Page 48 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Control of Two-Inertia Systems With LuGre Friction Model 39

The objective is to design a feedback control such that the tracking error
converges to a prescribed bound, while all signals in the closed-loop system
are bounded.
To facilitate the control design, we reformulate two-inertia system (3.1)
as

J m ˙ω m + J l ˙ω l = u − f (3.2)

where T f = f m + f l defines the combined friction force of the two-inertia
system.
Hence, we can denote the lumped unknown dynamics in (3.1)as F =
−J l ˙ω l + T f , which includes the frictions and the unknown dynamics, and
will be compensated on the motor side [27]. Moreover, it is noted that
the damping coefficient b f can be ignored due to the fact k f b f .Inthis
T
case, we can choose state vector and input variables as x =[ω l ,m s ,ω m ] =
T
[x 1 ,x 2 ,x 3 ] , z =−F,thenEq. (3.1)with(3.2)can bewrittenasfollows:
˙ x = Ax + Bu + Bz
(3.3)
y = Cx

1 ⎛ ⎞
⎛ ⎞
0 0 0 ⎛ ⎞
⎜ ⎟ 0
J l
⎜ ⎟ ⎜ 0 ⎟
⎜
⎟
where A = ⎜ −k f 0 ⎟ , B = ⎜ ⎟ , C = ⎝ 0 ⎠.
⎜ k f ⎟ ⎝ 1 ⎠
1
⎝ ⎠ 1
0 0 J m
J m
The above formulation is particularly suited for observer and control de-
sign. In fact, the frictions in the motor and load sides of two-inertia system
asshowninFig. 3.1, e.g., f m and f l , are lumped as an entire friction force
f in (3.2), which allows to compensate the effect of frictions on the mo-
tor side only by using the control action applied on the motor. Moreover,
this formulation also allows to consider the friction together with other
unknown dynamics in F, facilitating the subsequent observer and control
designs.
In order to describe the characteristics of the friction T f ,LuGre model
reported in [26] will be used, which is derived by using an internal friction
state z as
T f = σ 0z + σ 1 ˙z + σ 2 ω
|ω|z
˙ z = ω − (3.4)
g(ω)
2
2
g(ω) = f c + (f s − f c )e −(ω /ω s )