1 A Comparison of Sensorless Spee Estimation Methos for Inuction Motor Control Marc Boson an John Chiasson Abstract Many ifferent techniues have been propose to estimate the spee of an inuction motor without a shaft sensor. Three representative approaches are consiere in the paper. The methos are compare in terms of their sensitivity to parameter variations, their ability to hanle loas on the motor, an their spee tracking capability. Keywors Sensorless Control, Inuction Motor I. Sensorless Control Methos A. Mathematical Moel For reference, we begin with the mathematical moel of the inuction motor in terms of the variables θ, ω, ψ Ra, ψ Rb, i Sa an, as given by [17] ω/t = n p (M/JL R )( ψ Ra i Sa ψ Rb ) τ L /J ψ Ra /t = (R R /L R )ψ Ra n p ωψ Rb + M(R R /L R )i Sa ψ Rb /t = (R R /L R )ψ Rb + n p ωψ Ra + M(R R /L R ) u Sa = R S i Sa + σl S i Sa /t +(M/L R )ψ Ra /t u Sb = R S + σl S /t +(M/L R )ψ Rb /t (1) where σ, 1 M 2 /L R L S is the so-calle leakage parameter. The system (1) can also be written in explicit state-space form as ω/t = µ ( ψ Ra i Sa ψ Rb ) τ L /J ψ Ra /t = ηψ Ra n p ωψ Rb + ηmi Sa ψ Rb /t = ηψ Rb + n p ωψ Ra + ηm i Sa /t = ηβψ Ra + βn p ωψ Rb γi Sa + u Sa /σl S /t = ηβψ Rb βn p ωψ Ra γ + u Sb /σl S (2) with η, R R /L R = 1/T R, β, M/σL R L S,µ, 2n p M/n ph JL R, γ, M 2 R R /σl 2 R L S + R S /σl S. In inustrial settings where high-performance variable spee control is esire, fiel-oriente control (or one of its variants) is the algorithm of choice [17]. Fiel-oriente control is base on a coorinate system that rotates with the angle ρ =tan 1 (ψ Rb /ψ Ra ). One lets ρ =tan 1 (ψ Rb /ψ Ra ), ψ = ψ 2 Ra + ψ 2 Rb an the currents an voltages are then transforme into the new coorinate system as i cos(ρ) sin(ρ), i sin(ρ) cos(ρ) u cos(ρ) sin(ρ), sin(ρ) cos(ρ) u u Sb M. Boson is with the ECE Department, University of Utah, Salt Lake City, UT 84112, boson@ee.utah.eu. J. Chiasson is with the ECE Department, University of Tennessee, Knoxville, TN 37996, chiasson@utk.eu The usual implementation of fiel-oriente control assumes that the stator currents are measure, that a shaft sensor is use to obtain ω, an that the fluxes are estimate through integration of ˆψ Ra /t = η ˆψ Ra n p ω ˆψ Rb + ηmi Sa ˆψ Rb /t = ηψ Rb + n p ω ˆψ Ra + ηm (3) A straightforwar calculation shows that (ψ Ra ˆψ Ra ) 2 + (ψ Rb ˆψ Rb ) 2, so that (3) provies a stable estimate of the fluxes. The flux estimate is crucial to this approach even if only torue (as oppose to spee) control is reuire. This estimator epens on knowing the parameter η, R R /L R = 1/T R, that is, the reciprocal of the rotor time constant. Due to ohmic heating, this parameter varies, throwing off the flux estimate an conseuently the etermination of the angle ρ =tan 1 (ψ Rb /ψ Ra ). Quite a bit of research has been evote to making fiel-oriente control aaptive to the rotor time constant, assuming that a shaft sensor is available, as given in the work [18] [2] an the references therein. In contrast, the (ultimate) objective of sensorless control is to achieve aaptation to unknown spee as well as to motor parameters. B. Metho 1: Aaptive Metho One approach to the sensorless control problem is to consier the spee as an unknown constant parameter an to use the techniues of aaptive control to estimate this parameter [22] [23] [25]. The iea is that the spee changes slowly compare to the electrical variables. This approach was first formulate