Model Predictive Torque Control of a Switched

7/30/2019 Model Predictive Torque Control of a Switched

1/6

Model Predictive Torque Control of a Switched

Reluctance Motor

Helfried Peyrl

Automatic Control LaboratoryETH Zurich

Physikstrasse 3

CH-8092 Zurich, Switzerland

Email: [email protected]

Georgios Papafotiou

ABB Corporate ResearchSegelhof 1

CH-5405 Baden-Dattwil, Switzerland


Manfred Morari

Automatic Control LaboratoryETH Zurich

Physikstrasse 3

CH-8092 Zurich, Switzerland


AbstractThe strongly nonlinear magnetic characteristic ofSwitched Reluctance Motors (SRMs) makes their torque controla challenging task. In contrast to standard current-based controlschemes, we use Model Predictive Control (MPC) and directlymanipulate the switches of the dc-link power converter. At each

sampling time a constrained finite-time optimal control problembased on a discrete-time nonlinear prediction model is solvedyielding a receding horizon control strategy. The control objec-tive is torque regulation while winding currents and converterswitching frequency are minimized. Simulations demonstrate thata good closed-loop performance is achieved already for shortprediction horizons indicating the high potential of MPC in thecontrol of SRMs.

I. INTRODUCTION

Switched Reluctance Motors (SRMs) have evolved to repre-

sent interesting solutions for variable speed drive applications,

due to their low cost and high dynamic performance capabil-

ities. On the other hand, a number of less positive charac-

teristics, such as their inherent strongly nonlinear behavior,and the existence of a significant torque ripple in the output

(also accompanied by audible noise), make the torque control

problem associated with their operation a challenging task, and

have so far limited their deployment in practical applications.

By their construction, SRMs are doubly salient motors;

during their operation the windings of the stator poles are

excited by means of a power electronics converter, and torque

is produced by the tendency of its moveable part to move

to a position where the inductance of the excited winding

is maximized [1], i.e., to a position of alignment with the

excited stator pole. Rotor poles moving towards this position

contribute with a positive torque to the rotational movement,

while poles moving away from it produce a negative (breaking)

torque. This operation principle implies that for the torque

production unipolar phase currents are required to be switched

on and off when the rotor is at precise positions, which depend

on the strongly nonlinear magnetic dynamics of the machine.

The state-of-the-art method to achieve torque (and subse-

quently speed) regulation in SRMs, comprises the translation

of the desired torque reference into a suitable current reference

for the excited stator pole. The converter switches are driven

using a hysteresis- or PWM-based control logic with the aim

of keeping the winding current close to this reference, until

the rotor pole that is the closest is brought in alignment with

the excited stator pole. Subsequently, as the inertia of the

rotor movement drives the rotor pole away from the alignment

position, the winding current is switched off as quickly as

possible to demagnetize the stator pole and avoid the pro-

duction of negative (breaking) torque. A number of methods

have been reported in the literature, aimed at designing control

loops that achieve a minimization of the torque ripple. A

detailed overview of past work will not be provided here due

to space limitations, but the reader is referred to [2][4] and

the references therein for a more detailed coverage.

In this paper a different approach will be pursued. Specifi-

cally, Model Predictive Control (MPC) [5] is employed for

the torque control of a SRM. MPC has been traditionally

(and successfully) used in a large variety of industrial control

applications, and lately a number of publications have reported

on its possible application to the control of industrial electronic

systems, such as dc-dc converters [6], dc-ac inverters [7], [8],and induction motor drives [9][12]. Moreover, in [13] the

authors have already investigated the application of MPC for

the control of a SRM, using a set-up that keeps the hysteresis-

based stator current controller intact and employs MPC for

determining the proper current references. The controller is

then calculated off-line using the tools reported in [14], and

the result is a piecewise affine state-feedback control law that

is stored in a look-up table comprising a total of 19,000 entries

for the controller expressions.

The approach presented here uses a different problem set-up

and results in different controller computation requirements.

The problem is treated as a discrete-time control problem,where the complete converter switch positions are determined

by one central control algorithm, rather than by individual

controllers focusing on each stator winding. More specifically,

at each sampling time, all possible converter switch positions

are considered, and predictions of the motors behavior are

made over a finite prediction horizon of a few steps, using

a discrete-time nonlinear model of the system. The possible

time sequences of converter switch positions are then evaluated

by means of a cost function that aims at achieving motor

torque regulation, while keeping the winding currents to a

minimum and respecting the system constraints. Out of the


2/6

Fig. 1. Structure of a 6/4 SRM (6 stator poles, 4 rotor poles)

converter switching sequence that minimizes the cost function,

the first element is applied to the motor, and in the next

sampling instant the procedure is repeated in accordance with

the receding horizon policy.

