Sensorless vector control of psms drives wquipped with inverter output filter


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Sensorless vector control of psms drives wquipped with inverter output filter

  1. 1. Sensorless Vector Control of PMSM Drives Equipped With Inverter Output Filter Janne Salomäki, Antti Piippo, Marko Hinkkanen, and Jorma Luomi Power Electronics Laboratory Helsinki University of Technology P.O. Box 3000, FI-02015 TKK, Finland Abstract— The paper presents a sensorless vector control If the control method requires only the measurements of themethod for a permanent magnet synchronous motor when the inverter output current and the dc-link voltage, a filter can beoutput voltage of the PWM inverter is filtered by an LC filter. The added to an existing drive, and no hardware modifications aredynamics of the LC filter are taken into account in the designof the controller and adaptive full-order observer. The use of needed in the frequency converter. Full-order observers havethe output filter does not require additional current or voltage recently been proposed for induction motor drives equippedmeasurements. The speed adaptation is based on the estimation with an LC filter [13]–[15], thus avoiding additional currenterror of the inverter output current. Linearization analysis is or voltage measurements.used to design an observer that enables a wide operation region. In this paper, a sensorless control method is developed for aSimulation and experimental results show the functionality of theproposed control method. PMSM drive equipped with an LC filter. Cascaded controllers are used for controlling the inverter current, the stator voltage, and the stator current. An adaptive full-order observer is used I. I NTRODUCTION for estimating the stator voltage, the stator current, the rotor speed, and the rotor position. The observer gain is selected by Problems may be encountered in AC motor drives due to the using a linearized model. Finally, simulation and experimentalnon-sinusoidal voltage produced by a pulse-width modulated results are presented.(PWM) inverter. The high rate of change of the voltage(i.e. high du/dt) may cause excessive voltage stresses in thestator winding insulations. It may also cause leakage currents II. F ILTER AND M OTOR M ODELSthrough the parasitic capacitances of the stator winding and Fig. 1 shows a PMSM drive system equipped with an LCproduce bearing currents. Lower-order harmonics cause acous- filter. The inverter output voltage uA is filtered by the LCtic noise and power losses; the losses caused by eddy currents filter, resulting in a nearly sinusoidal stator voltage us . Theare a special concern in high-speed solid-rotor motors. inverter output current iA and the dc-link voltage udc are the A common approach to overcome these problems is to only measured quantities.use an inverter output filter [1]–[6]. An LC filter, having In the d-q reference frame fixed to the rotor, the model ofthe resonance frequency below the switching frequency, is a the three-phase LC filter and the PMSM can be written astypical choice for the filter topology if a nearly sinusoidaloutput voltage is required. However, this kind of heavy filtering T x = Ax + B uA ψ pm ˙ (1)affects the vector control of the motor. The filter replacements PSfrag dynamicsshould be taken into account in the control design. iA = Cx (2) Various methods have been proposed for the vector controlof variable-speed drives equipped with an LC filter. Methodsbased on a feedforward action and sliding mode control have ωm,ref iAbeen proposed for compensating the effects of the filter in Controla sensorless permanent magnet synchronous motor (PMSM) udc Lfdrive [4]. A model-based observer and an adaptive speed us isestimator have been implemented in the stator reference framefor estimating the rotor angle and speed in a sensorless PMSM PMSMdrive [5]. A feedforward current controller has been used in Grida speed-sensored synchronous reluctance motor drive with uA Diode bridge Cfan LC filter [6]. In these methods, stator current or stator Invertervoltage measurements are needed. Vector control methods forinduction motor drives with an LC filter in [7]–[12] alsorequire measurements from the motor side of the filter. Fig. 1. PMSM drive system equipped with three-phase LC filter. 1-4244-0136-4/06/$20.00 2006 IEEE 1059
