High performance IPMSM drives without rotational position sensors using reduced-order EKF - Energy Conversion, IEEE Transactions on

High performance IPMSM drives without rotational position sensors using reduced order EKF Energy Conversion, IEEE Transactions on 868 IEEE Transactions on Energy Conversion, Vol 14, No 4, December 199[.]

868 IEEE Transactions on Energy Conversion, Vol 14, No 4, December 1999

High Performance IPMSM Drives without Rotational Position Sensors

Using Reduced-Order EKF

Dept of Electrical Engineering, Chung-Ang University

22 1 HukSuk-Dong, DongJak-Ku, Seoul, 156-756 Korea Phone : +82-2-820-5290 Fax : +82-2-812-1407 Email : yhkim@cau.ac.kr

Keywords : IPMSM(1nterior Permanent Magnet Synchronous Motor), EKF(Extended Kalman Filter)

Absfracf - An extended Kalman filter(EKF) based approach

for position sensor elimination in interior permanent magnet

synchronous motor(1PMSM) drives is presented in this paper

The EKF is capable of estimating system parameters and state

variables for the IPMSM by eliminating virtually all influences

of structural noises in the vector control scheme This paper

presents a design method of a reduced-order EKF Position and

angular speed of the rotor are obtained through the reduced-

order EKF only by measuring stator currents Also, due to an

angle modification scheme with error tracking, the sensorless

drive system is robust to parameter variations Simulation and

experimental results are provided to verify the proposed

approach based on the reduced-order EKF

I , INTRODUCTION Recently, motor drive systems without electromechanical

sensors, so called ‘sensorless drives’, have gained increasing

popularity in industrial applications because of inherent

drawbacks of electromechanical sensors In general, electro-

mechanical sensors are used to obtain speed or position

information of motors A drawback of these sensors is

performance degradation due to vibration or humidity [I-21

Thus, variable control strategies for PMSM drives without

electromechanical sensors have been presented by many

authors in drives such as BDCM [I], SPMSM [2] and

IPMSM [3] Especially, the IPMSM has found wide

application on high performance motor drives because of

high speed operation These are high power density machines

capable of operating at high speed and inverter efficiencies

over wide speed ranges, including considerable ranges of

constant power operation The rotor magnetic circuit saliency preferentially increases the q-axis inductance and introduces

a reluctance torque term into IPMSM torque equations [4] Previous studies about sensorless drives for the IPMSM can be divided into three categories In the first one, the idea

is to manipulate the motor equations in order to express rotor position and speed as functions of terminal quantities [2] In this scheme, the sensitivity to motor parameters is a major drawback In the second scheme, sensorless drives have been developed on the basis of state observers The major disadvantage of this scheme is that linearization of the nonlinear equations describing system behaviors along the nominal state trajectoly does not guarantee the overall stability [ 5 ] In the last scheme, thanks to their ability to perform state estimation for nonlinear systems, EKF is adopted to estimate the rotor angle and speed The Kalman filter based on minimization of the estimation error covariance is suitable for obtaining high-accuracy estimates

of state variables and model parameters and eliminating measurement noise So far, however, computation requirements, parameter sensitivity, and initial conditions have unfavorably characterized this approach

As for computation load, however, the reduced-order EKF can easily bear real-time calculation using floating-point DSPs In this paper, the reduced-order EKF to estimate back- EMF and the angular speed as state variables only by measuring stator currents for IPMSM drives is proposed Furthermore, using the rotor position and speed estimation strategy, the actual rotor position, as well as the motor speed, can be estimated with little error even in the transient state such as speed fluctuation or sudden load variation Also,

since the angle error is newly modified with error tracking at each estimation step, the overall control system is robust to

PE-019-EC-0-10-1998 A paper recommended and approved by the

iEEE Electric Machinery Committee of the ,EEE Power Engineering

~ , ~ for aubiication ~ i in ~ the ~~ fiEEE ~ ti^^^ on E~~~~

Conversion Manuscript submitted January 22, 1998: made availabie parameter variations Therefore, regardless of detuned for printing November IO, 1998 parameters, the estimated angle always keeps track of the

actual angle To verify the feasibility of the proposed method, simulation and experiments are performed Main features of the proposed algorithm are :

