Page 68 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Servo Systems With Continuously Differentiable Friction Model 59

T
T
as x =[x 1 ,x 2 ] =[q, ˙q] , then the dynamics of servo mechanism (4.1)can
be simplified as:
⎧
⎪ ˙ x 1 = x 2
⎨
1
˙ x 2 = K 1u − K 2x 2 − f (x 1 ,x 2 ) − T l − T d − T f (4.2)
J
⎪
⎩
y = x 1
where K 1 = K T /R a ,K 2 = K T K E /R a are positive constants.
The objective of this chapter is to derive a control u so that: 1) the
output y tracks a given reference y d , and all signals in the closed-loop are
bounded; 2) both prescribed transient and steady-state performance of the
tracking error e = y − y d are preserved. Without loss of generality, the po-
sition x 1 and velocity x 2 are measurable, and the reference y d , ˙y d , ¨y d are
bounded.

4.2.2 Continuously Differentiable Friction Model

As stated in Chapter 1, conventional friction models (e.g., [17], [4], [18],
and [19]) are discontinuous or piecewise continuous, which may be prob-
lematic for deriving smooth control actions [15]. Moreover, the identi-
fication of friction model parameters is not a trivial task. In this chap-
ter, a newly developed continuously differentiable friction model [15]is
adopted, where the friction torque T f can be presented as the following
parameterized form

T f = α 1 (tanh(β 1 ˙q) − tanh(β 2 ˙q)) + α 2 tanh(β 3 ˙q) + α 3 ˙q (4.3)

where α 1, α 2, α 3, β 1, β 2, β 3 are positive parameters.
Unlike other friction models, Eq. (4.3) has a continuously differentiable
form to allow more flexible in adaptive control designs. For further details,
we refer to [15] and Chapter 1. In the subsequent control design, fric-
tion model (4.3) will be incorporated into a neural network to address the
friction and other unknown dynamics simultaneously, where offline mod-
eling is successfully avoided. Moreover, the modeling error of the friction
model (4.3) can be lumped into the additive disturbance T d ,which will be
compensated in the control design.