ARTICLE IN PRESS

Control Engineering Practice 17 (2009) 579–587

Contents lists available at ScienceDirect

Control Engineering Practice

0967-06

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/conengprac

Multi-phase current control using finite-state model-predictive control

M.R. Arahal a,�, F. Barrero b, S. Toral b, M. Duran c, R. Gregor b

a Dpto. Ing. de Sistemas y Automatica, Universidad de Sevilla, Spainb Dpto. Ing. Electronica, Universidad de Sevilla, Spainc Dpto. Ing. Electrica, Universidad de Malaga, Spain

a r t i c l e i n f o

Article history:

Received 4 April 2008

Accepted 9 October 2008Available online 26 November 2008

Keywords:

AC machines

Induction machines

Model-based control

Optimization problems

Predictive control

61/$ - see front matter & 2008 Elsevier Ltd. A

016/j.conengprac.2008.10.005

esponding author. Tel.: +34 954487353; fax:

ail address: arahal@esi.us.es (M.R. Arahal).

a b s t r a c t

The use of finite-state model-predictive controllers for current control of multi-phase machines is

investigated. The basic setup is comprised a predictive model and an exhaustive optimizer that

minimizes a predefined cost function for the next sampling period. The output of the predictive

controller is a vector of gating signals to be applied to a voltage source inverter. The inverter can

accommodate just a finite number of configurations and hence the name of finite-state. The use of

predictive controllers, already proposed for three-phase drives, is applied here to multi-phase drives.

Some implementation issues are discussed along, including the choice of the cost function, the

switching frequencies applied to the inverter and the computation time needed for optimization.

Simulation and experimental results are provided illustrating various aspects of the control scheme

using an asymmetrical dual three-phase AC motor drive as a test bed.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Multi-phase motor drives have some advantages over conven-tional three-phase such as lower torque pulsations, less DC linkcurrent harmonics, higher overall system reliability and betterpower distribution per phase. These superior characteristics haveprompted their use in ship propulsion, electric and hybrid electricvehicles and aircrafts (Levi, Bojoi, Profumo, Toliyat, & Williamson,2007). The most frequent control structure for AC drives is acascaded scheme with an inner loop for current control and anouter loop for flux and speed control. The fastest responsecorresponds to the inner loop requiring actuation signals to beissued in microseconds. Current control in conventional andmulti-phase motor drives is usually based on controllers withsub-harmonic voltage modulation techniques such as PWM andspace vector (Bojoi, Tenconi, Griva, & Profumo, 2007; Duran, Toral,Barrero, & Levi, 2007; Sing, Nam, & Lim, 2005). Other schemes fordrive control avoid the use of modulation, computing the gatingsignals to be commanded to the voltage source inverter (VSI). Forinstance, direct torque control (DTC) uses a switching table todetermine the VSI state. Some predictive schemes have beenproposed for DTC aimed at reducing switching frequency (Kennel& Linder, 2000) such as predictive direct mean torque control(Flach, Hoffmann, & Mutschler, 1997).

ll rights reserved.

+34 954487340.

Model-based predictive controllers (MBPC) (Camacho &Bordons, 2004) provide optimal control moves at the cost ofintensive computations. The increase in computing power ofdevices, like digital signal processors (DSP), makes predictive ideasa possibility for controlling modern VSI-driven systems (Cortes,Rodrıguez, Quevedo, & Silva, 2008; Muller, Ammann, & Rees, 2005;Rodrıguez, Pontt, Silva, Correa, & Lezana, 2007). Predictivecontrollers found their first applications in AC drives control toenhance a PWM-based current controller (Holtz & Stadtfeld, 1983;Zhang, Norman, & Shepherd, 1997). Later they have been used toeliminate the double control loop (Correa, Pacas, & Rodrıguez,2007; Geyer, Papafotiou, & Morari, 2005; Kennel & Linder, 2001).A taxonomy can be found in Kennel and Linder (2000).

Most reported applications deal with the use of predictiveideas for the inner loop of current control in three-phase DC/ACconverters (Aurtenechea, Rodrıguez, Oyarbide, & Torrealday,2007) and three-phase inverters (Rodrıguez et al., 2007). In thispaper, a finite-state model-predictive controller (FSMPC) isapplied to current control in multi-phase drives. The denomina-tion comes from the fact that the inverter admits just a handful ofconfigurations. The predictive model takes into account the wholedynamic of the machine, allowing the designer to cope withproblems posed by constraints in the form of saturation of signals,maximum switching frequency, etc. This raises the issue of thetrade-off between model complexity and sampling time attain-able by the computing device which is investigated through a casestudy. The paper includes simulation and experimental resultsusing a laboratory setup using a VSI-driven asymmetrical dualthree-phase AC motor drive as a test bed. The results provided

ARTICLE IN PRESS

Nomenclature

Variables

i currentv voltageC flux linkageo angular speedX state vectorY output vectoru input vector

Parameters

L inductanceR resistanceN inertia coefficientB friction coefficientn number of machine phases

P number of pole pairsp Laplace operator

Subscripts

r2s rotor–statora2b energy conversion-related subspace with sinusoidal

MMFx2y non-energy conversion-related subspace with sinu-

soidal MMF

Superscripts

� reference value^ predictionm measurable partu unmeasurable part

M.R. Arahal et al. / Control Engineering Practice 17 (2009) 579–587580

confirm the feasibility of the predictive control scheme for multi-phase AC machines and point out new research lines. The rest ofthe paper is organized as follows. Section 2 provides backgroundmaterial on the use of predictive controllers for AC drives. Section3 introduces the VSI-driven asymmetrical dual three-phase ACmotor drive used for simulations. The results obtained bysimulation and in a laboratory setup are shown in Section 4. Thepaper ends with Section 5 where the conclusions are presented.

