APPLICATION OF A SLIDING MODE OBSERVER FOR SENSORLESS OPERATION OF SWITCHED RELUCTANCE MOTORS. Stefan Brock

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1 Preprnt of the paper publshed n: Archves of Electrcal Engneerng Vol. 56, no 2 (2007) pp: DOI: Abstract APPLICATION OF A SLIDING MODE OBSERVER FOR SENSORLESS OPERATION OF SWITCHED RELUCTANCE MOTORS Stefan Brock Poznań Unversty of Technology, Insttute of Control and Informaton Engneerng Potrowo 3a, Poznań, Poland, e-mal: Stefan.Brock@put.poznan.pl The rotor speed and poston control of a swtched reluctance motor (SRM) needs an encoder or resolver to obtan nformaton about the rotor poston. These sensors may deterorate n harsh envronments, and ncrease the sze and cost of the drve. Ths paper presents an evaluaton of the slng-mode observer to estmate a swtched reluctance motor speed and rotor poston. A new concept of observer based on a smple but qute accurate model of SRM has been proposed. A non-lnear model of an SRM s used and a computer smulaton s developed based on data from an expermental motor system. The mplcaton of screte tme mplementaton wth varous sample rates was examned. Keywords: swtched reluctance motor, slng mode observer, real tme mplementaton I. INTRODUCTION The ponts n favour of a swtched reluctance motor (SRM) are smple and economcal constructon, hgh maxmum speed and the hghest possble relablty [5]. Generally SRM requres a rotor poston sensor for correct commutaton and current control. Ths poston sensor, however, not only adds complexty and cost to the system, but also tends to reduce the

2 relablty of the drve system. The use of nrect poston sensng can elmnate the mechancal complexty, reduce cost and ncrease relablty. Varous methods of sensorless poston estmaton have been nvestgated for SRM. The frst group of methods s the flux waveform detecton method that determnes the poston of the rotor usng the prncple of nductance measurement n one of the nactve phases. The drawbacks of these methods are the generaton of non-useful torque by njectng a test pulse and low resoluton for hgh-speed operaton due to tme delay by watng for fallng of prevous agnostc phase. The second group of methods uses the state observer theory, whch estmates the poston of the rotor by usng measurements of phase voltages and currents. The observer-based technques do not requre any adtonal pulse njecton and have no nherent speed lmtatons. Lumsdane and Lang [6] proposed a model-based Luenberger observer. The dynamcs of the SRM s modelled n state space and a model s computng n parallel wth the physcal machne. The model has the same nputs as the physcal machne, and the fference between ts outputs and the measured outputs of the SRM are used to force the model to converge to the real machne. Although a lnear observer theory to SRM drve had been successfully mplemented, the nonlnearty was neglected by means of very large rotor nerta. On the other hand, an accurate knowledge of machne and load parameters s requred by ths type of observers. Brosse and Henneberger [1] mplemented a Kalman flter for poston and speed estmaton n SRM. The pole placement desgn method was based on lnearzed model. The computatonal smplcty and robust stablty propertes of slng mode controllers prompted to study of desgnng observers wth slng surfaces [3, 9]. By structure, they are very smlar to the standard full order observers wth replacement of the lnear nnovatve term by a scontnuous functon. Slng mode observers are robust wth regard to the parameter varatons and load. McCann et al. [7,8] and Islam et al. [4] presented a slng mode observer rotor poston estmaton scheme for SRM. These observers are based on

3 a lnear magnetc model of SRM [7, 8], and on algebrac modellng of saturaton nonlnearty [4]. Smlar work was presented by Park et al. [10]. Zhan et al. [12] proposed a slng mode observer for SRM based on estmaton of the phase currents. Although the magnetc crcut s lnearzed, the practcal realsaton of the observer algorthm requres a great amount of computaton. Slng mode observer has chatterng problems n the rotor speed estmaton snce the scontnuous nputs enter the observer. Yang et al. [11] proposed bnary observer for SRM based on varable structure control theory wth nternal terms to exclude chatterng. Ths paper extends the prevous research be consderng the non-lnear model of a SRM. The model s based on data from an expermental motor system [2]. The mplcaton of hardware mplementaton usng a DSP and FPGA wth varable sample rate s examned. II. MODEL OF SWITCHED RELUCTANCE MOTOR In general, the nonlnear magnetc characterstcs of an SRM can be approprately modelled by two mensonal (2D) tables defnng the nonlnear flux-current-angle and torque-fluxangle characterstcs. 2D look-up tables are hard to mplement for real-tme applcaton due to nterpolaton and memory sze. Method of avodance of these problems was presented n paper [2]. The flux equaton for each phase governng the SRM dynamcs can be wrtten as follows: (, = Lc + L ( ) sat( ψ (1) where components of phase nductance: constant L c and varable L () can be dentfed through measurements. The value L c corresponds to the unalgned poston of the rotor, and the functon L () depends only on poston of the rotor, wthout saturaton phenomena. The magnetcally saturaton phenomena s taken nto account by nonlnear saturaton functon sat(. The results of the expermental dentfcaton of nonlnear functons L () and sat(, are presented n Fg. 1.