The closed loop performance of the proposed method

is studied by means of computer simulations for varying

prediction horizons and cost functions. The controller offersimpressive performance already for short prediction horizons,

and is easy to tune. The results indicate the high potential of

MPC in the control of SRMs.

The implied assumptions of the proposed approach are

that the motor winding currents are measurable, and that

information regarding the rotor position (either rotor speed or

angle) is available. Although the enumeration of all possible

converter switching sequences over the complete horizon im-

plies that the computational demand can increase significantly

when considering longer prediction horizons, the use of a

relatively simple motor prediction model and the fact that

a short prediction horizon is enough to render a satisfactoryclosed loop performance, make the actual implementation of

the presented method feasible with todays state-of-the-art

hardware.

The paper is organized as follows. Section II presents the

physical model of the SRM, as well as the discrete-time

model used for controller design. The MPC-based controller

is described in Section III, and simulation results are provided

in Section IV.

II. MODELLING

As already mentioned in the introduction, the switched

reluctance motor is a particular type of induction machine

where both rotor and stator have salient poles. Fig. 1 illustratesthe structure of a 6/4 SRM (6 stator poles, 4 rotor poles). The

phase windings reside at the stator poles, while the rotor has

no windings at all. Typically, the windings of diametrically

opposite stator poles are connected in series to form one phase.

In this paper, we will focus on the 6/4 SRM, noting that

an extension of the presented method to other SRM types is

straightforward.

A. Physical Model of the SRM

Because of the varying air-gap and the operation in a

saturated region, the flux linkage p of a phase p = {1, 2, 3}

ip

m

Im

unaligned

aligned

p(ip, 0)

p(ip, 45)

Fig. 2. Extremal magnetization curves of a SRM at aligned position ( = 0)and unaligned position ( = 45).

is a nonlinear function of the phase current ip and the rotorposition p:

p = p(ip, p).

The magnetization characteristics may be obtained from finite-

element computations, experimental measurements, or approx-

imated by analytical, nonlinear functions. We are using the

analytical model from Le-Huy et al. which gained widespread

use through its Simulink implementation in the SimPowerSys-

tems toolbox [3]. The basic assumption in this model is that

the mutual couplings between the phases can be neglected, and

that the effects of the phase current and the rotor position on

the flux linkage can be separated. The extremal magnetization

curves corresponding to the aligned and the unaligned rotor-

stator pole positions are approximated by analytical functions.

In the unaligned position (p = 45), the flux is assumed tobe a linear function of the stator current ip:

p(ip, 45) = Lqip

with inductance Lq. In the aligned position (p = 0), the fluxlinkage is described by a nonlinear function which captures the

saturation effects of the iron:

p(ip, 0) = Ldsatip + A(1 e

Bip),

where Ldsat denotes the saturated inductance, and A and Bare appropriately chosen constants:

A = m LdsatIm

and

B = (Ld Ldsat)/(m LdsatIm),where Ld is the non-saturated inductance in the alignedposition, and Im is the rated maximum current with corre-sponding flux linkage m. Fig. 2 shows the magnetizationcharacteristics of the SRM which we used for the simulations

presented in Section IV.

The magnetization curves for the intermediate positions

are obtained through interpolation between the two extremal

curves with an appropriate /2-periodic interpolation function:

f(p) =

128 3p/

3 48 2p/2 + 1 if p [0, /4]

f(/2 p) if p [/4, /2]


3/6

Vdc+

S1

D2

1

D1

S2

S3

D4

2

D3

S4

S5

D6

3

D5

S6

Fig. 3. Power converter topology of a three-phase SRM

Hence the magnetization characteristics of the 6/4 SRM are

described by the expression

p = Lqip +

Ldsatip + A(1 eBip) Lqip

f(p). (1)

The electromagnetic torque generated by a phase p is givenby the derivative of the machine co-energy:

Te,p =

pWp(ip, p),

where

Wp(ip, p) =

ip0

p(ip, p) dip.

Using (1), the electromagnetic torque is given by

Te,p =

Ldsat Lq

2i2p + Aip

A

B(1 eBip)

f(p).

