Page 187 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

Hence, there exist positive constants ι ϑ,1 ,ι ϑ,2 satisfying

| ϑ 1 − α r |≤ ι ϑ,1
(11.22)
| ϑ 2 −¨α r |≤ ι ϑ,2 ,
in finite-time t ≥ t T > 0.
It is shown that TD can improve the convergence performance in com-
parison to the first order filter used in conventional DSC designers, e.g.,
[2,19–21]. Hence, in this section, TD will be incorporated into the DSC
control to obtain the derivative of the intermediate control signals.

11.4.2 Dynamic Surface Control Design
In this subsection, based on the non-linear ESO (11.16), a modified ro-
bust DSC with TD (11.21) is developed for the system (11.14)withthe
unknown dead-zone input (11.2). The schematic diagram of the proposed
control system is shown in Fig. 11.1, in which TD is used to estimate the
intermediate control signals and their derivatives, such that the “explosion
of complexity” problem in the backstepping can be remedied, while the
use of TD can help improve the overall control performance.
Step 1: The first error surface is defined as

e 1 = ξ 1 − ϑ 1,1 (11.23)

with ϑ 1,1 being the filtered signal of the desired trajectory x d by TD as

⎧
⎨ ˙ = ϑ 1,2
ϑ 1,1
ϑ 1,2 | ϑ 1,2 | (11.24)
⎩ ˙ =−rsgn(ϑ 1,1 − x d + ).
ϑ 1,2
2r
Taking (11.14)and (11.15) into consideration, the derivative of e 1 satis-
fies

(11.25)
˙ e 1 ≤ ξ 2 − ϑ 1,2 + ι 1
where ι 1 = ι + ι ϑ,2 represents the filter error bound of ESO (11.15)and TD
(11.24).
Here, a Lyapunov function is considered as

1 2
V 1 = e (11.26)
2 1