  2. 2. T where x = iA us ψ s is the state vector consist- cascade control and the speed-adaptive full-order observer ing of the inverter output current iA = iAd iAq , T are implemented in the estimated rotor reference frame. The T ˆ estimated rotor position θm is obtained by integrating ωm . ˆ the stator voltage us = usd usq , and the stator flux T The inverter current, the stator voltage, and the stator current linkage ψ s = ψsd ψsq . The inverter output voltage T are controlled by PI controllers, and cross-couplings due to uA = uAd uAq and the permanent magnet flux ψ pm = the rotating reference frame are compensated. A simple one- T ψpm 0 are considered as inputs to the system. The matrix step-ahead current prediction is used in the inverter current transpose is denoted by superscript T . The system matrices in control in a fashion similar to [16]. A maximum torque per (1) and (2) are current method [17] is used for calculating the stator current   reference. The rotor speed is governed by a PI controller with −RLf L−1 I − ωm J −L−1 I f f 0 active damping. A= −1 Cf I −ωm J −Cf L−1  −1 s −1 0 I −Rs Ls − ωm J (3) IV. S PEED -A DAPTIVE F ULL -O RDER O BSERVER  −1  Lf I 0 A. Observer Structure B= 0 Cf L−1  −1 s (4) A speed-adaptive full-order observer has been successfully −1 0 R s Ls used in a sensorless induction motor drive equipped with an C= I 0 0 (5) LC filter [14], [15]. A similar observer structure is constructed for the PMSM drive in the following. where Lf is the inductance and RLf is the series resistance of The inverter current is the feedback signal for the observer, the filter inductor, Cf is the filter capacitance, Rs is the statorPSfrag replacements the electrical angular speed of the rotor, and and the electrical angular speed of the rotor is estimated using resistance, ωm is an adaptation mechanism. The observer is defined by 1 0 0 −1 T I= , J= ˙ 0 1 1 0 ˆ ˆ ˆ x = A x + B uA ˆ ψ pm + K(iA − ˆA ) i (6) The stator inductance matrix where the system matrix and the observer gain matrix are L 0   Ls = d −RLf L−1 I − ωm J −L−1 I ˆ 0 0 Lq f f ˆ A= −1 Cf I −ˆ m J ω −Cf L−1  −1 consists of the direct-axis inductance Ld and quadrature-axis s inductance Lq . 0 I −Rs L−1 − ωm J s ˆ (7)   III. C ONTROL S YSTEM k1d I + k1q J Voltage Fig. 2 control a simplified block diagram of the control shows K = k2d I + k2q J , (8) system (the estimated quantities being marked by ˆ). The k3d I + k3q J Estimated rotor Stator ˆs reference frame reference frame i is,ref us ωm,ref ˆ us,ref Speed iA,ref control Stator udc current Stator uA,ref control voltage Inverter control current ˆ eJ θm PWM control iA ˆ uA,ref e−Jθm ˆ ψR ˆ us ˆs i Adaptive ˆ θm ωm ˆ full-order observer PMSM Fig. 2. Simplified block diagram of the control system. Double lines indicate vector quantities whereas single lines indicate scalar quantities. The speed control includes the calculation of the stator current reference according to the maximum torque per current method. 1060
  3. 3. PSfrag replacements TABLE I iAq − ˆAq i Motor Parameters ωm + ωm ˆ Stator resistance Rs 3.59 Ω G(s) H(s) Direct-axis inductance Ld 36.0 mH Quadrature-axis inductance Lq 51.0 mH − Permanent magnet flux ψpm 0.545 Vs Total moment of inertia J 0.015 kgm2 Rated speed 1500 r/min Rated current (rms) 4.3 A Rated torque 14.0 Nm Fig. 3. Signal flow diagram of linearized model. LC Filter Parameters Inductance Lf 5.1 mH Capacitance Cf 6.8 µF respectively. The factors kid and kiq (i = 1, 2, 3) are scalar Series resistance RLf 0.1 Ω gain parameters. The adaptation law is ωm = −Kp (iAq − ˆAq ) − Ki ˆ i (iAq − ˆAq )dt i (9) The transfer function from the rotor speed estimation error ωm = ωm − ωm to the q component of the inverter current ˜ ˆ where Kp and Ki are nonnegative adaptation gains. The digital estimation error ˜Aq = iAq − ˆAq is i i implementation of the adaptive full-order observer is based on a symmetric Euler method [18]. G(s) = C (sI − A )−1 B (12) where C = 0 1 0 0 0 0 0 . Based on (9), the B. Linearization transfer function from iAq − ˆAq to the rotor speed estimate i ωm is ˆ Adaptive observers can be analysed via linearization [19], Ki [20]. The following linearization is carried out in the estimated H(s) = −Kp − . (13) s rotor reference frame in a fashion similar to [21]. Accurate pa- The resulting linearized model is illustrated in Fig. 3. rameter estimates are assumed for the analysis. The estimation ˜ ˆ error of the rotor position θm = θm − θm is taken into account in the system equation (1), and operating-point quantities are C. Observer Gain Selection marked by the subscript 0. The