2013 Eighth International Conference and Exhibition on Ecological Vehicles and Renewable Energies (EVER)

Speed Sensorless Flatness-Based Control of PMSM Using a Second Order Sliding Mode Observer

A FEZZANI, S. DRID and A. MAKOUF Laboratory, LSP-IE, University of Batna

Rue Cahid Med El-Hadi Boukhlof Batna 05000, Algeria amorfezzani@yahoo.fr, s _ drid@yahoo.fr, a _ makouf@yahoo.fr

Abstract- This paper deals with speed sensorless flatness

based control of permanent magnet synchronous motor (PM SM)

using a second order sliding mode observer. In the first part, the

flatness control is associated with PI controller, this solution

shows good robustness with respect to parameter variations and

guarantees torque and speed tracking. In the second part, the

high order sliding mode speed observer is used to overcome the

occurring chattering phenomena. The super twisting algorithm is

modified in order to design a speed and position observer for

PMSM. The simulation results confirm the effectiveness and the

good performance of the proposed control.

Keywords-flatness; nonlinear; permanents magnet

synchronous motor (PMSM); parameter variations; second-order

sliding mode.

I. INTRODUCTION

The permanent magnet synchronous machine becomes most popular servo motor due its high power density, large torque to inertia ratio and high efficiency. However, the PM SM model is nonlinear coupled and is subjected to parameter variations with temperature and saturation. Classical PI controller is a simple method used to control PMSM drives. But the main drawbacks of PI controller are the sensitivity of performances to the system parameter variations and inadequate rejection of external disturbances and load change. Some nonlinear control methods have been applied to control PM SM considering the nonlinear PMSM dynamics, such as the back-stepping control [1-4].

Recently, robust optimization methods for nonlinear differentially tlat systems have attracted many researchers. For instance; when the drive requirements include fast dynamic response and accurate speed or torque control, the motor may necessarily operate in closed loop mode with speed feedback. In this case, the tlatness based control (FBC) technique is used with the motor speed estimator. The flatness property of the system can be showed easily. The rotor position can by obtained by a shaft-mounted encoder or by a resolver. These components cause several disadvantages as the drive cost, encumbrance, reliability, and noise immunity. Tacking into ac count these problems, sensorless control of the PMSM has received more attention [4].

978-1-4673-5271-0/13/$31.00 ©2013 IEEE

L. CHRIFI ALAOUI Laboratoire des technologies innovantes (L. T.l), University of

Picardie, Jules Verne, Departement GEll, Cuffies, France larbi.alaoui@u-picardie.fr

In [5], the rotor tlux quantities can be determined analytically via machine terminal voltages integration. However, in this scheme, the poor precision and offset problem are the major drawbacks. The extended KaIman filter (EKF) is proposed for the rotor speed and position estimation by [7] and [8]. The approach uses a linear model of the system around operating points. The EKF is weIl known but the convergence of the estimated speed and position is difficult to guarantee. In [9], a nonlinear full state observer is proposed for the rotor speed and position estimation.

In this work sliding mode ob servers are used for estimation the rotor speed and position. These ob servers are designed and used in [11], [12], [32] and [34]. They are widely used due to their attractive features [41]:

• insensitivity (more than robustness) with respect to unknown inputs;

• possibilities to use the values of the equivalent output injection for the unknown inputs identification [34];

• finite time convergence to the reduced order manifold.

The robust exact differentiators were designed for ensuring finite time convergence to the real values of derivatives [29], [31], as an application of super-twisting algorithm [30]. New generation of ob servers based on the high order sliding mode differentiators are recently developed in [15], [25], [27] and [28]. Those observers:

• provide a convergence to the exact values of states variables;

• allow the exact identification the unknown inputs without fi Itrati on;

The sliding mode ob servers are widely used due to the finite-time convergence, robustness with respect to uncertainties and the possibility of uncertainty estimation [10], [11] and [12]. These observers require the proof of a separation principle theorem due to the asymptotic convergence of the estimated values to the real ones. In this paper, the simplified robust algorithm known super twisting algorithms have been implemented for estimation the rotor speed of the PMSM. The structure of the implemented sensorless control is based on the second-order sliding mode ob server using only the measurement of motor currents for on-line estimation of rotor speed [16]. In order to validate the proposed design approach, simulation investigation is performed.

