Underwater Robots - Gianluca Antonelli.pdf

7.8 Output Feedback Control 171
with �η d > 0, �η e > 0. It can be recognized that in the domain B ρ the following
inequalities hold
0 < � 1 − ρ 2 < �η e < 1 , (7.57)
�
�
� � �
�
ζ d � = � σ d − Λ d e de� ≤(1 +2λ max( Λ d ))ρ. (7.58)
By completing the squares in (7.56) and using (7.57),(7.58), itcan be
shown that there exists ascalar κ>0such that
˙V ≤−κ � x � 2
(7.59)
in the domain B ρ ,provided that the controller and observer parameters satisfy
the inequalities
�
λ min( K v ) >α1 C M + 3 D �
M
+ α 2 λ max( M )(1 + λ max( Λ d ))
2
λ min( Λ d K p ) > α 2 λ max( M ) λ max( Λ d )
� ,
1 − ρ 2
�
�
λ max( Λ d K p ) 2
λ min( L p ) > max α 2 λ max( M ) λ max( Λ d ) ,
λ min( Λ d K p ) � 1 − ρ 2
�
2
λ min( Λ v ) > � λ max( K v )+(2 α 1 + ρ ) C M +
1 − ρ 2
α �
1 D M
2
where α 1 =2(1 + λ max( Λ d ))ρ + ζ dM and α 2 = ζ dM +2λ max( Λ d ) ρ .
Therefore, given adomain B ρ characterized by any ρ