Te = xf (1 - 2(e2 +e3 )) + yf (e1 e2 +e3 n) + zf (e1 e3 - e2 n), (23)
where p and p are weighting factors. These weighting factors and k are computed by carrying out a multivariable optimization for the minimization of the terminal error function (12). Choice of the best control history for different values of these weighting factors leads to the determination of their values. When control constraint is not imposed on the system, it is obtained more than one admissible control history for different values of weighting factors. If control constraint is imposed on the system, there might still be admissible control histories.
In the optimal control law, (22), two points are of importance: the dissipation terms and the control-performance related terms. The terms coming from the power equation are the dissipation terms: 3Xuuu\u\ and 3Kppp\p\. There is not any other term reflecting system dynamics in the control law. The input to the UUV is effective on its two DOF. The surge and roll motions are controlled by the thrust. Therefore, the input related dissipation terms appear in (22). The second important term is Te and it is dominantly effective than dissipation terms in positioning the vehicle.
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