Mechanical parameters are not required, and problems

of parameter sensitivity are partially overcome

0885-8969/99/$10.00 0 1998 IEEE

from the initial value assumed by the algorithm

Measurements of the voltages applied to motors are no

longer required

II MATHEMATICAL MODEL OF THE IPMSM

The voltage equation of the IPMSM, based on the d-q

reference frame theory, is expressed in matrix fonn as :

where superscript 's' denotes stationary reference frame, and

'p' means differential operator

In this paper, by choosing back-EMF as a state variable,

state equations are simplified First, we assume that generally

the mechanical time constant is much larger than the

electrical time constant Here, the time constant is a measure

of rise time Therefore, if one time interval of a current

control loop is set smaller than a mechanical rise time, it can

be assumed that the angular speed is constant during one

estimation interval In this paper, by introducing this

assumption, an algorithm estimating the back-EMF

parameters is designed

The components of the back-EMF are defined as :

Then, from the assumption that the speed of a motor is

constant within one estimation interval, the derivative of

back-EMF is expressed in matrix form as :

(3)

Therefore, a dynamic model of IPMSM in the stationary

reference frame, by choosing stator currents i,,', id' and

back-EMF Eq', Ed' as state variables, is given as follows :

(4)

1 0 0 1

X(f) = Ax(t) t Bu(t)

The state equations of IPMSM are expressed as

(9

T

where x ( t ) = [iqd ids E," Et,"]' , U ( t ) = [vq,q Vd"]

In order to perform computer simulations and implement the Kalman Filter algorithm, continuous state equations can

be transformed into discrete state equations as follows

x ( k t l ) = F ( k ) x ( k ) + G ( k ) u ( k ) (7)

where F ( k ) = I + A T , , G ( k ) = B I ; , H ( k ) = C

FILTER ALGORITHM The EKF is an optimal recursive estimation algorithm based on the least-square sense for estimating the states of

dynamic nonlinear systems That is, it is an optimal estimator

for computing the conditional mean and covariance of the probability distribution of the state of a nonlinear stochastic system with uncorrelated Gaussian process and measurement noise

Since the state models are nonlinear, the EKF can be

applied to estimate state variables In this case, the back-EMF

is considered as a state variable Nonlinear discrete models with white noise are given as follows :

x(k t 1) = f ( x ( k ) , u ( k ) ) 4 k )

(9)

where w(k) and v(k) are zero-mean noise with covariance

Q and R respectively and are independent from the system state x(k) The system noise w ( k ) takes into account

system disturbances and model inaccuracies, while v(k)

represents the measurement noise The initial state vector

i(0) is a Gaussian random vector with mean "(0) and covariance matrix P(O), and u(k) is the deterministic input vector For linearization process in the model, the partial derivative is introduced and discrete state models are

870

From the dynamic model given from (9) to (12), the

hack-EMF can be estimated by the following EKF algorithm

1) estimation of an error covariance matrix

2) computation of a Kalmanfilter gain

K ( k + 1) = P - ( k + ])A7 (k)[A(k)P- (k + l)AT(k) + RI-' ( I 4)

3 ) update of a error covariance matrix

4) state estimation

P(k + 1) = [I - K(k + I)A(k)]P-(k t 1)

P(k + 1) = P(k)

(15)

where P - ( k ) is a priori error covariance matrix

The rotor speed w, is supposed to he constant within an

estimation period Then, in order to estimate the angular

speed and rotor position, the reduced-order EKF model is

rearranged by choosing hack-EMF and angular speed as state

variables instead of dq-axis stator currents As a result, the

dimension of EKF model in (4) is reduced as shown in (17),

which in turn saves computation time The rearranged system

and measurement models with noise are given as follows :

[ I , ( k t 1) - 6 ] I , ( k ) - Go,] = [ F,, O ] ~ ~ ~ ; $ ' ] t v( k ) ( 1 8)

The models are denoted by the following equations

Once the back-EMF is estimated, the angular speed and the rotor position are also easily obtained

The choice of initial values for matrices R , Q and

P(0) is very important Generally, P(0) determines the

initial transient characteristics of the filter On the contrary,

R and Q represent the dynamic charateristics during the transient-state and steady-state In this paper, to obtain the coefficients of the covariance matrix, a Gaussian noise generator is used Matrices are given as follows :

P(0) = 0 0.1 O :j (21) The initial state vector x(0) can be considered as a null vector

