Underwater Robots - Gianluca Antonelli.pdf

152 7. Dynamic Control of UVMSs
is apositive definite matrix, and sign(x )isthe vector function whose i -th
component is
�
1 ifx i ≥ 0
sign(x ) i =
− 1 ifx i < 0 .
In (7.10), s is the � (6 + n ) × 1 � sliding manifold defined asfollows
⎡
s = Λ ⎣ R B I ˜η ⎤ ⎡
1
˜ε ⎦ − ⎣
˜q
ν ⎤
1
ν ⎦ 2 = y − ζ , (7.11)
˙q
with Λ > O , ˜η 1 =[x d − x yd − y zd − z ] T , ˜q = q d − q where the subscript
d denotes desired values for the relevant variables.
7.6.1 Stability Analysis
In this Section it will be demonstrated that the discussed control law is
asymptotically stable in aLyapunov sense. Let us consider the function
V = 1
2 s T M ( q ) s , (7.12)
that is positive definite being M ( q ) > O .
Differentiating V with respect to time yields
˙V = 1
2 s T ˙ Ms+ s T M ˙s
that, taking into account the model (2.71), (7.11) and the skew-symmetry
of ˙ M − 2 C ,can be rewritten as
˙V = − s T Ds + s T [ M ˙y − Bu + Cy + Dy + g ] . (7.13)
Plugging (7.10) into (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 ,can be upper bounded
as follows
˙V ≤−λ min( K D + D ) � s � 2 − λ min( K S ) � s � +
+ � M ˙y +(C + D ) y + ˜g ��s � ,
where λ min denotes the smallest eigenvalue of the corresponding matrix.
By choosing K S such that
λ min( K S ) ≥�M ˙y +(C + D ) y + ˜g � , (7.14)
the time derivative of V is negative definite and thus s tends tozero asymptotically.
If an estimate of the dynamic parameters in (2.71) is available, itmight
be convenient to consider the control law