IP Library › Granted Patent US 12,348,161

Granted Patent B2

US 12,348,161 · App. 18/730,268 · Granted Jul 1, 2025

Motor control using piecewise affine model

Inventors: Matthias Preindl (New York, NY); Bernard William Steyaert (New York, NY); Ethan Bagget Swint (Redwood City, CA); Nick Marshall Vazquez (Redwood City, CA); Walter Wesley Pennington, III (Menlo Park, CA)

Assignees: TAU MOTORS, INC.; THE TRUSTEES OF COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK

H02P21/0017 H02P2207/01 H02P2207/05

View Patent ↗

Loading inventors, assignments & file history…

Monitor This Case

Get email alerts when status or documents change.

Order Certified Copies

Most orders are placed with the USPTO same day — all within 24 business hours.

Order via The Patent Place →

Pre-filled with this patent's details

Quick Facts

Patent No.: US 12,348,161
App. No.: 18/730,268
Granted: Jul 1, 2025
Kind: B2

External Links

USPTO Patent Center↗ Full Patent PDF↗

Abstract

Disclosed are systems and methods for motor control using piecewise affine modelling. An electronic controller may determine current values for a motor in a rotational reference frame. Each current value may be associated with a dimension of a set of dimensions of the rotational reference frame. The electronic controller may further determine, based on the current values, a flux linkage value for each of the set of dimensions of the rotational reference frame using a piecewise affine map. The electronic controller may further determine a target flux linkage value for each of the set of dimensions of the rotational reference frame. The electronic controller may then control a power switching network coupled between a power supply and the motor based on the flux linkage values and the target flux linkage values.

Claims (151)

1. A motor system comprising:

a power switching network configured to be coupled to a power supply and to a motor;

an electronic controller configured to:

determine current values for the motor in a rotational reference frame, each current value associated with a dimension of a set of dimensions of the rotational reference frame;

determine, based on the current values, a flux linkage or inductance value for each dimension of the set of dimensions of the rotational reference frame using a piecewise affine map;

determine a target value for each dimension of the set of dimensions of the rotational reference frame; and

control the power switching network based on the flux linkage or inductance values and the target values.

2. The motor system of claim 1 , wherein, to determine current values for the motor in a rotational reference frame, the electronic controller is configured to:

determine electrical operational characteristics of the motor in a stationary reference frame;

determine a rotational position of the motor; and

transform the electrical operational characteristics and the rotational position to the current values for the motor in the rotational reference frame.

3. The motor system of claim 1 ,

wherein the piecewise affine map includes a plurality of affine functions, each of the plurality of affine functions associated with a respective domain of a plurality of domains, and

wherein, to determine, based on the current values, a flux linkage or inductance value for each dimension of the set of dimensions of the rotational reference frame using a piecewise affine map, the electronic controller is configured to:

identify a first domain corresponding to the current values and selected from the plurality of domains, the first domain associated with a first affine function of the plurality of affine functions; and

apply the current values to the first affine function to determine the flux linkage or inductance value for each dimension of the set of dimensions of the rotational reference frame.

4. The motor system of claim 3 , wherein

each of the plurality of domains corresponds to a simplex that was provided by executing a domain decomposition algorithm on a data set of current and flux-linkage pairs for a plurality of operational points of the motor or

each of the plurality of domains corresponds to a simplex that was provided by executing a domain decomposition algorithm on a data set of current and inductance pairs for a plurality of operational points of the motor.

5. The motor system of claim 1 , wherein, to determine a target value for each dimension of the set of dimensions of the rotational reference frame, the electronic controller is configured to:

determine a desired control parameter for the motor;

transform the desired control parameter to target current values for the motor in the rotational reference frame; and

determine, based on the target current values, the target value for each dimension of the set of dimensions of the rotational reference frame using the piecewise affine map.

6. The motor system of claim 5 , wherein, to transform the desired control parameter to the target current values for the motor in the rotational reference frame, the electronic controller is further configured to:

determine, based on the desired control parameter, a target current value for each dimension of the set of dimensions of the rotational reference frame using a control parameter-to-current piecewise affine map.

7. The motor system of claim 6 ,

wherein the control parameter-to-current piecewise affine map includes a plurality of affine functions, each of the plurality of affine functions associated with a respective domain of a plurality of domains, and

wherein, to determine, based on the desired control parameter, a target current value for each dimension of the set of dimensions of the rotational reference frame using a control parameter-to-current piecewise affine map, the electronic controller is configured to:

identify a first domain corresponding to the desired control parameter and selected from the plurality of domains, the first domain associated with a first affine function of the plurality of affine functions; and

apply the desired control parameter to the first affine function to determine the target current value for each dimension of the set of dimensions of the rotational reference frame.

