1. Munmun Dutta, Balram Timande / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.1188-1192
A Relative Study Of Kalman Filtering For Control Of A
Permanent-Magnet Synchronous Motor Drive
Munmun Dutta*, Balram Timande**
*(Department of Electrical & Electronic Engineering, DIMAT, Raipur, India)
**(Department of Electronic & Telecommunication Engineering, DIMAT, Raipur, India)
ABSTRACT
There is increasing demand for derived from the Kalman filter based on the
dynamical systems to become more realizable successive linearization of the signal process and
and more cost-effective. These requirements observation map[3]. EKF is insensitive to parameter
extend new method of control and operation. changes and used for stochastic systems where
This paper presents a comparative study of the measurement and modeling noise is taken into
Extended Kalman Filter (EKF) for estimation of account.
the rotor speed and position of a permanent-
magnet synchronous motor (PMSM) drive[1]. 2. MATHEMATICAL MODEL OF
The system is highly nonlinear and one therefore PMSM
cannot directly use any linear system tools for The Permanent Magnet Synchronous Motor
estimation. However, if one can linearize the is rotating electric machine where the stator is a
system around a nominal (possibly time-varying) classic three phase coils like that of an induction
operating point then linear system tools could be motor and the permanent magnets are located on the
used for control and estimation. Extended rotor. A PMSM can be mathematically represented
Kalman Filter is generalized algorithm, which by the following equation in the d-q axis
can be used for non-linear systems such as synchronously rotating rotor reference frame for
PMSM. The estimation is done upon undisturbed assumed sinusoidal stator excitation [4].
input signals from overriding controller and 1
disturbed output signals of a real non-linear ia = −(R a L)ia + (λ L)ωr sinθr + ( )ua
L
plant, which are measured [2]. The entire state 1
estimated system has been modeled using ib = −(R a L)ib + (λ L)ωr cosθr + ( )ub
L
MATLAB. ωr = −(3λ 2J) ia sinθr + (3λ 2J) ib cosθr −
F J ωr − (1 J)TL
Keywords- Two-Phase Permanent Magnet
Synchronous Motor, Kalman Filter, Extended (2.1)
Kalman Filter, Matlab function. θr = ωr
where ia and ib are the currents through the
two windings, Ra and L are the resistance and
1. INTRODUCTION
High torque to inertia ratio, superior power inductance of the windings, θr and ωr are the angular
density, high efficiency and many other advantages position and velocity of the rotor, λ is the flux
made PMSM the most widely acceptable electrical constant of the motor, ua and ub are the voltages
motor in industrial applications. Compared with the applied across the two windings, J is the moment of
inverter-fed induction motor drive, the PMSM has inertia of the rotor and its load, F is the viscous
no rotor loss and hence it is more efficient and a friction of the rotor, and TL is the load torque [4, 5].
larger torque-to-weight ratio is achievable. To To avoid convergence problems at startup
reduce the cost and to improve the reliability, sensor and to simplify the motor equations, the rotor
less PMSM control strategies have been developed reference frame is chosen for evaluation of the
[2]. In these strategies, the motor position and speed Kalman filters [6]. The motor nonlinear state
is estimated and used as a feedback signal for equations can be expressed in the form:
closed-loop speed control. x = Ax + Bu + MTL
In this paper, we propose to estimate the (2.2)
motor quantities like speed, flux vector position, y = Cx
currents in direct and quadrature axes, from the (2.3)
measurements of three phase stator currents using an
Extended Kalman filter (EKF)[3]. The Kalman filter Where
is a special kind of observer which provides optimal x = ia ib ωr θr T is a state vector.
estimation of the system states based on least-square 𝑦 = 𝑖𝑎 𝑖𝑏 𝑇 is a output vector.
