Page 447 - Mathematical Techniques of Fractional Order Systems

432 Mathematical Techniques of Fractional Order Systems

while the slave system is given by:

α S s βS s I s
D S s 5 rS s 1 2 2 1 u 1 ðtÞ; ð14:65aÞ
k 1 1 aS s
α
D I s 5 βI s Z s 2 μI s 2 γI s 1 u 2 ðtÞ; ð14:65bÞ
1 1 aZ s
α
D Z s 5 1 ð S s 2 Z s Þ 1 u 3 ðtÞ: ð14:65cÞ
T
The control inputs were chosen to be:
r βS s I s βS m I m
2
2
u 1 ðtÞ 5 V 1 ðtÞ 1 ðS 2 S Þ 1 2 ; ð14:66aÞ
k s m 1 1 aS s 1 1 aS m
βI s Z s βI m Z m
u 2 ðtÞ 5 V 2 ðtÞ 2 1 ; ð14:66bÞ
1 1 aZ s 1 1 aZ m
u 3 ðtÞ 5 V 3 ðtÞ; ð14:66cÞ

where:
0 1
2ðr 1 1Þ 0 0
0 1 0 1
V 1 ðtÞ 0 0 S m 2 S s
B ðμ 11 γ 2 1Þ C
@ V 2 ðtÞ A 5 B 21 1 C@ I m 2 I s A : ð14:67Þ
0 2 1
@ A
V 3 ðtÞ Z m 2 Z s
T T
Fig. 14.11 shows the simulation results of the this synchronization
system where the control action is applied at t 5 5 seconds. The simulation
parameters are: r 5 2, k 5 5, a 5 0:01, β 5 0:5, μ 5 0:3, γ 5 0:2, and
T 5 0:85. The fractional order alpha 5 0:9. It was observed that the syn-
chronization time increases as the order approaches the integer case
ðα 5 1:0Þ.

14.7 MOTORS

Chaotic behavior is highly undesirable in motors as it causes many pro-
blems, such as low frequency oscillations in the current, torque ripples, and
even motor collapse. Hence, identifying the range of parameters at which
the motor exhibit chaos is important for motor protection. Once the motor
enters the chaotic region, chaos control must be applied to prevent system
collapse. This section discusses two fractional order nonlinear motor mod-
els introduced in literature: fractional order Brushless DC Motor model
(BLDCM) and fractional order Permanent Magnet Synchronous Motor
model (PMSM).