8. The motor system of claim 6 , wherein the desired control parameter is a target torque value for the motor.

9. The motor system of claim 1 , wherein, to determine a target value for each dimension of the set of dimensions of the rotational reference frame, the electronic controller is configured to:

determine a desired control parameter for the motor;

transform the desired control parameter to target current values for the motor in the rotational reference frame using a control parameter-to-current piecewise affine map, the target current values including a target current value for each dimension of the set of dimensions of the rotational reference frame; and

determine, based on the target current values, the target value for each dimension of the set of dimensions of the rotational reference frame.

10. The motor system of claim 1 , wherein, to determine a target value for each dimension of the set of dimensions of the rotational reference frame, the electronic controller is configured to:

determine a desired control parameter for the motor; and

transform the desired control parameter to target values for the motor in the rotational reference frame using a control parameter-to-flux linkage piecewise affine map, the target values including a target flux linkage value for each dimension of the set of dimensions of the rotational reference frame.

11. The motor system of claim 1 ,

wherein the electronic controller is configured to determine, based on the current values, the flux linkage value for each dimension of the set of dimensions of the rotational reference frame using the piecewise affine map;

wherein the target values are target flux linkage values; and

wherein, to control the power switching network based on the flux linkage values and the target flux linkage values, the electronic controller is configured to:

generate a voltage command for each dimension of the set of dimensions of the rotational reference frame based on a difference between the target flux linkage value and the flux linkage value for the dimension

transform the voltage commands in the rotational reference frame to the stationary reference frame;

generate a pulse width modulated control signal for each dimension of the stationary reference frame to control the power switching network to drive a stator of the motor; and

generate a rotor control signal to control driving of a rotor field winding.

12. The motor system of claim 1 ,

wherein the target values are target flux linkage values; and

wherein, to control the power switching network based on the flux linkage values and the target flux linkage values, the electronic controller is configured to:

generate control signals in the stationary reference frame to drive the motor based on a difference between the target flux linkage value and the flux linkage value for each dimension of the set of dimensions.

13. The motor system of claim 1 ,

wherein the electronic controller is configured to determine, based on the current values, the inductance value for each dimension of the set of dimensions of the rotational reference frame using the piecewise affine map;

wherein the target values are current values; and

wherein, to control the power switching network based on the inductance values and the target current values, the electronic controller is configured to:

generate control signals in the stationary reference frame to drive the motor based on a difference between the current value and the target current value for each dimension of the set of dimensions and on an inverse of the inductance values.

14. The motor system of claim 1 , wherein, to control the power switching network based on the flux linkage or inductance values and the target values, the electronic controller is configured to:

generate, based on a model predictive control algorithm, control signals to drive the motor based on the target values and the flux linkage or inductance values.

15. The motor system of claim 14 , wherein, to generate the control signals based on the model predictive control algorithm, the electronic controller is configured to solve linear state-space equations over a receding time window to select a next control parameter.

16. The motor system of claim 14 , wherein, to generate the control signals based on the model predictive control algorithm, the electronic controller is configured to access a second piecewise affine map defined by offline solving of the model predictive control algorithm using sample inputs.

17. The motor system of claim 1 , wherein the motor is a wound field synchronous motor comprising at least three stator phases and at least one rotor field winding.

18. The motor system of claim 1 , wherein the power switching network includes an inverter switch bridge including a plurality of power switching elements, the inverter switch bridge configured to receive DC power and output AC power to windings of the stator based on pulse width modulated control signals from the electronic controller.

19. The motor system of claim 1 , further comprising a DC/DC converter configured to receive input DC power and to provide output DC power to at least one rotor field winding in accordance with a pulse width modulated rotor control signal from the electronic controller.

20. The motor system of claim 1 , wherein the piecewise affine map maps a set of input values to an output flux linkage or inductance value for each dimension of the set of dimensions, the set of input values including the current values and at least one additional motor characteristic.

21. The motor system of claim 1 , wherein the motor is at least one selected from the group of a wound field synchronous motor, a hybrid synchronous motor, a permanent magnet synchronous motor, an induction motor, a universal motor, or a reluctance motor.