resulting linearized model is The linearized model is used for the observer gain selection. ˙ The parameters of a 2.2-kW six-pole interior-magnet PMSM ˜ x A0 − K 0 C D0 ˜ x 0 ˙ = ˜m + 1 ωm ˜ (10) (370 V, 75 Hz) and an LC filter, used for the following ˜m θ 0 0 θ analysis, are given in Table I. The base values of √ angular the A B frequency, current, and voltage are 2π · 75 rad/s, 2 · 4.3 A, and 2/3 · 370 V, respectively. The parameter values Kp = where   25 (As)−1 and Ki = 20 000 (As2 )−1 are used in the speed 0 adaptation.Sfrag replacements  PSfrag replacements  PSfrag replacements 0 Fig. 4(a) shows the poles of the linearized model as the D0 =  −1 −1 −1 Cf JLs (ψ pm − ψ s0 ) + Cf L−1 Jψ s0   (11) T s observer gain is zero, i.e. K = [0 0 0] . The angular speed Rs JL−1 (ψ pm − ψ s0 ) + Rs L−1 Jψ s0 s s of the rotor varies from −1 p.u. to 1 p.u. and the load torque (a) (a) (b) (b) (c) 15 (c) 15 15 0.01 0.01 Magnification Magnification Imaginary Axis (p.u.) Imaginary Axis (p.u.) Imaginary Axis (p.u.) 0 0 10 10 10 20 20 20 −0.01 −0.01 0.01 −0.01 0 0.01 −0.01 0 0.01 −0.01 5 5 5 −4 −4 −4 −3 −3 −3 0 0 0 −2 −1 0 1 −2 −1 0 1 −2 −1 0 1 Real Axis (p.u.) Real Axis (p.u.) Real Axis (p.u.) ωm0 ωm0 ωm0 (a) (b) (c) Magnification Fig. 4. Poles of linearized model as rotor angular speed ωm0 is varied from −1 p.u. to 1 p.u. and load torque is at positive rated value. Observer gain is (a) K = [0 0 0]T , (b) K = [(2000 s−1 )I 0 0]T , and (c) proposed gain. 1061
  4. 4. is at the positive rated value. The adaptive observer is unstable 0.1 ωm (p.u.)because of poles in the right half-plane. Due to the presence 0of the LC filter, zero gain cannot be used in the observer.Similar behavior has been reported for induction motor drives -0.1[13]–[15]. 0 1 2 3 4 2 The instability caused by the LC filter can be avoided by 1 Te /TNusing a simple constant gain: 0 T K = k1d I 0 0 (14) PSfrag replacements -1 0 1 2 3 4Fig. 4(b) shows the poles obtained using (14) with k1d = θm − θm (deg.) 1802000 s−1 . The poles originating from the LC filter are movedto the left half-plane. However, the saliency of the PMSM still 0 ˆcauses instability at low speeds in the motoring mode [21]. In -180this example, right half-plane poles appear at the rated load in 0 1 2 3 4the speed range between 0 and 0.08 p.u. t (s) (a) To further improve the stability, the gain   k1d I 0.1 ωm (p.u.) K= 0  (15) k3d I + k3q sign(ˆ m )J ω 0is proposed. Fig. 4(c) shows the poles obtained using this gain -0.1 0 1 2 3 4with k1d = 2000 s−1 and k3d = k3q = 4Rs . All poles stay 2in the left half-plane in the whole inspected operation region 1 Te /TN(except a pole in the origin at ωm = 0). It is to be noted that 0in practice, parameter and measurement errors cause stability -1problems at low speeds under load. 0 1 2 3 4 θm − θm (deg.) 10 V. S IMULATION R ESULTS 0 ˆ The system was investigated by computer simulations usingthe M ATLAB/Simulink software. Accurate motor and filter -10 0 1 2 3 4parameters, given in Table I, were used in the control. The t (s)sampling frequency was equal to the switching frequency of (b)5 kHz. The bandwidths of the controllers were 2π · 600 rad/s Fig. 5. Simulation results showing rated load torque step using (a) constantfor the inverter current, 2π · 400 rad/s for the stator voltage, observer gain (14) and (b) proposed observer gain (15). Speed reference is set2π · 200 rad/s for the stator current, and 2π · 4 rad/s for the to 0.067 p.u. (5 Hz). The first subplot shows the speed reference (dashed),rotor speed. the actual rotor speed (solid), and its estimate (dotted). The second subplot shows the electromagnetic torque of the PMSM. The third subplot shows the Fig. 5 shows a simulated comparison of the observer gains estimation error of the rotor position in electrical degrees.(14) and (15). The speed reference is kept constant, and ratedload torque is applied stepwise at t = 0.5 s. In Fig. 5(a),the constant gain (14) is used, and the system becomes PPC/DSP board. The motor and filter data are given in Table I.unstable after the load torque step. Fig. 5(b) shows the sameAt the startup, a dc voltage was applied for 0.4 s to force-sequence using the proposed gain (15). The system works now align the rotor with the stator-produced magnetic field beforesuccessfully. the controllers and the observer were enabled. Simple current Fig. 6 shows simulation results obtained for a sequence feedforward compensation for dead times and power deviceconsisting of a speed reference