Page 139 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

CHAPTER 9

Adaptive Neural Dynamic Surface

Control of Strict-Feedback

Systems With Non-linear

Dead-Zone

9.1 INTRODUCTION

Apart from dead-zone dynamics presented in the previous chapters, time-
delays are also unavoidable in the control systems, such as process control
and teleoperation, which could bring phase lag and thus may trigger in-
stability in the control systems. To address the effect of time-delays in the
control systems, Lyapunov-Krasovskii functions have been utilized [1–3]to
deal with delays in the system states. A novel integral Lyapunov function
was introduced to avoid the control singularity in [1,4]. For systems with
unknown control coefficients and time-delays, Nussbaum type functions
were effectively used [5] to guarantee the error convergence.
On the other hand, backstepping [6,7] has been proved to be a
powerful technique to design controllers for various systems, e.g., strict-
feedback systems, pure-feedback systems, or triangular systems, etc. The
over-parameterized problem was also overcome by introducing tuning
functions [7]. However, in the backstepping design, the “explosion of com-
plexity” caused by the repeated differentiation of virtual control functions,
as pointed in [8], becomes more significant as the order of the system in-
creases. A novel idea named as dynamic surface control (DSC) [8,9]has
subsequently been investigated by introducing a first-order filter at each re-
cursive step of the backstepping design procedure. Moreover, to address the
unknown non-linearities, neural networks (NNs) have been incorporated
into the control design [10–13]. However, in most of available adaptive
neural backstepping (or DSC) controllers, the number of adaptive param-
eters to be tuned online, i.e., the NN weight as a vector or matrix, will
rapidly grow with the dimension of functions to be approximated [2].
This chapter focuses on adaptive neural tracking control for a class of
non-linear systems with an unknown non-linear dead-zone input and mul-
tiple time-varying delays. The mean-value theorem is first applied to derive
Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics.
DOI: https://doi.org/10.1016/B978-0-12-813683-6.00012-X 135
Copyright © 2018 Elsevier Inc. All rights reserved.