22. A method of controlling a motor, the method comprising:

determining, by an electronic controller, current values for a motor in a rotational reference frame, each current value associated with a dimension of a set of dimensions of the rotational reference frame;

determining, based on the current values, a flux linkage or inductance value for each of the set of dimensions of the rotational reference frame using a piecewise affine map;

determining, by the electronic controller, a target value for each of the set of dimensions of the rotational reference frame; and

controlling, by the electronic controller, a power switching network coupled between a power supply and the motor based on the flux linkage or inductance values and the target values.

23. The method of claim 22 , wherein determining current values for the motor in a rotational reference frame includes:

determining electrical operational characteristics of the motor in a stationary reference frame;

determining a rotational position of the motor; and

transforming the electrical operational characteristics and the rotational position to the current values for the motor in the rotational reference frame.

24. The method of claim 22 ,

wherein the piecewise affine map includes a plurality of affine functions, each of the plurality of affine functions associated with a respective domain of a plurality of domains, and

wherein determining, based on the current values, a flux linkage or inductance value for each dimension of the set of dimensions of the rotational reference frame using a piecewise affine map includes:

identifying a first domain corresponding to the current values and selected from the plurality of domains, the first domain associated with a first affine function of the plurality of affine functions; and

applying the current values to the first affine function to determine the flux linkage or inductance value for each dimension of the set of dimensions of the rotational reference frame.

25. The method of claim 24 , wherein

26. The method of claim 22 , wherein determining a target value for each dimension of the set of dimensions of the rotational reference frame includes:

determining a desired control parameter for the motor;

transforming the desired control parameter to target current values for the motor in the rotational reference frame; and

determining, based on the target current values, the target value for each dimension of the set of dimensions of the rotational reference frame using the piecewise affine map.

27. The method of claim 24 , wherein transforming the desired control parameter to the target current values for the motor in the rotational reference frame includes:

determining, based on the desired control parameter, a target current value for each dimension of the set of dimensions of the rotational reference frame using a control parameter-to-current piecewise affine map.

28. The method of claim 27 , wherein the control parameter-to-current piecewise affine map includes a plurality of affine functions, each of the plurality of affine functions associated with a respective domain of a plurality of domains, and

wherein determining, based on the desired control parameter, a target current value for each dimension of the set of dimensions of the rotational reference frame using a control parameter-to-current piecewise affine map includes:

identifying a first domain corresponding to the desired control parameter and selected from the plurality of domains, the first domain associated with a first affine function of the plurality of affine functions; and

applying the desired control parameter to the first affine function to determine the target current value for each dimension of the set of dimensions of the rotational reference frame.

29. The method of claim 27 , wherein the desired control parameter is a target torque value for the motor.

30. The method of claim 22 , wherein determining a target flux linkage value for each dimension of the set of dimensions of the rotational reference frame includes:

determining a desired control parameter for the motor;

transforming the desired control parameter to target current values for the motor in the rotational reference frame using a control parameter-to-current piecewise affine map, the target current values including a target current value for each dimension of the set of dimensions of the rotational reference frame; and

determining, based on the target current values, the target value for each dimension of the set of dimensions of the rotational reference frame.

31. The method of claim 22 , wherein determining a target value for each dimension of the set of dimensions of the rotational reference frame includes:

determining a desired control parameter for the motor; and

transforming the desired control parameter to target values for the motor in the rotational reference frame using a control parameter-to-flux linkage piecewise affine map, the target values including a target flux linkage value for each dimension of the set of dimensions of the rotational reference frame.

32. The method of claim 22 ,

wherein the determining, based on the current values, the flux linkage or inductance value for each dimension of the set of dimensions includes determining, based on the current values, the flux linkage value for each dimension of the set of dimensions;

wherein the target values are target flux linkage values; and

wherein controlling the power switching network based on the flux linkage values and the target flux linkage values includes:

generating a voltage command for each dimension of the set of dimensions of the rotational reference frame based on a difference between the target flux linkage value and the flux linkage value for the dimension

transforming the voltage commands in the rotational reference frame to the stationary reference frame;

generating a pulse width modulated control signal for each dimension of the stationary reference frame to control the power switching network to drive a stator of the motor; and

generating a rotor control signal to control driving of a rotor field winding.

33. The method of claim 22 ,

wherein the target values are target flux linkage values; and

wherein controlling the power switching network based on the flux linkage values and the target flux linkage values includes:

generating control signals in the stationary reference frame to drive the motor based on a difference between the target flux linkage value and the flux linkage value for the dimension.

