Flatness-based hierarchical control of the PM synchronous motor Emmanuel Delaleau* Universit´ e Paris-sud Laboratoire des signaux et syst` emes Gif-sur-Yvette, France Emmanuel.Delaleau@lss.supelec.fr Aleksandar M. Stankovi´ c Northeastern University ECE Department Boston, USA astankov@ece.neu.edu Abstract— The paper describes a flatness-based control scheme for the permanent magnet synchronous motor. A hierarchical control is developed and the scheme is shown to be compatible with the flatness properties via a detailed time-scale analysis. The proposed control strategy also achieves copper loss minimization at all operating points. Detailed simulations are presented to illustrate stability and robustness properties of the control scheme. Index Terms— Permanent magnet synchronous motor control, hier- archical control, differential flatness, robust control I. I NTRODUCTION The industrial importance of permanent magnet synchronous motor drives has been increasing for a number of years. This class of motors achieves very good efficiency that equals and sometimes even surpasses the efficiency of induction motor drives which have been dominating the industry for a long time. Permanent magnet machines also maximize torque per unit volume and weight, resulting in numerous applications in vehicles and autonomous systems. While a number of control schemes have been proposed for permanent magnet drives in the literature, our aim in this paper is to explore a new class of nonlinear control algorithms based on the concept of differential flatness. We show that an appropriate choice of the “direct current” allows to minimize copper losses in every mode of operation. Moreover, we established that the flatness-based control is compatible with standard hierarchical control schemes. The stability of the new control scheme, which involve a load torque observer for on-line re-parameterization of trajectories, is established via a precise singular and regular perturbation study of the tracking error equation, coupled with the application of an advance stability result. Since we are interested in industrial appli- cations, we will also evaluate stability robustness of our scheme, in particular with respect to large parametric perturbations that are typical for high-performance applications. II. SHORT REVIEW OF DIFFERENTIAL FLATNESS The differential flatness is an important structural property of many control systems [1]. Consider a nonlinear control system given by a state-variable representation: ˙ x = f (x, e) (1) where e =(e1,...,em) ⊤ is the input and x =(x1,...,xn) ⊤ is the state. System (1) is said to be (differentially) flat if and only if there exists a set of m variables z =(z1,...,zm) ⊤ having the following 3 properties: 1) z = h(x, e, . . . , e (α) ); *Work performed when E. Delaleau was visiting professor in the ECE Department at Northeastern University with a grant from “Direction g´ en´ erale pour l’armement”. 2) every variable of (1) can be expressed in terms of z and a finite number of its time derivatives, in particular: x = A(z, ˙ z,...,z (β) ) (2a) e = B(z, ˙ z,...,z (β+1) ); (2b) 3) the components of z are differentially independent. Such a set of variables z =(z1,...,zm) is called a flat output or linearizing output of the system (1). The synthesis of control laws using differential flatness or flatness-based control is done in two steps: a) Design of an open-loop nominal control corresponding to the predicted trajectory of the flat output; b) Application of feedback law in order to stabilize the real trajectory around the predicted trajectory of the flat output. A complete flatness-based control methodology is presented in [2], [3] with stabilization and robustness results. III. MODEL OF THE MOTOR IN DQ FRAME AND FLATNESS Consider the DQ model of a synchronous motor with permanent magnets (see for example [6]): L d di d dt = v d − Rsi d + npLq Ωiq (3a) Lq diq dt = vq − Rsiq − npL d Ωi d − npΦ f Ω (3b) ˙ θ = Ω (3c) J ˙ Ω = np [Φ f +ΔLi d ] iq | {z } Te −TL (3d) where notation are usual and ΔL = L d − Lq is the difference between the direct and quadrature inductances. The model (3) of the synchronous motor is flat with flat output z = (θ, i d ,TL). The proof is straightforward and consists of checking the relevant conditions. The main point is the possibility to express every variable of (3) —namely, Ω,i d ,v d and vq — in terms of θ, i d ,TL and their derivative from algebraic manipulation of the equations (3). IV. EFFICIENCY OPTIMIZATION This section is devoted to the study of copper losses (Joule’s effect) minimization by the control law. The flatness property allows us to derive a simple and elegant solution. The minimum of copper losses is achieved by the appropriate choice of i d value. This is a component of the flat output and, consequently, one can achieve a control in order to make this current to track a nominal desired trajectory. An interesting and useful property of the differentially flat systems is that every variable can be expressed in terms of the flat output components and their derivatives. Therefore, one can Proceeding of the 2004 American Control Conference Boston, Massachusetts June 30 - July 2, 2004 0-7803-8335-4/04/$17.00 ©2004 AACC WeA03.2 65