Page 451 - Mathematical Techniques of Fractional Order Systems

436 Mathematical Techniques of Fractional Order Systems

14.7.2 Permanent Magnet Synchronous Motor
Generalization of the integer model of PMSM to the fractional order domain
was discussed by Chun-Lai et al. (2012). The dimensionless form of the sys-
tem is given by (Chun-Lai et al., 2012):
D i d 52 i d 1 ωi q ; ð14:72aÞ
α 1
D i q 52 i q 2 ωi d 1 γω; ð14:72bÞ
α 2
D ω 5 σði q 2 ωÞ; ð14:72cÞ
α 3
where i d and i q are the stator currents, ω is the angular velocity, and
α i ; i 5 1; 2; 3 are the fractional orders. The system has three equilibrium
p ffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi
points: E 1 5 ð0; 0; 0Þ and E 2;3 5 ðγ 2 1; 6 γ 2 1; 6 γ 2 1Þ. The authors
then proposed an adaptive control strategy where the controlled system has
the form (Chun-Lai et al., 2012):
D i d 52 i d 1 ωi q 2 d 1 ; ð14:73aÞ
α 1
D i q 52 i q 2 ωi d 1 γω 1 d 2 2 ρði q 2 i q Þ; ð14:73bÞ
α 2

D ω 5 σði q 2 ωÞ 1 d 3 ; ð14:73cÞ
α 3
2
D ρ 5 ηði q 2i q Þ ; ð14:73dÞ
α j

where d i is a random disturbance such that jd 1 j # 0:5, jd 2 j # 1, and

jd 3 j # 0:3and i is a constant. The variable ρ is part of the adaptive feed-
d
back control scheme. Fig. 14.15 shows the simulation result of the con-
trolled system with and without external disturbance where the control
action was applied at t 5 5 seconds. The simulation parameters are:
α 1 5 0:98, α 2 5 1:0, α 3 5 0:99, α j 5 0:5, i d ð0Þ 5 2:5, i q ð0Þ 5 3, and ω 5 1.

The control parameter i is set to be equal to 7.0 to force the system to the
q
equilibrium point E 2 . The simulations show how the system is robust
against external disturbances.

14.8 FPGA IMPLEMENTATIONS
There are two main approaches for FPGA implementation of FOCS in lit-
erature: HDL coder-based (Rajagopal et al., 2017; Shah et al., 2017; Rana
et al., 2016), and designing from scratch (Tolba et al., 2017). Although
the former approach is easier, it gives the designer less involvement in the
optimization of the generated HDL code. The latter approach is rather
tedious but it permits the designer to fully control the details of each
HDL block in the system. The next subsections discuss examples from
both approaches.