Page 115 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

dead-zone-input systems is given in [3]. Moreover, a new smoothly ap-
proximated inverse of the dead-zone was suggested in [7], which can avoid
chattering problems that may occur in the non-smooth inverse approaches
when this new approximated dead-zone model is used in the backstepping
control design to compensate the effect of the dead-zone. To further ad-
dress the inherent non-linearities in the dead-zone characteristics, a novel
description of a general non-linear dead-zone model was also developed
in [8], where the potential non-linear behavior embedded in the actua-
tor dead-zone can be described by using non-linear functions. This new
model can be reformulated as a combination of a time-varying gain and a
bounded disturbance-like term as the linear counterpart in [3]byusing the
mean value theorem, and thus making the control system design possible
without necessarily constructing a dead-zone inverse [9,10].
In this chapter, we will introduce several well-known dead-zone mod-
els, which will be used in the subsequent control designs in this book. This
includes the linear dead-zone model [1] and also the generic non-linear
dead-zone model [8]. Moreover, several typical systems with dead-zone
input will also be briefly introduced.

7.2 DEAD-ZONE MODELS

Dead-zone is a static relationship DZ(·) between the actuator input v(t) and
the actuator output u(t), in which for a range of input values v(t) the output
u(t) is zero, while for the input v(t) is outside of this band, the output
u(t) appears and is a function of the input, where the slope between the
input and the output is constant (linear model) or time-varying (non-linear
model) [1]. Hence, these two different models will be introduced in this
section.

7.2.1 Linear Dead-Zone Model
The analytical expression of the linear dead-zone characteristic is given by

⎧
⎪ m r (v(t) − b r ) v(t) ≥ b r
⎨
u(t) = DZ(v(t)) = 0 b l < v(t)< b r (7.1)
m l (v(t) − b l ) v(t) ≤ b l
⎪
⎩
where v is the input and u is the output, b r ≥ 0,b l ≤ 0 are the break-points,
and m r ,m l > 0 are the slopes. Without loss of generality, it is assumed that