Page 217 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

singular terminal sliding mode control (NTSMC) to make the tracking
error fall within a prescribed bound. Sliding mode control (SMC) and the
modified NTSMC have been widely used to deal with system uncertainties
and bounded disturbances [16–18]. Neural network (NN) is also used to
cope with other unknown system dynamics. Hence, no prior knowledge
of the input saturation bounds is required in the proposed method. The
effectiveness is demonstrated by simulation results.

14.2 PROBLEM FORMULATION AND PRELIMINARIES
14.2.1 System Description and Problem Formulation

The mechanical dynamics of the studied servo system can be described as
follows:
m¨x + f (x,t) + d(x,t) = k 0v(u)
(14.1)
y = x
2
T
where x =[x, ˙x] ∈ R , u(t) ∈ R, y ∈ R are the state variables, the con-
trol input voltage to the motor and the system output, respectively; x is the
angular position, m is the inertia, k 0 is a positive control gain (the force con-
stant), f (x,t) is the friction force; d(x,t) represents a bounded disturbance
including non-linear elastic forces generated by coupling and protective
covers, measurement noise, and other uncertainties. v(u) ∈ R is the con-
strained control input given by the following saturation non-linearity

v maxsgn(u), |u|≥ v max
v(u) = sat(u) = (14.2)
u, |u| < v max
where v max is the upper bound of the input saturation.
To facilitate the controller design, we define x 1 = x,and x 2 =˙x,then
the dynamics of the motor servo system in (14.2) can be rewritten in a
state-space form given by

⎧
⎪ ˙ x 1 = x 2
⎨
f (x,t)+d(x,t)
˙ x 2 =− + k 0 v(u). (14.3)
m m
⎪
⎩
y = x 1
The objective is to find a control u, such that the system output y can
track a given desired trajectory y d . Without loss of generality, the desired
position trajectory y d and its derivatives ˙y d , ¨y d are all bounded. Moreover,
the angular position and velocity, x 1 and x 2, are measurable.