2026 Theses Doctoral

Optimal Piecewise Affine Current-Flux and Piecewise Quadratic Torque Models for Synchronous Motors with Conservation of Energy Bounds

Steyaert, Bernard

Synchronous machines are central to electrification across automotive, aerospace, and industrial sectors. As machine designs grow more compact and magnetically saturated, accurate nonlinear flux modeling becomes critical for high-performance field-oriented control. Conventional magnetic models face a fundamental trade-off: linear inductance maps preserve desirable control properties such as invertibility and energy conservation but cannot capture saturation, while nonlinear models such as look-up tables and spline interpolants can represent saturation but sacrifice these guarantees.

This dissertation develops optimal piecewise-affine (PWA) current-flux maps for synchronous machines that resolve this trade-off. The proposed maps are locally linear and globally continuous, retain invertibility, model magnetic saturation and cross-saturation, and, uniquely among nonlinear offline magnetic models, guarantee conservation of energy. Energy conservation is established by requiring each local inductance matrix to be positive definite, yielding a strictly pseudoconvex piecewise-quadratic (PWQ) energy function from which the PWA map is derived as its gradient. It is shown that the PWQ electromagnetic torque is continuous for any PWA flux map, whereas PWQ energy continuity requires symmetric local inductance matrices. A scaling analysis demonstrates that enforcing exact symmetry becomes infeasible in three dimensions in large or irregularly gridded vertex formations, motivating penalization and bounding strategies based on a Stokes-theorem energy-error metric. An asymptotic analysis confirms that the energy error of unconstrained PWA fits decreases monotonically as vertex density increases, demonstrating that inductance matrix skew is an artifact of the piecewise-affine approximation itself rather than a property of the underlying physical system.

The optimal PWA map is obtained by solving a convex semidefinite program (SDP) that minimizes the least-squares flux error with respect to finite-element analysis (FEA) or experimental data subject to positive-definite constraints. The formulation supports non-gridded vertex placement, and a heuristic for irregularly gridded vertex placement via K-means clustering in a stacked current–inductance space is proposed, concentrating vertices where the magnetic nonlinearity is greatest. Three regularization strategies, skew penalization, global energy budgets, and local energy budgets, are developed to control the trade-off between fit accuracy and energy conservation. The approach is validated on a 10 kW permanent-magnet synchronous machine (PMSM) using experimental data and a 65 kW wound-rotor synchronous machine (WRSM) using FEA data.

Compared to conventional linear inductance maps, the optimal PWA maps achieve a 90% reduction in average flux error for the PMSM and 39% for the WRSM. Comparisons against industrystandard linear and cubic-spline look-up tables show competitive accuracy with the added guarantee of energy conservation. Plant simulations confirm that non-energy-conservative models produce residual power and torque errors on the order of 1–2% per unit, while the proposed PWA maps maintain these residuals near zero. Real-time microcontroller implementation demonstrates that PWA map evaluation requires comparable floating-point operations to existing look-up table methods, confirming suitability for embedded motor control.