model predictive current control of switched reluctance motors...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2015.2497301, IEEETransactions on Industrial Electronics

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Model Predictive Current Control of SwitchedReluctance Motors with Inductance

Auto-CalibrationXin Li, Pourya Shamsi, Member, IEEE,

Abstract—This paper investigates application of an uncon-strained Model Predictive Controller (MPC) known as a finitehorizon Linear Quadratic Regulator (LQR) for current controlof a Switched Reluctance Motor (SRM). The proposed LQRcan cope with the measurement noise as well as uncertaintieswithin the machine inductance profile. This paper utilizes MPCto generate the optimal duty cycles for drive of SRMs usingPulse Width Modulation (PWM) in oppose to delta-modulation.In this paper, first a practical MPC scheme for embeddedimplementation of the system is introduced. Afterwards, Kalmanfiltering is used for state estimation while an adaptive controlleris used to dynamically tune and update both MPC and Kalmanmodels. Hence, the overall control structure is considered as astochastic MPC with adaptive model calibration. Lastly, simula-tion and experimental results are provided to demonstrate theeffectiveness of the proposed method.

Index Terms—SRM, MPC, Kalman filter, adaptive, predictivecontrol, motor drive, current control.

I. INTRODUCTION

CONVENTIONALLY, induction machines (IM) and per-manent magnet motors (PM) are widely employed inelectric vehicles (EV) and industrial applications. SRMs aremostly used for specialty applications including safety criticalapplications and high speed drives [1]. Due to recent reduc-tions in the cost of power electronics and superior properties ofSRMs including rugged construction, low manufacturing costs,and wide speed range SRMs are becoming viable candidatesfor replacing IMs and PMs in variable speed drives and hybridelectric vehicles [2]–[5]. In particular, double stator switchedreluctance motor (DSSRM) has shown superior performance inpower density which exceeds IM benchmarks while maintain-ing a low acoustic noise operation which makes this machinea candidate for the traction drive of future EVs [6].

SRMs operate based on complete switching of the magneticfield within each phase of the motor. Therefore, phase currentsare in the form of train of pulses. In order to provide therequired sharp edges in this pulse train, a sufficiently large

Manuscript received March 31, 2015; revised August 6, 2015; acceptedOctober 4, 2015.

Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

The research is supported in part by the U.S. Department of Energy (DOE,ARPA-E) under grant no. DE-AR0000210.

Xin Li is currently a graduate student at Missouri University of Scienceand Technology, Rolla, Missouri 65409 USA (email: [email protected]).

Pourya Shamsi is currently with Missouri University of Science andTechnology, Rolla, Missouri 65409 USA (email: [email protected]).

ratio between the dc bus voltage and the phase inductanceis demanded. Maximum dc bus voltage is limited to theavailability of a high voltage source as well as limitationsinduced by insulation classes and existing standards. Hence,in many practical applications, the phase inductance is used asa design parameter for controlling the maximum speed of themachine and maintaining the required pulse edge sharpness.Based on this introduction, it is expected to observe lowphase inductances in high-power high-speed machines. Unfor-tunately, in high power applications, the maximum switchingfrequency of the semiconductor switches is limited by theswitch technology. Hence, the drive system encounters atechnical challenge in offering low phase current ripples forlow inductance SRMs under limited switching frequencies.

Drive of SRMs has been a significant research topic with avariety of drive objectives such as position sensorless drives[7], [8], torque ripple reduction [9], and automotive driveswith a wide speed range [10]. To ensure accurate tracking ofthe reference torque and current signals, an accurate currentcontroller is of interest. As a widely employed solution forSRMs, hysteresis controller (or bang-bang control) [11] hasbeen effective in regulating the phase currents with a good dy-namic response. However, this controller suffers from variableswitching frequencies which leads to higher electromagneticinterferences. Also, ideal hysteresis control is not applicableas the maximum switching frequency is limited by the thermalresponse of the semiconductor technology. Hence, a practicalhysteresis control has an upper cap for this frequency whichis known as the delta-modulation. However, this approach willintroduce high ripples in low inductance applications [12].

