Aalborg Univeritet Senorle Speed Control including zero peed of Non Salient PM Synchronou Drive Ramuen, Henrik Publication date: 2 Document Verion Publiher' PDF, alo known a Verion of record Link to publication from Aalborg Univerity Citation for publihed verion (APA): Ramuen, H. (2). Senorle Speed Control including zero peed of Non Salient PM Synchronou Drive. Aalborg: Department of Control Engineering, Aalborg Univerity. General right Copyright and moral right for the publication made acceible in the public portal are retained by the author and/or other copyright owner and it i a condition of acceing publication that uer recognie and abide by the legal requirement aociated with thee right.? Uer may download and print one copy of any publication from the public portal for the purpoe of private tudy or reearch.? You may not further ditribute the material or ue it for any profit-making activity or commercial gain? You may freely ditribute the URL identifying the publication in the public portal? Take down policy If you believe that thi document breache copyright pleae contact u at vbn@aub.aau.dk providing detail, and we will remove acce to the work immediately and invetigate your claim. Downloaded from vbn.aau.dk on: november 2, 28
Senorle Speed Control including zero peed of Non Salient PM Synchronou Drive H. Ramuen Aalborg Univerity, Aalborg, Denmark e-mail: hr@control.aau.dk 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 8 W motor. u Ri ( S ) ( S ) e jθˆ θˆ p c Re{} + jc2 p + jωˆ ωˆ ( ) ψˆ R Im{} ( ) ψˆ R r Li Fig.. Rotor angle oberver (R) Im PI ψ M ψˆ rq NOMENCLATURE a complex patial operator e j2π/3 i A, B, C tator phae current A,B and C u A, B, C tator phae voltage A,B and C i tator current complex pace vector u tator voltage complex pace vector ψ tator flux ψ r rotor flux R tator reitance L tator inductance ω ynchronou angular frequency ω mech rotor peed p time derivative operatord/dt Z p number of pole pair θ rotor poition I. INTRODUCTION 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 996 can be found in the reference [4]. 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 [] or by voltage injection method exciting aliency or aturation effect in the motor ee [2], [6], [7]. In [] enorle alient-pole PM ynchronou motor drive in all peed range are obtained by witching between a backemf (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 8 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. A. Voltage equation II. OBSERVER DESIGN 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 with ω = pθ In the coordinate ytem e j ˆθ of the etimated rotor angle
Inertion of ω = ˆω ω in the equation now give p( ˆψ rd ψ M ) = c ( ˆψ rd ψ M ) + ω ˆψ rq p ˆψ rq = c 2 ˆψrq ω( ˆψ rd ψ M ) ωψ M () C. Lyapunov analyi Stability of the error equation may be obtained by uing a Lyapunov function candidate P = {( 2 ˆψ rd ψ M ) 2 + ( ˆψ rq ) 2 + γ } ω2 The time derivative of thi function i ˆθ the voltage equation i u ( ˆR) = Ri ( ˆR) Fig. 2. Rotor Field Oriented Control + (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 give u = B. Oberver candidate Ri + (p + j ˆω)Li + j ˆωψ M +(ω in θ + j(ω co θ ˆω))ψ M From thi equation the tator flux linkage i given by ψ M + Li = p+j ˆω { u Ri (ω in θ + j(ω co θ ˆω))ψ M } { 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 = p + j ˆω {u Ri + v} where equation for v and ˆω have to be calculated. From the above equation the error dynamic are given by 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 ( ˆψ rd ψ M ) c 2 ˆψrq and taking the real and imaginary part of the equation give p( ˆψ rd ψ M ) = c ( 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 ( 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 ) dp dt = c ( ˆψ rd ψ M ) 2 c 2 ( ˆψ rq ) 2 + ω( ψ M ˆψrq + d ω γ dt ) Chooing deω dt = γψ M ˆψ rq then give dp dt = c ( ˆψ rd ψ M ) 2 c 2 ( ˆψ rq ) 2 For c > and c 2 > 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 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 If ω i aumed contant the adaptation condition become dˆω dt = γψ M ˆψ rq 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. The above equation then give the following oberver ˆψ r = ˆψ Li d dt = u Ri c ( ˆψ rd ψ M ) jc 2 ˆψrq j ˆω ˆψ d dt ˆω = γψ ˆψ M rq d dt = ˆω 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. D. PI adjutment The adjutment of ω given by the Lyapunov method i an integral controller ω(t) = γψ M t ˆψ rq (τ)dτ
