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### Thrust Allocation

Ole Jakob Sørdalen, PhD

Counsellor Science & Technology

The Royal Norwegian Embassy, Singapore

Controller architecture

- Sensor signal processing
- Signal QA
- Filtering and weighting

- Vessel Model
- Separate LF/WF model
- Kalman filter estimator
- Mooring model

- Optimal Control
- Positioning and damping
- Reduce fuel, tear and wear
- Mooring line break compensation

- Feedforward control
- Wind load compensation
- Reference model tracking

- Optimal thrust allocation
- Adaptive control

Problem statement

Given desired forces and moment from the controller, tc =[txc, tyc, tyc]T.

Determine thrusts T=[T1, T2,..., Tn]T and azimuth angles a=[a1, a2,..., an]T so that

- ||A(a) T - tc|| is minimal to minimize the error
- ||T|| is minimal to minimize fuel consumption
- ai(t) is slowly varying to reduce wear and tear
Assumption here: Thrusters are bi-directional

Challenges

- Singularities: the singular values of A(a) can be small;
A(a) T = t , simple pseudo inversion can give high gains and high thrust

- An azimuth thruster cannot be considered as two independent perpendicular thrusters since the rotation velocity is limited
- If the thruster is not symmetric, how should the azimuth respond to 180o changes of desired thrust directions?
- Forbidden zones

3

t

t

T

y

x

1

T

t

2

y

SingularitiesThere is an azimuth angle where det A(ais) = 0

A(ais) cannot be inverted

Example of a singular configuration:

Singular Value Decomposition

Any m x n matrix A can be factored into

A = U S VT

Where U snd V are orthogonal matrices.

S is given by

About SVD ...

- Coloumns of U: orthonormal eigen vectors of AAT
- Coloumns of V: orthonormal eigen vectors of ATA
- si = sqrt (eig(ATA) i)
- Pseudo inverse of A:
A+ = V S+ UT

- The least square solution to Ax = y is
x = A+y

i.e. either min ||Ax – y||2 or min ||x|| 2 Can use weighted LS.

Example: plot of smallest singular value

Bow azimuth fixed 90o. Aft azimuts rotate

Area where s < 0.05

Area where s < 0.015

Fixed angle between aft azimuths

s < 0.05

How to determine angles a?

- Consider azimuth thrusters as two perpendicar fixed thrusters
- New (expanded) relation: AeTe = t
- desired ”expanded” thrust vector Ted:
Ted = A+etc

How to determine thrust T?

- Note: T = A+(af)tc large T close to singular configurations!
- Modified pseudo inverse:
Ad+ = V Sd+ UT T = V Sd+ UTtc

Geomtrical interpretation

- Commanded thrust in directions representing small singular values are neglected
- This is GOOD
- Azimuth angles are always oriented towards the mean environment forces & torques
- Other commanded forces typically due to noise
efficient ”geometrical” filtering of this noise

Features

- Automatic azimuth control
- Automatic avoidance of forbidden sectors: not shown here
- Optimal direction control
- Smooth turning

- Optimal singularity handling
- Avoidance of unnecessary use of thrust
- Reduced wear and tear of propulsion devices

- Optimal priority handling
- Among thruster devices
- Among surge, sway, yaw

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