International Journal of Computer Architecture and Mobility (ISSN 2319-9229) Volume 1-Issue 6, April 2013

Available Online at: www.ijcam.com

Review of Sensorless Vector Control of Induction Motor

Based on Comparisons of Model Reference Adaptive System

and Kalman Filter Speed Estimation Techniques

Payal Thakur1, Rakesh Singh Lodhi

2

Indore,India

1 payal.thakur87@gmail.com 2 rakeshlodhigs@gmail.com

Abstract - In recent year, sensorless vector control of induction

motor has been most popular research topic. Continuing

research has to concentrate on elimination of the problem of

parameter variation of induction motor drive. This technical

review will be of assistance in reaching a dynamic modeling of

induction motor and sensorless vector control of induction

motor using efficient speed estimation approach : Model

Reference Adaptive System and Kalman Filter and its

application in area of electrical vehicles (EV’s) propelled by an

induction motor drive; compare the results of both methods

Keywords - Vector Control, Sensorless Control, Model

Reference Adaptive System (MRAS), Kalman Filter (KF),

Speed Estimation, Induction Motor Drives.

I.INTRODUCTION

Induction motors have been widely used in industry because of

the advantage of simple construction, ruggedness, reliability, low

cost and minimum maintaince.[1]

Mathematical model of induction motor is non linear and

presents strong coupling between the input, output and

intervariable, such as torque, speed or flux. So, its control is very

complex. Many schemes have been proposed for control of

induction motor drive, among which the field oriented control [2, 3]

or vector control has been accepted as most effective methods. In

vector control, the knowledge of the rotor speed is necessary, this

necessity requires additional speed sensor which add the cost and

complexity of the drive system.

Over the past few year, ongoing research has concentrated on the

elimination of speed sensor at the machine shaft without

deteriorating the dynamic performance of the drive control system

[4]. In order to achieve good performance of sensorless vector

control, different speed estimation scheme have been proposed and

a variety of speed estimator’s exists now a day’s [5], such as Direct

Calculation, Model Reference Adaptive System, Extended

Luenberger Observer, Kalman filter etc.

Out of various methods, Model Reference Adaptive System,

and Kalman filter based sensorless speed estimation has been used.

In MRAS, speed is estimated using difference between the

reference model output and adaptive model output. The biggest

problem of MRAS approach is the integration of pure voltage

signals. This problem is solved by modify the pure integration in

voltage model to the low pass filter. The Kalman filter has a good

dynamic behavior and it can work even in standstill position. The

filter implementation required model of AC motor must be

calculated in real time which is very complex problem .The

Extended Kalam Filter is a full order stochastic observer for the

estimation of a non linear dynamic system in real time by using

signals that are corrupted by noise.

This paper is organized as follows: In section 2, Dynamic

modeling of induction motor is reviewed. The principle of vector

control and sensorless speed estimation technique: MRAS and KF

are proposed in section 3, 4 and 5.Vechile model is given in section

6. Simulation results are shown in section 7. Finally, concluding

remark is provided in section 8.

II. DYNAMIC MODELING OF INDUCTION MOTOR

Mathematical model of three phase induction motor referring to

rotating reference frame (d-q) can be expressed as follows. [6]

𝑝𝑖𝑑𝑠𝑝𝑖𝑞𝑠

= −𝐴1 𝜔𝑠−𝜔𝑠 −𝐴1

𝑖𝑠𝑑𝑖𝑠𝑞

+

𝐿𝑚

𝜎𝐿𝑠𝐿𝑟𝑇𝑟𝐴2𝜔𝑟

−𝐴2𝜔𝑟𝐿𝑚

𝜎𝐿𝑠𝐿𝑟𝑇𝑟

−𝛹𝑟𝑑−𝛹𝑟𝑞

+ 𝐴3 00 𝐴3

𝑉𝑠𝑑𝑉𝑠𝑞

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𝑝Ψ𝑑𝑟𝑝Ψ𝑞𝑟

=

𝐿𝑚

𝑇𝑟0

0𝐿𝑚

𝑇𝑟

𝑖𝑠𝑑𝑖𝑠𝑞

+

−1

𝑇𝑟(𝜔𝑠−𝜔𝑟)

−(𝜔𝑠−𝜔𝑟)−1

𝑇𝑟

−𝛹𝑟𝑑−𝛹𝑟𝑞

(1)

