Page 140 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

a formulation of the perturbed non-linear dead-zone, such that it can be
taken into account together with other system non-linearities. The DSC
design is then extended to this general non-linear time-delay system such
that the differentiation calculation of the virtual control and the corre-
sponding “explosion of complexity” can be avoided. At each recursive
step, novel high-order neural networks (HONNs) with a simpler structure
and less adaptive parameters are established to approximate unknown non-
linear functions. Moreover, the control singularity problem and unknown
time-delays are handled by introducing an improved Lyapunov-Krasovskii
function including an exponential term. The salient features of the pro-
posed control are that, first, the conventional dead-zone inverse model
compensation is not needed to avoid the dead-zone identification [14]; sec-
ond, only two scalar parameters, independent of the number of NN hidden
nodes, are updated online at each step, and thus the computational burden
of the algorithm can drastically be reduced; third, some design difficulties
(e.g., control singularity, discontinuous control) are resolved without using
the information on the bounds of delayed functions and control functions.
Numerical simulations are given to verify above claims.

9.2 PROBLEM FORMULATION AND PRELIMINARIES
9.2.1 Problem Statement
Consider the following general non-linear systems with time-varying delays

⎧
⎪ ˙ x i (t) = f i (¯x i (t), ¯x i (t − τ ij (t))) + g i (¯x i (t), ¯x i (t − τ ij (t)))x i+1 (t)
⎪
⎪
⎪ .
.
⎨
. (9.1)
⎪ ˙ x n (t) = f n (x(t),x(t − τ nj (t))) + g n (x(t),x(t − τ nj (t)))u(t)
⎪
⎪
⎪
y(t) = x 1 (t)
⎩
T
i
n
T
where ¯x i =[x 1 ,x 2 ···x i ] ∈ R ,i = 1,···n, x =[x 1 ,x 2 ···x n ] ∈ R are the
i
system states and y(t) ∈ R is the output, f i (·),g i (·) : R → R ∈ C(s),i =
1,···n are unknown non-linear smooth functions of the corresponding
variables. The values of time-varying delays τ ij (t),i = 1,···n;j = 1,···m i
τ
are unknown and bounded by positive constants τ im and ¯ i , i.e. τ ij (t) ≤ τ im
and ˙ ij (t) ≤¯ i < 1. (Notice τ im and ¯ i are only used in the analysis.) The
τ
τ
τ
scalar u(t) ∈ R is the output of the following non-linear dead-zone
⎧
D r (v(t))
⎪ if v(t) ≥ b r
⎨
u(t) = DZ(v(t)) = 0 if b l <v(t)< b r (9.2)
D l (v(t)) if v(t) ≤ b l
⎪
⎩