory ofS.A. Kitagorodskii', /. Geophys. Res., 1964, 69, pp. 5181-5190

15 PATTON, R.J., and GRIMBLE, M.J.: 'An investigation into theKalman filtering methods in dynamic ship positioning'. SheffieldCity Polytechnic Research Report EEE/14/1978

16 GRIMBLE M.J., PATTON, R.J., and WISE, D.A.: The design ofdynamic positioning control systems using extended Kalmanfiltering techniques'. Presented at the Oceans '79 Conference,Sept. 17-19 1979, San Diego, California

17 PATTON, R.J.: 'An adaptive Kalman filter for dynamic shippositioning control systems'. Sheffield City PolytechnicResearch Report EEE/32, 1979

18 GELB, A.: 'Applied optimal estimation' (MIT Press, 1974).19 RHODES, I.B.: 'A tutorial introduction to estimation and

filtering', IEEE Trans., 1971, AC-16, pp. 688-70620 SORENSON, H.W.: TCalman filtering techniques' (Academic

Press), pp. 219-29221 GRIMBLE, M.J.: 'Solution of the linear-estimation problem in

the s-domain', Proc. IEE, 1978, 125, (6), pp. 541-54922 ANDERSON, B.D.O., and MOORE, J.B.: 'Linear optimal

control' (Prentice-Hall, 1971) p. 23923 GRIMBLE, M. J.: 'Optimal control of linear systems with cross-

product weighting',Proc. IEE, 1979, 126, (1), pp. 95-10324 DREYFUS, S.E.: 'Dynamic programming and the calculus of

variations' (Academic Press, New York, 1965)25 WONHAM, W.M.: 'On the separation theorem of stochastic

control', SIAMJ. Control & Optimiz., 1968, 6, pp. 312-32626 KWAKERNAAK, H., and SIVAN, R.: 'Linear optimal control

systems' (Wiley Interscience, 1972) chaps. 1 and 527 GRIMBLE, M.J.: The design of optimal stochastic regulating

systems including integral action', Proc. IEE, 1979, 26, (9),pp. 841-848

28 SCHWEPPE, F.C.: 'Uncertain dynamic systems' (Prentice Hall,1973), pp. 121 and 172

29 DAVENPORT, A.G.: The spectrum of horizontal gustiness nearthe ground in high winds', Q. J. R. Meteorol. Soc. 1961, 87,pp. 194-211

11 Appendixes

/ /. / Calcula tion o f Kalman gain ma trix

The position measurements are not defined in continuousform, but are sampled at regular intervals. The systemsimulation and the Kalman filter have both been modelledusing their discrete forms. The resulting discrete equationsare as follows:

= f '4>(T)Z)dr•In

(39)

x(k + 1) = <t>(k + l,k)x(k)

z(k) = Cxik) + v(k)

(34)

(35)

with the following first and second moments for theGaussian-noise sequences u (k) and v(k):

*{«(*)} = 0 E{<*(k)<*T(m)} = Q8km (36)

E{v(k)} = 0 E{v(k)vT{m)} = R8km (37)

and where 5fem is the Dirac function. The matrices ^ and Fare related to their continuous-time counterparts by

J n

and

(40)

where TX is the sampling interval.The state estimate is given by calculating the predictedstate

x(k + l\k) = \\k)x(k\k) (41)

(38)

and then calculating the estimated state at the instant(k + 1), using

JC(* + 1|A:+ 1) = x(k + l\k)+K(k+l)(y(k+\)

-CxQc+1 \k)) + *u (k) (42)

The Kalman gain matrix K(k + 1) can be obtained, firstby calculating the predicted error covariance matrix

P(k + l\k) = Q(k+l\k)P(k\k)QT(k+l\k) + rQrT

(43)

for some initial error covariance P {k \k), and then calculating

K(k + l) = P(k+ l\k)CT[CP(k+ 1 \k)CT + R] " !

(44)

Finally, the error covariance matrix is obtained9 using

P(k + \\k+l) = (I-K(k+l)C)

P(k + l\k)(I - K(k + l)C)T

+ K(k+\)RKT(k+\) (45)

The above equations can be used iteratively to obtain thestate estimate at any future sampling time, given the initialstate and error covariance.

7 7.2 Compu ta tional aspec ts

In some applications of the Kalman filter, the errorcovariance matrix is ill-conditioned, which results from thecomputer's finite word length and the resulting round-offerror in the computations. Schweppe28 discusses thisproblem and suggests the use of square root factorisation tocombat such an error build-up. However, in the ship-positioning system, this is not a problem, since the errorcovariance and gain calculation are offline, on a relativelyaccurate large computer. These precomputed gains arestored in a 'look-up' table on the smaller on-board controlcomputer. As the weather conditions change, the constantgain matrices are switched to maintain optimum perform-ance.

Computer error build-up may also cause errors in theestimator-difference equations, and where this is trouble-some, a weighting sequence implementation of the filtercan be used. There are also many different ways toimplement the algorithm which should be chosen tominimise computation time, computer storage andcomputation error. The algorithm given above is notnecessarily the best in this respect, but is recommended byJazwinski,9 and has proved successful in our application.

102 IEE PROCEEDINGS, Vol. 127, Pt. D, No. 3, MAY 1980