氫氣泵自適應無跡卡爾曼濾波無傳感器控制

Sheianov Aleksandr 康爾良

摘?要：提出了一種氫氣泵用PMSM轉(zhuǎn)子位置新型非線性估算方法。卡爾曼濾波常用于在非線性系統(tǒng)，在計算量相同的情況下，無跡卡爾曼濾波器（UKF）較擴展卡爾曼濾波器（EKF）的計算結(jié)果更為準確，因為應用較多。在燃料電池系統(tǒng)中，氫氣泵用PMSM的負載常發(fā)生持續(xù)變化或突變;由于UKF采用固定過程噪聲協(xié)方差（Q矩陣），無法響應負載變換造成的過程噪聲變化，傳統(tǒng)卡爾曼濾波器的性能可能下降。提出了一種自適應UKF方法，通過計算UKF的自適應增益，以補償實際殘差協(xié)方差與濾波器導出值之間的不匹配，從而在負載變換的情況下，保證了轉(zhuǎn)子位置估算的精度。最后，搭建氫氣泵用PMSM控制系統(tǒng)，并進行了自適應UKF轉(zhuǎn)子位置估算。實驗結(jié)果驗證了該算法的可行性和有效性。

關鍵詞：無傳感器控制;自適應卡爾曼濾波;氫氣泵

DOI：10.15938/j.jhust.2020.03.005

中圖分類號：?TM351

文獻標志碼：?A

文章編號：?1007-2683（2020）03-0025-08

Abstract：This?paper?demonstrates?a?special?and?new?type?of?non-linear?observer?for?a?permanent?magnet?synchronous?motor?used?in?hydrogen?pump?applications.?For?nonlinear?systems，?the?unscented?Kalman?filter?（UKF）?is?a?very?popular?approach?for?the?controller?design.?Some?researches?have?shown?that?the?UKF?is?usually?more?accurate?than?the?extended?Kalman?filter?（EKF）?whereas?the?computation?burden?is?the?same?in?both?cases.?However，?the?performance?of?traditional?Kalman?filter?may?degrade?when?process?noise?and?load?are?constantly?changing?or?a?sudden?disturbance?occurs?due?to?the?fixed?process?noise?covariance?（Q?matrix）?in?the?filter，?which?is?the?case?in?fuel?cell?systems?where?a?hydrogen?pump?is?used.?An?adaptive?gain?is?calculated?to?compensate?for?the?mismatch?between?the?actual?residual?covariance?and?the?deduced?value?from?the?filter，?ensuring?that?the?sequence?of?residual?is?uncorrelated.?Also，?the?derivation?of?the?proposed?adaptive?UKF?is?explained?in?details.?Finally，?experimental?tests?under?sensorless?position?control?for?hydrogen?pump?are?carried?out?with?the?proposed?method，?in?which?the?feasibility?and?effectiveness?of?the?algorithm?is?shown.

Keywords：sensorless?control;?adaptive?unscented?kalman?filter;?hydrogen?pump

0?INTRODUCTION

Recently，?sensorless?permanent-magnet?synchronous?motor?（PMSM）?drive?systems?have?gained?much?interest?from?industries?as?well?as?researchers?because?of?the?elimination?of?fragile?and?noisy?rotor?position?sensors?[1-3]，?[19-21].?In?these?drive?systems，?the?speed?and?rotor?position?play?a?key?role?in?the?good?performance?of?a?closed-loop?system，?but?these?values?cannot?be?directly?estimated.?Therefore，?to?obtain?these?values?and?then?feed?them?into?a?controller?and?reference?frame?transformations，?position?estimator?in?the?feed-back?path?is?used.?Usually?the?inputs?to?the?estimator?are?the?measured?voltages?and?currents，?and?the?outputs?are?the?estimated?speed?and?rotor?position.

