Sensorless speed control including zero speed of non salient PM synchronous drives

Transcription

1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol. 54, No. 3, 2006 Senorle peed control including zero peed of non alient PM ynchronou drive H. RASMUSSEN Aalborg Univerity, Fredrik Bajer Vej 5, DK-9100 Aalborg, Denmark Abtract. Thi paper preent a poition enorle drive of non alient pole PM ynchronou motor for all peed including zero peed. Uing adaptive Lyapunov deign a new approach for the deign of an oberver i developed. The reulting cheme lead to a nonlinear full order oberver for the motor tate including the rotor peed. Auming motor parameter known the deign achieve tability with guaranteed region of attraction even at zero peed. The control method i made robut at zero and low peed by changing the direct vector current component to a value different from zero. In order to verify the applicability of the method the controller ha been implemented and teted on a 800 W motor. Key word: PM motor, low peed operation, peed enorle control.. Nomenclature: a i A, B, C u A, B, C i u ψ ψ r R L ω ω mech p Z p θ 1. Introduction complex patial operator e j2π/3 tator phae current A,B and C tator phae voltage A,B and C tator current complex pace vector tator voltage complex pace vector tator flux rotor flux tator reitance tator inductance ynchronou angular frequency rotor peed time derivative operator d/dt number of pole pair rotor poition The permanent magnet ynchronou motor (PMSM) offer high power denity implying reduced ize and le material to tranport. PM motor cannot be operated in open loop due to the highly untable behavior of the motor dynamic. Thi mean that PM motor need a meaurement of the rotor poition in order to control the motor in a robut way. The traditional method to determine the rotor poition i to ue an encoder or reolver, but thee component are expenive and will add additional cot to the motor. In more than ten year there ha been an extenive reearch in finding reliable poition enorle method to etimate the rotor poition from the applied voltage and the conumed current. The mot ignificant paper of the reearch up to 1996 can be found in the reference [1]. In thi paper focu will be on PMSM type of motor due to the fact that the reult are mend for low noie emiion application like pump for dometic ue. In PMSM olution the rotor poition i normally determined by an open loop or cloed loop oberver ee [2] or by voltage injection method exciting aliency or aturation effect in the motor ee [3,5,6]. In [8] enorle alient-pole PM ynchronou motor drive in all peed range are obtained by witching between a back-emf (BEMF) method in the medium and high peed range and an injection method in the low and zero peed range. For a non alient PM ynchronou motor the injection method decribed are not valid and no BEMF i preented at zero peed. Thi i normally handled by a tart-up procedure operating the motor in open loop up to a given minimum peed where BEMF i reliable, and after thi point a jump to oberver baed field-oriented control take place. Thi jump can be noie full and can give extreme peed tranient and pull out can in evere ituation occur. In thi paper a new method baed on Lyapunov tability will be preented operating from zero peed without changing the control tructure. The idea of the method choen i to force direct current into the machine in the faulty poition the oberver etimate at low peed; thi will force the rotor poition to the incorrectly etimated poition, and the difference between the real poition and the etimated poition will be reduced. The method i implemented and verified experimentally on a 800 W motor. The reult demontrate that the method work uccefully from zero to full peed. The method i able to produce etimate of poition and peed with a preciion good enough to replace a haft enor. 2. Oberver deign 2.1. Voltage equation. In the rotor oriented coordinate ytem e jθ the voltage equation for the motor i u (R) = Ri (R) + (p + jω)li (R) + jωψ M (1) with ω = pθ. Where p i the time derivative operator. hr@control.aau.dk 293