Where 𝑝 = 𝑑

𝑑𝑡 and

𝑇𝑒𝑚 − 𝑇𝑙 = 𝑗𝑑𝜔𝑟

𝑑𝑡+ 𝐷𝜔𝑟 2

Where 𝑇𝑒𝑚 =3

2 𝑝

𝐿𝑚

𝑇𝑟 𝛹𝑟𝑑 𝑖𝑠𝑞 − 𝛹𝑟𝑞 𝑖𝑠𝑑 and 3

𝐴1 = 𝑅𝑠

𝜎𝐿𝑠+

1−𝜎

𝜎𝑇𝑟 ; 𝐴2 =

𝐿𝑚

𝜎𝐿𝑠𝐿𝑟 ; 𝐴3 =

1

𝜎𝐿𝑠

𝜎 = 1 − 𝐿𝑚

2

𝐿𝑠𝐿𝑟 ; 𝑇𝑟 =

𝐿𝑟

𝑅𝑟 ; 𝜔𝑔 = 𝜔𝑠 - 𝜔𝑟

𝜔𝑔 = Slip frequency;

𝜔𝑠 = Electrical synchronous stator speed; 𝜔𝑟 = Electrical rotor speed; 𝜎 = Linkage coefficient; 𝑇𝑟 = Rotor time constant; 𝑇𝑒𝑚 ,𝑇𝑙 = Electromagnetic torque and mechanical load or

disturbance torque;

J, D = Moment of inertia and viscous coefficient of motor;

𝐿𝑠 , 𝐿𝑚 , 𝐿𝑟 = Stator inductance, mutual inductance and rotor inductance.

III. PRINCIPLE OF FIELD ORIENTED CONTROL OR

VECTOR CONTROL

In 1972, Blaschke has introduced the principle of Field Oriented

Control to realize D.C motor characteristics in an induction motor.

In D.C motor, mmf produced by the armature current is

perpendicular to the field flux produced by stator. Being

orthogonal, there is no net interaction on one another is produced by

these two fluxes. DC motor flux can be controlled by adjusting the

field current and by adjusting armature current torque can be

controlled independently of flux [7].

In AC machine, the interaction between stator and rotor fields

whose orientation is not held at 90 degrees. So, AC machine is not

simple. For DC machine like performance, the field and armature

fields is orthogonal oriented and holding fixed in an AC machine by

orienting the stator current with respect to rotor flux so as to attain

independently controlled torque and flux. Such a scheme is called

vector control. [2]

Fig.1 : Basic block diagram of vector control

A. Basic Theory

Fig.2: Block diagram of Sensorless Induction Motor control

Sensorless vector control of induction motor block diagram

shown in fig 2, Sensorless control of induction motor means the

control of speed of induction motor without speed encoder. A speed

encoder in a drive is undesirable because it add cost and reliability

problem, besides the need for mounting arrangement and shaft

extension. The inverter is used for controlling the motor by

providing switching pulses. The speed and flux estimators are used

to estimate the speed and flux respectively. PI controller is used for

controlling such signals and compared with reference values.

IV. THE MODEL REFRENCE ADAPTIVE SYSTEM

In 1987, Tamai [8] has proposed speed estimation technique

based on Model Reference Adaptive System (MRAS).Two year

later, Schauder [9] presented an more effective and less complex

alternative MRAS scheme. Adaptive control has emerging potential

solution for implementing high performance control system. The

MRAS approach having two model one is reference model which

does not involve the estimated quantity (𝜔𝑟 ) and another is adaptive

model which involve the quantity to be estimated. The output of

adaptive model is compare with the output of reference model

difference is produced which is used to drive the adaptive

mechanism whose output is the quantity to be estimated (rotor

speed) .A number of MRAS based speed sensorless scheme have

been described in the literature for field oriented induction motor

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drive[10-11],[12-13]. The block diagram of MRAS is shown in fig

3.

