## Explicit solutions based on minimum norm and nullspace methods nonrotatable actuators

In flight and aerospace control systems, the problems of control allocation and saturating control have been addressed by Durham (1993, 1994a, 1994b). They also propose an explicit solution to avoid saturation referred to as the direct method. By noticing that there are infinite combinations of admissible controls that generate control forces on the boundary of the closed subset of attainable controls, the direct method calculates admissible controls in the interior of the attainable forces as scaled down versions of the unique solutions for force demands. Unfortunately it is not possible to minimize the norm of the control forces on the boundary or some other constraint since the solutions on the boundary are unique. The computational complexity of the algorithm is proportional to the square of the number of controls, which can be problematic in real-time applications.

In Bordignon and Durham (1995) the null space interaction method is used to minimize the norm of the control vector when possible, and still access the attainable forces to overcome the drawbacks of the direct method. This method is also explicit but much more computational intensive. For instance 20 independent controls imply that up to 3.4 billon points have to be checked at each sample. In Durham (1999) a computationally simple and efficient method to obtain near-optimal solutions is described. The method is based on prior knowledge of the controls' effectiveness and limits such that pre-calculation of several generalized inverses can be done.

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