2 H. Ramuen Fig. 1. Rotor angle oberver Fig. 2. Rotor field oriented control In the coordinate ytem e j ˆθ of the etimated rotor angle ˆθ the voltage equation i u ( ˆR) = Ri ( ˆR) + (p + j ˆω)Li ( ˆR) + (p + j ˆω)ψ M e j(θ ˆθ) with ˆω = pˆθ Omitting ( ˆR) in the following and introducing θ = ˆθ θ (2) give u = Ri + (p + j ˆω)Li + j ˆωψ M +(ω in θ + j(ω co θ ˆω))ψ M Oberver candidate. From (3) the tator flux linkage i given by ψ M + Li = 1 p+j ˆω { u Ri (ω in θ + j(ω co θ ˆω))ψ M } (4) (2) (3) { Becaue the term ω in θ + j(ω co θ } ˆω) ψ M i unknown a feedback oberver may be given the form ˆψ r + Li = 1 p + j ˆω {u Ri + v} (5) where equation for v and ˆω have to be calculated. From (4) and (5) the error dynamic are become p( ˆψ r ψ M ) = j ˆω( ˆψ r ψ M ) + v +(ω in θ + j(ω co θ ˆω))ψ M. Stabilizing with v = c 1 ( ˆψ rd ψ M ) c 2 ˆψrq and taking the (6) 294 Bull. Pol. Ac.: Tech. 54(3) 2006

3 Senorle peed control including zero peed of non alient PM ynchronou drive real and imaginary part of Eq. (6) give p( ˆψ rd ψ M ) = c 1 ( ˆψ rd ψ M ) + ˆω ˆψ rq +ω in θψ M p ˆψ rq = c 2 ˆψrq ˆω( ˆψ rd ψ M ) (ˆω ω co θ)ψ M p( ˆψ rd ψ M ) = c 1 ( ˆψ rd ψ M ) + ˆω ˆψ rq ω ˆψ rq p ˆψ rq = c 2 ˆψrq ˆω( ˆψ rd ψ M ) (ˆω ω)ψ M + ω( ˆψ rd ψ M ) Inertion of ω = ˆω ω in (8) now give p( ˆψ rd ψ M ) = c 1 ( ˆψ rd ψ M ) + ω ˆψ rq p ˆψ rq = c 2 ˆψrq ω( ˆψ rd ψ M ) ωψ M Lyapunov analyi. Stability of the error equation may be obtained by uing a Lyapunov function candidate P = 1 {( 2 ˆψ rd ψ M ) 2 + ( ˆψ rq ) 2 + 1γ } ω2. (10) Uing (9) in the time derivative of (10) give dp dt = c 1( ˆψ rd ψ M ) 2 c 2 ( ˆψ rq ) 2 + ω( ψ M ˆψrq + 1 d ω γ dt ). (11) Chooing d ω dt = γψ ˆψ M rq in (11) then give dp dt = c 1( ˆψ rd ψ M ) 2 c 2 ( ˆψ rq ) 2. (12) For c 1 > 0 and c 2 > 0 the Lyapunov function candidate i hown to be a Lyapunov function and the following convergence i obtained ˆψ r = ˆψ rd + j ˆψ rq ψ M. (13) Becaue we have ˆψ rq = ψ M in θ it i alo hown that ˆθ θ. Now we only have to examine the adaptation condition d ω dt = γψ M ˆψ rq. (14) If ω i aumed contant the adaptation condition become dˆω dt = γψ ˆψ M rq. (15) The aumption of contant ω mean in practice that the variation of ω ha to be low compared to the adaptation time contant, which depend of γ. With the above aumption the following oberver i obtained ˆψ r = ˆψ Li d ˆψ dt = u Ri c 1 ( ˆψ rd ψ M ) jc 2 ˆψrq j ˆω ˆψ d = γψ ˆψ. M rq dt ˆω d ˆθ dt = ˆω (16) For known value of the initial rotor poition, (R, L, ψ M ), no offet due to drift and perfect dead time compenation the oberver converge to the correct rotor angle. Becaue thi i not to be expected in practice a robutne analyi ha to be performed. (7) (8) (9) 2.4. PI adjutment. The adjutment of ω given by the Lyapunov method i an integral controller ω(t) = γψ M t ˆψ rq (τ)dτ. (17) It may be expected that a quicker adaptation can be achieved by uing a PI controller ω(t) = γ 1 ˆψrq (t) + γ 2 t Since a ytem with the tranfer function ˆψ rq (τ)dτ. (18) H() = γ 1 + γ 2 (19) i output trictly paive for poitive γ 1 and γ 2 it follow from the paivity theorem [7], that PI adjutment i table if a tranfer function from ω to ˆψ rq i trictly poitive real. From (9) a tranfer function G() i found (20) p + c 1 ˆψ rq = p 2 + (c 1 + c 2 )p + c 1 c 2 + ω 2 ψ M ω = G(p)ψ M ω. (20) For c 1 > 0 and c 2 > 0 the tranfer function G() i trictly poitive real becaue it ha no pole in the right half-plane, no pole and zero on the imaginary axi and Re{G(jω)} = c 1 (c 1 c 2 + ω 2 )+c 2 ω 2 ( ω 2 +jω(c 1+c 2)+c 1c 2+ ω 2 ) > 0. The oberver with PI adjutment given by (21) i hown in Fig. 1 ˆψ r = ˆψ Li ˆψ = t (u Ri c 1 ( ˆψ rd ψ M ) jc 2 ˆψrq j ˆω ˆψ )dτ. (21) t ˆω = γ 1 ˆψrq + γ 2 ˆψrq dτ ˆθ = t ˆωdτ 2.5. Robutne. Becaue the Lyapunov analyi ha hown tability, the robutne i analyzed in teady tate with ˆω = ω. Let u aume that uncompenated dead time compenation and error in R give and voltage error δu. The oberver equation then give 0 = δu + jωψ M c 1 ( ˆψ rd ψ M ) jc 2 ˆψrq jω ˆψ r. (22) For ω c 1 and ω c 2 the real and imaginary part of thi equation give leading to ˆψ rd ψ M δu q /ω ˆψ rq δu d /ω (23) in(ˆθ θ) δu d ωψ M. (24) Becaue thee inequalitie call for mall value of (c 1, c 2 ) and the dynamic of the oberver call for (c 1, c 2 ) a big a poible, the practical choice i a compromie between fat oberver dynamic and mall teady tate etimation error. Experiment and imulation how that bet performance i obtained for c 2 c 1 and that the performance i inenitive for the choice of the value. In the experiment hown the value (c 1, c 2 ) = (50, 1) are ued. Bull. Pol. Ac.: Tech. 54(3)