In?the?present，?there?are?a?number?of?types?of?observers?in?Kalmans?family.?A?Kalman?filter?originally?is?a?linear?observer，?also?known?as?a?linear?quadratic?estimator?（LQE）.?The?filter?is?named?after?Rudolf?E.?Kálmán，?the?developer?of?this?theory.?Kalman?filter?is?widely?used?in?applications?such?as?guidance，?navigation，?and?control?of?vehicles，?particularly?aircraft?and?spacecraft，?because?it?gives?the?best?performance?among?all?of?the?linear?estimators?and?observers.?Also，?for?non-linear?problems，?the?Kalman?filters?family?provide?a?number?of?different?approaches?[4].?The?most?popular?and?most?studied?until?now?is?the?extended?Kalman?filter，?which?is?used?to?compute?the?predicted?state?from?the?previous?estimate?first?and?then?in?the?same?manner，?the?output?function?h?computes?the?predicted?measurement?from?the?forecasted?state.?However，?in?this?type?of?filters?f?and?h?cannot?be?applied?to?the?covariance?directly.?Therefore，?a?matrix?of?partial?derivatives?（the?Jacobian）?is?calculated?at?each?time?step?[5].?Although?the?extended?Kalman?filter?（EKF）?is?relatively?simple?to?apply?for?the?motor?control?application，?it?has?a?number?of?major?drawbacks：

1）relatively?complex?derivation?of?the?Jacobian?matrices?in?the?linearization?process.

2）it?has?only?first-order?accuracy，?because?higher-order?terms?are?neglected.

3）Filter?instability?because?of?linearization.

Therefore，to?help?eliminate?this?problem?and?therefore?make?non-linear?Kalman?filters?family?be?more?accurate?and?superior?to?other?non-linear?algorithms，?Julier?and?Uhlmann[6-7]proposed?a?new?approach?to?improve?the?performance?of?non-linear?Kalman?filters.?Instead?of?linearizing?non-linear?system?at?each?time?step（sovling?Jacobian?matrices）?the?UKF?uses?a?minimal?set?of?sample?points，?which?are?fed?to?the?nonlinear-state?equations?to?obtain?the?mean?and?covariance?of?a?non-linear?system?in?the?prediction?step?and?then?use?them?to?correct?the?predicted?estimates?according?to?the?new?measurements.?The?UKF?has?been?proven?to?be?more?accurate?than?EKF?with?similar?computation?burden.?It?also?has?been?successfully?implemented?in?a?few?practical?applications[7-8]such?as：?navigation，?radar?tracking，?signal?processing，?neural?networks，?and?robotics.?The?first?implementation?of?the?UKF?in?motor?control?was?reported?by?Akin[9].?Tze-Fun?Chan，?Pieter?Borsje?and?Weimin?Wang?[10]，?reported?the?application?of?the?UKF?for?sensorless?vector?control?of?PMSM?in?d-q?reference?frame，?whereas?A.Titaouine，?D.?Taibi?[11]，?realized?this?algorithm?together?with?non-linear?control?of?inverter?[16]，?also?Cheol?Moon，?Kee?Hyun?Nam?[12]，?reported?the?results?of?the?implementation?of?the?UKF?in?alpha-beta?reference?frame.?However，?UKF?still?suffers?some?drawbacks?from?traditional?Kalman?filter.?For?the?state?estimation?of?a?hydrogen?pump，?the?pm?motor?may?work?in?a?various?load?conditions.?Thus，?a?mismatch?between?the?real?process?noise?characteristic?and?the?one?in?the?filter?usually?occurs?and?the?performance?of?the?filter?deteriorates.?Besides，?UKF?might?react?slowly?in?cases?when?sudden?disturbances?occur?in?the?system.?Therefore，?the?adaptive?UKF[13-15]is?needed?to?overcome?the?drawbacks?listed?above.?Xia，?Rao?et?al.?developed?an?adaptive?fading?Kalman?filter?（AFKF）?for?linear?systems，?which?ensured?the?Kalman?gain?was?optimal?by?making?the?auto-covariance?of?residual?equal?zero[17，22].

In?this?paper，?an?adaptive?UKF?is?designed?by?adjusting?the?process?noise?covariance?matrix?Qk?at?each?sample?with?an?adaptive?gain.?The?gain?is?calculated?based?on?the?optimal?feature?in?[17]?and?real-time?estimated?k?inspired?by?[18].?The?proposed?method?is?tested?on?the?hydrogen?pumps?permanent?magnet?motor.?Experimental?results?show?that?the?proposed?algorithm?has?a?good?performance?in?sensorless?control?and?hydrogen?pumps?state?estimation.

