Robust Position And Speed Estimation Algorithm For Permanent Magnet Synchronous Drives

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Robust Position And Speed Estimation Algorithm For Permanent Magnet Synchronous Drives

Fateh Benchabane, Abdenacer Titaouine, Ouafae Bennis, Abderazak Guettaf, Khaled Yahia, Djamel Taibi

To cite this version:

Fateh Benchabane, Abdenacer Titaouine, Ouafae Bennis, Abderazak Guettaf, Khaled Yahia, et al..

Robust Position And Speed Estimation Algorithm For Permanent Magnet Synchronous Drives. Eu-

ropean Journal of Scientific Research, EuroJournals, 2011, 57 (1), pp.PP 6-14. �hal-00657627�

ISSN 1450-216X Vol.57 No.1 (2011), pp.6-14

© EuroJournals Publishing, Inc. 2011 http://www.eurojournals.com/ejsr.htm

Robust Position and Speed Estimation Algorithms for Permanent Magnet Synchronous Drives

F. Benchabane

MSE Laboratory, University of Biskra B.P.145, 07000, Biskra, Algeria E-mail: fateh_benchabane @ yahoo.fr

A. Titaouine

MSE Laboratory, University of Biskra B.P.145, 07000, Biskra, Algeria

O. Bennis

PRISME Institute, University of Orléans 21 rue Loigny La Bataille, 28000 Chartres, France

A. Guettaf

GEB Laboratory, University of Biskra B.P.145, 07000, Biskra, Algeria

K. Yahia

GEB Laboratory, University of Biskra B.P.145, 07000, Biskra, Algeria

D. Taibi

MSE Laboratory, University of Biskras B.P.145, 07000, Biskra, Algeria

Abstract

This paper deals with the design of sliding mode controllers (SMC) for a vector- controlled of permanent magnet synchronous motor (PMSM). The advantages of the sliding mode controller are disturbance rejection, strong robustness and simple implementation. Speed and torque control of permanent magnet synchronous motors are usually attained by the application of position and speed sensors. However, speed and position sensors require the additional mounting space, reduce the reliability in harsh environments and increase the cost of motor. Many studies have been performed for the elimination of speed and position sensors. Our proposed control strategy is based on an accurate Extended Kalman Filter (EKF) observer which estimates speed, position, rotor – fixed current i ,i

and load torque.

The investigations show that the EKF is capable of tracking the actual rotor speed,

position and load torque provided that the elements of the covariance matrices are properly

selected. Moreover, the performance of the EKF is satisfactory even in the presence of

noise or when there are variations in permanent magnet synchronous motor load.

Robust Position and Speed Estimation Algorithms for Permanent Magnet Synchronous Drives 7 Keywords: PMSM, EKF, SMC, SSMC, Speed, Position, Load torque.

1. Introduction

The permanent magnet synchronous motor offer many advantages over induction motor, such as overall efficiency, effective use of reluctance torque, smaller losses and compact motor size [1]. The main advantage of the sliding mode technique is that the controlled system presents robustness to parameters variations and rejection to external disturbances. The sliding mode control of a permanent magnet synchronous motor is usually implemented through measuring the speed and position, .however speed and position sensors require the additional mounting space, reduce the reliability in harsh environments and increase the cost of motor [1, 2]. Various control algorithms have been proposed for the elimination of speed and position sensors: estimators using state equations, artificial intelligence, direct control of torque and flux, Model Reference Adaptive System (MRAS) and so on.

This paper proposes the SMC strategy based on the EKF in the sensorless control of a PMSM.

Kalman filter is a special kind of observer which provides optimal filtering of the noises in measurement and inside the system if the covariances of these noises are known [3, 4, 5]. If rotor speed and rotor position (as extended states) are included in the dynamic model of PMSM, the extended Kalman filter EKF can be used to relinearize the nonlinear state model for each new state estimate as it becomes available. Consequently, the EKF is considered to be the best solution for the speed and position estimation.

