Page 173 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

APPC of Strict-Feedback Systems With Non-linear Dead-Zone 169

Integrating (10.51)over [0,t],itcan be obtained

c
c
n n
ϑ + i=1 i1M i /2 ϑ + i=1 i1M i /2 −γt
V ≤ + V(0) − e
γ γ
t
n

γς
˙
+ e −γt α[G i (x)N(ξ i ) + 1]ξ ie dς
i=1
0
n t
c
ϑ + i=1 i1M i /2 −γt −γt n γς
˙
≤ + V(0)e + e α[G i (x)N(ξ i ) + 1]ξ ie dς
γ i=1 0
(10.52)
Consequently, one can rewrite (10.52)as
t
n
γς
˙
V(t) ≤ δ + e −γt α[G i (x)N(ξ i ) + 1]ξ ie dς (10.53)
i=1
0
n
c
where δ = ϑ + i=1 i1M i /2 /γ + V(0) is a positive constant. Then
according to Lemma 10.2, it can be concluded that V i, ξ i ,and
t ˙ γς ∗ ∗
0 G i (x)N(ξ i )ξ ie dς are all bounded on [0,t f ). In addition, since θ , ε ,
i
i
and y d , ˙y d are all bounded, ˆ , ˆε i, ˙ ,and x i, z i are then bounded, which
ξ i
θ i
further implies that the control signals α i and v are bounded. According to
Proposition 2 in [28], if the solution of the closed-loop system is bounded
on the interval [0,t f ) for any t f > 0, it is also true as t f →∞ as discussed
in [13,28]. Consequently, we get that all signals in the control system are
bounded. Moreover, as stated in Lemma 10.1, the boundedness of z 1 is
sufficient to guarantee the PPF condition (10.4)via theproposederror
transformation. This means that the tracking control of system (10.1)with
guaranteed performance (10.4)isachieved.
It should be noted that the estimated vector ˆ W i is replaced by a scalar
∗
∗
θ i through introducing an unknown scalar θ = W ∗T W as the adaptive
ˆ
i i i
parameter of HONN, such that the computational cost of the proposed
control can be reduced significantly. Moreover, with the help of Nussbaum-
type function, the unknown signs of g i (¯x i ) and unknown dead-zone non-
linearity are all handled. The functions k ij (·) and parameters g 0i, g 1i are not
used in the control implementation.
10.4 SIMULATIONS
To illustrate the validity of the proposed control, consider the following
non-linear time-delay system