34. The method of claim 22 ,

wherein the determining, based on the current values, the flux linkage or inductance value for each dimension of the set of dimensions includes determining, based on the current values, the inductance value for each dimension of the set of dimensions;

wherein the target values are current values; and

wherein controlling the power switching network based on the inductance values and the target current value includes:

generating control signals in the stationary reference frame to drive the motor based on a difference between the current value and the target current value for each dimension of the set of dimensions and on an inverse of the inductance values.

35. The method of claim 22 , wherein controlling the power switching network based on the flux linkage or inductance values and the target values includes:

generating, based on a model predictive control algorithm, control signals to drive the motor based on the target values and the flux linkage or inductance values.

36. The method of claim 22 , wherein generating the control signals based on the model predictive control algorithm includes solving linear state-space equations over a receding time window to select a next control parameter.

37. The method of claim 22 , wherein generating the control signals based on the model predictive control algorithm includes accessing a second piecewise affine map defined by offline solving of the model predictive control algorithm using sample inputs.

38. The method of claim 22 , wherein the motor is a wound field synchronous motor comprising at least three stator phases and at least one rotor field winding.

39. The method of claim 22 , wherein the power switching network includes an inverter switch bridge including a plurality of power switching elements, the inverter switch bridge receiving DC power and outputting AC power to windings of the stator based on pulse width modulated control signals from the electronic controller.

40. The method of claim 22 , further comprising:

receiving, by a DC/DC converter, input DC power; and

providing, by the DC/DC converter, output DC power to at least one rotor field winding in accordance with a pulse width modulated rotor control signal from the electronic controller.

41. The method of claim 22 , wherein the piecewise affine map maps a set of input values to an output flux linkage or inductance value for each dimension of the set of dimensions, the set of input values including the current values and at least one additional motor characteristic.

42. The method of claim 22 , wherein the motor is at least one selected from the group of a wound field synchronous motor, a hybrid synchronous motor, a permanent magnet synchronous motor, an induction motor, a universal motor, or a reluctance motor.

43. A non-transitory computer-readable medium storing computer-executable instructions, the instructions for causing a processor to:

determine current values for a motor in a rotational reference frame, each current value associated with a dimension of a set of dimensions of the rotational reference frame;

determine, based on the current values, a flux linkage or inductance value for each dimension of the set of dimensions of the rotational reference frame using a piecewise affine map;

determine a target value for each dimension of the set of dimensions of the rotational reference frame; and

control a power switching network coupled to the motor and a power supply based on the flux linkage or inductance values and the target values.

44. The non-transitory computer-readable medium of claim 43 , wherein, to determine current values for the motor in a rotational reference frame, the instructions are for further causing a processor to:

determine electrical operational characteristics of the motor in a stationary reference frame;

determine a rotational position of the motor; and

transform the electrical operational characteristics and the rotational position to the current values for the motor in the rotational reference frame.

45. The non-transitory computer-readable medium of claim 43 ,

wherein the piecewise affine map includes a plurality of affine functions, each of the plurality of affine functions associated with a respective domain of a plurality of domains, and

apply the current values to the first affine function to determine the flux linkage or inductance value for each dimension of the set of dimensions of the rotational reference frame.

46. The non-transitory computer-readable medium of claim 43 , wherein, to determine a target value for each dimension of the set of dimensions of the rotational reference frame, the instructions are for further causing a processor to:

determine a desired control parameter for the motor;

transform the desired control parameter to target current values for the motor in the rotational reference frame; and

determine, based on the target current values, the target value for each dimension of the set of dimensions of the rotational reference frame using the piecewise affine map.

47. The non-transitory computer-readable medium of claim 43 , wherein, to determine a target flux linkage value for each dimension of the set of dimensions of the rotational reference frame, the instructions are for further causing a processor to:

determine a desired control parameter for the motor;

determine, based on the target current values, the target value for each dimension of the set of dimensions of the rotational reference frame.

Assignments (6)

RELEASE OF SECURITY INTEREST Recorded Apr 17, 2026

From: DNS-MOTOR BRIDGE 2025, LLC

To: TAU MOTORS, INC.

Reel/Frame 074406/0694 →

RELEASE OF SECURITY INTEREST Recorded Oct 31, 2025

From: DNS-MOTOR BRIDGE 2025, LLC

To: TAU MOTORS, INC.

Reel/Frame 072747/0844 →

SECURITY INTEREST Recorded Oct 31, 2025

From: TAU MOTORS, INC.

To: DNS-MOTOR BRIDGE 2025, LLC

Reel/Frame 072748/0986 →

SECURITY INTEREST Recorded May 19, 2025

From: TAU MOTORS, INC.

