Page 126 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

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

The control objective is to design an adaptive robust finite-time con-
troller v(t) for system (8.7), such that all signals in the closed-loop system
are bounded, and the tracking error e canconvergetozerowithinafinite
time.

8.3 ADAPTIVE FINITE-TIME CONTROL DESIGN AND
STABILITY ANALYSIS

In this section, an adaptive robust finite-time control scheme is designed
for the PMSM servo system (8.7).

8.3.1 Fast Terminal Sliding Mode Manifold

The linear sliding mode (LSM) and terminal sliding mode (TSM) can be
described by the following differential equations [25], [29]:
LSM:

s =˙e + λ 0e (8.9)

TSM:

γ
s =˙e + λ 0 |e| sgn(e) (8.10)

where λ 0 > 0 is a positive constant, and γ = q/p, p,q > 0are positive odd
numbers satisfying q < p.
Once the sliding mode manifold s = 0 is reached, the expressions of
γ
LSM and TSM are reduced to ˙e =−λ 0e and ˙e =−λ 0 |e| sgn(e), respectively.
γ
According to the fact 0 <γ < 1, we have |e| < |e| for any |e| > 1, which
implies that TSM has a slower convergence speed than LSM when the
system position is far away from the desired trajectory. Otherwise, when
the system position is very close to the desired trajectory, TSM has a faster
γ
convergence speed than LSM due to |e| > |e| for any |e| < 1.
Hence, by introducing the linear term of (8.9) into the TSM design
(8.19), a fast terminal sliding mode (FTSM) surface is defined as:

γ
s =˙e + λ 1e + λ 2 |e| sgn(e) = x r −¨x (8.11)
with

γ
x r =˙y d + λ 1e + λ 2 |e| sgn(e) (8.12)