Page 266 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Inverse Model Based Control of Uncertain Systems With Backlash 267

which implies that

= w(t) − u d (t) = 0 (17.35)

3) When the second condition of (17.27)istrue, wehave

ˆ
ˆ
w(t) = l(u(t) + d s ) (17.36)
ˆ ˆ

d 1 d 2
ˆ
where d s ∈ , ,then
ρ ρ
= w(t) − u d (t) = ρW(t) + d s (17.37)
which shows that | |=|ρW(t) + d s |≤|d 1 |+|d 2 |, and thus is bound-
ˆ
ˆ
ed.
17.4.2 Controller Design With Inverse Compensation
To design the feedback control u d , we define an intermediate error related
to the output tracking error as

r = e n−1 + λ n−1e n−2 + ··· + λ 1e (17.38)

where e = y − y d ,and λ 1 ,··· ,λ n−1 are appropriately selected parameters,
such that the polynomial s n−1 +λ n−1s n−2 +···+λ 1 is stable, i.e., all poles have
a negative real part. Hence, if r is bounded and exponentially converges to
zero, then the tracking error e is also bounded and exponentially converges
to zero.
From (17.26)and (17.38), the error dynamics can be obtained as

n n

(n) (n−1)
˙ r = − a ix n−j+1 + w(t) − (y − λ n−1e + ··· + λ 1 ˙e) + ˜ a jx n−j+1
d
j=1 j=1
(17.39)

Then the following feedback control u d can be designed
n

u d (t) =−k pr + ˆ a jx n−j+1 + y eq + u r (17.40)
j=1

where k p > 0 is the feedback gain, y eq = y (n) −λ n−1e (n−1) +···+λ 1 ˙e,and u r is
d
a robust term to address the identification error of a i, which can be selected