To: DNS-MOTOR BRIDGE 2025, LLC

Reel/Frame 071154/0530 →

ASSIGNMENT OF ASSIGNOR'S INTEREST Recorded Dec 23, 2024

From: SWINT, ETHAN BAGGET; VAZQUEZ, NICK MARSHALL; PENNINGTON, WALTER WESLEY, III

To: TAU MOTORS, INC.

Reel/Frame 069668/0725 →

ASSIGNMENT OF ASSIGNOR'S INTEREST Recorded Dec 23, 2024

From: PREINDL, MATTHIAS; STEYAERT, BERNARD WILLIAM

To: THE TRUSTEES OF COLUMBIA UNIVERSITY IN THE CITY OF NEW YORK

Reel/Frame 069668/0854 →

Continuity (6)

Provisional Application 63479352 · Jan 10, 2023

Provisional Application 63420360 · Oct 28, 2022

Provisional Application 63409418 · Sep 23, 2022

Provisional Application 63345879 · May 25, 2022

Provisional Application 63304527 · Jan 28, 2022

Related Publication 20250096709A1 · Mar 20, 2025

References Cited (84)

US 5907228A · Thomas et al. · 1999 [cited by applicant]

US 7243006B2 · Richards · 2007 [cited by applicant]

US 7772790B2 · Nashiki · 2010 [cited by applicant]

US 8786230B2 · Lee et al. · 2014 [cited by applicant]

US 8829831B2 · Yoo · 2014 [cited by applicant]

US 9007004B2 · Hunter · 2015 [cited by applicant]

US 9219441B2 · Yasui et al. · 2015 [cited by applicant]

US 9323254B2 · Furutani et al. · 2016 [cited by applicant]

US 9854012B2 · Na et al. · 2017 [cited by applicant]

US 10725440B1 · Grand et al. · 2020 [cited by applicant]

US 11074378B1 · Yamada et al. · 2021 [cited by applicant]

US 20060055363A1 · Patel et al. · 2006 [cited by applicant]

US 20080129243A1 · Nashiki · 2008 [cited by applicant]

US 20140306641A1 · Igarashi et al. · 2014 [cited by applicant]

US 20200177117A1 · Preindl · 2020 [cited by examiner]

US 20200280276A1 · Thyagarajan et al. · 2020 [cited by applicant]

Arora et al., “Review of Design and performance of permanent magnet synchronous motor”, IJIEEE. Oct. 2015; 3 (10): pp. 20-28. [cited by applicant]

Ayala et al., “Modelling the Flux-Linkage in Permanent Magnet Synchronous Machines,” 2019 IEEE International Electric Machines & Drives Conference (IEMDC), San Diego, CA, USA, 2019, pp. 112-118, doi: 10.1109/IEMDC.2019.… [cited by applicant]

Bedetti et al., “Stand-still self-identification of flux characteristics for synchronous reluctance machines using novel saturation approximating function and multiple linear regression”, IEEE Trans Ind Appl. Jul. 2016;… [cited by applicant]

Brosch et al., “Data-driven recursive least squared estimation for Model predictive current control of permanent magnet synchronous motors”, IEEE TPEL. Feb. 2021; 36(2): pp. 2179-2190. [cited by applicant]

Constaintin Carpiuc et al., “Modeling of synchronous electric machines for real-time simulation and automotive applications”, Journal of the Franklin Institute, vol. 354, Issue 14, Sep. 2017, pp. 6258-6281. [cited by applicant]

Doria-Cerezo et al., “Passivity-based control of a wound-rotor synchronous motor” The Institution of Engineering and Technology, vol. 4, Issue 10, Oct. 2010, p. 2049-2057. [cited by applicant]

Doria-Cerezo et al., “Control of a Stand-Alone Wound Rotor Synchronous Generator: Two Sliding Mode Approaches via Regulation of the d-Voltage Component,” in IEEE Transactions on Control Systems Technology, vol. 20, No. … [cited by applicant]

Haghgooei et al., “Magnetic Model Identification of Wound Rotor Synchronous Machine Using a Novel Flux Estimator”, IEEE. Jun. 2021; 57(5): pp. 5389-5399. [cited by applicant]

Jebai et al., “Sensorless position estimation and control of permanent-magnet synchronous motors using a saturation model”, International Journal of Control, 2015, 89(3), 535-549. [cited by applicant]

