273081

Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2011, Article ID 273081, 11 pagesdoi:10.1155/2011/273081

Research Article

Application of Quadratic Linearization for the Control ofPermanent Magnet Synchronous Motor

Parvathy Ayalur Krishnamoorthy,1 Kamaraj Vijayarajan,2 and Devanathan Rajagopalan1

1Hindustan Institute of Technology and Science, Chennai 600 016, India2 SSN College of Engineering, Chennai 600 004, India

Correspondence should be addressed to Parvathy Ayalur Krishnamoorthy, [email protected]

Received 30 June 2011; Accepted 8 August 2011

Academic Editor: Henry S. H. Chung

Copyright 2011 Parvathy Ayalur Krishnamoorthy et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Many of the existing control methods for the permanent magnet synchronous motor (PMSM) either deal with steady state modelsor consider dynamic models under particular cases. A dynamic model of the PM machine allows powerful control-theoretictechniques such as linearization to be applied to the system. Existing exact feedback linearization of dynamic model of PMSMsuers from singularity issues. In this paper, we propose a quadratic linearization approach for PMSM based on the approximatelinearization technique which does not introduce singularities. A MATLAB simulation is used to verify the eectiveness of thelinearization technique proposed. Also, to account for higher-order and unmodelled dynamics of PMSM, tuning of the linearizingtransformation is proposed and verified using simulation.

1. Introduction

Permanent magnet (PM) machines, particularly at lowpower range, are widely used in the industry because oftheir high eciency. They have gained popularity in variablefrequency drive applications. The merits of the machine areelimination of field copper loss, higher power density, lowerrotor inertia, and a robust construction of the rotor [1].

Steady state models, such as in [2], have been used todevelop control strategies for the permanent magnet syn-chronous motor. Vector control [3] enables independentcontrol over the magnitude and angle of the current withrespect to the rotor such that instantaneous control overtorque is possible. But vector control implemented, in prac-tice [4], assumes a constant flux in the closed loop and usesstatic models to eect the control scheme. The steady statemodels used to describe the machine behaviour do notcapture explicitly the dynamics of the machine involved.Decoupled control [5], while considering the dynamics ofthemachine, corresponds to application of particular controlstrategies for machine control.

A dynamic model of a PM machine using direct andquadrature axis variables [1] allows more powerful general

control theories to be brought to bear on the problem of thecontrol of PM machine. One such control-theoretic ap-proach is quadratic linearization of the PM machine model.Linearization of a nonlinear system allows a simple fixedcontroller to be applied for the control of the linearizedsystem and yet get a uniform closed loop response fordierent reference and load conditions.

Bodson and Chiasson [6] have designed a controllerusing dierential geometric method based on exact feedbacklinearization. The main drawback with this method and itsvariations [7] is that, even if the system is linearizable, thelinearizability is subject to certain function of the state beingnonsingular in a region of operation of the machine. Thisputs a constraint on the practical implementation of thecontrol strategy. Zhu et al. [8] provide a static feedback lin-earization of a PM model (see [8, equation (25)]) which hasa simpler quadratic term compared to our model. With theassumption that Ld=Lq, the solution of quadratic lineariza-tion becomes trivial. Our method is more general and isapplicable to PMSM where Ld /=Lq.

Krener [9] formulated an approximate feedback lin-earization technique based on Taylor series expansion; Kangand Krener [10, 11] extended the work of Poincare [12] on

2 Advances in Power Electronics

the lines of Krener to consider quadratic linearization ofcontrol ane systems. Quadratic linearization can be appliedto a PMSM model [1] as the model has a predominantquadratic nonlinearity. But being an approximate technique,higher-order terms are introduced. Consideration of coreloss involves fourth-order terms involving product of squaresof both current and angular velocity. These together withthe stray losses and unmodelled dynamics coupled with thethird- and higher-order nonlinearities introduced due toquadratic linearization are best accounted for by tuning thetransformations against an actual PM machine on the linessimilar to those proposed by Levin and Narendra [13].