2. Mathematical Model of PMSM

The Kalman filter is based on a model of the permanent magnet synchronous motor PMSM in discrete time state space [2, 3]. The dynamic model for the PMSM in rotor reference frame, choosing rotor fixed current i

i

, the angular velocity Ω , the rotor position θ and load torque T

as state variables, the fundamental voltages u

, u

as inputs, is described by equations (1) and (2).

k 1 k k k k

x

A(x ).x

B(x ).u (1)

Where:

s q

s s

d d

d s f

s s s

q q q

k d q f

s qs s s s

s

R L

1 T T 0 0 0

L L

L R

T 1 T T 0 0

L L L

A(x ) L L f 1

pT i pT 1 T 0 T

J J J J

0 0 T 1 0

0 0 0 0 1

− ω

−ω − − φ

= − φ

− −

s d

s q k

T 1 0

L 0 T 1

L B(x )

0 0

0 0

0 0

=

( )

k ds qs l k

k q k

x i i T , u u

u

= Ω θ =

With: R

= stator résistance, L , L

= d,q axis inductance,

= permanent magnet flux linkage. T

=sampling time.,

p .

The resulting output vector y

consists of the estimated motor current in a rotor reference frame being compared to the measured current. The difference is used to correct the state vector of the system model.

ds

k k

qs

i 1 0 0 0 0

y x

i 0 1 0 0 0

= =

(2)

3. Design of Sliding Mode Controller

The sliding mode control can be justified and designed using the notion of lyapunov stability.

Whatever the application, the design of sliding mode control can summarized in 3 steps:

a. The choice of the number of the sliding surfaces: generally the number of the sliding surfaces is equal to the dimension of the input control vector.

b. The choice of the sliding surface equation form: it must satisfy the convergence of the control and the stability of the system. This goal can be reached if the control variable u

permits to satisfy the Lyapunov function:

s.s 0

•

<

Based on this condition slotine [2] propose a general form of sliding surface:

r 1

x ref

s(x) ( ) (x x)

dt

∂ −

= + λ −

(3)

c. The control law design: the control variable is decomposed in two parts: U

and U

:

c equ n

U

U

U (4)

The dynamic while in sliding mode can be written as:

s 0

•

=

(5)

By solving this equation ( s 0

•

= ), the equivalent control U

can be obtained. The U

component satisfies s.s 0

•

<

and is given by:

U

K.sign(s(x)) (6)

The common used form of U

is a constant relay control:

Figure 1: Relay function of

U

S(x) U

Nevertheless, this solution introduces a chattering. In order to eliminate or to reduce the

chattering which is due to the presence of finite time delay for control computation and the limitation

Robust Position and Speed Estimation Algorithms for Permanent Magnet Synchronous Drives 9 of physical actuators[6], we have chosen to smooth the term u

by an appropriate selection of sign function [6] indicated in equation (7).

n

KS(x ) si S(x )

u K s i gn(S(x )) si S(x ) 0

< ε ε

= > ε

ε >

(7)

4. Selection of Switching Surfaces and Determination of the Control Inputs

Surfaces are chosen in order to determine the behaviour of the motor in the transient period. For the speed control, we propose a switching law which depends on the difference between reference speed and real speed, presented in (8):

S( )

(8)

The derivative of this surface is given by the expression:

r ref

1 2 ds 3 qs

S( ) c C (c I c )I

J

• •

Ω = − Ω + + Ω − +

(9)

The associated control input is given by (12):

r ref

1 qsref

2 ds 3

c C K SingS( )

I J

(c I c )

•

− Ω + + Ω + Ω Ω

= +

(10)

The components i

and i

are independently controlled as described by (12) and (14):

ds dsref ds

S(I )

I

I , I

0 (11)

Frequently I

is made equal to zero, because its contribution to the motor torque is almost insignificant [6, 7]. Flux and torque control are independently made through the surfaces S(I )

and

S(I )

respectively.

