Page 263 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

ˆ p j

ˆ
l = (17.22)
j − 1
j
Consider that the point (u m ,w m ) is on the curve of w l , hence it follows
w m
ˆ
d 1 = u m − (17.23)
ˆ l
Similarly, the point (u M ,w M ) is on the curve of w r as showninthe right
half plane of Fig. 17.1, then the estimation of d 2 can be obtained as

w M
ˆ
d 2 = u m − (17.24)
ˆ l
It can be seen from (17.15)–(17.24) that the parametric piecewise linear
expressions can be used to estimate the backlash characteristic parameters
and the unknown coefficients in the linear transfer function simultaneously
without the need to estimate the non-linear parameters and the linear pa-
rameters.

Theorem 17.1. [11] For system (17.6), for any hT < k <(h + 1)T, with
the input signal shown in (17.10), the parameter estimation given by (17.18)and
(17.22)–(17.24) is uniformly convergent, i.e., θ N → θ,as N →∞.
ˆ
Proof. For any k ∈[hT,hT + 4n + 1],since u 0 (k) ∈{0,m 1 + δ},wehave

p 0 + p 1 σ 1 (0,u 0 − α 1 ,β 1 − α 1 ) k = k h + 2n
ˆ
ˆ w l (k) = BI(u 0 (k)) =
0 others
(17.25)

According to (17.25)and thelemmasin[12], it is known that the re-
gressor vector ϕ(k) is persistently excited, and thus based on the property of
LS method, the uniform convergence of the estimated θ N can be obtained.
ˆ
Consequently, the convergence of the other estimated parameters based on
θ N can be claimed.
ˆ

17.4 INVERSE COMPENSATION BASED CONTROL DESIGN
AND STABILITY ANALYSIS

After conducting system identification, we can rewrite system (17.1)asthe
following form