  The optimal feedforward torque control problem is tackled and solved
analytically for synchronous machines while stator resistance and
cross-coupling inductance are explicitly considered. Analytical solutions for
the direct and quadrature optimal reference currents are found for all major
operation strategies such as Maximum Torque per Current (MTPC) (or often called
as Maximum Torque per Ampere (MTPA)), Maximum Current (MC), Field Weakening
(FW), Maximum Torque per Voltage (MTPV) and Maximum Torque per Flux (MTPF).
Numerical methods (approximating the optimal solutions only) or simplifying
assumptions (neglecting stator resistance and/or cross-coupling inductance) are
no longer necessary. The presented results are based on one simple idea: all
optimization problems (e.g. MTPC, MTPV or MTPF) with their respective
constraints (e.g. current or voltage limit) and the computation of the
intersection point(s) of voltage ellipse, current circle, or torque, MTPC, MTPV
and MTPF hyperbolas are reformulated implicitly as quadrics (quadratic
surfaces) which allow to solve the feedforward torque control problem by
invoking the Lagrangian formalism and by finding the roots of a fourth-order
polynomial analytically. The proposed solutions are applicable to any
anisotropic (or isotropic) synchronous machine independent of the underlying
current control strategy.