Kaimori et al., Behavior Modeling of Permanent Magnet Synchronous Motors Using Flux Linkages for Coupling with Circuit Simulation, IEEJ Journal of Industry Appl., vol. 7 (2018) Issue 1. [cited by applicant]

Nasar, Research Initiation: Analysis and Design of Flux Reversal Machines, Award Abstract #9802454, Aug. 7, 1998. [cited by applicant]

Nie et al., “Deadbeat-Direct Torque and Flux Control of Wound Field Synchronous Machine at Low Sampling to Fundamental Frequency Ratios”, IEEE Transactions on Industry Applications, vol. 55, No. 4, pp. 3813-3822, Jul.-A… [cited by applicant]

Ramu, Research Initiation: Sensor-less Operation of Permanent Magnet Synchronous Motor Drives for Variable Speed Applications in Robotics, Aerospace Actuators and Spindle Drives, Award Abstract # 8809551, Jul. 1, 1988. [cited by applicant]

Roy et al., “Development of Non-linear State-space Affine Model of PMSM and Analysis of Its Dynamic Behavior Encompassing the Possibility of Chaos”, 2022 IEEE International Conference on Electronics, Computing and Commu… [cited by applicant]

Uzel et al., “Gradient descent maximum torque control up to the base speed of wound rotor synchronous motors,” 17th International Conference on Mechatronics—Mechatronika (ME), Prague, Czech Republic, 2016, pp. 1-6. [cited by applicant]

Yang et al., “Multiple-vector-based predictive direct current control for a wound rotor synchronous machine drive”, IEEE. Oct. 2016; 16447462. [cited by applicant]

Pillay et al., “Application Characteristics of Permanent Magnet Synchronous and Brushless dc Motors for Servo Drive”, IEEE, Transactions on Industry Applications. vol. 27. No. 5., Oct. 1991. [cited by applicant]

International Search Report and Written Opinion as issued in corresponding PCT Application No. PCTUS2023/011786, issued Jun. 15, 2023. [cited by applicant]

Alessandri et al., “Receding-horizon estimation for switching discrete-time linear systems”, IEEE Trans. Autom. Control 2005, 50, 1736-1748. https://doi.org/10.1109/TAC.2005.858684. [cited by applicant]

Armando et al., “Experimental Identification of the Magnetic Model of Synchronous Machines,” IEEE Trans. Ind. Appl., vol. 49, No. 5, pp. 2116-2125, Sep. 2013. [cited by applicant]

Armando et al., “Flux linkage maps identification of synchronous AC motors under controlled thermal conditions,” in 2017 IEEE Int. Electr. Mach. Drives Conf. (IEMDC), pp. 1-8, May 2017. [cited by applicant]

Aurenhammer, “Voronoi diagrams, a survey of a fundamental geometric data structure,” ACM Comput. Surv., vol. 23, No. 3, p. 345.405, Sep. 1991. [cited by applicant]

Bilgin et al., “Making the Case for Electrified Transportation”, IEEE Trans. Transp. Electrif. 2015, 1, 4-17. https://doi.org/10.1109/TTE.2015.2437338. [cited by applicant]

Borrelli et al., “Predictive control for linear and hybrid systems,” 2017. [cited by applicant]

Brelje et al., “Electric, hybrid, and turboelectric fixed-wing aircraft: A review of concepts, models, and design approaches”, Prog. Aerosp. Sci. 2019, 104, 1-19. https://doi.org/10.1016/j.paerosci.2018.06.004. [cited by applicant]

Brent, “An algorithm with guaranteed convergence for finding a zero of a function,” Comput. J., vol. 14, pp. 422-425, 1971. [cited by applicant]

Canay, “Causes of discrepancies on calculation of rotor quantities and exact equivalent diagrams of the synchronous machine,” vol. PAS-88, pp. 1114-1120. [cited by applicant]

Chen et al., “Analysis of Flux-Linkage Characteristics of Switched Reluctance Linear Generator,” IEEE Trans. Appl. Supercond., vol. 24, No. 3, pp. 1-5, Jun. 2014. [cited by applicant]

CHEN et al., “Electromagnetic Analysis of Flux Characteristics of Double-Sided Switched Reluctance Linear Machine,” IEEE Trans. Appl. Supercond., vol. 26, No. 4, pp. 1-7, Jun. 2016. [cited by applicant]

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed. 1em plus 0.5em minus 0.4em The MIT Press, 2001. [cited by applicant]