For this reason, research has been investigated to find alter-native controllers with a fixed switching frequencies mostlyby utilizing a PWM unit. A common approach for PWMcurrent control is by incorporating a PI controller, however, asSRM winding inductance is inherently dependent on the phasecurrent and rotor position, it is difficult to design PI parameterswhich well fit all operation conditions without any additionaleffort. Furthermore, a PI controller is not sufficiently fast todeliver sharp current pulse edges. Research for current controlof SRMs include improved hysteresis control [13], improvedPI control [14], [15], sliding mode control [11], model pre-dictive control [16]–[18], and non-conventional methods suchas neural networks [19]. Among these, MPC is a promisingmethods to handle the non-linear magnetic characteristic ofSRMs, provide fast response, and maintain a fixed switchingfrequency.




MPC offers fast tracking capabilities. However, it has ahigh computational burden and requires accurate knowledgeof the model. Various implementations of MPC have beenreported in the power electronic literatures. Some commonmethods are Finite Control Set Model Predictive Control(FCS-MPC) [20] or Direct MPC [21]. Another approach isthe single-step predictive control, or so-called the deadbeatcontrol with PWM where the input vector is selected bysolving the receding horizon problem for only a single step.This approach offers a straightforward closed-form expressionand is easy to combine with other control strategies. However,a drawback of the deadbeat control is that it is sensitive toload variations and is vulnerable to measurement noise. Inaddition to these common MPC methods, some other MPCapproaches have been introduced in the literature such as theexplicit MPC which is solved by multiple offline optimizationsand is enforced using a lookup table [22], and the adaptiveMPC which uses self-tuning techniques for model correctionand calibration [23].

One of the main challenges in industrial deployment ofa MPC current controller is the uncertainty of the modeland the measurement noise [24]. Measurement noise caninduce large transients on a predictive-controlled system asthe controller aims at reaching the reference in small numberof cycles. Also, model variations will change the trajectory ofthe output and will render the MPC ineffective. In this paper, amodel predictive control with Kalman filtering and inductanceprofile auto-calibration is presented, which aims to addressthe introduced issues surrounding SRM current control. Inparticular, a tracking MPC is applied for current controland regulation in the machine. Kalman filters are used forstate estimation by minimizing the influence of measurementnoise. In addition, both MPC and Kalman filter parameters areupdated dynamically using adaptive techniques to maintain anaccurate model of the system. After analytical modeling, theproposed strategy is simulated. Lastly, experimental results areprovided to validate the proposed techniques.

II. MODEL PREDICTIVE CONTROL OF SRMS

Without loss of generality, a three phase 12/8 SRM is con-sidered as the plant. However, the discussion results can bereadily used in other SRMs as well. Figure 1 demonstratesthe topology of a widely used asymmetric bridge for SRMdrives. Unlike [25], in this topology, each phase of the SRMis controlled individually which eliminates any concerns re-garding the deactivation of other phases. In this section, firsttwo standard implementations of MPC in the form of LQRsare introduced, then a stochastic LQR is incorporated for SRMcurrent control.

A. Dynamic Model of a SRM

In various SRM structures, the mutual inductance betweenadjacent phases are negligible. In order to derive a simple MPCscheme with low computational cost, the mutual inductancebetween phases are neglected. With this assumption, the fluxlinkage of a single phase of the machine is dψ (t)/dt =

S1

A

S2

S3

B

S4

S5

C

S6

Fig. 1. The topology of an asymmetric bridge inverter.