1?MODEL?OF?PMSM

The?Kalman?filter?requires?a?mathematical?model?of?the?system?in?order?to?estimate?the?states?of?the?system.?Formulation?of?the?mathematical?model?of?a?controlled?system?is?an?important?task?during?the?design?stage?of?the?estimator.?The?correct?model?can?simplify?the?solution?of?the?estimation?problem?as?well?as?simplify?the?computational?cost?of?the?whole?algorithm.

To?avoid?convergence?problems?at?startup?and?to?simplify?the?motor?equations，?the?d-q?reference?frame?is?chosen?for?evaluation?of?the?Kalman?filters?[2].?The?motor?nonlinear?state?equations?can?be?expressed：

The?UKF?consists?of?four?state?variables：?stationary?reference?frame?currents，?estimated?speed?and?estimated?angle.?Since?the?mechanical?variables?have?a?tendency?to?change?rapidly?and?are?very?hard?to?measure?correctly，?the?state?variables?consists?of?only?electrical?variables.?The?state?variable?vector?is?then?x=[id?iq?ωe?θe]T.?By?taking?the?partial?derivatives?by?x?we?can?write?the?system?state?matrices?（3）?and?（4）：

2?UNSCENTED?KALMAN?FILTER

For?sensorless?control?of?PMSM?using?UKF，?the?motor?nonlinear?state?equations?（1）?and?（2）?should?be?expressed?in?the?discretized?form?（5）～（6）：

The?state?model?represented?by?（5）?and?（6）?also?includes?the?statistical?description?for?the?inaccuracies，?where?wk～N（0，Qk）?and?vk～N（0，Rk）?are，?respectively，?the?zero-mean?Gaussian?process?noise?and?measurement?noise?vectors?with?covariance?matrices?Qk?and?Rk.

2.1?Unscented?Transformation?（UT）

Unscented?Transformation?（UT）?is?designed?on?the?fact?that?it?is?easier?to?approximate?a?probability?distribution?than?to?approximate?a?nonlinear?function（later?is?the?principle?the?EKF?is?based?on）.?But?the?most?important?is?that?the?approximations?are?accurate?up?to?the?third?order?in?case?of?Gaussian?inputs.?For?non-Gaussian?inputs，?approximations?are?accurate?to?at?least?a?second?order.?Therefore，?the?UKF?is?expected?to?give?better?performance?and?accuracy?than?the?EKF?which?has?only?first-order?accuracy.

Unscented?Kalman?filters，?as?in?a?classical?form?of?the?linear?Kalman?filter?is?based?on?two?cycles?which?include?prediction?and?correction.?However，?in?the?case?of?the?UKF?a?set?of?sample?points?around?the?last?state?is?taken?and?propagated?through?a?nonlinear?function?（the?PMSM?model?or?PMSM?nonlinear?state?transition?and?measurement?functions）.?With?these?results?a?mean?and?covariance?can?be?approximated?using?weighted?sample?mean?and?covariance?of?the?transformed?sample?points.?These?weighted?sample?points?are?generated?as?follows.?Consider?the?state?variable?x?with?dimension?L?having?mean?and?covariance?Px.?We?now?choose?a?set?of?2L+1?weighted?samples?χi?（sigma?points）?deterministically?so?that?they?completely?represent?the?true?mean?and?covariance?of?state?x.?Following?is?how?sigma?points?and?weights?determined：

where?=α2（L+κ）-L?indicates?a?scaling?parameter.?The?superscripts?m，c，?express?the?weighted?point?for?mean?and?covariance?calculation.?The?value?α?determines?the?spread?of?the?sigma?points?around?x?and?usually?it?equals?10-4<α≤1.?The?constant?κ?is?an?another?scaling?parameter?which?is?usually?set?to?（3-L），?and?β?is?a?prior?knowledge?of?the?distribution?of?x?（for?Gaussian?distribution?β=2?is?optimal）.