It may be expected that a quicker adaptation can be achieved by uing a PI controller ω(t) = γ ˆψrq (t) + γ 2 t Since a ytem with the tranfer function ˆψ rq (τ)dτ H() = γ + γ 2 i output trictly paive for poitive γ and γ 2 it follow from the paivity theorem [8], that PI adjutment i table if a tranfer function from ω to ˆψ rq i trictly poitive real. From equ.() a tranfer function i found p + c ˆψ rq = p 2 + (c + c 2 )p + c c 2 + ω 2 ψ M ω = G(p)ψ M ω G() i for c > and c 2 > hown to be SPR becaue: The function ha no pole in the right half-plane, It ha no pole and zeroe on the imaginary axi and Re{G(jω)} = c (c c 2 +eω 2 )+c 2 ω 2 ( ω 2 +jω(c +c 2 )+c c 2 +eω 2 ) > With the above aumption of contant ω the equation for the oberver are given by ˆψ r = ˆψ Li ˆψ = t (u Ri c ( ˆψ rd ψ M ) jc 2 ˆψrq j ˆω ˆψ )dτ ˆω = γ ˆψrq + γ 2 t ˆψrq dτ ˆθ = t ˆωdτ Thee equation are hown in fig. E. 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 = δu + jωψ M c ( ˆψ rd ψ M ) jc 2 ˆψrq jω ˆψ r For ω c 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 in(ˆθ θ) δu d ωψ M Becaue thee inequalitie call for mall value of (c, c 2 ) and the dynamic of the oberver call for (c, 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. Fig. 3. Laboratory etup F. Zero and very low peed control Fig.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 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. 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 i d,ref = i ref e ˆω r /ω (2) with i ref determined by the maximal load torque expected at zero peed and ω mall due to the fact that etimation error of the rotor angle i only ignificant at low peed. A. Laboratory etup III. EXPERIMENTS The laboratory etup hown in Fig.3 i baed on Real Time Workhop, Simulink and DSpace. The drive ytem i via a ignal conditioner connected to a DSP board in the computer. The control oftware i Simulink block written in C. The nominal motor parameter are R L ψ M Z p Nom. 4..3.3 3 B. Initial tartup Fig.4.a and Fig.4.b 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.4.a i due to the fact that the etimation error of the rotor angle ˆθ θ > 9 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. C. Rotor peed tep repone Fig.4.c 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 rpm.
2 2 dth 8 6 [deg] 4 2 2... 2 a) Firt Rotor peed tep repone with rotor angle initialized to an 8 degree error at tartup... 2 b) Convergence of the etimated rotor angle with rotor angle initialized to an 8 degree error at tartup 2 6 2 rotor peed rotor peed 4 3 2... 2 time... 2 time c) Very low rotor peed tep repone 2rpm d) Low rotor peed tep repone rpm 9 8 7 6 4 3 rotor peed 2... 2... 2 time e) Rotor peed tep repone rpm f) High rotor peed tep repone rpm Fig. 4. Experiment
2 2 8 6 4 2 98 96 94 92 2 2 2 9 2 2 a) Rotor peed b) Rotor peed i d & i q i d & i q 6 4 4 3 3 [A] [A] 2 2 2 2 2 2 c) i d and i q d) i d and i q 2 theta real theta etimated. theta real theta etimated [degree] [degree].. 2 2 2 2 e) Etimation error ˆθ θ in degree f) Etimation error ˆθ θ in degree Fig.. Torque tep at rotor peed rpm (a,c,e) and rpm (b,d,f)
Fig.4.d, 4.e and 4.f 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. [7] 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. July/Aug 998, pp 784-789 [8] Åtröm K.J. and Wittenmark Björn, Adaptive Control - Second Edition, Addion - Weley Publihing Company, 99. D. Load tep repone Fig. how the repone of tep in the load torque both at rpm and at rpm. Fig..a and b how the peed variation for tep in the load torque. Even at zero peed reference the peed i equal to zero in teady tate. Fig..c and d how the i d and i q current. Fig..d how the normal repone at rpm where i d = and i q i the torque producing current. fig..c how the i d and i q at zero peed. The i d i given a reference value equal to A and when the load torque trie to turn the rotor axi the angle between thi current in the etimated rotor angle and the real rotor angle produce a torque equal to the load torque and the peed i kept equal to zero a een from fig..a Fig..e and f how the error between the etimated and real rotor angle. 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). Fig..f how the error at high peed and the cale of the axi ha to be noticed. IV. CONCLUSION 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 8 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. REFERENCES [] H. akehita, A. Uui, and N. Matui, Senorle Salient-Pole PM Synchronou Motor Drive in All Speed Range, Electrical Engineering in Japan, Vol. 3, No. 3, (2). [2] Holtz Joachim, Method for Speed Senorle Control of AC Drive, Senorle Control of AC Motor, IEEE Pre Book, (996). [3] Holtz Joachim, Senorle Vector Control of Induction Motor at Very Low Speed uing a Nonlinear Inverter Model and Parameter Identification, IEEE-IAS 2, Chicago, Illinoi, (2). [4] K. Rajahekara, A. Kawamura and K. Matue. Senorle Control of AC Motor Drive, IEEE Pre, 996. [] N. Matui, Senorle PM Bruhle DC Motor Drive, IEEE Tran. Ind. Electron. vol 43, no.2, pp. 3-38, Apr. 996 [6] P.L. Janen, R.D. Lorenz, Tranducerle Poition and Velocity Etimation in Induction and Salient AC Machine, IEEE Tran. on Ind. Appl., March/April 99, pp. 24-247