Page 34 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Sliding Mode Control of Non-linear Servo Systems With LuGre Friction Model 25

Figure 2.1 Identified Stribeck curve for static parameters.

2.3.3 Dynamic Parameters Identification
Dynamic parameters σ 0 ,σ 1 describe the friction effect at the low velocity.
When the system begins to move, we have ˙z ≈ ω and z ≈ θ in (2.4). Hence,
the friction torque T f can be rewritten as

T f = σ 0 θ + (σ 1 + σ 2 )ω (2.16)

Substituting (2.16)into(2.1), we have

˙
T = Jθ + (σ 1 + σ 2 )θ + σ 0 θ (2.17)
¨
Similarly, define a parameter vector δ m for each glowworm individual as
T
δ m =[σ 0 ,σ 1 ,J] (2.18)
where m = 1,2,...,M and M is the size of glowworms’ population, δ m is a
set of values represented by the m-th glowworm.
The estimation of torque T is given by
ˆ i
θ + (ˆσ 1 + σ 2 )θ +ˆσ 0 θ
T = ˆ J ¨ ˙ (2.19)
m
ˆ i
where i = 1,2,...,N is the number of selected velocity signals, T denotes
m
i
the estimation of T for the i-th velocity signal of the m-th glowworm.
m
The fitness function of dynamic parameters is set as
N
1 i 2
J m = (e d_m ) , m = 1,2,...,M (2.20)
2
i=1