3）Each?point?is?propagated?through?the?nonlinear?state?transition?and?measurement?functions?to?yield?a?set?of?transformed?sigma?points.

4）The?mean?and?covariance?of?y?are?approximated?by?the?weighted?average?mean?and?covariance.

2.2?UKF?Algorithm

The?UKF?is?a?straightforward?application?of?the?UT?to?the?recursiveKalman?filter?equations.?Fig.1?shows?the?flowchart?of?the?proposed?algorithm，?which?involves?the?following?steps：

In?this?paper，?the?author?designs?an?adaptive?UKF?（AUKF）for?state?estimation?of?permanent?magnet?synchronous?motor.?In?the?application?of?a?hydrogen?pump?system，?load?conditions?varies?according?to?the?demand?of?the?fuel?cell?system.?Thus?the?characteristic?of?process?noise?is?hard?to?predict?before?the?implementation.?Its?possible?for?standard?UKF?that?the?estimator?would?have?an?unsatisfactory?performance?due?to?the?model?error?or?the?difference?between?the?real?noise?characteristic?and?the?one?used?in?the?filter.?Moreover，?standard?UKF?filter?may?respond?slowly?if?the?disturbance?occurs?and?measurements?have?a?sudden?change.?The?solution?to?the?problems?above?is?to?adjust?process?noise?covariance?adaptively.?In?this?paper，?equation?（17）?is?modified?as：

where?η（k）?is?a?positive?gain，?and?Qa.k?is?the?adjusted?process?noise?covarianve?in?discrete-time?domain.?The?advantage?of?using?Qa.k?instead?of?original?noise?covariance?matrix?Qk?is?that?the?characteristic?of?each?noise?signal?can?be?estimated?adaptively.

The?adaptive?gain?η（k）?is?derived?first.?In?[17]?Xia?et?al.?proposed?an?adaptive?fading?linear?Kalman?filter?which?guaranteed?that?the?sequence?of?residuals?was?uncorrelated?by?the?following?equation：

where?Hk?is?the?output?matrix，?and?C0，k?is?the?covariance?of?the?residual.?This?idea?is?extended?in?the?UKF?adaptive?scheme?in?this?paper.?Define?the?residual：

If?weknow?Qa，k，?then?adaptive?gain?can?be?computed.?In?the?following，?an?estimation?algorithm?for?process?noise?covariance?matrix?is?designed.?In?[18]，?process?noice?covariance?matrix?is?estimated?on-line?by：

whereФ?is?Jacobian?matrix?and?Δxj?is?the?difference?between?a?posterior?and?a?priori?estimated?state：

Using?（32）?in?UKF?scheme?and?use?another?fading?factor?ρ2?instead?of?the?average?operation?in?（32）?to?overweight?the?recent?values，?the?real-time?k?is：

However，?（34）?may?yield?a?k?which?is?not?positive-definite.?To?cope?with?this?issue，?some?constraints?must?be?introduced.?When?the?sample?time?is?small，?the?estimated?c?can?be?obtained?if?B′wBw?is?non-singular：

Since?a?relatively?large?process?noise?covariance?matrix?is?needed?to?overcome?the?model?error?and?sudden?disturbances，?very?small?diagonal?elements?in?k?is?not?suitable，?which?means?that?minimum?values?constraints?must?be?set.?Besides，?the?maximum?values?also?need?to?be?constrained?because?（34）?is?only?an?estimation.?It?is?not?realistic?if?the?estimated?covariance?is?very?large.?As?a?result，?the?constraints?for?the?diagonal?terms?in?estimated?covariance?matrix?c?are?as?follows：

where?ξ?is?a?large?positive?tuning?parameter，?d?is?the?number?of?disturbances?in?（5）?and?C（i，i）?is?the?ith?diagonal?element?of?c.?Normalize?the?constrained?c?so?that：

If?we?define?this?adjusted?matrixas?′c，?then?the?process?noise?covariance?matrix?Qa，k?in?（30）?can?be?calculated?as：

Substitute?（38）?into?（31），?the?gain?η（k）?can?be?obtained.?Since?large?process?noise?covariance?can?contribute?to?a?quick?response?which?is?one?of?the?main?requirements?in?hydrogen?pump?application，?therefore?η（k）?is?set?to?be?larger?than?1，

Thediagram?of?the?proposed?sensorless?control?system?is?presented?in?Fig.2.