step from zero speed to voltage drops was applied [22].0.67 p.u., a rated load torque step, a load removal, and a Fig. 8 shows experimental results corresponding to thedeceleration to standstill. The proposed observer and controlsimulation shown in Fig. 6. The measured performance is inmethod work fine. It is to be noted that the q components of accordance with the simulation results. An oscillation at thethe inverter and stator currents are nearly equal in steady state, sixth harmonic can be seen in the currents under load. Thisbut the d components differ from each other at higher speeds.oscillation originates mainly from the inductance harmonicsAfter the deceleration, a steady-state error exists in the rotor of the motor; it exists even when the drive is used without theposition estimation because the system is not observable at filter.standstill. Fig. 9 shows experimental results obtained for a sequence consisting of an acceleration to 0.27 p.u., a rated load torque VI. E XPERIMENTAL R ESULTS step, a speed reversal, a load removal, and a deceleration to The experimental setup is illustrated in Fig. 7. The fre- standstill. The fast speed reversal is successful. The inverterquency converter was controlled by a dSPACE DS1103 and stator voltage and current waveforms are shown in detail in 1062
  5. 5. 1 1 ωm (p.u.) ωm (p.u.) 0.5 0.5 0 0 -0.5 -0.5 0 1 2 3 4 0 1 2 3 4 2 2 θm − θm (deg.) iAd , isd (p.u.) iAq , isq (p.u.) θm − θm (deg.) iAd , ˆsd (p.u.) iAq , ˆsq (p.u.) 0 0 i -2 -2 0 1 2 3 4 0 1 2 3 4 0.5 0.5 0 0 i -0.5 -0.5 0 1 2 3 4 0 1 2 3 4 20 PSfrag replacements 20 0 0 ˆ ˆ -20 -20 0 1 2 3 4 0 1 2 3 4 t (s) t (s) Fig. 6. Simulation results showing a sequence with speed and load changes. Fig. 8. Experimental results showing a sequence with speed and load changes. The first subplot shows the speed reference (dashed), the actual rotor speed The first subplot shows the speed reference (dashed), the actual rotor speed (solid), and its estimate (dotted). The second subplot shows the q components (solid), and its estimate (dotted). The second subplot shows the q components of the stator (solid) and inverter (dashed) currents. The third subplot shows of the estimated stator current (solid) and actual inverter current (dashed). The the d components of the stator (solid) and inverter (dashed) currents. The third subplot shows the d components of the estimated stator current (solid)replacements fourth subplot shows the estimation error of the rotor position in electrical and actual inverter current (dashed). The fourth subplot shows the estimation degrees. error of the rotor position in electrical degrees. Freq. LC filter PM Freq. converter PMSM servo converter Speed for 0.5 ωm (p.u.) monitoring 0 -0.5 0 1 2 3 4 5 1.5 θm − θm (deg.) iAd , ˆsd (p.u.) iAq , ˆsq (p.u.) PC with dSPACE DS1103 board Fig. 7. Experimental setup. The permanent magnet (PM) servo motor is used 0 i as the loading machine. -1.5 0 1 2 3 4 5 0.5 Fig. 10. The stator voltage and current are close to sinusoidal. 0 Fig. 11 shows experimental speed-torque curves as the i speed reference is kept constant and the load torque is -0.5 slowly changed from rated torque to negative rated torque. 0 1 2 3 4 5 20 The duration of each torque reversal was 60 seconds. The figure illustrates the operation range of the drive. The control 0 ˆ works fine at high speeds. At low speeds, the inverter output -20 voltage is close to zero and the nonidealities of the inverter 0 1 2 3 4 5 deteriorate the performance. When the speed is 0.025 p.u. (2 t (s) Hz), operation in the regeneration mode is not possible, but the drive withstands loads up to the rated torque in the motoring Fig. 9. Experimental results showing a sequence with speed and load changes. The explanations of the curves are as in Fig. 8. mode. VII. C ONCLUSION Nested control loops and an adaptive full-order observer can be used for the sensorless vector control of a PMSM drive Linearization analysis is a suitable method for designing an equipped with an LC filter at the inverter output. Only the observer gain for the system. Simulation and experimental inverter output current and the dc-link voltage need to be mea- results show that the performance of the proposed control sured. Hence, it is possible to add a filter to an existing drive method is good, comparable to that of the drive without the without any hardware modifications in the frequency converter. filter. 1063
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