techniques. The extended Kalman filter (EKF) is With this definition, “equation (2.1)” can be written
widely used for nonlinear filter problems. It is compactly[7]
1188 | P a g e
2. Munmun Dutta, Balram Timande / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.1188-1192
as x = x1 x2 x3 x4 T OR
−R a λ xk+1 = Ak xk + Bk uk + Mk TL + wk
L
0 − L sinx4 0
−R a λ
(3.7)
0 cosx4 0 where,
A= L L
−3λ 3λ −F T L ∆ua
sinx4 cosx4 0
2J 2J j T L ∆ub
wk =
0 0 1 0 −T J ∆TL
1
() 0 0 1 0
T
0
L
1 0 0 1 (3.8)
B= 0 ( ) ,M= −1 andC= Similarly, if the measurements ia and ib are
L 0 0
0 0 J
0 0 distorted by noises ∆ia and ∆ib respectively, then
0 0 0 “equation (3.2)” becomes
ia + ∆ia
3. ESTIMATION OF SPEED AND ib + ∆ib ∆ia
yk = Ck =Ck xk+ = Ck xk + vk
ROTOR POSITION USING ROBUST ωr ∆ib
EXTENDED KALMANFILTER θr
The Kalman filter is often applied during (3.9)
dissolving state estimation of dynamical system, where,
disturbed by the known signals. Kalman filter ∆ia
vk =
algorithm is used for estimating the parameters of ∆ib
linear system, but the PMSM model is non-linear, so (3.10)
we cannot use that filter in this case. Extended where the vectors wk and vk are called the
Kalman Filter is generalized algorithm, which can be process and measurement noises respectively[12].
used for non-linear systems The Kalman filter solution does not apply unless
The EKF is an optimal estimator in the least certain assumptions about the noise that affects the
square sense for the estimation of nonlinear dynamic system under study must be satisfied[13]:
systems. It is derived from the Kalman filter based It is firstly to assume that the average
on successive linearization of the signal process and 1. It is firstly to assume that the average value of
observation map [8]. both wk and vk are zero.
For this application the motor nonlinear state 2. One has to further assume that no correlation
“equations (2.1)” are expressed in the discretized exists between wk and vk . That is, at any time k,
form wk and vk are independent random variables.
xk+1 = Ak xk + Bk uk + Mk TL Then the noise covariance matrices Sw and Sv
(3.1) are defined as:
yk = Ck xk Process noise covarianc[6]:
T
(3.2) Sw = E wk wk
Ak and Bk are the discretized system and (3.11)
input matrices, respectively. They are [9,2,10] Measurement noise covariance:
AT 2 Sv = E vk vk T
Ak = eAT = I + AT + +⋯
2! (3.12)
≅ I+AT where wT and vT indicate the transpose of w
(3.3) and v random noise vectors, and E(.) means the
T
expected value.
Bk = eAζ Bdζ = [eAT − I]A−1 B Substituting “equation (3.8) ”into“ equation
0
ABT 2 (3.11)” and “equation (3.10)” into “equation
= BT + 2!
+ ⋯ ≅ BT (3.12)”,[14] one can get the following process and
(3.4) measurement noise covariance matrices:
Mk ≅ MT
(3.5) Sw
Ck = C T 2 T 2 T 2
(3.6) ∆ua 2 ∆ua ∆ub − ∆ua ∆TL 0
where T is the sampling time and I is an L L JL
identity (4X4) matrix. T 2 T 2 T 2
= ∆ua ∆ub ∆ub 2 − ∆ub ∆TL 0
If the noises ∆ua , ∆ub have corrupted the L L JL
inputs ua and ub respectively, and the noise ∆α has T 2 T 2 T 2
been admitted to account for uncertainties in the load − ∆ua ∆TL − ∆ub ∆TL ∆TL 2 0
JL JL J
torque[11], then a noise vector will arise in
0 0 0 0
“equation.(3.1)”. (3.13)
u + ∆ua
xk+1 = Ak xk + Bk a + Mk TL + ∆TL
ub + ∆ub k
1189 | P a g e
3. Munmun Dutta, Balram Timande / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.1188-1192
∆ia 2 ∆ia ∆ib Robust EKF estimates the instantaneous motor
Sv = E speed, rotor position, quadrature and direct axis
∆ia ∆ib ∆ib 2
(3.14) currents and voltages.
