Page 230 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

put saturation function. Then, a neural dynamic surface control is designed
to achieve output tracking, in which an NN with single hidden layer is
employed for approximating the lumped uncertainties. Hence, the complex
and tedious backstepping design procedure can be avoided and the associ-
ated ‘complexity explosion’ problem is remedied. Simulations are given to
show the validity of the proposed scheme.

15.2 PROBLEM FORMULATION AND PRELIMINARIES
15.2.1 System Description
Consider a class of non-linear systems in the following pure-feedback form:

⎧
⎪ ˙ x i = f i (¯ x i ,x i+1 ) , 1 ≤ i ≤ n − 1
⎨
˙ x n = f n (¯x n ,v(u)) (15.1)
⎪
⎩
y = x 1
T i
where ¯x i = [x 1 ,...,x i ] ∈ R is the vector of system states, and ¯x n =
T
n
[x 1 ,...,x n ] ∈ R ; f i (·), i = 1,··· ,n − 1, are unknown smooth functions of
(x 1 ,...x i+1 ) satisfying f i (0,...,0) = 0; y ∈ R is the system output; v(u) ∈ R
is the control input subject to the following saturation non-linearity as

v maxsgn(u), |u|≥ v max
v(u) = sat(u) = (15.2)
u, |u| < v max
where v max is a positive but unknown constant denoting the maximum
actuator power.
The dynamics of input saturation is shown in Fig. 12.3, where the con-
trol input v(u) ∈ R is the output of the saturation and u(t) ∈ Ris the input
of the saturation (controller output). Then following the discussion pre-
sented in Section 12.3 in Chapter 12, the saturation can be approximated
by a smooth non-affine function defined as [10]

g (u) = v max × tanh u = v max × e u/v max −e −u/v max (15.3)
v max e u/v max +e −u/v max
Then, the saturation dynamics v(u) = sat(u) in (15.2) can be expressed
as

u + d 1 (u) (15.4)
v (u) = sat(u) = g (u) + d 1 (u) = g u ξ