Page 89 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

RISE Based Asymptotic PPC of Servo Systems With Continuously Differentiable Friction Model 81

5.3.2 Adaptive Control Design With RISE
˙
In the practical control design, the estimation of ¯ F d is used, which is calcu-
lated by
ˆ ˙ ¯ F d = (Z) (5.18)
T
ˆ
where is the estimate of the augmented weight .
ˆ
Then based on the system dynamics in (5.17), the control u is designed
as

1
u = − (k s + 2)z 2 (t) + (k s + 2)z 2 (0)
ρθ
t
T
ˆ
− [ (Z(σ)) + (k s + 2)k 2z 2 (σ) + β 1sgn(z 2 (σ))]dσ
(5.19)
0
t
1 T
ˆ
= − μ s − (Z(σ))dσ
ρθ 0
where k s is a positive feedback gain, β 1 is a positive constant, sgn(·) is the
signum function, and μ s denotes the robust feedback term, which is given
as
t
μ s = (k s + 2)z 2 (t) − (k s + 2)z 2 (0) + [(k s + 2)k 2z 2 (σ) + β 1sgn(z 2 (σ))]dσ
0
(5.20)

ˆ
The ESN weight can be updated by using the following adaptive law
˙
ˆ
ˆ
= [ (Z)z 2 − σ ] (5.21)
where > 0 is the learning gain, and σ> 0 is the forgetting factor. Note
the ESN weight is online updated depending on the tracking error z 2.
From (5.19), the derivative of control u with zero initial condition is
given by
1
T
ˆ
˙ u = [− (Z) −¨μ s ] (5.22)
ρθ
Moreover, one may verify that ˙μ s fulfills
˙ μ s = (k s + 2)r + β 1sgn(z 2 ) (5.23)

Consequently, by substituting (5.23)into(5.17), one can obtain the
closed-loop tracking error system dynamics as: