Page 50 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Control of Two-Inertia Systems With LuGre Friction Model 41

Hence, the function f (x) can be expressed as
∗T
f (x) = X(x) + ε ∗ ∀x ∈ ⊂ R n (3.7)
0
∗ ∗ ∗
where ε is the ESN error fulfilling |ε |≤ ε m, is theideal valueof 0
0
that minimizes the approximation error ε . Therefore
∗

T
= arg min sup|f (x) − X(x)| (3.8)
∗
0 0
0 ∈R L×(K+N+L) x∈
Because is unknown, the estimation value ˆ of can be used, which
∗
∗
0
0 0
will be online updated to minimize the approximation error. Then, the
estimation error of ESN weight can be written as
˜ = 0 − ∗
ˆ
0 (3.9)
0
By setting C = 1,a = 1,G = 1, it can be obtained from (3.6)that
in out
X = ψ u + X + y (3.10)
T
when ˙ X = 0. In this chapter, we choose X(Z) =[ϕ 1 (Z),ϕ 2 (Z),...,ϕ l (Z)]
as Gaussian functions with l being the node number of ESNs output layer.
That is
(Z − ς) (Z − ς)

T
ϕ k (Z) = exp − (3.11)
η 2
T
with Z =[z 1 ,...,z i ] , i = 1,...,n being the number of input variables, ς and
η are the center and radius of the Gaussian function. For more details on
ESNs, we refer to [28].

3.2.3 Prescribed Performance Function
To study the transient and steady-state performances of tracking error e(t) =
+
+
[e 1 (t),e 2 (t),...e n (t)], a smooth decreasing function μ i (t) : R → R with
lim μ i (t) = μ i∞ will be used as the prescribed performance function (PPF).
t→∞
In this chapter, μ i (t) is given as
μ i (t) = (μ i0 − μ i∞ )e −κ i t + μ i∞ (3.12)

where μ i0 >μ i∞ and κ i are design parameters.
According to [29], the prescribed error performance is given as

−δ μ i (t)< e i (t)< δ i μ i (t),∀t > 0 (3.13)
i