Underwater Robots - Gianluca Antonelli.pdf

Let us consider the scalar function
V = 1
2 s T M ( q ) s + 1
2 ˜ θ T K θ ˜ θ +
+ 1
⎡ ⎤ T ⎡
˜η 1
⎣ ˜z ⎦
2
˜q
⎣ k p I 3 O 3 × 4 O 3 × n
O 4 × 3 2 k o I 4 O 4 × n
O n × 3 O n × 4 K q
⎤ ⎡
⎦ ⎣
7.7 Adaptive Control 159
⎤
˜η 1
˜z ⎦ (7.27)
˜q
where ˜z = [1 0 T ] T − z = [1 − ˜η − ˜ε T ] T . V ≥ 0in view of positive
definiteness of M ( q ), K θ , k p , k o and K q .
Differentiating V with respect to time yields
˙V = 1
2 s T ˙ Ms+ s T M ˙s + ˜ θ T K θ ˙˜ θ +
+ k p ˜η T 1 R I B ˜ν 1 − 2 k o ˜z T J k,oq( z ) ˜ν 2 + ˜q T K q ˙˜q . (7.28)
Observing that, in view of (7.25) and (7.26) it is
˜ν 1 = s p − λ p R B I ˜η 1 , (7.29)
˜ν 2 = s o − λ o ˜ε , (7.30)
˙˜q = s q − Λ q ˜q , (7.31)
and taking into account (2.71), (2.16), (7.29)–(7.31), and the skew-symmetry
of ˙ M − 2 C ,(7.28) can be rewritten as
˙V = − s T Ds − ˜ θ T K θ ˙ ˆ θ + s T [ M ˙
ζ r − Bu + C ( ζ ) ζ r + D ( ζ ) ζ r + g ]+
+ 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 +
˜q T K q s q − ˜q T K q Λ q ˜q (7.32)
where ˙˜ ˙
θ = − ˆθ was assumed, i.e., the dynamic parameters are constant or
slowly varying.
Exploiting (2.73), (7.32) can be rewritten incompact form:
⎡
˙V = − ⎣
˜η 1
˜ε
˜q
⎤
⎦
T ⎡
⎣ k p λ p I 3 O 3 × 3 O 3 × n
O 3 × 3 k o λ o I 3 O 3 × n
O n × 3 O n × 3 K q Λ q
�
T
+ s Φ ( q , R B I , ζ , ζ r , ˙ ζ r ) θ − Bu + K P ˜y
⎤ ⎡
⎦ ⎣
�
˜η 1
˜ε
˜q
⎤
⎦ +
− s T Ds − ˜ θ T K θ ˙ ˆ θ . (7.33)
Plugging the control law (7.22)–(7.23) into (7.33), one finally obtains:
˙V = − ˜y T K � ˜y − s T ( K D + D ) s
that is negative semi-definite over the state space { ˜y , s , ˜ θ } .
It is now possible to prove the system stability inaLyapunov-Like sense
using the Barbălat’s Lemma. Since V is lower bounded, ˙ V ( ˜y , s , ˜ θ ) ≤ 0and
˙V ( ˜y , s , ˜ θ )isuniformly continuous then ˙ V ( ˜y , s , ˜ θ ) → 0as t → ∞.Thus
˜y , s → 0 as t →∞.However itisnot possible to prove asymptotic stability
of the state, since ˜ θ is only guaranteed to be bounded.