Similarly, another short period with the same symptoms and causes takes place in the horizontal thruster 6 at about 404(s), also in the form of an oscillation with a separation less that 4% (see f6 and n6 in Fig. 7, bottom).

Fig. 7. Evolution of f vs. fdeal and n vs. g3Uideal in thruster 3 at about 10(s) (top) and in thruster 6 at about 404(s) (bottom)

The sudden mass changes are absorbed above all by thrusters 2 and 3 (vertical thrusters in the bow) where jumps are also noticed in the evolutions of thrusts. However they have retained an exact coincidence between f and fdmi. Jumps are noticed in all four horizontal thrusters too, with the same amplitude, however to a lesser degree. The coincidence between f andfckai also persists during periodic parameter changes in all thrusters, see Fig. 6, left. The performance of the disturbance/state observer can be seen in Fig. 6, right, where the true shaft rate n versus the filtered ideal shaft rate g3nd£al are depicted for all thrusters. One notices a good concordance between both evolutions in almost the whole period of the mission. Contrary to the thrust evolutions, the convergence transients of n to g3nd£al at the start phase take a very short time less than 1(s). However, the evolutions begin with strong excursions and remain in time only a few seconds.

Similarly as in the thrusts f and fidaai, there exist additionally two significant periods with short transients of non coincidence between n and g3ndeal. These occur at about 10(s) and 404(s) by thrusters 3 and 6, respectively (see Fig. 7, top and bottom). All of them are related to crosses around the zero value under a relatively large value of its axial velocity va (cf. Fig. 2). One notices that the evolution of n is more jagged than that of g3nd£al due to the discontinuities at the short transients and due to the fact that g3nideai is a smoothed signal.

8. Conclusions

In this chapter a complete approach to design a high-performance adaptive control system for guidance of autonomous underwater vehicles in 6 degrees of freedom was presented. The approach is focused on a general time-varying dynamics with strong nonlinearities in the drag, Coriolis and centripetal forces, buoyancy and actuators. Also, the generally rapid dynamics of the actuators is here in the design not neglected and so a controller with a wide working band of frequencies is aimed.

The design is based on a adaptive speed-gradient algorithm and an state/disturbance observer in order to perform the servo-tracking problem for arbitrary kinematic and positioning references. It is shown that the adaptation capability of the adaptive control system is not only centered in a selftuning phase but also in the adaptation to time-varying dynamics as long as the rate of variation of the system parameter is vanishing in time. Moreover, bounded staggered changes of the system matrices are allowed in the dynamics. By means of theorem results it was proved that the path-tracking control can achieve always asymptotically vanishing trajectory errors of complex smooth geometric and kinematic paths if the thruster set can be described through its nonlinear static characteristics, i.e., when its dynamics can be assumed parasitic in comparison with the dominant controlled vehicle dynamics and therefore neglected. This embraces the important case for instance of vehicles with large inertia and parsimonious movements. On the other side, when the actuators are completely modelled by statics and dynamics, an observer of the inverse dynamics of the actuators is needed in order to calculate the setpoint inputs to the thrusters. In this case, the asymptotic path tracking is generally lost, though the trajectory errors can be maintain sufficiently small by proper tuning of special ad-hoc high-pass filters. It is also shown, that the transient performance under time-varying dynamics can be setup appropriately and easily with the help of ad-hoc design matrices. In this way the adaptive control system can acquire high-performance guidance features.

A simulated case study based on a model of a real underwater vehicle illustrates the goodness of the presented approach.

9. References

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