−Rsψ (t)/L (t, ψ (t) , θ) + v (t) where Rs is the phase re-sistance. To include saturation, L(t, ψ(t), θ) is the nonlinearflux and position dependent inductance of each phase. v(t)is the input voltage and the current can be calculated asi(t) = ψ(t)/L(t, ψ(t), θ). For a digital implementation, theSRM model is derived in the discrete-time domain using theforward method as{

ψk+1 = akψk + bkdkik = ckψk

(1)

where ak = 1 − TsRs/Lk, bk = TsVdc, and ck = 1/Lk. Tsis the sampling time. In using the converter shown in Fig. 1,only one switch of each bridge is controlled using Pulse WidthModulation (PWM) to enforce soft-chopping. Hence, in (1),v(t) = d(t)Vdc and hence, bk = TsVdc. Also, since the controlis much faster than the variations due to the rotor speed, it isassumed that θ̄k = θk and L(k, ψk, θ̄k) ' L(k, ψk) which issimply denoted as Lk (for the time between two control steps).This inductance is inherently periodic due to the mechanicalstructure of the machine. Depending on the number of rotorpoles, the period will change. For standard machines, thisperiod is often π/4 or π/3. Due to the periodic nature, theinductance of the machines can be represented using Fourierseries such as [12]

Lk = l0(ik) + l1(ik)cos(8θ̄k) + l2(k)cos(16θ̄k) (2)

for a 12/8 machine where the Fourier coefficients l0(ik)through l2(ik) are calculated using l0(ik)l1(ik)

l2(ik)

= 0.25 0.5 0.25−0.5 0 0.5

0.25 −0.5 0.25

la(ik)lm(ik)lu(ik)

(3)where la(ik), lm(ik), and lu(ik) are the inductance at fullyaligned, midway, and unaligned positions of the rotor underphase current of ik, respectively. Unfortunately, these param-eters change during operation of the machine as a resultof temperature variations and aging. Hence, a method foradaptive estimation of these parameters is proposed in the nextsection. An example plot of these parameters for the machineunder study is depicted in Fig. 2.

B. The Matrix Form of the Controller

MPC is often used in tracking applications. In general, objec-tives are in the form of quadratic cost functions and constraintsinclude the model of the system and boundaries on the inputand states. Output of the MPC is the duty cycle for soft-chopping of the asymmetric bridge converter. Hence, the input




0 50 100 150 200 250

4

8

12

16

Current [ A ]

Indu

ctan

ce [

mH

]

LaLmLu

Fig. 2. Variations of the base inductance parameters as the function of phasecurrent.

has an inherent clamping function which limits it to the set of[0, 1]. Since the goal of this paper is a practical implementationof the MPC, some assumptions are made. First, the objectivefunction is selected as a sum of a quadratic function ofthe tracking error and a quadratic input cost function. Also,to reduce the computational burden, no additional input orstate constraints are considered. Traditionally, the matrix formof LQR was used as the control scheme. Examples includedynamic matrix control and a wide range of problems weresolved using this approach [26]. Given a predictive horizon ofHp and a control horizon of Hu ≤ Hp, the flux linkage modelcan be expressed as a known portion ψk and an unknown dutycycle vector Dk = [dk|k,· · · , dk+Hu−1|k]T . The behavior ofthe system during the prediction horizon can be expressed asan augmented system of

ψ̂k+1|k...

ψ̂k+Hu|k...

ψ̂k+Hp|k

=

ak...

aHuk+Hu−1...

aHpk+Hp−1

ψk+

bk 0 · · · 0...

. . . . . ....

bkaHu−1··· bka

Hu−2··· · · · bk

......

. . ....

bkaHp−1··· bka

Hp−2··· · · ·

∑Hp−Huj=0 bka

j···

Dk (4)

This augmented system can be modeled as Ψ̂k = Akψk +BkDk. Additionally, the quadratic cost function considered isJHp|k = (Îk−I∗)TQ(Îk−I∗)+DTk RDk where Q and R arethe weight factors and I∗ is the reference signal. In trackingapplications, the input often has a non-zero mean. Hence,costing the input can reduce the accuracy of the control.Therefore, in some applications, JHp|k = (Îk − I∗)TQ(Îk −I∗) + ∆DTk R∆Dk where ∆Dk = Dk −Dk−1. In this paper,Dk is directly used for calculating the cost and observedresults are satisfactory.

