## Criteria function

As was explained above, the criteria function consists of the control-performance (performance criterion), the performance measure describing the control effort, and the energy equation of the controlled system is given by

where the second term on the right-hand side of the equation is the power delivered to the actuator, and r' is the weighting factor which is determined as equal to the number of actuators. This control effort term can be denoted as Je and written as:

The last term in (13) is the energy equation. The total power equation, P, is the sum of all power flows in the system:

P Pin + P generated - (Pd issipated + Pstored + Pout ) =0. (15)

Since each equation corresponding to each DOF in the equations of motion, (6), represents a force or moment equality, each equation can be also formulated so as to represent a power equality. If each term of each equation in (6) is multiplied by the corresponding velocity, the power equalities are obtained. The sum of all equations yields the total power equation:

+ Xuuu2 |u| + YvvV2 |v| + Zwww2 |w| + Kppp2 |p|+Mqqq2 | q | + Nrrr2 |r|=0

In the total power equation, the remaining terms after summation are the input, stored, and dissipated power terms. Among these terms, the dissipation terms are of great importance since they appear in the control law and have contribution on the stability of the system.

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