A. Reference Model

The stator voltage in d-q equivalent circuit is given by

𝑉𝑑𝑠 = 𝑅𝑠 𝑖𝑑𝑠 + 𝐿𝑠 𝑑𝑖𝑑𝑠𝑑𝑡

+𝑑Ψ𝑑𝑚𝑑𝑡

4

Where Ψ𝑑𝑚 = 𝐿𝑚

𝐿𝑟 𝛹𝑑𝑟 − 𝐿𝑟 𝑖𝑑𝑠

Put Ψ𝑑𝑚 in (4) gives

𝑉𝑑𝑠 = 𝐿𝑚

𝐿𝑟 𝑑Ψ𝑑𝑟

𝑑𝑡+ 𝑅𝑠 + 𝜎𝑠𝐿𝑠 𝑖𝑑𝑠 or

𝑑Ψ𝑑𝑟

𝑑𝑡=

𝐿𝑟

𝐿𝑚𝑉𝑑𝑠 −

𝐿𝑟

𝐿𝑚 𝑅𝑠 + 𝜎𝑠𝐿𝑠 𝑖𝑑𝑠 5

Similarly,

𝑑Ψ𝑞𝑟

𝑑𝑡=

𝐿𝑟

𝐿𝑚𝑉𝑞𝑠 −

𝐿𝑟

𝐿𝑚 𝑅𝑠 + 𝜎𝑠𝐿𝑠 𝑖𝑞𝑠 6

The equation (5) and (6) together give stationary frame equations

for reference model.

B. Adaptive Model

The rotor in d-q equivalent circuit equation in d-q equivalent is

given by

𝑑Ψ𝑑𝑟

𝑑𝑡+ 𝑅𝑟 𝑖𝑑𝑟 + 𝜔𝑟 Ψ𝑞𝑟 = 0

𝑑Ψ𝑞𝑟

𝑑𝑡+ 𝑅𝑟 𝑖𝑞𝑟 − 𝜔𝑟 Ψ𝑑𝑟 = 0 7

Adding 𝐿𝑚 𝑅𝑟

𝐿𝑟 𝑖𝑑𝑠 and

𝐿𝑚 𝑅𝑟

𝐿𝑟 𝑖𝑞𝑠 on both side of equation (7)

and substitutes

Ψ𝑑𝑟 = 𝐿𝑚 𝑖𝑑𝑠 + 𝐿𝑟 𝑖𝑑𝑟

Ψ𝑞𝑟 = 𝐿𝑚 𝑖𝑞𝑠 + 𝐿𝑟 𝑖𝑞𝑟

We have

𝑑Ψ𝑑𝑟

𝑑𝑡=

𝐿𝑚

𝑇𝑟𝑖𝑑𝑠 − 𝜔𝑟 Ψ𝑞𝑟 −

1

𝑇𝑟 Ψ𝑑𝑟 8

𝑑Ψ𝑞𝑟

𝑑𝑡=

𝐿𝑚

𝑇𝑟𝑖𝑞𝑠 − 𝜔𝑟 Ψ𝑑𝑟 −

1

𝑇𝑟 Ψ𝑞𝑟 9

The above equation (8) and (9) gives the rotor flux values as a

function of stator current and rotor speed. In above fig.3 adaptation

algorithms is used to tune the speed so that error 𝜉 = 0.we can

drive the following speed estimation relation using Popov criteria

for hyper stability for a asymptotically stable system[14].

𝜔𝑟 = 𝜉 𝐾𝑝 +𝐾𝐼𝑠

Where 𝜉 = Ψ𝑑𝑟^ Ψ𝑞𝑟 − Ψ𝑞𝑟

^ Ψ𝑑𝑟

The estimated speed from MRAS control is feedback to the speed

controller where it is compared with reference speed to get the

commanded output.

Fig.3: Block diagram of MRAS speed estimation.

V. THE KALMAN FILTER

The Kalman Filter is a deterministic type linear observer,

derived to meet a optimality stochastic condition. The kalman filter

has two forms: basic and extended. The extended kalman filter is

basically a full order stochastic observer can be used for non linear

system. The Kalman filter allows obtaining no measured state

variables with usage measured state variable and as well noise and

measurements statistics. [16]

The block diagram of extended kalman filter for speed

estimation shown in fig.4, where machine model is indicated at the

top. The extended kalman filter algorithms use the full machine

dynamic model, where speed is considered a parameter as well a

state. The augmented machine model [14] can be given by

𝑑𝑋

𝑑𝑡= 𝐴𝑋 + 𝐵𝑉𝑠 (10)

Y = CX (11)