by Shauer [25], with important moifications propose by Peng an Fukao [22]. Specifically, using the last two euations of (1), one efines v ma, (M/L R ) ψ Ra /t = R S i Sa σl S i Sa /t + u Sa (4) v mb, (M/L R ) ψ Rb /t = R S σl S /t + u Sb where v ma,v mb are known (measure/calculate) uantities. Multiplying the secon an thir euations of (1) by M/L R, an then ifferentiating with respect to time, one has v ma /t = ηv ma n p ωv mb + η M 2 R R /t (5) v mb /t = ηv mb + n p ωv ma + η M 2 R R /t where ω/t =, as the spee is assume constant in this approach. 376
2 An estimator for v ma,v mb (erivative of the fluxes) is efine through ˆv ma /t = ηˆv ma n p ˆωˆv mb + η M 2 R R /t (6) ˆv mb /t = ηˆv mb n p ˆωˆv ma + η M 2 R R isb /t With e ma = v ma ˆv ma,e mb = v mb ˆv mb, the error ynamics are foun by subtracting (6) from (5) to get ema t e mb η np ω = n p ω η ema e mb ˆvmb n p (ˆω ω) ˆv ma The erivative of the Lyapunov function V (e ma,e mb ) = µ i 1 Sa 2 e 2 ma + e 2 mb gives = u Sa + i Sa u Sb σl S + i Sa t t V/t = η e 2 ma + e 2 mb + np (ˆω ω)(e maˆv mb e mbˆv ma ). Choosing (9) oes not epen on R S. With ˆω, the spee estimate, one solves Z t ˆω = K P (e maˆv mb e mbˆv ma ) K I (e maˆv mb e mbˆv ma ) τ î ma /t = ηî ma n p ˆωî mb + ηi Sa (7) î mb /t = ηî mb n p ˆωî ma + η (1) results in for the estimate î m of the magnetizing current. Then, this V/t = η e 2 ma + e 2 mb np K P (e maˆv mb e mbˆv ma ) 2 estimate is use to compute n p K 1 I f(t)f (t) ˆ m, M 2 ηîma n i p ˆωî ma + ηi Sa Sa R L R ηî mb + n p ˆωî mb + η t where f(t), K I (e maˆv mb e mbˆv ma ) τ + ω (again with ω/t =). As R (11) t f(t)f (t) 1 2 f 2 (), it follows from which also oes not epen on the value of R S. In [22], ˆω Popov s criterion that as t is chosen as V (e ma,e mb ) = (1/2) Z e 2 ma + e 2 t mb = (1/2) ˆω = K p ( m ˆ m )+K I ( m ˆ m )t (12) (v ma ˆv ma ) 2 +(v mb ˆv mb ) 2 In summary, the algorithm consists in computing v ma,v mb from (4), ˆv ma, ˆv mb from (6) an then using (7) to obtain the spee estimate ˆω. The parameters K p,k I > are ajuste by the esigner. Important issues with this algorithm are the sensitivity to parameter variation an the ability to work uner full loa torue. The metho reuires knowlege of R S, σl S to compute v ma,v mb from (4), an of η,m to compute ˆv ma, ˆv mb from (6). The primary concern is with R S an T R, which vary ue to ohmic heating. Peng an Fukao [22] moifie the approach to make it insensitive to the stator resistance R S. Tooso,themagnetizing current is efine as ima i m = i mb ψra /M, ψ Rb /M so that iviing the secon an thir euations in (1) by M gives i ma /t = ηi ma n p ωi mb + ηi Sa i mb /t = ηi mb + n p ωi ma + η (8) Also, note that v m = vma = M L R t, M 2 i m L R t v mb ψra ψ Rb satisfies (4). The measurable variable m, v i ma Sa v mb = M 2 L R ima t i mb = i Sa R S i Sa σl S i Sa /t + u Sa R S σl S /t + u Sb Although [22] presents some inication as to why this metho might work, an actual proof of convergence is not given. In summary, this estimator consists of (9)(1)(11)(12). The flux estimate is simply given by ˆψ Ra = Mî ma, ˆψ Rb = Mî mb. An open problem is to fin the conitions uner which this estimator converges an to make it aaptive to the changing value of the rotor time constant T R =1/η. C. Metho 2: Least-Suares Metho Verghese an his collaborators have approache the sensorless control problem from a parameter ientification point of view [19] [28] [29]. The