The dynamics of the phase currents are governed by thedifferential equation (cf. e.g., [15])

dipdt

=1

pip

Up Rip

pp

,

where Up denotes the phase voltage, R the stator windingresistance, and the rotor speed.

The mechanical part of the motor is described by

d

dt=

1

J[Te TL D],

with rotor and load inertia J, friction coefficient D, load torqueTL, and total electromagnetic torque Te =

p Te,p.

To sum up, the dynamics of the SRM are described by the

differential equations

dipdt

=1

pip

Up Rip

pp

, p = 1, 2, 3

d

dt= 1

J[Te TL D]

d

dt= , p = + (p 1)/6.

(2)

B. Model of the Converter

The power converter topology of a three-phase 6/4 SRM

with two controlled switches per phase is shown in Fig. 3.

When both switches of a phase are closed, the dc-link voltage

Vdc is supplied to the phase windings, and the flux willincrease. If both switches are turned off, the voltage will

be reversed and the flux rapidly decays to zero. However, if

just one switch is open, and the other one remains closed,no voltage will be supplied from the dc-link, and a flux in

the inductance decreases more slowly. We will describe these

three different switch configurations by three integer variables

u1, u2, u3 {1, 0, 1}, one for every phase. We use up = 1to denote the configuration in which both switches are open,

up = 1 when both are closed, and up = 0 when one switch isopen and the second one is closed. In total, the power converter

admits 33 = 27 switch combinations.

C. Modelling for Controller Design

Since the time constant of the rotor speed dynamics is by

orders of magnitudes greater than the length of the predictioninterval, we can neglect the rotational dynamics and consider as constant over the horizon. Using a forward Euler discretiza-

tion, the continuous-time model (2) of the motor is replaced

by a discrete-time model which can be posed in the standard

formx(k + 1) = f(xk(k), u(k))

y(k) = g(x(k))

(3)

with the overall state vector x

x(k) =

i1(k) i2(k) i3(k) (k)T

and the output

y(k) = Te(k).The model inputs are the integer variables u1, u2, and u3which denote the switch configurations of the converter:

u(k) =

u1(k) u2(k) u3(k)T

{1, 0, 1}3.

Furthermore, we assume the all states are measurable.

III . MODEL PREDICTIVE TORQUE CONTROL

A. Control Problem

Usually the main objective in control of an induction

machine is to regulate and keep its torque close to a reference

value which is typically set by an outer control loop. Further

aims include the minimization of the winding currents and

the operation within the rated values, e.g., keeping the phase

current below the specified maximum.

Clearly, a finite switching frequency makes it impossible to

regulate the torque of a motor driven by discrete voltages arbi-

trarily close to the reference value. As every switch transition

also causes a heat loss in the converter, a further objective

in the controller design is the minimization of the average

switching frequency. Consequently, there is an inherent trade-

off between achieving a low torque ripple and operating at a

low switching frequency.


4/6


5/6


6/6

t [s]

Te

[N

m]

ip[A]

p

[Wb]

Flux

Current

Torque

0

0

00

00

0.1

0.2

0.3

0.4

0.005

0.005

0.005

0.01

0.01

0.01

0.015

0.015

0.015

0.02

0.02

0.02

200

150

100

100

50

50

Fig. 5. Simulation results with N = 2 and qsw(0) = 0. The phase fluxesare shown at the top, the phase currents in the middle, and the electromagnetic

torque of the three phases and their sum are shown at the bottom.

t [s]

Te

[N

m]

ip[A]

p

[Wb]

Flux

Current

Torque

0

0

00

00

0.1

0.2

0.3

0.4

0.005

0.005

0.005

0.01

0.01

0.01

0.015

0.015

0.015

0.02

0.02

0.02

200

150

100

100

50

50

Fig. 6. Simulation results with N = 2, qsw(0) = 300, and qsw(1) = 60.The phase fluxes are shown at the top, the phase currents in the middle, andthe electromagnetic torque of the three phases and their sum are shown at thebottom.

only slightly improved performance but comes at the price of216 different switching law scenarios.