If the noises ∆ua ( ∆ub ), ∆TL , ∆ia ( ∆ib ) are The parameters of the motor are listed in “Table -
white, zero mean, uncorrelated, and have known 4.1”,[16]
variances σ2 , σ2 and σ2 , 2Ts and 2Ms , respectively,
i T M
then the covariance matrices Sw and Sv will become TABLE-4.1: PARAMETERS OF TWO-PHASE
MOTOR
T 2
σ2 0 0 0
L i Winding Resistance Ra 2Ω
T 2
Sw = 0 σ2
i 0 0
L
T 2
0 0 J
σ2
T 0 Winding Inductance L 3mH
0 0 0 0
(3.15)
Motor flux constant λ 0.1
σ2
M 0 Standard deviation of
Sv = (3.16) control input noises
0 σ2M
The timing diagram of the various σi 0.001A
quantities involved in the discrete optimal filter Standard deviation of σT 0.05 rad/ sec2
equations is shown in “Fig.(3.1)”. load torque noise
Standard deviation of σM 0.1A
measurement noise
Moment of Inertia J 0.002
Frequency f 1Hz
Coefficient of viscous B 0.001
friction
Figure 3.1 timeline showing a priori and a posteriori state
estimates and estimation- error covariance.
The “Fig (3.1)” shows that after we process
the measurement at time (k-1), we have an estimate
+
of xk+1 (denoted xk−1 ) and the covariance of that
+
estimate (denoted Pk−1 )[15]. When time k arrives,
before we process the measurement at time k we
compute an
estimate of xk (denoted 𝐱 − ) and the covariance of
𝐤
that estimate (denoted 𝐏 − ).Then the measurement
𝐤
is processed at time k to refine our estimate of x .
k
The resulting estimate of xk is denoted 𝐱 + and its
𝐤
covariance is denoted 𝐏 +.
𝐤 Figure-4.2 extended Kalman filter simulation results
for a two-phase permanent magnet synchronous
4. EXPERIMENTAL RESULTS & motor
SIMULATIONS
Simulation of the given PMSM has been “Figure 4.2” show, respectively, the
carried out using MATLAB[12]. The three phase simulation results of EKF state estimation
stator currents are taken from the motor model, and performance for the sensor less PMSM drive. The
given to the robust extended Kalman filter. The system was simulated at sampling time (T=1 ms).
1190 | P a g e
5. Munmun Dutta, Balram Timande / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.1188-1192
Munmun Dutta, born in Rajasthan, Prof. Balram Timande, Born in
India, in 1985. He received the 1970, currently working as Reader in
Bachelor degree in Electrical department of E&TC at DIMAT
Engineering from the NIT Raipur, Raipur. He is having 9.5 years
India, in 2007 and pursuing Master Industrial experience and 7.5 years
degree in Power System Engineering teaching experience. He obtained B.E.
with the Department of Electrical & Electronic (Electronics) from Nagpur University (INDIA) in
Engineering from the DIMAT,Raipur, India. Her 1993 and M. Tech. form B.I.T. Durg (affiliated to
main area of interest are Control system C.S.V.T.U. Bhilai) in 2007. He published 02 papers
engineering, Power system engineering , Testing in National Journal LAB EXPERIMENTS, (LE)
& commissioning of electrical installations. ISSN no. 09726055KARENG/2001/8386 and 02
Papers in International Journal IJECSE – ISSN-
2277-1956. His area of research/interest is
Embedded system design and Digital Image
Processing.
