## Linear quadratic unconstrained control allocation

The simplest allocation problem is the one where all control forces are produced by thrusters in fixed directions alone or in combination with rudders and control surfaces such that a = constant, T = T(a) = constant.

Assume that the allocation problem is unconstrained-i.e., there are no bounds on the vector elements f and u and their time derivatives. Saturating control and constrained control allocation are discussed in Sections 4-5.

For marine craft where the configuration matrix T is square or non-square (r > n), that is there are equal or more control inputs than controllable DOF, it is possible to find an optimal distribution of control forces f, for each DOF by using an explicit method. Consider the unconstrained least-squares (LS) optimization problem (Fossen & Sagatun, 1991), mm {/ = f T Wf}

Here W is a positive definite matrix, usually diagonal, weighting the control forces. For marine craft which have both control surfaces and propellers, the elements in W should be selected such that using the control surfaces is much more inexpensive than using the propellers. 