iea is to consier the spee ω as an unknown constant parameter, an to fin the value ˆω that best fits the measure/calculate ata (i.e., u Sa, u Sb, i Sa,, i Sa /t, /t) to the ynamic euations of the motor. For this approach, it is convenient to use the representation of the inuction motor in explicit state-space form (2). As in all the sensorless control techniues, the first euation of (2) is of no use because the spee, position, an loa torue are all unknown. Thus, the least-suares approach fins the value of ˆω that best fits the last four euations of (2). However, the fluxes are 377
3 unknown, so that the four euations must be reuce to two by eliminating the fluxes. We refer to the 2 n an 3 r euations of (2) as the current euations, an to the 4 th an 5 th euationsasthevoltage euations. To procee, we rewrite the voltage euations as t = β γ η n p ω n p ω η + 1 σl S ψra ψ Rb u Sb (13) Differentiating (13), with the assumption that ω/t =, results in 2 η n t 2 = β p ω ψra n p ω η t γ t + 1 σl S t ψ Rb u Sb (14) In this set of euations, the erivative of the fluxes are not known. However, they can be eliminate by using the current euations. The voltage euations (13) are solve for the fluxes as ψra = 1 1 µ η n p ω ψ Rb β n p ω η t +γ 1 (15) σl S u Sb an this is then substitute into the current euations (2 n an 3 r euations of (2)) to obtain ψra = 1 1 η np ω η n p ω t ψ Rb β n p ω η n p ω η µ + γ 1 t σl S u Sb +ηm = 1 µ + γ 1 β t σl S u Sb +ηm (16) the unknown spee ω. The efinitions γ + η = (R S + L S /T R )/σl S, γ ηmβ = R S /σl S, η =1/T R were use. The least-suares approach reuires that one measure u Sa, u Sb, i Sa,, compute i Sa /t, /t, 2 i Sa /t 2, 2 /t 2, an take the spee estimate ˆω that best fits (17) for the present an past values of time. With obvious efinitions for y(t) an c(t), the least-suares regressor euation (17) is simply y1 (t) y 2 (t) c1 (t) = c 2 (t) n p ω (18) To account for the fact that the spee is actually changing, the value n p ˆω(NT) is chosen that best fits (18) for the present an immeiate past values of time, rather than for all previous time values. To o this, the earlier values are weighte less than the present values by choosing a forgetting factor λ satisfying < λ < 1. The performance criterion J N (λ,n p ˆω) is thus chosen as n=n X J N (λ,n p ˆω), λ N n ky(nt ) c(nt )n p ˆωk 2 (19) n= The solution to (19) is the stanar least-suares algorithm with forgetting factor, which is given by ˆω(NT)=ˆω((N 1)T ) y1 (NT) c +K(NT) 1 (NT)ˆω((N 1)T ) y 2 (NT) c 2 (NT)ˆω((N 1)T ) where K(NT)=p((N 1)T ) c 1 (NT) c 2 (NT) µ 1 λ + 1 c1 (NT) p((n 1)T ) c1 (NT) c c 2 (NT) 2 (NT) 1 p(nt)= µ p((n 1)T ) 1 k 11 (NT) k 12 (NT) c 1 (NT) λ c 2 (NT) To fin the fluxes, a iscrete-time version of (15) is use. Of course, this approach assumes that the parameters are known an fixe in time. The approach woul have to be moifie to take into account variations in R S,T R. Such a moification is not straightforwar because the regressor (17) is nonlinear in R S, T R, ω. A similar issue was ientifie in the authors work on the ientification of inuction motor parameters [26]. Substituting (16) into (14) an rearranging results in the least-suares regressor euation 2 σl S t 2 +(R i S + L S ) Sb T R t + R S 1 T R t u Sb T R u Sb D. Metho 3: Nonlinear Metho µ isb isb usb Yoo an Ha [13] [31] have propose a metho to estimate the spee an flux of an inuction motor without = n p ω σl S + R t i S Sa i Sa u Sa (17) assuming that the spee is slowly varying compare to the electrical variables. Their approach uses a polar coorinate moel of the fluxes (ρ, ψ ) rather than the Cartesian The system of euations (17) is in terms of the measure/calculate u Sa, u Sb, i Sa,, i Sa /t, /t, coorinates moel ψ Ra, ψ Rb. Letting ψ Ra = ψ cos(ρ), 2 i Sa /t 2, 2 /t 2, the known motor parameters an ψ Rb = ψ sin(ρ), so that ρ = tan 1 (ψ Rb /ψ Ra ), ψ = 378