4 H. Ramuen Fig. 3. Laboratory etup 2.6. Zero and very low peed control. For known value of the initial rotor poition, (R, L, ψ M ), no offet due to drift and perfect dead time compenation the oberver converge to the correct rotor angle. Becaue thi i not to be expected in practice the robutne analyi ha hown that the oberver i table but give large error of the etimated angle ˆθ for mall value of ω. The idea of the method choen i to force direct current into the machine in the faulty poition the oberver etimate at low peed; thi will force the rotor poition to the incorrectly etimated poition, and the difference between the real poition and the etimated poition will be reduced. Figure 2 how the rotor field oriented control ytem. At zero and very low peed the reference value for i d i given a value different from zero. If the rotor angle i etimated correctly a value for i d 0 give no torque. If the rotor angle i etimated with an error the rotor i forced in the direction of ˆθ ued in the controller. The error at zero peed depend on i d,ref due to the fact that we have m e = 3/2Z p ψ M i d in( θ). Thi new principle mean that no manual mode at zero and low peed i neceary, cloed loop control i obtained for all value of the peed reference. The function for i d,ref i 3.2. Initial tartup. Figure 4a and 4b how the tart-up repone with the rotor angle initialized to the wort condition, which i to be oppoite to the initial value given the etimate of the rotor poition. The initial negative peed in Fig. 4a i due to the fact that the etimation error of the rotor angle ˆθ θ > 90 degree. Thi ituation only occur the very firt time the control ytem i tarted. Next time the reference i et to zero peed the angle i etimated with a mall etimation error due to the i d,ref value different from zero Rotor peed tep repone. Figure 4c how a tep repone for a very low peed. The cale of the peed axi ha to be noticed howing that the variation of the rotor peed i le than 5 rpm. Figure 4d, 4e and 4f how tep repone for rpm, rpm and rpm repectively. The curve form of the repone are a expected and the ound of the motor alo indicate field orientation during the tranient Load tep repone. Figure 5 how the repone of tep in the load torque both at 0 rpm and at 1000 rpm. Figure 5a and 5b how the peed variation for tep in the i d,ref = i ref0 e ˆω r /ω 0 (25) load torque. Even at zero peed reference the peed i equal to with i ref0 determined by the maximal load torque expected at zero peed and ω 0 mall due to the fact that etimation error of the rotor angle i only ignificant at low peed. zero in teady tate. Figure 5c and 5d how the i d and i q current. Figure 5d how the normal repone at 1000 rpm where i d = 0 and i q i the torque producing current. Figure 5c how the i d and i q 3. Experiment at zero peed. The i d i given a reference value equal to 5A and when the load torque trie to turn the rotor axi the angle 3.1. Laboratory etup. The laboratory etup hown in Fig. 3 between thi current in the etimated rotor angle and the real i baed on Real Time Workhop, Simulink and DSpace. The rotor angle produce a torque equal to the load torque and the drive ytem i via a ignal conditioner connected to a DSP peed i kept equal to zero a een from Fig. 5a board in the computer. The control oftware i Simulink block Figure 5e and 5f how the error between the etimated and written in C. The nominal motor parameter are real rotor angle. The error at zero peed depend on i d,ref due R L ψ M Z p N mech (26) to the fact that we have m e = 3/2Z p ψ M i d in( θ). Figure 5f Nom. 4.0 Ω H 0.3 V rpm how the error at high peed and the cale of the axi ha to be noticed. 296 Bull. Pol. Ac.: Tech. 54(3) 2006

