Page 74 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Servo Systems With Continuously Differentiable Friction Model 65

ii) We have proved that the transformed error z 1 is bounded, i.e.,
z 1 ∈ L ∞. Then according to the properties of function S(z 1 ), we know that
−δ < S(z 1 )< δ, which further implies −δμ(t)< e(t)< δμ(t) according to
¯
¯
(4.7). Then one can conclude based on Lemma 4.1 that the tracking control
of system (4.2) with prescribed error performance (4.6)isachieved.

4.3.3 Practical Implementation
The implementation of the proposed APPC can be implemented step-by-
step as:
1) Determine the parameters k 1, , a, σ 1 ,σ 2 ,σ 3,and μ 0 ,μ ∞ ,κ,δ,δ,and
¯
ˆ
initial condition θ(0) ≥ 0, ˆε(0) ≥ 0;
2) Derive the tracking error e = y − y d and z 1 , ˙z 1 based on (4.10)–(4.11),
and obtain the filtered error (4.13);
3) Calculate the practical control effort u according to (4.18), and update
the adaptive parameters based on (4.19)–(4.20);
4) Apply the derived control on the realistic system and record the in-
put/output measurements;
5) Go back to Step 2) for next sampling interval.
In practical applications, a preliminary parameter tuning procedure
needs to be conducted. All parameters can be taken into two groups: 1) the
PPF parameters μ 0 ,μ ∞ ,κ,δ,δ mainly determine the control performance
¯
and can be selected offline; 2) the control parameters k 1 , , a and σ 1 ,σ 2 ,σ 3
are determined online based on a trial-and-error method to stabilize the
transformed error system (4.14). Here, the detailed tuning guidelines for all
parameters are summarized as:
1) The PPF parameters μ 0, δ,and δ should be selected to ensure −δμ(0)<
¯
e(0)< δμ(0), which depend on the initial condition of systems, i.e., we
¯
set them adequately large in the initial phase, and then decrease them if
possible.
2) The parameter κ determines the tracking error convergence speed and
thus can be set small at the beginning and then increased via a trial-and-
error method. μ ∞ defines the final steady-state error bound, which can
be set large initially and then reduced in the subsequent tuning. The fi-
nal choice of these two parameters should make a tradeoff between the
demand of users and the realistic system operation conditions. In gen-
eral case, small μ ∞ and large κ will obtain good tracking performance
but in the cost of large control actions.
3) A large k 1 will lead to faster error convergence, while the resulting
control action may be oscillated. High gains , a will improve the pa-