Page 206 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ESO Based Adaptive Sliding Mode Control of Servo Systems With Input Saturation 205

The objective is to design a control u such that the system output θ m can
track a given desired trajectory, while all signals in the closed-loop system
are bounded.

13.2.2 Saturation Model
Similar to the discussion presented in Chapter 12 and shown in Fig. 12.3,
the saturation function sat(u) can be approximated by the following hyper-
bolic tangent function as [11]

g(u) = v max × tanh( u ) = v max × e u/vmax −e −u/vmax (13.4)
v max e u/vmax +e −u/vmax
Then, based on the statements in Chapter 12, the saturation formulation
(13.2) can be reexpressed as

v(u) = sat(u) = g(u ξ )u + d 1 (u) (13.5)

where g(u ξ ) is a function of intermediate variable u ξ , d 1 (u) = sat(u) − g(u),
and d 1 (u) satisfies

(13.6)
|d 1 (u)|≤ D 1

where D 1 = v max (1 − tanh(1)) is the maximum value of d 1 (u).
Substituting (13.5)into(13.3)yields

⎧
dθ m
⎪
⎨ = ω m
dt (13.7)
dω m K t g u ξ D 1 K t d 1 (u)
= u − ω m − T f +
⎪
⎩
dt J J J J
As shown in (13.7), the control input u is in an affine form with a
for any u ξ and a bounded disturbance d 1.Hence,the
bounded gain g u ξ
control design to be presented in this chapter is to design a non-linear ESO
to estimate and compensate for the friction T f , the saturation approxima-
tion error d 1 (u) and other external disturbance. Based on the estimated
dynamics, an adaptive sliding mode controller is designed to guarantee the
system states θ m and ω m track the desired given trajectory.