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. Keywords—Maximum Torque per Ampere (MTPA), Maximum Torque per Current (MTPC), Maximum Torque per Voltage (MTPV), Maximum Torque per Flux (MTPF), Maximum Current (MC), Field Weakening (FW), analytical solution, optimal feedforward torque control, anisotropy, synchronous machine, interior permanent-magnet synchronous machine, reluctance synchronous machine, permanent-magnet assisted or enhanced synchronous machine, quadrics, quartics, Lagrangian optimization NOTATION N,R,C: natural, real, complex numbers. x := (x1, . . . , xn) ∈ R: column vector, n ∈ N where “>” and “:=” mean “transposed” (interchanging rows and columns of a matrix or vector) and “is defined as”, resp., 0n ∈ R: zero vector. a>b := a1b1 + · · · + anbn: scalar product of the vectors a := (a1, . . . , an) and b := (b1, . . . , bn). ‖x‖ := √ x>x = √ x1 + · · ·+ xn: Euclidean norm of x. A ∈ Rn×n: (square) matrix with n rows and columns. A−1: inverse of A (if exists). det(A): determinant of A, spec(A): spectrum of A (the set of the eigenvalues of A). In ∈ Rn×n := diag(1, . . . , 1): identity matrix. On×p ∈ Rn×p: zero matrix, n, p ∈ N. Tp(φk) = [ cos(φk) − sin(φk) sin(φk) cos(φk) ] : Park transformation matrix (with electrical angle φk) and J := Tp(π/2) = [ 0 −1 1 0 ] : rotation matrix (counter-clock wise rotation by π2 ; see e.g. [1], [2]). “s.t.”: subject to (optimization with constraints). : imaginary unit with  = √ −1, “ ! =”: must equal. X ∩ Y : intersection of the sets X and Y (in this paper: X, Y ⊂ R).