1192 | P a g e
![Munmun Dutta, Balram Timande / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.1188-1192
A Relative Study Of Kalman Filtering For Control Of A
Permanent-Magnet Synchronous Motor Drive
Munmun Dutta*, Balram Timande**
*(Department of Electrical & Electronic Engineering, DIMAT, Raipur, India)
**(Department of Electronic & Telecommunication Engineering, DIMAT, Raipur, India)
ABSTRACT
There is increasing demand for derived from the Kalman filter based on the
dynamical systems to become more realizable successive linearization of the signal process and
and more cost-effective. These requirements observation map[3]. EKF is insensitive to parameter
extend new method of control and operation. changes and used for stochastic systems where
This paper presents a comparative study of the measurement and modeling noise is taken into
Extended Kalman Filter (EKF) for estimation of account.
the rotor speed and position of a permanent-
magnet synchronous motor (PMSM) drive[1]. 2. MATHEMATICAL MODEL OF
The system is highly nonlinear and one therefore PMSM
cannot directly use any linear system tools for The Permanent Magnet Synchronous Motor
estimation. However, if one can linearize the is rotating electric machine where the stator is a
system around a nominal (possibly time-varying) classic three phase coils like that of an induction
operating point then linear system tools could be motor and the permanent magnets are located on the
used for control and estimation. Extended rotor. A PMSM can be mathematically represented
Kalman Filter is generalized algorithm, which by the following equation in the d-q axis
can be used for non-linear systems such as synchronously rotating rotor reference frame for
PMSM. The estimation is done upon undisturbed assumed sinusoidal stator excitation [4].
input signals from overriding controller and 1
disturbed output signals of a real non-linear ia = −(R a L)ia + (λ L)ωr sinθr + ( )ua
L
plant, which are measured [2]. The entire state 1
estimated system has been modeled using ib = −(R a L)ib + (λ L)ωr cosθr + ( )ub
L
MATLAB. ωr = −(3λ 2J) ia sinθr + (3λ 2J) ib cosθr −
F J ωr − (1 J)TL
Keywords- Two-Phase Permanent Magnet
Synchronous Motor, Kalman Filter, Extended (2.1)
Kalman Filter, Matlab function. θr = ωr
where ia and ib are the currents through the
two windings, Ra and L are the resistance and
1. INTRODUCTION
High torque to inertia ratio, superior power inductance of the windings, θr and ωr are the angular
density, high efficiency and many other advantages position and velocity of the rotor, λ is the flux
made PMSM the most widely acceptable electrical constant of the motor, ua and ub are the voltages
motor in industrial applications. Compared with the applied across the two windings, J is the moment of
inverter-fed induction motor drive, the PMSM has inertia of the rotor and its load, F is the viscous
no rotor loss and hence it is more efficient and a friction of the rotor, and TL is the load torque [4, 5].
larger torque-to-weight ratio is achievable. To To avoid convergence problems at startup
reduce the cost and to improve the reliability, sensor and to simplify the motor equations, the rotor
less PMSM control strategies have been developed reference frame is chosen for evaluation of the
[2]. In these strategies, the motor position and speed Kalman filters [6]. The motor nonlinear state
is estimated and used as a feedback signal for equations can be expressed in the form:
closed-loop speed control. x = Ax + Bu + MTL
In this paper, we propose to estimate the (2.2)
motor quantities like speed, flux vector position, y = Cx
currents in direct and quadrature axes, from the (2.3)
measurements of three phase stator currents using an
Extended Kalman filter (EKF)[3]. The Kalman filter Where
is a special kind of observer which provides optimal x = ia ib ωr θr T is a state vector.
estimation of the system states based on least-square 𝑦 = 𝑖𝑎 𝑖𝑏 𝑇 is a output vector.
