Page 107 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

other estimation schemes should be incorporated into the LS method [14].
Different to this idea, in the next subsection, we will incorporate the pa-
rameter estimation of D f into adaptive control design, where an adaptive
law is proposed to online update the estimate of D f by minimizing the
tracking control error.

6.4 ADAPTIVE CONTROL WITH FRICTION COMPENSATION
AND STABILITY ANALYSIS

To achieve output tracking of system (6.1) and the online compensation for
friction, we define a filtered tracking error as

e v =˙e + γe (6.8)

where γ> 0 is a constant.
Then by differentiating e v and using (6.1), the error dynamics can be
written as

M(x)˙e v = τ − C(x, ˙x)e v − H(˙x d , ¨x d ,e, ˙e) (6.9)
where H(x d , ˙x d , ¨x d ,e, ˙e) = M(x d +e)(¨x d −γ ˙e)+C(x d +e, ˙x d +˙e)(˙x d −γe)+T f
defines the lumped unknown non-linearities, which includes the friction
T f and other dynamics of e,x d .
Because the desired velocity ˙x d and acceleration ¨x d are uniformly
bounded, we know that the function H(x d , ˙x d , ¨x d ,e, ˙e) is also piecewise
continuous. Hence, it can be taken as a generalized and augmented friction
torque, which can be approximated by using the above mentioned DPPR.
Thus, the ideal discontinuous piecewise parametric representation can be
used to estimate H(x d , ˙x d , ¨x d ,e, ˙e),which isgivenby
∗T
H(χ) = D φ(χ) +
(6.10)
where χ =[x d , ˙x d , ¨x d ,e, ˙e] is the input of the augmented DPPR, and
D ∈ R L×n is the optimal augmented coefficient vector, L is the number of
∗
subintervals used in the DPPR formulation, φ(χ) ∈ R L×1 is the regressor
function,
is the approximation error satisfying
è ε with ε> 0[13].
∗
∗
∗
The ideal parameter vector D is assumed to be bounded by D è L on
the compact set ,where L > 0 is an unknown constant.
∗
The control of system (6.9)isnowgivenby
T
τ =−K pe v + D (t)φ(χ) (6.11)
where K p > 0 is the feedback gain, and D(t) is the estimate of D at time t.
∗