5 Senorle peed control including zero peed of non alient PM ynchronou drive Fig. 4. Experimental waveform of etimated peed (a,c,d,e,f) and rotor angle (b) during motor tarting proce: (a) firt rotor peed tep repone with rotor angle initialized to an 180 error at tartup, (b) convergence of the etimated rotor angle with rotor angle initialized to an 180 error at tartup, (c) very low rotor peed tep repone 0 20 rpm, (d) low rotor peed tep repone rpm, (e) rotor peed tep repone rpm, (f) high rotor peed tep repone rpm 4. Concluion A new oberver for the rotor angle i preented. Stability of the oberver i proven by the Lyapunov method and robutne i analyzed in teady tate. Variou paper concerning method for tarting PMSM without poition enor have been preented. Mot method have a pecial mode for tart-up and operation at low peed. The propoed method operate in the ame mode from zero peed to maximum peed, which implifie the control algorithm and eliminate the lag of robutne when a controller hift mode. The method make it poible to tart from zero peed in cloed loop and produce a contant torque at very low peed by changing the direct vector current component a a function of the peed. The method i implemented and verified experimentally on a 800 W motor. The reult demontrate that the method work uccefully over a wide peed range. The method i able to produce etimate of poition and peed with a preciion good enough to replace a haft enor. Bull. Pol. Ac.: Tech. 54(3)

6 H. Ramuen Fig. 5. Torque tep at rotor peed 0 rpm (a,c,e) and 1000 rpm (b,d,f), (a) rotor peed N mech, (b) rotor peed N mech, (c) i d and i q, (d) i d and i q, (e) etimation error ˆθ θ in degree, (f) etimation error ˆθ θ in degree REFERENCES [1] K. Rajahekara, A. Kawamura and K. Matue, Senorle Control of AC Motor Drive, IEEE Pre, [2] N. Matui, Senorle PM bruhle DC motor drive, IEEE Tran. Ind. Electron. 43 (2), (1996). [3] J. Holtz, Method for peed enorle control of AC drive, in Senorle Control of AC Motor, IEEE Pre Book, [4] J. Holtz, Senorle vector control of induction motor at very low peed uing a nonlinear inverter model and parameter identification, IEEE-IAS 2001, Chicago, Illinoi, [5] P.L. Janen and R.D. Lorenz, Tranducerle poition and velocity etimation in induction and alient AC machine, IEEE Tran. on Ind. Appl., (1995). [6] M.Corley and R. Lorenz, Rotor poition and velocity etimation for alient pole permanent magnet ynchronou machine at tandtill and high peed, IEEE Tran. on Ind. Appl., (1998). [7] K.J. Åtröm and B. Wittenmark, Adaptive Control, Addion Weley Publihing Company, [8] H. Takehita, A. Uui, and N. Matui, Senorle Salient pole PM ynchronou motor drive in all peed range, Electrical Engineering in Japan 135 (3), 8 15 (2001). 298 Bull. Pol. Ac.: Tech. 54(3) 2006