The derivative of the surface S(I )

is given by the expression:

ds dsref 1 ds 2 qs d

S(I ) I a I a I 1 u

Ld

•

= − − Ω −

(12)

The associated control input is given by (15):

dsref 1 ds 2 qs d ds

dref

[I a I a I ] k signS(I )

u Ld

•

− − Ω +

=

(13)

and:

qs qsref qs

S(I )

I

I (14)

The derivative of this surface is given by the expression:

qs qsref 1 qs 2 ds 3 q

S(I ) I b I b I b 1 u

Lq

•

= − − Ω − Ω −

(15)

The associated control input is given by (18):

qsref 1 qs 2 ds 3 q qs

qref

[I b I b I b k signS(I )

u Ld

•

− − Ω + Ω +

=

(16)

With:

q

1 2 1 2

d d q q

R pL R pLd

a ,a , b , b ,

L L L L

− − −

= = = =

d q

f r f

3 1 2 3

q

p(L L )

p f p

b , c , c ,c

L J J J

− φ − − φ

= = = =

The suitable choice of the parameters k k

and k

[7, 8]:

• Ensures the rapidity of the reaching mode,

• Imposes the dynamic of the convergence and sliding mode,

• Allow to the drive to work with maximum energy during transient state.

Hence, k , k

and k

are positives gains, given as followed:

ds qs

ds qs

ds qs

d ds q qs

I ,I ,

q qs ds ds f

I ,I ,

r I ,I ,

f

K max RI L I

K max RI L I

C f

K max

p

ω

ω

ω ω

< − − ω

< − − ω + ωφ

< − − ω

φ

(17)

The configuration of SMC structure control is presented in fig.2.

Figure 2: Configuration of the cascade structure control

udref

uqref

ω

dsref

0

i =

S(ω) S(i

)

E

P

P

P PWM

Inverter

PMSM

GT

+ - + -

+ -

θ θ

Iqs

Ids

5. Design of EKF Observer

Accurate and robust estimation of motor variables which are not measured is crucial for high performance sensor less drives. A multitude observers have been proposed, but only a few are able to sustain persistent and accurate wide speed range sensor less operation. At very low speed, their performances are poor. One of the raisons is high sensitivity of the observers to unmodeled nonlinearities, disturbance and model parameters detuning.

The Kalman filter provides a solution that directly cares for the effects of disturbance noises including system and measurement noises. The errors in parameters will normally also handled as noise.

The dynamic state model for non linear stochastic machine is as follows where all symbols in the formulations denote matrices or vectors [9, 11]:

x(t) f (x(t), u(t), t) w(t) y(t) h(x(t), t) v(t)

•

= +

= +

(18) w(t) : System noise vector.

v(t) : Measurement noise vector

w, v : Are unrelated and zero mean stochastic processes.

Robust Position and Speed Estimation Algorithms for Permanent Magnet Synchronous Drives 11 A recursive algorithm is presented for the discrete time case. For the given sampling time T

, both the optimal estimate sequence x

and its covariance matrix P

generated by the filter go through a tow step loop.

The first step (prediction) performs a prediction of both quantities based on the previous estimates x

and the mean voltage vector actually applied to the system in the period from T

to

Tk

.F is system gradient matrix (Jacobean matrix).

T

x (t ) x(t )

f (x(t), u(t), t) F(x(t), t)

x (t)

= ∂

∂ ⁽¹⁹⁾

k / k 1 k 1/ k 1 s k 1/ k 1 k 1

x ₋ = x _{− −} + T .f (x _{− −} , u ₋ ) ₍₂₀₎

T

k / k 1 k / k 1 k 1/ k 1 k 1/ k 1 s

P ₋ = P ₋ + (FP _{− −} + P _{− −} F ).T + Q (21) The second step (innovation) corrects the predicted state estimate and its covariance matrix trough a feedback correction scheme that makes use of the actual measured quantities; this is realized by the following recursive relations:

k / k k / k 1 k k k / k 1

x = x ₋ + K (Y − Hx ₋ ) ₍₂₂₎

k / k k / k 1 k k / k 1

P = P ₋ − K HP _{− (23)}

Where the filter gain matrix is defined by:

T T 1

k k / k 1 k / k 1

k

P

H (HP

H

R)

(24)

H is transformation matrix.

x (t) x (t)

H(x(t), t) h

x

= ∂

∂ ⁽²⁵⁾

The proposed EKF observer is designed in rotor reference frame ( d,q frame).

