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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 1, JANUARY 1997 119

Long-Range Predictive Control of Current Regulated PWM for Induction

Motor Drives Using the Synchronous Reference FrameL. Zhang, R. Norman, and W. Shepherd

Abstract— This paper presents the application of long-rangepredictive control to the problem of induction motor statorcurrent regulation. The particular controller used is the general-ized predictive controller in a synchronous reference frame. Thetheory and implementation of this controller are discussed andsimulation results are presented which compare favorably withthose for proportional integral compensated controllers.

Index Terms— Generalized predictive control, current regu-lated PWM, voltage source inverter, induction machine drive.

NOMENCLATURE

Supply frequency rad/s.

V Stator voltage vector in the stationary reference

frame.V Stator voltage vector in the synchronous reference

frame.

I Stator current vector in the stationary reference

frame.

I Stator current vector in the synchronous reference

frame.

R Stator resistance.

L Stator leakage inductance.

L Rotor leakage inductance referred to primary turns.

L Mutual magnetizing inductance.

L Total leakage inductance.

E Machine back EMF vector in the stationary refer-

ence frame.E Machine back EMF vector in the synchronous

reference frame.

K Proportional gain factor of the PI controller.

K Integral factor of the PI controller.

K Equivalent gain factor of a voltage source inverter.

I. INTRODUCTION

AN induction motor vector control scheme decouples the

stator input current into flux- and torque-producing com-

ponents corresponding to those of a dc motor. The generated

motor torque is proportional to the product of the two compo-

nents. When implementing the vector control scheme the statorcurrents are commonly controlled using a current-regulated

pulse width modulator inverter (CRPWM). Although generally

sensitive and robust this does not, however, guarantee that the

stator current level is completely aligned with the value desired

for fast transient response and zero steady-state error. An exact

Manuscript received December 19, 1994. Recommended by AssociateEditor, J. Smith.

The authors are with the Department of Electronic and Electrical Engineer-ing, University of Bradford, Bradford BD7 1DP U.K.

Publisher Item Identifier S 1063-6536(97)00245-5.

knowledge of the rotor parameters is vital to the performanceof the vector controller, as is accurate control of the stator

current.

There are two main types of CRPWM inverters, the hys-

teresis regulator and the sine-triangle comparison controller

[1]–[3]. A hysteresis current controller gives a near-sinusoidal

motor current with small ripple but requires a high and

nonconstant switching frequency in the inverter. A sine-

triangle controller uses a natural sampled PWM scheme, in-

corporated with a proportional integral (PI) current controller,

where the parameters are adjusted to minimize the magnitude

and the phase errors in the ac currents [2]–[4]. With fixed

switching frequency and linear characteristics, a PI controller

generates less acoustic noise and inverter switching loss thanits hysteresis counter-part. However, a sinusoidally varying

steady-state error is always present in the stator current [4]–[6]

of magnitude dependent on the machine operating frequency.

Various approaches to solving the current regulation prob-

lem have been reported, most of which have attributed the

problem to a loss of either gain or bandwidth [7]–[10]. Only

Schauder has recognized that the problem stems from the

inherent structure of the controller [5]. Rowan and Kerkmansuggested the design of a PI-sine-triangular comparison reg-

ulator in the synchronous rotating reference frame, where the

input and output are dc quantities [6] rather than the stationary

reference frame. A synchronous PI controller relies on the

accurate tuning of the proportional gain and the integration

time, which is difficult because of changes in the operating

point and variations in the machine parameters.

Among various forms of predictive controller, the gener-

alized predictive controller (GPC) is particularly robust due

to its use of model structure [11]–[13]. This paper presents a

PWM current regulator using a generalized predictive control

scheme is developed in a synchronously rotating reference

frame. The controller predicts the stator current over several

sample intervals, namely long range. The control output, to

generate the natural sampled PWM waveform, is calculated

by minimizing the sum of the squares of the stator current

errors. It will be shown that this scheme outperforms any of theexisting current control methods. It provides high-performance

transient regulation, zero steady-state error, and the control

parameters are easily tuned.

The present paper analyzes the problems of PI-based

CRPWM in both synchronous and stationary reference frames.

It then presents the theory and implementation details of

the GPC method as applied to the stator current control

problem. The results of simulation tests on a transient model

of the induction motor stator are compared with those for

synchronous and stationary PI regulators.

1063–6536/97$10.00 © 1997 IEEE

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120 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 1, JANUARY 1997

(a)

(b)

Fig. 1. Transient model of a three-phase induction motor for stator currentcontroller design.

