233

Sensorless high order sliding mode control of permanent magnet

synchronous motor

Marwa Ezzat, Alain Glumineau and Franck Plestan

Abstract— In this paper, a robust sensorless speed observer-controller scheme for a permanent magnet synchronous motor(PMSM) is proposed. To estimate both position and speed,without any mechanical sensors and only from the electricalmeasurements, a back-EMF-based sliding mode is designed.The convergence of this observer is proved. Then a highorder sliding mode controller is developed. By using thistechnique, the drawback of the sliding mode technique called"chattering phenomenon" is strongly attenuated. After that,the convergence of the overall system is analyzed. Moreover,the robustness of this method with respect to the parametersuncertainties is verified by significant simulation results in theframework of an industrial benchmark.

Index Terms— permanent magnet synchronous motor, sen-sorless control, non linear observer, benchmark.

Nomenclature

ω Rotor angular speed.

Ω Rotor mechanical speed.

θ Rotor angular position.

ψf Magnet flux.

θe Rotor electrical position.

Rs Stator-winding resistance.

Ls Stator-winding inductance.

fv The viscous damping coefficient.

J The rotor moment of inertia.

e Electromotive force(emf).

p Pole pairs.

I. INTRODUCTION

The permanent magnet synchronous motors attract the

industrial world attention thanks to their superior advan-

tages, for instance their higher efficiency, low inertia, high

torque to current ratio and almost no maintenance. On the

other hand, the necessity for the rotor position information

leads to install position sensors "encoders, tachometers and

resolvers". Unfortunately, these mechanical sensors increase

the overall cost as well as the size of the machine, reduce the

reliability, the noise immunity of the system and moreover

in some occasions are even not permitted. That is why

the sensorless techniques are preferred. Therefore many ap-

proaches for speed/position estimation have been investigated

in the literature. In general, these estimation approaches

can be categorized into two basic groups. The first is the

model-based approach and the second is the non-model-

based approach. Regarding to the first approach, most of the

methods proposed are based on the back electromotive force

The authors are with IRCCyN, UMR 6597, Ecole Centrale de Nantes,France alain.glumineau@irccyn.ec-nantes.fr

"e.m.f" or the flux linkage. For the surface permanent magnet

synchronous motors "SPMSM", most of the speed/position

estimation is based on the back electromotive force [7]. But

this method does not perform well at both standstill and

low speed because of the non-observability phenomenon.

Moreover, this technique is not suitable for the internal

permanent magnet synchronous motors "IPMSM". This due

to the fact that the position information is contained not only

in the back electromotive force but also in the inductances.

Thus the extended electromotive force is used instead [2],

[6].

Nearly all the published observers have been tested with

classical vector control. Whereas, the nonlinearities of the

system could not be neglected except a partially neglection

could be achieved through the feedback linearization

technique. So the object of the control design is to develop

a robust control which overcomes uncertainties due to

parameter variations and load torque application. From this

point of view, the nonlinear control is proposed such as

sliding mode and backstepping [8], [14], [10].

In this paper, a robust sensorless speed observer-controller

scheme for a PMSM is proposed. A back-emf observer based

on sliding mode technique in order to estimate both position

and speed is designed. Its convergence is studied. Then a

high order sliding mode controller is developed. The conver-

gence of the whole system is proved. The overall system is

tested by Simulink/Matlab simulation with significant tests

of robustness with respect to parameters deviations in the

framework of an industrial benchmark.

II. MATHEMATICAL MODEL

The SPMSM model in the synchronous reference frame

(d-q) reads as [3]

dθ

dt= Ω

dΩ

dt=

p

Jψf iq −

fv

JΩ −

Tl

Jdid

dt= −

Rs

Ls

id + pΩiq +1

Ls

vd

diq

dt= −p

ψf

Ls

Ω − pΩid −Rs

Ls

iq +1

Ls

vq

(1)

Usually, this model (1) is used for the vector control design

whereas, for the observer design, the model will be written

in stationary reference frame (α − β) as the speed and

the position information are ready to be extracted in this

reference frame [5]. Then the model can be written as follows

978-1-4244-5831-8/10/$26.00 ©2010 IEEE

2010 11th International Workshop on Variable Structure Systems

Mexico City, Mexico, June 26 - 28, 2010

234

diα

dt=

−Rs

Ls

iα −1

Ls

eα +1

Ls

uα

diβ

dt=

−Rs

Ls

iβ −1

Ls

eβ +1

Ls

uβ

(2)

whereeα = −ψfω sin θe eβ = ψfω cos θe. (3)

III. OBSERVER DESIGN

Currently, different approaches of observation have been

studied. The sliding mode technique attracts the attention [5],

[12], [11], [13]. Thanks to its robustness properties face up

to parameter uncertainties. This technique is selected here to

design an observer in order to estimate both the speed and

the position only from the electrical states measurements.