State vector is chosen to be:

T

ds qs l

X

[i i

T ] Input:

T

d q

U

[u u ] ; And output:

ds qs

Y

[i i ]

ds qs

i ,i and u , u ,T

are motor stator currents, voltages in rotor reference frame and load torque.

The critical step in the EKF is the search for the best covariance matrices Q and R have to be set-up based on the stochastic properties of the corresponding noise. The noise covariance R accounts for the measurement noise introduced by the current sensors and quantization errors of the A/D converters [9, 10]. Increasing R indicates stronger disturbance of the current. The noise is weighted less by the filter, causing also a slower transient performance of system.

The noise covariance Q reflects the system model inaccuracy, the errors of the parameters and the noise introduced by the voltage estimation [12, 13]. Q has to be increased at stronger noise driving the system, entailing a more heavily weighting of the measured current and a faster transient performance.

An initial matrix P

represents the matrix of the covariance in knowledge of the initial condition. Varying P

affects neither the transient performance nor the steady state condition of the system. In this study, the value of these elements is tuned “manually”, by running several simulations.

This is maybe one of the major drawbacks of the kalman filter.

Fig.3. Show the proposed Sensor less SMC using EKF. In this study, the outputs of a PWM voltage source inverter are used as the control inputs for the EKF. These signals contain components at high frequencies, which are used as the required noise by the kalman filter. Thus, no additional external signals are then needed

Figure 3: Speed control of a PMSM using the SMC with an EKF for speed and position estimation

dsref 0

i =

Iqs

Ids

ωref

P⁻1

S(i_ds)

S(ω)

+

+

- _PWM

Inverter

+ -

uqref

udref

PMSM

EKF +

ω

θ

P ^a I

Ib Ids

Iqs

θ

Tl

Cr

6. Simulation Results

Extensive simulations have been performed using Matlab/ Simulink Software to examine control algorithm of the SSMC applied for PMSM presented in fig.4 and based on parameters of 3Kw motor (Appendix).

Fig.4 shows the responses of speed, position, currents

and load torque with errors between the actual and estimated states for step reference with 75% load at t=0.1. speed reversion around (+100rad/s: -100rad/s) at 0.2s and the comportement at very low speed at 0.4s.

These responses illustrate high performance of the proposed EKF observer during transients and steady state.

However, high accuracy and strong robustness of the SSMC when speed reversion around (+100rad/s: -100rad/s) are applied and it can be noticed that the proposed EKF works in very low speed region, where many speed estimators or observers have poor performances.

7. Conclusion

In this study, a sensorless sliding mode control system has been designed; this proposed SSMC has the advantage of less torque ripple in steady state with selection of sign function presented in equation (7).

The EKF observer is able to increase the performance of the SSMC in terms of low speed behaviour, dynamical and statical comportment, speed reversion and load rejection.

References

[1] G. Sturtzer, E. Smigiel, Modélisation et commande des moteurs triphasés, commande vectorielle des moteurs synchrones, commande numérique par contrôleurs DSP, Edition Ellipses (2000).

[2] A. Titaouine, A. Moussi, Sensorless nonlinear control of permanent magnet synchronous motor

using the Extended Kalman Filter, Asian Journal of Information Technology, Vol. 5 (2006)

1416-1422.

Robust Position and Speed Estimation Algorithms for Permanent Magnet Synchronous Drives 13 [3] F. Benchabnae, A. Titaouine, Sensorless Control Strategy For Permanent Magnet Synchronous Motor Fed By AC/DC/AC Converter, IEEE International Conference on Electrical Machines , Italy, (2010). ICEM.

[4] M.T.Benchouia, S.E. Zouzou, A. Golea, A. Ghamri, A.Modeling and simulation of variable speed system with adaptive fuzzy controller application to PMSM, IEEE International Conference on Industrial Technology ICIT, Vol. 2 (2004) 683-687.