V. CONCLUSION AND OUTLOOK

In this paper we present an MPC based control scheme for

the torque control of switched reluctance motors. In contrast

to other approaches which rely on current controllers, the pro-

posed method operates at the level of the power converter and

directly manipulates its switches. We use a nonlinear state of

the art model from the literature to predict the highly nonlinear

behavior of the motor. The main objectives in torque control,

i.e., keeping the torque close to its reference, minimizing the

winding currents, and the switching frequency, are encoded

in the objective function of a constrained nonlinear optimal

control problem which is solved at every time instance. Several

heuristics account for the requirement of a controller with

tractable complexity by keeping the number of switching

law scenarios at a reasonable level. The good performance

obtained in simulations already for short horizons paired with

MPCs simplicity and transparency points to the high potentialof the method in the control of SRMs. Application of the

controller to a real motor, investigation of its robustness, and

an improved MPC scheme that takes machine symmetries into

account is subject of future work.

REFERENCES

[1] T. J. E. Miller, Ed., Electronic Control of Switched Reluctance Motors.Newnes Power Engineering Series, 2001.

[2] I. S. Manolas, A. X. Kaletsanos, and S. N. Manias, Nonlinear currentcontrol technique for high performance switched reluctance machinedrives, in Power Electronics Specialists Conference. PESC 2008. IEEE,Rodos, Greece, Jun. 2008, pp. 12291234.

[3] H. Le-Huy and P. Brunelle, A versatile nonlinear switched reluctancemotor model in Simulink using realistic and analytical magnetizationcharacteristics, in Industrial Electronics Society, 2005. IECON 2005.31st Annual Conference of IEEE, Nov. 2005, pp. 15561561.

[4] C. Mademlis and I. Kioskeridis, Performance optimization in switchedreluctance motor drives with online commutation angle control, IEEETrans. Energy Convers., vol. 18, no. 3, pp. 448457, Sep. 2003.

[5] J. Maciejowski, Predictive Control with Constraints. Prentice Hall,2001.

[6] T. Geyer, G. Papafotiou, R. Frasca, and M. Morari, ConstrainedOptimal Control of Switch-Mode DC-DC Converters, IEEE Trans.Power Electron., in press.

[7] P. Corts, J. Rodriguez, D. E. Quevedo, and C. Silva, Predictive currentcontrol strategy with imposed load current spectrum, IEEE Trans.Power Electron., vol. 23, no. 2, pp. 612618, Mar. 2008.

[8] J. Rodriguez, J. Pontt, P. Correa, P. Lezana, and P. Cortes, Predictivepower control of an AC/DC/AC converter, in Industry ApplicationsConference, 2005. Fourtieth IAS Annual Meeting. Conference Record of

the 2005, vol. 2, Oct. 2005, pp. 934939.[9] T. Geyer, G. Papafotiou, and M. Morari, Model Predictive Direct

Torque Control Part I: Algorithm, Concept & Analysis, IEEE Trans.Ind. Electron., Special Issue on Predictive Control of Electric MotorDrives, in press.

[10] G. Papafotiou, J. Kley, K. Papadopoulos, P. Bohren, and M. Morari,Model Predictive Direct Torque Control Part II: Implementation andPerformance Evaluation, IEEE Trans. Ind. Electron., Special Issue onPredictive Control of Electric Motor Drives, in press.

[11] J. Rodriguez, J. Pontt, C. A. Silva, P. Correa, P. Lezana, P. Cortes, andU. Ammann, Predictive Current Control of a Voltage Source Inverter,

IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 495503, Feb. 2007.[12] A. Linder and R. Kennel, Model predictive control for electrical drives,

in Power Electronics Specialists Conference. PESC 2005. IEEE, 2005,pp. 17931799.

[13] M. Vasak, D. Zarko, N. Peric, F. Kolonic, and C. Hao, Bounding thetorque ripple in switched reluctance motors using polyhedral invariant

set theory, in Proceedings of the 46th IEEE Conference on Decisionand Control, New Orleans, LA, USA, Dec. 2007, pp. 61066111.

[14] M. Kvasnica, P. Grieder, M. Baotic, and M. Morari, Multi parametrictoolbox (MPT), in HSCC (Hybrid Systems: Computation and Control),ser. Lecture Notes in Computer Science, R. Alur and G. Pappas, Eds.Springer Verlag, 2004, vol. 2993, pp. 448462, http://control.ee.ethz.ch/mpt.

[15] M. Stiebler and K. Liu, An analytical model of switched reluctancemachines, IEEE Trans. Energy Convers., vol. 14, no. 4, pp. 11001107,Dec. 1998.

[16] D. Torrey, X. Niu, and E. Unkauf, Analytical modelling of variable-reluctance machine magnetisation characteristics, in IEE Proc. - Electr.Power Appl., vol. 142, no. 1, Jan. 1995, pp. 1422.

Documents

Model Predictive Torque Control of a Switched