Page 133 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Adaptive Finite-Time Neural Control of Servo Systems With Non-linear Dead-Zone 129

Constructing the following Lyapunov as
1
V 2 = e 2 (8.44)
2
and differentiating V 2 along (8.43)yields

γ +1
2 γ +1 γ +1 2 (8.45)
˙ V 2 =−λ 1e − λ 2 |e| =−2λ 1V 2 − λ 22 2 V 2 .
γ +1 γ +1
Setting β 1 = 2λ 1, β 2 = λ 22 2 ,and β 3 = , it can be obtained from
2
(8.45)as

˙ V 2 + β 1V 2 + β 2V β 3 ≤ 0. (8.46)
2
According to Lemma 8.1, we can conclude that the tracking error e will
converge to zero within a finite time t 2 given by

1 β 1V 2 1−β 3 (t 1 ) + β 2
t 2 = ln . (8.47)
β 1 (1 − β 3 ) β 2
This completes the proof.

From (8.42)and (8.47), it can be concluded that in order to reduce the
reaching time t 1 and t 2, we need to increase the value of parameters k 1, λ 1,
r,and γ .However,too large k 1 and λ 1 may lead to a high control gain.
Consequently, the parameters k 1, λ 1, r,and γ should be chosen properly.

8.4 EXPERIMENTAL VALIDATION
8.4.1 Experimental Setup
AsshowninFig. 4.1 and Fig. 4.2, the same turntable servo system with
one degree of freedom (DOF) is used as the test-rig to validate the pro-
posed control, which comprises a permanent magnet synchronous motor
(PMSM, HC-UFS13), an encoder, and pulse width modulation (PWM)
amplifiers in the motor drive card (MR-J2S-10A), a digital signal process-
ing unit (DSP, TMS3202812) performing as the controller. We refer to
Chapter 4 (Section 4.1) for detailed description of this test-rig.
It is noted in Chapter 4 and previous chapters, the friction dynamics are
the dominant non-smooth behaviors in this platform. Hence, in order to
show the efficacy of the proposed control to compensate for the dead-zone