Page 93 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

RISE Based Asymptotic PPC of Servo Systems With Continuously Differentiable Friction Model 85

2
where U(y) = c z is a continuous positive semi-definite function over
¯
¯
an arbitrary compact set , which is determined by ≡{y(t)| y à
−1

η (2 λ 3k s )}.
Therefore, the inequalities (5.35)and (5.43) can be used to show that
V L ∈ L ∞ over ¯ which further implies that z 1, z 2, P(t), Q(t), r(t) ∈ L ∞.
,
Given e 1 (t), z 1 (t), z 2 (t),and r(t) ∈ L ∞, the standard linear analysis methods
.
canbeusedtoprove that ˙z 1, ˙z 2 ∈ L ∞ in ¯ Since z 1 (t), z 2 (t),and r(t) ∈ L ∞,
the assumption that x d , ˙x d ,and ¨x d are bounded can be used to conclude that
,
x 1, ˙x 1 ∈ L ∞.Given that r(t) ∈ L ∞ in ¯ it can be shown that ˙u(t) ∈ L ∞ in ¯
.
Hence, we can also verify that ˙r(t) ∈ L ∞. Since ˙e(t), ˙z 1, ˙z 2,and ˙r(t) ∈ L ∞,
the definitions for U(y) and z(t) canbeusedtoprove U(y) is uniformly
continuous in . In this case, since the transformed error z 1 is bounded,
¯
we can recall Lemma 5.1 and then claim that the tracking control error
e 1 (t) can be retained within the prescribed performance bound (5.5).
In the following, we further prove asymptotic convergence of the track-
ing error e 1 to zero. To this end, the definitions of U(y) and y can be used
to conclude that ˙z 1,and ˙z 2 are uniformly continuous, i.e., z 1, z 2 ∈ L 2.
Thus, according to Barbalat’s lemma, we can conclude that

(5.44)
r(t) → 0as t →∞ ∀y(0) ∈ S 0
where S 0 ⊂ is a compact set defined as
¯

−1 2

S 0 ≡{y(t) |U 2 (y)| <λ 1 (η (2 λ 3k s )) } (5.45)

Moreover, based on the definition for r, one has

(5.46)
|z 1 (t)|→ 0, |z 2 (t)|→ 0, as t →∞ ∀y(0) ∈ S 0

Thus, based on the property of the error transform function (4.10), we can
verify that

|e 1 (t)|→ 0as t →∞ ∀y(0) ∈ S 0 (5.47)
This implies that the tracking error can converge to zero asymptoti-
cally with guaranteed transient convergence bound. This completes the
proof.