By substitution and expansion of the augmented model inthe cost function, the cost function can be written as

JHp|k = DTk

(BTk C

Tk QCkBk +R

)Dk + δ

Tk Qδk

− 2Tr(DTk B

Tk C

Tk Qδk

)(5)

where δ = I∗ − CkAkψk is the tracking error. Since theproblem is convex, the solution can be simply calculated

by setting ∇DkJHp|k = 0 as 2(BTk C

Tk QCkBk +R

)Dk −

Tr(2BTk C

Tk Qδk

)= 0 and the optimal vector is

Dk = [BTk C

Tk QCkBk +R]

−1BTk CTk Q[I

∗ − CkAkψk] (6)

and lastly, dk|k is applied to the system. This is equivalent to asingle step recursive LQR on the augmented system. Althoughthe derivation of the problem was simple and no dynamicprograming was needed, the matrix inversion required for thisinput cannot be calculated easily.

C. The Importance of R

In many tracking applications, it is recommended that thereshould be no cost for the input to let the input vary freely.However, this is not the case for power electronic applications.In such applications, the model of the power converter isaveraged. Hence, such model does not contain the rippleinformation. If R = 0, the controller will apply the maximumcontrol in the first step of the LQR to get to the reference inone cycle. This acts similar to a dead-beat controller. However,this is exactly what we are avoiding in this paper. If thecontroller applies the maximum control, it means that the dutycycle is either 100% or 0% which converts the LQR into adelta-modulation and leads to large ripples. This is due tothe averaged form of the model that does not contain rippleinformation. Hence, a good ratio of Q/R is of interest whichallows free variations of the input but prohibits the controllerto try to reach the reference in the first control cycle.

For R = 0, a better deadbeat control of the SRM withartificial sampling steps was studied in [12]. In this approach,inputs are assumed to be constant over a number of m steps(or in the signal processing language, the actual samplingfrequency is reduced by a ratio of m). If m = Hp, dk|k =(1−ak)(I∗−cka

Hpk ψk)/(bkck(1−akHp)). The main purpose

of this method is to minimize the calculation burden whilemaintaining robustness to the noise. However, this approachdoes not perform as well as the proposed technique in thispaper. Later, results demonstrate the importance of R.

D. Recursive Implementation of the Controller

In practice, one can utilize standard dynamic programmingto solve the problem backward in time. Results from thisapproach is similar to the original matrix form while thecomputational burden is reduced. Additionally, a simplifiedversion is proposed for the SRM model. If the control horizonis small or the motor is driving at low speeds, the relativechange in the inductance within the LQR horizon is negligibleand hence, ak and ck can be considered constant to the valuecalculated at the time step k. This can slightly reduce theperformance at high speeds but will help with the reductionof the computational burden in a practical implementation. Inthis approach, for the time period kTs to (k +Hp − 1)Ts theinputs dj for j ∈ {0,· · · , Hp − 1} are calculated as

dk+j|k = Mj [uj+1 − Sj+1akψk+j|k] (7)Mj = [b

Tk Sj+1bk +R]

−1bTk (8)

Sj = cTkQck + a

Tk Sj+1[I − bkMjSj+1]ak (9)




uj = aTk [I − bkMjSj+1]Tuj+1 + ckQi∗k+j (10)

SHp = cTkQck, uHp = c

TkQi

∗k+hp (11)

where the sequences of Mj and Sj can be calculated fordifferent angles and currents (for which the matrices ak andck change) and stored in a table. These sequences can bere-calculated every time the inductance profile is updated.This approach requires more memory and in return, providesless computational burden. Two simplifications are possible inSRM drives.

