A nonlinear state observer for the sensorless control of a permanent-magnet AC machine

A Nonlinear State Observer for the Sensorless Control

of a Permanent-Magnet AC Machine Guchuan Zhu,Azeddine Kaddouri,Member,IEEE,Louis-A.Dessaint,Senior Member,IEEE,and

Ouassima Akhrif,Member,IEEE

Abstract—This paper presents a sensorless speed regulation scheme for a permanent-magnet synchronous motor(PMSM) based solely on the motor line currents measurements.The proposed scheme combines an exact linearization-based controller with a nonlinear state observer which estimates the rotor position and speed.Moreover,the stability of the closed-loop system, including the observer,is demonstrated through Lyapunov sta-bility theory.The proposed observer has the advantage of being insensitive to rotation direction.It is shown how a singularity at zero velocity appears in the scheme and how it can be avoided by switching smoothly from the observer-based closed-loop control to an open-loop control at low velocity.The system performance is tested with an experimental setup consisting of a PMSM servo drive and a digital-signal-processor-based controller for both unidirectional and bidirectional speed regulation.

Index Terms—Digital signal processors,Lyapunov methods, nonlinear systems,observability,observer,permanent-magnet synchronous motors.

I.I NTRODUCTION

A LOT OF attention has been paid recently to the elim-

ination of the position sensor in permanent-magnet (PM)ac machines.These drives need a precise knowledge of rotor position to synchronize the stator currents with the back electromotive force(EMF)of the motor.This requirement known as self-synchronization is particularly stringent with sinusoidally excited PM machines,also called PM synchronous motors(PMSMs).Many approaches have been proposed for the sensorless operation of a PMSM.One method is based on the estimation of flux linkage[7],[19],but the rotor must be brought to a known position to establish the initial value of flux linkage.A second method is based on an extended Kalman filter(EKF)applied to a nonlinear model of the PMSM[1], [6].The method based on EKF gives good results in open loop, but the closed-loop performance and stability were not inves-tigated.Finally,a third method is based on the nonlinear state observer theory.In[5],the authors have proposed a nonlinear Manuscript received July8,2000;revised June1,2001.Abstract published on the Internet October24,2001.

G.Zhu was with the Groupe de Recherche enélectronique de Puissance et Commande Industrielle,Electrical Engineering Department,Ecole de Tech-nologie Supérieure,Montréal QC H3C1K3Canada.He is now with SR Telecom Inc.,Saint-Laurent,QC H4T1V8Canada.

A.Kaddouri was with the Groupe de Recherche enélectronique de Puis-sance et Commande Industrielle,Electrical Engineering Department,Ecole de Technologie Supérieure,Montréal QC H3C1K3Canada.He is now with the Electrical Engineering Department,University of Moncton,Moncton,NB E1A 3E9Canada.

L.-A.Dessaint and O.Akhrif are with the Groupe de Recherche enélectron-ique de Puissance et Commande Industrielle,Electrical Engineering Depart-ment,Ecole de Technologie Supérieure,Montréal QC H3C1K3Canada. Publisher Item Identifier S0278-0046(01)10278-9.observer to estimate only the speed based on measurements of the position and the phase currents.In[17]and[18],the authors design a nonlinear observer that estimates both the rotor position and speed and use this observer to implement a state feedback linearization of the PMSM.Their design and stability analysis are demonstrated,however,for an observer that uses the derivatives of the phase currents which,of course, is noise prone.The authors get around this disadvantage by introducing a practical version of their observer that was tested by simulation.It is important to point out,however,that this practical observer retains the same high sensitivity to noise because it depends directly on the currents that are multiplied with the observer high gains(see[18,Eq.(26)]).On the other hand,an observer design that uses directly the actual currents instead of their derivatives yields a state estimation law that filters out the noise component of the currents by an integration. This paper presents,therefore,the design and the stability analysis of a full-order nonlinear observer that directly uses the currents and eliminates the recourse to a practical observer.A nonlinear state feedback linearization controller is applied and the stability of the closed-loop system including the observer is demonstrated through Lyapunov stability theory.Moreover,the proposed algorithm is tested with an experimental setup con-sisting of a PMSM and a digital-signal-processor(DSP)-based controller.The control scheme is very simple to implement and is suitable for unidirectional or bidirectional speed control. This paper is organized as follows.In Section II,the mathe-matical model of the PMSM is described and the observability is verified.In Section III,the observer’s equations are derived. In Section IV,the state observer is associated with a nonlinear controller,and the stability of the closed-loop system is demon-strated.This sensorless control scheme is tested with an exper-imental setup,and the results are shown in Section V.Finally, some conclusions are presented in Section VI.

II.T HEORETICAL B ACKGROUND

A.Mathematical Model of the PMSM

The dynamical model of a PMSM in a stator-fixed reference

frame

where ,

,,

and are the currents and the voltages in

phase

.are,respectively,the resistance,inductance,and PM

linkage,

the rotor

inertia,the rotor speed,

and

frame can be

easily obtained from the actual three phases quantities by a linear transformation

[4]

(4)

where

:

:

is a smooth vector

field

at

can be instantaneously distinguished

by a judicious choice of

input

of .An observability test criterion is given in [9],which states that for an admissible

input

of

such

that

(5)

for

all along the vector

field

is de-fined in (1).Applying the above criterion to the PMSM system (6)with (2)as measurement

yields

,

,

,

and

and given

by

(7)

is full rank if and only

if

.Hence,one can conclude that system (1)and (2)is locally observable for

all

.

Generally,the unobservable points (or sets)are the singular points (or sets)that should be avoided in the observer design.This is the reason why a thorough analysis of the observability of the PMSM is important since it helps explain why the sensorless operation performs poorly in low-speed environment.In fact,any observer-based algorithm works only in the case where the rotor speed is high enough,while in the low-speed region,an open-loop control strategy must be considered [19].

III.S TATE O BSERVER D ESIGN

Consider the following dynamic

system:

(8)

where

denotes

is the same function as in (6),

and

and

is the observer gain,

and

such that the error dynamics (9)are asymptotically stable

at the

origin

.Unfortunately,this problem is in most cases unsolvable in the original coordinates.A solution for overcoming this difficulty is to transform the original nonlinear system into a suitable form.In this case,the following change of coordinates [18]is

used:

or

which transfers into the following

form:

is

then

is given

by

,

let

,for

any

vector

matrix

.The time derivative

of

is locally Lipschitz,thus,there

exists a positive

constant

(18)

is

satisfied,

,

where

is not available as this,and other

questions related to Lyapunov equations are still being inves-

tigated theoretically by several researchers[8],[11].For this

paper,was chosen and the

matrix

and

(19)

We remark,however,that by processing this way,the

points

will produce the same point

for arbitrary

integer

obtained by(19)are used for bidirec-

tional speed control.It is necessary therefore to implement the

observer in the original coordinates.

To this aim,

if

(21)

where the Jacobian

matrix

in terms

of

where

where

,which is precisely the singular

point

of

–

(26)

and an identity diffeomorphism.The closed-loop system is then

given

by

(27)

where

and are the so-called auxiliary inputs.The re-

sulting linearized system(27)is a completely controllable

linear system.

B.Rotor Speed Regulation

Recall that the paper’s objective is to regulate the rotor speed.

This can be taken into account in the design of the auxiliary