Underwater Robots - Gianluca Antonelli.pdf

56 3. Dynamic Control of 6-DOF AUVs
3.6.1 Stability Analysis
Let us consider the following scalar function
V = 1
2 s T v M v s v + 1
2 ˜ θ T
v K θ ˜ θ v + 1
2 k p ˜η T 1 ˜η 1 + k o ˜z T ˜z , (3.30)
where ˜z =[1 0 T ] T − z =[1 − ˜η − ˜ε T ] T .Due to the positive definiteness
of M v , K θ ,and K ,the scalar function V ( ˜η 1 , ˜z , s v , ˜ θ v )ispositive definite.
Let us define the following partition for the variable s v that will be useful
later:
� �
s p
s v = , (3.31)
s o
with s p ∈ IR 3 and s o ∈ IR 3 .Inview of (3.31) and the definition of s v ,itis
˜ν 1 = s p − λ p R B I ˜η 1
(3.32)
˜ν 2 = s o − λ o ˜ε . (3.33)
Differentiating V with respect to time yields:
˜θ v + k p ˜η T 1 R I B ˜ν 1 − 2 k o ˜z T J k,oq ˜ν 2
�
˙ν d − ˙ν + Λ ˙˜y
�
+ ˜ θ T
v K θ ˙ ˜θ v +
+ k p ˜η T 1 R I �
B s p − λ p R B I ˜η �
1 +
− k o [1− ˜η − ˜ε T �
�
]
( s o − λ o ˜ε ) .
˙V = s T v M v ˙s v + ˜ θ T
v K θ ˙
= s T v M v
Then, defining
− ˜ε T
˜η I 3 + S ( ˜ε )
ν a = ν d + Λ ˜y (3.34)
yields (dependencies are dropped out to increase readability):
˙V = s T v [ M v ˙ν a − τ v + C v ν + D RBν + g RB + τ v,C ]+
− ˜ θ T
v K θ ˙
ˆθ v + k p ˜η T 1 R I B s p − k p λ p ˜η T 1 ˜η 1 +
+ k o ˜ε T s o − λ o k o ˜ε T ˜ε , (3.35)
that can be rewritten as:
˙V = s T v [ Φ v,T θ v − τ v ] − s T v D RBs v − ˜ θ T
v K θ ˙
ˆθ v +
+ s T v K ˜y − k p λ p ˜η T 1 ˜η 1 − k o λ o ˜ε T ˜ε .
By considering the control law (3.28) and the parameters update (3.29),
it is:
˙V = − s T v ( K D + D RB) s v − k p λ p ˜η T 1 ˜η 1 − k o λ o ˜ε T ˜ε
that is negative semi-definite over the state space { ˜η 1 , ˜z , s v , ˜ θ v } .