Page 164 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

160 Adaptive Identification and Control of Uncertain Systems with Non-smooth Dynamics

Definition 10.1. The Nussbaum-type function N(·) is defined as specific
even smooth function fulfilling
1 t 1 t
lim sup N(ς)dς =+∞, lim inf N(ς)dς =−∞. (10.11)
t→∞ t 0 t→∞ t 0
Nussbaum functions (10.11) have infinite gains and infinite switching
2
2
2
frequencies. The functions ξ cos(ξ), ξ sin(ξ), exp(ξ )cos(πξ/2) have been
2
verified [27,28]. For simplicity, ξ cos(ξ) is used in this paper. Then the
following Lemma holds:
Lemma 10.2. [12]: Let V(·),ξ(·) be smooth functions defined on [0,t f ) with
V(t) ≥ 0,∀t ∈[0,t f ) and N(ξ) be an even smooth Nussbaum-type function. If
the following property holds:

t
V(t) ≤ c 0 + e −c 1 t [G(·)N(ξ) + 1]ξe −c 1 ς dς, ∀t ∈[0,t f ) (10.12)
˙
0
with c 0 and c 1 being positive constants, and G(·) is a time-varying function, then
t
˙
V(t), ξ(t),and G(·)N(ξ)ξdς are bounded on [0,t f ) with t f < +∞.
0
It should be noted that Lemma 10.2 guarantees the forward com-
pleteness property (boundedness up to finite time [0,t f )) of Nussbaum
parameters and the associated Lyapunov functions.

10.3 CONTROL DESIGN AND STABILITY ANALYSIS

10.3.1 Adaptive Prescribed Performance Control
In this section, an adaptive control is provided for system (10.9) to guarantee
the boundedness of z 1, and consequently to achieve the convergence of
tracking error e within the PPF bound (10.4). For notation conciseness,
the time variable t will be omitted except for the terms with unknown
time-varying delays τ i (t).
Define the coordinate error as z i = x i − α i−1 ,i = 2,··· ,n,where α i is
the virtual control for each sub-system, then the control design procedure
can be presented as:
Step 1. To stabilize sub-system z 1, the virtual control α 1 can be specified
as

2
ˆ
k 1z 1 θ 1sgn(z 1 ) T ˆ ε z 1 e ˙μ
1
α 1 = N(ξ 1 ) + (Z 1 ) 1 (Z 1 ) + −
1
r 2η 2 ˆ ε 1 |z 1 |+ σ 11 μ
1
(10.13)