2. Overview of predictive current control in AC drives

Current control in multi-phase motor drives is usually basedon controllers with sub-harmonic voltage modulation techniquesin a way similar to the three-phase case. However, adequatemethods are necessary in order to avoid large harmonic statorcurrents (Zhao & Lipo, 1995). A predictive controller for currentcontrol does not require modulation techniques. The controller (C)computes gating signals u to be forwarded to a VSI as shown inFig. 1. The objective of the controller is to track reference statorcurrents given by i�s . To this end it uses an optimizer that selectsthe most adequate control signal in order to minimize a costfunction. Exhaustive search over the possible control signalsprovides the optimal solution. The above described predictivecontroller is a model-based nonlinear control strategy withprediction and control horizons equal to one. There is more thanone way to implement this control scheme for multi-phase drives.Some of these alternatives are considered next. To proceed orderlythe following subsections review the components of the pre-dictive controller for current control in AC drives.

2.1. Predictive model

The phase variable model of a multi-phase machine withdistributed windings can be obtained from the general theory of

C VSI EM

is

i s u ω

Fig. 1. Simplified diagram of the predictive current controller (C) for a VSI-driven

electrical machine (EM).

electrical machines (White & Woodson, 1959). From the equationsin phase variables, the vector space decomposition (VSD) approach(Zhao & Lipo, 1995) leads to the decomposition of the n-dimensional space of a n-phase machine into a subspace involvedin the energy conversion process (a2b components) and othersubspaces that only generate losses in the stator of the machine(x2y1; . . . ; x2yðn�4Þ=2). The equations of the n-phase machine withan even number of phases after the VSD can be written as

vabsðtÞ ¼ RsiabsðtÞ þ pCabsðtÞ

0 ¼ RriabrðtÞ þ pCabrðtÞ � jorðtÞCabrðtÞ

CabsðtÞ ¼ LsiabsðtÞ þ LmiabrðtÞ

CabrðtÞ ¼ LmiabsðtÞ þ LriabrðtÞ

vwxysðtÞ ¼ Rsi

wxysðtÞ þ pCw

xysðtÞ

CwxysðtÞ ¼ Llsi

wxysðtÞ (1)

where p is the derivative operator and w an integer ranging from 1to ðn� 4Þ=2. This set of equations links stator voltages vsðtÞ, statorand rotor currents isðtÞ, irðtÞ, fluxes CsðtÞ, CrðtÞ and rotor electricalangular speed orðtÞ in a2b and x2yw subspaces. Note thatvariables are indicated using a complex notation, for instance,the stator voltage in the a2b subspace is decomposed in vabsðtÞ ¼

vasðtÞ þ jvbsðtÞ with j ¼ffiffiffiffiffiffiffi�1p

. The equations also include thefollowing machine parameters, stator and rotor resistances Rs, Rr ,stator and rotor inductances Ls, Lr , stator leakage inductance Lls

and mutual inductance Lm.The mechanical part of the drive is given by the following

equations:

TeðtÞ ¼ PLm

Lr½CarðtÞibsðtÞ �CbrðtÞiasðtÞ�

NporðtÞ þ BorðtÞ ¼ P½TeðtÞ � TLðtÞ� (2)

where TeðtÞ is the generated torque, TLðtÞ the load torque, P thenumber of pair of poles, N the inertia coefficient and B the frictioncoefficient.

The drive includes not only the electrical machine but also thepower electronics. Usually the VSI dynamic is neglected. There are,however, a number of reasons to take it into account, first the powerelectronic must be used within safety limits, second the dynamicinclude a dead time that can affect negatively the control loop, thirdthe losses in the VSI are influenced by the number of simultaneousswitch changes and the voltage drops across the switches that affectthe control performance (Bolognani, Peretti, & Zigliotto, 2007).

ARTICLE IN PRESS

Table 1Number of combinations of the gating signals t, and non-redundant combinations

� as a function of the number of legs of the inverter f.

f t �

3 8 7

5 32 31

6 64 49

7 128 127

9 512 343

M.R. Arahal et al. / Control Engineering Practice 17 (2009) 579–587 581

The inverter model produces link voltages to gating signals. Inthe generic case of a f-legged inverter the gating signals can beaccommodate in a row vector

u ¼ ðK1;K2; . . . ;Kf Þ 2 Bf (3)

with B ¼ f0;1g and being Kj the j-th gating signal for j ¼ 1; . . . ; f .Each gating signal can be either active Kj ¼ 1 or inactive Kj ¼ 0,producing t ¼ 2f different possible control actions at eachsampling period.

An ideal inverter converts gating signals into stator voltagesthat can be projected to abxy axes and gathered in a row vectorvabxys computed as

vabxys ¼ ðvas; vbs; vxs; vysÞ ¼ VdcuTM (4)

being Vdc the DC link voltage, T a connectivity matrix that takesinto account how the VSI gating signals are distributed and M acoordinate transformation matrix accounting for the spatialdistribution of machine windings.