In this paper, input and state transformations are derivedfor a 4-dimensional PM machine model in order to linearizePMSM machine model. The PMSM model is quadraticlinearized using the approximate technique. Tuning rules arederived for the linearizing transformations to account forhigher-order terms, by back propagation of error betweenthe outputs of quadratic linearized system with a normalform output.

The linearization technique is verified using SIMULINKmodel which is developed for interior permanent magnet(IPM) machine. The core loss which consists of higher-orderterms is included in the SIMULINK model of PMSM, andtuning rules are also simulated. The closed loop responseand open loop gain for the system before and after tuningare obtained.

The simulation results after linearization indicate auniform closed loop response for dierent reference and loadconditions, thus verifying the theory. The simulation resultsafter tuning also verify the eectiveness of tuning.

To summarise the rest of the paper, in Section 2,background required for quadratic linearization is given. InSection 3, the model of PM synchronous motor is derivedand is reduced to normal form. In Section 4, PMSM modelis linearized. In Section 5, a SIMULINK model of PMSMis constructed and simulation is presented to show theeectiveness of the proposed method. In Section 6, tuning ofthe transformation coecients is derived by including coreloss in the PMSMmodel and also simulation results are givento verify the eectiveness of tuning. In Section 7, the paper isconcluded.

2. Background

Consider a single input control ane system of the form [10]

x = Ax + Bu + f (2)(x) + f (3)(x) + + f (m)(x) +

+ g(1)(x) + g(2)(x) + + g(m1)(x) + ,(1)

where A and B are matrices in the controller normal form

A =

0 1 0 00 0 1 0 0 0 10 0 0 0

; B =

0

0

1

, (2)

A is an n n constant matrix and B is an n 1 constantmatrix.

x = [x1 x2 xn]T and u is a scalar input. f (m)(x),g(m1)(x) are vector homogeneous polynomials of order mand (m 1), respectively, m = 2, 3, . . ..

In order to linearize the system, a change of coordinatesand feedback [10, 11] of the following form is considered:

y = x + (x), (3)u = (1 + (x))v + (x), (4)

where y = [y1 y2 yn]T , v is a scalar, and (x), (x),and (x) are polynomials given by

(x) = (2)(x) + (3)(x) + + (m)(x) + ,

(x) = (2)(x) + (3)(x) + + (m)(x) + ,

(x) = (1)(x) + (2)(x) + + (m1) (x) + .

(5)

Applying the transformations (3) and (4), (1) is reducedto

y = Ay + Bv (6)provided that the following equations, called generalizedhomological equations [10], are satisfied:

A(m)(x) + B(m)(x) + f (m)(x) + (m)(x)x

Ax = 0,

B(m1)(x)v +(m)(x)

xBv + g

(m1)(x)v = 0; v,(7)

where f(m)(x) = f (m)(x) form = 2 and f (m)(x) is expressed

in terms of f (mi)(x), i = 0, 1, 2 . . . (m 2) and (m j)(x),j = 1, 2 . . . (m 2), m > 2. g (m1)(x) = g(1)(x), m = 2 andg(m1)(x) is expressed in terms of g(mi)(x), i = 1, 2 . . . (m

1) and (m j)(x), j = 1, 2 . . . (m 2),m > 2.

Remark 1. Quadratic linearization involves specialization ofthe above result for m = 2. Consider

x = Ax + Bu + f (2)(x) + g(1)(x) +O(3)(x,u), (8)where A and B are in controller normal form [10] andO(3)(x,u) corresponds to terms of order 3 ormore. Equations(3) and (4) for m = 2 can be written as

y = x + (2)(x), (9)

u =(1 + (1)(x)

)v + (2)(x). (10)

Applying (9) and (10), (8) can be written as

y = Ay + Bv +O(3)(y, v) (11)provided that the homological equations

A(2)(x) + B(2)(x) + f (2)(x) + (2)(x)x

Ax = 0,

B(1)(x)v +(2)(x)

xBv + g(1)(x)v = 0; v

(12)

can be solved for (2)(x),(2)(x), and (1)(x).