[5] S. Sayeef, M.F. Rahman, Comparison of proportional + integral control and variable structure control of interior permanent magnet synchronous motor drives, 38th IEEE Power Electronics Specialists Conference, Orlando, USA, (2007) 1645-1650.

[6] L.R. You, Z.Y. Ma, X.H. Wang, H. Zhangl, Sliding mode control for permanent magnet synchronous motor based on a double closed-loop decoupling method, Proceedings of the Fourth International Conference on Machine Learning and Cybernetics, Guangzhou, 18-21 August (2005) 1291-1296.

[7] G.A. Capolino, A. Goléa, H. Hénao, Modélisation et simulation d'un asservissement à vitesse variable avec mode glissant, Journées d’études Avertissement Electromécaniques Rapides, Modélisation et Régulation Avancées, Metz (1992).

[8] Y.Y. He, W. Jiang, A new variable structure controller for direct torque controlled interior permanent magnet synchronous motor drive, Proceedings of the IEEE International Conference on Automation and Logistics, (2007) 2349-2354.

[9] M.S. Sayed Ahmed, Z. Ping, Y.J. Wu, Modified sliding mode controller with extended Kalman filter for stochastic systems, IEEE International Conference on Control and Automation, Guangzhou, (2007) 630-635.

[10] D. Janiszewski, Extended Kalman Filter based speed sensorless PMSM control with load reconstruction, The 23nd Annual Conference of the IEEE Industrial Electronics Society, (2006) 1465-1468.

[11] A. Titaouine, F. Benchabane, Application of Ac/Dc/Ac converter for sensorless nonlinear control of permanent magnet synchronous motor, IEEE Internatinal Conference on Systems, Man and Cybernetics, Turkey, (2010) 2282-2287.

[12] A. Titaouine, A. Moussi, Sensorless nonlinear control of permanent Magnet synchronous Motor using the extended kalman filtre, Asian Journal of Information Technology, Vol. 5 (2006)1416-1422.

[13] F. Benchabane, A. Titaouine, Systematic fuzzy sliding mode approach combined with extended Kalman filter for permanent magnet synchronous motor control", IEEE Internatinal Conference on Systems, Man and Cybernetics, Turkey (2010). 2169-2174.

Appendix

Motor parameters

P

=3Kw, C

=8.5Nm, I

=20A L

=1.4mH L

=2.8mH, φ

=0.12wb, p =4, J =1.110

kgm

f =1.410

3

Nm/rad

R =0.6 Ω .

Figure 4: Simulation results of the sensor less sliding mode control with extended Kalman Filter.

0 0.1 0.2 0.3 0.4 0.5

-100 -50 0 50 100

0 0.1 0.2 0.3 0.4 0.5

0 5 10 13

0 0.1 0.2 0.3 0.4 0.5

0 10 20

0 0.1 0.2 0.3 0.4 0.5

0 0.01 0.02

0 0.1 0.2 0.3 0.4 0.5

-5 0 5

0 0.1 0.2 0.3 0.4 0.5

0 5 10

0 0.1 0.2 0.3 0.4 0.5

-35 -20 0 20

0 0.1 0.2 0.3 0.4 0.5

0 0.5 1 1.5

0 0.1 0.2 0.3 0.4 0.5

-5 0 5

0 0.1 0.2 0.3 0.4 0.5

0 0.5 1 1.2

Real speed Estimated speed

Real position Estimated position

Imposed load torque Estimated load torque (r d / s)

Ω Error (rd / s)

θ(rd) Error (rd)

C (Nm)r Error (Nm)

I (A)q Error (A)

I (A)d Error (A)

t(s) t(s)

t(s) t(s)

t(s) t(s)

t(s) t(s)

t(s) t(s)

0.08 0.1 0.12 0.14 0.16 90

95 100 105 110 115

Real Iqs

E stimated I _qs

Real Ids

Estimated I _ds