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An Adaptive Flux and Position Observer for Interior Permanent Magnet Synchronous Motors⁎
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An An An An
Adaptive Flux and Position Observer Adaptive Flux and Position Observer Adaptive Flux and Position Observer Adaptive Flux and Position Observer for Interior Permanent Magnet for Interior Permanent Magnet for Interior Permanent Magnet for Interior Permanent Magnet Synchronous Motors Synchronous Motors Synchronous Motors Synchronous Motors ∗ ∗,∗∗ ∗
∗ Anton Pyrkin ∗,∗∗ Alexey Bobtsov ∗ Madina Sinetova Madina Sinetova Pyrkin ∗ ∗,∗∗ ∗ ∗ Anton ∗,∗∗ Alexey ∗ ∗∗∗,∗ ∗ Madina Romeo Sinetova Pyrkin Alexey Bobtsov Bobtsov ∗ Anton ∗,∗∗ ∗ ∗∗∗,∗ ∗ Ortega Alexey Vedyakov Madina Romeo Sinetova Anton Pyrkin Alexey Bobtsov Ortega Alexey Vedyakov ∗ ∗∗∗,∗ ∗ Romeo Ortega ∗∗∗,∗ Alexey Vedyakov ∗∗∗,∗ ∗ Romeo Ortega Alexey Vedyakov ∗ ∗ ITMO University, Faculty of Control Systems and Robotics, ITMO University, Faculty of Control and Robotics, ∗ ∗ ITMO University, Faculty of Control Systems Systems and Robotics, ∗ 49 Kronverksky Pr., St. 197101, Russia ITMO University, Faculty ofPetersburg, Control Systems and Robotics, 49 Kronverksky Pr., St. Petersburg, 197101, Russia 49 Kronverksky Pr., St. Petersburg, 197101, Russia (e-mail: [email protected]). 49 Kronverksky Pr., St. Petersburg, 197101, Russia (e-mail: [email protected]). ∗∗ (e-mail:[email protected]). ∗∗ Center for Technologies Robotics and Mechatronics Components, (e-mail:[email protected]). for Technologies and Components, ∗∗ ∗∗ Center in Robotics RoboticsInnopolis, and Mechatronics Mechatronics Components, ∗∗ Center for Technologies Innopolis University, Russia Center for Technologies in Robotics and Mechatronics Components, Innopolis University, Innopolis, Russia ∗∗∗ Innopolis University, Innopolis, Russia ∗∗∗ Laboratoire des Signaux et CNRS-SUPELEC, Plateau Innopolis University, Innopolis, Russia Signaux et Systmes, Systmes, CNRS-SUPELEC, Plateau du du ∗∗∗ ∗∗∗ Laboratoire des Signaux et CNRS-SUPELEC, Plateau ∗∗∗ Laboratoire des Moulon, 91192, Gif-sur-Yvette, France Laboratoire des Signaux et Systmes, Systmes, CNRS-SUPELEC, Plateau du du Moulon, 91192, Gif-sur-Yvette, France Moulon, Moulon, 91192, 91192, Gif-sur-Yvette, Gif-sur-Yvette, France France Abstract: The The design design of of an an adaptive active active flux flux observer observer is is an an open problem, problem, which which attracts attracts Abstract: Abstract: The design of an adaptive adaptive active flux observer is an open open problem, which attracts a lot of scientists from adaptive control and electric drive societies. The design of new observer The design of an adaptive active flux observer is an open problem, which attracts aAbstract: lot of scientists from adaptive control and electric drive societies. The design of new observer a lot of scientists from adaptive control and electric drive societies. The design of new observer for the interior permanent magnet synchronous motor is societies. based on on The the design ”linear ofregression regression plus a lotthe of interior scientistspermanent from adaptive control and electric drive new observer for magnet synchronous motor is based the ”linear plus for the synchronous motor is on ”linear plus gradient search”permanent approach. It Itmagnet is proved proved that the the flux flux observer provides asymptotic convergence of for the interior interior permanent magnet synchronous motor is based based on the the ”linear regression regression plus gradient search” approach. is that observer provides asymptotic convergence of gradient search” approach. It is proved that the flux observer provides asymptotic convergence of estimation errors to zero. In comparison with known solutions proposed approach gives improved gradient search” approach. Itcomparison is proved that the flux observer provides asymptotic convergence of estimation errors to zero. In with known solutions proposed approach gives improved estimation errors to zero.performance In comparison with known solutions proposed approach gives improved results with guaranteed (monotonicity, convergence rate regulation) and allows to estimation errors to zero.performance In comparison with known solutions proposed approach gives improved results with guaranteed (monotonicity, convergence rate regulation) and allows to results with guaranteed performance (monotonicity, convergence rate regulation) and allows to regulate the guaranteed convergenceperformance rate selection adaptation gains. gains. The main main feature feature of the the results with rate regulation) and allows to regulate the convergence rate via via the the (monotonicity, selection of of the theconvergence adaptation The of adaptation gains. The main feature of the regulate the convergence rate via the selection of the proposedthe observer, compared to existing existing ones, of is the thatadaptation it is well well defined in allmain operating regulate convergence rate via the selection gains. The featuremodes, of the proposed observer, compared to ones, is that that it it is is well defined defined in in all all operating operating modes, modes, proposed compared to improving its robustness robustness properties. proposed observer, observer, compared to existing existing ones, ones, is is that it is well defined in all operating modes, improving its properties. improving its robustness properties. improving its robustness properties. © 2019, IFAC (International Federation of Automatic Control) synchro Hosting bymotors, Elsevierflux Ltd.estimator, All rights reserved. Keywords: Nonlinear control systems, robust observers, observers, speed Keywords: Nonlinear control systems, robust synchro motors, flux estimator, speed Keywords: Nonlinear control systems, robust observers, synchro motors, flux estimator, speed estimator, sensorless approach. Keywords: sensorless Nonlinear control systems, robust observers, synchro motors, flux estimator, speed estimator, estimator, sensorless approach. approach. estimator, sensorless approach. 1. INTRODUCTION INTRODUCTION The self 1. The model model of of IPMSMs IPMSMs most most incorporate incorporate the the effect effect of of of self self 1. INTRODUCTION The model of IPMSMs most incorporate the effect and mutual inductances, which vary with an electrical 1. INTRODUCTION The model ofinductances, IPMSMs most incorporate the an effect of self and mutual which vary with electrical and mutual inductances, which vary with ancan electrical angle between phases and rotor axis, as you see in Existence mutual inductances, which vary with ancan electrical angle between phases and rotor axis, you in Existence of of permanent permanent magnets magnets (PMs) (PMs) exclusion exclusion the the use use and angle betweenNam phases and and rotorOrtega axis, as as you(2011). can see seeThe in the literature (2010) et al. Existence of permanent magnets (PMs) exclusion the use of field exciting coils and slip rings for current conduction. betweenNam phases and and rotorOrtega axis, as you(2011). can seeThe in the literature (2010) et al. Existence of permanent exclusion the use angle of field exciting coils andmagnets slip rings(PMs) for current conduction. the literature Nam (2010) and Ortega et al. (2011). The dynamic equations that describe the behavior of IPMSMs of field exciting coils and rings current conduction. The absence of field the rotor provides low the literature Nam that (2010) and Ortega et al. (2011). The dynamic equations describe the behavior of IPMSMs of field exciting coils winding and slip slip inside rings for for current conduction. The absence of field winding inside the rotor provides low dynamic equations that describe the behavior of IPMSMs are far more complicated than those of SPMSMs. The absence of field winding inside the rotor provides low inertia of The is so the equations that describe the behavior of IPMSMs are The absence field winding insidestrength the rotor inertia of PM PMof motors. motors. The field field strength is provides so high high low the dynamic are far far more more complicated complicated than than those those of of SPMSMs. SPMSMs. inertia of PM motors. The field strength is so high the motor be Also, there no copper far more complicated than those of SPMSMs. inertia of PM can motors. The field strength high loss the are A review of the first used methods of sensorless control was motor volume volume can be reduced. reduced. Also, there is is is noso copper loss A review of used methods of control motor volume be Also, is copper loss of the secondary that’s whythere the PM PM motors have review of the the first firstand used methods of sensorless sensorless control was was motor volume can canwinding, be reduced. reduced. Also, there is no nomotors copperhave loss A given in Acarnley Watson (2006), then a Luenberger of the secondary winding, that’s why the review of the firstand used methods of sensorless control was given in (2006), then aa Luenberger of the secondary winding, that’s why the PM motors have A higher efficiency compared induction motors. Permanent in Acarnley Acarnley and Watson Watson (2006), then Luenberger of the secondary winding, that’s why the PM motors have given observer was proposed in Poulain et al. (2009). There is higher efficiency compared induction motors. Permanent given in Acarnley and Watson (2006), then a Luenberger observer was proposed in et (2009). There is higher efficiency compared induction