Assuming machine voltages derived from (4) is widely used inthe context of modeling and control of AC drives (Duran et al., 2007).A more detailed model can be obtained considering an equivalentinductance and resistance between the VSI and the load or directlyincluding the delay provided by the dead time effect (Bolognaniet al., 2007). The delay caused by power electronics adds to thecomputation time needed by the optimizer. This can amount to awhole sampling time which should be taken into account by thepredictive model as done in Antoniewicz, Kazmierkowski, Cortes,Rodrıguez, and Sikorski (2007). Another aspect to be considered isthe resulting switching frequency. Limits should be imposed both oncurrents and switching frequencies. In high power/current applica-tions, where the multi-phase drives have been mostly considered,the maximum current limitations lead to the use of parallel VSIs or,alternatively, to an increased number of phases for power splittingpurposes. Maximum switching frequencies for such applications aretypically reduced down to the range of 1–5 kHz, allowing for ahigher harmonic content but limiting the switching losses. Areduced number of switch changes can be used to increase theswitching period for the same switching frequency as it isdemonstrated on discontinuous PWM techniques (Hadiouche,Baghli, & Rezzoug, 2006). Furthermore, a large number of switchchanges leads to lower performance in multi-dimensional PWMtechniques (Duran et al., 2007). This fact has motivated researchersto include the number of switch changes into the cost function of thecontroller, so that the switching losses are minimized together withthe current error as done in Vargas, Cortes, Ammann, Rodrıguez, andPontt (2007) for a three-phase neutral-point-clamped inverter.

2.2. Cost function

The cost function should include all aspects to be optimized. Incurrent control the most important figure is the tracking error instator currents predicted for next sampling time. To minimize thismeasure at each sampling time k it suffices to use a simple termsuch as

J ¼ keðkþ 1Þk2 ¼ ki�ðkþ 1Þ � iðkþ 1Þk2 (5)

where k � k denotes vector modulus, i� is a vector containingthe reference for the stator currents and iðkþ 1Þ is a vectorcontaining the predictions based on the actual state and controlmove. More complicated cost functions can be devised forinstance to minimize harmonic content and/or VSI losses. Also,in multi-phase drives stator current can be decomposed insubspaces in different ways. An appropriate decomposition allowto put more emphasis on harmonic reduction as will be shown inthe case study for a six-phase motor drive.

2.3. Optimizer

The optimization is done by exhaustive search over all possiblerealizations of the control action. This guarantees optimality butrequires running the model t times to produce the correspondingvalues of the cost function. Notice that, for electrical machines,there are some combinations of gating signals that produce thesame stator voltages. This means that, for prediction purposes,they are equivalent. This reduces the effective number of gatingcombinations to � ¼ 2f

� r being r the number of redundantconfigurations. Table 1 shows the total and effective number ofgating combinations for common drives. It can be seen that thetotal number of possible gating combinations t increasesexponentially with the number of phases.

For a generic multi-phase machine, the matrix composed of allnon-redundant vectors is denoted as U 2 B��f . The optimizationalgorithm produces the optimum gating signal combination uo asfollows:

�

Assign initial values Jo 1, i 1 � While ip�� Take uj Ui;j 8j ¼ 1; . . . ; f .� Compute stator voltages corresponding to gating combina-

tion u using the inverter model.� Use the predictive model to compute a prediction of the

machine state for the next sampling period.� Compute the cost function J.� If JoJo, then Jo J, uo u.� Increment counter i.

�

End.

Taking care of redundancy does reduce the number of gatingcombinations, however, this number remains large for multi-phase machines. It will be shown in the case study that it ispossible to further reduce this number by neglecting some gatingcombinations. This way of proceeding increases the speed atwhich the optimizer can be run, allowing to decrease thesampling time at the cost of losing optimality.

2.4. Parameter and state estimation

The electrical parameters of the drive have to be measured insome way. Unfortunately their values are not constant but rathervary during operation due to temperature and/or operating pointchanges. In the predictive controller an incorrectly estimated rotorresistance and/or mutual inductance degrades the predictions givenby the model and also the state estimation. Many methods havebeen proposed to perform off-line as well as on-line identificationbeing the latter preferred for high performance drives (Fang, Lin, &Wang, 2005). Rotor resistance estimation is the subject of manystudies being most of them reviewed in Toliyat, Levi, and Raina(2003) and Roncero-Sanchez, Garcıa-Cerrada, and Feliu-Batlle(2007). In this paper the sensitivity of the proposed method withrespect to parameter variation will be empirically assessed.

ARTICLE IN PRESS

M.R. Arahal et al. / Control Engineering Practice 17 (2009) 579–587582

3. Current control of an asymmetrical dual three-phase AC drive

The system considered as a case study of the application of thepredictive controller is an asymmetrical dual three-phase ACdrive. The components of the drive are schematically shown inFig. 2. The VSI with isolated neutrals is depicted in subplot (a),being the gating signals represented by ðKa; . . . ;Kf Þ and theircomplementary values by ðKa; . . . ; Kf Þ. The machine windings aredistributed as shown in subplot (b). All voltage vectors, in the a2bsubspace, that can be generated by the inverter by some gatingcombination are shown in subplot (c).

