## Nonlinear constrained control allocation rotatable actuators

The control allocation problem for vessels equipped with azimuth thrusters is in general a non-convex optimization problem that is hard to solve. The primary constraint is t = T(a)f, (1.42)

where a e Rp denotes the azimuth angles. The azimuth angles must be computed at each sample together with the control inputs u e Rp which are subject to both amplitude and rate saturations. In addition, rotatable thrusters may only operate in feasible sectors a < at < a, max at a limiting turning rate a. Another problem is that the inverse,

may not exist for certain a -values due to singularity. The consequence of such a singularity is that no force is produced in certain directions. This may greatly reduce dynamic performance and maneuverability as the azimuth angles can be changed slowly only. This suggests that the following criterion should be minimized (Johansen et al., 2004),

13/2

subject to T(a)f = t + s fmin < f < fmaxx amin < a < amax

• Ei=1p|./i| 1 represents power consumption where P > 0 (i = 1,...,r) are positive weights.

• srQs penalizes the error s between the commanded and achieved generalized force. This is necessary in order to guarantee that the optimization problem has a feasible solution for any t and a0. The weight Q > 0 is chosen so large that the optimal solution is s » 0 whenever possible.

• fmin < f < fmax is used to limit the use of force (saturation handling).

• amin < a < amax denotes the feasible sectors of the azimuth angles.

• Aamin < a - a0 < Aamax ensures that the azimuth angles do not move to much within one sample taking a0 equal to the angles at the previous sample. This is equivalent to limiting | a -i.e. the turning rate of the thrusters.

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