According to the work cited above, as it is classically

assumed, we suppose that the speed is varying slowly, so

ω ≈ 0 and then the emf dynamics will be written as

deα

dt= −ωeβ ,

deβ

dt= ωeα. (4)

The sliding mode observer is based on a stator current

estimator as the stator currents and voltages are the only

measured states in a PMSM drive system. Then a sliding

mode observer can be designed as follows

˙iα =

−Rs

Ls

iα −eα

Ls

+1

Ls

uα +K1sgn(iα)

˙iβ =

−Rs

Ls

iβ −eβ

Ls

+1

Ls

uβ +K1sgn(iβ)

˙eα = −eβω +K2sgn(iα)˙eβ = eαω +K2sgn(iβ)

(5)

with iα = iα − iα, iβ = iβ − iβ and K1 and K2 are the

observer gains.

The estimated speed Ω can be calculated from (3) as

follows:|ω| =

1

ψf

√

e2α + e2β Ω =1

pω

with the sign of the speed (ω) is calculated from the

quadrature EMF. For the electrical position θe, it can be

estimated either from the integration of the speed as

θe =

∫

ωdt+ θo

with θo is position the initial condition or from the emf as

θe = arctan−eα

eβ

.

The first method based on the speed integration is applied to

get the estimated position assuming that the initial condition

θo is known. Then, the rotor angular position is obtained

from θ = 1pθe. It yields the currents estimated errors

dynamics

˙iα =−Rs

Ls

iα −eα

Ls

−K1sgn(iα)

˙iβ =−Rs

Ls

iβ −eβ

Ls

−K1sgn(iβ)(6)

with eα = eα − eα, eβ = eβ − eβ .

The analysis of the observer convergence will be carried out

using the step-by-step method [1] and using the following

candidate Lyapunov function.

• First step :

V =1

2(iα

2+ iβ

2). (7)

Its time derivative reads as

V = iα˙iα + iβ

˙iβ . (8)

It yields

V =−Rs

Ls

i2α −eαiα

Ls

− iαK1sgn(iα)

−Rs

Ls

i2β −eβ iβ

Ls

− iβK1sgn(iβ).(9)

To guarantee the convergence, the time derivative of the

candidate Lyapunov function is forced such that V < 0.

Knowing that−Rs

Lsi2α < 0 −Rs

Lsi2β < 0,

it is sufficient to choose K1 such that

−eαiα

Ls

−K1 |iα| −eβ iβ

Ls

−K1 |iβ | < 0. (10)

Thus, K1 will be tuned such as

K1 > max(

∣

∣

∣

∣

eα

Ls

∣

∣

∣

∣

,

∣

∣

∣

∣

eβ

Ls

∣

∣

∣

∣

).

After a finite time t0, the current errors dynamics will

reached the sliding surfaces. At this stage ˙iα = 0 and ˙iβ = 0,

so from (6) it is possible to apply the equivalent control

during the sliding on their surfaces.

• Second step :

eα = −LsK1sgneq(iα)eβ = −LsK1sgneq(iβ),

(11)

and the emf errors dynamics are

˙eα = −ωeβ + eβω −K2sgneq(iα)˙eβ = ωeα − eαω −K2sgneq(iβ).

(12)

These equations can be rewritten according to (11) as

˙eα = −ωeβ + ωeβ +eα

K1Ls

K2

˙eβ = ωeα − ωeα +eβ

K1Ls

K2.(13)

Applying a new candidate Lyapunov function, it yields

V ′ = eα ˙eα + eβ ˙eβ (14)

then

V ′ = −ωeαeβ + ωeαeβ +e2α

K1Ls

K2

+ωeβeα − ωeβ eα +e2

β

K1LsK2.

