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ANDSC of Strict-Feedback Systems With Non-linear Dead-Zone 139

9.3 CONTROL DESIGN AND STABILITY ANALYSIS
In this section, the dynamic surface control (DSC) technique originally
proposed in [8,9] is incorporated into the neural control design for system
(9.1) such that the “explosion of complexity” and extra assumptions caused
by the repeated differentiation of the virtual control in the traditional back-
stepping design [6,2] can be removed. For notation conciseness, the time
variable t will be omitted except appearing with an unknown time-varying
delay as x(t − τ ij (t)),and g i will denote the unknown control gain function
g i (¯x i (t), ¯x i (t − τ ij (t))).

9.3.1 Adaptive Neural Dynamic Surface Control

Define the coordinate transformations as: z 1 = x 1 −y d and z i = x i −s i−1 ,i =
2,··· ,n,where s i−1 is the output of a first order filter with the input α i−1
as

s
μ i ˙ i + s i = α i ,s i (0) = α i (0),i = 1,··· ,n − 1, (9.6)
where μ i is the constant filter parameter, α i is the intermediate control for
the i-th sub-system designed later.
The major difference to conventional backstepping methods [6,2]is
to replace, at each recursive step, ˙α i−1 by ˙s i−1 in determining the virtual
control α i . As a result, the differentiation operation ˙α i−1 can be replaced by
asimpler filter(9.6), and thus the assumption that ˙x i should be measurable
[1,2,4] is removed.
Define the filter errors e i as

e i = s i − α i , i = 1··· ,n − 1. (9.7)

Step 1. Consider the definition z 1 = x 1 − y d and z 2 = x 2 − s 1,thenfrom
(9.4)and (9.7), it follows

˙ z 1 =˙x 1 −¨y d = f 1 (x 1 ,0) + h 1 (x 1 ,x 1 (t − τ 1j (t))) + g 1 (x 1 ,x 1 (t − τ 1j (t)))x 2 −¨y d
(9.8)

The adaptive virtual control α 1 can be specified for the first subsystem as

ˆ
θ 1 T z 1
α 1 =−k 1z 1 − z 1 (Z 1 ) 1 (Z 1 ) − ε 1 tanh( ), (9.9)
1
2 ω 1
˙
1 2 T
ˆ
ˆ
θ 1 = [z (Z 1 ) 1 (Z 1 ) − σ 1 θ 1 ], (9.10)
1
1
2