Abstract: FPSO (Floating Production Storage and Offloading) is required to maintain the specified position during oil producing and offloading under the external forces such as ocean current, wind or wave. In this position keeping process, it is important to measure the exact position of FPSO. Recently GPS (Global Positioning System) is mainly used to measure ship's position. However the position measurements by the GPS contain noise, which generates the position error of ships. For example, in case of DGPS (Differential GPS), the position error is generated about 2〜10m. Because the position error has large influence on DPS (Dynamic Positioning System) of FPSO, it is necessary to minimize the position error by removing the noise of GPS.

 In this paper, Inverse Linear Quadratic (ILQ) optimal servo theory using Kalman filter is applied to design control system of DPS. Generally, Kalman filter is to estimate the low-frequency motions of the vessel so that control can be applied to minimize the position error. But in this paper, because we already consider the motion of FPSO as low-frequency motion, Kalman filter is used to remove the measurement noise. In addition, we investigated the influence of the rate of the thruster power change per second on the control performance.

 The numerical simulations show the performance of the combined Kalman filter and ILQ optimal servo system.

　

 FPSO drew up oil using a flexible riser from the oil field deep beneath the seabed. She also has the capability to store vast quantities of oil in a similar way to an oil tanker. The oil transported from FPSO to a shuttle tanker for transportation to the mainland. In this processes, FPSO is required to maintain her position under external forces such as current, wind and wave.

　

 In this paper, ILQ optimal servo theory is applied to design the control system of DPS under external disturbances. The control system of a vessel, which performs position keeping control gives a control command on a basis of the position measured by various methods. Therefore, it is important to measure the exact position of a vessel. The position of vessel can be measured by a variety of methods. As latest method, GPS using satellites has mainly been used to measure a vessel's position. Although accuracy of the GPS is better than other methods, it is surely accompanied by the noise, which generates the position error of a vessel. For example, the position error of general GPS will be about 100m. Also in case of DGPS, the position error will be at least about 2〜10m. The position error is quite large compared with the horizontal displacement of FPSO, which can be controlled. Consequently, the position error has a large influence in the DPS of FPSO. Therefore, it is necessary to minimize the influence of position error by filtering the noise of GPS. Filtering means that a noise ingredient is separated from the time history of a signal by which a noise was added to a signal ingredient, and only a meaningful signal ingredient is taken out. Of the various methods of filtering, in this paper, the Kalman filter to minimize the influence of measurement error was used.

　

 Numerical calculations were carried out changing the magnitude of standard deviation (SD) of measurement error. Also we carried out calculation changing the rate of the thruster power change per second (δTh) of maximum thruster power.

　

 We consider a FPSO with no forward velocity, on the surface of an ideal incompressible fluid and in the presence of external disturbances such as current, wind and wave. As shown in Fig.1, let ο - x0y0 be earth-fixed coordinate and G - xy be body-fixed coordinate with origin at the center of gravity of EPSO. α,υ and χ represent the incident angles of current, wind and wave respectively. External forces consist of the current force (XC, YC NC), the wind force ( XW ,YW ,NW ), the wave drifting force (XD,YD,ND) and the thruster force (XTh, YTh,NTh). In this case, the manoeuvnng equations are given by,

　

　

where,

　

m : ship mass,

mx ,my : added mass components in x - y plane,

ψ : heading angle of ship,

Izz : moment of inertia,

izz : added moment of inertia,

u,v : velocity components in x-y plane,

r : angular velocity.

　

　

 The current forces are calculated by the results of free-running model tests concerning ship manoeuvring motions[1]. The wind forces are determined by using approximate formulae derived from the experimental data of various new ships, for example VLCC, PCC, LNG, and so on[2]. And the wave drifting forces are calculated by momentum relation in terms of Kochin function of far-field velocity potential of the ship[3].

　

3.1 ILQ design method [4]

　

 In order to design the control system, the differential equation (1) are linearized around the target value by using Taylor series expansions. By the linearization, the linearized form of the equations of ship motion have the following state equation form which are assumed to be completely observable and controllable.

　

　

where,

　

τM : thrust force of Main thruster,

τS : thrust force of Stem thruster,

τB : thrust force of Bow thruster.

　

 We considered design problem of optimal servo system that the output y(t) of the state equation (2) is approached the step response r(t). This system has no zeros at the origin; that is, the following matrix is regarded as a real nonsingular matrix.

　

　

 From the above assumption, we explained the ILQ design method in more detail.

　

[1] Compute di and D as defined by following equation and confirm whether D is a real nonsingular matrix or not.

　

　

[2] Compute the zeros of control system and confirm whether they are stable or not.

　

[3] The multinomial φi(s) which prescribed the Gyr^∞(s) can be expressed in the form.