(15)

It is sufficient to choose K2 such that

−ωeαeβ + ωeβeα +e2α

K1Ls

K2 +e2β

K1Ls

K2 < 0. (16)

Thus, K2 will be tuned such as

K2 > max(|ωeαeβ | , |ωeβeα|). (17)

Mexico City, Mexico, June 26 - 28, 2010

2010 11th International Workshop on Variable Structure Systems

235

IV. CONTROLLER DESIGN

In this section, a high order sliding mode controller is

designed. Based on [10] and supplied by the estimations

of the speed and the position, a high order sliding mode

controller is designed for the speed control purpose. First-

of-all, in the system (1), the parameter uncertainties are

presented aspψf

J= k1 = k01 + δk1, −

fv

J= k2 = k02 + δk2

−Rs

Ls

= k3 = k03 + δk3,1

Ls

= a4 = k04 + δk4

−pψf

Ls

= k5 = k05 + δk5,

with k0i (1 ≤ i ≤ 5) the nominal value of the concerned

parameter, δki the uncertainty on the concerned parameter

such that

|δki| ≤ δk0i < |k0i|

with δk0i a known positive bound. Then, the system (1) reads

as θ = Ω

Ω = k1iq + k2Ω

id = k3id + pΩiq + k4ud

iq = k5Ω + pΩid + k3iq + k4uq

. (18)

To fulfill speed and current control, the sliding vector is

defined as

s =

[

s1s2

]

=

[

Ω − Ωref (t)id − idref (t)

]

(19)

with Ωref (t) the rotor angular speed reference and idref (t)the current reference. This latter equals zero in order to

avoid the reluctance and ripple effects in the electromagnetic

torque.

The relative degree vector of the sliding variables s1 and s2is [2 1]T respectively. In order to design a robust controller

and to attenuate the chattering phenomenon, the proposed

controller is chosen as a 3-2 order sliding mode one, which

yields that the discontinuity is acting on the first time

derivative of the control inputs.

A. Control design

The control design follows the high order sliding mode

technique given in Appendix I. All the details are given in

this latter section. The second and first time derivatives of

s1 and s2 are respectively

s(2)1 = χ1 + Γ12u2, s

(1)2 = χ2 + Γ21u1 (20)

with [u1 u2]T := [ud uq]

T and

χ1 = −x(2)2ref + k1(k5x2 + px2x3 + k3x4)

+k2(k1x4 + k2x2)

=: χ10 + χ1∆,

χ2 = k3x3 + px2x4

=: χ20 + χ2∆,

Γ12 = k1k4 =: Γ120 + Γ12∆,

Γ21 = k4 =: Γ210 + Γ21∆

The terms χ10, χ20, Γ120 and Γ210 correspond to the well

known nominal terms while χ1∆, χ2∆, Γ12∆, and Γ21∆

represent the uncertainties, i.e. parameters and load torque

variations. By denoting

Γ0 =

[

0 Γ120

Γ210 0

]

, χ0 =

[

χ10

χ20

]

(21)

and from the possible values of Γ120 and Γ210, the invert-

ibility of Γ0 is ensured. Applying the following preliminary

static state feedback[

u1

u2

]

= Γ−10 · [−χ0 + v] (22)

with v := [v1 v2]T the new control vector. The dynamics of

(20) are controlled by (22) is now given by:[

s1(3)

s2(2)

]

=

[

χ1

χ2

]

+

[

0 Γ12

Γ21 0

] [

v1v2

]

(23)

with the uncertain residual terms are bounded such that:

|χi| ≤ C0i,∀x ∈ χ,

0 ≤ Km1 ≤ |Γ12| ≤ KM1,

0 ≤ Km2 ≤ |Γ21| ≤ KM2.

(24)

The problem now consists in defining the both control laws

v1 and v2, in order to ensure the establishment of 3−2 order

sliding mode control. The design is made in two steps

• First-of-all, a switching vector S based on the sliding

variable s1 and s2 is defined.

• Then, the discontinuous control law can be designed.

The first property of the switching vector S is such as, early

from t = 0, the system is evolving on switching domain

S = 0. The second property is that, if the system is evolving

on S = 0 in spite of the uncertainties thanks to the control

law, a 3 − 2 order sliding mode is established in a a priori

defined finite time.

