Page 30 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Sliding Mode Control of Non-linear Servo Systems With LuGre Friction Model 21

f c are the static friction torque and the Coulomb friction torque, respec-
tively; ω s denotes the Stribeck velocity and g(ω) is a non-linear function
representing different friction effects.
By using (2.3), then system (2.2) can be rewritten as

˙ x = F(x) + G(x,T) (2.6)

0 1 0
T x 1
where x = [x 1 ,x 2 ] , F(x) = , G(x,T) = [−z,−¨z,
0 0 x 2 1
T
σ 0 σ 1 σ 2 T d 1
−x 2 ,−1], = , , , , .
J J J J J
The problem to be addressed is to identify the unknown parameters
and friction coefficients, and then design a control to make the system
output x 1 track a given trajectory.

2.3 OFFLINE FRICTION IDENTIFICATION

In this section, an intelligent glowworm swarm optimization algorithm is
presented to offline identify the friction model parameters σ 0, σ 1, σ 2, f c ,
f s, ω s. The estimates of these parameters will be used as the initial values
of the online adaptation in the control design to be presented in the next
section.

2.3.1 Glowworm Swarm Optimization
Glowworm swarm optimization is a swarm intelligence algorithm based
on the release of luciferin by glowworms. This luciferin attracts glowworms
creating a movement toward another glowworm in the neighborhood. The
luciferin level depends on the fitness of each glowworms’ location, which
is evaluated by using the objective fitness function [7,9]. The introduction
of optimization mechanism is given as follows.

Definition 2.1. Update luciferin:

l i (t) = (1 − ρ)l i (t − 1) + γJ(x i (t)) (2.7)

where l i (t), l i (t − 1) are the luciferin level of glowworm i at iteration t and
t − 1; x i (t), J(x i (t)) are the position and objective fitness function value
associated with position x i (t) of glowworm i at iteration t; ρ is the luciferin
decay constant (0 <ρ < 1) and γ is the luciferin enhancement constant
(0 <γ < 1).