Advances in Power Electronics 3

3. Machine Model

The PM machine model can be derived [1, 14, 15] as

x = Ax + Bu + f (2)(x), (13)

x =[x1 x2 x3 x4

]T =[ e iq id

]T,

u =[u1 u2

]T =[vq vd

]T,

(14)

where vq, vd, iq, and id represent the quadrature and directaxis voltages and currents, respectively, and and erepresent rotor position and angular velocity, respectively.

A =

0 1 0 0

0 01.5pJ

0

0pLq

RLq

0

0 0 0RLd

,

B =

0

0

1Lq0

0

0

0

1Ld

, f (2)(x) =

0

1.5p(Ld Lq

)idiq

JLd peid

LqLq peiq

Ld

,

(15)

is the flux induced by the permanent magnet of the rotorin the stator phases. Ld and Lq are the direct and quadratureinductances, respectively. R is the stator resistance, p is thenumber of pole pairs, and J is the system moment of inertia.

The model (13) can be reduced, in a standard wayusing linear change of coordinates and feedback [16], to theBrunovsky form [17] for two inputs as in (16) below (wherex,u,A, and B are retained for simplicity of notation):

x = Ax + Bu + f (2)(x), (16)where

A=

0 1 0 0

0 0 1 0

0 0 0 0

0 0 0 0

, B=

0 0

0 0

1 0

0 1

, f (2)(x)=

0

k1x3x4

k2x2x4

k3x2x3

,

k1=1.5p

(LdLq

)a4

Ja1, k2= Ld pa1a4

Lq,

k3=LqC1

2a1Lda4

,

a1 = 1.5pJ

, a4 = RLd

, C1 = 1Lq

.

(17)

4. Linearization of Interior Permanent Magnet(IPM) Synchronous Motor

The quadratic linearization given in Section 2 is extended tothe case of two inputs in a straightforward way and appliedto a PM machine model in this section [14, 15].

Theorem 2. Given the 4-dimensional model of a PM syn-chronous motor (IPM model) of the form (16), the system canbe linearized using the following transformations:

y = x + (2)(x), (18)

u =(I2 + (1)(x)

)v + (2)(x), (19)

where u = [u1 u2]T , v = [v1 v2]T ,

(2)(x) =

0

0

k1x3x4

0

, (20)

(2)(x) =k2x2x4k3x2x3

, (21)

(1)(x) = (1)(BT

(2)(x)x

B

)=

k1x4 k1x3

0 0

, (22)

where I2 is the identity matrix of order 2.The system then reduces to

y = Ay + Bv +O(3)(y, v), (23)

where O(3)(y, v) represents third- and higher-order nonlinear-ities.

Proof. Applying transformations (18) and (19) (which corre-spond to a natural extension of (9) and (10) to two inputs)to (16), the homological equations to be considered can bewritten as

A(2)(x) + B(2)(x) + f (2)(x) + (2)(x)x

Ax = 0, (24)

B(1)(x) +(2)(x)

xB = 0. (25)


1

1

VM1

v

+

+

+

VM

Poles Poles

Out1

Speed

Poles

Scope

1/s

1/s

Int

s

CM1i

+

++

s

+

s +

s+

v+

iCM

3

-C-

0

0.0008

24

vqVq

vd

Te

YqIq

Yd

YdId

Te

Tl

TlJ

JBB

Rq

Rd

Product1

Product

Lqs

LdsInt 1

Yf

Scope1

Scope2

rYd

rYq

r

m

Figure 1: PMSM design using SIMULINK.

By choosing (2)(x) and (2)(x) as in (20) and (21), (24)reduces to

0

k1x3x4

0

0

+

0

0

k2x2x4k3x2x3

+

0

k1x3x4

k2x2x4

k3x2x3

+

0 0 0 0

0 0 0 0

0 0 k1x4 k1x3

0 0 0 0

x2

x3

0

0

= 0.

