Page 141 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

ANDSC of Strict-Feedback Systems With Non-linear Dead-Zone 137

where v(t) ∈ R is the input of the dead-zone (practical control signal),
D l (v), D r (v) are unknown non-linear smooth functions and b l , b r are un-
known width parameters of the dead-zone. Without loss of generality, it
is assumed that b l < 0, b r > 0. The input-output profile of dead-zone (9.2)
can be found in Fig. 7.2.
The non-linear dead-zone (9.2) can cover more general cases including
linear and symmetrical dead-zones previously stated in Chapter 7. More-
over, the dead-zone functions D l (v), D r (v) and characteristic parameters b l ,
b r are not necessarily known in the following control design. Following the
statements presented in Chapter 7.2, the above non-linear dead-zone can
be represented as

u(t) = D(v(t)) = (χ l (t) + χ r (t))v(t) + ρ(t) = d(t)v(t) + ρ(t) (9.3)

where d(t) and ρ(t) areallgivenin(7.10)and (7.11), which are all bounded
as stated in Chapter 7.
The objective is to obtain a control v(t) for system (9.1) such that
the output y(t) follows a specified trajectory y d (t), while all signals in
the closed-loop are bounded. In system (9.1), the current states ¯x i (t) and
the delayed states ¯x i (t − τ ij (t)) are involved in the non-linear functions
f i (¯x i (t), ¯x i (t − τ ij (t))) and g i (¯x i (t), ¯x i (t − τ ij (t))) simultaneously, to cover more
general systems.
To facilitate the control design, one can represent f i (¯x i (t), ¯x i (t − τ ij (t)))
into a delay free function together with a delayed function, and then sub-
stituting (9.3)into(9.1)yields:
⎧
⎪ ˙ x i (t) = f i (¯x i (t),0) + h i (¯x i (t), ¯x i (t − τ ij (t))) + g i (¯x i (t), ¯x i (t − τ ij (t)))x i+1 (t)
⎪
⎪
˙ x n (t) = f n (x(t),0) + h n (x(t),x(t − τ nj (t)))
⎨
+ g n (x(t),x(t − τ nj (t)))[d(t)v(t) + ρ(t)]
⎪
⎪
⎪
y(t) = x 1 (t)
⎩
(9.4)
where h i (¯x i (t), ¯x i (t −τ ij (t))) = f i (¯x i (t), ¯x i (t −τ ij (t)))−f i (¯x i (t),0) are unknown
non-linear functions.
Assumption 9.1. There exist non-negative functions ϕ ij (¯x i (t − τ ij (t))) ≥ 0,i =
1,··· ,n; j = 1,··· ,m i, such that the unknown functions h i (¯x i (t), ¯x i (t − τ ij (t)))

m i
in (9.4) are bounded by h i (¯x i (t), ¯x i (t − τ ij (t))) ≤ j=1 ϕ ij (¯x i (t − τ ij (t))),where

ϕ ij (·) are bounded on any compact set C i.
Assumption 9.2. The signs of unknown control functions g i (·) are known, and

there exist unknown positive constants g 0i and g 1i ,suchthat 0 < g i0 ≤ g i (·) ≤

g i1 ,i = 1,···n.