## Explicit solution for a constant using piecewise linear functions nonrotatable actuators

An explicit solution approach for parametric quadratic programming has been developed by T0ndel et al. (2003) while applications to marine vessels are presented by Johansen et al. (2005). In this work the constrained optimization problem is formulated as min f ,s,f

subject to:

-f < fvf2,...fr < f where s e R" is a vector of slack variables and forces f = f, f2,..., fr ]T G Rr

The first term of the criterion corresponds to the LS criterion (1.25), while the third term is introduced to minimize the largest force f = max_ | f | among the actuators. The constant P > 0 controls the relative weighting of the two criteria. This formulation ensures that the constraints fmin < f < fm™ (i = 1,..., r) are satisfied, if necessary by allowing the resulting generalized force Tf to deviate from its specificationt . To achieve accurate generalized force, the slack variable should be close to zero. This is obtained by choosing the weighting matrix Q » W > 0. Moreover, saturation and other constraints are handled in an optimal manner by minimizing the combined criterion (1.35). Let p = [tt , fm denote the parameter vector and,

R"

Hence, it is straightforward to see that the optimization problem (1.35) can be reformulated as a QP problem:

where:

subject to:

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