(26)

Premultiplying (25) by BT and noting that BTB = I2, (25)reduces to

(1)(x) = (1)(Bt

(2)(x)x

B

). (27)

Also, substitution of (27) satisfies (25). Hence, the homo-logical equations are satisfied and quadratic linearization isachieved, hence the proof.

5. Experimental Simulation Results

Given the parameters R = 2.875, Lq = 9mH, Ld =7mH, e = 3500 rpm, p = 4, J = 0.0008 kgm2 and =175mWb turns, of an actual PM machine, Figure 1 showsthe SIMULINK model of the PMSM which is constructedbyusing speed and torque blocks and control circuit. vq andvd are taken as inputs to the motor. The model is reducedto normal form in a standard way by using the lineartransformations. The PM model in Figure 1 is especially

Table 1: Steady state gain of e versus vq for the system in openloop (prior to linearization).

vq e k = de/dvq5 2.55

10 4.7709 0.44418

15 6.4706 0.33994

20 7.6082 0.22752

25 8.2567 0.1297

30 8.5326 0.05518

configured for the IPM where Ld /=Lq. N1 and N2 blocksin Figure 2 include the above linear transformation and thenonlinear transformation (18) and (19) as well.

Prior to linearization, the open loop steady state gainof e versus vq of the PMSM model is investigated and theresults are given in Table 1. In this table, it is observed thatthe open loop steady state gain of e versus vq (keeping vdconstant) is not constant because of the system nonlinearity.To verify the linearity of the system after linearization, weinvestigated the variation of its gain of y2 (a scaled versionof e as can be seen from (18) and (20)) with input v1(see Figure 2) and the results are given in Table 2. The tablereveals that the gain of the system is nearly constant, thusverifying that by applying the homogeneous linearizingtransformation, the PMSM model is made nearly linear forthe given set of inputs.

We now proceed to show that the nearly constant gainof the linearized model results in a uniform closed loopresponse on a range of set point and load inputs with a fixedcontroller. This is in contrast to the case before linearizationunder the corresponding conditions.


Input2

Input1

Vq

VdIq

IqId

Id

N2

Z1

Z2

Z3

Z4

vq1

vd1

1iq

iq idid

vq

vd

L2

V2

y1

y2

y3

y4

L1

N1

V1vq1vd1

1iqid1

Iq

Id

PMSM

vq

vd

1

1

Coordinate 1 Coordinate 2

Figure 2: Linearization of PMSM.

0 020. 00. 4 00. 6 00. 8 0.10

0.5

1

1.5

2

2.5

3

Time (s)

Angu

larvelocity

Figure 3: Time response of angular velocity in closed loop whenvq = 5; kp = 50; ki = 2 (before linearization).

Table 2: Steady state gain of y2 versus v1 for the linearized systemin open loop.

v1y2

106k = dy2/dv1 106

5 2.176

10 4.366 0.438

15 6.585 0.4438

20 8.845 0.452

25 11.162 0.4634

30 13.55 0.4776

Figures 3 and 4 show the time response of angularvelocity e by closing the loop around PMSM model beforelinearization when vq = 5 and 30 units, respectively. It is

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5Angu

larvelocity

Time (s)

Figure 4: Time response of angular velocity in closed loop whenvq = 30; kp = 50; ki = 2 (before linearization).

observed that the dynamic response for vq = 5 is moreoscillatory compared to the case of vq = 30 with a fixedcontroller of proportional gain = 50 and integral constant =2. This is to be expected since the loop gain is higher in theformer case with a higher static gain in the motor as can beseen from Table 1.

Figures 5 and 6 show the time response of y2 of the trans-formed PMSM system (Figure 2) in closed loop when v1 = 5and 30 units, respectively. It is observed that a uniform out-put response is obtained in the closed loop after linearizationwhen the reference is varied. Since the static gain in Table 2 isnearly uniform, the loop gain is also nearly constant for theextreme points in the operating range, thus resulting in theuniform dynamic responses as in Figures 5 and 6.