11. PM SM MODE L

The stator equations of the PMSM in the rotor reference frame are [22].

Where the tlux expression are given by

<Pd=LdId+<Pf <Pq=Lqlq

(1)

Considering 1 d and 1 q are stats variables (1) can be written as

The electromagnetic torque is given by

and the associated equation of motion is

Form (2), (3) and (4), the state model is rewritten as:

where

and

x = Ir(x)+ gd (x)vd + gq(x)Vq Y = h(x)

gi+[':] g" (') � [�"l

(2)

(3)

(4)

(5)

where Vd and vq are the stator voltages, Id and Iq are the stator currents, <Pd and <Pq are flux linkages, <Pf is the magnetic flux linkage, p is the number of poles pairs, Tr is the load torque, Te is the electromagnetic torque, Jm is the moment of inertia, f,n is the viscous friction coefficient and Q is the rotor speed.

111. F LA TNESS-BASED CONTRO L

A. Short Review 01 FEC 1) Definition 1: differentially flat system:

x = I (x, t) is a system with has states x E Rn, and input

U E R m , the system is said to be differentially tlat if there

exist outputs yE Rm of the form y = y(x, u, ü ..... u(p) ) such that,

x = x& , y, ji. .... y(q) ) and u = u& , y,ji. .... y(q) ) . [17]

A system is tlat if conditions are satisfied

• Finite set Y= (Yl,Y2 ..... Y"' ) of variables witch are

differentially independent.

• Components Yi are differential functions of the variables

system (state, input) and their derivatives.

• Any system variable is a differential function of the

Yi component.

ThereY= (Yl'Y2 . ... Ym ) is a tlat or linearizing output. Its

components number is equals to the number of independent input channels.

B. Controller Design 1) Flatness Model: The design of a feed forward irIput is

based on a planned motion of the flat outputs by simply combining values of the flat outputs and their derivatives. Therefore, without perturbation, nominal control is sufficient to move the system form one state to another, once a compatible trajectory with the irIitial and fmal positions which has been supposed. The PMS motor model is a flat system, SirIce all state variables and inputs can be parameterized in terms of so-called flat outputs and a fmite number of their successive time derivatives [16] and [17].

The outputs to be controlled are as followirIg:

(6)

The states variables of the PMS motor can be parameterized in terms of the speed, direct current and time derivatives.

Equation (1) can also be written as:

with

and

Xl = Yl X3 = Y2 eh -a33Y2 -a34T;· ) X2 =

.. .

(a31YI + a32)2 (Y2 -a33Y2 -a34Tr )

(a31YI + a32?

where, using (8), (9) and (7), we have:

compactly we can write

(7)

(8)

(9)

(10)

Relation (10) can be used for feed forward and feedback tracking controller design.

Practically, only the stator currents and the rotor speed are available for control purposes. To estimate the load torque we use:

(11)

The method suggested by le Pioufle permits to estimate in real time the load torque [24]. The scheme of the proposed estimator load torque is shown in Fig. 1.

PI Fig. 1. Bloc diagram of torque estimator.

The error between measured speed and estimated speed is presented as an input of regulator PI. The constants of the PI are determined by the poles placement method.

The load torque estimation is quite similar to the signal delivered by the output of the load torque sensor. In fact a major drawback of the FBC is that it requires exact knowledge of the motor parameters and any variation in the parameters or the load torque will reduce the controller performance. In order to overcome this problem a feedback stabilization control is proposed.

20

I 15

� 2" 10

� iil 5

.s 13 0

" E -5 .� "

l' -10 "

1l @ -15

cE � -20

Ir (

I

I I I I I I

:\ o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Time (s) Fig. 2. Load torque estimation.

