IDENTIFICATION OF DYNAMICALLY POSITIONED

model(cid:1)based dynamic positioning (cid:2)DP(cid:3) systems require that the ship and thruster dynamics are known with some accuracy in order to use linear quadratic optimal control theory(cid:4) However(cid:5) it is di(cid:6)cult to identify the mathematical model of a dynamically positioned (cid:2)DP(cid:3) ship(cid:5) since the ship is not persistently excited under DP(cid:4) In addition(cid:5) the ship parameter(cid:1)estimation problem is nonlinear and multivariable(cid:5) with only position and thruster state measurements avail(cid:1) able for parameter estimation(cid:4) The process and measurement noise must also be modeled in order to avoid parameter drift due to environmental disturbances and sensor failure(cid:4) This article discusses an o(cid:7)(cid:1)line parallel extended Kalman (cid:8)lter (cid:2)EKF(cid:3) algorithm utilizing two measurement series in parallel to estimate the parameters in the DP ship model(cid:4) Full(cid:1)scale experiments with a supply vessel are used to demonstrate the convergence and robustness of the proposed parameter estimator(cid:4)


INTRODUCTION
Modern dynamic positioning (DP) systems are based on model-based feedback c o n trol.The state estimator and control law are designed by applying a low-frequency (LF) mathematical model of the ship motions caused by currents, wind and 2nd-order wave loads, and a high-frequency (HF) model of the 1st-order ship motions caused by 1storder wave disturbances see (Fossen, 1994).
Model-based control systems utilizing stochastic optimal control theory and Kalman ltering techniques were rst employed with the DP problem by (Balchen et al., 1976).Later extensions and modi cations of this work have been reported by Balchen et al. (1980a, b), Grimble et al. (1980a, b), Fung andGrimble (1983) andS lid et al. (1983).
In order to achieve good performance of the control system it is necessary to have a su ciently detailed mathematical model of the ship.ABB Industri AS in Oslo has marketed a new self-tuning model-based DP system based on the results presented in this article.The identi ed model is used as basis for the control system design.This simpli es the tuning of the control law.The control system design is discussed in detail by (S rensen et al., 1995).

SHIP AND THRUSTER MODELS
This section describes the mathematical model of the thrusters and the LF motion of the ship.

Thruster Model
Most DP ships use thrusters and main propellers to maintain their position and heading.The thrust force of a pitch-controlled thruster can be approximated by: where the force coe cient K(n) is assumed to be constant for constant propeller revolution n, P is the \traveled distance per revolution", D is the propeller diameter and: is the pitch ratio.p 0 is pitch ratio o -set de ned such that p = p 0 yields zero thrust, that is: F(n p 0 ) = 0 for the supply vessel in Figure 1 can be written: where u 2 IR r is a control variable de ned as: u = jp 1 ; p 10 j(p 1 ; p 10 ) jp 2 ; p 20 j(p 2 ; p 20 ) ::: jp r ; p r0 j(p r ; p r0 )] T (5) where p i0 (i = 1 :::r) are the pitch ratio o -sets for thruster No. i and r is the maximum number of thrusters.
Thrust Force C o e cient Matrix.The thrust force coe cient matrix K is a diagonal matrix of thrust force coe cients de ned as: K = diagfK 1 (n 1 ) K 2 (n 2 ) ::: K r (n r )g (6) where n i (i = 1 ::::r) is the propeller revolution of propeller number i.The thrust forces K i (n i )u i are distributed to the surge, sway a n d y aw m o d e s by a 3 r thruster con guration matrix T.
Thruster Con guration Matrix.Consider the ship in Figure 1, which is equipped with two m a i n p r opellers, three tunnel thrusters and one azimuth thruster which can be rotated to an arbitrary angle .The control variables are assigned according to: u 1 = port main propeller u 2 = starboard main propeller u 3 = aft tunnel thruster I u 4 = aft tunnel thruster II u 5 = bow tunnel thruster u 6 = bow azimuth thruster.
It is also seen that l 2 = l 1 (symmetrical location of main p r o p e llers).The thrust d emands a r e de ned such that positive thrust force/moment results in positive motion according to the vessel's parallel axis system, de ned such that positive x-direction is forwards, positive y-direction is starboard and positive z-direction is downwards.The origin is located in the center of buoyancy.

LF Ship Dynamics
The LF ship model in surge, sway and yaw can be described by ( F ossen 1994): where = u v r] T denotes the LF velocity v ector, c = u c v c r c ] T is a vector of current v elocities, is a vector of control forces and moments and w = w 1 w 2 w 3 ] T is a vector of zero-mean Gaussian white noise processes describing unmodelled dynamics and disturbances.Notice that r c does not represent a p h ysical current v elocity, b u t can be interpreted as the e ect of currents in yaw.
The current states are useful in the parameter estimator since they represent slowly-varying nonzero bias terms.
The inertia matrix including hydrodynamic added mass terms is assumed to be positive de nite M = M T > 0 for a dynamically positioned ship, whereas D > 0 is a strictly positive matrix representing linear hydrodynamic damping.Nonlinear damping is assumed to be negligible for stationkeeping of ships, whereas the assumption of star- The Coriolis and centrifugal matrix C( ) is included in the model to improve the convergence of the parameter estimator.Moreover, this matrix may be signi cant for a ship operating at some speed, whereas C( ) = 0 for a ship at rest.It should be noted that inclusion of C( ) in the model will not increase the number of parameters to be estimated, since C( ) is only a function of the elements m ij of the inertia matrix see Theorem 2.2. on page 27 in (Fossen, 1994).In fact M = fm ij g yields: C( ) =