4 Fg. 1. Inductance L () and effectve current sat( dentfcaton results The dervatve of flux can be derved from the flux expresson of (1) as: dψ (, dl ( ) dsat = Lc + ω sat( + L ( ) (2) d ( where d ω = s the speed of the motor. Table values dsat ( and dl ( ) d can be calculated based on expermental dentfcaton of L ( ) and sat(. The dervatve of flux can be calculated based on appled phase voltage u, accorng to equaton (3): dψ = u R (3) From the equaton (2) the dervatve of the phase current can be expressed as: L c ( dψ (, dsat dl ( ) + L ( ) = ω sat( (4) d From eq. (3) and (4) the state equaton (5) for phase currents can be wrtten: dl ( ) u R ω sat( = d dsat ( ) ( Lc + L (5) Equaton (5) s used to mathematcal modellng of SRM wth phase currents as state varables.

5 The electromagnetc torque generatng by each phase can be calculated from electromechancal prncples. The electromagnetc torque T ph s calculated as dervatve of mechancal work wth respect to poston of rotor (5): Wm T ph = (6) For the case of constant exctaton, the ncremental mechancal work done Wm s equal to the rate of change of coenergy W f. The coenergy W f, accorngly to the flux expresson of (1), can be expressed as: 0 W = Table value sat( ) f 0 2 Lc ψ (, dι = ( Lc ι + L ( ) sat( ι) ) dι = + L ( ) sat( ι) dι (7) 2 0 ι dι can also be calculated based on expermental dentfcaton of sat(. Based on dervatve of coenergy, the torque T ph generated by every phase can be descrbed as equaton 8: 0 T ph W f dl ( ) (, = = sat( ι)dι d 0 = const (8) The whole electromechancal torque generated by SRM T el (, s the sum of elementary torque generatng by each phase (9): el (, = Tph( N T, (9) When the load torque T L s taken nto account, the moton equatons of the drve can be expressed as:

6 u dsat( sat( sat( ι)dι 0 L ( ) d L d ( ) T ph (, sequencer ω dω T el (, T L Fg. 2 Block agram of the swtchng reluctance motor dω 1 = el, J d = ω ( T ( T ) L (10) where J s the rotor nerta. From equaton (1) to (10), the smulaton model of SRM can be bult. A detaled computer smulaton was developed n Matlab/Smulnk usng tabulated emprcal data wth nterpolaton. The model s presented n Fg. 2. Because the smulaton s calbrated wth expermental data, the desgn and evaluaton of the slng mode observer may be carred out wth confdence based on smulaton results. III. PROPOSED SLIDING MODE OBSERVER ALGORITHM For an SRM descrbed by equaton (1)-(10) the slng mode observer of rotor speed and poston s proposed. The acton of observer s based on comparson of two estmated flux

7 values: frst one ψ ˆ ( u, s calculated from the ntegral of appled phase voltage u, accorng u to equaton (11) and the second ψˆ (, s calculated as a functon of phase current and estmated rotor poston, accorng to equaton (12). t ( u( τ ) R ( τ ) ψ ˆ ( u, = ) dτ (11) u o ( ˆ, = Lc + L ( ˆ ) sat( ψˆ (12) The values (11) and (12) are calculated for each phase of SRM. The global estmaton error e est s expressed as (13): ( ψˆ ( ˆ,, ψˆ ( u, ) = ( ψˆ ( ˆ, ψˆ ( u, ) sgn( ˆ ) e = F ) (13) est u N The global error functon F( ) s chosen as the sum of estmaton errors for each phase. The factor sgn of estmated poston sgn( ˆ) s used to compensate even functon L () (Fg. 1). The global estmaton error s used as nput sgnal for speed and poston observer (Fg. 3). The observer equatons can be wrtten as follows: d ˆ u ω = K sgn( ) (14) ω e est d ˆ = ˆ ω + K sgn( e est ) (15) The estmated poston s sequencng to nterval <-π/6,π/6> (for four phase SRM wth 8/6 topology). The observer errors can be defned as: e e ω = ω( t) ˆ( ω t) (16) = ( t) ˆ( t) (17) where (t) and ˆ ( t ) are the actual and estmated postons, and ω (t) and ˆ ω ( t) are the actual and estmated speeds. The rato of change of poston observer error can be expressed as: de ( e ) = e K sgn( ) = ω ˆ( ω (18) ( t) t) K sgn est ω e est

8 u ^ ψ u (u, + - e ψ e est F() sgn() K ω + ω^ + LPF ^ ^ ^ ψ (, ^ K sequencer Fg. 3. Block agram of the slng mode observer The one mensonal slng manfold e = 0 s attractve as long as: de e < 0 (19) A stronger conton, whch guarantees reachng the slng surface wthn a fnte tme, s expressed as: de e < η e (20) where η s a postve scalar constant. Because sgn(e est )=sgn(e ), from eq. (18) and (20) the value of gan K can be calculated as: eω + max η K (21) On the slng surface, the speed error converges exponentally to zero. A large value of K ω ensures faster convergence, but ncreased ampltude of chatterng of the estmated speed sgnal. Gan K ω s selected to be large enough to compensate the dynamc of speed equaton (10). The chatterng n the estmated speed was mnmzed usng a low-pass flter. In order to avod negatves effects of bg low pass flter, for practcal mplementaton the sgn functons are replaced by hgh gan blocks.

