Page 182 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Output Feedback Control of Pure-Feedback Systems 179

of generality, the dead-zone parameters, b r and b l , are unknown bounded
constants, and their signs are known.
To accommodate the dead-zone dynamics in system (11.1), as detailed
in Chapter 7, the non-linear dead-zone model (11.2) can be reformulated
as

u(t) = (χ l (t) + χ r (t))v(t) + ρ(t) = d(t)v(t) + ρ(t) (11.3)
where the definitions of χ l (t),χ r (t),d(t),ρ(t) can be found in (7.10)–(7.11).
Moreover, it is verified that = min(d l0 ,d r0 ) ≤ d(t) ≤ d l1 + d r1 and |ρ(t)| ≤ p
with positive constants 0 < < +∞ and p = (d l1 + d r1 )max{b r ,−b l },where
the scalars d l0, d l1, d r0, d r1,and , p are only used for analysis. We refer to
Chapter 7 for more details of dead-zone reformulation.
From (11.3), it can be seen that the first part of u in (11.3) is a time-
varying gain, and the second term can be taken as a bounded disturbance,
which can be handled by ESO [11,12] to be designed in the following
section.
The objective of this chapter is to design an adaptive controller v(t) for
system (11.1), such that all signals involved in the closed-loop system are
bounded, and the tracking error e 1 = x 1 − x d for a given desired trajectory
x d can be guaranteed.

11.3 COORDINATE TRANSFORMATION AND OBSERVER
DESIGN

11.3.1 Coordinate Transformation
To facilitate the control design, we will apply the mean-value theorem on
the pure-feedback system (11.1) to reformulate it into a strict-feedback
form, which allows to tailor a coordinate transformation in [8]torepresent
the system as a canonical form.
According to [1,3], the functions f i (·,·) in (11.1) can be represented by
using the mean-value theorem as

f i (¯x i ,x i+1 ) =f i (¯x i ,x 0 ) + ∂f i (¯x i ,x i+1 ) | λ i ,×(x i+1 − x 0 ),1 ≤ i ≤ n − 1
i+1 ∂x i+1 x i+1 =x i+1 i+1
0 ∂f n (¯x n ,u) 0
f n (¯x n ,u) =f n (¯x n ,u ) + | u=u n (u − u ) (11.4)
λ
∂u
where x λ i = λ ix i+1 + (1 − λ i )x 0 ,with 0 <λ i < 1,1 ≤ i ≤ n − 1, and u =
λ n
i+1 i+1
0
λ nu + (1 − λ n )u ,with 0 <λ n < 1.