1) Option 1: In SRM applications, the input referenceneeds not to vary during a control step as the firing angles canbe adjusted to achieve the desired behavior. Variations of thereference will occur during the turn-off and turn-on periods.Therefore, instead of having a time varying sequence of i∗j , thereference input can be constant during one control step and beupdated for the next step. Using this simple approach, the ujsequence can be calculated at a lower frequency (every timethe inductance profile is updated) and only dk|k is calculatedfor every control step.

2) Option 2: In an additional level of sub-optimality, thetracking problem can be reduced to a settling problem. Definethe error as εk+j|k = i∗/ck − ψk+j|k then εk+j+1|k =akεk+j|k − bkdk+j|k − (1 − ak)i∗/ck. Now, in a particularcase of SRM applications, one can notice that (1− ak)/ck =RsTs ' 0. Then, the new model has only one input and canbe studied as a settling LQR problem. Hence, the control se-quence uj is eliminated from (7)-(11) and the processing bur-den is reduced. Hence, dk+j|k = MjSj+1ak(i∗/ck − ψk+j|k)where Mj and Sj are calculated as before. In practice, nosignificant difference between the performance of this simpli-fication and the original LQR implementation is measurable.

E. State-Estimation

A deterministic MPC was developed for SRMs in the priorsection. In order to convert this MPC to a stochastic MPCboth in model and measurement noise, state estimators andalso model adaptation are required. In the first step, a Kalmanestimator is developed to reduce the influence of stochasticsampling noise and improve the state estimation. Later, in thenext section, a system identification approach will improve ro-bustness of the proposed control to stochastic model variations.Model of the SRM can be extended to include the stochasticnoise as

ψk+1 = akψk + bkdk + gkWk (12)ik = ckψk + hkMk (13)

where Wk and Mk are zero-mean process and measurementnoises, respectively. It should be noted that although this modelis scalar, the following analytical studies are written in thegeneral matrix form. Therefore, in that case of the matrix form,the gain matrices are written as Gg and Hk where Gk =Ḡk⊗gk and Ḡk = [Gij ]k = ai−jk Ii≥j where Ix is the indexerfunction (Ix = 1 if x is satisfied) and Hk = IHp ⊗ hk whereIHp is an identity matrix of size Hp. Based on this model,the optimal input in (6) is no longer promising since ψk isa random variable and can contain the measurement noise.

04.5

913.5

1822.5

0100

200300

400500

0

5

10

15

20

Rotor Po

sition [ D

eg. ] Current [ A ]

Ind

ucta

nce

[mH

]

Fig. 3. Inductance profile of a 12/8 SRM as a function of rotor position andphase current.

To ensure high performance operation of this controller, theexpected value of the initial state should be used as E[ψk].

In this paper, a Kalman filter is incorporated to maintainan estimation of the mean of ψk. Using Kalman filter andassuming that the last measurement ik is available and thenoise process is a Wiener process, the flux can be correctedas

ψk|k = ak−1ψk−1|k−1 + bk−1dk−1

+Kk[ik − ck(ak−1ψk−1|k−1 + bk−1dk−1)] (14)

where

Kk = P−k c

Tk [ckP

−k c

Tk + hkσ

2mh

Tk ]−1 (15)

P−k = akPk−1aTk + gkσ

2pg

Tk (16)

Pk = (I−Kkck)P−k (17)

which can be calculated easily for the model of this paper(it will be reduced to a set of scalar equations). Kalmanparameters Kk and Pk are calculated recursively and willquickly converge to their steady state values. σm and σp arethe measurement and process noise variances, respectively. Itis assumed that the added noise is zero-mean. Hence, Kalmanfilter cannot compensate for any drifts in the model. In the nextsection, drifts in the model are compensated using a separateinductance estimator.

III. INDUCTANCE AUTO-CALIBRATION

MPC will predict the behavior of the system and correct theoptimal control input in every step to ensure fast and accuratedynamic response. However, it is assumed that the modelof the system is accurate. Otherwise, the prediction itself isnot valid and each control output from the MPC guides thephysical system to a different path than the reference trackingsignal. For this reason, a system identification approach isused to estimate the motor inductance and tune the controlparameters for optimal performance.

