Nonlinear Optimal Control of Magnetically Geared Induction Motors

Parameter	Definition
ω_m, ω_L	Angular speed of the motor, load
J_m, J_L, J_g	Moment of inertia of the rotor, load, gear
T_e, T_L, T_g	Torque of the rotor, load, gear
ϕ	Angle denoting the speed difference between rotor and load
G_r	Transmission ratio of the magnetic gear
B_m, B_L, B_g	Friction coefficient at rotor, load and gear
p_o, p_m	Number of ferromagnetic pole pieces and air gaps
n_L	Sum of ferromagnetic pole pieces and air gaps
i_{s_d}, i_{s_q}	d,q axis components of the IM stator currents
R_s, R_r	Resistance of the IM’s stator, rotor
ψ_{r_d}, ψ_{r_q}	d,q axis components of the IM rotor flux
L_s, L_r	Inductance of the IM’s stator, rotor
M	Mutual inductance between IM’s stator and rotor
n_p	Number of poles of the IM’s stator
ρ	Orientation of the IM’s magnetic field
α, β, γ	$α = \frac{R_{r}}{L_{r}}, β = \frac{M}{σ L_{s} L_{s}}, γ = \frac{M^{2} R_{r}}{σ L_{s} {L_{r}}^{2}} + \frac{R_{s}}{σ L_{s}}$
μ, σ	$μ = \frac{n_{p} M}{J}, σ = 1 - \frac{M^{2}}{L_{s} L_{r}}$

[i] IM, Induction Motor; MGIM, magnetically-geared induction motor.

Diagram of the control scheme for the MGIM. 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.

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 *e_i*, i = 1,…,6 associated with the state variables *x_i*, i = 1,…,6. 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.

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 *e_i*, i = 1,…,6 associated with the state variables *x_i*, i = 1,…,6. MGIM, magnetically geared induction motor.

Table 2.

Tracking RMSE for the MGIM in the disturbance-free case

	RMSE_x1	RMSE_x2	RMSE_x3	RMSE_x4	RMSE_x5	RMSE_x6
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

[i] MGIM, magnetically geared induction motor; RMSE, Root Mean Square Error.

Table 3.

Tracking RMSE for the MGIM in the case of disturbances

Δa%	RMSE_x1	RMSE_x2	RMSE_x3	RMSE_x4	RMSE_x5	RMSE_x6
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

[i] MGIM, magnetically geared induction motor; RMSE, Root Mean Square Error.

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DOI: https://doi.org/10.2478/pead-2025-0009 | Journal eISSN: 2543-4292 | Journal ISSN: 2451-0262

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Language: English

Page range: 227 - 240

Submitted on: Feb 19, 2025

Accepted on: May 7, 2025

Published on: Aug 19, 2025

Published by: Wroclaw University of Science and Technology

In partnership with: Paradigm Publishing Services

Keywords:

magnetically geared induction motors,

non-linear optimal control,

Taylor series expansion,

Lyapunov analysis,

differential flatness

Related subjects:

Computer sciences,

Artificial intelligence,

Engineering,

Electrical engineering,

Electronics

© 2025 G. Rigatos, P. Siano, M. Abbaszadeh, G. Cuccurullo, K. Ouahada, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 10 (2025): Issue 1 (January 2025)

Nonlinear Optimal Control of Magnetically Geared Induction Motors

Figures & Tables

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Table 1.

Figure 2.

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Figure 5.

Figure 6.

Table 2.

Table 3.

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