State Augmented E x t e n d e d Kalman Filter
Consider the following nonlinear system: where x 2 IR n is the state vector, u 2 IR r is the input vector, 2 IR p is the unknown parameter vector to be estimated and w 1 2 IR n are zeromean Gaussian white noise processes.This model can be expressed in augmented state-space form as: where = x T T ] T is the augmented state vector, w = w T 1 T ] T and: Furthermore, it is assumed that the measurement equation can be written: where z 2 IR m and m is the number of sensors.The discrete-time extended Kalman lter algorithm in Table 1 can then be applied to estimate = x T T ] T in ( 17) by means of the measurement (19).For details on the implementation issues, see (Gelb et al., 1 9 8 8 ) .

O -Line EKF for Parallel Processing
In order to improve t h e convergence and performance of the parameter estimator, the same quan- System model Error covariance update tity can be measured N 2 times for di erent excitation sequences.Moreover, let the input u i 2 IR r correspond to the state vector x i 2 IR n and measurement v ector z i 2 IR m for (i = 1 :::N).
Under the assumption of constant parameters, the parameter vector 2 IR p will be the same for all these subsystems.This can be expressed mathematically as: x with measurements: Hence, this system can be written in augmented state-space form according to: where u = u T 1 : : : u T N ] T , z = z T 1 : : : z T N ] T , = x T 1 : : : x T N T ] T and: It is observed that dim x = N n + p, dim u = N r and dim z = N m .It is then evident that more information about the system is obtained by using multiple measurement sequences.Increased information improves parameter identi ability and reduces the possibility for parameter drift.However, it should be noted that parallel processing implies that the parameter estimation must be performed o -line.

IDENTIFICATION OF A SUPPLY VESSEL
Full-scale experiments with the supply vessel in Figure 1 will be used to demonstrate the convergence of the proposed parameter-estimation algorithm.

Sea T rials
In order to improve the convergence of the parameter estimator it is proposed to use several oline measurement series generated by a n umberof carefully prede ned maneuvers.For instance, it is advantageous to decouple the surge mode from the sway and yaw modes in order to improve t h e convergence of the parameter estimator.This is motivated by the block diagonal structure of M 00 and D 00 .
Decoupled Ship Maneuvers.The following three decoupled ship maneuvers are proposed: (1) uncoupled surge: the ship is only allowed to move in surge (constant heading) by means of the main propellers u 1 and u 2 .The heading is controlled by means of one of the bow thrusters.At l e a s t t wo maneuvers should be performed see Figure 5.
(2) coupled sway and yaw: the ship should performtwo coupled maneuvers in sway and yaw b y means of the three tunnel thrusters, u 3 u 4 and u 5 .Two maneuvers should be performed see Figure 6.
(3) azimuth test: the last test involves running the azimuth thruster u 6 alone.Two measurement series are required see Figure 7.

Implementation Issues
This implies that at least 6 sea trials must be performed for N = 2 .The rst two sea trials are used to identify the parameters K 00 1 = K 00 2 and X 00 u in the decoupled surge equation: (1 ; X 00 _ u ) _ u 00 ; X 00 u u 00 = K 00 1 u 00 1 + K 00 2 u 00 2 (31) _ x 00 = u 00 (32) where X 00 _ u is computed by using strip theory (Faltinsen, 1990).The parameter vector corresponding to this system is denoted as 00 1 = K 00 1 X 00 u ] T .The estimated parameter vector ^ 00 1 in surge is frozen and used as input for the second system identi cation scheme (SI2), that is the coupled sway and yaw identi cation.Similarly, the output from the second parameter-estimation scheme ^ 00 2 is frozen and used as input for the last parameter-estimation scheme (SI3), that is ^ 00 3 .The last scheme is used to estimate only one parameter, that is 00 3 = K 00 6 whereas the second parameter-estimation scheme is used to estimate the coupling terms in sway and yaw see Figure 4.In the last two parameter-estimation schemes the ship is commanded to change heading during the maneuvers, which implies that the nonlinear kinematic equation: _ 00 = J 00 ( 00 ) 00 (33) where 00 = x 00 y 00 00 ] T , should be used together with the dynamic model ( 26).Hence the unknown parameter vector corresponding to sea trial 2 is 00 2 = Y 00 v Y 00 r N 00 v N 00 r K 00 3 K 00 5 ] T .In this example the tunnel thrusters at the stern are of same type (K 00 4 = K 00 3 ).It is convenient to rewrite ( 26

Fig. 2 .
Fig. 1.Picture showing the supply vessel which w as used during the sea trials in the North Sea (L = 7 6 :2 m).
where the non-zero elements m ij = ;m ji are de-ned according to (9) such t h at: -LINE PARAMETER ESTIMATOR The o -line parameter estimator is based on the state augmented extended Kalman lter (EKF).

Fig. 6 .
Fig. 6.Sea Trial 2: Full-scale experiment with a supply vessel (coupled sway a n d y aw).

Table 1
Summary of d i screte-time extended K alman lter (EKF).