Underwater Robots - Gianluca Antonelli.pdf

48 3. Dynamic Control of 6-DOF AUVs
Let now consider the positive semi-definite function:
V = 1
2 s T M � v s > 0 , ∀ s �= α
� 0
Q
�
, α ∈ IR (3.5)
after straightforward calculation, its time derivative isgiven by:
˙V = − s T [ Λ + D � RB] s < 0 , ∀ s �= 0 (3.6)
Since V is only positive semi-definite and the system is non-autonomous
the stability can not bederived by applying the Lyapunov’s theorem. By
further assuming that ˙p r is twice differentiable, then ¨ V is bounded and ˙ V
is uniformly continuous. Hence, application of the Barbălat’s Lemma allows
to prove global convergence of s → 0 as t →∞.Due to the definition of
the vector s ,its convergence tozero also implies convergence of ˜p to the null
value.
In case of perfect knowledge ofthe dynamic model, moreover, the convergence
ofthe error tozero can be demonstrated even for K I = O ,i.e.,
without integral action.
Compensation of the persistent effects. If areduced version of the controller
is implemented, e.g., by neglecting the model-based terms in (3.4), the
restoring moment is not compensated efficiently. Infact, let consider avehicle
in the two static postures shown in Figure 3.1 and let suppose that thevehicle,
starting from the left configuration, is driven to the right configuration and,
after awhile, back tothe left configuration. Inthe left configuration the integral
action in(3.3) does not give any contribution to the control moment as
expected, because the vectors of gravity and buoyancy are aligned. Furthermore,
in the right configuration the integral action will compensate exactly at
the steady state for the moment generated by the misalignment between gravity
and buoyancy. When the vehicle is driven back to the left configuration,
anull steady-state compensation error ispossible after the integral action is
discharged; this poses asevere limitation tothe control bandwidth that can
be achieved. Asimilar argument holds inthe typical practical situation in
which the compensation implemented through the vector g � RB
is not exact.
On the other hand, since the error variables are defined inthe earth-fixed
frame, the controller is appropriate to counteract the current effect. This
point will be clarified in next Subsections, when discussing the drawbacks of
the controllers C and D with respect to the current compensation.
Finally, anadaptive version of this controller is not straightforward. In
fact, since the dynamic model (2.53) does not depend on the absolute vehicle
position, asteady null linear velocity ofthe vehicle with anon-null position
error would not excite acorrective adaptive control action. As aresult, null
position error at rest cannot be guaranteed in presenceofocean current. From
the theoretical point ofview, this drawback can be avoided by defining the
velocity error using the current measurement. However, from the practical
point ofview, this approach cannot achieve fine positioning ofthe vehicle
since local vortices can make current measurement too noisy.