Switching vector S. Assuming that s1(0) = Ω(0) −Ωref (0) 6= 0 and s2(0) = id(0) 6= 0 (Hypothesis H3 in

Appendix fulfilled), the switching vector S = [S1 S2]T is

written as

S1 = s(2)1 −F

(2)1 + λ11

[

s1 − F1

]

+ λ10 [s1 −F1]

S2 = s2 −F(2)2 + ω2

n [s2 −F2](25)

with

F1(t) = K1F1eF1tT1s1(0) and F2(t) = K2e

F2tT2s2(0).

As described by equation (38) in Appendix, if the sliding

variables s1 and s2 and their first- and second-order time

derivatives exactly track the functions F1(t) and F2(t) (and

their time derivatives), it yields

• S1(0) = 0 and S2(0) = 0,

• s1(tF1) = 0, s1(tF1) = 0 , s1(tF1) = 0,

• s2(tF2) = 0 , s2(tF2) = 0.

Then, high order sliding mode behavior is established for

the both sliding variables. From (40), in case of S1, the

convergence time is set as tF1= 50 msec, whereas s1(0) =

Mexico City, Mexico, June 26 - 28, 2010

2010 11th International Workshop on Variable Structure Systems

236

0 rad.s−3, s1(0) = 0 rad.s−2, s1(0) = 0.1 rad.s−1, with

K1, F1 and T1 are defined following [9].

For S2, the convergence time is set as tF2= 50 msec,

s2(0) = 0 A.s−1, s2(0) = .1 A with K2, F2 and T2 are

defined following [9].

Discontinuous control input v. The input v is defined as

v = [v1 v2]T such that

vi = −αi sign(Si) (26)

where

αi ≥C0i + Θi + ηi

Kmi

, (27)

with C0i, Θi, ηi and Kmi are delivered in Appendix for a

general system. The gains α1 and α2 have been tuned by

taking into account the uncertainties and load torque varia-

tions. The controller is synthesized so that the performances

are kept despite the load torque applying and parameters

variations.

V. STABILITY ANALYZIS OF THE WHOLE SYSTEM

In order to prove the stability of the whole system (ob-

server+controller), the speed and the current id are replaced

by their estimated values in the control law (22). Then, the

sliding vector is

s1 = Ω − Ω∗, s2 = id − i∗d

From equation (20), one gets

s(2)

1= χ1 + Γ12u2, s

(1)

2= χ2 + Γ21u1 (28)

with

χ1 = −Ω(2)ref + k1(k5Ω + pΩid + k3iq)

k2(k1iq + k2Ω)

χ2 = k3id + pΩiq,Γ12 = k1k4,Γ21 = k4.

Consider the following estimation errors

Ω = Ω − Ω, id = id − id, iq = iq − iq. (29)

From the equations (29) and (28), one gets

χ1 = χ1 + ∆χ1

χ2 = χ1 + ∆χ2,(30)

in which ∆χ1 and ∆χ1 contain the parametric uncertain-

ties, disturbances (measurement noise) and estimation errors.

Then, the control law can be written as

[

u1

u2

]

= Γ−10

[

−

[

χ01

χ02

]

+

[

v1v2

]

]

(31)

with[

v1v2

]

=

[

−α1.sign(SΩ)−α2.sign(Sid

)

]

.

Then, it is easy to find a gain αi (eqn 26) such that

S1 · S1 < −η1|S1|, S2 · S2 < −η2|S2| (32)

where ηi > 0 is a positive real number as shown in [9]

VI. SIMULATION RESULTS

The overall control system has been simulated by

Simulink/MatLab tool. The parameters of the tested SPMSM

are given in Table (I). The system is tested according to a

sensorless industrial benchmark [15]. The measured speed is

used only for comparison purpose.

Figure 1.a shows the measured speed Ω and the estimated

speed as well as the speed error for the nominal case (see

Figure 1.b). It is clear that the estimation error is small.

This result confirms the good estimation on the speed. Figure

2.a displays the measured and reference speed, whereas the

tracking error is plotted in Figure 2.b. This result attests the

good tracking of the proposed observer-controller.

Some robustness tests are carried out. Figure 3 shows the es-

timated speed and its error for the stator inductance variations

of −20%. Then, the viscous damping coefficient variation of

+20% then −20% are plotted in Figure 4.

0 2 4 6 8 10 12−50

0

50

100

150

200

250

300

350

a

Time (Sec.)

