Page 553 - Mathematical Techniques of Fractional Order Systems
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532 Mathematical Techniques of Fractional Order Systems
q 1
d i d
52 i d 1 ωi q 1 u d
dt
q 2
d i q
52 i d 2 ωi q 1 γω 1 u q ð18:1Þ
dt
d ω
q 3
5 σði q 2 ωÞ 2 T L
dt
where q i (i 5 1,2,3) are the noninteger derivative orders, i d and i q are the sta-
tor current in d, q axis, respectively. ω is the rotor angular velocity. u d and
u q are the stator voltage in the d, q axis. σ, γ are the dimensionless operating
parameters of the system. T L is the load torque.
If the external inputs of the system are equal to 0 where T L 5 u d 5 u q 5 0,
the previous system of equations will be rewritten as follows 18.2 (Li et al.,
2012).
q 1
d i d
52 i d 1 ωi q
dt
q 2
d i q
52 i d 2 ωi q 1 γω
dt ð18:2Þ
d ω
q 3
5 σði q 2 ωÞ
dt
The chaotic behavior in the PMSM occurs when the parameters of the
commensurate fractional order PMSM model σ, γ are equal to 10, 100,
respectively, and q 1 5 q 2 5 q 3 5 q 5 0.95 as mentioned in Xue et al. (2015).
While, with respect to the data reported in Li et al. (2012), the chaotic
behavior in the PMSM occurs when the parameters of the incommensurate
fractional order PMSM model σ, γ, q 1 , q 2 , and q 3 are equal to 4, 50, 0.99, 1,
and 0.98, respectively. The obtained chaotic behaviors from the system 18.2
at the two cases of commensurate and incommensurate fractional order mod-
els with the initial conditions [i d , i q , ω] 5 [2.5, 3, 1] are plotted in Fig. 18.1A
and B, respectively
18.3 PROBLEM FORMULATION
Practically, the dimensionless unknown parameters of the commensurate and
the incommensurate fractional order PMSM models are (σ, γ, q 1 5 q 2 5 q 3 5 q)
and (σ, γ, q 1 , q 2 , q 3 ), respectively. These parameters are necessary to be identi-
fied accurately because at these parameters the PMSM exhibits unexpected per-
formance which is known as a chaotic behavior. Hence, the main target of the
introduced optimization techniques is to extract these parameters via compari-
son between the measured response of the system and the estimated one
(Tofighi et al., 2013).