II. THE INDUCTION MOTOR MODEL

An induction motor supplied through a voltage source

inverter has an input signal that is nonsinusoidal. This can be

decomposed to a set of sine waves of different frequencies

by Fourier analysis and for each there is a corresponding

equivalent circuit. Motor current response via superposition

is complex so a simplified approach is usually adapted, under

the following assumptions [1], [15]:

1) The total leakage inductance per phase can be repre-

sented as L , where

and are stator and rotor leakage inductances and

is the mutual leakage inductance.

2) The harmonic slip is unity, hence for harmonic currents

the motor behaves as a leakage inductance.3) The fundamental current loop under normal conditions,

with a slip close to zero, is presented with a back EMF

approximated as whererotor flux and is the angular frequency of

the ac supply. The simplified motor dynamic model is

shown in Fig. 1.

4) Machine speed is assumed to be constant because the

circuit time response constant is very much shorter than

that of the mechanical system.

The axis voltage matrix in a stationary reference frame

may be written

(1)

where the superscript denotes that the axes are

stationary and denotes the differential operator . Termsrepresent supply voltage vectors and I and I are stator

current vectors in the stationary reference frame. Its equivalent

in a synchronous reference frame is

(2)

In (2) the superscript is used for the synchronous reference

frame. Note that (2) represents an interactive two-input/two-

output system, whereas the input and output in (1) are com-

pletely decoupled and the system can therefore be represented

by two independent processes. Furthermore, the variables in

(2) are all dc quantities, whereas those in (1) are sinusoidal

variables. This leads to a difference in the controller structure

that is discussed in Section III below. To transform quantities

in the stationary reference frame to those in a synchronous

reference frame, a transformation matrix

(3)

may be used, where , the stator angular position, equals

.

For example, the measured voltages and currents from an

induction motor drive are ac variables in a stationary reference

frame. Using T , from (3), their synchronous equivalents

can be derived as

(4)

III. STATOR CURRENT REGULATOR USING A PI CONTROLLER

The objective of the stator current controller is to ensure

that the measured stator currents track the required values

accurately and with as short a transient interval as possible.

This is hard to achieve because:

1) the stator circuit is nonlinear due to speed-dependent

back EMF’s in the stator windings;

2) the controlled variables are three-phase ac quantities, re-

quiring controller parameters suitable for both amplitude

and phase angle regulation;

3) the PWM voltage source inverter introduces dead times

into the system which result in waveform distortion in

the power stage.

In a PI type current control loop in a stationary reference

frame, the three-phase sinusoidal stator currents are comparedwith the reference currents generated by the speed/torque

control loop and the errors are used by three identical PI

controllers. The controller outputs are then compared with a

fixed-frequency triangular carrier wave. The resulting PWM

signals, whose duty cycle is proportional to the control output,

control the inverter switching. If the coordinate is used,

the three-phase variables are converted into axis forms

and two identical controllers are used. The control

outputs are converted back to their three-phase equivalent

before proceeding to the PWM waveform generation process,

as shown in Fig. 2. If the controller parameters are chosenadequately, this control scheme provides good quality control

during transient periods. The main problem is that, in thesteady state, there is always a phase error between the stator

phase currents and the reference values. Moreover, the size of

this error varies with the machine operating frequency. The

cause of this lies in the structure of the controller and is

analyzed below.

Consider the transfer function of a PI controller

(5)

where K is the proportional gain factor and K the integration

gain factor.

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IEEE TRANSCATIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 1, JANUARY 1997 121

Fig. 2. Closed-loop stator current control system in the stationary reference frame.

As back EMF is assumed constant, the stator circuit transfer

function for either I or I may be written

(6)

where represents the net circuit gain and T is . The

corresponding closed-loop transfer function is given by

(7)

In (7), K represents the gain factor caused by the PWM

inverter, K equals the product of K and A, and the current

outputs can be expressed as

(8)

It is known that the objective of the current controller is

to drive I (S) as close as possible to I (S) until they are

equal in the steady state at all operating frequencies, when the

magnitude and phase-angle of the closed-loop transfer function

are

(9)

In the steady state, the modulating (drive) frequency needs

to be in the range 0 2 100. It can be seen from (7) that

the conditions of (9) can only be achieved when equals

zero. For any other values, both the amplitude and phase

angle of are different. The parameters of the controllershould, therefore, be tuned to meet the conditions defined

in (9) as closely as possible, but they cannot completely

satisfy them. This, in turn, means that the controller has to be

retuned when changes to avoid steady-state performance

degradation. Consequently, an unlimited number of regulators

with different characteristics are required. Clearly, this is

unrealistic to achieve and hence the PI control scheme in the

stationary reference frame is not ideal for high-performance

machine drives.

