Page 36 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Sliding Mode Control of Non-linear Servo Systems With LuGre Friction Model 27

where α> 0, υ> 0 are the tuning parameters, and exp(·) is the exponential
function.
The constant N denotes the linear part of the non-linear surface (2.23)
chosen to guarantee that (a 11 − a 12N) is negative, in which its amplitude
determines the damping ratio to achieve fast response; P and R are positive
constants satisfying

T
(a 11 − a 12N) P + P(a 11 − a 12N) =−R (2.25)

According to (2.23), we define an auxiliary variable

˙
δ = θ ref − (N − ψ(θ)P)e = θ ref − λe. (2.26)
˙
From (2.1), (2.21), (2.23), and (2.26), we can obtain

s = ω − δ (2.27)

Since the bristle deformation z is an immeasurable variable, we design
an observer by using the tracking error s as

˙ |ω|
ˆ z 0 = ω − ˆ z 0 − k 0s
g(ω) (2.28)
˙ ˆ z 1 = ω − |ω| |ω| s
g(ω) ˆ z 1 + k 1 g(ω)
where ˆz 0, ˆz 1 are the estimation of z,and k 0 > 0, k 1 > 0 are constant ob-
server gains.
Let ˆσ 0, ˆσ 1, ˆσ 2, ˆ J and T d be the estimation of the parameters σ 0, σ 1, σ 2,
ˆ
J and T d . Combining the system dynamics in (2.1) and dual-closed loop
observer (2.28), we can design a controller as

|ω|
ˆ
ˆ
T =−ks +ˆσ 0 ˆz 0 − σ 1 ˆ z 1 + ζè + T d + ˆ Jδ ˙ (2.29)
g(ω)
ˆ
where k > 0 is a constant feedback gain and ζ =ˆσ 1 +ˆσ 2.
Substituting (2.27)and (2.29)into(2.1), we have

˙ |ω| |ω| ˜ ˜ ˙
J˙s = J ˙ω − Jδ =−ks +˜σ 0 ˆz 0 + σ 0 ˜z 0 −Èσ 1 ˆ z 1 − σ 1 ˜ z 1 + ζè + T d + ˜ Jδ
g(ω) g(ω)
(2.30)
where ˜σ 0 =ˆσ 0 − σ 0, ˜σ 1 =ˆσ 1 − σ 1, ˜z 0 =ˆz 0 − z 0, ˜z 1 =ˆz 1 − z 1, ζ = ζ − ζ,
˜
ˆ
ˆ
˜
T d = T d − T d , ˜ J = ˆ J − J are the estimation errors.