Page 160 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

functions g(x) are strictly positive or negative. Hence, these methods are
not suitable for systems with unknown control gain directions.
This chapter focuses on the adaptive tracking control design for a
class of non-linear systems with an unknown non-linear dead-zone in-
put and time-delays. The main idea is to further tailor the principle of
prescribed performance control (PPC) that has been introduced in the
previous chapter of this book for the studied systems. After representing
the non-linear dead-zone as a linear time-varying system with a bounded
disturbance term, we can lump the dead-zone dynamics into unknown sys-
tem dynamics. The unknown control directions and non-linear dead-zone
are also handled by means of Nussbaum-type function [19]. By employ-
ing a prescribed performance function (PPF) as [21,22], an output error
transformed system is then derived. Consequently, the tracking error con-
vergence within prescribed bound of the original system can be guaranteed
provided the transformed error system is stable. To achieve this, an adaptive
control derived based on backstepping is designed so that both the transient
and steady-state tracking error performance including the convergence rate
and maximum overshoot of original system are all ensured. To accommo-
date unknown non-linearities, high-order neural networks (HONNs) [23]
with a simpler structure are established, where only a scalar parameter, in-
dependent of the number of hidden nodes in the neural network [15], is
updated online.

10.2 PROBLEM FORMULATION AND PRELIMINARIES

Consider the following non-linear systems with dead-zone input and time-
delays

⎧
⎪ ˙ x i = f i (¯x i ) + g i (¯x i )x i+1 + h i (t, ¯x i (t − τ i (t))), 1 ≤ i ≤ n − 1
⎨
˙ x n = f n (x) + g n (x)u + h n (t,x(t − τ n (t))) (10.1)
⎪
y = x 1
⎩
n
T
i
T
where ¯x i =[x 1 ,x 2 ···x i ] ∈ R ,i = 1,···n, x =[x 1 ,x 2 ···x n ] ∈ R , y(t) ∈ R
are the system state and output, respectively; f i (·),g i (·),h i (·),i = 1,···n are
] are unknown time-
unknown smooth functions; and τ i (t) =[τ i1 ,τ i2 ,···τ im i
varying delays, which fulfill τ ij (t) ≤ τ im and ˙τ ij (t) ≤¯τ i < 1,i = 1,···n,j =
1,···m i with τ im, ¯ i being positive constants. The real control u(t) ∈ Ris
τ