Page 205 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

can be estimated by designing an extended state observer (ESO). Inspired
by the pioneer idea of [7], many research results have been recently reported
concerning the application of ESO and the associated ADRC [8–10].
In this chapter, an adaptive sliding-mode control scheme based on the
ESO is proposed for an electro-mechanical servo system with unknown
friction and input saturation constraint. First of all, the non-smooth sat-
uration is transformed into a smooth affine function according to the
differential mean value theorem. Then the unknown friction, saturation
constraint, and external disturbance are estimated and compensated by us-
ing ESO. A pole placement technique is employed to determine the ESO
parameters. Finally, by combining an adaptive law and the sliding mode
control theory, an adaptive sliding mode control is designed to guarantee
that the system output can rapidly track a given desired trajectory with re-
duced chattering. Comparative simulation results are provided to show the
superior performance of the proposed method.

13.2 SYSTEM DESCRIPTION AND SATURATION MODEL

13.2.1 System Description
The model of electro-mechanical servo system is given by
2
d θ m dθ m
J + D + T f = K t v(u) (13.1)
dt 2 dt
where θ m is the motor position; J and D are the equivalent inertia and
damping coefficients on the motor shaft side; K t denotes the motor torque
constant; T f represents the friction torque and the external disturbance;
u is the control input to the actuators, and v is the output of the saturation
sat(u) given by

v maxsgn(u), |u|≥ v max
v(u) = sat(u) = (13.2)
u, |u| < v max
where v max is the maximum output power of the actuator. The dynamics
of input saturation can be found in Fig. 12.2.
dθ m
Define ω m = ,andthen (13.1) can be rewritten as
dt
⎧
⎪ dθ m
⎪ = ω m
dt
⎨
D 1 (13.3)
dω m K t
⎪
⎪ = v(u) − ω m − T f
dt J J J
⎩