Page 33 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Table 2.2 The parameters for GSO algorithm
l 0 γ β n t s ρ r s m iter
Static parameter settings 5 0.6 0.08 5 0.001 0.4 0.5 50 2000000
Dynamic parameter settings 5 0.6 0.08 5 0.001 0.4 10 50 2000000

Table 2.3 The identification result of static parameters
Parameters f c (Nm) f s (Nm) σ 2 (Nm s/rad) ω s (rad/s)
True value 0.28 0.34 0.01 0.2
Estimate value 0.2736 0.3138 0.0097 0.2885

In (2.12), the static parameters σ 2, f c , f s,and ω s can be obtained by using
GSO algorithm based on the measurable data of the input torque T and the
output velocity ω. To simplify the notation, the following estimated vector
ˆ v m is defined

ˆ ˆ T
ˆ v m =[ f c f s ˆ σ 2 ˆ ω s ] , (2.13)
where m = 1,2,...,M; M is the size of glowworms’ population, and ˆv m is a
set of estimated parameters represented by the m-th glowworm.
Then the estimation of the friction T f is obtained by

ˆ i
ˆ
T =[f c + (f s − f c )e −(ω i ˆω s ) 2 ]sgn(ω i ) +ˆσ 2 ω i (2.14)
ˆ
ˆ
f _m
where i = 1,2,...,N is the number of selected velocity signals, T ˆ i denotes
f _m
the estimation of friction torque corresponding to the i-th velocity signal
of the m-th glowworm.
The fitness function of the static parameters is chosen by
N
1 i 2
J m = (e s_m ) , m = 1,2,...,M (2.15)
2
i=1
ˆ i
i
where e i s_m = T f _m − T f _m represents the static identification error for the
i-th velocity signal of the m-th glowworm, and T i represents the friction
f _m
torque for the i-th velocity signal of the m-th glowworm.
Hence, identifying the optimal static parameters of LuGre model is
equivalent to minimization of the objective function J m. Following the
above mentioned implementation steps in Table 2.1 and parameters in Ta-
ble 2.2, we carried out numerical simulations. The identification result of
static parameters in the LuGre model is given in Table 2.3, and the Stribeck
curve is shown in Fig. 2.1, which shows very satisfactory performance.