Page 109 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

102 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

Therefore, we have
1 1
2 T ∗ 2 ∗2
˙ V ≤−λ min (K p ) e v − e
− σ( ˜ D F − L ) e v + σL e v
v
2 4
1 ∗ 2 1 ∗2
(6.18)
≤− e v λ min (K p ) e v + σ( ˜ D F − L ) − ε − σL
2 4
1
Since ε + σL ∗2 is a constant, ˙ V < 0 istrueaslongas
4
1
ε + σL ∗2 .
e v ø 4 = B e v (6.19)
λ min (K p )
and

1 ε L ∗2 .
∗
˜ D F ≥ L + + = B ˜ D (6.20)
2 σ 4
Now, since M(x) is an inertia matrix, we can see that
1 1
2 2
V ≤ μB + B = B V (6.21)
2 e v 2 ˜ D
where μ is the maximum singular value of m(x) and is the minimum
.
singular value of
. Thus, V > B V implies either e v ø B e v or ˜ D F ≥ B ˜ D
implies V ≤
Suppose this is not the case, then e v < B e v and ˜ D F < B ˜ D
or
B V , which is a contradiction. Therefore, if V > B V ,wehave e v ø B e v
, which means ˙ V < 0 according to (6.18). Hence, the system is
˜ D F ≥ B ˜ D
uniformly ultimately bounded [15,11] with practical bounds on e v , ˜ D F
. Since e v =˙e + γe is a stable system, we obtain
given respectively by B e v , B ˜ D
that e(t) satisfies
1
e v ε + σL ∗2
4
e è ≤ . (6.22)
γ λ min (K p )γ
This completes the proof.

6.5 SIMULATIONS

To illustrate the performance of the proposed DPPR based modeling and
adaptive control with friction compensation, a 3-link manipulator is uti-
lized in the simulations. It is noted that there is only rotation but no
translational motion on the gripper installed in the end of the arm [14].
In other words, the motion of the end-gripper has no effect on its position.