motors. Permanent magnet synchronous motors (PMSMs) advantageous in was proposed in Poulain Poulain et al. al. (2009). There is higher compared motors. Permanent aa simple gradient observer, proposed in Lee et al. (2010) magnetefficiency synchronous motors induction (PMSMs) are are advantageous in observer observer was proposed in Poulain et al. (2009). There is simple gradient observer, proposed in Lee et al. (2010) magnet synchronous motors (PMSMs) are advantageous in incorporating the torque the field-weakening a simple gradient observer, proposed in Lee et shown al. (2010) magnet synchronous motors (PMSMs) in and analyzed in Ortega et al. (2011). It has been the incorporating the reluctance reluctance torque in in are the advantageous field-weakening a simple gradient observer, proposed in Lee et al. (2010) and analyzed in Ortega et al. (2011). It has been shown the incorporating the torque in the range, so that they can be designed wide constant and analyzed in Ortega et al. (2011). It has been shown the incorporating the reluctance reluctance torqueto inhave the afield-weakening field-weakening observer is effective estimates rotor position in practice. A range, that they be to have wide constant analyzed in Ortega et al. (2011). It has been shown the observer is estimates rotor position A range, so so thatrange they can can be designed designed to have aapower wide densities constant and power speed (CPSR). In the result, is effective effective estimates rotor positionisin ininpractice. practice. A range, so thatrange they can be designed to have apower wide densities constant observer very minor modification of thise observer the paper power speed (CPSR). In the result, observer is effective estimates rotor position in practice. A very minor modification of thise observer is in the paper power speed range (CPSR). In the result, power densities of PMSMs is higher than any other types of motors. very minor modification of thise observer is in the paper power speedis range the result, power densities very Malaiz et al. al. (2012) and andofit itthise can observer be globally globally convergent of PMSMs higher(CPSR). than anyInother types of motors. minor modification is inconvergent the paper Malaiz et (2012) can be of PMSMs is higher than any other types of motors. Malaiz to et convexity al. (2012) and it can be globally convergent of PMSMs is higher than any other types ofcost, motors. thanks Due et al. (2012)properties. and it can be globally convergent thanks Due to to recent recent reduction reduction in in PM PM material material cost, PMSMs PMSMs Malaiz thanks to to convexity convexity properties. properties. Due to recent reduction in PM material cost, PMSMs are widely used in home appliances such as refrigerators, thanks to convexity properties. Due to recent in PM material PMSMs Also Also the the challenging problem problem of of sensorless sensorless control control is is conconare widely used reduction in home appliances such ascost, refrigerators, are widely home such refrigerators, air vacuum cleaners, Likewise, Also theinchallenging challenging problem of sensorless control is conare widely used used in in home appliances appliances such as as etc. refrigerators, Bobtsov et al. (2015); Marino et al. (2010), air conditioners, conditioners, vacuum cleaners, washers, washers, etc. Likewise, sidered Also the challenging problem of sensorless control is considered in Bobtsov et al. (2015); Marino et al. (2010), air conditioners, vacuum cleaners, washers, etc. Likewise, hydraulic actuators in vehicles and airplanes are being residered in Bobtsovvariables et al. (2015); Marino et al. (2010), air conditioners, vacuum cleaners, washers, etc. Likewise, where mechanical of the drive are reconstructed hydraulic actuators in vehicles and airplanes are being residered in Bobtsov et al. (2015); Marino et al. (2010), where mechanical variables of the drive are reconstructed hydraulic actuators in vehicles and airplanes are being replaced PMSMs PMSMs mechanicalofvariables offlux. the That’s drive are reconstructed hydraulic in higher vehiclesfuel andefficiency. airplanesAlso, are being re- where using knowledge the total why usually the placed by by actuators PMSMs for for higher fuel efficiency. Also, PMSMs where mechanicalofvariables offlux. the That’s drive are reconstructed using knowledge the total why usually the placed by PMSMs for higher fuel efficiency. Also, PMSMs are popularly used as propulsion motors for hybrid electric using knowledge of the total flux. That’s why usually the placed by PMSMs forpropulsion higher fuelmotors efficiency. Also, PMSMs magnetic flux observation observation is aaflux. key problem, problem, which attracts are popularly used as for hybrid electric using knowledge of the total That’s why usually the magnetic flux is key which attracts are popularly used as propulsion motors for hybrid electric vehicles and ships. flux observation is a keycontrol problem, which attracts are popularly used as propulsion motors for hybrid electric magnetic amagnetic lot of scientists from adaptive and electric drive vehicles and ships. flux observation is a key problem, which attracts a lot of scientists from adaptive control and electric drive vehicles asocieties, lot of scientists from adaptive control andR.electric drive vehicles and and ships. ships. including L. Praly, Ortega, Marino, P. asocieties, lot of scientists from andR.electric drive including L. adaptive Praly, R. R.control Ortega, Marino, P. societies, including L. Praly, R. Ortega, R. Marino, P. Tomei, K. Nam, A. Stankovic, and many other researchers. societies, including L. Praly, R. Ortega, R. Marino, P. Tomei, K. Nam, A. Stankovic, and many other researchers. article is supported by Government of Russian Federa This article is supported by Government of Russian FederaTomei, K. Nam, A. Stankovic, and many other researchers. This An observer for a class of electromechanical systems has article is supported by Government of Russian FederaTomei, K. Nam, A. Stankovic, and many other systems researchers. tion (GOSZADANIE 2.8878.2017/8.9, grant (sections 1, An observer for a class of electromechanical has This tion (GOSZADANIE 2.8878.2017/8.9, grant 08-08) 08-08) (sections 1, 3, 3, This article is supported by Government of Russian FederaAn observer forina Pyrkin class ofetelectromechanical systems been proposed al. (2018). Although a lothas of tion (GOSZADANIE 2.8878.2017/8.9, grantof08-08) (sections 1, 3, 4, and by Science and An observer forina Pyrkin class ofetelectromechanical systems has been proposed al. (2018). Although a 4, 6 6 (GOSZADANIE and appendix) appendix) and and by the the Ministry Ministry Science and Higher Higher tion 2.8878.2017/8.9, grantof08-08) (sections 1, 3, been proposed in Pyrkin etand al. even (2018). Although a lot lot of of 4, 6 and appendix) and by Federation, the Ministry of Science andidentifier Higher different approaches exist, already implemented Education of the Russian project unique been proposed in Pyrkin etand al. even (2018). Although a lot of different approaches exist, already implemented Education of the Russian project unique 4, 6 and appendix) and by Federation, the Ministry of Science andidentifier Higher different approaches exist, and even already implemented Education of the Russian Federation, project unique identifier as feature distributed actuators, the RFMEFI57818X0271 ”Adaptive Sensorless Control for different and even already however, implemented as preset presetapproaches feature in in exist, distributed actuators, however, the RFMEFI57818X0271 ”Adaptive Sensorlessproject Controlunique for Synchronous Synchronous Education of the Russian Federation, identifier RFMEFI57818X0271 ”Adaptive Sensorless Control for Synchronous as preset feature in distributed actuators, however, the Electric Robotics and Systems” (secElectric Drives Drives in in Intelligent Intelligent Robotics and Transport Transport Systems” (secas preset feature in distributed actuators, however, the RFMEFI57818X0271 ”Adaptive Sensorless Control for Synchronous Electric Drives in Intelligent Robotics and Transport Systems” (sec-
tions Electric Drives in Intelligent Robotics and Transport Systems” (sections 2, 2, 5) 5) Electric Drives in Intelligent Robotics and Transport Systems” (sections 2, 5) tions 2, 5) 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.619
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performance of such control strategy is still an open problem. The main purpose of this paper is to present the adaptive flux observer for IPMSMs. The problem formulation is in the Section 2, where is presented the classical two phase model of the IPMSM. In the Section 3 the linear regression equation and active flux observer are proposed. The simulations for the IPMSM are shown in the Section 4. 2. PROBLEM FORMULATION Consider the classical, two phase αβ model of the IPMSM described by Nam (2010) λ˙ = v − Ri,
L0 Q(2θ)]−1 [λ − ψm C(θ)] (1) i = [Ls I2 + 2 where λ ∈ R2 is the total flux, i ∈ R2 are the currents, v ∈ R2 are the voltages, R > 0 is the stator windings resistance, I2 is an identity matrix, ψm is PM flux linkage constant, θ ∈ S := [0, 2π] is the rotor flux angle. Ls is averaged inductance, L0 is inductance difference, that are given by d and q-axis inductances Ld and Lq 1 Ls : = (Ld + Lq ) , L0 : = (Ld − Lq ) . 2 We defined
cos(2np θ) sin(2np θ) Q(2θ) = sin(2np θ) − cos(2np θ)
and C(θ) :=
cos(np θ), , sin(np θ)
where np is the number of pole pairs.