Figures 7 and 8 show the time response of angularvelocity e of the PMSMmodel before linearization in closed


0 0.002 0.004 0.006 0.008 0.010.5

0

0.5

1

1.5

2

2.5106

Time (s)

Angu

larvelocity

Figure 5: Time Response of y2 for the linearized system in closedloop when v1 = 5; kp = 50; ki = 2.

0 0.002 0.004 0.006 0.008 0.012

0

2

4

6

8

10

12

14

16106

Time (s)

Angu

larvelocity

Figure 6: Time Response of y2 for the linearized system in closedloop when v1 = 30; kp = 50; ki = 2.

loop when vq = 15 units with the load torque T variedas T = 0 and 1 unit, respectively. Note that the dynamicresponses in Figures 7 and 8 are dierent, with the responseof Figure 8 corresponding to an inverse response (i.e., speedreduces first before increasing). The latter corresponds to aresponse of a nonminimum phase system.

Figures 9 and 10 show the time response of y2 of thetransformed PMSM system in closed loop when v1 = 15units and T = 0 and 1 unit, respectively. It is observed thata uniform output response is obtained in the closed loopafter linearization when load torque is varied. Hence, it isverified that a linearized system gives a uniform closed loopresponse for the dierent reference and load conditions usedfor testing as above.

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Angu

larvelocity

Figure 7: Time response of angular velocity in closed loop whenvq = 15 units, T = 0N-m; kp = 50; ki = 2 (before linearizaton).

0 0.02 0.04 0.06 0.08 0.10.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Angu

larvelocity

Figure 8: Time response of angular velocity in closed loop whenvq = 15 units, T = 1N-m; kp = 50; ki = 2 (before linearization).

6. Tuning

6.1. Core Loss. The core loss or iron loss, caused by thepermanentmagnet (PM) flux and armature reaction flux, is asignificant component in the total loss of a PMSM, and, thus,it can have a considerable eect on the PMSM modeling andperformance prediction.

The net core loss Plc [2] for the machine is computed asfollows:

Plc =1.5r2

(LqIq

)2

Rc+

1.5r2(a f + LdId

)2

Rc

, (28)

where Rc represents core loss resistance, a f representsmagnet flux linkage and r denotes rotor electrical speed.


0 0.002 0.004 0.006 0.008 0.011

0

1

2

3

4

5

6

7106

Time (s)

Angu

larvelocity

Figure 9: Time Response of y2 for the linearized system in closedloop when v1 = 15 units, T = 0N-m; kp = 50; ki = 2.

The mechanical torque equation including core losses isgiven by

Te =(2P

)Jdedt

+ Tl + Tlc, (29)

Tlc = Plcr

. (30)

6.2. Tuning Formula. To account for the core loss (28),unmodelled dynamics, and higher-order nonlinearities, tun-ing of the quadratic linearization transformations N1 andN2 can be done [13]. Figure 11 shows the block diagram fortuning.

Error (E) can be calculated as

E =(T)=[(y y)T(y y)

]1/2, (31)

where = (1 2 3 4)T .The error can be written as

E = (21 + 22 + 23 + 24)1/2

. (32)

Since (2)(x) and (1)(x) are both functions of k1, we shallredefine

(2)(x) =

0

0

k1x3x4

0

,

(1)(x) = k

1x4 k

1x3

0 0

(33)

so that (2)(x) and (1)(x) can be independently tuned bytuning k1 and k1, respectively. (2)(x) is not varied.

0 0.002 0.004 0.006 0.008 0.011

0

1

2

3

4

5

6

7106

Time (s)

Angu

larvelocity

Figure 10: Time Response of y2 for the linearized system in closedloop when v1 = 15 units, T = 1N-m; kp = 50; ki = 2.

v(m)

u(m) x(m)

y(m)

y(m)

N1 N2

Updatinglaw

Normalform

SIMULINKmodel

Figure 11: Block diagram for tuning of transformation.