Dai et al., “Development of PMSM Drives for Hybrid Electric Car Applications”, IEEE Trans. Magn. 2007, 43, 434-437. https://doi.org/10.1109/TMAG.2006.887718. [cited by applicant]

De Santiago et al., “Electrical Motor Drivelines in Commercial All-Electric Vehicles: A Review,” IEEE Trans. Veh. Technol., vol. 61, No. 2, pp. 475-484, Feb. 2012. [cited by applicant]

Eull et al., “A review of virtual-flux model predictive control and receding horizon estimation in motor drives,” in 2021 IEEE Transportation Electrification Conference and Expo (ITEC), pp. 439-444, 2021. [cited by applicant]

Eull et al., “Bidirectional nonisolated fast charger integrated in the electric vehicle traction drivetrain,” IEEE Transactions on Transportation Electrification, vol. 8, http://dx.doi.org/10.1109/TTE.2021.3124936DOI 10… [cited by applicant]

Fernanez et al., “Permanent Magnet Temperature Estimation in PM Synchronous Motors Using Low-Cost Hall Effect Sensors”, IEEE Trans. Ind. Appl. 2017, 53, 4515-4525. https://doi.org/10.1109/TIA.2017.2705580. [cited by applicant]

Fernanez et al., “Impact of Machine Magnetization State on Permanent Magnet Losses in Permanent Magnet Synchronous Machines”, IEEE Trans. Ind. Appl. 2019, 55, 344-353. https://doi.org/10.1109/TIA.2018.2867326. [cited by applicant]

Haghgooei et al., “Online Stator Flux Estimation for a Wound Rotor Synchronous Machine (WRSM),” in Ind. Appl. Soc. IEEE-IAS Annu. Meet., pp. 1-6, Oct. 2020, ISSN: 2576-702X. [cited by applicant]

He et al., “A divide-and-conquer approach to determine the pareto frontier for optimization of protein engineering experiments,” Proteins, vol. 80, http://dx.doi.org/10.1002/prot.23237DOI 10.1002/prot.23237, pp. 790-806… [cited by applicant]

Herceg et al., “Multi-parametric toolbox 3.0,” in 2013 European Control Conference (ECC), pp. 502-510, 2013. [cited by applicant]

Ichikawa et al., “Sensorless control of permanent-magnet synchronous motors using online parameter identification based on system identification theory”, IEEE Trans. Ind. Electron. 2006, 53, 363-372. https://doi.org/10.… [cited by applicant]

Jones et al., “Multiparametric linear complementarity problems,” in Proceedings of the 45th IEEE Conf. on Decision and Control, pp. 5687-5692, 2006. [cited by applicant]

Kang et al., “Analysis of irreversible magnet demagnetization in line-start motors based on the finite-element method”, IEEE Trans. Magn. 2003, 39, 1488-1491. https://doi.org/10.1109/TMAG.2003.810330. [cited by applicant]

Kim et al., “Copper-Loss-Minimizing Field Current Control Scheme for Wound Synchronous Machines,” IEEE Trans. Power Electron, vol. 32, No. 2, pp. 1335-1345, Feb. 2017. [cited by applicant]

Kutkut et al., “Analysis of winding losses in high frequency foil wound inductors,” in IAS '95. Conference Record of the 1995 IEEE Industry Applications Conference Thirtieth IAS Annual Meeting, vol. 1, http://dx.doi.org… [cited by applicant]

Li et al., “General Formulation of Kalman-Filter-Based Online Parameter Identification Methods for VSI-Fed PMSM,” IEEE Trans. Ind. Electron., vol. 68, No. 4, pp. 2856-2864, Apr. 2021. [cited by applicant]

Liu et al., “Identification of Flux Linkage Map of Permanent Magnet Synchronous Machines Under Uncertain Circuit Resistance and Inverter Nonlinearity,” IEEE Trans. Industr. Inform., vol. 14, No. 2, pp. 556-568, Feb. 201… [cited by applicant]

Luo et al., “Elimination of harmonic currents using a reference voltage vector based-model predictive control for a six-phase pmsm motor,” IEEE Transactions on Power Electronics, vol. 34, No. 7, pp. 6960-6972, 2019. [cited by applicant]

Miao et al., “Mtpa fitting and torque estimation technique based on a new flux-linkage model for interior-permanent-magnet synchronous machines,” IEEE Transactions on Industry Applications, vol. 53, http://dx.doi.org/10… [cited by applicant]

Nachtsheim et al., “Physically inspired neural network for modeling induction machine nonlinear magnetic saturation,” in 2022 IEEE Transportation Electrification Conference and Expo (ITEC), pp. 874-881, 2022. [cited by applicant]