3.1. Predictive model

According to Zhao and Lipo (1995) the machine can berepresented with three stator–rotor pairs of windings in ortho-gonal subspaces. The first, termed a2b engages with electro-mechanical energy conversion. The x2y pair represent supplyharmonic of the order 6k 1 being k an odd integer. Finally, thezero sequence harmonic components do not appear if neutral

Ka Kd Kb Ke Kc Kf

Ka Kd Kb Ke Kc Kf

a

NN’

+Vdc

0

a

b

d

e

f

ar

br crdr

er

fr

α

β

α

β

d b e c f

30°

c

Fig. 2. Schematic diagrams for the dual three-phase drive with isolated neutrals.

(a) Six-phase inverter with isolated neutrals. (b) Machine windings. (c) Voltages

generated by the inverter in a2b space.

points are isolated. According to this approach, the original six-dimensional space of the machine is decomposed into twoorthogonal subspaces, a2b and x2y, by means of a 6� 6transformation matrix. Eliminating the fluxes, six current andone mechanical equation are found. Using the state components

x1 ¼ ias; x2 ¼ ibs; x3 ¼ iar ; x4 ¼ ibr

x5 ¼ ixs; x6 ¼ iys; x7 ¼ or (6)

the resulting equations can be cast in the form

_x1 ¼ �Rsl2x1 þ l3ðRrx3 þ x7x4Lr þ x7x2LmÞ þ l2v1

_x2 ¼ �Rsl2x2 þ l3ðRrx4 � x7x3Lr � x7x1LmÞ þ l2v2

_x3 ¼ Rsl3x1 þ l4ð�Rrx3 � x7x4Lr � x7x2LmÞ � l3v1

_x4 ¼ Rsl3x2 þ l4ð�Rrx4 þ x7x3Lr þ x7x1LmÞ � l3v2

_x5 ¼ �Rsl5x5 þ l5v3

_x6 ¼ �Rsl5x6 þ l5v4

_x7 ¼ l6ðx2x3 � x1x4Þ � l7x7 � l8v5 (7)

with coefficients given by

l1 ¼ LsLr � L2m; l2 ¼

Lr

l1; l3 ¼

Lm

l1; l4 ¼

Ls

l1; l5 ¼

1

Lls

l6 ¼P2Lm

J; l7 ¼

B

J; l8 ¼

P

J(8)

and being the input signals the applied stator voltages and theexternal load TL

v1 ¼ vas; v2 ¼ vbs; v3 ¼ vxs; v4 ¼ vys; v5 ¼ TL (9)

Stator voltages are related to the control input signals through theinverter model. The simplest model has been selected for this casestudy for the sake of speeding up the optimization process. Then ifthe gating signals are arranged in vector u ¼ ðKa;Kb; . . . ;Kf Þ 2 B

6,with B ¼ f0;1g the stator voltages are obtained from (4) with

T ¼1

3

2 �1 �1 0 0 0

�1 2 �1 0 0 0

�1 �1 2 0 0 0

0 0 0 2 �1 �1

0 0 0 �1 2 �1

0 0 0 �1 �1 2

0BBBBBBBB@

1CCCCCCCCA

M ¼1

3

1 0 1 0

c4 s4 c8 s8

c8 s8 c4 s4

c1 s1 c5 s5

c5 s5 c1 s1

c9 s9 c9 s9

0BBBBBBBBB@

1CCCCCCCCCA

(10)

being ci ¼ cosðip=6Þ and si ¼ sinðip=6Þ for i ¼ 1;4;5;8;9.Combining Eqs. (7)–(9) and (4) a nonlinear set of equations

arises that can be written in state space form

_XðtÞ ¼ FðXðtÞ;uðtÞÞ

YðtÞ ¼ CXðtÞ (11)

with state vector XðtÞ ¼ ðx1; . . . ; x7ÞT, input vector uðtÞ ¼ ðK1;K2;

. . . ;Kf Þ and output vector YðtÞ ¼ ðx1; x2; x5; x6; x7ÞT. The compo-

nents of vectorial function F ¼ ðf 1; f 2 . . . f 6ÞT and matrix C are

obtained in a straightforward manner from (7) and the definitionsof state and output vector.

Model (11) must be discretized in order to be of use for thepredictive controller. A forward Euler method is used to keepa low computational burden. As a consequence the resul-ting equations will have the needed digital control form,with predicted variables depending just on past values and not

ARTICLE IN PRESS

Table 2Parameters of the machine used to obtain the experimental results as well as the

simulations.

Parameter Value

Stator resistance Rs (O) 1.63

Rotor resistance Rr (O) 1.08

Stator inductance Ls (H) 0.2792

Rotor inductance Lr (H) 0.2886

Mutual inductance Lm (H) 0.2602

Stator leakage inductance Lls (H) 0.0189

Inertia coefficient N (kg m2) 0.109

Friction coefficient B (kg m2=s) 0.0221

Pairs of poles P 3

M.R. Arahal et al. / Control Engineering Practice 17 (2009) 579–587 583

on present values of variables. This leads to the followingequations:

Xðkþ 1jkÞ ¼ XðkÞ þ TsFðXðkÞ;uðkÞÞ (12)

denoting by ðkÞ the current sample, Ts the sampling time andbeing Xðkþ 1jkÞ a prediction of the future next-sample state madeat sample time ðkÞ.

