Page 56 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Control of Two-Inertia Systems With LuGre Friction Model 47

Step 3: In the final step, the controller u will be obtained based on the
error variable
s 3 = x 3 −¯χ 2 . (3.36)

Thetimederivativeof z 3 is obtained by

˙ μ 3 ˙ μ 3

˙
˙ z 3 = r 3 ˙ s 3 − s 3 = r 3 x 3 − ¯χ 2 − s 3
μ 3 μ 3 (3.37)
1 1 1
˙ μ 3
˙
= r 3 − ˆ x 2 + u − F − ¯χ 2 − s 3
J m J m J m μ 3
where r 3 = (1/2μ 3 )[1/(ρ 3 + δ ) − 1/(ρ 3 − δ 3 )],and ρ 3 = s 3 /μ 3.
¯
3
Again, let ¯χ 2 go through the following HGTD

˙
ϑ 1,3 = ϑ 2,3 (3.38)
α
ϑ 2,3 = H 2 − ρ 1,3 [ϑ 1,3 −¯χ 2 ] − ρ 2,3 [ϑ 2,3 /H] β
˙
where ρ 1,3 and ρ 2,3 are the design parameters.
Then substituting (3.38)into(3.37), we have
1 1 1 ˙ μ 3
˙ z 3 = r 3 u − ˆ x 2 − F − ϑ 2,3 − s 3 (3.39)
J m J m J m μ 3

Finally, the control signal u is chosen to be

˙ μ 3

u = J m − k 3z 3 + ϑ 2,3 + s 3 +ˆ x 2 + ˆ F (3.40)
μ 3
where k 3 > 0 is a design parameter, and ˆ F is the estimation of unknown
dynamics F, which will be designed in the following subsection.

3.3.3 Friction Compensation With ESN

To compensate for the effect of ˆ F including friction T f ,anESNwill be
used. Specifically, to address the friction dynamics, we define = z − z 0
and z 0 = g(ω)sgn(ω) [33], such that:

F = T f − J l ˙ω l = σ 0z + σ 1 ˙z + σ 2 ω m − J l ˙ω l
= σ 2 ω m + f c + (f s − f c )e −(ω m /ω s ) 2 sgn(ω m )

(3.41)
1

+ σ 0 1 − 2 |ω m | − J l ˙ω l .
f c + (f s − f c )e −(ω m /ω s )