Underwater Robots - Gianluca Antonelli.pdf

7.8 Output Feedback Control 169
C T ( q , σ d ) ζ o + D ( q , ζ ) ζ − 1
2 D ( q , ζ r )(ζ r + ζ o ) . (7.50)
Astate vector for the closed-loop system (7.49),(7.50) isthen
⎡
⎢
x = ⎣
σ d
e d
σ e
e e
⎤
⎥
⎦ .
Notice that perfect tracking of the desired motion together with exact estimate
of the system velocities results in x = 0 .Therefore, the controlobjective
is fulfilled if the closed loop system (7.49), (7.50) is asymptotically stable at
the origin of its state space. This is ensured by the following theorem:
Theorem. There exists achoice ofthe controller gains K p , K v , Λ d and of
the observer parameters L p , L v , Λ e such that the origin of the state space of
system (7.49),(7.50) islocally exponentially stable.
Consider the positive definite Lyapunov function candidate
V = 1
2 σ T d M ( q ) σ d + 1
+ 1
2 k pP �η
T
B
1 ,d
2 σ T e M ( q ) σ e +
�
B
�η 1 ,d + k pO (1 − �η d ) 2 + �ε T
d �ε d
�
B
�η 1 ,e + l pO (1 − �η e ) 2 + �ε T
e �ε �
e
�
+ 1 T
�q d K pQ�q d +
2
+ 1
2 l T
B
pP �η 1 ,e + 1 T
�q e L pQ�q e . (7.51)
2
The time derivative of V along the trajectories ofthe closed-loop system
(7.49),(7.50) isgiven by
˙V = − σ T d K v σ d − e T deΛ d K p e d − e T e Λ e L p e e +
− σ T e
�
L v A ( � �
Q e ) − K v σ e − σ T d C ( q , σ e ) ζ r +
− σ T e C ( q , ζ ) σ e + σ T e C T ( q , σ d ) ζ o +
+(σ d + σ e ) T D ( q , ζ ) ζ + − 1
2 ( σ d + σ e ) T D ( q , ζ r )(ζ r + ζ o )+
− σ T d M ( q ) S PO( �ν 2 ,d) ζ d − σ T d M ( q ) Λ d S P ( �ν 2 ,d) e de . (7.52)
In the following it is assumed that �η d > 0, �η e > 0; in view of theangle/axis
interpretation of the unit quaternion, the above assumption corresponds to
considering orientation errors characterized by angular displacements inthe
range ]− π, π [.
From the equality � Q de = � − 1 Q e ∗ � Q d ,the following equality follows
�ε eT
de �ε d = �η e �ε T
d �ε d − �η d �ε T
d �ε e ,
where �η d and �η e are the scalar parts of the quaternions � Q d and � Q e ,respectively.
The above equation, in view of �η B
1 ,de = �η B
1 ,d − �η B
1 ,e and �q de = �q d − �q e ,
implies that