Page 118 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Dead-Zone Dynamics and Modeling 113

D r (v) = D r (v) − D r (b r ) = D (ζ )(v − b r ),∀v ∈[b l ,+∞) (7.8)
r
r

where ζ ∈ (v,b l ),if v ≤ b l or ζ ∈ (b l ,v) if b l < v ≤ b r ;and ζ ∈ (b r ,v) if v ≥ b r
l l r
or ζ ∈ (v,b r ),if b l ≤ v < b r .

r
Consequently, the dead-zone (7.2) can be represented as [10]
u(t) =[χ l (t) + χ r (t)]v(t) + ρ(t) = d(t)v(t) + ρ(t) (7.9)
where

D (ζ ) if v < b r D (ζ ) if v > b l

r
l
r
l
χ l (t) = , χ r (t) = (7.10)
0 if v ≥ b r 0 if v ≤ b l
⎧

⎪ −D (ζ )b r if v ≥ b r
r r
⎨

ρ(t) = −(D (ζ ) + D (ζ ))v if b l <v < b r (7.11)
r r l l
⎪
⎩
−D (ζ )b l if v ≤ b l
l l
As shownin(7.9), the non-linear dead-zone model (7.2) is represented
as a linear time-varying system with a time-varying gain d(t) and a bounded
disturbance ρ(t), thus conventional inverse dead-zone compensation used
in [1,11,3] can be avoided. Moreover, compared with other studies with a
linear dead-zone or even symmetric dead-zone (i.e., b r = b l ), the non-linear
dead-zone (7.2) may cover more realistic cases. It is verified from (7.4)–(7.9)
that
= min(d l0 ,d r0 ) ≤ d(t) ≤ d l1 + d r1 and |ρ(t)| ≤ p (7.12)
with positive constants 0 < < +∞ and p = (d l1 + d r1 )max{b r ,−b l }.How-
ever, in the following control designs, the scalars d l0, d l1, d r0, d r1,and , p
are only used for analysis and not used in the control implementation.
7.3 EXAMPLES WITH DEAD-ZONE
The dead-zone dynamics appear in numerous systems converging wide va-
riety of phenomena, but not limited to man-made systems. We briefly
present three typical examples in this section.

7.3.1 Upper-Limb Model
In functional neuromuscular stimulation, a controlled electrical stimulus v
is applied to the intact nerve in an attempt to replace upper motor neuron
control which may be lost through cerebral stroke, brain injury, tumor, or