Page 31 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Definition 2.2. The probability of glowworm moving toward a neighbor
is given by
l j (t) − l i (t)
(2.8)
(l k (t) − l i (t))
p ij =
k∈N i (t)
i
where j ∈ N i (t), N i (t) ={j : d ij (t)< r (t);l i (t)< l j (t)} is the set of neighbors
d
of glowworm i at the iteration t; d ij (t) is the distance between glowworms
i
i and j; r (t) represents the dynamic decision region radius of glowworm i,
d
i
and 0 < r (t) ≤ r s,where r s is the perception maximum radius.
d
Definition 2.3. Update position:
x j (t) − x i (t)
(2.9)
x i (t + 1) = x i (t) +
x j (t) − x i (t)
where x i (t + 1) is the position of glowworm i at iteration t + 1.
Definition 2.4. Update decision region radius:

i

i
r (t + 1) = min r s ,max 0,r (t) + β(n t − |N i (t)|) (2.10)
d
d
i
where r (t + 1) is the dynamic decision region radius of glowworm i at
d
iteration t + 1; β is the changing rate of neighborhood; n t is a threshold to
control the number of neighbors of glowworm i.
The implementation procedure of the GSO algorithm is presented in
Table 2.1, and the parameters of the GSO are given in Table 2.2,inwhich
l 0 is the value of initial luciferin; m is the size of the glowworms population,
and iter is the maximum number of iterations.

2.3.2 Static Parameters Identification
When system is running at the high speed, the static parameters f c ,f s ,ω s can
describe the steady-state characteristics, such that ˙z ≈ 0and ˙ω ≈ 0 hold. In
this case, (2.4) can be reduced to
z = g(ω)sgn(ω) (2.11)

Substituting (2.3)–(2.5)and (2.11)into(2.1), and ignoring the system
disturbance T d , we can get the relationship between the steady-state input
torque and the friction torque as:
2
T ≈ T f =[f c + (f s − f c )e −(ω ω s ) ]sgn(ω) + σ 2 ω (2.12)