3.2. State estimation

In the state space description of (11) just stator currents,voltages and mechanical speed are measurable. Stator voltages areeasily predicted from gating commands issued to the VSI, fluxes,however, are not directly measured except in laboratory setups.This difficulty can be overcome in two ways, estimating the fluxesor lumping all non-measurable terms in one or two factors thatare later tracked and updated. The first approach benefits from thefact that flux estimators are normally used in high performancecontrol of AC drives (Alonge & D’Ippolito, 2007; Castaldi, Geri,Montanari, & Tilli, 2005; Menaa, Touhami, Ibtiouen, & Fadel, 2008;Toliyat et al., 2003). The second approach aggregates all EMF-related components into some terms and uses an estimationprocedure to obtain appropriate values to be used for prediction(Rodrıguez et al., 2007). These EMF-related terms constitute newvariables that can be estimated using past values of the measuredvariables. The estimated terms are projected into the future andused in the predictive model. For the case of the dual three-phasedrive this can be achieved splitting the state vector into ameasurable part Xm

¼ ðx1; x2; x5; x6; x7ÞT and an unmeasurable

part Xu¼ ðx3; x4Þ

T. The evolution of the measurable part can beobtained from (11) as

_XmðtÞ ¼ HðXm

ðtÞ;uðtÞÞ þ GðXuðtÞÞ (13)

with H ¼ ðf 1; f 2; f 5; f 6; f 7ÞT. Euler discretization of this submodel

produce the following predictive expression:

Xmðkþ 1jkÞ ¼ Xm

ðkÞ þ Ts½HðXmðkÞ;uðkÞÞ þ GðXu

ðtÞÞ� (14)

Now, for (14) to be used it is necessary to provide an estimation ofthe term GðXu

ðkÞÞ. It must be noted that the previous valueGðXuðk� 1ÞÞ can be computed at time k as

GðXuðk� 1ÞÞ ¼

XmðkÞ � Xm

ðk� 1Þ

Ts� HðXm

ðk� 1Þ;uðk� 1ÞÞ (15)

This a posteriori estimated value is projected to the actual sampletime producing GðkjkÞ ¼ GðXu

ðk� 1ÞÞ. In this way the finalexpression for the predictive model is arrived at

Xmðkþ 1jkÞ ¼ Xm

ðkÞ þ Ts½HðXmðkÞ;uðkÞÞ þ GðkjkÞ� (16)

3.3. Cost function

The proposed cost function takes into account predicteddeviations from current references in the a2b and x2y subspaces.For ease of notation the predicted deviations are gathered in twovectors

eab ¼1

Aabði�aðkþ 1Þ � iaðkþ 1jkÞ; i�bðkþ 1Þ � ibðkþ 1jkÞÞ

exy ¼ �ðixðkþ 1jkÞ; iyðkþ 1jkÞÞ (17)

being Aab the amplitude of the reference signals i�a and i�b, a factorthat must be included to make the errors in the differentsubspaces comparable. With these definitions the cost functioncan be expressed as

Jabxy ¼ keabk2 þ lxykexyk

2 (18)

where k � k denotes vector modulus and lxy is a tuning parameterthat allows to put more emphasis on a2b or x2y subspaces. It willbe shown in the next section that a reduction in computing time isobtained if the cost function takes a simpler form given by

Jab ¼ keabk2 (19)

3.4. Delays

The computation of the control signal does take a significantamount of time which is comparable with the sampling time. Alsothe peripherals that must be used to generate the gating signalsare likely to produce a delay. In this situation it is best to waituntil the next sampling time to release the computed controlsignals. By doing so a one-sampling time delay is includedbetween the controller and the system (Antoniewicz et al., 2007).The algorithm of the optimizer must be changed to reflect the factthat the system output at time kþ 1 is a function of uðk� 1Þ andthe control action being computed uðkÞ will not affect the outputof the system until time ðkþ 2Þ. To solve the problem apreliminary prediction must be issued to produce X

mðkþ 1jkÞ

using the already computed uðk� 1Þ. This prediction is used tocompute the expected output for ðkþ 2Þ as

Xmðkþ 2jkÞ ¼ X

mðkþ 1jkÞ þ Ts½HðX

mðkþ 1jkÞ;uðkÞÞ þ GðkjkÞ� (20)

The optimizer then uses (20) to obtain the best value of uðkÞaccording to the selected cost function (Jab or Jabxy) using thepredicted outputs for ðkþ 2Þ.

4. Results for the case study

A Matlab/Simulink simulation environment has been designedfor the asymmetrical dual three-phase induction machine, andsome simulations have been performed to prove the effectivenessof the proposed control method. Table 2 shows the parameters ofthe six-phase machine used in this case study.

4.1. DSP computing time

A real-time implementation analysis has been done to provethe viability of the FSMPC. Two 32 bit Texas Instruments DSP havebeen used to assess the capability of modern computing devicesto implement the control algorithm. It must be emphasized thatactual execution times obtained by practitioners may be differentdepending on several factors such as code optimization. Theresults reported here are aimed at providing evidence of thepossibility of using FSMPC for multi-phase machines.

The exhaustive search performed by the optimizer amounts torun the predictive model � ¼ 49 times according to Table 1, whichcan take a substantial amount of time. It is interesting to note that

ARTICLE IN PRESS

Table 4Simulation results for current tracking with different configurations of the control

algorithm.