4 ψ 2 Ra + ψ 2 Rb, one efines the computable uantities α a, i Sa /t + γi Sa u Sa /σl S α b, /t + γ u Sb /σl S allowing the last two euations of the state-space moel (2) to be rewritten as α a = ψ (ηβ cos(ρ)+βn p ω sin(ρ)) = ηβψ Ra + βn p ωψ Rb α b = ψ (ηβ sin(ρ) βn p ω cos(ρ)) (2) = ηβψ Rb + βn p ωψ Ra Solving for cos(ρ), sin(ρ) results in cos(ρ) = (ηβα a βn p ωα b ) / ψ η 2 β 2 + β 2 n 2 pω 2 (21) sin(ρ) = (ηβα b + βn p ωα a ) / ψ η 2 β 2 + β 2 n 2 pω 2 where α 2 a + α 2 b = ψ 2 η 2 β 2 + β 2 n 2 pω 2 Solving this last expression for ω gives the spee estimation euation ω = sign(ω) µα 2a + α 2b ψ2η 2 β 2 /βn p (22) which can be use for spee estimation, reuiring the knowlege of the sign of the spee an the magnitue of the flux ψ. Substitution of (22) into (21) results in µηβα a ψ sign(ω)α b α 2a + α 2b ψ2η 2 β 2 This estimator reuires only the knowlege of the sign of the spee. However, it turns out that the estimator (consisting of (25) combine with (22)(23)) cannot be guarantee to converge for every operating conition of the motor [31]. In orer to get a spee estimate when the sign of the spee is unknown, or a spee estimator when the flux estimator is not guarantee to converge, Yoo an Ha evelope a complementary flux estimator. Rewriting the 3 r an 4 th euations of (2) using (2) results in However, one also has ψ Ra /t = α a /β + ηmi Sa ψ Rb /t = α b /β + ηm ψ Ra /t = (ψ cos(ρ))/t = cos(ρ)ψ /t ψ sin(ρ)ρ/t ψ Rb /t = (ψ sin(ρ))/t = sin(ρ)ψ /t + ψ cos(ρ)ρ/t so that solving for ρ/t gives ρ/t = (cos(ρ)ψ Rb /t sin(ρ)ψ Ra /t) /ψ ³ = ( α b /β + ηm )cos(ρ) ³ α a /β + ηmi Sa sin(ρ) /ψ The complementary flux estimator consists of the following set of euations ³ ˆρ/t = ( α b /β + ηm )cos(ˆρ) ( α a /β + ηmi Sa )sin(ˆρ) /ˆψ cos(ρ) = (α 2 a + α 2 b ) (23) µηβα b ψ sign(ω)α a α 2a + α 2b ψ2η 2 β 2 ˆψ /t = η ˆψ + ηm (cos(ˆρ)i Sa +sin(ˆρ) ) with the spee estimate given by ˆω =(α a sin(ˆρ) α b cos(ˆρ)) /ηβˆψ sin(ρ) = (α 2 a + α 2 b ) If the magnitue of the flux ψ is known (along with the sign of the spee), then ω an ρ can be etermine from (22) an (23) respectively. Otherwise, to etermine ψ 2 ψ = Ra + ψ 2 Rb, one notes that ψ satisfies the flux magnitue euation given by ψ /t = ηψ + ηm (cos(ρ)i Sa +sin(ρ) ) (24) an substitution of (21) into (24) results in the main flux estimator t ˆψ = η η2 βm (α a i Sa + α b ) (α 2 a + α 2 b ) ˆψ +βmsign(ω) (α a α b i Sa ) (α 2 a + α 2 b ) α 2 a + α 2 b ˆψ 2 η 2 β 2 (25) an the fluxes are foun by solving (2). It has been shown [31] that the conitions uner which the two estimators converge are base on the ratio κ of the electrical freuency ρ/t an the rotor electrical spee n p ω,thatis, κ, ρ/t n p ω =1+ηM τ µ n p ωψ 2 Specifically, it was shown in [31] that the main estimator converges for κ > while the complementary estimator converges for κ < or κ > 1, but not necessarily for < κ < 1. At start-up, κ >> 1, (i.e., τ an ω have the same sign with ω small), an the complementary estimator is use. Though this metho oes not assume that the spee is constant, it is heavily parameter-epenent. In [13], an approach is given to estimate the stator resistance an rotor time constant without a sensor, but it epens on the motor being at constant spee an in sinusoial steay-state. 379