6.2.1. Updation of N2 Transformation Coecients. Tuning ofN2 transformation implies the tuning of (2)(x). As (2)(x)is a function of only k1x3x4, the coecient k1 has to beupdated based on the error between the outputs of quadraticlinearized system and normal form. The updation law isderived as follows:

k1 = Ek1

= Ey

y

k1. (34)

From (31), it is seen that E/yi = i/E, i = 1, 2, 3, 4.Hence, E/y = [1/E 2/E 3/E 4/E],

k1 =[1E

2E

3E

4E

]

0

0

x3x4

0

= 3

Ex3x4. (35)

Updation of (2)(x) is done by using the formula

k1(m) = k1(m 1) 1k1(m); 0 < 1 < 1, (36)

where m corresponds to the updating step and 1 corre-sponds to the accelerating factor.


Loss torque

VM1

v

+

VM

Speed

Poles

PolesPoles

ScopeProduct

Int

1s

+

+

+++

v+

s

+

s

+

s

CM1i

+iCM

24

-C-

1

0.0008

0

1/s

1/s

Lq

Ld.76e3

1.2e3LdIq

r

Lq

Id

rYd

rYq

TlTl

Yd

vq

vd

Te

Te

YqIq

IqYd

J

BJB

r

m

Scope2

Scope1

3ScopeRq

Rd

Rc1

Rc

Product1

Lqs

LdsInt 1

Vq

Y fTlc

Tlc

Out 1

Figure 12: PMSM model including core loss.

6.2.2. Updation of N1 Transformation Coecients. Tuning ofN1 transformation is achieved by tuning of (1)(x). As (1)(x)is a function of k1x3 and k

1x4, the coecient k

1 has to be

updated based on the error between the outputs of quadraticlinearized system and normal form. The updation law isderived as follows:

k1 =E

k1= E

y

y

x

x

u1

u1k1

, (37)

where

E

y=[1E

2E

3E

4E

],

y

x=

1 0 0 0

0 1 0 0

0 0 (1 + k1x4) k1x3

0 0 0 1

,

x

u1=

0

1k2x40

1k2x2

,

(38)

assuming that the steady state of the SIMULINK model isreached within the tuning period:

u1k1

= v1x4 v2x3. (39)

Thus, k1 = ((v1x4 + v2x3)/E)(2/k2x4 + (3k1x3 + 4)/k2x2).Updation of (1)(x) is done by using the formula

k1(m) = k1(m 1) 2k1(m), 0 < 2 < 1, (40)

where m corresponds to the updating step and 2 corre-sponds to the accelerating factor.

6.3. Tuning Simulation Results. PMSM model including coreloss is given in Figure 12 where the block due to loss torque

(30) is also included. The tuning of the transformationsis done using the updations for (2)(x) and (1)(x) as performulae (36) and (40) as given in Figure 13. The simulationdiagram for tuning is done by using memory blocks tostore the updated values of (2)(x) and (1)(x). It is seenthat the error after tuning is reduced to 0.01. Table 3 showsthe variation of y2 versus v1 of the PMSM model afterlinearization including core loss prior to tuning of thecoecients. It is observed that the open loop steady stategain is not constant due to the eect of core loss. Table 4shows the variation of y2 versus v1 for the linearized systemafter incorporating tuning of the transformation coecients.The table reveals that the gain of the system is nearly constantthus verifying the eectiveness of tuning. Figures 14 and 15show the closed loop time response of y2 of the linearizedsystem including core loss before tuning when v1 = 5 unitsand 30 units, respectively; kp = 50; ki = 2. Figures 16 and17 show the closed loop time response of y2 of the linearizedsystem including core loss after tuning when v1 = 5 unitsand 30 units, respectively; kp = 50; ki = 2. It is observedthat the dynamic response before tuning for v1 = 5 unitsis more oscillatory compared to the case of v1 = 30 units.This is to be expected since the loop gain is higher in theformer case with a higher static gain in the plant or motoras can be seen from Table 3. Since the static gain in Table 4is nearly uniform, the loop gain is also nearly constant forthe extreme points in the operating range, thus resulting inthe uniform dynamic responses in Figures 16 and 17. Thus,the eectiveness of tuning the linearizing transformations toovercome the unmodelled core loss is verified.