Where,

A =

−

𝐿𝑚2 𝑅𝑟+ 𝐿𝑟

2 𝑅𝑠

𝜎𝐿𝑠𝐿𝑟2 0

𝐿𝑚𝑅𝑟

𝜎𝐿𝑠𝐿𝑟2

𝐿𝑚𝜔𝑟

𝜎𝐿𝑠𝐿𝑟0

0 − 𝐿𝑚

2 𝑅𝑟+ 𝐿𝑟2 𝑅𝑠

𝜎𝐿𝑠𝐿𝑟2

−𝐿𝑚𝜔𝑟

𝜎𝐿𝑠𝐿𝑟

𝐿𝑚𝑅𝑟

𝜎𝐿𝑠𝐿𝑟2 0

𝐿𝑚𝑅𝑟

𝐿𝑟

00

0𝐿𝑚𝑅𝑟

𝐿𝑟

0

−𝑅𝑟

𝐿𝑟 −𝜔𝑟 0

𝜔𝑟 −𝑅𝑟

𝐿𝑟 0

0 0 0

:

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B =

1

𝜎𝐿𝑠0

01

𝜎𝐿𝑠

000

000

: C = 1 0 0 0 00 1 0 0 0

X = 𝑖𝑑𝑠 𝑖𝑞𝑠 Ψ𝑑𝑟 Ψ𝑞𝑟 𝜔𝑟 𝑇

𝑉𝑠 = 𝑉𝑑𝑠 𝑉𝑞𝑠 𝑇 Is an input vector.

Fig.4 : The block diagram of extended kalman filter for speed estimation

equation (10) is of 5th order and speed 𝜔𝑟 is a state.If variation in

speed is negligible then 𝑑ω𝑟

𝑑𝑡= 0. With 𝜔𝑟 as a constant parameter,

the machine model used in extended kalman filter is linear. For

digital implementation of an extended kalman filter, the model

[17]must be discretized in following form:

X (K +1) = f ( X(K), U(K), K ) + W(K)

Y(K) = h (X(K), K) + V(K)

Where

Y(K) : is a vector containing the d-q component of

stator current space vector.

U(K) : is a vector of excitation signals.

X(K) : is a vector containing the state.

W(K) and V(K) : are the process and the measurement noise

vectors at time K.

E { W(K) } = 0, E{ W(K) W(j)T } = Q 𝛿kj ,Q ≥ 0

E { V(K) } = 0, E{ V(K) V(j)T } = R 𝛿kj , R ≥ 0

Where Q and R are respectively the process and measurement

covariance matrices.

The extended kalman filter equation is [6]

K(k) = F(k) P(k) HT [ H P(k) HT + R]-1

𝑋 = (k+1) = f (𝑋 (k) ,U(k)) + K(k) [Y(k) - H𝑋 (k)]

P(k+1) = F(k) P(k) FT(k) + Q – K(k) [H P(k) HT + R] KT(k)

Where 𝑋 (k) is state estimate;

P(k) is estimated error covariance matrix;

K(k) is Kalman gain matrix.

F(k) and H is given by

F(k) = 𝜕

𝜕𝑋 { f (X(k), U(k), K) } X ̂(k) ,U(k)

H = 𝜕

𝜕𝑋 { h (X(k), K) } R ̂(k) ,U(k)

VI.VECHILE MODEL

The vehicle model is based on mechanics and aerodynamics

principle [18-19].The total tractive effort is given by

Fte = Frr + Fad + Fhc + Fla + Fwa

Where Frr : is the rolling resistance force;

Fad : is the aerodynamic drag;

Fhc : is the hill climbing force;

Fla : is the force required to give linear acceleration;

Fwa : is the force required to give angular acceleration to the

rotating motor.

The power required to drive a vehicle at a speed v is given by:

Pte = v Fte = v (Frr + Fad + Fhc + Fla + Fwa )

VII.CONCLUSION

In this reveiw paper, sensorless vector control of induction

motor uses two different approaches: MRAS and KF have been

proposed. Sensorless control gives the benefits of vector control

without using any shaft encoder. This paper also elaborates the

dynamic model of induction motor and principle of vector control.

The comparative study between MRAS and KF response based on

speed reversal and step change in load conclude that MRAS is

better than extended KF response. But, extended KF shows a stable

behavior after a certain time has passed for settling. Torque

disturbances are reduced in extended KF as compare to MRAS.The

application of MRAS and KF approach in area of EV’s is also

explained in this review paper .The advantage of automotive speed

sensorless drive are increased reliability, lower cost, reduced size of

drive system and elimination of sensor cables. At low speed, speed

estimation methods responsible for poor drive performance due to

parameter variation. Such problem is over come by KF and MRAS.

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BUT MRAS shows better results than KF approach in the field of

EV’s propelled by an induction motor drive.

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