0 0 0.001

A Rotor Angle Estimation Strategy

To achieve high performance vector control, the accurate rotor angle, that is, the angle transformed in a synchronous reference frame is required But when the angular speed is close to zero, a lot of enor is included in the back-EMF Therefore, the estimated angle is largely fluctuating and the system doesn't converge to the steady state

In this paper, the trigonometric function of rotor angle is not used directly Instead, the arc-tangent function of the back-EMF is used The reason for this is that stability of the system is guaranteed in the low speed or start-up The rotor angle is obtained from (2) as follows :

I

In the case of reverse speed direction, 180 degree phase

difference is generated For this, it is necessary to compensate the difference Therefore, the rotor angle is given

as follows :

8, = 0,

~1 ' (23) between the magnitude I , and phase angle p of armature

current to obtain the maximum torque per armature current is derived as follows

if w, > 0,

B Speed Estimation Strategy

from (2) The angular speed can be derived as follows :

The direction of the motor speed is required It is

obtained from angular difference A@+) Thus, by the

influence of the estimated error of back-EMF, the ripple may

be included in the estimated speed Such a ripple speed can

be a factor limiting the response time of a speed control

system It is desirable to use low-pass filters

C Compensation Algorithm for fhe estimated angle

In this paper, by introducing a limit in the changes of the

estimated rotor angle, the proposed sensorless scheme has

good start-up or speed reversal ability

If we assume that the angular speed is constant within a

estimation interval, the rotor angle is the integral of the

angular speed That is :

A t ( , < ) = e.(,,) - %(,v-l) = C,Y, ' 4 (25)

In this algorithm, the rotor angle is limited by a

maximum allowable value of the difference as follows :

A@,,,, = 2.x/LF(&,)l' 5,

where LF means a first-order low-pass filter and its

frequency response is 20[Hz] This means that when the

estimated rotor angle 8e(,3, has some estimation errors by

system noises, the angle is limited by the control angle

A@+) The compensation algorithm is shown in fig 1

V , SIMULATION RESULTS

In the maximum torque control, the current phase angle

p is actively controlled according to (27), which is a

function of I, The speed attainable at the maximum torque

is limited by the available maximum output voltage of the inverter As the power factor is improved by the maximum

torque control, the limited speed also increases As a result,

the maximum output power becomes large, and power capability is greatly expanded Fig 2 shows the block diagram of a high performance speed control system with

maximum torque control The magnitude command I,' of

the armature current is determined through the speed controller The current phase command p is calculated

according to (27) based on motor parameters The current

commands and I<;* are calculated as (28)

1;' = +Ip' cosp

t

Fig I The block diagram of the sensorless controller using reduced-order

EKF

In this paper, a synchronous frame current regulator is used, and reference voltages are calculated from the current regulator The stationary reference frame voltages are used in reduced-order EKF algorithm as stator voltages without

As quick transient response and large torque are desired

be

measwment The estimated rotor angle is used for the

transformation between stationary reference frame and the synchronously rotating reference frame In PWM techniques, for IPMSM drives, the current phase angle

872

the space vector voltage modulator is used to obtain constant

switching frequency The nominal parameters used for the

simulation and experiment are given as follows :

Table 1 IPMSM parameters

,*, ’c””]

i , i A- .T , , , a,

The proposed sensorless control of IPMSM is shown in

fig 2

Fig 2 l h e block diagram ofthe overall control algorithm

159

R s i l 5Rs

T,ms(WC,

Fig 3 Parameter dependency

(detuned R,? to 150% of its nominal value)

(a) Actual and estimated angular velocity

(b) Actual and estimated rotor angle

Fig 3 shows influence of parameter uncertainties Actual

and estimated values of the speed and angle quantities are

shown when the stator resistance is detuned to 150% of

actual value The angle estimation is not affected by stator

resistance variation

For the high performance IPMSM drives, the overall IPMSM drive system in fig 4 is implemented with a

TMS320C31 DSP control board and a PWM IGBT inverter The switching frequency of the inverter is 1OkHz and space vector PWM algorithm is used for the maximum utilization

of DC link voltage Since the machine terminal voltage generated by the inverter has the same value of the command voltage from the current controller, the command voltage can

be used as a voltage information without direct terminal voltage measurement

A l l k W 4-pole IPMSM is used And for actual load

emulation, the dynamo meter is coupled to the IPMSM Actual rotor angle and machine speed are measured from an incremental encoder with 2OOO[ppr] resolution for monitoring The sampling time of the current controller loop

is loo[ ja ] and that of the outer voltage regulating loop and speed loop is l[ms] The control algorithm including the proposed scheme was fully implemented with the software

