Underwater Robots - Gianluca Antonelli.pdf

170 7. Dynamic Control of UVMSs
and thus
e T T
B
deΛ d K p e d = k pP �η 1 ,d Λ dP �η B
1 ,d + λ dOk pO�η e � �ε d � 2 + �q T
d Λ dQK pQ�q d +
T
B
− k pP �η 1 ,d Λ dP �η B
1 ,e − λ dOk pO�η d �ε T
d �ε e + �q T
d Λ dQK pQ�q e ,
e T deΛ d K p e d ≥ λ min( Λ d K p ) �η e � e d � 2 − λ max( Λ d K p ) � e d ��e e � , (7.53)
where λ min( Λ d K p )(λ max( Λ d K p )) is the minimum (maximum) eigenvalue
of the matrix Λ d K p .Moreover, in view of the block diagonal structure of
the matrix L v and ofthe skew-symmetry of the matrix S ( · ), the following
inequality holds
σ T e L v A ( � Q e ) σ e ≥ 1
2 λ min( Λ v ) �η e � σ e � 2 , (7.54)
where λ min( Λ v )isthe minimum eigenvalue of the matrix Λ v .
Moreover, the following two terms in (7.52) can be rewritten as:
( σ d + σ e ) T D ( q , ζ ) ζ − 1
2 ( σ d + σ e ) T D ( q , ζ r )(ζ r + ζ o )=
− 1
2 ( σ d + σ e ) T D ( q , ζ )(σ d + σ e )+
− 1
2 ( σ d + σ e ) T ( D ( q , ζ r ) − D ( q , ζ ))(ζ r + ζ o ) . (7.55)
In view of the properties of the model (2.71) and equations (7.45), (7.46),
(7.53), (7.54), by taking into account that ζ = ζ d − � ζ d with � ζ d �≤ζ dM and
� e de� ≤�e d � + � e e � ,the function ˙ V can be upper bounded as follows
˙V = − λ min( K v ) � σ d � 2 − λ min( Λ d K p ) �η e � e d � 2 − λ min( Λ v )
�η e � σ e �
2
2 +
+ λ max( K v ) � σ d � 2 − λ min( L p ) � e e � 2 + λ max( Λ d K p ) � e d ��e e � +
� �
�
+ C M � σ d ��σ e � 2 � � �
�
�
ζ d � +2ζ dM + � σ d � + � σ e � +
+ C M � σ e � 2 � � � �
�
� ζ � �
d � + ζ dM +
+ D M
2 ( � σ d � 2 �
�
+ � σ d ��σ e � )(2 � � �
�
ζ d � +2ζ dM + � σ d � + � σ e � )+
�
�
+ λ max( M ) � σ d � � � �
�
ζ d � ( ζ dM + λ max( Λ d )(� e d � + � e e � )) (7.56)
where λ min( K v )(λ max( K v )) denotes the minimum (maximum) eigenvalue
of the matrix K v , λ min( L p ) denotes the minimum eigenvalue of L p and
λ max( Λ d )denotes the maximum eigenvalue of the matrix Λ d .
Consider the state space domain defined as follows
B ρ = { x : � x �