Using the adaptive flux observer for PMSM that was proposed in Bobtsov et al. (2015) allows to get the equation L0 Q(2θ)]i|2 − λ2m = 0. |λ − [Ls I2 + 2 The problem of adaptive flux observer construction for IPMSM is considered in this paper. It’s assumed that only the electrical signals are available for measurement, namely i and v, and all electrical parameters, i.e.,R, L0 , Ls , ψm are exactly known. The goal is to design an observer that asymptotically reconstructs the active flux λ and position θ: ˆ ˆ = 0, lim |θ(t) − θ(t)| = 0. (2) lim |λ(t) − λ(t)| t→∞
t→∞
3. PRELIMINARIES As indicated in the previous section, the electrical dynamics of the IPMSM is given by Faraday’s Law (1), together with the constitutive relation L0 Q(2θ)]i + ψm C(θ). (3) λ = [Ls I2 + 2 We used some simple calculations from the paper Choi et al. (2018) and rewrote (3) as λ = Lq i + L0 i C(θ) + ψm C(θ). (4)
Following the idea from the new paper Ortega et al. (2019) we would like to start from the definition of the state x (Equation (6)), that is (5) x := λ − Lq i. Additionally we have the equalities x = (L0 i C(θ) + ψm )C(θ) and d x˙ = v − Ri − Lq i. dt Besides we obtain |x|2 = (L0 i C(θ) + ψm )2 Also the rotor angle is reconstructed from x via x2 1 arctan , θ= np x1
(6) (7) (8)
(9)
The next Lemma gives a simpler parametrization. Lemma 1. (Ortega et al. (2019)). The electrical dynamics of the IPMSM (1) satisfies the following (perturbed) linear regression equation (10) yo = Φ o x + do + t where x is the active flux defined in (5), the unknown perturbing signal d is given by αp x i do := −l p+α |x| and the measurable signals y and Ψ are generated as 1 1 α i Ω Ω1 , Ω1 + |Ω1 |2 + yo := L0 p+α α p+α 2 (11) Φo := Ω1 + Ω2 , d where l := ψm L0 , p = dt is the differentiation operator, and α > 0 is a tuning parameter and α Ω1 := (v − Ri − Lq pi) , p+α αp Ω2 := Ω1 − L0 i. p+α Proposition 1. (Ortega et al. (2019)). Consider the electrical dynamics of the IPMSM with the signal Φo , defined in (11). Define the active flux/position observer ˆ˙ = v − Ri λ αp ˆ x + γΦo y − Φo x i , ˆ+l p+α |ˆ x| ˆ − Lq i, x ˆ=λ x ˆ2 1 arctan θˆ = , (12) np x ˆ1 with yo defined in (11), γ > 0 and α are tuning parameters.
There exist αmax > 0 and γmax > 0 such that for all α ≤ αmax and γ ≤ γmax we have ˜ ˜ ∀t ≥ 0, |λ(t)| ≤ me−ρt |λ(0)|, for some m > 0, ρ > 0. Moreover ˜ lim θ(t) = 0. t→∞
4. MAIN RESULT
In this section we present our flux observer. And the next Lemma gives a parametrization.
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Lemma 2. The electrical dynamics of the IPMSM satisfies the following (perturbed) linear regression equation y = Φ x + d + t
(13)
where x is the active flux defined in (5), the unknown perturbing signal αp x C(θ), (14) d := ψm p+α d is the differentiation operator, and the where p = dt measurable signals y and Φ are generated as y : = (2R − αL0 )φ3 − 2φ4 + 2Lq φ˙ 3 − Lq (Lq + L0 )φ5 2Lq ˙ 2 − φ˙ φ2 + Lq φ˙ φ˙ 1 φ − φ 1 1 α 2 1 ˙ + L 0 Lq φ 1 φ1 , (15) Φ : = αL0 i + (2R − αL0 )φ1 − 2φ2 + 2Lq φ˙ 1 ,
where α > 0 is a tuning parameter and α i, φ1 := p+α α v, φ2 := p+α 1 φ3 := (v − Ri) φ1 , p+α 1 φ4 := (v − Ri) φ2 , p+α 1 ˙ ˙ φ5 := φ φ1 , p+α 1 where φ˙ j , j = 1, 5 are measurable as well.
45
αp ˆ x C(θ) − x ˆ C(θ) (19) d˜ = ψm p+α αp ˆ + x C(θ) − C(θ) ˆ = ψm x ˜ C(θ) (20) p+α αp x ˜ w(x, x ˜) . (21) = p+α The function w(x, x ˜) : R2 ×R2 → R2 is analytical and welldefined mapping, where a constant ψm is involved inside the filter ˆ x ˜))+ w(x, x ˜) := ψm C(θ(x, 2ψm sin δ(x, x ˜) − sin σ(x, x ˜) x + x ˜, cos σ(x, x ˜) x ˜ x ˜ x2 − x ˜2 cos arctan x1 − x ˜1 ˆ x C(θ(x, ˜)) := x2 − x ˜2 sin arctan x1 − x ˜1 1 x2 x2 − x ˜2 σ(x, x ˜) := arctan + arctan 2 x1 x1 − x ˜1 1 x2 x2 − x ˜2 δ(x, x ˜) := arctan − arctan . 2 x1 x1 − x ˜1 Combining (18) and (19), let us rewrite the full model of the observation error in a state-space representation x ˜˙ = −γΦ(Φ x ˜ + z − αw x ˜) z˙ = −αz + α2 w x ˜.
(22)
(16)
The latter coincides with the proof of Proposition 1 in the work Ortega et al. (2019) which with the same arguments allow us to complete this proof.
Proof of the lemma will be established in Appendix. Proposition 2. Consider the electrical dynamics of the IPMSM with the signal Φ, defined in (15). Define the active flux/position observer ˆ˙ = v − Ri λ αp ˆ + γΦ y − Φ x x ˆ C(θ) , ˆ − ψm p+α ˆ − Lq i, x ˆ=λ x ˆ2 1 ˆ arctan θ= , (17) np x ˆ1
Note that a new parameterisation of model of IPMSM as well as the flux/position observer doesn’t contain a division by |x|.
There exist αmax > 0 and γmax > 0 such that for all α ≤ αmax and γ ≤ γmax we have ˜ ˜ ∀t ≥ 0, |λ(t)| ≤ me−ρt |λ(0)|,
Table 1. Parameters of the IPMS
with y and Φ defined in (15), γ > 0 and α are a tuning parameters.
for some m > 0, ρ > 0. Moreover ˜ lim θ(t) = 0. t→∞
Proof.
The observation error dynamics for x ˜ := x − x ˆ is x ˜˙ = x˙ − x ˆ˙ where
˜ ˜ + d), = −γΦ(Φ x
(18)
5. NUMERICAL EXAMPLE In this section we consider the motor as in the paper Ortega et al. (2019), which parameters are listed in Table 1. Parameter Pairs of poles np (–) PM flux linkage constant ψm (Wb) d-axis inductance Ld (mH) q-axis inductance Lq (mH) Stator resistance R (Ω) Drive inertia j (kgm2 )
Motor 6 0.11 5.74 8.68 0.43 0.01
The general structure of the sensorless control algorithm of IPMSM with active flux/position observer is shown in the 1. The initial conditions of the observer are selected as ˆ = [0.5, 2]. Simulations were conducted to evaluate the λ performance of the proposed flux observer with three pairs of different parameters of the observer that are selected as α1 = 10, γ1 = 20, α2 = 20, γ2 = 25, α3 = 30, γ3 = 30 and input voltage v = col (sin t, cos t). Fig. 2 and Fig. 3 show ˜ = λ ˆ−λ the flux and position observer errors, where λ and θ˜ = θˆ − θ. Fig. 4 and Fig. 5 show comparison of two observers Ortega et al. (2019) and the new one, which
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L0
Fig. 1. Structure of the sensorless control algorithm of IPMSM with observer illustrate that the new proposed observer ( Proposition 2, (17)) slightly improved. 6. CONCLUSION In this paper we propose the new adaptive observer design algorithm for IPMSM. The flux observer provides asymptotic convergence of estimation errors to zero. In comparison with well-known proposed approach gives a improved results with guaranteed performance (monotonicity, convergence rate regulation) and allows to regulate the convergence rate via adaptation gains. A new parametrization of model of IPMSM is proposed that improves the results of Ortega et al. (2019), since it does not contain a division by |x| which potentially generates a numerical problem due to division by a small number. APPENDIX Proof of Lemma 2 Multiplying (6) on C (θ) from the left yields C (θ)x = L0 i C(θ) + ψm .