Sp

ee

d (

ra

d/s

ec

.)

Wobs.

Wmeas.

0 2 4 6 8 10 12−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

b

Time (Sec.)

Sp

ee

d e

rro

r (

ra

d/s

ec

.)

Fig. 1. a) Measured and Estimated speeds b) Observed speed error

0 2 4 6 8 10 12−50

0

50

100

150

200

250

300

350

a

Time (Sec.)

Sp

eed

(rad

/sec.)

Wmeas.

Wref.

0 2 4 6 8 10 12−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

b

Time (Sec.)

Sp

eed

erro

r (

rad

/sec.)

Fig. 2. a) Measured and reference speeds b) Tracking speed error

VII. CONCLUSION

A sensorless robust control of a permanent magnet syn-

chronous motor has been presented. The sensorless scheme

is achieved via a sliding mode observer that estimates the

speed and the position of a SPMSM. The controller is

based on the higher order sliding mode technique. This

controller is associated with the sliding mode observer and

the whole stability is proved. The whole system was tested

within the framework of a specific industrial benchmark.

Mexico City, Mexico, June 26 - 28, 2010

2010 11th International Workshop on Variable Structure Systems

237

0 2 4 6 8 10 12−50

0

50

100

150

200

250

300

350

a

Time (Sec.)

Sp

ee

d (

rad

/sec.)

Wobs.

Wmeas.

0 2 4 6 8 10 12−1.5

−1

−0.5

0

0.5

1

1.5

b

Time (Sec.)

Sp

ee

d e

rro

r (r

ad

/sec

.)

Fig. 3. a) Reference and estimated speeds for −20%Ls b) Speed error

0 2 4 6 8 10 120

50

100

150

200

250

300

350

a

Time (Sec.)

Sp

ee

d (

rad

/se

c.)

Wobs. −20% fv

Wobs. +20% fv

0 2 4 6 8 10 12−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

b

Time (Sec.)

Sp

ee

d e

rro

r (r

ad

/se

c.)

err. −20% fv

err. +20% fv

Fig. 4. Viscous damping coefficient variation: a)Estimated speed b)Speederror

Many robustness tests have been completed successfully.

The results proved the efficiency and the robustness of the

proposed sensorless controller.

TABLE I

MOTOR PARAMETERS

Current 9.67A Torque 9NmSpeed 3000 rpm ψf 0.1814 WbRs 0.295 Ω Ls 3.425 mHJ 0.00679 kg.m2 fv 0.0034 kg/msp 3

APPENDIX I

HIGH ORDER SLIDING MODE TECHNIQUE

The used technique is based on [9], [10] and the design

the controller for a large class of uncertain nonlinear systems

consists in two steps

1) Design of the switching variable

2) Design of a discontinuous control input

Consider the uncertain nonlinear SISO system

x = f(x, t) + g(x, t)vy = s(x, t)

(33)

where x ∈ IRn is the state variable, v ∈ IR is the input con-

trol and s(x, t) ∈ IR is a measured smooth output function

(sliding variable) defined to satisfy the control objectives.

f(x, t) and g(x, t) are uncertain smooth functions. Assuming

that the relative degree r of this system (33) w.r.t. s is known,

constant and the associated zero dynamics are stable. Let

this system (33) to be closed by some possibly-dynamical

discontinous feedback. Then, provided that 1 s, s, · · · , s(r−1)

are continuous functions, and the set

Sr = x | s(x, t) = s(x, t) = · · · = s(r−1)(x, t) = 0,

called “rth order sliding set”, is non-empty and is locally an

integral set in the Filippov sense [4]; the motion on Sr is

called “rth order sliding mode” with respect to the sliding

variable s. The rth order sliding mode control approach

allows the finite time stabilization to zero of the sliding

variable s and its r − 1 first time derivatives by defining

a suitable discontinuous control function. From [9], there

exists χ-vector and Γ-matrix such that

s(r) = χ(·) + Γ(·)v (34)

H1. The solutions of (34) are understood in the Filippov

sense [4], and system trajectories are supposed to be in-

finitely extendible in time for any bounded Lebesgue mea-

surable input.