No. NGC lxy SS eRMSab (%) eRMS

xy (A) THD (%)

m–s m–s m–s m–s

A1 49 0.00 1445–428 3.49–2.40 4.48–2.01 80.12–35.51

A2 49 0.01 1559–354 5.18–1.75 0.39–0.07 17.51–8.14

A3 49 0.10 1415–368 7.67–1.81 0.21–0.02 12.71–4.59

A4 13 0.01 1489–359 7.19–1.96 0.41–0.08 20.50–9.97

A5 13 0.00 1442–436 6.73–2.76 1.24–0.32 42.50–17.35

A6 49 0.12 1700–451 6.64–1.75 0.17–0.02 10.19–3.63

A7 13 0.08 1745–441 6.45–1.61 0.18–0.02 10.82–4.53

B1 49 0.01 1560–355 5.18–1.75 0.39–0.07 17.51–8.14

B2 49 0.01 1578–355 5.25–1.78 0.39–0.07 17.61–8.33

B3 49 0.01 1599–359 7.26–3.30 0.39–0.07 17.24–7.73

B4 49 0.01 1589–355 5.83–2.36 0.39–0.07 17.34–8.11

C1 – – 2000–0 9.35–3.64 0.11–0.06 5.92–2.02

C2 – – 3000–0 8.69–3.56 0.09–0.05 4.92–1.66

C3 – – 4000–0 8.27–3.49 0.08–0.05 4.20–1.47

M.R. Arahal et al. / Control Engineering Practice 17 (2009) 579–587584

the number of gating combinations NGC considered by theoptimizer can be less than � ¼ 49. For instance, the outer ring ofFig. 2(c) corresponds to some gating combinations that producethe largest amplitudes in a2b subspace and minimum amplitudein x2y subspace. It is possible to use these 12 outer vectors andone null combination to produce the desired currents (Bojoi et al.,2007), yielding NGC ¼ 13. If such strategy is used, a suboptimalsolution is then issued by the controller but at a faster rate.A further reduction in computing time is obtained if the costfunction is evaluated only for the a2b subspace. In terms ofoptimality, the results of using Jab are equivalent to those obtainedwith Jabxy setting lxy ¼ 0, but the controller can run at a faster ratebecause some computations are avoided.

Table 3 shows the real-time implementation results obtainedfor two DSP, the TMS320F2812 (single ALU fixed point) and theTMS320C6711 (multiple ALUs floating point). These results showthe execution time once the algorithm has been stored in thecache memory. Notice that the TMS320F2812 runs at a slowerpace but it does include peripherals for A/D conversion that aremissing in the TMS320C6711. The gating combinations have beenselected according to Fig. 2 considering vectors within some ringsand including a null combination. The first row of Table 3corresponds to vectors in the outermost ring (12) and the origin.The second row corresponds to vectors in the first and secondoutermost rings (24) and the origin. The third row includes thethree outermost rings (36) and the origin. The fourth rowcorresponds to all non-redundant active combinations (48) andthe origin. It can be seen that the clock cycles required by thealgorithm, and, consequently, the maximum attainable samplingfrequency, vary with NGC. The cost function used has a lessereffect in this case since Jab avoids just a few computations.

It is worth remarking that, if the maximum switching frequency

of the VSI is limited by technical restrictions to f VSImax1 each switch

must change once on and once off each TVSImin ¼ ðf

VSImax1Þ

�1 seconds. If

the time to compute the whole process of the predictive controller is

TDSPoTVSImin then the limitation is caused by the VSI and consequently

one may consider a more elaborated cost function or look for theoptimal solution. On the other hand, if the necessary time for

the DSP to calculate the firing signal is TDSP4TVSImin, then it is the

calculation process that is limiting the switching frequency that

needs to be set to f VSImax2 ¼ ðT

DSP�1o f VSI

max1. This case will probably

not arise in high power applications that use IGCT, but is commonwith fast power electronic devices such as IGBT. In this case thecomputing time is a relevant variable and suboptimal solutions at

f VSImax1 can be preferable over optimal solutions at f VSI

max2.

4.2. Simulation results

The simulated motor drive has been subject to a number oftests using different configurations of the controller. The tests

Table 3Computing time of the implementation on DSP of the control algorithm.

J NGC Clock cycles–sampling

freq. (kHz)

TMS320F2812

Clock cycles–sampling

freq. (kHz)

