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Nonlinear Optimal Control of Magnetically Geared Induction Motors

G. Rigatos

G. Rigatos

Unit of Industrial Automation, Industrial Systems Institute, Athena RCRion Patras, Greece
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Rigatos, G.
, 
P. Siano

P. Siano

  • Department of Management and Innovation Systems, University of SalernoFisciano, Italy
  • Department of Electrical Engineering, University of JohannesburgJohannesburg, South Africa
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Siano, P.
, 
M. Abbaszadeh

M. Abbaszadeh

Department of ECS Engineering, Rensselaer Polytechnic InstituteNew York, USA
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Abbaszadeh, M.
, 
G. Cuccurullo

G. Cuccurullo

Depatment of Industrial Engineering, University of SalernoFisciano, Italy
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Cuccurullo, G.
 e 
K. Ouahada

K. Ouahada

Department of Electrical Engineering, University of JohannesburgJohannesburg, South Africa
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Ouahada, K.
  
19 ago 2025
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Categoria dell'articolo: Research paper
Pubblicato online: 19 ago 2025
Pagine: 227 - 240
Ricevuto: 19 feb 2025
Accettato: 07 mag 2025
DOI: https://doi.org/10.2478/pead-2025-0009
Parole chiave
© 2025 G. Rigatos et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1.

Diagram of the traction system of an EV based on a MGIM. EV, electric vehicle; MGIM, magnetically geared induction motor.
Diagram of the traction system of an EV based on a MGIM. EV, electric vehicle; MGIM, magnetically geared induction motor.

Figure 2.

Diagram of the control scheme for the MGIM. MGIM, magnetically geared induction motor.
Diagram of the control scheme for the MGIM. MGIM, magnetically geared induction motor.

Figure 3.

Tracking of setpoint 1 by the MGIM with the use of non-linear optimal control: (a) convergence of state variables x1 to x3 (blue lines) to the associated setpoints (red lines) and estimated values provided by Kalman Filtering (b) convergence of state variables x4 to x6 (blue lines) to the associated setpoints (red lines) and estimated values provided by Kalman Filtering. MGIM, magnetically geared induction motor.
Tracking of setpoint 1 by the MGIM with the use of non-linear optimal control: (a) convergence of state variables x1 to x3 (blue lines) to the associated setpoints (red lines) and estimated values provided by Kalman Filtering (b) convergence of state variables x4 to x6 (blue lines) to the associated setpoints (red lines) and estimated values provided by Kalman Filtering. MGIM, magnetically geared induction motor.

Figure 4.

Tracking of setpoint 1 by the MGIM with the use of non-linear optimal control: (a) variations of the control inputs u1 and u2 (blue lines) (b) variation of the tracking error variables ei, i = 1,…,6 associated with the state variables xi, i = 1,…,6. MGIM, magnetically geared induction motor.
Tracking of setpoint 1 by the MGIM with the use of non-linear optimal control: (a) variations of the control inputs u1 and u2 (blue lines) (b) variation of the tracking error variables ei, i = 1,…,6 associated with the state variables xi, i = 1,…,6. MGIM, magnetically geared induction motor.

Figure 5.

Tracking of setpoint 2 by the MGIM with the use of non-linear optimal control: (a) convergence of state variables x1 to x3 (blue lines) to the associated setpoints (red lines) and estimated values provided by Kalman Filtering (b) convergence of state variables x4 to x6 (blue lines) to the associated setpoints (red lines) and estimated values provided by Kalman Filtering. MGIM, magnetically geared induction motor.
Tracking of setpoint 2 by the MGIM with the use of non-linear optimal control: (a) convergence of state variables x1 to x3 (blue lines) to the associated setpoints (red lines) and estimated values provided by Kalman Filtering (b) convergence of state variables x4 to x6 (blue lines) to the associated setpoints (red lines) and estimated values provided by Kalman Filtering. MGIM, magnetically geared induction motor.

Figure 6.

Tracking of setpoint 2 by the MGIM with the use of non-linear optimal control: (a) variations of the control inputs u1 and u2 (blue lines) (b) variation of the tracking error variables ei, i = 1,…,6 associated with the state variables xi, i = 1,…,6. MGIM, magnetically geared induction motor.
Tracking of setpoint 2 by the MGIM with the use of non-linear optimal control: (a) variations of the control inputs u1 and u2 (blue lines) (b) variation of the tracking error variables ei, i = 1,…,6 associated with the state variables xi, i = 1,…,6. MGIM, magnetically geared induction motor.

Tracking RMSE for the MGIM in the disturbance-free case

RMSEx1 RMSEx2 RMSEx3 RMSEx4 RMSEx5 RMSEx6
Test1 0.0052 0.0026 0.0064 0.0037 0.0001 0.0002
Test2 0.0041 0.0020 0.0064 0.0063 0.0002 0.0003

Tracking RMSE for the MGIM in the case of disturbances

Δa% RMSEx1 RMSEx2 RMSEx3 RMSEx4 RMSEx5 RMSEx6
0% 0.0052 0.0026 0.0064 0.0037 0.0001 0.0002
10% 0.0057 0.0029 0.0064 0.0014 0.0001 0.0003
20% 0.0062 0.0031 0.0064 0.0007 0.0001 0.0003
30% 0.0066 0.0033 0.0064 0.0027 0.0002 0.0001
40% 0.0069 0.0035 0.0065 0.0046 0.0002 0.0003
50% 0.0073 0.0036 0.0065 0.0064 0.0002 0.0003
60% 0.0075 0.0038 0.0065 0.0081 0.0002 0.0003

Parameters of the MGIM dynamic model_

Parameter Definition
ωm, ωL Angular speed of the motor, load
Jm, JL, Jg Moment of inertia of the rotor, load, gear
Te, TL, Tg Torque of the rotor, load, gear
ϕ Angle denoting the speed difference between rotor and load
Gr Transmission ratio of the magnetic gear
Bm, BL, Bg Friction coefficient at rotor, load and gear
po, pm Number of ferromagnetic pole pieces and air gaps
nL Sum of ferromagnetic pole pieces and air gaps
isd, isq d,q axis components of the IM stator currents
Rs, Rr Resistance of the IM’s stator, rotor
ψrd, ψrq d,q axis components of the IM rotor flux
Ls, Lr Inductance of the IM’s stator, rotor
M Mutual inductance between IM’s stator and rotor
np Number of poles of the IM’s stator
ρ Orientation of the IM’s magnetic field
α, β, γ α=RrLr,β=MσLsLs,γ=M2RrσLsLr2+RsσLs \alpha = {{{R_r}} \over {{L_r}}},\beta = {M \over {\sigma {L_s}{L_s}}},\gamma = {{{M^2}{R_r}} \over {\sigma {L_s}{L_r}^2}} + {{{R_s}} \over {\sigma {L_s}}}
μ, σ μ=npMJ,σ=1−M2LsLr \mu = {{{n_p}M} \over J},\sigma = 1 - {{{M^2}} \over {{L_s}{L_r}}}
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