H2. Functions χ(·) and Γ(·) are bounded uncertain functions,

and, without loss of generality, let also the sign of the control

gain χ be constant and strictly positive. Thus, there exist

Km ∈ IR+∗, KM ∈ IR+∗, C0 ∈ IR+ such that

0 < Km < Γ < KM |χ| ≤ C0. (35)

for x ∈ X ⊂ IRn, X being a bounded open subset of IRn

within which the boundedness of the system dynamics is

ensured. Furthermore, control input v is bounded.

With Z1 = [Z01 Z

11 · · · Zr−2

1 ]T := [s s · · · s(r−2)]T , Z2 =s(r−1), the rth order sliding mode control of (33) w.r.t. s is

equivalent to the finite time stabilization of

Z1 = A11Z1 +A12Z2, Z2 = χ+ Γv (36)

which satisfies the global boundedness conditions (35),

where A11 and A12 are previously defined [9].

Switching variable design.

Let S denote the switching variable defined as

S = s(r−1) −F (r−1)(t) + λr−2

[

s(r−2) − F (r−2)(t)]

+ · · · + λ0 [s(x, t) −F(t)] ,(37)

with λr−2, · · · , λ0 defined such that P (z) = z(r−1) +λr−2z

(r−2)+· · ·+λ0 is a Hurwitz polynomial in the complex

variable z. The function F(t) is a Cr-one defined such that

S(t = 0) = 0 and s(k)(x(tF ), tF )−F (k)(tF ) = 0 (0 ≤ k ≤r − 1). Then, from initial and final conditions the problem

consists in finding the function F(t) such thats(x(0), 0) = F(0), s(x(tF ), tF ) = F(tF ) = 0,

s(x(0), 0) = F(0), s(x(tF ), tF ) = F(tF ) = 0,...

s(r−1)(x(0), 0) = F (r−1)(0),s(r−1)(x(tF ), tF ) = F (r−1)(tF ) = 0

(38)

1All over this paper,s(·)(k) (k ∈ IN ) denotes the kth time derivative ofthe function s(·). This notation is also applied for every function.

Mexico City, Mexico, June 26 - 28, 2010

2010 11th International Workshop on Variable Structure Systems

238

A solution for F(t) reads as (1 ≤ j ≤ r)

F(t) = KeFtTs(r−j)(0) (39)

with F a 2r × 2r-dimensional Hurwitz matrix (strictly

negative eigenvalues) and T a 2r × 1-dimensional vector.

H3. The integer j is such that s(r−j)(0) 6= 0 and bounded.

Lemma 1: Given Hypothesis H3 and tF > 0 bounded,

there exists a Hurwitz matrix F2r×2r and a matrix T2r×1

such that matrix K defined as

K =

F r−1Ts(r−j)(0)F r−1eFtF T

F r−2Ts(r−j)(0)F r−2eFtF T

...

Ts(r−j)(0)eFtF T

T

is invertible.

K is a 1 × 2r-dimensional gain matrix tuned such system

(38) is fulfilled. Then, one gets

K =

s(r−1)(0)0

s(r−2)(0)0...

s(0)0

T

· K−1 (40)

Then, the switching variable S reads as

S = s(r−1) −KF r−1eFtTs(r−j)(0)+λr−2

[

s(r−2) −KF r−2eFtTs(r−j)(0)]

+ · · · + λ0

[

s(x, t) −KeFtTs(r−j)(0)]

.

(41)

H4. There exists a finite positive constant Θ ∈ IR+ such that∣

∣KF reFtTs(r−j)(0)−λr−2

[

s(r−1) −KF r−1eFtTs(r−j)(0)]

− · · ·−λ0

[

s(x, t) −KFeFtTs(r−j)(0)]∣

∣ < Θ

(42)

Equation S = 0 describes the desired dynamics which satisfy

the finite time stabilization of vector [s(r−1) s(r−2) · · · s]T

to zero. Then, the switching manifold on which system (36)

is forced to slide on, via a discontinuous control v, is defined

as

S = x | S = 0 (43)

Given equation (38), one gets S(t = 0) = 0: at the initial

time, the system still evolves on the switching manifold.

Discontinuous control design. The design of the discontin-

uous control law u allows to force the system trajectories of

(36) to slide on S, to reach in finite time the origin and to

maintain the system at the origin.

