Page 69 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

4.2.3 Neural Network Approximation
To approximate unknown non-linearities, a neural network (NN) with a
single hidden layer [18]isusedoveracompactset as
T
Q(Z) = W (Z) + ε, ∀Z ∈ ⊂ R n (4.4)

where Q(Z) is the unknown function to be approximated, W =[w 1 ,w 2 ···
L
T
w L ] ∈ R is the bounded NN weight vector and ε ∈ R is a bounded
approximation error, i.e., ||W || ≤ W N , |ε|≤ ε N with W N ,ε N being posi-
∗
T
L
tive constants. (Z) =[ 1 (Z),··· , L (Z)] ∈ R is the NN basis vector.
In this chapter, a high-order neural network (HONN) [20], [12]with
d k (j)
basis functions k (Z) = [σ(Z j )] ,k = 1,...,L is used with J k being
j∈J k
collections of L-non-ordered subsets of {0,1,...,n},and d k (j) being non-
negative integers. σ(·) is a sigmoid function σ(x) = a/(1 + e −bx ) + c for
+
∀a,b ∈ R ,c ∈ R, where the positive parameters a,b and real number c are
the bound, slope, and the bias of sigmoidal function, respectively.

4.3 ADAPTIVE PRESCRIBED PERFORMANCE CONTROL
DESIGN
4.3.1 Prescribed Performance Function and Error Transform
To study the transient and steady-state performance of tracking error e(t) =
y(t) − y d (t), a positive decreasing smooth function μ(t) : R → R with
+
+
lim μ(t) = μ ∞ > 0 will be used as the prescribed performance function
t→∞
(PPF). Similar to Chapter 2, we select μ(t) as
μ(t) = (μ 0 − μ ∞ )e −κt + μ ∞ (4.5)

where μ 0 >μ ∞ and κ> 0 are design parameters.
Then as presented by [13]and [14], it is sufficient to achieve the control
objective if the following condition (4.6) holds:

¯
−δμ(t)< e(t)< δμ(t), ∀t > 0 (4.6)
where δ,δ> 0 are constants selected by the designer.
¯
In (4.5)and (4.6), we know that δμ 0 defines the upper bound of the
¯
maximum overshoot and −δμ 0 defines the lower bound of the undershoot,
the decreasing rate κ introduces a lower bound on the convergence speed,
and μ ∞ denotes the allowable steady-state tracking error [14]. Hence the