Inductance variations in a SRM have fast and slow dy-namical terms. Fast variations are caused by changes in themagnetic circuit as a result of rotor movement and changesin permeability as a result of saturation. These variations aredeterministic. Therefore, the inductance can be representedas a function lk = L(θk, ik) where θk is the rotor positionat time k and i is the phase current. The surface generatedby this function in illustrated in Fig. 3. Slow variations ofthe inductance which correspond to slow variations of the




function L(., .) are stochastic and are due to aging of bearings,deformations in the motor magnetic structure, and chemicalreactions such as rusting as well as faults such as inter-turn short circuits within the windings. Various methods havebeen studied for inductance profile estimation of SRMs [27],[28]. In this section, an inductor estimation and identificationscheme is introduced to maintain a dynamic knowledge of thefunction L(., .).

A discrete time adaptive estimation algorithm known asRecursive Least-Squares (RLS) is employed to estimate the in-ductance and eliminate the requirements for a matrix inversionof the standard Least-Squares Estimator (LSE). To perform theestimation, a flux model of the motor is used for derivationof the inductance profile (this flux model is in addition tothe models used in MPC and Kalman filtering). To estimateR̂s and l̂k one can use open-loop estimation based on theobserved signals. Here, we utilize the flux model to developa closed loop adaptive estimator using the measured signalsfrom the machine and the inputs to the machine. It should benoted that this loop is separated from the MPC and Kalmanfilter loops to ensure stability. After acquiring a new estimate,L(θk, ik) is updated with the new estimate for l̂k. To do so,it is assumed that the shape of the inductance profile willnot change. Hence, the inductance profile is still calculatedusing (2) and (3). However, a gain coefficient is added toenforce the impacts of variations in the total inductance asl̂k = α̂klk = α̂kL(θk, ik). Similarly, the resistance is estimatedas R̂s = β̂kRs. By defining the estimation error ek = ik − îkwhere ik is the measured signal (which is often down-sampledwith the addition of a finite impulse response (FIR) filter toacquire a more accurate sample), then

γ̂k+1 = γ̂k +Gkek (18)

Gk = Fkφk/(1 + φTk Fkφk) (19)

Fk+1 = (I−GkφTk )Fk/ρ (20)

where γk = [αk, βk]T . ρ ∈ (0, 1) is a discount (or forgetting)factor which controls the trade-off between the speed ofconvergence and robustness to noise. Fk is a weight factormatrix which is calculating the inverse of the training sequenceiteratively and F0 > 0. I ∈ R2×2 is the unity matrix. φkis the regression vector and contains the derivatives of theestimated output with respect to the estimation parameters asφk = [∂ψ̂k/∂α̂k, ∂ψ̂k/∂β̂k]

T or

φk = [ikLk, Ts

k−1∑n=1

inRs]T (21)

Using this method, both inductance and resistance are cali-brated on-line. In order to ensure stability, variations of γk islimited to a closed set containing unity. If Rs is small, then thisterm can be neglected and the estimator is reduced to a scalarform which improves computational burden. Since the currenthas to be filtered, it is better to perform this estimation onlywhen the current has been kept constant for a few samplingsteps. In another word, this estimation should be performedwhile the phase current is regulated at a certain referencevalue and during the active region of each SRM phase. This

RLS FIR filter

ukiref

θk,

ikˆ

k ,k ki

SRMPWMMPC

Kalman Filter

[ , ]

kd

Fig. 4. Control block diagram of the overall system.

requirement is easily achieved due to the nature of a SRMdrive.

Lastly, this estimator can dynamically tune the model pa-rameters in the MPC and Kalman filter controllers. The overallblock diagram of the proposed scheme is depicted in Fig.4. It should be noted that this block is needed per eachphase of the SRM. For instance, in the case of a three phaseSRM, three sets of control blocks are needed to drive themachine. Also, in this approach, the overall inductance profileis gained assuming that the shape of this profile does notchange. In practice, the shape will slightly change as well andone can use a more accurate estimation method proposed in[29]. However, our proposed method requires a much smallermemory requirement and lower computational burden.