Theorem 1: [9] Consider the nonlinear system (33) with

a relative degree r with respect to s(x, t). Suppose that it is

minimum phase and that hypotheses H1, H2, H3 and H4 are

fulfilled. Let r be the sliding mode order and 0 < tF < ∞the desired convergence time. Define S ∈ IR by (41) with

K unique solution of (39) given by (40) and that assumption

H5 is fulfilled. The control input v defined by

v = −α sign(S) (44)

with

α ≥C0 + Θ + η

Km

, (45)

C0, Km defined by (35), Θ defined by (42), η > 0, leads to

the establishment of a rth order sliding mode with respect

to σ. The convergence time is tF .

REFERENCES

[1] T. Boukhobza et J-P. Barbot, " Step by step sliding mode observer forimplicit triangular observer form”, in Proceeding of l’IFAC NOLCOS98, Enschede, the Netherland, pp 233-238, 1998.

[2] Z. Chen, M. Tomita, S. Doki, and S. Okuma, "an extended electromo-tive force model for sensorless control of interior permanent-magnetsynchronous motor", IEEE TRANSACTIONS ON INDUSTRIALELECTRONICS, VOL. 50, NO. 2, APRIL 2003.

[3] J. Chiasson, "Modeling and High-Performance Control of ElectricMachines", WILEY-INTERSCIENCE A JOHN WILEY & SONS,INC., PUBLICATION 2005.

[4] A.F. Filippov, Differential Equations with Discontinuous Right-Hand

Side, Kluwer, Dordrecht, The Netherlands, 1988.[5] Changsheng LI, and Malik Elbuluk, "A sliding mode observer for

sensorless control of permanent magnet synchronous motors", IndustryApplications Conference, 2001. Thirty-Sixth IAS Annual Meeting.Conference Record of the 2001.

[6] S. Morimoto, K. Kawamoto, M. Sanada, and Y. Takeda, "SensorlessControl Strategy for Salient-Pole PMSM Based on Extended EMF inRotating Reference Frame", IEEE TRANSACTIONS ON INDUSTRYAPPLICATIONS, VOL. 38, NO. 4, JULY/AUGUST 2002.

[7] B. Nahid Mobarakeh, F. Meibody-Tabar, and F.-M. Sargos, "Mechani-cal Sensorless Control of PMSM With Online Estimation of Stator Re-sistance", IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS,VOL. 40, NO. 2, MARCH/APRIL 2004.

[8] M. Ouassaid, M. Cherkaoui, and Y. Zidani, "New robust positioncontrol of a synchronous motor by high order sliding mode", IEEEInternational Conference on Industrial Technology (ICIT), Vol. 3,pp.1287 - 1292, 2004.

[9] F. Plestan, A. Glumineau, and S. Laghrouche, "A new algorithm forhigh order sliding mode control", International Journal of Robust and

Nonlinear Control, 2007.[10] F. Plestan, A. Glumineau, and G. J. Bazani, "New robust position

control of a synchronous motor by high order sliding mode", 46thIEEE Conference on Decision and Control, pp. 3697 - 3702, Dec.12-14, 2007, New Orleans, LA, USA.

[11] Z.Xu, and M.F. Rahman, "An Improved Stator Flux Estimation for aVariable Structure Direct Torque Controlled IPM synchronous MotorDrive Using a Sliding Observer", Industry Applications Conference,2005. Fourtieth IAS Annual Meeting. Conference Record of the 2005.

[12] Zhang Yan, and Vadim Utkin, "Sliding Mode Observers for ElectricMachines-An Overview", IECON 02 [IEEE 2002 28th Annual Con-ference of the Industrial Electronics Society], Vol.3, pp.1842 - 1847,5-8 Nov. 2002.

[13] Sheng Zhao and Xiafu Peng, "A modified direct torque control usingspace vector modulation (DTC-SVM) for surface permanent magnetsynchronous machine (PMSM) with modified 4-order sliding modeobserver", Proceedings of the 2007 IEEE International Conference onMechatronics and Automation, Harbin, China, August 5 - 8, 2007.

[14] J. Zhou, and Y. Wang, "Adaptive backstepping speed controller designfor a permanent magnet synchronous motor", IIEE Proc.-Electr. PowerAppl.. Vol. 149, No. 2, pp. 165 - 172, March 2002.

[15] www.irccyn.ec-nantes.fr/hebergement/BancEssai/

Mexico City, Mexico, June 26 - 28, 2010

2010 11th International Workshop on Variable Structure Systems