Introduce sλ = sign(L0 i C(θ) + ψm ). Since C(θ) = we have for (6) x x x s λ = L0 i s λ + ψ m |x| |x| and after avoiding division by zero we get x x − L0 i x − ψm |x|sλ = 0.
(23) x sλ |x| (24)
(25)
α p+α
to the equation (25) α α xT i = xT x L0 p+α p+α α −ψm |x|sλ + t p+α and using Swapping Lemma from Sastry and Bodson (2011) we get 1 α α i − L0 x˙ i L0 x p+α p+α p+α 2 x x˙ = x x − p+α α − ψm |x|sλ + t (26) p+α Applying filter
Consider separately two terms of (26) for simplifying the expression. Find that the following term is computable if we invoke filters (16):
1 p+α
x˙
α i p+α 1 ˙ α ˆ = L0 (v − Ri − Lq i) i p+α p+α 1 (v − Ri) φ1 = L0 p+α 1 ˙ α ˆ − L0 Lq i i p+α p+α L0 Lq ˙ 1 − 1 φ˙ φ˙ 1 φ = L 0 φ3 − φ 1 α p+α 1 L0 Lq ˙1 . φ 5 − φ (27) φ = L0 φ 3 + 1 α
Consider the term α α ˙ x x˙ = x (v − Ri − Lqˆi) p+α p+α α ˙ (v − Ri − Lqˆi) = x p+α α 1 ˙ ˙ ˆ ˆ − (v − Ri − Lq i) (v − Ri − Lq i) p+α p+α (28) = x (φ2 − Rφ1 − Lq φ˙ 1 ) − ya . where ya is also computable signal α ˙ 1 (v − Ri − L ˆi) ˙ (v − Ri − Lqˆi) ya : = q p+α p+α 1 (v − Ri) (φ2 − Rφ1 − Lq φ˙ 1 ) = p+α 1 ˆ˙ − Lq i (φ2 − Rφ1 − Lq φ˙ 1 ) p+α L2q φ5 = φ4 − Rφ3 − Lq φ˙ 3 + 2 Lq ˙ ˙ Lq ˙ ˙ φ 1 φ2 − φ 1 φ 2 + φ φ1 , (29) + α 2 1 where we used the following straightforward calculations: 1 (v − Ri) (φ2 − Rφ1 − Lq φ˙ 1 ) = p+α = φ4 − Rφ3 1 v˙ − Rˆi˙ φ1 − Lq φ˙ 3 − p+α = φ4 − Rφ3 − Lq φ˙ 3 Lq ˙ 2 − Rφ˙ 1 ) − 1 (φ˙ (φ˙ 2 − Rφ˙ 1 )) φ + ( φ 1 α p+α 1 = φ4 − Rφ3 − Lq φ˙ 3 Lq 1 ˙ ˙ ˙ ˙ φ 1 ( φ2 − R φ1 ) − φ φ2 + Rφ3 + α p+α 1 and 1 ˆ˙ i (φ2 − Rφ1 − Lq φ˙ 1 ) = p+α 1 = (φ2 − Rφ1 − Lq φ˙ 1 ) φ˙ 1 α L 1 1 ˙ 1 ¨ ˙ q (φ2 − Rφ˙ 1 ) φ˙ 1 + φ φ1 − αp+α α p+α 1 1 1 1 ˙2 = (φ2 − Rφ1 − Lq φ˙ 1 ) φ˙ 1 − φ˙ φ α αp+α 1 R Lq ˙ ˙ Lq + φ5 + φ φ1 − φ5 . α 2α 1 2
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2
2
1.5
1.5
1
1
0.5
0.5
0
0 0
100
200
300
400
0
(a) Observer (17)
100
47
200
300
400
(b) Observer Ortega et al. (2019)
Fig. 2. Error of total flux estimation with 1) α1 = 30, γ1 = 20, 2) α2 = 100, γ2 = 25, 3) α3 = 200, γ3 = 30 6
6
5
5
4
4
3
3
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1
1 0
0 0
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Fig. 3. Error of rotor angle estimation with 1) α1 = 30, γ1 = 20, 2) α2 = 100, γ2 = 25, 3) α3 = 200, γ3 = 30 5 2
4 3
1.5
2 1 1 0.5 0 0 0
100
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(a) Error of total flux estimation
600
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(b) Error of rotor angle estimation
Fig. 4. Comparison of two observers for the case of adaptation gains α = 30, γ = 30, voltage v = col (0.1 sin t, 0.1 cos t) 1) for observer from the paper Ortega et al. (2019) 2) for the proposed observer (17)
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5
2
4 1.5 3 1 2 0.5
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(a) Error of total flux estimation
0
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(b) Error of rotor angle estimation
Fig. 5. Comparison of two observers for the case of adaptation gains α = 100, γ = 30, voltage v = col (sin t, cos t) 1) for observer from the paper Ortega et al. (2019) 2) for the proposed observer (17)
Rewrite (26) using previous derivations and additionally multiplying everything on α L0 Lq α ˙1 αL0 x = i − α L 0 φ3 + φ 5 − φ φ 1 p+α α = αx x − 2x (φ2 − Rφ1 − Lq φ˙ 1 ) + 2ya α2 |x|sλ + t − ψm (30) p+α Making substitution x x = L0 i x + ψm |x|sλ from (25), ya from (29) and recombining terms in (30), we finally get: (2R − αL0 )φ3 − 2φ4 + 2Lq φ˙ 3 − Lq (Lq + L0 )φ5 2Lq ˙ 2 − φ˙ φ2 + Lq φ˙ φ˙ 1 + L0 Lq φ φ˙ 1 = φ − φ 1 1 1 α 2 1 = αL0 i + (2R − αL0 )φ1 − 2φ2 + 2Lq φ˙ 1 x αp |x|sλ + t . + ψm (31) p+α Finally, noticing that |x|sλ = x C(θ), we get the equation of the form (10), which completes the proof. REFERENCES P.P. Acarnley and J. F. Watson. Review of positionsensorless operation of brushless permanent-magnet machines. Industrial Electronics, IEEE Transactions on, 53:352 – 362, 2006. A.A. Bobtsov, A. A. Pyrkin, R. Ortega, S. N. Vukosavic, A. M. Stankovic, and E. V. Panteley. A robust globally convergent position observer for the permanent magnet synchronous motor. Automatica, 61:47–54, 2015. J. Choi, K. Nam, A. A. Bobtsov, and R. Ortega. Sensorless control of ipmsm based on regression model. IEEE Transactions on Power Electronics, pages 1–1, 2018. J. Lee, J. Hong, K. Nam, R. Ortega, L. Praly, and A. Astolfi. Sensorless control of surface-mount permanentmagnet synchronous motors based on a nonlinear observer. Power Electronics, IEEE Transactions on, 25: 290 – 297, 2010.
J. Malaiz, L. Praly, and N. Henwood. Globally convergent nonlinear observer for the sensorless control of surfacemount permanent magnet synchronous machines. pages 5900–5905, 2012. R. Marino, P. Tomei, and C. Verrelli. Induction Motor Control Design. Springer-Verlag, 2010. K. Nam. AC motor control and electrical vehicle applications. CRC press, 2010. R. Ortega, L. Praly, A. Astolfi, J. Lee, and K. Nam. Estimation of rotor position and speed of permanent magnet synchronous motors with guaranteed stability. Control Systems Technology, IEEE Transactions on, 19: 601 – 614, 2011. R. Ortega, B. Yi, K. Nam, and J. Choi. A globally exponentially stable position observer for interior permanent magnet synchronous motors. CoRR, abs/1905.00833, 2019. URL http://arxiv.org/abs/1905.00833. F. Poulain, L. Praly, and R. Ortega. An observer for permanent magnet synchronous motors with currents and voltages as only measurements. pages 5390 – 5395, 2009. A.A. Pyrkin, A.A. Vedyakov, R. Ortega, and A.A. Bobtsov. A robust adaptive flux observer for a class of electromechanical systems. International Journal of Control, pages 1–11, 2018. S. Sastry and M. Bodson. Adaptive control: stability, convergence and robustness. Courier Corporation, 2011.