IV. SIMULATION RESULTS

In this section various simulations are provided to evaluateindividual control blocks shown in Fig. 4. The machine understudy is a simulation model of the 12/8 SRM which will beused in the experimental studies. This machine has an unsatu-rated unaligned inductance of 6mH and aligned inductanceof 15mH. The nominal current of the machine is 4A andthe phase resistance is 2Ω. During these simulation studies,the PWM switching frequency for the MPC is selected to be10kHz.

A. Drive Using Delta-Modulation

In this scenario, the conventional delta-modulation is appliedto the SRM as the reference. To be fair, delta-modulationsampling frequency is selected to be 20kHz. In this manner,the maximum switching frequency that the modulator cangenerate is 10kHz which is consistent with the following MPCtests. In this test, the machine is running at 500RPM and thereference phase current is set to 4A. Figure 5 illustrates thesimulation results for delta-modulation. It can be observedthat the maximum ripples are about 1A. Also, the left axisillustrates the status of the switch. It can be observed that theswitch is either on or off for one complete control cycle.

B. Drive Using MPC with no Calibration

In this scenario, the horizon is set to 3 steps and no cali-bration is considered. To demonstrate the importance of thecalibration, the inductance of the model is set to 0.75% ofthe inductance of the machine. As a result, it is expected toobserve currents that are unable to meet the reference. Figure




Time

Pha

se C

urre

nt [

A ]

0

1

2

3

4

5 Current

Sw

itch

Stat

us

0

1

Switch status

Fig. 5. Simulation results for delta-modulation.

Time

Pha

se C

urre

nt [

A ]

0

1

2

3

4

5 Current

Sw

itch

Stat

us

0

1

Switch status

Fig. 6. Simulation results with no calibration.

6 shows the results from this scenario. The current has failedto follow the reference.

C. Inductance Profile Adaptation

In this scenario, the calibration mechanism is activated. Resultsfor this scenario are shown in Figure 7. As it was expected,the adaptive regulator has detected the mismatch and has com-pensated for the difference. Details regarding the compensationare shown in Figure 8. In this figure, the RLS coefficient γk isconverging to around 1.33 which compensates the differencebetween the pre-loaded and the actual inductance profiles (i.e.0.75 × 1.33 ' 1). Consequently, the inductance profile iscorrected by the RLS estimator for MPC and Kalman filter.

D. Ratio between Q and R

A very important aspect of the proposed MPC is the ratiobetween Q and R which was discussed earlier. If R is large,

Time

Pha

se C

urre

nt [

A ]

0

1

2

3

4

5 Current

Sw

itch

Stat

us

0

1

Switch status

Fig. 7. Simulation results with calibration.

0 0.5 1 1.5 2 2.5 3

x 10−3

0.9

1

1.1

1.2

1.3

1.4

Time [ sec ]

γk Reference

Fig. 8. Convergence of the RLSE.

Time

Pha

se C

urre

nt [

A ]

0

1

2

3

4

5 Current

Sw

itch

Stat

us

0

1

Switch status

Fig. 9. Results with a large R (i.e. a small Q/R ratio).

Time

Pha

se C

urre

nt [

A ]

0

1

2

3

4

5 Current

Sw

itch

Stat

us

0

1

Switch status

Fig. 10. Results with a very large Q/R ratio.

then there is a large cost associated with the input whichprohibits the LQR from tracking the reference. Figure 9illustrates such conditions.

Also, if the Q/R ratio is very large or if R = 0, then thecontroller becomes similar to a dead-beat controller and tendsto track the reference in one step. Therefore, it will use 0%or 100% duty cycles only which leads to large ripples. Suchscenarios are illustrated in Figure 10. As it can be observedfrom this figure, the LQR has degraded to a delta-modulator.Hence, optimal selection of the Q/R is a design technique.