To overcome the problem of controlling ac quantities, a

synchronous PI regulator is defined which takes the same

form as that given by (5) but operates on the stator circuit

model in a synchronous reference frame given by (2). Theinput and output variables are all dc quantities in the steady

state. The diagonal elements of the model (2) are identical

first order processes but the plant should be viewed as a

multiple-input/multiple-output system. For a fixed value of

the eigenvalues of this second-order system lie at /Lj and at high values of the inherent damping becomes

very low, with possible stability problems. For low and

medium values of , the synchronous PI controller is likelyto give stable closed-loop responses for currents I and I . Italso eliminates completely the steady-state error. Nevertheless,

the two nonzero off-diagonal elements, reactances and

will inevitably affect the performance of the transient

response and hence the parameters of the controller must be

carefully chosen to minimize the effect of interaction.

The synchronous controller can also be implemented in

a stationary reference frame. This involves transforming the

controller (5) to a stationary reference frame, yielding the

controller outputs V expressed by

(10)

The two-input/two-output control loop is shown in Fig. 3.

The controller consists of two parts, the conventional PI

regulator and the cross-coupled control states .

In transient periods the PI part dominates the control action in

order to force the current error to zero. When the steady state

is reached, the current errors are zero and the cross-coupling

states provide sinusoidal voltage output to the machine at the

operating frequency. Hence this controller effectively solves

the problem of nonzero steady-state error under ac operatingconditions. Nevertheless, the transient performance is gener-

ally worse than that of the stationary regulator, indicated by

the cross-coupling states.

IV. LONG-RANGE PREDICTIVE

CONTROL OF STATOR CURRENT

A. System Equations

Predictive control is a type of control scheme which, given

a model of the system and a knowledge of past inputs

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Fig. 3. Closed-loop stator current control system using the synchronous reference frame controller.

Fig. 4. Closed-loop stator current control system using the GPC controller.

and outputs, allows the future outputs to be expressed in

terms of future inputs. The long-range prediction forecasts

several sample steps ahead and the most suitable control

signal is derived at each step by minimizing a quadratic costfunction. Various types of predictive control algorithm have

been proposed in the past decade and of these the GPC is

potentially the most capable of providing good performance

and robust control. Fig. 4 shows the stator current regulation

system using the GPC. The inverter supplies the induction

motor, the stator currents of which are sampled, fed back

through a three-phase-to-two-axis conversion, and transferred

to the synchronously rotating reference frame for comparison

with the reference currents to generate the input to the GPC.

Control signals from the GPC are transformed back from the

synchronously rotating reference frame to the stationary frame

by the inverse transformation matrix T and then changed

from two-axis to the equivalent three-phase quantities. These

three-phase values are then compared with a high-frequency

triangular carrier wave in order to generate triggering signals

for the inverter.

To consider the design of the GPC controller for the

induction motor drive, the dynamic model in a synchronous

reference frame given by (2) is used. Using the Z-Transform

method the discrete-time version of this model is

(11)

where ,

, and represents disturbance.Assuming that at arbitrary time the values of the stator

voltages V and V are defined, these will be keptconstant for future sample steps. The GPC predicts these

steps ahead to give and as

...

(12)

where

If the GPC operates in incremental form, the relationships

in (12) become

...

(13)

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(a)

(b)

(c)

Fig. 5. Responses of phase A stator current when command current changes. (a) Stationary reference frame PI controller. (b) Synchronous referenceframe PI controller. (c) GPC controller.

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IEEE TRANSCATIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 1, JANUARY 1997 125

...

...

(22)

where

The new control voltages from the GPC are therefore

(23)

B. Control Algorithm

The control algorithm is implemented in the following way.

Step 1) Choose the prediction length N and compute

G(z and S(z based on the discrete-time

model of the stator circuit.Step 2) Calculate the 2 2 matrix (M M + I) .

Step 3) At , calculate (1) and (1) using (15) by

assuming that both and are set to

zero.Step 4) Take the results of Step 3 as the measured currents

and calculate predictions of I and I where

, iteratively.

Step 5) Calculate the next voltage increments V (t) and

V (t) using the reference currents I and I , the

result from Steps 2 and 3 and (22).

Step 6) Calculate the new voltage outputs using (23).

Step 7) Convert the voltage output from Step 6 to a three-

phase signal and generate the PWM output tocontrol the inverter.

Step 8) Return to Step 3 for the next voltage output cal-

culation.

V. SIMULATION RESULTS

The GPC strategy was implemented to control the current

of an induction machine model on a Sun operating system. In

order to compare the response of the new controller with thatof a conventional current regulator, PI compensated controllers

in both synchronous and stationary reference frames were also

simulated.

Machine ratings are given for a 3 hp induction motor, 208V, 4-Pole, 60 Hz, 12 Nm. Its parameters are listed in the

Appendix.

