VECTOR CONTROLLED RELUCTANCE SYNCHRONOUS

MOTOR DRIVES WITH PRESCRIBED CLOSED-LOOP

SPEED DYNAMICS

Model of Reluctance Synchronous Motor

Non-linear differential equations formulated in rotor-fixed d,q co-ordinate system describe the reluctance synchronous motor and form the basis of the control system development.

Control Structure for Reluctance Control Structure for Reluctance Synchronous MotorSynchronous Motor

Master Control Law Master Control Law

Linearising function

1 1

15T J

c L L i id r d q d q L

Demanded dynamic behavior

d

d t Tr

d r

1

1

Dynamic torque equation

d

dt Jc L L i ir

d q d q L

1

5

Vector control condition for maximum torque

a) per unit stator current

b) for a given stator flux

baserr

basedKd

baserdKd

forii

forii

tanLLc

T

J

iqd5

Lrd1

d

i

J

Tc i

cq dem

d r L q dK

d

15

5

*

*

a) b)

i i sign Tq dem d dem d tan

SET OF OBSERVERS FOR STATE ESTIMATION

AND FILTERING

Pseudo-Sliding Mode Observer for Rotor Speed

id 1

s

Ksm

id*

ud

1

Ld

R

Ls

d

vd eq

Ksm

R

Ls

q

1

s

1

Lq

iq uq

iq*

vq eq

d

d t

i

i

R

LR

L

i

i

L

L

uu

v

vd

q

s

d

s

q

d

q

d

q

d

q

d eq

q eq

*

*

*

*

0

0

10

01

d

d t

ii

R

Lp

L

L

pL

L

R

L

ii

L

L

uu

d

q

s

dr

q

d

rd

q

s

q

d

q

d

q

d

q

10

01

a)

*

*

d d d

q q q

i i

i i

d d d

q q q

i i

i i

*

*

Motor equations

Model system

definition of error

Angular velocity extractor

d

q

s

d

s

q

d

q

q

d

d

q

d

q

d eq

q eq

R

LR

L

L

LL

L

ii

v

v

0

0

0

0

Error system

v V sign i i

v V sign i i

d eq d d

q eq q q

max*

max*

( )

( )

0

Condition for Sliding Motion

Sliding-Mode Observer

v

v

L

LL

L

ii

d eq

q eq

q

d

d

q

d

q

0

0

Pseudo-SMC Observerv K i i

v K i i

d eq sm d d

q eq sm q q

( )

( )

*

*

Estimate of rotor speedEquivalent variables

rq q eq

d d

L v

pL i*

The Filtering ObserverThe Filtering Observer

r

L

1

s

1

s

K K

r

15~

~ ~ ~

Jc L L i id q d q

VJ

where design of:

needs adjustment of the one parameter only or as two different poles:

k J Ts 9 0

~k J Ts 81 4 0

2~

k J ~ 1 2 k J ~1 2

Electrical torque of SRM is treated as an external model input

Filtered values of and are produced by the observer based on Kalman filter

r

e

Jc L L i i k e

k e

r

r d q d q L

L

~

15

L

Load torque is modeled as a state variable

Original control structure of speed Original control structure of speed controlled RSMcontrolled RSM

q

rotorpositionsensor

external loadtorque L

r

UqUd

I2- I 3

I q

I d

Id dem

demanded d_qstator currents

demanded three-phase voltages

vd_eq

Iq dem

U 1

U2

U3

I 1

ReluctanceSynchronous

Motor

MasterControl

Law

Angularvelocityextractor

Powerelectronic

drivecircuit

d_q

transf.

Rotor fluxcalculator

demandedrotor speed

Sliding-modeobserver

Slavecontrol law

Filteringobserver

r

I dUdI q

d_q&

a,b,ctransf

Switchingtable

sr

d

Udc

Measured variables:rotor position,stator current,DC circuit voltage

Uq

d

vq_eq

L

r

r r

Reference Model (of closed-loop system)

Inner & Middle Loop(real system)

correction loop

mrK

Ts

Kd1

Ts1

1

dr̂d

id

Model TF

r

d

s

s sT

1

1

Parameter mismatch increases a correction

Kmr r id

Ts

KK

sTK

Ts

K

s

s

dmr

mrd

d

r

11

11

11ˆ

Mason’s rule

Kmr

r

d

s

s sT

1

1

MRAC outer loop

Simulation results a1) id=const without MRAC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.5

1

1.5

2

2.5

3

3.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

10

20

30

40

50

60

70

80

90

100

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-20

0

20

40

60

80

100

a) id, iq = f(t) b) d, q = f(t) c) Ld = f(t)

d) id, est = f(t) e) L, Lest = f(t) f) id, r = f(t)

Simulation results a2) id=const with MRAC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4010

20

304050

60

7080

90100

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-20

0

20

40

60

80

100

120

a) id, iq = f(t) b) d, q = f(t) c) Ld = f(t)

d) id, est = f(t) e) L, Lest = f(t) f) id, r = f(t)

Simulation results (without MRAC)

b1) dq-current angle control

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.5

0

0.5

1

1.5

2

2.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-20

0

20

40

60

80

100

120

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

10

20

30

40

50

60

70

80

90

100

a) id, iq = f(t) b) d, q = f(t) c) Ld = f(t)

d) id, est = f(t) e) L, Lest = f(t) f) id, r = f(t)

Simulation results (with MRAC) b2) dq-current angle control

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.5

1

1.5

2

2.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-20

0

20

40

60

80

100

120

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40102030405060708090

100

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1

0

1

2

3

4

5

6

a) id, iq = f(t) b) d, q = f(t) c) Ld = f(t)

d) id, est = f(t) e) L, Lest = f(t) f) id, r = f(t)

Effect of MRAC on Various Types of Prescribed Dynamics

a) constant torquea) constant torque

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

b) first order dyn.b) first order dyn.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

c) second ord. dyn.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

0

50

100

150

200

250Ideal, Estim. & Real Speed

Conclusions and Recommendations

The simulation results of the proposed new control method for electric drives employing SRM show a good agreement with the theoretical predictions.

The only departure of the system performance from the ideal is the transient influence of the external load torque on the rotor speed.

This effect is substantially reduced if MRAC outer loop is applied.

It is highly desirable to employ suggested control strategy experimentally.