ScienceDirect IFAC PapersOnLine 52-29 (2019) 43–48
An An An An
Adaptive Flux and Position Observer Adaptive Flux and Position Observer Adaptive Flux and Position Observer Adaptive Flux and Position Observer for Interior Permanent Magnet for Interior Permanent Magnet for Interior Permanent Magnet for Interior Permanent Magnet Synchronous Motors Synchronous Motors Synchronous Motors Synchronous Motors ∗ ∗,∗∗ ∗
∗ Anton Pyrkin ∗,∗∗ Alexey Bobtsov ∗ Madina Sinetova Madina Sinetova Pyrkin ∗ ∗,∗∗ ∗ ∗ Anton ∗,∗∗ Alexey ∗ ∗∗∗,∗ ∗ Madina Romeo Sinetova Pyrkin Alexey Bobtsov Bobtsov ∗ Anton ∗,∗∗ ∗ ∗∗∗,∗ ∗ Ortega Alexey Vedyakov Madina Romeo Sinetova Anton Pyrkin Alexey Bobtsov Ortega Alexey Vedyakov ∗ ∗∗∗,∗ ∗ Romeo Ortega ∗∗∗,∗ Alexey Vedyakov ∗∗∗,∗ ∗ Romeo Ortega Alexey Vedyakov ∗ ∗ ITMO University, Faculty of Control Systems and Robotics, ITMO University, Faculty of Control and Robotics, ∗ ∗ ITMO University, Faculty of Control Systems Systems and Robotics, ∗ 49 Kronverksky Pr., St. 197101, Russia ITMO University, Faculty ofPetersburg, Control Systems and Robotics, 49 Kronverksky Pr., St. Petersburg, 197101, Russia 49 Kronverksky Pr., St. Petersburg, 197101, Russia (e-mail: [email protected]). 49 Kronverksky Pr., St. Petersburg, 197101, Russia (e-mail: [email protected]). ∗∗ (e-mail:[email protected]). ∗∗ Center for Technologies Robotics and Mechatronics Components, (e-mail:[email protected]). for Technologies and Components, ∗∗ ∗∗ Center in Robotics RoboticsInnopolis, and Mechatronics Mechatronics Components, ∗∗ Center for Technologies Innopolis University, Russia Center for Technologies in Robotics and Mechatronics Components, Innopolis University, Innopolis, Russia ∗∗∗ Innopolis University, Innopolis, Russia ∗∗∗ Laboratoire des Signaux et CNRS-SUPELEC, Plateau Innopolis University, Innopolis, Russia Signaux et Systmes, Systmes, CNRS-SUPELEC, Plateau du du ∗∗∗ ∗∗∗ Laboratoire des Signaux et CNRS-SUPELEC, Plateau ∗∗∗ Laboratoire des Moulon, 91192, Gif-sur-Yvette, France Laboratoire des Signaux et Systmes, Systmes, CNRS-SUPELEC, Plateau du du Moulon, 91192, Gif-sur-Yvette, France Moulon, Moulon, 91192, 91192, Gif-sur-Yvette, Gif-sur-Yvette, France France Abstract: The The design design of of an an adaptive active active flux flux observer observer is is an an open problem, problem, which which attracts attracts Abstract: Abstract: The design of an adaptive adaptive active flux observer is an open open problem, which attracts a lot of scientists from adaptive control and electric drive societies. The design of new observer The design of an adaptive active flux observer is an open problem, which attracts aAbstract: lot of scientists from adaptive control and electric drive societies. The design of new observer a lot of scientists from adaptive control and electric drive societies. The design of new observer for the interior permanent magnet synchronous motor is societies. based on on The the design ”linear ofregression regression plus a lotthe of interior scientistspermanent from adaptive control and electric drive new observer for magnet synchronous motor is based the ”linear plus for the synchronous motor is on ”linear plus gradient search”permanent approach. It Itmagnet is proved proved that the the flux flux observer provides asymptotic convergence of for the interior interior permanent magnet synchronous motor is based based on the the ”linear regression regression plus gradient search” approach. is that observer provides asymptotic convergence of gradient search” approach. It is proved that the flux observer provides asymptotic convergence of estimation errors to zero. In comparison with known solutions proposed approach gives improved gradient search” approach. Itcomparison is proved that the flux observer provides asymptotic convergence of estimation errors to zero. In with known solutions proposed approach gives improved estimation errors to zero.performance In comparison with known solutions proposed approach gives improved results with guaranteed (monotonicity, convergence rate regulation) and allows to estimation errors to zero.performance In comparison with known solutions proposed approach gives improved results with guaranteed (monotonicity, convergence rate regulation) and allows to results with guaranteed performance (monotonicity, convergence rate regulation) and allows to regulate the guaranteed convergenceperformance rate selection adaptation gains. gains. The main main feature feature of the the results with rate regulation) and allows to regulate the convergence rate via via the the (monotonicity, selection of of the theconvergence adaptation The of adaptation gains. The main feature of the regulate the convergence rate via the selection of the proposedthe observer, compared to existing existing ones, of is the thatadaptation it is well well defined in allmain operating regulate convergence rate via the selection gains. The featuremodes, of the proposed observer, compared to ones, is that that it it is is well defined defined in in all all operating operating modes, modes, proposed compared to improving its robustness robustness properties. proposed observer, observer, compared to existing existing ones, ones, is is that it is well defined in all operating modes, improving its properties. improving its robustness properties. improving its robustness properties. © 2019, IFAC (International Federation of Automatic Control) synchro Hosting bymotors, Elsevierflux Ltd.estimator, All rights reserved. Keywords: Nonlinear control systems, robust observers, observers, speed Keywords: Nonlinear control systems, robust synchro motors, flux estimator, speed Keywords: Nonlinear control systems, robust observers, synchro motors, flux estimator, speed estimator, sensorless approach. Keywords: sensorless Nonlinear control systems, robust observers, synchro motors, flux estimator, speed estimator, estimator, sensorless approach. approach. estimator, sensorless approach. 1. INTRODUCTION INTRODUCTION The self 1. The model model of of IPMSMs IPMSMs most most incorporate incorporate the the effect effect of of of self self 1. INTRODUCTION The model of IPMSMs most incorporate the effect and mutual inductances, which vary with an electrical 1. INTRODUCTION The model ofinductances, IPMSMs most incorporate the an effect of self and mutual which vary with electrical and mutual inductances, which vary with ancan electrical angle between phases and rotor axis, as you see in Existence mutual inductances, which vary with ancan electrical angle between phases and rotor axis, you in Existence of of permanent permanent magnets magnets (PMs) (PMs) exclusion exclusion the the use use and angle betweenNam phases and and rotorOrtega axis, as as you(2011). can see seeThe in the literature (2010) et al. Existence of permanent magnets (PMs) exclusion the use of field exciting coils and slip rings for current conduction. betweenNam phases and and rotorOrtega axis, as you(2011). can seeThe in the literature (2010) et al. Existence of permanent exclusion the use angle of field exciting coils andmagnets slip rings(PMs) for current conduction. the literature Nam (2010) and Ortega et al. (2011). The dynamic equations that describe the behavior of IPMSMs of field exciting coils and rings current conduction. The absence of field the rotor provides low the literature Nam that (2010) and Ortega et al. (2011). The dynamic equations describe the behavior of IPMSMs of field exciting coils winding and slip slip inside rings for for current conduction. The absence of field winding inside the rotor provides low dynamic equations that describe the behavior of IPMSMs are far more complicated than those of SPMSMs. The absence of field winding inside the rotor provides low inertia of The is so the equations that describe the behavior of IPMSMs are The absence field winding insidestrength the rotor inertia of PM PMof motors. motors. The field field strength is provides so high high low the dynamic are far far more more complicated complicated than than those those of of SPMSMs. SPMSMs. inertia of PM motors. The field strength is so high the motor be Also, there no copper far more complicated than those of SPMSMs. inertia of PM can motors. The field strength high loss the are A review of the first used methods of sensorless control was motor volume volume can be reduced. reduced. Also, there is is is noso copper loss A review of used methods of control motor volume be Also, is copper loss of the secondary that’s whythere the PM PM motors have review of the the first firstand used methods of sensorless sensorless control was was motor volume can canwinding, be reduced. reduced. Also, there is no nomotors copperhave loss A given in Acarnley Watson (2006), then a Luenberger of the secondary winding, that’s why the review of the firstand used methods of sensorless control was given in (2006), then aa Luenberger of the secondary winding, that’s why the PM motors have A higher efficiency compared induction motors. Permanent in Acarnley Acarnley and Watson Watson (2006), then Luenberger of the secondary winding, that’s why the PM motors have given observer was proposed in Poulain et al. (2009). There is higher efficiency compared induction motors. Permanent given in Acarnley and Watson (2006), then a Luenberger observer was proposed in et (2009). There is higher efficiency compared induction motors. Permanent magnet synchronous motors (PMSMs) advantageous in was proposed in Poulain Poulain et