Øving 8: Utvikling av regulator for dynamisk posisjonering av skip ("DP")

I denne øvingen skal du utvikle en regulator for dynamisk posisjonering. Regulatoren skal baseres på Feedback linearization.

Dere skal arbeide videre med DP-systemet i en større prosjektoppgave senere i semesteret.

Technical information about DP

Dynamic positioning means position control of ships using thrusters and propellers as actuators to keep the ship at a reference position relative to the seafloor or relative to another ship or platform. DP is a crucial technology in sea operations.

Kongsberg Maritime (KM) is one of the world's leading producers of DP systems. The following links available from KM's home page give information about DP (you should now browse the information):

• Dynamic positioning - DP systems, K-Pos
• Dynamic positioning - Basic principles

Figure 1 (below) shows the main components of a DP system. In this project we will apply ordinary feedback PID control + feedforward control in the controller part of Figure 1, which is the SDP frame in the figure. (In real implementation the controller consists of a state-estimator in the form of a Kalman-filter algorithm which is based on a mathematical model of ship. The controller uses state estimates of ship position, ship speed, and water current speed to produce the control signals to the thrusters. As a part of the more comprehensive controller there is actually a PID controller.)

Figure 1. (Source: Kongsberg Maritime)

Mathematical model

(Kongsberg Maritime has approved using the information presented in this document for teaching purposes.)

Definition of coordinates used in the mathematical model

In the document Dynamic positioning - Basic principles the ship-fixed cartesian coordinates surge, sway, heave, and the rotational coordinates roll, pitch, yaw are defined.

Figure 2 (below) shows the relation between the earth-fixed coordinate system and the ship-fixed coordinate system. (However, this information will not be used in this assignment.)

Figure 2

Nonlinear model

Below is a simplified mathematical model of a ship based on force balance (Newton's 2. Law) and torque balance. The model is based on force balances along the surge axis and the sway axis, and torque balance about the yaw axis (rotation). (The nomenclature is according to Kongsberg Maritme.)

M is mass. I is inertia. D is damping coefficient. u, v, and r are ship velocities (speed variables). u_c and v_c are water current velocities. X and Y are forces. N is torque. The first vector at the right side of the equality sign are hydrodynamic forces or torque.

The model does not contain any of the ship positions as variables (only velocities). But the relation between position and velocity is of course that the time-derivative of position is velocities. In particular, the relation between surge position and speed is

dx/dt = u

and similar for sway and yaw positions and velocities.

Positive direction for ship surge velocity and speed, water current speed, wind force, and thruster force is "forward".

Although not needed in the tasks in this assignment, the coordinate transformations from ship-fixed velocities to earth-fixed velocities are presented here, cf. Figure 2 (above):

For simplicity, this project considers only surge movement, assuming there is no movement in the other directions.

Parameter values

Although numerical values are not needed in the present assignment, here are values for a given ship (in the traditional "DP" terminology the unit "ton" represents a force of 1 kN, but in this project tons represents 1000 kg, as normal):

• M_x = 71164 tons
• D_x = -8.4 kN/(m/s)².
• Ship length is L_pp = 233 m. Width is 42 m. Depth is 10 m. (However, these parameters are not needed in this project.)
• u_c typically varies in the range 0 - 3 m/s.
• The longitudal thruster force X_Thrust (which is the actuator force) has a limit of 552 kN forwards and 467 kN backwards.
• Longitudal wind force (in the surge direction) is

  X_Wind = V_w²[c_Wx1 cos(fi) + c_Wx2 cos(3*fi)]

  where V_w is the wind speed relative to the ship. fi is wind angle relative to the ship. If the wind comes from the front (of the ship) fi = 180 degrees. c_Wx1 = 0.1838 and c_Wx2 = -0.0068 are so-called first and second order wind coefficients. One example: With V_w = 20 m/s and fi = 180 deg, we have X_Wind = -70.8 kN (you should verify this example with your own calculations).
  Figure 4 shows the definitions of the different wind speeds.

Figure 4: Wind speeds

It is assumed that position measurement signal contains noise (in every practical system there is measurement noise). The noise is assumed to be uniformly distributed random signal which is added to the (pure) position. The noise has zero mean and standard deviation 0.1 m.

Tasks

1. Design a positional controller for the ship based on Feedback linearization (develop the controller function) using the following information:

   • For simplicity, only surge position shall be concidered (hence, the positional reference is the desired surge position).

   • The control variable is X_Thrust.

   • The approximate time constant (response time) of the control system is Tc.

   • The transformed process model will be a double-integrator. You can use Skogestad's PID tuning formulas for the double-integrator.

   • The "internal" PID controller in the total controller is assumed to be a parallel PID controller. Skogestad's formulas assumes a serial PID controller (PI controller in series with a PD controller). Hence, you need to transform from serial PID settings to parallell PID settings. The transformation formulas are given in the compendium for this course.
      

2. Which measurements (or estimates provided by e.g. a Kalman filter if the measurements are not available) are needed to implement the controller?

[Emnets hjemmeside]

Oppdatert 19.1.09 av Finn Haugen. E-post: finn@techteach.no.