The inverter switch frequency was set at 2 kHz. The

controller settings are = 5.13 and 0.006 88 s for

the PI regulators in both stationary and synchronous frames.

The discrete-time model polynomials are

for the GPC with a sampling rate of 500 s and a control cost,

.

Fig. 5 shows the phase A stator current responses to the

command current change when different controllers were

used. In Fig. 5(a) a phase error between the reference and

the measured currents is clearly present and this cannot be

eliminated by controller gain adjustment. The single phase

current response using the synchronous PI regulator is shown

in Fig. 5(b). The phase error is cancelled but the poor quality

of the transient control, indicated by the cross-coupling terms,

is clearly visible when changing the command current. The

superior performance of the GPC regulator is illustrated in

Fig. 5(c) which shows that there is a zero phase shift between

the stator currents and their respective reference signal. Also,

the quality of the command current following during transient

state is good. Relative transient performances of the two

controllers are the subject of an ongoing investigation, to

be reported later. Preliminary results indicate that with the

GPC controller the transient response is significantly improved

compared with the PI controller.

VI. CONCLUSION

The application of GPC to the problem of stator current

regulation for high-performance control systems has been

presented. Results prove superior to those of the conventional

current regulation method of PI control, in terms of the

removal of phase and magnitude errors which degrade the

transient performance of motor control systems. The GPC

method is also superior to that of the synchronous PI regulator

which, although accurate in the steady state, has a poor

dynamic performance.

APPENDIX

r stator resistance, 0.6 ;r rotor resistance referred to primary turns, 0.4 ;

L stator leakage inductance, 0.0021H;

L rotor leakage inductance referred to primary turns,

0.0021H;

L magnetizing inductance, 0.059H.

REFERENCES

[1] A. B. Plunkett, “A current controlled PWM transistor inverter drive,”in IEEE Ind. Applicat. Soc. Annu. Mtg. Conf. Rec., 1979, pp. 785–792.

[2] Y.-T. Kao and C.-H. Liu, “Analysis and design of microprocessor-basedvector-controlled induction motor drives,” IEEE Trans. Ind. Electron.,vol. 39, pp 46–54, 1992.

[3] J. M. D. Murphy and F. G. Turnbull, Power Electronic Control of acMotors. New York: Pergamon, 1988, pp 139–141.

[4] P. N. Ejeti, P. D. Ziogas, and J. F. Lindsay, “A new current controlscheme for AC motor drives,” IEEE Trans. Ind. Applicat., vol. 28,July/Aug. 1992.

[5] C. D. Schauder and R. Caddy, “Current control of voltage-sourceinverters for fast four-quadrant drive performance,” IEEE Trans. Ind.Applicat., vol. 1A-18, pp. 163–171, Mar./Apr. 1982.

[6] T. M. Rowan and R. J. Kerkman, “A new synchronous current regulatorand an analysis of current-regulated PWM inverters,” IEEE Trans. Ind.Applicat., vol. 1A-22, pp. 678–690, July/Aug. 1986.

[7] S. Meshkat and E. K. Persson, “Optimum current vector control of abrushless servo amplifier using microprocessors,” in IEEE Ind. Applicat.

Soc. Annu. Mtg. Conf. Rec., 1984, pp. 451–457.[8] H. Ikejima, M. Nomura, H. Sugimoto, and E. Ohno, “Microprocessor-

based ac motor drive control for elevator,” in Proc. IEEE Power Electron. Specialists’ Conf., 1983, pp 64–69.

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[9] D. M. Brod and D. W. Novotny, “Current control of VSI-PWM invert-ers,” IEEE Trans. Ind. Applicat., vol. IA-21, pp. 562–570, May/June1985.

[10] H. Nagase, Y. Matsua, K. Ohmishi, H. Ninomiya, and T. Koike, “High-performance induction motor drive system using a PWM inverter,” inIEEE Ind. Applicat. Soc. Annu. Mtg. Conf. Rec., 1983, pp 596–603.

[11] D. W. Clarke and L. Zhang, “Long-range predictive control usingweighting-sequence models,” IEE Proc. Pt. D, vol. 136, no. 3, 1987,pp. 187–185.

[12] D. W. Clarke, C. Mohtadi, and P. S. Tuffs, “Generalized predictivecontrol—Part 1: The basic algorithm,” Automatica, vol. 23, no. 2, pp137–148, 1987.

[13] C. R. Cutler and B. L. Ramaker, “Dynamic matrix control—A computercontrol algorithm,” in Proc. Int. Symp. Advanced Process Supervi-sion and Real-Time Knowledge-Based Contr., Newcastle, U.K., Nov.1988.

[14] D. M. Brod, “Current control of VSI-PWM inverters,” M.S. thesis, Univ.Wisconsin—Madison, 1984.

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