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2. i/Fwhere estimated quantities are marked by ^ and Us,ref is thestator voltage reference. The estimate of the stator current and Us,refAd- +/mthe estimation error of the stator current are ==>justableL-(,Os -,pm) --A1tmodel js is = (5) ALis~ ~ 1is = il *1 - is (6)respectively, where iV is the measured stator current expressed Fig. 1. Block diagram of the adaptive observer.in the estimated rotor reference frame. The feedback gainmatrix A is proportional to the rotor speed up to the nominalspeed [17].II. PMSM MODEL The speed adaptation is based on an error term The PMSM is modeled in the d-q reference frame fixed tothe rotor. The d axis is oriented along the PM flux, whoseFe =Clis(7)angle in the stator reference frame is 0m in electrical radians.The stator voltage equation iswhere Ci = [O Lq], i.e., the current error in the estimated qdirection is used for adaptation. The estimate of the electrical Us Rsis +s WmJ+5+(1)angular speed of the rotor is obtained using a PI speedwhere us [ud Uq ]T is the stator voltage, is [id iq ]T theadaptation mechanismstator current, sb= [bd /q ]T the stator flux, Rs the statorresistance, Wm =m the electrical angular speed of the rotor,Wm=-kpF -kiJ Fedt (8)andThe stator flux is JK0 -1I where kp and ki are nonnegative gains. The gain selection isdescribed in [17]. The estimate Om for the rotor position isevaluated by integrating Winsb= Lis+ bpm (2)B. High-Frequency Signal Injectionwhere 14pm= [/pm O]T is the PM flux andThe adaptive observer described above is augmented with aHF signal injection method to stabilize the speed and positionL L Lq Ld fq] estimation at low speeds [17]. By using the HF signal injectionmethod with an alternating voltage u, as a carrier excitationis the inductance matrix, Ld and Lq being the direct- and signal [18], an error signal E 2K Om proportional to thequadrature-axis inductances, respectively. The electromagneticposition estimation error 0m = m 0-m is obtained, KStorque is given bybeing the signal injection gain. The error signal is used forcorrecting the estimated position by influencing the directionTe=3p2TJTis(3)of the stator flux estimate of the adjustable model. For thecombined observer, the adjustable model (4) is modified towhere p is the number of pole pairs.III. SPEED AND POSITION ESTIMATION Qs = Us,ref -Rss- (m - )JQs + Als(9)A. Adaptive Observerwhere An adaptive observer [17] is used for the estimation of thestator current, rotor speed, and rotor position. The speed andSE: = apE + -aj+dt(10)position estimation is based on the estimation error betweenis the speed correction term, -yp and tYj being the gains of thetwo different models; the actual motor can be considered as aPI mechanism driving the error signal E to zero. In accordancereference model and the observer including the rotor speedwith [6], these gains are selected asestimate Wm as an adjustable model. The error term usedin a speed adaptation mechanism is based on the estimation2Ki 2 (1 1)error of the stator current. The estimated rotor speed, obtained/yP 2K6KEusing the adaptation mechanism, is fed back to the adjustablemodel.where ai is the approximate bandwidth of the PI mechanism. The adaptive observer is formulated in the estimated rotorAt low speeds, the combined observer relies both on thereference frame. The block diagram of the adaptive observer signal injection method and on the adaptive observer. Theis shown in Fig. 1. The adjustable model is based on (1) andinfluence of the HF signal injection is decreased linearly as the(2), and defined by speed increases by decreasing both the HF excitation voltageand the bandwidth ai. At speeds higher than a threshold speed ,bs = Us,ref -Rsis- mJQs + Ais(4)WoA, the estimation is based only on the adaptive observer. 176 3. TABLE I IV. PARAMETER ADAPTATIONMOTOR DATA The current estimation error is of the adaptive observer Nominal voltage UN370 Vcontains information that can be used for the parameter Nominal current IN 4.3 Aadaptation. In [7], [10], the components of is are used for Nominal frequency fN75 HzNominal torque TN14.0 Nmthe adaptation of two parameters in a PMSM drive equipped Stator resistance RS3.59 Qwith a motion sensor. Solving the parameters from the steady- Direct-axis inductance Ld36.0 mHstate voltage equations as in [11] would require filtering to Quadrature-axis inductance Lq51.0 mHPM flux WmoLdLqiq the result iscan be driven to zero by adjusting the parameter estimates.R,WmiqL 2L q[.td Rlk ;qqiqRsMl(17)A. Steady-State Analysis Stator Resistance Adaptation L ejfpm L Som -L opm At low speeds, the stator resistance estimate plays an im-It can be seen that both variables on the left-hand side dependportant role in the speed and position estimation, particularlyboth on Rs and