al. al. (2009). There is higher compared motors. Permanent aa simple gradient observer, proposed in Lee et al. (2010) magnetefficiency synchronous motors induction (PMSMs) are are advantageous in observer observer was proposed in Poulain et al. (2009). There is simple gradient observer, proposed in Lee et al. (2010) magnet synchronous motors (PMSMs) are advantageous in incorporating the torque the field-weakening a simple gradient observer, proposed in Lee et shown al. (2010) magnet synchronous motors (PMSMs) in and analyzed in Ortega et al. (2011). It has been the incorporating the reluctance reluctance torque in in are the advantageous field-weakening a simple gradient observer, proposed in Lee et al. (2010) and analyzed in Ortega et al. (2011). It has been shown the incorporating the torque in the range, so that they can be designed wide constant and analyzed in Ortega et al. (2011). It has been shown the incorporating the reluctance reluctance torqueto inhave the afield-weakening field-weakening observer is effective estimates rotor position in practice. A range, that they be to have wide constant analyzed in Ortega et al. (2011). It has been shown the observer is estimates rotor position A range, so so thatrange they can can be designed designed to have aapower wide densities constant and power speed (CPSR). In the result, is effective effective estimates rotor positionisin ininpractice. practice. A range, so thatrange they can be designed to have apower wide densities constant observer very minor modification of thise observer the paper power speed (CPSR). In the result, observer is effective estimates rotor position in practice. A very minor modification of thise observer is in the paper power speed range (CPSR). In the result, power densities of PMSMs is higher than any other types of motors. very minor modification of thise observer is in the paper power speedis range the result, power densities very Malaiz et al. al. (2012) and andofit itthise can observer be globally globally convergent of PMSMs higher(CPSR). than anyInother types of motors. minor modification is inconvergent the paper Malaiz et (2012) can be of PMSMs is higher than any other types of motors. Malaiz to et convexity al. (2012) and it can be globally convergent of PMSMs is higher than any other types ofcost, motors. thanks Due et al. (2012)properties. and it can be globally convergent thanks Due to to recent recent reduction reduction in in PM PM material material cost, PMSMs PMSMs Malaiz thanks to to convexity convexity properties. properties. Due to recent reduction in PM material cost, PMSMs are widely used in home appliances such as refrigerators, thanks to convexity properties. Due to recent in PM material PMSMs Also Also the the challenging problem problem of of sensorless sensorless control control is is conconare widely used reduction in home appliances such ascost, refrigerators, are widely home such refrigerators, air vacuum cleaners, Likewise, Also theinchallenging challenging problem of sensorless control is conare widely used used in in home appliances appliances such as as etc. refrigerators, Bobtsov et al. (2015); Marino et al. (2010), air conditioners, conditioners, vacuum cleaners, washers, washers, etc. Likewise, sidered Also the challenging problem of sensorless control is considered in Bobtsov et al. (2015); Marino et al. (2010), air conditioners, vacuum cleaners, washers, etc. Likewise, hydraulic actuators in vehicles and airplanes are being residered in Bobtsovvariables et al. (2015); Marino et al. (2010), air conditioners, vacuum cleaners, washers, etc. Likewise, where mechanical of the drive are reconstructed hydraulic actuators in vehicles and airplanes are being residered in Bobtsov et al. (2015); Marino et al. (2010), where mechanical variables of the drive are reconstructed hydraulic actuators in vehicles and airplanes are being replaced PMSMs PMSMs mechanicalofvariables offlux. the That’s drive are reconstructed hydraulic in higher vehiclesfuel andefficiency. airplanesAlso, are being re- where using knowledge the total why usually the placed by by actuators PMSMs for for higher fuel efficiency. Also, PMSMs where mechanicalofvariables offlux. the That’s drive are reconstructed using knowledge the total why usually the placed by PMSMs for higher fuel efficiency. Also, PMSMs are popularly used as propulsion motors for hybrid electric using knowledge of the total flux. That’s why usually the placed by PMSMs forpropulsion higher fuelmotors efficiency. Also, PMSMs magnetic flux observation observation is aaflux. key problem, problem, which attracts are popularly used as for hybrid electric using knowledge of the total That’s why usually the magnetic flux is key which attracts are popularly used as propulsion motors for hybrid electric vehicles and ships. flux observation is a keycontrol problem, which attracts are popularly used as propulsion motors for hybrid electric magnetic amagnetic lot of scientists from adaptive and electric drive vehicles and ships. flux observation is a key problem, which attracts a lot of scientists from adaptive control and electric drive vehicles asocieties, lot of scientists from adaptive control andR.electric drive vehicles and and ships. ships. including L. Praly, Ortega, Marino, P. asocieties, lot of scientists from andR.electric drive including L. adaptive Praly, R. R.control Ortega, Marino, P. societies, including L. Praly, R. Ortega, R. Marino, P. Tomei, K. Nam, A. Stankovic, and many other researchers. societies, including L. Praly, R. Ortega, R. Marino, P. Tomei, K. Nam, A. Stankovic, and many other researchers. article is supported by Government of Russian Federa This article is supported by Government of Russian FederaTomei, K. Nam, A. Stankovic, and many other researchers. This An observer for a class of electromechanical systems has article is supported by Government of Russian FederaTomei, K. Nam, A. Stankovic, and many other systems researchers. tion (GOSZADANIE 2.8878.2017/8.9, grant (sections 1, An observer for a class of electromechanical has This tion (GOSZADANIE 2.8878.2017/8.9, grant 08-08) 08-08) (sections 1, 3, 3, This article is supported by Government of Russian FederaAn observer forina Pyrkin class ofetelectromechanical systems been proposed al. (2018). Although a lothas of tion (GOSZADANIE 2.8878.2017/8.9, grantof08-08) (sections 1, 3, 4, and by Science and An observer forina Pyrkin class ofetelectromechanical systems has been proposed al. (2018). Although a 4, 6 6 (GOSZADANIE and appendix) appendix) and and by the the Ministry Ministry Science and Higher Higher tion 2.8878.2017/8.9, grantof08-08) (sections 1, 3, been proposed in Pyrkin etand al. even (2018). Although a lot lot of of 4, 6 and appendix) and by Federation, the Ministry of Science andidentifier Higher different approaches exist, already implemented Education of the Russian project unique been proposed in Pyrkin etand al. even (2018). Although a lot of different approaches exist, already implemented Education of the Russian project unique 4, 6 and appendix) and by Federation, the Ministry of Science andidentifier Higher different approaches exist, and even already implemented Education of the Russian Federation, project unique identifier as feature distributed actuators, the RFMEFI57818X0271 ”Adaptive Sensorless Control for different and even already however, implemented as preset presetapproaches feature in in exist, distributed actuators, however, the RFMEFI57818X0271 ”Adaptive Sensorlessproject Controlunique for Synchronous Synchronous Education of the Russian Federation, identifier RFMEFI57818X0271 ”Adaptive Sensorless Control for Synchronous as preset feature in distributed actuators, however, the Electric Robotics and Systems” (secElectric Drives Drives in in Intelligent Intelligent Robotics and Transport Transport Systems” (secas preset feature in distributed actuators, however, the RFMEFI57818X0271 ”Adaptive Sensorless Control for Synchronous Electric Drives in Intelligent Robotics and Transport Systems” (sec-
tions Electric Drives in Intelligent Robotics and Transport Systems” (sections 2, 2, 5) 5) Electric Drives in Intelligent Robotics and Transport Systems” (sections 2, 5) tions 2, 5) 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2019.12.619
Madina Sinetova et al. / IFAC PapersOnLine 52-29 (2019) 43–48
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performance of such control strategy is still an open problem. The main purpose of this paper is to present the adaptive flux observer for IPMSMs. The problem formulation is in the Section 2, where is presented the classical two phase model of the IPMSM. In the Section 3 the linear regression equation and active flux observer are proposed. The simulations for the IPMSM are shown in the Section 4. 2. PROBLEM FORMULATION Consider the classical, two phase αβ model of the IPMSM described by Nam (2010) λ˙ = v − Ri,
L0 Q(2θ)]−1 [λ − ψm C(θ)] (1) i = [Ls I2 + 2 where λ ∈ R2 is the total flux, i ∈ R2 are the currents, v ∈ R2 are the voltages, R > 0 is the stator windings resistance, I2 is an identity matrix, ψm is PM flux linkage constant, θ ∈ S := [0, 2π] is the rotor flux angle. Ls is averaged inductance, L0 is inductance difference, that are given by d and q-axis inductances Ld and Lq 1 Ls : = (Ld + Lq ) , L0 : = (Ld − Lq ) . 2 We defined
cos(2np θ) sin(2np θ) Q(2θ) = sin(2np θ) − cos(2np θ)
and C(θ) :=
cos(np θ), , sin(np θ)
where np is the number of pole pairs.