7. Conclusion

As the PMSM is inherently nonlinear, to design a controllerthat can provide a predictable uniform performance of thedrive under varying operating conditions, it is necessary to


Updating

In1

Updating

In2

State-space

PMSM

Data storememory1

+

+

+

+

x = Ax + Buy = Cx + Du

Data storememory

Scope5

Scope3

Scope2

Scope6

Scope1

y1y2y3y4

iqid

iqid

4

1L2

Z1

del K1

Z2Z3

Z4

IqId

vq

vdvq

vd

vq1vd1

1iqid

vq1vd1

1iq

e1e2e3e4Z2Z3Z4Vq

Vq

Vd

Vd

A

B

Vq

Vd

N2

1

1

id1

N1

L1

Figure 13: Simulation diagram for controller tuning.

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

Time (s)

Angu

larvelocity

Figure 14: Time response of angular velocity in closed loop whenvq = 5 unit; kp = 50; ki = 2 before tuning.

Table 3: Steady state gain of y2 versus v1 for the linearized systemin open loop before tuning.

v1y2

106k = dy2/dv1 106

5 2.5317

10 4.673 0.4286

15 6.2126 0.30792

20 7.147 0.18688

25 7.594 0.0894

30 7.7001 0.02122

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (s)

Angu

larvelocity

Figure 15: Time response of angular velocity in closed loop whenvq = 30 unit; kp = 50; ki = 2 before tuning.

Table 4: Steady state gain of y2 versus v1 for the linearized systemin open loop after tuning.

v1y2

106k = dy2/dv1 106

5 2.0033

10 4.0264 0.40462

15 6.0925 0.41322

20 8.193 0.4201

25 10.295 0.4204

30 12.434 0.4278


0 0.002 0.004 0.006 0.008 0.010.5

0

0.5

1

1.5

2

2.5106

Time (s)

Angu

larvelocity

Figure 16: Time response of y2 for the linearized system in closedloop when vq = 5 unit; kp = 50; ki = 2 after tuning.

0 0.002 0.004 0.006 0.008 0.012

0

2

4

6

8

10

12

14106

Time (s)

Angu

larvelocity

Figure 17: Time response of y2 for the linearized system in closedloop when vq = 30 unit; kp = 50; ki = 2 after tuning.

linearize the PMSM. In this paper, the dynamic model of aPM synchronous motor involving quadratic nonlinearity islinearized. The technique used is based on the control inputextension of Poincares work due to Kang and Krener whichis in line with the approximate linearization technique ofKrener [9, 18]. The existing techniques of exact feedbacklinearization introduce singularities in the system, whichmay cause diculties in the implementation of the closedloop control. In the proposed method, the problem ofsingularities does not arise.

The PMSM machine model, together with the state andinput transformations, are simulated using SIMULINK. Thesimulation results show that the quadratic linearizing trans-formations eectively linearize the system thus, supportingthe theory. The simulation verifies that a uniform response

under a fixed controller is obtained for the linearized systemfor variations of reference speed and load conditions, incontrast to the case before linearization.

Further, to account for the core loss, unmodelled dynam-ics and third- and higher-order nonlinearities, tuning of thetransformation parameters is proposed by comparing theoutput of the linearized system with a normal form output.After tuning, it is shown that the linearized system showsimproved linearity in terms of static gain compared to thecondition before tuning. Closed loop system response forthe linearized system is also shown to be uniform due tothe eect of tuning of linearization transformations, undervarying reference inputs.

The proposed linearization method can be extended toinduction motor and wound synchronous motor models aswell [19].

References

[1] B. K. Bose, Modern Power Electronics and AC Drives, PearsonEducation, 2002.

