Page 161 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Strict-Feedback Systems With Non-linear Dead-Zone 157

the output of the following non-linear dead-zone
⎧
⎪ D r (v(t)) if v(t) ≥ b r
⎨
u(t) = D(v(t)) = 0 if b l <v(t)< b r (10.2)
⎪
⎩
D l (v(t)) if v(t) ≤ b l
where v(t) ∈ R is the input (real control to be determined), D l (v), D r (v) are
unknown dynamics and b l < 0, b r > 0 are unknown deadband parameters.
The profile of dead-zone (10.2)can be viewed from Fig. 7.2.
The objective of this chapter is to determine a control v(t) for system
(10.1), such that: 1) the output y(t) tracks a given trajectory y d (t), and all
signals in the closed-loop are bounded; 2) both transient and steady-state
performance of the tracking error e(t) = y(t) − y d (t) should be preserved.
The following assumptions on non-linear system (10.1) are introduced:

Assumption 10.1. The functions h i (t, ¯x i ) can be represented as h i (t, ¯x i (t −

m i

h
τ i (t)))= j=1 ij (t, ¯x i (t−τ ij (t))), and fulfill h ij (t, ¯x i ) ≤ k ij (¯x i ),where k ij (¯x i ) ≥ 0

are unknown bounded functions on any compact set C i.
Assumption 10.2. The functions g i (¯x i ) and their signs are unknown, and there

exist positive constants g 0i and g 1i ,suchthat 0 < g i0 ≤ g i (¯x i ) ≤ g i1 ,i = 1,···n.

Assumptions 10.1–10.2 have been widely used in the literature [11–19,
24]. However, the assumptions used in this chapter are less stringent com-
pared to their counterparts in the literature. For instance, the delayed func-
tions h i (t, ¯x i (t − τ i (t))) in (10.1) contain multiple unknown varying delays,
(t), which further improves the results concern-
i.e., τ i1 (t) = τ i2 (t) =,···τ im i
ing single or constant delay [11,25,26]. Moreover, it should be noticed that
the functions k ij (¯x i ) and constants g 0i , g 1i are only used for analytical pur-
pose. In comparison to the results presented in the previous Chapter 9,
the information on the signum of the control gain functions g i (·) is not
assumed to be either positive or negative, which could cover more realistic
plants.
To accommodate the dead-zone dynamics in system (10.1), as detailed
in Chapter 7, the non-linear dead-zone model (10.2) can be reformulated
as
u(t) = (χ l (t) + χ r (t))v(t) + ρ(t) = d(t)v(t) + ρ(t) (10.3)
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