Page 192 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Dynamic Surface Output Feedback Control of Pure-Feedback Systems 189

we have the following inequality

˙ V ≤−2ρ vV + ς. (11.50)

Solving the inequality (11.50)yields

ς ς −2ρ v t
0 ≤ V(t) ≤ + (V(0) − )e . (11.51)
2ρ v 2ρ v
The above inequality implies that V(t) is eventually bounded by ς/2ρ v .
Consequently, all signals in the closed-loop system including e i (t),η i (t),i =
1,··· ,n are ultimately uniformly bounded. Furthermore, it follows from
(11.51)that

ς
lim V(t) ≤ μ ∞ (11.52)
2ρ v
t→∞
which means that the tracking errors e i ,i = 1,··· ,n, can converge to a
compact set around zero defined by
√
{e i :|e i |≤ 2μ ∞ }. (11.53)
∞
This completes the proof.

11.5 SIMULATIONS

In this section, in order to illustrate the performance of the proposed con-
trol algorithm, we consider the following second-order system [3]:
x
⎧ 3
⎪ = x 1 + x 2 + 2
⎪ ˙ x 1
5
⎪
⎨ 3
u (11.54)
⎪ ˙x 2 = x 1x 2 + + u
7
⎪
⎪
⎩
y = x 1
with
⎧
⎪ (1 − 0.3sin(v))(v − b r ) if v ≥ b r
⎨
u = DZ(v) = 0 if b l < v < b r (11.55)
(0.8 − 0.2cos(v))(v − b l ) if v ≤ b l .
⎪
⎩
The initial state values are x 1 (0) = 0.6and x 2 (0) = 0.5, and the reference
signal is given by x d (t) = sin(t)+cos(0.5t). Then a third-order ESO as given
in (11.15) can be used, where b 0 = 1 is the estimation of b(¯x n ),andthe