Page 235 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Neural Dynamic Surface Control for Pure-Feedback Systems With Input Saturation 235

Lemma 15.1 means that the equalities are kept in two-sliding mode,
and thus the states ˆz =[ˆz 1 , ˆz 2 ,··· , ˆz n ] of HOSM (15.21) can precisely esti-
mate unknown states z =[z 1 ,z 2 ,··· ,z n ] of system (15.17) with guaranteed
convergence in finite time. It is stated in [16] that the finite time conver-
gence of HOSM observer (15.21) makes it attractive in the control design
and synthesis since it allows the separation principle to be trivially fulfilled.
Moreover, the estimated error will be diminished by selecting sufficiently
large gains μ i in the HOSM observer. Hence, for the ease of notation sim-
plicity, we will use y,z 2 ,··· ,z n in the subsequent control designs.

15.3 SLIDING MODE DYNAMIC SURFACE CONTROL DESIGN
AND STABILITY ANALYSIS

15.3.1 Sliding Mode Dynamic Surface Control Design
In this section, we will adopt the dynamic surface control and integral
sliding mode techniques to design an adaptive control for the n-th order
system described by (15.17). Similar to the traditional backstepping design,
the recursive design procedure contains n steps. From step 1 to step n − 1,
the virtual controls z i+1, i = 1,··· ,n−1 are obtained, and an integral sliding
mode surface is proposed in the first step. Finally, the practical control u is
obtained at step n, where the saturation error d 1 will be addressed together
with a n.
Step 1: In this step, we consider the first equation of (15.17), i.e., ˙z 1 = z 2.
Then we define the tracking error and its sliding surface as

e = y − y d
(15.22)
s 1 = e + λ edt

where y d is the desired reference signal and λ is a positive constant.
The derivatives of e and s 1 are

˙ e =˙y −¨y d = z 2 −¨y d
(15.23)
s ˙ 1 =˙e + λe = z 2 −¨y d + λe

Choose a virtual control ¯z 2 as

¯ z 2 =−k 1s 1 +˙y d − λe (15.24)

where k 1 > 0 is a positive constant.