Spee (ra/s) Spee (ra/s) Spee (ra/s) Spee (ra/s) 5 II. Comparison of the Sensorless Control Methos The sensorless control problem may be state as follows (a) achieve vector control of the inuction motor for spee an/or torue tracking; (b) be robust to parameter uncertainty; (c) hanle full rate loa-torue on the motor at start-up; () achieve rate spee of the motor in steaystate. These conitions are typically reuire of a fieloriente control system with asensor. To emonstrate the capabilities an limitations of the existing approaches, simulation moels of the above sensorless control schemes were evelope using Simnon R. The motor uner consieration was a 5 Hp, 3-phase inuction motor (see [15], p.19), with the motor parameters n p =2, n ph =3, R S =.262Ω, R R =.187Ω, J =11.6 kg-m 2, L S =.1465 H, L R =.1465 H, M =.143 H, V line line max = 23 Volts. The stator leakage inuctance is l S = L S M =.35 H an the rotor leakage inuctance is l R = L R M =.35 H. The synchronous spee is 2π(6/n p ) = 188 ra/sec, giving a rate spee of 18 ra/sec at 3% rate slip. The base spee is 8 ra/sec (i.e., the spee at which fiel weakening is starte) an the rate loa torue is τ L = 2 Nm. In all the simulations, the machine flux was allowe to buil up to its nominal value ψ o = Mi, with i =5Amps. The trajectory an loa torue τ L were applie at t =4secons. The plots show that the motor spee actually becomes negative at first ue to the application of the loa torue at t =4secons. This effect woul also be seen using a fiel-oriente controller with a shaft sensor, an coul be lessene by estimating the loa torue an fee-forwaring the estimate to the controller. A. Simulations of the Three Methos The simulation plots for the least-suares case are given in Fig. 1. The full loa torue τ L = 2 Nm is assume. The sampling perio for the spee estimator was.1 millisec an the forgetting factor λ was.99. The left-han plot is the spee an its reference, with the rotor resistance in the motor moel change from its nominal value of R R =.187Ω to R R =.3Ω. This results in a 5% spee tracking error. The plot of the right-han sie is uner the same conitions (with the rotor resistance returne to its nominal value R R =.187Ω), except that the stator resistance in the motor moel was change from its nominal value of R S =.262Ω to R S =.5Ω (about 1% increase, possibly ue to ohmic heating). The eterioration in performance is uite pronounce. This simulation shows the significant impact of parameter variations an the nee for joint estimation of the spee an of the motor parameters. Fig. 2 shows the simulation results for the nonlinear scheme given in [31]. It was foun that the loa torue ha to be reuce to τ L = 17 Nm in orer to achieve tracking with the nominal parameters. The plot on the lefthan sie shows the spee ω an spee reference ω ref,assuming nominal parameters in both the estimator an the controller (perfect parameter match). The right-han sie 15 1 5 2 4 6 8 1 12 14 16 15 1 5 2 4 6 8 1 12 14 16 Fig. 1. Spee an spee reference (least-suares scheme). Left: R R =.187Ω. Right:R R =.3Ω. 18 16 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16-2 -4-6 -8-1 -12-14 -16 2 4 6 8 1 12 14 16 Fig. 2. Spee