ICBTinrrmr

~~

-

L ~ ~ ,

Fig 4 The ovcrall IPMSM drive system

Experiments are conducted to evaluate the performance

of the new position sensor elimination algorithm based on the reduced-order EKF

Fig S Back-EMF estimation characteristics

(a) Estimated rotor angle (b) Estimated angular velocity (c) Estimated q-axis back-EMF (d) Estimated d-axis back-EMF

VI Experimental Results

of modeling and measurement noises

W CONCLUSION

In this paper, the reduced-order EKF approach for a speed sensorless vector control is proposed to estimate the rotor position and speed The performance of the algorithm was investigated over a wide range of speeds The reduced- order EKF scheme has an advantage in reducing computational time Also, due to the angle compensation scheme, the sensorless drive system has robustness to parameter variations The experimental results show the validity of the proposed algorithm for the IPMSM drives

Vm REFERENCES

[I] S Ogasawara, H Akagi, "An Approach to Position Scnsoclers Drivc

for Brushless DC Motors." ILLL Trsans on Ind A 0 0 l vol 27 no 5

-18001

Id) 180]

-1800

[ b J ' ' ' ' ] - 0 -

1

0 h i c l IS mridiv]

Fig 1 The start-up capability characteristics response

(a) Actiial and estimated rotor angle

(b) Actual and estimated angular velocily

CO,

n -

-n

Fig 5 shows the traces of the estimated back-EMF

quantities on the stationary d-q frame at the startup instance

As shown in the figure, the proposed algorithm works well in

spite of the variation of the machine speed

The step response of the proposed sensorless algorithm is

shown in fig 6 when the speed reference is changed from

O[rpm] to 1500[rpm] and with full load torque

The starting capability of the proposed sensorless

algorithm is depicted in the fig 7 For any position, a stable

starting can be obtained with the help of the angle

compensation algorithm

Fig 8 shows the influence of the parameter uncertainties

to the reduced-order EKF The actual and estimated values of

the speed angle quantities are shown when the stator

resistance is detuned to 150% of its actual value As shown in

these figures, the angle estimation is not affected by the

parameter uncertaintities and a stable machine drives can be

obtained It can be seen that this scheme is robust against the

o,

c

~~

1991, pp 928-933

R Wu, G R Slemon, "A Permancnt Magnet Motor Drive without a I21

Shaft Sensors." IEEE Trans on Ind A m i ' , vol 21 ~, no S Seo./Oct ,

1991,pp IOOS-1011

A B Kulkarni, M Bhsani, "A Novel Position Scnsor Elimination [3]

Technique for the Interior Permanent Magnet Syechronous Motor

Drive,"IEEE Trans on hid Appl.,vol 28, nu 1, 1992, pp 128-135

S Morimoto, Y Takeda, K Hatanaka, Y Tong, T Mirasa, "Design and Control System OC Inverter-Driven Permancnt magnel Synchronous Motors for High Torque Operation," /EL.? Trans on Ind Appl., vol 29, no 6,Nov./Dec., 1993; pp 11SO-11S4

L A Jones J H Lane, "A State Observer for Permanent Magnet

[4]

IS1

Synchronous Motor," IfLE Trans on Ind E l e c ~ , vol 36, no 3, Aug

1989, pp 374-382

BIOGRAPHY

Yoou 110 Kim received a B.S degree from Scoul National University, Korca, a

M.S degree from State University of New York at Hoffalo, and a Ph D degree from Texas A & M University, all in electrical engineering Sincc 1987, he has been with Chung-Ang University, Seoul, Korca, is inow

professor in electrical engineering His main

interests arc industrial clectronics and

industrial drives Dr Kim is a member o f t h c IEEE and The Korean Institule of Electrical Engineers

Yoan Sang Kook was born in ChunBuk,

Korea, on February 26, 1972 Ile received a

B.S degree and a M.S degree from Chung- Ang University He is presenlly in doctoral

co~irse o f electrical engineering at Chung-

Ang University His main interests are power

electronics and modern control theory He is

a student inember of KlEE and KIPE