2) Control Structure:

2

The Fig.3 presents the flatness-based control structure with the two-degree-of-freedom synthesis structure. This structure includes a feed forward part (nominal control) and a feedback controller (PI) and is potentially more powerful to achieve strong performance requirements than the usual unity feedback structure.

To stabilize the reference trajectory, additional Feedback controller can be realized using flat feedback [18], [38].

with

where : kp> 0 and k{ > O.

u : Input applied to the system.

Un: Reference control (nominal control).

Öu: Feedback stabilizing contro\.

PI controller has been used in order to stabilize the output system around a desired trajectory.

Reference Generation

Fig. 3. Structure of Flatness-based Contro!

y

C. Reference Trajectory Generation

Nonlinear system

y

1) Re{erence control (nominal contro!): The theoretical study of (FBC) has been discussed above, the flat outputs are defmed as

The states and the input control trajectories in open-Ioop (reference control or nominal) can be rewritten as a function of the flat outputs and their derivatives. A feedforward tracking references are obtained by inserting the references trajectories XI ref, X2re( and X3 ref, into the differential parameterization eq.(10).

X3_rel = Y2r X2 ref Y2r - a33Y2r - a34Tr

than eq.(lO) is rewritten as:

- � I' �2r -a33Y2r -a34Tr )]

- ""d Ylr -al l Ylr -al2Ylr a31Ylr + a32 = A hr(al lYlr -a33Y2r -a34TrXa3lYlr +a32)

q a31Ylr + a32 a31Ylt° �2r -a33Y2r -a34 Tl' )

a31Ylt° + a32 _ [(Y2r -a33Y2r -a34Tr) a22YlrY2r -a23Y2r ll a31Ylr + a32

(13)

2) Reference Trajectory: a) Speed: The first objective is to bring the motor speed

form 121' (t;) = 0 to 121' VI ) = 121' In order to have smooth

dynamics without discontinuity and torque ripple, the following additional constraints are chosen:

Thus, the reference trajectory can be chosen as a fifth order polynomial equation [39]:

with A = (t - t; ) t = Os I t t ' I - f

b) Direct current: The second objective is to control direct current. From (3) it appears that the motor torque is a function of x2 . In order to obtain maximal torque, it is suitable

to choose a direct current reference trajectory such as Ylr=/cFO. The speed reference trajectory is given in Fig. 4.

200 F.(����������������i������---------- �-------- -150 - - - - - - - - _1 - _______ _ 1

1 100 - - - - - - - - -1--------- ---------,-------- -

1

50 - --------: - - - - - - - - - - - - - - - - - - � ---------

J 0 1

- - - - - - - - -1- -------- ---------,-------- -1 1 1

-50 - --------: - - - - - - - - - i - - - - - - - - - � -------- -

-100

-150

1 1 1 ---------,---------, ---------, -------- -1 1 1 ---------:- - - - - - - - - � - - - - - - - - � -------- -

-200�����������'L'\���������� o 0.5 Tim� (s) 1.5 2

Fig. 4. Speed reference trajectory.

In this context the inputs command are given in Fig. 5.

350,----------�1--------�----------�1---------, 1 1

300 - - - - - - - - -1-- - - - - - - - - - - - - - - - - t- - ------- -1 1

250 - - - - - - - - -1--------- ---------,-------- -1

200 - - - - - - - - -1- _______ _ _________ L ________ _ �

1 1

"0 150 - - - - - - - - -1--------- ---------,-------- -1 --------- �-------- -f!'. - - - - - - - - _1 - _______ _ 100 1 1 1

50 - - - - - - - - -1- - - - - - - - - - - - - - - - - - +- - ------- -

o � :� -50 �--------�----------�--------�--------�

o O� 1 Time (s) 1.5

200 - - - - - - - - - - - - - - - - � - - - - - - - - - 1- ---------

100 If 1 ---------,---------

� 1 0" 0 - --------,--------- ---------,---------

f!'. 1

-100 - - - - - - - - -1--------- \- - - - - - - - � ---------

1 -200 - - - - - - - - -1- - - - - - - - - - - - - - - - - - L - - - - - - - --

1

o 0.5 1 Time (s) 1.5

Fig. 5. Reference trajectories of inputs control (nominal control).