[2] R. Monajemy, Control strategies and parameter compensationof permanent magnet synchronous motor drives, Ph.D. thesis,Virginia Polytechnic Institute and State University, Blachsberg,Va, USA, 2000.

[3] J. K. Seok, J. K. Lee, and D. C. Lee, Sensorless speed controlof nonsalient permanent-magnet synchronous motor usingrotor-position-tracking PI controller, IEEE Transactions onIndustrial Electronics, vol. 53, no. 2, pp. 399405, 2006.

[4] P. Pillay and R. Krishnan, Modeling, simulation, and anal-ysis of permanent-magnet motor drives. I. The permanent-magnet synchronous motor drive, IEEE Transactions onIndustry Applications, vol. 25, no. 2, pp. 265273, 1989.

[5] H. Zhu, X. Xiao, and Y. Li, PI type dynamic decoupling con-trol scheme for PMSMhigh speed operation, in Proceedings ofthe 25th Annual IEEE Applied Power Electronics Conference andExposition (APEC 10), pp. 17361739, Palm Springs, Calif,USA, February 2010.

[6] M. Bodson and J. Chiasson, Dierential-geometric methodsfor control of electric motors, International Journal of Robustand Nonlinear Control, vol. 8, no. 11, pp. 923954, 1998.

[7] J. Chiasson, Nonlinear controllers for an induction motor,Control Engineering Practice, vol. 4, no. 7, pp. 977990, 1996.

[8] G. Zhu, A. Kaddouri, L. A. Dessaint, and O. Akhrif, Anonlinear state observer for the sensorless control of apermanent-magnet AC machine, IEEE Transactions on Indus-trial Electronics, vol. 48, no. 6, pp. 10981108, 2001.

[9] A. J. Krener, Approximate linearization by state feedback andcoordinate change, Systems and Control Letters, vol. 5, no. 3,pp. 181185, 1984.

[10] W. Kang and A. J. Krener, Extended quadratic controllernormal form and dynamic state feedback linearization of non-linear systems, SIAM Journal on Control and Optimization,vol. 30, no. 6, pp. 13191337, 1992.

[11] A. J. Krener and W. Kang, Extended normal forms ofquadratic systems, in Proceedings of the 29th IEEE Conferenceon Decision and Control, pp. 20912096, IEEE, New York, NY,USA, December 1990.

[12] V. I. Arnold, Geometric Methods in the Theory of OrdinaryDierential Equations, Springer, New York, NY, USA, 1983.

[13] A. U. Levin and K. S. Narendra, Control of nonlineardynamical systems using neural networks. Controllability and


stabilization, IEEE Transactions on Neural Networks, vol. 4,no. 2, pp. 192206, 1993.

[14] A. K. Parvathy, V. Kamaraj, and R. Devanathan, A newlinearisation technique for permanent magnet synchronousmotor model, in Proceedings of the Joint International Con-ference on Power System Technology and IEEE Power IndiaConference (POWERCON 08), pp. 15, New Delhi, India,October 2008.

[15] A. K. Parvathy, R. Devanathan, and V. Kamaraj, Applicationof quadratic linearization to control of Permanent Magnetsynchronous motor, in Proceedings of the 1st InternationalConference on Electrical Energy Systems (ICEES 11), pp.158163, 2011.

[16] B. C. Kuo, Automatic Control Systems, Prentice-Hall, NewDelhi, India, 2001.

[17] P. Brunovsky, A classification of linear controllable systems,Kybernetika, vol. 6, no. 3, pp. 173188, 1970.

[18] G. S. Cardoso and L. Schnitman, Analysis of exact lineari-zation and aproximate feedback linearization techniques,Mathematical Problems in Engineering, vol. 2011, Article ID205939, 17 pages, 2011.

[19] A. K. Parvathy, V. Kamaraj, and R. Devanathan, Completequadratic linearisation of machine models, in Proceedings ofthe IEEE International Conference on Control Applications, pp.11301133, Singapore, October 2007.

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