Using the adaptive flux observer for PMSM that was proposed in Bobtsov et al. (2015) allows to get the equation L0 Q(2θ)]i|2 − λ2m = 0. |λ − [Ls I2 + 2 The problem of adaptive flux observer construction for IPMSM is considered in this paper. It’s assumed that only the electrical signals are available for measurement, namely i and v, and all electrical parameters, i.e.,R, L0 , Ls , ψm are exactly known. The goal is to design an observer that asymptotically reconstructs the active flux λ and position θ: ˆ ˆ = 0, lim |θ(t) − θ(t)| = 0. (2) lim |λ(t) − λ(t)| t→∞
t→∞
3. PRELIMINARIES As indicated in the previous section, the electrical dynamics of the IPMSM is given by Faraday’s Law (1), together with the constitutive relation L0 Q(2θ)]i + ψm C(θ). (3) λ = [Ls I2 + 2 We used some simple calculations from the paper Choi et al. (2018) and rewrote (3) as λ = Lq i + L0 i C(θ) + ψm C(θ). (4)
Following the idea from the new paper Ortega et al. (2019) we would like to start from the definition of the state x (Equation (6)), that is (5) x := λ − Lq i. Additionally we have the equalities x = (L0 i C(θ) + ψm )C(θ) and d x˙ = v − Ri − Lq i. dt Besides we obtain |x|2 = (L0 i C(θ) + ψm )2 Also the rotor angle is reconstructed from x via x2 1 arctan , θ= np x1
(6) (7) (8)
(9)
The next Lemma gives a simpler parametrization. Lemma 1. (Ortega et al. (2019)). The electrical dynamics of the IPMSM (1) satisfies the following (perturbed) linear regression equation (10) yo = Φ o x + do + t where x is the active flux defined in (5), the unknown perturbing signal d is given by αp x i do := −l p+α |x| and the measurable signals y and Ψ are generated as 1 1 α i Ω Ω1 , Ω1 + |Ω1 |2 + yo := L0 p+α α p+α 2 (11) Φo := Ω1 + Ω2 , d where l := ψm L0 , p = dt is the differentiation operator, and α > 0 is a tuning parameter and α Ω1 := (v − Ri − Lq pi) , p+α αp Ω2 := Ω1 − L0 i. p+α Proposition 1. (Ortega et al. (2019)). Consider the electrical dynamics of the IPMSM with the signal Φo , defined in (11). Define the active flux/position observer ˆ˙ = v − Ri λ αp ˆ x + γΦo y − Φo x i , ˆ+l p+α |ˆ x| ˆ − Lq i, x ˆ=λ x ˆ2 1 arctan θˆ = , (12) np x ˆ1 with yo defined in (11), γ > 0 and α are tuning parameters.
There exist αmax > 0 and γmax > 0 such that for all α ≤ αmax and γ ≤ γmax we have ˜ ˜ ∀t ≥ 0, |λ(t)| ≤ me−ρt |λ(0)|, for some m > 0, ρ > 0. Moreover ˜ lim θ(t) = 0. t→∞
4. MAIN RESULT
In this section we present our flux observer. And the next Lemma gives a parametrization.
Madina Sinetova et al. / IFAC PapersOnLine 52-29 (2019) 43–48
Lemma 2. The electrical dynamics of the IPMSM satisfies the following (perturbed) linear regression equation y = Φ x + d + t
(13)
where x is the active flux defined in (5), the unknown perturbing signal αp x C(θ), (14) d := ψm p+α d is the differentiation operator, and the where p = dt measurable signals y and Φ are generated as y : = (2R − αL0 )φ3 − 2φ4 + 2Lq φ˙ 3 − Lq (Lq + L0 )φ5 2Lq ˙ 2 − φ˙ φ2 + Lq φ˙ φ˙ 1 φ − φ 1 1 α 2 1 ˙ + L 0 Lq φ 1 φ1 , (15) Φ : = αL0 i + (2R − αL0 )φ1 − 2φ2 + 2Lq φ˙ 1 ,
where α > 0 is a tuning parameter and α i, φ1 := p+α α v, φ2 := p+α 1 φ3 := (v − Ri) φ1 , p+α 1 φ4 := (v − Ri) φ2 , p+α 1 ˙ ˙ φ5 := φ φ1 , p+α 1 where φ˙ j , j = 1, 5 are measurable as well.
45
αp ˆ x C(θ) − x ˆ C(θ) (19) d˜ = ψm p+α αp ˆ + x C(θ) − C(θ) ˆ = ψm x ˜ C(θ) (20) p+α αp x ˜ w(x, x ˜) . (21) = p+α The function w(x, x ˜) : R2 ×R2 → R2 is analytical and welldefined mapping, where a constant ψm is involved inside the filter ˆ x ˜))+ w(x, x ˜) := ψm C(θ(x, 2ψm sin δ(x, x ˜) − sin σ(x, x ˜) x + x ˜, cos σ(x, x ˜) x ˜ x ˜ x2 − x ˜2 cos arctan x1 − x ˜1 ˆ x C(θ(x, ˜)) := x2 − x ˜2 sin arctan x1 − x ˜1 1 x2 x2 − x ˜2 σ(x, x ˜) := arctan + arctan 2 x1 x1 − x ˜1 1 x2 x2 − x ˜2 δ(x, x ˜) := arctan − arctan . 2 x1 x1 − x ˜1 Combining (18) and (19), let us rewrite the full model of the observation error in a state-space representation x ˜˙ = −γΦ(Φ x ˜ + z − αw x ˜) z˙ = −αz + α2 w x ˜.
(22)
(16)
The latter coincides with the proof of Proposition 1 in the work Ortega et al. (2019) which with the same arguments allow us to complete this proof.
Proof of the lemma will be established in Appendix. Proposition 2. Consider the electrical dynamics of the IPMSM with the signal Φ, defined in (15). Define the active flux/position observer ˆ˙ = v − Ri λ αp ˆ + γΦ y − Φ x x ˆ C(θ) , ˆ − ψm p+α ˆ − Lq i, x ˆ=λ x ˆ2 1 ˆ arctan θ= , (17) np x ˆ1
Note that a new parameterisation of model of IPMSM as well as the flux/position observer doesn’t contain a division by |x|.
There exist αmax > 0 and γmax > 0 such that for all α ≤ αmax and γ ≤ γmax we have ˜ ˜ ∀t ≥ 0, |λ(t)| ≤ me−ρt |λ(0)|,
Table 1. Parameters of the IPMS
with y and Φ defined in (15), γ > 0 and α are a tuning parameters.
for some m > 0, ρ > 0. Moreover ˜ lim θ(t) = 0. t→∞
Proof.
The observation error dynamics for x ˜ := x − x ˆ is x ˜˙ = x˙ − x ˆ˙ where
˜ ˜ + d), = −γΦ(Φ x
(18)
5. NUMERICAL EXAMPLE In this section we consider the motor as in the paper Ortega et al. (2019), which parameters are listed in Table 1. Parameter Pairs of poles np (–) PM flux linkage constant ψm (Wb) d-axis inductance Ld (mH) q-axis inductance Lq (mH) Stator resistance R (Ω) Drive inertia j (kgm2 )
Motor 6 0.11 5.74 8.68 0.43 0.01
The general structure of the sensorless control algorithm of IPMSM with active flux/position observer is shown in the 1. The initial conditions of the observer are selected as ˆ = [0.5, 2]. Simulations were conducted to evaluate the λ performance of the proposed flux observer with three pairs of different parameters of the observer that are selected as α1 = 10, γ1 = 20, α2 = 20, γ2 = 25, α3 = 30, γ3 = 30 and input voltage v = col (sin t, cos t). Fig. 2 and Fig. 3 show ˜ = λ ˆ−λ the flux and position observer errors, where λ and θ˜ = θˆ − θ. Fig. 4 and Fig. 5 show comparison of two observers Ortega et al. (2019) and the new one, which
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L0
Fig. 1. Structure of the sensorless control algorithm of IPMSM with observer illustrate that the new proposed observer ( Proposition 2, (17)) slightly improved. 6. CONCLUSION In this paper we propose the new adaptive observer design algorithm for IPMSM. The flux observer provides asymptotic convergence of estimation errors to zero. In comparison with well-known proposed approach gives a improved results with guaranteed performance (monotonicity, convergence rate regulation) and allows to regulate the convergence rate via adaptation gains. A new parametrization of model of IPMSM is proposed that improves the results of Ortega et al. (2019), since it does not contain a division by |x| which potentially generates a numerical problem due to division by a small number. APPENDIX Proof of Lemma 2 Multiplying (6) on C (θ) from the left yields C (θ)x = L0 i C(θ) + ψm .
