Page 29 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

tification. First of all, GSO algorithm is employed to achieve the offline
identification of both static and dynamic parameters of the LuGre friction
model. With the GSO identification result, a finite-time online parametric
estimation method is further proposed to improve the identification accu-
racy and the transient response. Moreover, it is noted that among different
control methods, sliding mode control (SMC) has strong robustness with
respect to the external disturbance and internal system uncertainties. In
particular, the recently proposed non-linear sliding mode control (NSMC)
technique is able to improve the tracking error convergence rate compared
to linear sliding mode control [15–17], and thus will be tailored in this
chapter to achieve a satisfactory tracking performance.

2.2 SYSTEM DESCRIPTION AND PROBLEM FORMULATION

The dynamics of the mechanical servo system are described [18–20]as

¨
Jθ = T − T f − T d (2.1)
where θ denotes the actual position of the motor output shaft, J is the rotary
inertia equivalent to the motor shaft, T is the control input torque of the
motor, T f is the total friction torque, and T d is the external disturbances.
To facilitate control design, we define x 1 = θ, x 2 = ω = θ,thensystem
˙
(2.1) can be transformed into the following state-space form

˙ x 1 = x 2 (2.2)
1
˙ x 2 = (T − T f − T d )
J
where ω denotes the motor velocity, and the friction T f is described by
LuGre model as

T f = σ 0z + σ 1 ˙z + σ 2 ω (2.3)
|ω|
˙ z = ω − z (2.4)
g(ω)
2
σ 0g(ω) = f c + (f s − f c )e −(ω ω s ) (2.5)
where σ 0, σ 1,and σ 2 are the bristle stiffness coefficient, bristle damping
coefficient, and viscous damping coefficient, respectively; z is the internal
friction state, which denotes the average deformation of the bristles; f s and