Underwater Robots - Gianluca Antonelli.pdf

7.6 Sliding Mode Control 153
u = B † [ K D s + ˆg + ˆ M ˙y +( Ĉ + ˆ D ) y + K S sign(s )] (7.15)
in lieu of (7.10). Starting from the function in(7.12) and plugging (7.15)
in (7.13) gives
˙V = − s T ( D + K D ) s + s T [ � M ˙y +( � C + � D ) y + ˜g − K S sign(s )]
that, in view of positive definiteness of K D and D ,leads to negative definiteness
of ˙ V if
�
�
λ min( K S ) ≥ � � M ˙y +( � C + � �
�
D ) y + ˜g � . (7.16)
It is worth noting that condition (7.16) is weaker than condition (7.14) in
that the matrix K S must overcome the sole model parameters mismatching.
Stability ofthe sliding manifold. It was demonstrated that the discussed
control law guarantees convergence of s to the sliding manifold s = 0 .Inthe
following it will be demonstrated that, once the sliding manifold has been
reached, the error vectors ˜η 1 , ˜ε , ˜q converge asymptotically to the origin, i.e.,
that regulation of the system variables totheir desired values is achieved.
By taking Λ =blockdiag { Λ p , Λ o , Λ q } where Λ p ∈ IR 3 × 3 , Λ o ∈ IR 3 × 3 ,
Λ q ∈ IR n × n ,from (7.11) it is possible to notice that the stability analysis can
be decoupled in 3parts as follows.
Vehicle position error dynamics. The vehicle position error dynamics on
the sliding manifold is described bythe equation
− ν 1 + Λ p R B I ˜η 1 = 0 .
Notice that the rotation matrix R B I is afunction ofthe vehicle orientation.
By considering
V = 1
2 ˜η T 1 ˜η 1
as Lyapunov function candidate and observing that ˙η 1 = R I B ν 1 ,itiseasily
obtained
˙V = − ˜η T 1 R I B Λ p R B I ˜η 1 ,
that is negative definite for Λ p > O .Hence, ˜η 1 converges asymptotically to
the origin.
Vehicle orientation error dynamics. The vehicle orientation error dynamics
on the sliding manifold is described bythe equation
− ν 2 + Λ o ˜ε = 0 ⇒ ν 2 = Λ o ˜ε . (7.17)
Further, by taking into account the quaternion propagation and (7.17), it
can be recognized that
˙˜η = 1
2 ˜ε T ν 2 = 1
2 ˜ε T Λ o ˜ε . (7.18)