Page 117 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

based on the following non-linear dead-zone model

⎧
⎪ D r (v(t)) v(t) ≥ b r
⎨
u(t) = DZ(v(t)) = 0 b l < v(t)< b r (7.2)
⎪
D l (v(t)) v(t) ≤ b l
⎩
where D l (v(t)),D r (v(t)) are unknown non-linear smooth functions, b l and
b r are the unknown break-points of the dead-zone.
Compared with linear dead-zone model (7.1), non-linear smooth func-
tions D l (v) and D r (v) are used to replace the linear terms m lv and m r v,soas
to cover more realistic non-linear dynamics embedded in the actuators.
To facilitate the control system design, the above dead-zone model can
be represented in a combination of a linear term (with time-varying gain)
and a disturbance-like term. We refer to [8,9]formore detailsonthisre-
formulation. As stated in Fig. 7.2, the dead-zone functions D l (v), D r (v) are
continuous over (− ,b l ] and [b r ,+∞), and there exist unknown positive
constants d l0, d l1, d r0,and d r1 such that

0 < d l0 ≤ D (v) ≤ d l1 , ∀v ∈ (− ,b l ),

l
(7.3)

0 < d r0 ≤ D (v) ≤ d r1 , ∀v ∈ (b r ,+∞)
r
where D (v) = d(D i (z))/dz| z=v ,i = l,r is the derivative of D i (v),i = l,r.

i
Consider the fact D r (b r ) = D l (b l ) = 0 and apply the differential mean-
value theorem on D l (v), D r (v), it follows

D l (v) = D l (v) − D l (b l ) = D (ζ l )(v − b l ),∀v ∈ (− ,b l ] with ζ l ∈ (v,b l )
l
(7.4)
D r (v) = D r (v) − D r (b r ) = D (ζ r )(v − b r ),∀v ∈[b r ,+∞) with ζ r ∈ (b r ,v)

r
(7.5)
In addition, as explained in [8], the properties in (7.4)and (7.5)can be
extended to the interval (b l ,b r ) as

D l (v) = D (ζ l )(v − b l ),∀v ∈ (b l ,b r ] with ζ l ∈ (b l ,v)
l
(7.6)

D r (v) = D (ζ r )(v − b r ),∀v ∈[b l ,b r ) with ζ r ∈ (v,b r )
r
From (7.4)–(7.6), one may obtain

D l (v) = D l (v) − D l (b l ) = D (ζ )(v − b l ),∀v ∈ (− ,b r ] (7.7)
l l