3?EXPERIMENTAL?RESULTS

Experiments?were?carried?out?to?confirm?the?effectiveness?of?the?proposed?design.?The?experimental?setup?shown?in?Fig.?3?consists?of?an?IPMSM?（1.5?kW）?coupled?with?a?hydrogen?pump.?The?parameters?of?the?motor?are?given?below：

Rs=0.22Ω：stator?resistance?（phase?to?phase）;

Lq=0.00217H：motor?q-axis?inductance;

Ld=0.00107H：motor?d-axis?inductance;

J=0.0001605N·m/rad/s：moment?of?inertia;

p=8：number?of?the?poles;

nN=5000r/min：rated?speed;

ψf=0.2614Wb：flux?of?the?permanent?magnet;

Vdc=300V：rated?voltage.

The?carrier?frequency?of?the?inverter?is?10kHz，?and?Bw?is?an?identity?matrix.?Tuning?parameter?κ，α?and?β?in?are?0，?0.01?and?2?respectively.?Fading?factor?ρ1?in?（32）?and?ρ2?in?（34）?are?0.4.

The?tests?were?carried?out?under?the?condition?of?a?half?of?the?rated?torque.?During?the?steady-state?operation?of?the?motor?some?random?disturbances?were?introduced?to?the?system?which?emulated?the?real?working?conditions?of?hydrogen?pump?in?a?real?fuel?cell?system.?Fig?4.?illustrates?the?speed?sensorless?control?response?under?the?reference?speed?ωref=600r/min，?ωref=1000r/min?and?the?changes?of?the?load?demand.?It?can?be?noticed?that?the?system?is?completely?robust?against?these?changes?as?well?as?disturbances?occured?during?the?steady?state?operation，?also?the?transient?time?is?very?short，?which?confirms?the?effectiveness?of?the?proposed?scheme.

Fig.5?shows?the?real?and?estimated?rotor?position.?Themaximum?value?of?the?rotor?position?estimation?error?is?0.02?which?is?equivalent?to?7°in?the?whole?operation?range.?The?phase?current?of?the?proposed?method?under?a?half?of?the?rated?torque?and?ωref=1000r/min?are?shown?in?Fig?6.

The?only?drawback?of?the?proposed?method?is?that?it?cant?be?implemented?on?a?fixed-point?processor，?because?it?doesnt?guarantee?the?convergence?of?the?state?covariance?matrix.?But?it?can?be?readily?realized?on?a?floating?point?processor?with?frequency?more?than?70MHz?which?has?enough?capability?to?calculate?the?whole?sensorless?control?algorithm?during?one?interrupt?and?the?problem?with?convergence?will?not?occur.

4?CONCLUSION

In?this?paper，?the?sensorless?control?of?permanent?magnet?motor?drive?system?using?an?adaptive?Unscented?Kalman?filter?（UKF）?has?been?studied?to?improve?the?state?estimation?accuracy，?robustness?of?the?system?and?the?overall?performance?of?the?motor?drive?system.?The?process?noise?covariance?matrix?is?first?estimated?and?restricted?to?maintain?its?positive?definitiveness，?and?then?an?adaptive?gain?is?introduced?to?ensure?the?sequence?of?output?residuals?is?uncorrelated.?Notice?that?the?algorithm?requires?that?the?measurement?model?is?linear.?For?the?nonlinear?case，?the?Jacobian?matrix?of?measurement?model?can?be?used.?Experimental?test?results?validate?that?the?proposed?filter?has?the?best?performance?compared?with?EKF，?and?standard?UKF?with?the?presence?of?disturbances，?load?changes?and?process?uncertainty?mismatch?for?sensorless?control?algorithm?of?permanent?magnet?motor?used?in?hydrogen?pump.

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（編輯：溫澤宇）