Page 102 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Control for Manipulation Systems 95

Figure 6.1 The proposed control scheme.

trajectory x d , and the unknown friction torque T f can be modeled and
compensated by using the idea of feedforward control.
Fig. 6.1 shows the overall control structure for tracking and friction
feedforward compensation, where C F is the feedforward compensator to
address the unknown friction. Hence, the first objective of this chapter is
to design a feedforward compensator C F by introducing a new discontinu-
ous piecewise parametric representation (DPPR) of friction, which can be
incorporated into the feedback control systems.
n
We denote the desired position as x d ∈ R and the position tracking
error as e = x − x d . Without loss of generality, it is assumed that the desired
trajectories x, ˙x d and ¨x d are bounded [11].

6.3 MODELING AND IDENTIFICATION OF FRICTION
6.3.1 Discontinuous Piecewise Parametric Representation
(DPPR)
This chapter will further tailor the idea of continuous piecewise-linear neu-
ral networks reported in [13] to address the dynamics of frictions. Hence,
the idea of this configuration of the parameterized model for a piecewise
linear function will be introduced first. A canonical continuous represen-
tation was discussed in [13] for an arbitrary continuous piecewise linear
function in any dimension, in which the coefficients appear linearly and
can be determined effortlessly using the least squares methods. There is no
stringent assumptions on the unknown functions in this formulation. We
refer to [13] for more theoretical analysis of the configuration of continuous
piecewise-linear neural networks. Here, only the necessary presentation is
provided.
Given an arbitrary continuous piecewise linear function f (x) of x ∈ R,
whichisdefinedonthe interval I =[¯x,x]. We can divide I into N non-
overlapping subintervals I r (r = 1,··· ,N) and I r = I. Then, the func-
1≤r≤N