Page 200 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

By using the mean-value theorem, for any u 0 there exists a constant
0 <ξ < 1, such that

(u − u 0 ) (12.7)
g(u) = g(u 0 ) + g u ξ
∂g(u)
= is a bounded function of u ξ given by u ξ = ξu + (1 −
∂u
where g u ξ | u=u ξ
ξ)u 0.
Specifically, when choosing u 0 = 0, we can obtain g(u 0 ) = 0, and thus
the approximated function g(u) can be represented in a linear form as

u (12.8)
g(u) = g u ξ
Hence, the saturation dynamics (12.5) can be described by

u + d 1 (u) (12.9)
v(u) = sat(u) = g u ξ
Clearly, one can find from (12.9) that the saturation can be mathemat-

ically formulated as a linear-like system of u with time-varying gain g u ξ
and a bounded disturbance d 1 (u). This new description is more suitable
for control design and implementation, in particular for adaptive control of
non-linear systems, and thus will be used in the subsequent control designs.

12.4 EXAMPLES WITH SATURATIONS

The actuator saturation is a kind of non-smooth non-linearities encoun-
tered in the control designs due to the physical limit of actuators. In this
section, we briefly introduce several typical control systems with satura-
tions.

12.4.1 Active Micro-Gravity Isolation System

As explained in [10], the actuator force used in an active micro-gravity
isolation system is limited by saturation. The schematic of this system can be
found in [2]. The control objective is to achieve a level of isolation between
the base acceleration and the inertial acceleration of the isolated platform.
The isolated platform must operate in a limited rattle space. Hence, to
prevent the platform from bumping into its hard stops, an additional design
constraint is that the relative displacement between the base acceleration
and the isolated platform acceleration should not exceed a given limit.