Page 90 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

˙
T
˙ r = (Z) − (k s + 2)r + β 1sgn(z 2 ) + S + + ε (5.24)
˜
˜
˙
where = − is the weight estimation error.
ˆ
˜
To facilitate the subsequent closed-loop stability analysis, inspired
by [11], we further represent (5.24)as
˙ r = ˜ N + N − z 2 − (k s + 1)r + β 1sgn(z 2 ) (5.25)

where the unmeasurable auxiliary terms ˜ N and N are defined as

˙ ˜ T ˙ ˜ (5.26)
˜ N = S + z 2 − r, N = N B + N d , N B = (Z), N d = + ε
Similar to the analysis shown in [11], one can apply the Mean Value The-
orem on the continuously differentiable function ˜ N, and then from (5.10),
(5.16), and (5.17), it can be verified that ˜ N can be upper bounded by

˜ N à η( z ) z (5.27)

where η is a positive globally invertible non-decreasing function, and z is
defined as

T
z =[z 1 ,z 2 ,r] (5.28)
Moreover, following the above inequalities and the analysis in [11], N B , N d
and their time derivatives ˙ N B , ˙ N d are bounded by

N B à ζ , ˙ N B à ζ + ζ N B2 z 2 , N d à ζ , ˙ N d à ζ
N B0 N B1 N d1 N d2
(5.29)

are positive constants.
where ζ N B0 , ζ N B1 , ζ N d1 ,and ζ N d2
In contrary to conventional RISE control designs, e.g., [11], the trans-
formed variables z 1 ,z 2 of tracking error e 1 with PPF are used in the control
implementation. Consequently, the transient error constraint (5.5)withthe
proposed control can be guaranteed. On the other hand, the proposed
control in this chapter can not only guarantee that the control error z 1
is bounded, but also show that z 1 converges asymptotically to zero. In this
case, the original tracking error e 1 will converge to zero as well. This can
be achieved by using the robust term μ s, which compensates for the effect
of ESN error ε and disturbance d.