Page 220 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Non-singular Terminal Sliding Mode Funnel Control of Servo Systems With Input Saturation 219

To ensure that the tracking error e(t) evolves inside the funnel boundary
F φ (t), the expression of ρ(·) can be chosen as
1
ρ(t) = (14.11)
F φ (t)−|e(t)|
From (14.11), we can see that when the gain ρ(t) increases, the error
e(t) approaches to the boundary F φ , and when the gain ρ(t) decreases con-
versely, the error e(t) becomes small. A proper funnel boundary to prescribe
the performance is selected as

F φ (t) = δ 0e −a 0 t + δ ∞ (14.12)

where δ 0 ≥ δ ∞ > 0, a 0 > 0, δ ∞ = lim F φ (t),and |e(0)| < F φ (0).
t→∞
According to (14.9)and (14.11), we define a new funnel error variable
s 1 as

e(t)
s 1 = (14.13)
F φ (t)−|e(t)|
where the funnel boundary F φ (t) satisfies the condition given in (14.12),
and this variable will be employed to ensure the prescribed output perfor-
mance.
The derivative of (14.13) can be calculated as

F φ ˙e− ˙ F φ e
s ˙ 1 = (F φ −|e|) 2 = F φ F ˙e − ˙ F φ F e (14.14)
2
where F = 1/(F φ −|e|) and

s ¨ 1 = F φ F ¨e + F φ F ˙e + ˙ F φ F ˙e − ¨ F φ F e − ˙ F φ F e − ˙ F φ F ˙e
˙
˙
(14.15)
= F φ F ¨e + H 1 (x,e,t)
where H 1 (x,e,t) = F φ F ˙e + ˙ F φ F ˙e − ¨ F φ F e − ˙ F φ F e − ˙ F φ F ˙e is a lumped
˙
˙
unknown dynamics to be addressed.
14.3.2 Controller Design
Considering (14.14)and (14.15), the sliding mode manifold s 2 is designed
as

s 2 =˙s 1 + αs 1 (14.16)

where α> 0 is a positive constant.