On the optimal feedforward torque control problem of anisotropic   synchronous machines: Quadrics, quartics and analytical solutions
  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. 
Hisham Eldeeb, Christoph M. Hackl, Lorenz Horlbeck and Julian Kullick
cs.SY