TMS320C6711

Jab 13 23711–6.33 5146–29.15

Jab 25 45370–3.31 8607–17.43

Jab 37 66336–2.26 12612–11.89

Jab 49 87853–1.71 16003–9.37

Jabxy 13 26175–5.73 6154–24.37

Jabxy 25 50152–2.99 10947–13.70

Jabxy 37 73495–2.04 16254–9.23

Jabxy 49 96388–1.56 20970–7.15

have been organized into cases for ease of presentation. Table 4summarizes the conditions for each case, or group of tests, andpresents the results. The three first columns on the left indicatethe case identifier, the value of lxy used in the cost function, andthe number of gating combinations (NGCp�) considered by theoptimizer. A number of tests have been performed for each case,being the references i�a, i�b sinusoidal variables with differentvalues of amplitude and frequency. The mean (m) and standarddeviation (s) over these tests have been computed and reported inthe right side of Table 4. The actual measures are the number ofchanges in the VSI legs per phase and per second (SS), the root-mean-squared (RMS) error in the current tracking for the a2bexpressed as a percentage of the amplitude of the reference signal,the RMS of the x2y components of the stator current (thereference signal here is zero) and the total harmonic distortion(THD) in the resulting phase currents computed in the usual wayas the percent of the RMS of all harmonics to the RMS of thefundamental current. In all cases a sampling time of Ts ¼ 2:5�10�4 s is used except in cases A6 and A7 where Ts ¼ 2� 10�4 s forcomparison purposes. A whole sampling time is allowed to passby before applying the computed control signal. The optimizeruses the strategy discussed in Section 3.4 to take into account theforced delay of one sampling period. In all cases the cost functionis Jabxy except in A5.

Cases A1–A7 correspond to the nominal case presented in theprevious section. The model parameters are assumed to be equalto the real drive. It can be stated from cases A1 to A3 that the useof lxya0 allows to diminish the x2y current componentsproducing larger values of tracking error in a2b. If the hardwaredoes not allow the computation of the control signal on time onecan resort to issuing a suboptimal solution using a lower NGC as inA4. A further reduction in computing time is obtained in A5, usingJab instead of Jabxy. An increase in the x2y current components isobserved. The effect of the attainable sampling time is assessed bymeans of cases A6 and A7. For both NGC ¼ 49 and 13 the resultsare better for lower Ts at the cost of a moderate increase in thenumber of switches changes per second.

Cases B1–B4 have been performed using a predictive modelwith parameters that do not match those of the drive. For eachcase a 50% error is introduced in Rr , Rs and Lm, respectively. Inthis way, the sensitivity of the algorithm with respect to changesin machine parameters is assessed. Mismatches in N and B have

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no noticeable effect in current tracking since the mechanicaldynamic evolves in much larger time scales. Notice that the mainrequirement of the predictive controller is that the parameters ofthe electrical machine must be well-known to define theprediction model. Although these parameters can be obtainedusing self-commissioning processes previous to normal machineoperation, some of them do change significantly during the normaloperation of the machine. Case B4 corresponds to a 20% errorintroduced in all parameters simultaneously. The results show thatthere is a low sensitivity with respect to changes in the stator androtor resistance (cases B1 and B2), but a high sensitivity toinductance Lm (case B3). It must be noted, however, that such largevariations are unlikely to be present in a well commissioned drive.

Cases C1–C3 correspond to a PWM control scheme with acarrier frequency of 1, 1.5 and 2 kHz, respectively. These tests areincluded for comparison purposes. The selected carrier frequencyproduce a switching comparable to that of the predictivecontroller although higher. Notice that the PWM control techni-que is continuous, and on/off switching occurs within everysampling period, for all inverter legs and all sectors. The predictivecontroller is discontinuous since on/off switching is not guaran-teed, within every sampling period, for all inverter legs and allsectors. Therefore, a sampling frequency of 2 kHz in the predictivecontroller will produce a switching frequency much lower than2 kHz, and slightly worse results are assumed compared withPWM control with a carrier frequency of 2 kHz. The results shownin Table 4 indicate that the predictive controller produce less

0 0.02 0.04 0.06 0.08 0.1−5

−4

−3

−2

−1

0

1

2

3

4

5

Time (s)

i*

iαix

−

−

−

−

0 0.002 0.004 0.006 0.008 0.01

4.5

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

Time (s)

i*A1A2A4 4

4

4

4

4

5

5

5

Fig. 3. Simulation results. (a) Current trajectories for case A2. (b) Close-up on voltages fo

currents for tests A5, B4 and C1.

current tracking error at the cost of higher harmonic content. It isinteresting to notice that the variability of the tracking error withthe operating point is lower for the predictive controller. This isindicated by the smaller value of the standard deviation s for thereported figures in Table 4.

Fig. 3 presents the evolution of currents in some of the testspreviously introduced. For clarity of presentation just a and x

components are shown. It must be noted that b and y componentsof currents are, respectively, very similar to a and x components.Plot (a) corresponds to case A2 which is taken as a nominal case todraw comparisons with. It can be seen that the tracking of acurrent is tight with acceptable values for the x component.A close up for this test is presented in plot (b) showing the avoltage produced by the gating combination issued by thecontroller. It can be seen that, in order to synthesize the referencecurrent the controller must commute between different activevectors. This illustrates the importance of the sampling time, forhigher switching frequency can be attained producing betterresults. Again, this choice must be balanced with the require-ments of the power electronics of the inverter.