ÜBSERVER DESIGN

A. Theory Part

2

2

The observer proposed is based on a second-order sliding mode approach_ Tt is robust versus to parametric variations, modeling errors and disturbances. The sensorless control structure used is shown in Fig. 6.

r--------------

I I

I .---

•

Reference • Contro' • • n'·'f : - I • Reference : '-"""'!"-: trajectory I 1: I I I

- I

l�'d

Super Twisting Speed aod position Observer & Torque

estimation Flatness control I ----..

.. - - - - - - - - - - - - - _I

Fig. 6. Block diagram of the FBC speed control with second-order sliding mode observer (Super Twisting algorithm observer)

This technique consists on the use of the available measurements in order to reconstruct the no measured state variables. In this part the observer is based on the so-called super twisting algorithm presented in [15] as folIows:

X4=X3+Zj �3 = !o(t,X4,X3,U)+Z2

(14)

Where x3 and x4 are the states estimations, 10 is a

nonlinear function, U = U (t,X4,X3) is the control input (may

be computed in function of the system states or their estimates),

z, and Z2 are the correction factors based on the super twisting

algorithm having the following forms:

Zl = Alx3 -x311!2 sign(x3 -x3) Z2 = asign(x3 -x3)

We consider initially that .\:4 = x4 and.\:3 = o.

(15)

Taking into account e3 = x3 -.\:3 and e4 = X4 -x4 we

deduce the following error equations

e4 = e3 -Alel2 sign(eJ e3 = F(t,x4 ,x3 ,X3)-Cl.sign(e3)

(16)

Where

Suppose that the system states can be assumed bounded as it is shown by following inequality

Holds for any possiblet,x3,x4 andsup lx31 ::; 2sup lx31 , and the control input u = U(t, X4' X3) is bounded, c; is the

uncertainties.

Let CI. and A satisty the inequalities.

(17)

Where P is some chosen constant 0 < P < 1 as it is

described in [16], it is sufficient to choose CI. = I. If+ andA. = 1.5(r )12.

B. Speed Observer In this section, the sliding mode observer is proposed to

estimate speed and error speed.

Using equation (3) and (4) we obtain compactly the following form:

X4 = X3 x3 = !c)(t,X4,X3,U)+ ;;(t,X4,X3,U)

Where x4 and x3 are respectively e ,Q ,u is the torque, ; is

the uncertainties.

The super twisting second order sliding mode ob server is designed as follows:

(19)

Where 8 and Q are the states estimations and the

correction variables

form:

z\ and Z2 are output injections of the

Z\ = Aln - 6112 sign(n - 6) Zz = asign(n - 6)

, , We consider initially that 8 = 8 and Q = O . Taking into

account es = 8 - 8 and eQ = Q - n we obtain the following

error equations

(20)

We assume that:

Holds for any possible t, 8, Q and sup ln l :s; 2 suplQI. Let a and A satisfy the inequalities cited above in theory

part section in equation (17). The use of this super twisting

algorithm ensures the fmite time convergence of , ,

state 8 � 8 , Q � Q and observation error to zero.

V. SIMU LATION

In order to validate our approach, simulation tests were carried out using the proposed control PMSM scheme. The parameters of the PMSM used in the simulation are given as: rated voltage V=511 V, number of poles p=3, armature resistance R,=lAQ, Stator inductances Ld= 0.0066H, Lq=0.0058H, viscous damping fm=0.00038N.m.s/rad, moment of inertia Jm=O.OO 16 kg.m2, rotor flux <Df =0.1546 Wb, rated torque Tn =10 N.m.