V. EXPERIMENTAL RESULTS

In this section, various experimental results are providedto evaluate the effectiveness of the proposed method. Theexperimental setup is developed using a TI TMS320F28377Dmicro controller and an asymmetric bridge inverter. Using this2-core 100MHz controller, floating point add and multiplyoperations are calculated in 1 processor cycle while a floatingpoint divide is calculated in 5 cycles. Hence, one core ofthis controller can easily perform the processing required fora LQR with a horizon of 3 running at 10kHz. The dc busvoltage is 100V and machine parameters are as described inthe simulation section. Additionally, a dc machine is used asthe mechanical load. This test bed is illustrated in Figure 11.

Fig. 11. The experimental setup.




Fig. 12. Delta-modulation at 500RPM.

Fig. 13. LQR at 500RPM.

A. Comparing with the Simulation Results at 500RPM

In the first set of examples, the SRM is operating at 500RPMand the aim is to compare the experimental results withthe simulations. First, the delta-modulation is applied to themachine with a sampling frequency of 20kHz (to be fairwhile comparing with a 10kHz LQR). Figure 12 illustratesthe experimental measurements. Also, the phase voltage ofthe machine demonstrate the status of the signals.

Next, the LQR is tested at this speed. Figure 13 shows theimproved performance of the system in comparison with thedelta-modulation.

B. Results at 100RPM

Now, the performance of the system at a lower speed of100RPM is investigated. Figure 14 shows the behavior of thedelta-modulation at 100RPM. Using LQR, the large ripplesinduced by delta-modulation is reduced as shown in Figure15.

Also, if the adaptive estimator is deactivated, one can noticethe inferior performance of the LQR due to the incorrectassumption on the model of the machine. Figure 16 shows

Fig. 14. Delta-modulation at 100RPM.

Fig. 15. LQR at 100RPM.

Fig. 16. LQR with no calibration at 100RPM.

the measured signals from the LQR while operating withoutthe adaptive calibrator.

C. Results at 1000RPM

In this part, LQR is studied at higher speeds. In 1000RPM,the 4A reference signal can be easily tracked by the LQR asshown in Figure 17. Also, if the current is increased to 5A, themachine is saturated more than the nominal rating. However,the LQR equipped with the proposed inductance profile modeland calibration can easily track the input signal as shown inFigure 18.

VI. CONCLUSION

In this paper, a model predictive current controller for appli-cations in switched reluctance motor drives was introduced.This controller is equipped with Kalman filter state estimators.Additionally, to cope with model variations, two adaptivegains were dynamically calculated to compensate for theinductance and resistance mismatch between the model and thephysical system. In conclusion, the proposed control scheme

Fig. 17. LQR with at 1000RPM with a reference of 4A.




Fig. 18. LQR with at 1000RPM with a reference of 5A.

is successful in providing low current ripples and ensuringsuccessful tracking of the reference current signal in SRMdrives.

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Xin Li received the B.Sc. and M.Sc. degree inelectrical engineering from the Tianjin University,Tianjin, China, in 2005 and 2007 respectively. Healso received the M.Sc. degree in electrical engi-neering from the Missouri University of Scienceand Technology, Rolla, USA in 2015. From 2008to 2013, he was a motor engineer with EmersonClimate Technologies, Suzhou, China. He is cur-rently a Sr. research engineer in Emerson ClimateTechnologies, St. Louis, USA.

Pourya Shamsi (M13) received his B.Sc. and Ph.D.in Electrical Engineering from the University ofTehran, Iran in 2007, and The University of Texas atDallas, USA in 2012, respectively. He is currentlyan assistant professor of Electrical Engineering atMissouri University of Science and Technology(formerly UMR). His research interests are micro-grids, reliability and reachability, hybrid systems,networked control systems, power electronics, andmotor drives.

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