Page 218 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Non-singular Terminal Sliding Mode Funnel Control of Servo Systems With Input Saturation 217

14.2.2 Saturation Model
As showninFig. 12.2, the relationship between the realistic control action
v(t) and the controller output u(t) has a sharp corner when |u(t)|= v max.
Hence, as presented in Chapter 12 and shown in Fig. 12.3, this saturation
can be approximated by a smooth non-affine function defined as

g(u) = v max × tanh( u ) = v max × e u/vmax −e −u/vmax (14.4)
v max e u/vmax +e −u/vmax

In this case, the saturation output v = sat(u) in (14.2) can be expressed
in the following form

(u)u + d 1 (u) (14.5)
v(u) = sat(u) = g(u) + d 1 (u) = g u ξ
is a bounded gain for any u ξ ,and d 1 (u) = sat(u) − g(u) is a
where g u ξ
bounded function and its bound can be obtained as

|d 1 (u)|=|sat(u) − g(u)|≤ v max (1 − tanh(1)) = D 1
where D 1 is the upper bound of |d 1 (u)|.
Then, we know that the system (14.3) can be presented as

⎧
⎪ ˙ x 1 = x 2
⎨
˙ x 2 = h(x,t) + b 0u (14.6)
⎪
⎩
y = x 1
f (x,t)+d(x,t) k 0
where h(x,t) =− + d 1 is the lumped unknown dynamics, and
m m
k 0 g u ξ
b 0 = is the input gain.
m
14.2.3 Neural Network Approximation
In this chapter, the following neural network (NN) with one hidden layer
only will be used to approximate the unknown continuous function as

h(X) = W ∗T φ(X) + ε (14.7)

n
n
where W ∈ R is the ideal weight matrix, φ(X) ∈ R is the basis function
∗
of the neural network, ε is the NN approximation error bounded by |ε|≤
T
ε N for a positive constant ε N , φ(X) =[φ 1 (X),··· ,φ n (X)] can be chosen
as the commonly used sigmoid function vector, which is in the following