Pang et al., “Calculation of Cross-Coupling Inductance and Electromagnetic Torque in Wound-Rotor Synchronous Starter/Generator,” IEEE Trans. Ind. Electron, vol. 66, No. 7, pp. 5115-5123, Jul. 2019. [cited by applicant]

V. Pareto, Manual of political economy (manuale di economia politica). 1em plus 0.5em minus 0.4em New York: Kelley, 1971 (1906), translated by Ann S. Schwier and Alfred N. Page. [cited by applicant]

Preindl et al., “Model predictive direct torque control with finite control set for pmsm drive systems, part 1: Maximum torque per ampere operation,” IEEE Transactions on Industrial Informatics, vol. 9, No. 4, pp. 1912-… [cited by applicant]

Preindl et al., “Optimal state reference computation with constrained mtpa criterion for pm motor drives,” IEEE Trans. Power Electron., vol. 30, No. 8, pp. 4524-4535, 2015. [cited by applicant]

Preindl et al., “Robust control invariant sets and lyapunov-based mpc for ipm synchronous motor drives,” IEEE Transactions on Industrial Electronics, vol. 63, http://dx.doi.org/10.1109/TIE.2016.2527722DOI 10.1109/TIE.20… [cited by applicant]

Retif et al., “New hybrid direct-torque control for a winding rotor synchronous machine,” in 2004 IEEE 35th Annual Power Electronics Specialists Conference (IEEE Cat. No. 04CH37551), vol. 2, http://dx.doi.org/10.1109/PE… [cited by applicant]

Rodriguez et al., “State of the art of finite control set model predictive control in power electronics,” IEEE Transactions on Industrial Informatics, vol. 9, No. 2, pp. 1003-1016, 2013. [cited by applicant]

Rossi et al., “Wound Rotor Salient Pole Synchronous Machine Drive for Electric Traction,” in Conf. Rec. IEEE Ind. Appl. Conf., IAS Annu. Meet., vol. 3, pp. 1235-1241, Oct. 2006, iSSN: 0197-2618. [cited by applicant]

P. Samuel and S. Levy, Projective Geometry, ser. Undergraduate Texts in Mathematics. 1em plus 0.5em minus 0.4em Springer New York, 1988. [cited by applicant]

Sayed et al., “Review of Electric Machines in More-/Hybrid-/Turbo-Electric Aircraft”, IEEE Trans. Transp. Electrif. 2021, 7, 2976-3005. https://doi.org/10.1109/TTE.2021.3089605. [cited by applicant]

Seilmeier et al., “Identification of steady-state inductances of PMSM using polynomial representations of the flux surfaces,” in IECON—39th Annu. Conf. IEEE Ind. Electron. Soc., pp. 2899-2904, Nov. 2013, iSSN: 1553-572X. [cited by applicant]

Steyaert et al., “Piecewise affine maximum torque per ampere for the wound rotor synchronous machine,” in 2022 IEEE Transp. Electr. Conf. and Expo (ITEC), pp. 356-363, 2022. [cited by applicant]

S. D. Sudhoff, Transformer Design, pp. 205-253, 2014. [cited by applicant]

Wang et al., “Offline Parameter Self-Learning Method for General-Purpose PMSM Drives With Estimation Error Compensation,” IEEE Trans. Power Electron., vol. 34, No. 11, pp. 11 103-11 115, Nov. 2019. [cited by applicant]

Wang et al., “Fast model predictive control using online optimization,” IEEE Trans. Control Syst. Technol., vol. 18, No. 2, pp. 267-278, 2010. [cited by applicant]

Wlas et al., “Neural-Network-Based Parameter Estimations of Induction Motors,” IEEE Trans. Ind. Electron., vol. 55, No. 4, pp. 1783-1794, Apr. 2008. [cited by applicant]

Zhang et al., “Maximum Torque per Ampere Control with Inductance and Field Current Estimation for Wound-Rotor Synchronous Starter/Generator,” in 23rd Int. Conf. Electr. Mach. Syst. (ICEMS), pp. 172-176, Nov. 2020. [cited by applicant]

Nasar, Award Abstract #9802454, Analysis and Design of Flux Reversal Machines, Aug. 7, 1998. [cited by applicant]

Boldea et al., “Analysis and Design of Flux—Reversal Linear Permanent Magnet Oscillating Machine,” IEEE Xplore 1998. [cited by applicant]