9 IV. TEST RESULTS The desgn method presented n secton III has been verfed through extensve smulatons. The smulaton model of SRM, presented n secton II, was used. The hysteress type current controllers are used. The smulaton also accounts for scretzaton and processng delays due to the mcroprocessor based motor controller. The screte tme formulaton of the observer follows from (11) (15). The drve was tested under varous operatng scenaros. In frst case, the drve wth encoder (4000 nc/rev) was used. Ths case can be use as reference pont for next scenaros. In second Fg. 4. Speed and poston estmaton, phase currents and total torque wthout observer, wth sem-contnuous observer and wth screte-tme observer.

10 case, the control system becomes the poston feedback sgnal from fast, sem-contnuous observer (samplng tme 1 µs). In last case the screte tme observer wth samplng tme 50 µs s tested. The sem contnuous observer can be mplemented n Programmable Logc Devce lke FPGA, when screte observer can be mplemented n Dgtal Sgnal Processor. The smulaton results usng the proposed scenaros are shown n Fg. 4. The left column s for the system wth poston feedback, the mddle column s for the system wth a fast, semcontnuous observer and the rght s for a screte tme observer. For the observer there s the ntal poston error =π/12. The step change of reference speed from standstll to 35 rad/s s smulated. From these fgures, the proposed observer shows a good performance for fast mplementaton, but t s senstve to larger samplng tmes. V. CONCLUSION In ths paper, a slng mode observer was appled to elmnate the poston encoder of a swtched reluctance motor drve. The observer requres termnal measurements of phase voltages and currents. The observer-based poston and speed estmates elmnate the need for any adtonal crcutry, whch makes the mplementaton smple and nexpensve. The mplcaton of selectng the sample rate was examned. The next study wll be focused on the fast observer real-tme mplementaton usng DSP and FPGA Acknowledgement: The paper was supported by grant No. 3 T10A of the Polsh Mnstry of Scence and Hgher Educaton. REFERENCES [1] A. Brosse and G. Henneberger: Sensorless control of a swtched reluctance motor usng Kalman flter, Proc. of 7 th EPE Conference, Trondhem-Norway, pp , 1997

11 [2] J. Deskur and A. Macejuk: Modellng of swtched reluctance motor drve, Proc. of Symposum on Electromagnetc Phenomena n Nonlnear Crcuts, Marbor, Slovena 2006 [3] S. Drakunov and V. Utkn: Slng Mode Observers. Tutoral. Proc. of the 34 th Conference on Decson & Control, New Orleans, LA - December 1995 [4] M.S. Islam, I. Husan, R.J. Vellette, and C. Batur: Desgn and Performance Analyss of Slng-Mode Observers for Sensorless Operaton of Swtched Reluctance Motors IEEE Trans. on Control Systems Tech., Vol. 11, No. 3, pp , May, 2003 [5] R. Krshnan: Swtched Reluctance Motor Drves: Modellng, Smulaton, Analyss, Desgn, and Applcatons, CRC Press, 2001 [6] A. Lumsdane and J.H. Lang: State observers for varable reluctance motors, IEEE Trans. on Ind. Electrontoncs, vol. 37, No. 2, pp.: , 1990 [7] R. McCann and I. Husan: Applcaton of a slng mode observer for swtched Sformatowane: Punktory numeracja reluctance motor drves Proc. of IEEE Applcatons Conference, New Orleans, USA, pp , vol.1, 1997 [8] R. McCann, M.S. Islam, and I. Husan: Applcaton of Slng Mode Observer for Swtched Reluctance Motor Drves IEEE Trans. on Industry Applcatons, Vol. 37, No. 1, pp , Jan.-Feb [9] E.A. Msawa and J.K.Hedrck: Nonlnear Observers A State-of-the-art Survey, ASME J. Dynamc Syst. Measurement Contr., v. lll, No. 3, 1989 [10] H.-W. Park, J.-H. Park, S.-J. Kwon, J.-G. Park, and C.-U Km: A Novel Hgh Performance Poston Sensorless Control of Swtched Reluctance Motor Drves Proc. of IEEE Internatonal Symposum on. Industral Electroncs, Pusuan, Korea, pp , 2001

12 [11] I.-W. Yang and Y.-S. Km: Rotor speed and poston sensorless control of a swtched reluctance motor usng the bnary observer IEE Proceengs - Electrc Power Applcatons -,Volume 147, Issue 3, pp , 2000 [12] Y.J. Zhan, C.C. Chan, and K.T. Chau: A novel poston and velocty observer for robust control of swtched reluctance motors Proc. of IEEE Power Electroncs Specalsts Conference, pp , 1998