an spee reference (nonlinear scheme). Left: R S =.262Ω. Right:R S =.29Ω. shows the spee plots with the stator resistance in the motor moel change from its nominal value of R S =.262Ω to R S =.29Ω (a change of only 1%). Thesystemcoul not hanle this small variation in the stator resistance. Again, a significant sensitivity to parameter uncertainty is observe with this metho. In view of these observations, the aaptive sensorless scheme propose in [22], esigne to be insensitive to the value of the stator resistance R S, woul seem uite attractive. It oes reuire knowlege of the rotor time constant T R (R R ), but so oes the stanar fiel-oriente controller with a position sensor. However, in our simulations, we were unable to get this sensorless scheme to work unless the loa torue was very low (τ L = 4 Nm or 2% of rate torue) an the motor was not allowe to ecelerate. TheplotsshowninFig. 3giveacomparisonwithaloa of τ L = 4 Nm on the left an with a loa of τ L = 5 Nm on the right. Interestingly, it was foun that the estimator itself worke fine if it was not use for feeback control, that is, if the true spee was fe back rather than its estimate. This observation inicates that, in aition to aressing the joint spee/parameter estimation problem, any practical sensorless control methos shoul also consier the joint control/estimation problem. We simulate the other sensorless scheme given in [22] (escribe above), which is a moification of the scheme presente by Shauer [25]. In our simulations, this scheme 38
Spee (ra/s) Spee (ra/s) 6 18 16 14 12 1 8 6 4 2 1 2 3 4 5 6 7 8 9 1 8 6 4 2-2 -4-6 -8 1 2 3 4 5 6 7 Fig. 3. Spee an spee reference (aaptive sensorless scheme). Left: τ L =4Nm. Right: τ L =5Nm. worke for eceleration, but still coul only accommoate small loa torues (τ L < 5 Nm) even uner nominal conitions (no parameter uncertainty). We also simulate the original aaptive approach of Shauer [25], an it was able to hanle a loa τ L = 17 Nm, a variation in the stator resistance from.262ω to.4 (5%), but only a variation in the rotor resistance of 15% (from.187ω to.2ω). III. Conclusions In summary, the above simulations inicate the ifficulty in trying to o sensorless control an perhaps the nee to not limit oneself to anyone particular algorithm. Algorithms shoul provie sensorless control uner full loa (to avoi over sizing the motor) while allowing for variations in stator resistance an in rotor time constant. A satisfactory solution may be obtaine through the combination of above techniues. For example, the least-suares approach coul be moifie to estimate the stator resistance R S an the rotor time constant T R in aition to the spee. In essence, this approach woul be the combination of the parameter ientification in [26], together with the metho of Verghese et al [19] [28] [29]. Another iea woul be to combine the metho of [22], which is insensitive to R S,with another metho; for example, the spee estimate of [22] woul not be fe back, but instea woul be use to calibrate the errors in another estimator that is sensitive to R S. References [1] Baaer, U. an M.Depenbrock, Direct Self Control (DSC) of Inverter-Fe Inuction Machine A Basis for Spee Control without Spee Measurement, IEEE Trans. on Inustry Applications, vol.28,no.3,may/june1992. [2] Bolea, I., an S.A. Nasar, Electric Drives, CRC Press, Boca Raton, FL, 1999. [3] Boson, M. an J. 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