The typical step references of the speed and load torque are given in Fig.7a and Fig.7b, respectively. All with Td reference current elose to zero.

a) 200F.(��������������������­

, 150 - --------,--------- ---------r---------, ,

� 100 - - - - - - - - -1- - ------- ---------r---------

� , , , , al 50 ---------: - - - - - - - - - � - - - - - - - - - � - --------

Q) CL

(fJ o - --------:- - - - - - - - - � �----+ - --------

, -50 - ------- - - - - - - - - � - - - - - - - - -

I - - - - - - - - -

:\ -100����������������======� o 0.5 1

Time (s) 1.5

10�����========�========�����-b) I ! I

E 6

5 r------.I- -,- -------- , --------- ---------

� 0 - - - - - - - - -1- - - - - - - - - � - -------- ---------

f! �

-5 ---------,--------- , --------- ---------

, -10 �

2

0 0.5 1.5 2 Time (s)

Fig. 7. The typical step references

Fig. 8 shows the simulation result in terms of estimate speed. Additionally, the real speed is compared with estimate one. lt can be seen that very good performances are obtained.

To test the low speed tracking, we applied a sinusoidal speed reference of magnitude 10 radis with frequency 5 Hz. The comparison between the real speed and estimation one shown by Fig.9a demonstrate the effectiveness of the method. Also, to track a reference torque, we applied a sinusoidal torque reference of magnitude 10 Nm with rrequency 5 Hz.

� al Ql Cl.

(fJ

150

100

50

o

-50

I········· Re!

_ _ _ _ _ _ _. __ .- Real speed - Estimed speed

1 - - - - - - - - -1- - - - - - - - - - - - - - - - - -1 - ------- -

1 1 1 - - - - - - - - -1- - - - - - - - - --t - - - - - - - - - t- - --------

1 1 1

' \. - - - - - - - - -1- ________ � � _____ -+ ________ _

_________ 1 _________ J _________ ________ _

:\ -100 �������������������� o 0.5 1 Time (s) 1.5 2

20 r---------,----------r---------,�======� b) _ L I 1 J = ��:�ue

1 O .\.- - - - - - -Ir---+-.---I �---"i" Ql :J

o - - - - - - - - -------- -

f! -10 - - - - - - - - -1- - ------- --------- .�------� �

-20 - - - - - - - - _1 - _______ _ -------- -

-30 - --------,--------- , --------- -------- -

_40L-----------L-----------�----------�----------� o 0.5 1 1.5 2 Time (s)

10 .-----------,------------,------------.------------, c)

1 I I 5 --------- T --------- , --------- ,--------- -

50 ,----------,-----------,-----------.-----------,

� c o 1ii E

e)

.� 0 ,... ....... ____ L,-_--I'" __ ---, .. _------, -�----__1 al � � (/) ö 2 w

-500L-----------0�.5------------1L-----------1�.5----------�2 Time (s) 400.--------r-,-----------,-----------,--�======�

f) 300 ------!-;_� ,;-...;;.;;-.J--_.-- '--------- � --1----- �:� 1 ,,-_______ r I 1

2 200 --------�------------------�--------iß 100 -------- --t---------

E � o - - - - - - - - � - - - - - - - - -t------1.;;.;;..;;.;;,.;;.;;..;;.;;� j -100 1f- - - - - - -1,- � - - -_.I''' - -------

�OO --------�------------------1

-300 --------�--------�---------0.5 1

Time (8) 1.5 2

Fig. 8. Simulation results of FBe _super twisting observer: a) motor Speed reference, real and estimation b) motor torque c) stator current Id d) stator current Ia e) Speed estimation error f) Voltage vsd, vsq

10.--------,---------,--------,-------------,-----�

a)

5

o

1 ! -5 -5 - - - - - - - - - +- - - - - - - - - - --t - - - - - - - - - -I ----------1 I I

-10 L-----------�----------�------------�----------�

c � :J U

o 0.5 1 1.5 Times (s)

1 1 ---------�-------- --------�---------

1 --------�---------

(; -10

� �O ---------r--------

�o ---------r-------- -------- ,---------o 0.5 1 Time (s) 1.5 2

b) 15 .--------,---------,---------,---------,--------�

0.1 0.2 Time (s) 0.3 0.4 0.5

Fig. 9. Behavior at low motor speed: a) Sinusoidal speed reference, real and estimation b) Torque-tracking response reference and real.