Introduce sλ = sign(L0 i C(θ) + ψm ). Since C(θ) = we have for (6) x x x s λ = L0 i s λ + ψ m |x| |x| and after avoiding division by zero we get x x − L0 i x − ψm |x|sλ = 0.
(23) x sλ |x| (24)
(25)
α p+α
to the equation (25) α α xT i = xT x L0 p+α p+α α −ψm |x|sλ + t p+α and using Swapping Lemma from Sastry and Bodson (2011) we get 1 α α i − L0 x˙ i L0 x p+α p+α p+α 2 x x˙ = x x − p+α α − ψm |x|sλ + t (26) p+α Applying filter
Consider separately two terms of (26) for simplifying the expression. Find that the following term is computable if we invoke filters (16):
1 p+α
x˙
α i p+α 1 ˙ α ˆ = L0 (v − Ri − Lq i) i p+α p+α 1 (v − Ri) φ1 = L0 p+α 1 ˙ α ˆ − L0 Lq i i p+α p+α L0 Lq ˙ 1 − 1 φ˙ φ˙ 1 φ = L 0 φ3 − φ 1 α p+α 1 L0 Lq ˙1 . φ 5 − φ (27) φ = L0 φ 3 + 1 α
Consider the term α α ˙ x x˙ = x (v − Ri − Lqˆi) p+α p+α α ˙ (v − Ri − Lqˆi) = x p+α α 1 ˙ ˙ ˆ ˆ − (v − Ri − Lq i) (v − Ri − Lq i) p+α p+α (28) = x (φ2 − Rφ1 − Lq φ˙ 1 ) − ya . where ya is also computable signal α ˙ 1 (v − Ri − L ˆi) ˙ (v − Ri − Lqˆi) ya : = q p+α p+α 1 (v − Ri) (φ2 − Rφ1 − Lq φ˙ 1 ) = p+α 1 ˆ˙ − Lq i (φ2 − Rφ1 − Lq φ˙ 1 ) p+α L2q φ5 = φ4 − Rφ3 − Lq φ˙ 3 + 2 Lq ˙ ˙ Lq ˙ ˙ φ 1 φ2 − φ 1 φ 2 + φ φ1 , (29) + α 2 1 where we used the following straightforward calculations: 1 (v − Ri) (φ2 − Rφ1 − Lq φ˙ 1 ) = p+α = φ4 − Rφ3 1 v˙ − Rˆi˙ φ1 − Lq φ˙ 3 − p+α = φ4 − Rφ3 − Lq φ˙ 3 Lq ˙ 2 − Rφ˙ 1 ) − 1 (φ˙ (φ˙ 2 − Rφ˙ 1 )) φ + ( φ 1 α p+α 1 = φ4 − Rφ3 − Lq φ˙ 3 Lq 1 ˙ ˙ ˙ ˙ φ 1 ( φ2 − R φ1 ) − φ φ2 + Rφ3 + α p+α 1 and 1 ˆ˙ i (φ2 − Rφ1 − Lq φ˙ 1 ) = p+α 1 = (φ2 − Rφ1 − Lq φ˙ 1 ) φ˙ 1 α L 1 1 ˙ 1 ¨ ˙ q (φ2 − Rφ˙ 1 ) φ˙ 1 + φ φ1 − αp+α α p+α 1 1 1 1 ˙2 = (φ2 − Rφ1 − Lq φ˙ 1 ) φ˙ 1 − φ˙ φ α αp+α 1 R Lq ˙ ˙ Lq + φ5 + φ φ1 − φ5 . α 2α 1 2
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Fig. 2. Error of total flux estimation with 1) α1 = 30, γ1 = 20, 2) α2 = 100, γ2 = 25, 3) α3 = 200, γ3 = 30 6
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Fig. 3. Error of rotor angle estimation with 1) α1 = 30, γ1 = 20, 2) α2 = 100, γ2 = 25, 3) α3 = 200, γ3 = 30 5 2
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Fig. 4. Comparison of two observers for the case of adaptation gains α = 30, γ = 30, voltage v = col (0.1 sin t, 0.1 cos t) 1) for observer from the paper Ortega et al. (2019) 2) for the proposed observer (17)
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Fig. 5. Comparison of two observers for the case of adaptation gains α = 100, γ = 30, voltage v = col (sin t, cos t) 1) for observer from the paper Ortega et al. (2019) 2) for the proposed observer (17)
Rewrite (26) using previous derivations and additionally multiplying everything on α L0 Lq α ˙1 αL0 x = i − α L 0 φ3 + φ 5 − φ φ 1 p+α α = αx x − 2x (φ2 − Rφ1 − Lq φ˙ 1 ) + 2ya α2 |x|sλ + t − ψm (30) p+α Making substitution x x = L0 i x + ψm |x|sλ from (25), ya from (29) and recombining terms in (30), we finally get: (2R − αL0 )φ3 − 2φ4 + 2Lq φ˙ 3 − Lq (Lq + L0 )φ5 2Lq ˙ 2 − φ˙ φ2 + Lq φ˙ φ˙ 1 + L0 Lq φ φ˙ 1 = φ − φ 1 1 1 α 2 1 = αL0 i + (2R − αL0 )φ1 − 2φ2 + 2Lq φ˙ 1 x αp |x|sλ + t . + ψm (31) p+α Finally, noticing that |x|sλ = x C(θ), we get the equation of the form (10), which completes the proof. REFERENCES P.P. Acarnley and J. F. Watson. Review of positionsensorless operation of brushless permanent-magnet machines. Industrial Electronics, IEEE Transactions on, 53:352 – 362, 2006. A.A. Bobtsov, A. A. Pyrkin, R. Ortega, S. N. Vukosavic, A. M. Stankovic, and E. V. Panteley. A robust globally convergent position observer for the permanent magnet synchronous motor. Automatica, 61:47–54, 2015. J. Choi, K. Nam, A. A. Bobtsov, and R. Ortega. Sensorless control of ipmsm based on regression model. IEEE Transactions on Power Electronics, pages 1–1, 2018. J. Lee, J. Hong, K. Nam, R. Ortega, L. Praly, and A. Astolfi. Sensorless control of surface-mount permanentmagnet synchronous motors based on a nonlinear observer. Power Electronics, IEEE Transactions on, 25: 290 – 297, 2010.
J. Malaiz, L. Praly, and N. Henwood. Globally convergent nonlinear observer for the sensorless control of surfacemount permanent magnet synchronous machines. pages 5900–5905, 2012. R. Marino, P. Tomei, and C. Verrelli. Induction Motor Control Design. Springer-Verlag, 2010. K. Nam. AC motor control and electrical vehicle applications. CRC press, 2010. R. Ortega, L. Praly, A. Astolfi, J. Lee, and K. Nam. Estimation of rotor position and speed of permanent magnet synchronous motors with guaranteed stability. Control Systems Technology, IEEE Transactions on, 19: 601 – 614, 2011. R. Ortega, B. Yi, K. Nam, and J. Choi. A globally exponentially stable position observer for interior permanent magnet synchronous motors. CoRR, abs/1905.00833, 2019. URL http://arxiv.org/abs/1905.00833. F. Poulain, L. Praly, and R. Ortega. An observer for permanent magnet synchronous motors with currents and voltages as only measurements. pages 5390 – 5395, 2009. A.A. Pyrkin, A.A. Vedyakov, R. Ortega, and A.A. Bobtsov. A robust adaptive flux observer for a class of electromechanical systems. International Journal of Control, pages 1–11, 2018. S. Sastry and M. Bodson. Adaptive control: stability, convergence and robustness. Courier Corporation, 2011.