Page 125 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Finite-Time Neural Control of Servo Systems With Non-linear Dead-Zone 121

where v(t) ∈ R is the input of the dead-zone (i.e., practical control signal),
D l (v(t)), D r (v(t)) are unknown non-linear smooth functions, and b l , b r are
unknown width parameters of the dead-zone. Without loss of generality,
we assume b l < 0and b r > 0.
Because the maximum and minimum slope values b l , b r of the dead-
zone are difficult to obtain practically in the non-linear dead-zone (8.3),
a model-independent compensation scheme is developed, in which the
functions D l (v(t)), D r (v(t)) and the characteristic parameters b l , b r are not
necessarily known. To facilitate the control design, the following assump-
tion is needed.

Assumption 8.1. [13–15]: The functions D l (v(t)) and D r (v(t)) are smooth,
and there exist unknown positive constants d l0,d l1,d r0,and d r1 such that

0 < d l0 ≤ D l (v(t)) ≤ d l1 ,∀v(t) ∈ (− ,b l ) (8.4)

0 < d r0 ≤ D (v(t)) ≤ d r1 ,∀v(t) ∈ (b r ,+∞) (8.5)
r

where D r (b r ) = D l (b l ) = 0,D (v(t)) = dD l (z)/dz| z=v(t),and D (v(t)) =
l r
dD r (z)/dz| z=v(t).
Then according to the statements presented in Section 7.2 of Chapter 7,
we can have

u(t) = d(t)v(t) + ρ(t),∀t ≥ 0 (8.6)

where d(t) and ρ(t) areallgivenin(7.10)and (7.11), which are all bounded
as stated in Chapter 7.
Substituting (8.6)into(8.2), we have

¨ x =−h(¯x,t) + k 0 [d(t)v(t) + ρ(t)]
(8.7)
y = x

From Assumption 8.1 and the statements shown in Chapter 7 (Sec-
tion 7.2.2), we can verify that d(t) ∈[ϕ 0 ,ϕ 1 ]⊂ (0,+∞) with ϕ 0 =
min(d l0 ,d r0 ) and ϕ 1 = d l1 + d r1, |ρ(v(t))|≤ p with p = (d l1 + d r1 )max{b r ,−b l }
being a positive constant, and thus k 0d(t)
= 0 isalwaystrue.
Let y d be a given desired trajectory, and then the tracking error e is
defined as

e = y d − y (8.8)