Page 199 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Saturation Dynamics and Modeling 197

Figure 12.3 Saturation sat(u) (solid-line) and smooth function g(u) (dot-line).

Note that the saturation formulation given in (12.2) can cover those speci-
fied linear, symmetric actuator saturation considered in [2,3]. In ideal case
with unity ratio m = 1 and symmetric amplitude limits |v min |=|v max |,then
the saturation dynamics can be further rewritten as

v maxsgn(u), |u|≥ v max
v(u) = sat(u) = (12.3)
u, |u| < v max

12.3 SATURATION APPROXIMATION

The above non-smooth saturation dynamics (12.1)or(12.3) cannot be di-
rectly used in the control design and synthesis, in particular for adaptive
control. Hence, to facilitate adaptive control design, we will introduce a
smooth approximation of such saturation dynamics as [2]. As shown in
Fig. 12.3, the non-smooth saturation behavior can be approximated by a
smooth function tanh(·), such that

u e u/vmax −e −u/vmax
g(u) = v max × tanh( ) = v max × (12.4)
v max e u/vmax +e −u/vmax
Consequently, Eq. (12.1) can be rewritten as

v(u) = sat(u) = g(u) + d 1 (u) (12.5)

where d 1 (u) = sat(u) − g(u) is a bounded function satisfying

(12.6)
|d 1 (u)|=|sat(u) − g(u)|≤ v max (1 − tanh(1)) = D 1

with D 1 > 0 being a bounded positive constant.