Nonlinear PI current control of reluctance synchronous machines

This paper discusses nonlinear proportional-integral (PI) current control with anti-windup of reluctance synchronous machines (RSMs) for which the flux linkage maps are known. The nonlinear controller design is based on the tuning rule “Magnitude Optimum criterion” [1]. Due to the nonlinear flux linkage, the current dynamics of RSMs are highly nonlinear and, so, the parameters of the PI controllers and the disturbance compensation feedforward control must be adjusted online (e.g., for each sampling instant). The theoretical results are illustrated and validated by simulation and measurement results. NOTATION N,R: natural, real numbers. x := (x1, . . . , xn) > ∈ R: column vector, n ∈ N. 0n ∈ R: zero vector. ‖x‖ := √ x > x: Euclidean norm of x. A ∈ Rn×m: real matrix, n,m ∈ N, det(A): determinant of A. In ∈ Rn×n: identity matrix. C(I;Y ): space of continuously differentiable functions mapping I → Y . For modeling of electrical machines, a signal ξ may be represented in the three-phase (a, b, c)reference frame ξ := ( ξ a , ξ b , ξ c )> , the stator-fixed sreference frame ξ := ( ξ α , ξ β )> and the arbitrarily rotating k-reference frame ξ := ( ξ d , ξ q )> , which are related by ξ = Tp(φk) −1ξs = Tp(φk) Tcξ . φk [rad] is the (electrical) angle of the k-reference frame with respect to the s-reference frame and Tp(φk) = [ cos(φk) − sin(φk) sin(φk) cos(φk) ] , J = [ 0 −1 1 0 ] and Tc = 2 3 [ 1 − 1 2 − 1 2 0 √ 3 2 − √ 3 2 ] are Park, rotation (by π2 ) and (amplitude correct) Clarke transformation matrix, respectively (see [2, 3]).