techniques. The extended Kalman filter (EKF) is With this definition, “equation (2.1)” can be written
widely used for nonlinear filter problems. It is compactly[7]
1188 | P a g e](https://image.slidesharecdn.com/gn2511881192-121002063836-phpapp01/85/gn2511881192-1-320.jpg)
![Munmun Dutta, Balram Timande / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.1188-1192
as x = x1 x2 x3 x4 T OR
−R a λ xk+1 = Ak xk + Bk uk + Mk TL + wk
L
0 − L sinx4 0
−R a λ
(3.7)
0 cosx4 0 where,
A= L L
−3λ 3λ −F T L ∆ua
sinx4 cosx4 0
2J 2J j T L ∆ub
wk =
0 0 1 0 −T J ∆TL
1
() 0 0 1 0
T
0
L
1 0 0 1 (3.8)
B= 0 ( ) ,M= −1 andC= Similarly, if the measurements ia and ib are
L 0 0
0 0 J
0 0 distorted by noises ∆ia and ∆ib respectively, then
0 0 0 “equation (3.2)” becomes
ia + ∆ia
3. ESTIMATION OF SPEED AND ib + ∆ib ∆ia
yk = Ck =Ck xk+ = Ck xk + vk
ROTOR POSITION USING ROBUST ωr ∆ib
EXTENDED KALMANFILTER θr
The Kalman filter is often applied during (3.9)
dissolving state estimation of dynamical system, where,
disturbed by the known signals. Kalman filter ∆ia
vk =
algorithm is used for estimating the parameters of ∆ib
linear system, but the PMSM model is non-linear, so (3.10)
we cannot use that filter in this case. Extended where the vectors wk and vk are called the
Kalman Filter is generalized algorithm, which can be process and measurement noises respectively[12].
used for non-linear systems The Kalman filter solution does not apply unless
The EKF is an optimal estimator in the least certain assumptions about the noise that affects the
square sense for the estimation of nonlinear dynamic system under study must be satisfied[13]:
systems. It is derived from the Kalman filter based It is firstly to assume that the average
on successive linearization of the signal process and 1. It is firstly to assume that the average value of
observation map [8]. both wk and vk are zero.
For this application the motor nonlinear state 2. One has to further assume that no correlation
“equations (2.1)” are expressed in the discretized exists between wk and vk . That is, at any time k,
form wk and vk are independent random variables.
xk+1 = Ak xk + Bk uk + Mk TL Then the noise covariance matrices Sw and Sv
(3.1) are defined as:
yk = Ck xk Process noise covarianc[6]:
T
(3.2) Sw = E wk wk
Ak and Bk are the discretized system and (3.11)
input matrices, respectively. They are [9,2,10] Measurement noise covariance:
AT 2 Sv = E vk vk T
Ak = eAT = I + AT + +⋯
2! (3.12)
≅ I+AT where wT and vT indicate the transpose of w
(3.3) and v random noise vectors, and E(.) means the
T
expected value.
Bk = eAζ Bdζ = [eAT − I]A−1 B Substituting “equation (3.8) ”into“ equation
0
ABT 2 (3.11)” and “equation (3.10)” into “equation
= BT + 2!
+ ⋯ ≅ BT (3.12)”,[14] one can get the following process and
(3.4) measurement noise covariance matrices:
Mk ≅ MT
(3.5) Sw
Ck = C T 2 T 2 T 2
(3.6) ∆ua 2 ∆ua ∆ub − ∆ua ∆TL 0
where T is the sampling time and I is an L L JL
identity (4X4) matrix. T 2 T 2 T 2
= ∆ua ∆ub ∆ub 2 − ∆ub ∆TL 0
If the noises ∆ua , ∆ub have corrupted the L L JL
inputs ua and ub respectively, and the noise ∆α has T 2 T 2 T 2
been admitted to account for uncertainties in the load − ∆ua ∆TL − ∆ub ∆TL ∆TL 2 0
JL JL J
torque[11], then a noise vector will arise in
0 0 0 0
“equation.(3.1)”. (3.13)
u + ∆ua
xk+1 = Ak xk + Bk a + Mk TL + ∆TL
ub + ∆ub k
1189 | P a g e](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)