IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 6, DECEMBER 2007 3205 Design and Low-Cost Implementation of an Optimally Robust Reduced-Order Rotor Flux Observer for Induction Motor Control Francesco Alonge, Member, IEEE, Filippo D’Ippolito, Member, IEEE, Giuseppe Giardina, and Tonino Scaffidi Abstract—The aim of this paper is to design and analyze reduced-order observers of the rotor flux of induction motors. The design is carried out in two steps. In the first step, a bound- ary of the stability region of the observation error is obtained corresponding to a chosen Lyapunov function. In the second step, the boundary is translated into a performance index that is minimized with respect to stator and rotor resistance variations and differences of voltages supplying the motor and those sup- plying the observer in order to obtain the largest stability region. Implementation of the observer on a low-cost fixed-point digital signal processor using look-up tables is described. Experimental results are shown with reference to a prototype consisting of a simple proportional–integral controller, the proposed observer, and a 2.2-kW induction motor; the implementations of both the controller and the observer are carried out on a DS1104 dSpace microcontroller using the fixed-point option. Index Terms—Induction motors, low-cost implementation, rotor flux observers. NOMENCLATURE i a (i b ) Stator current component along the a-axis (b-axis) fixed to the stator (in amperes). ν a (ν b ) Stator voltage component along the a-axis (b-axis) (in volts). φ e Rotor flux vector. φ (= (L m /L r )φ). Scaled rotor flux vector. φ a (φ b ) Component of φ along the a-axis (b-axis) (in webers). ω Speed (in radians per second). R s (L s ) Stator resistance (inductance) [in ohms (in henry)]. L m (L r ) Mutual (rotor) inductance (in henry). τ r Rotor time constant (in seconds). L e (= L s - (L 2 m /L r )). Stator equivalent inductance (in henry). “0” Subscript for nominal parameters. λ i (Q) ith eigenvalue of matrix Q. I. I NTRODUCTION Processing control laws for induction motors based on rotor flux feedback requires knowledge of the rotor flux vector in the Manuscript received November 9, 2005; revised July 23, 2007. This work was supported in part by the University of Palermo (ex 60%). The authors are with the Dipartimento di Ingegneria dell’Automazione e dei Sistemi, Università di Palermo, 90128 Palermo, Italy (e-mail: alonge@ unipa.it). Digital Object Identifier 10.1109/TIE.2007.905632 frame fixed with the stator. In order to estimate the rotor flux components, full- and reduced-order observers can be imple- mented. Full-order observers give estimates of the stator current and rotor flux components from measurements of the stator voltage and stator current components [1]–[3]. The principal advantage of these observers is due to the availability of stator current estimates for processing the control law, which are less noisy than the measured ones. A full-order observer for sensorless direct field-oriented control has been proposed in [4] using a proportional–integral (PI)-type adaptation law for speed estimation; the effects of sta- tor and rotor resistance variations on speed estimation are also analyzed, and a heuristic updating law for these parameters is given. The gains of the observer are chosen so that its poles are proportional to those of the motor. The same speed adaptation law has been considered in [5] and [6], where a method is given for determining both the gains of the observer and the parame- ters of the speed adaptation law. In [7], a full-order observer is considered in which the gain matrix is parameterized with the rotor speed, estimated using Model Reference Adaptive Systems (MRAS) techniques, and adjusted online according to a given algorithm derived from a pole placement technique. In [8] and [9], methods for designing adaptive full-order observers are designed with the aim of overcoming the stability problems that arise in sensorless vector control of induction motors based on adaptive full-order observers in the regenerating mode at low-speed operations. The reduced-order observers only give estimates of the rotor flux components starting from the same measurements as for full-order observers [10]–[14]. Processing of the control law is carried out using estimated rotor flux and measured stator currents. Filters of suitable bandwidth are employed in order to reproduce current signals with suitable signal-to-noise ratio and negligible delay. In view of using DSP boards, the same filters can also be used for avoiding aliasing. It follows that the use of reduced-order observers is a valid alternative to that of full-order observers. Rotor flux estimation algorithms have also been proposed and theoretically and experimentally tested in the contest of par- ticular control laws. For example, in [15], an interesting output feedback control law is studied for a current-fed induction mo- tor considering a third-order model that describes the dynamics of the rotor flux components and the speed in which the control variables are the stator current components; in this case, in fact, the dynamics of the stator current components can be neglected. Obviously, the dynamic system that gives estimates of the rotor 0278-0046/$25.00 © 2007 IEEE