The second test is carried out with parameter variations at 150% of inductance Lq, Ld and resistance Rs at OAs 0.5s and 0.6s respectively (see Fig.lO). The control shows the good speed tracking without effect of parameters deviations.

2'2,---------,--------,---------,---------,---------, a) 2,1 _______ , _______ -' _______ +--________ L-______ ---1

'" c

2 -------:--------: ------- , -------l------� '� 1,9

, -------f-------� , -------1- - - - - - - ---1 - ------.� Ql � �

_______ L ______ � , , -------r------� ,

1,8 - - - - - - _1 - _______ I ______ _

1,7 -------1- - - - - - - I - ------

� -------�------� , -------�------� 1,6 -------1- - - - - - - ---.j - ------

, --------------, 1,5 ,

1 ,4 f-----------r----------,-----------f ------- , ------�

0,2 0,4 Time (5) 0,6 0,8

b)260 - - - - - - -: - - - - - - - r�-�-��-�-�-�-L�-�-��-�-�-�-�-�-�---�-�-�-= c 0 '� .� Ql U c m t3 ::J

'0 E

240

220

200

180

I , ------i------- -------�-------I ______ J __ _ I I

, I -------,-- - - - - - 1- - -I

, I -------1- - - - - - - i - ---------------..j.-------------!

, ------- , -------, ------- , -------

I 1/Ld I - - - � - - - - - - ----- 1/Lq --, ,

160 -------,- - - - - - - -j - - - - - - + - - - - - - - f- - ------, , 140

0L-------�

0�

,�2--------

0�

,�4--------

0�,6--------�OL,8

-------- �

Time (5)

160

c) 140 f -----: ------ : - - - - - - - t ------- : ------�

� TI Ql Ql Cl, (f)

120

100

80

60

40

20

-------1- - - - - - - --j - - - - - - - + - - - - - - - I- ------� I j I I ------ -------;-------+-------�------� I j I I -------1- - - - - - - ---j - - - - - - - -+ - - - - - - - I- - ----- -I j I j I j I j -------1--------1 - - - - - - -

T - - - - - - - j - ----- -

I j I j -------1- - - - - - - ---.j - - - - - - - -+ - - - - - - - I- - - - - - - _ I j I j I j I j -------1- - - - - - - -I - - - - - - - T - - - - - - - I ------�

°0L-------�0� ,� 2

------ �0� ,4�T�

im--

e--

(5�

)�0�,6�

------ �OL,8�------ �

10.--------,---------.---------,---------.--------� d)

1 ! 1 1 � 5 - - - - - - - ,- - - - - - - -, - - - - - - - I - ------ , ------ -31 'E § O ��,������HM����� .. � .. �� __ ���� U

j 1 1 1 1 -5 -------1- - - - - - - ---i - - - - - - - -+ - - - - - - - 1- ------�

1 1 1 1

-10L-------�--------�--------�--------�--------� o 0,2 0,4 0,6 0,8

Time (5)

Fig, 10, Simulation results of FBC�super twisting observer accorrding parameters variations: a) resistance vaiation, b) inductances variations, c) Speed reponse and d) stator current Id

VI. CONC LUSION

The flatness-based control (FBC) technique is used with the motor speed estimation. This paper presents the design and implementation of a sensorless speed control of PM SM drives based on the second-order mode ob server. This observer based on an exact differentiator (super-Twisting algorithm) was used for two main reasons: the final time convergence and the ability to take into account naturally the variable structure of the system. The only measurements of the stator currents are required for the on-line rotor speed estimation. The second order sliding mode observer achieves good real time estimation of motor speed. Simulation results illustrate the efficiency of the second order sliding mode ob server.

[I]

[2]

[3]

[4]

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