Continuing with the results of Fig. 3, plot (c) shows a close-upon a currents for tests A1, A2 and A4. The quality of tracking of A1is the best, at the cost of higher x2y components as shownpreviously. The differences are, however, small. Plot (d) showscases A5, B4 and C1. The mismatches between prediction modeland simulation model in the B4 case cause a degradation incurrent tracking characterized by a variability around the

0 0.005 0.01 0.015 0.02 0.0254

3

2

1

0

1

2

3

4

Time (s)

i*

iαvα/50

0 0.002 0.004 0.006 0.008 0.01

.5

.6

.7

.8

.9

5

.1

.2

.3

Time (s)

i*A5B4C1

r case A2. (c) Comparison of a currents for tests A1, A2 and A4. (d) Comparison of a

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0

5

10

15 i*αiα Pred

iα PWM

M.R. Arahal et al. / Control Engineering Practice 17 (2009) 579–587586

reference signal. The PWM represented by case C1 shows lessvariability but the measured current lags behind the reference. Itcould be argued that a different PI tuning would lead to betterresults for the PWM. It must be noted, however, that theimprovement at some frequencies does produce a deteriorationat others. The selected PI setting has been obtained by extensivesimulation searching for a balanced solution at different frequen-cies. Finally, case A5 in subplot (d) shows a degradation withrespect to case A4 in subplot (c) for this particular test. The meanvalue of tracking error for A5 is, as shown in Table 4, lower for a2band higher for x2y.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

−10

−5

time (s)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−3

−2

−1

0

1

2

3

4

5

time (s)

ix Pred

ix PWM

4.3. Experimental results

An experimental test rig has been used for implementing thepredictive controller as shown in Fig. 4. It includes a 10 kW dualthree-phase induction machine with three pairs of poles andparameters shown in Table 2. Two Semistack-IGBT modules fromSemikron Inc. (series SKS21F) have been used to drive themachine allowing a maximum switching frequency of 15 kHz.Speed and current measurements are available through adequatesensors. The control code is written in C, for a control board basedon the TMS320F2812 that is well suited for real-time control ofthe power system.

Fig. 5 shows the current tracking performance obtainedexperimentally by a predictive controller with Jab, NGC ¼ 13 andTs ¼ 3:4� 10�4 s. The results are less clean than those obtained bysimulation due to various factors, the existence of noise, the factthat the model is just an approximation of the physical systemand the use of a higher sampling time. Nevertheless the referencecurrent is tracked at various conditions given by electricalfrequency, amplitude and drive load. The results of the predictivecontroller are superimposed in Fig. 5 to those obtained with aPWM with similar switching frequency using a carrier frequencyof 1.5 kHz. Current tracking in a2b is similar in both cases. In thex2y subspace the PWM presents a better behavior. It has to benoted that the comparison is not all that fair because theswitching frequency of PWM is much higher than that of thepredictive controller. Consequently, for similar performance it ispreferable the use of the predictive controller because it leads tolower losses and better overall efficiency.

11 22

33

44

Fig. 4. Photographs of the experimental setup including (1) the power electronics,

(2) the windings, (3) the stator connection grid and (4) the machine mechanically

connected to a load.

Fig. 5. Experimental results for the predictive controller and for a PWM. (a)

Tracking of a reference current. (b) Evolution of the x component of current.

The reported results show that the use of predictive controllersfor multi-phase drives is a possibility even with a modesthardware. It must be, however, remarked that the performanceof the predictive scheme, compared with classical approaches,depends largely on the availability of computing devices. It hasbeen shown that the lower the sampling time the better theresults. On the other hand, the predictive controller offers a greatflexibility and ease of tuning as is made apparent by the results ofthe different cases commented in this section.

5. Conclusions

The area of multi-phase induction motor drives has experienceda substantial growth since the beginning of this century. Particu-larly, the asymmetrical dual three-phase AC machine is the subjectof current research activities. Predictive control has recentlyreceived some attention in power electronics, limited as it is bythe high computational cost. In this context, a FSMPC strategy formulti-phase electrical drives has been presented. The practical

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implementation of the control system has been discussed. Inparticular, the computing time needed to deliver the optimizedcontrol move has been analyzed for two DSPs. It has been shownthat conventional switching frequencies can be obtained providedthat the number of gating combinations is not excessive, which isthe case for three-, five- and six-phase machines.

Simulation and experimental results for an asymmetrical dualthree-phase induction motor drive have been obtained. The resultsshow that the proposed method provides acceptable performanceeven employing modest computing devices. All of this make thepresented method an alternative to other classical methods, likeSVPWM current regulation. The experimental results for thepredictive controller are promising compared with standardPWM. Performance figures are similar for both methods, however,there are a number of factors supporting the predictive scheme.

�

Lower switching frequencies can be obtained for similarperformance compared with PWM. � The predictive formulation allows many possible cost func-

tions that can be designed to cope with issues particular foreach application, for instance copper losses, switching losses inVSI, current THD, etc.

� The proposed algorithm avoids the use of modulation schemes.

This aspect is especially important in multi-phase drivesbecause modulation is more complex and less flexible thanin standard three-phase drives. As an example, the modulationschemes for six- and five-phase drives are complex andcompletely different whereas predictive controllers are basi-cally the same for both. Also, for the six-phase drive, the PWMscheme requires the tuning of four PI controllers whereas thepredictive controller needs none.

� The design of a proper control strategy is shifted to a higher

level compared with standard PI+PWM schemes. This makesan easier task for the designer to go from objectives to controldesign just acting on the cost function.

� The predictive controller allows a straightforward general-

ization to applications with non-sinusoidal and/or unbalancedcurrents.

The field is open for theorist and practitioners to provide newevidence supporting the use of predictive controllers to multi-phase drives and to explore the different possibilities allowed bythe predictive control scheme.

Acknowledgments

The authors gratefully acknowledge the Spanish Governmentfor the economical support provided by Grant DPI2005/04438.Also, they wish to express their gratitude to the anonymousreviewers for their helpful comments and suggestions.

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