Page 130 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

In practical control implementation, since the discontinuous switching
function sgn(·) shown in (8.24)and (8.27) may result in the chattering
phenomenon, the following continuous function η(·) couldbeusedto
replace sgn(·) in the practical control

sgn(s),|s|≥ ζ
η(s) = 2s (8.28)
|s|+ζ ,|s| <ζ
where ζ> 0 is the boundary layer thickness.

8.3.3 Stability Analysis
In this section, the boundedness of all signals and the finite-time conver-
gence of the tracking error for system (8.7) in both the reaching phase and
the sliding phase will be addressed.
Lemma 8.1. [20] Suppose that there exists a continuous, positive-definite func-
tion V(t) satisfying the following differential inequality

γ
˙ V(t) + αV(t) + βV (t) ≤ 0, ∀t ≥ t 0 , V(t 0 ) ≥ 0 (8.29)
where α,β > 0, 0 <γ < 1 are constants. Then, for any given t 0,V(t) satisfies the
following inequality
V 1−γ (t) ≤ (αV 1−γ (t 0 ) + β)e −α(1−γ)(t−t 0 ) − β, t 0 ≤ t ≤ t s (8.30)

and
(8.31)
V(t) ≡ 0, ∀t ≥ t s

with t s given by
1 αV 1−γ (t 0 ) + β
t s = t 0 + ln . (8.32)
α(1 − γ) β
Now, we will summarize the main results of this chapter as the following
theorem:

Theorem 8.1. Consider the PMSM servo system (8.7) with unknown non-
linear dead-zone (8.3), terminal sliding manifold (8.20), feedback control (8.24),
and NN weight adaptive law (8.25), then:
i) All signals in the closed-loop system are semi-globally uniformly ultimately
bounded (SGUUB).